titelei 1..4

Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
Deutsche
Forschungsgemeinschaft
Plasticity of Metals:
Experiments, Models,
Computation

Deutsche
Forschungsgemeinschaft
Plasticity of Metals:
Experiments, Models,
Computation
Final Report of the
Collaborative Research Centre 319,
Stoffgesetze fÏr das inelastischeVerhalten metallischer
Werkstoffe – Entwicklung und technischeAnwendung
1985–1996
Edited by
Elmar Steck, Reinhold Ritter, Udo Pfeil and Alf Ziegenbein
Collaborative Research Centres
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

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ISBN 3-527-27728-5
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
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Contents
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
Preface ................................................... XV
1 Correlation between Energy and Mechanical Quantities
of Face-Centred Cubic Metals, Cold-Worked and Softened
to Different States ..................................... 1
Lothar Kaps, Frank Haeßner
1.1 Introduction .......................................... 1
1.2 Experiments .......................................... 1
1.3 Simulation ........................................... 11
1.4 Summary ............................................ 14
References ........................................... 15
2 Material State after Uni- and Biaxial Cyclic Deformation ....... 17
Walter Gieseke, K. Roger Hillert, Günter Lange
2.1 Introduction .......................................... 17
2.2 Experiments and Measurement Methods ...................... 18
2.3 Results .............................................. 19
2.3.1 Cyclic stress-strain behaviour .............................. 19
2.3.2 Dislocation structures ................................... 24
2.3.3 Yield surfaces ......................................... 28
2.3.3.1 Yield surfaces on AlMg 3 ................................. 28
2.3.3.2 Yield surfaces on copper ................................. 30
2.3.3.3 Yield surfaces on steel ................................... 30
2.4 Sequence Effects ....................................... 31
2.5 Summary ............................................ 34
Acknowledgements ..................................... 35
References ........................................... 35
V

Contents
3 Plasticity of Metals and Life Prediction in the Range
of Low-Cycle Fatigue: Description of Deformation
Behaviour and Creep-Fatigue Interaction ................... 37
Kyong-Tschong Rie, Henrik Wittke, Jürgen Olfe
Abstract ............................................. 37
3.1 Introduction .......................................... 37
3.2 Experimental Details .................................... 38
3.2.1 Experimental details for room-temperature tests ................ 38
3.2.2 Experimental details for high-temperature tests ................. 39
3.3 Tests at Room Temperature: Description
of the Deformation Behaviour ............................. 40
3.3.1 Macroscopic test results .................................. 40
3.3.2 Microstructural results and interpretation ..................... 43
3.3.3 Phenomenological description of the deformation behaviour ....... 45
3.3.3.1 Description of cyclic hardening curve, cyclic stress-strain curve
and hysteresis-loop ..................................... 45
3.3.3.2 Description of various hysteresis-loops with few constants ......... 47
3.3.4 Physically based description of deformation behaviour ........... 47
3.3.4.1 Internal stress measurement and cyclic proportional limit .......... 47
3.3.4.2 Description of cyclic plasticity with the models
of Steck and Hatanaka .................................. 50
3.3.5 Application in the field of fatigue-fracture mechanics ............ 51
3.4 Creep-Fatigue Interaction ................................. 53
3.4.1 A physically based model for predicting LCF-life
under creep-fatigue interaction ............................. 53
3.4.1.1 The original model ..................................... 53
3.4.1.2 Modifications of the model ............................... 54
3.4.1.3 Experimental verification of the physical assumptions ............ 55
3.4.1.4 Life prediction ........................................ 55
3.4.2 Computer simulation and experimental verification
of cavity formation and growth during creep-fatigue ............. 57
3.4.2.1 Stereometric metallography ............................... 57
3.4.2.2 Computer simulation .................................... 58
3.4.2.3 Results .............................................. 59
3.4.3 In-situ measurement of local strain at the crack tip
during creep-fatigue .................................... 61
3.4.3.1 Influence of the crack length and the strain amplitude
on the local strain distribution ............................. 61
3.4.3.2 Comparison of the strain field in tension and compression ......... 62
3.4.3.3 Influence of the hold time in tension on the strain field ........... 63
3.5 Summary and Conclusions ............................... 64
References ........................................... 65
VI

Contents
4 Development and Application of Constitutive Models
for the Plasticity of Metals .............................. 68
Elmar Steck, Frank Thielecke, Malte Lewerenz
Abstract ............................................. 68
4.1 Introduction .......................................... 68
4.2 Mechanisms on the Microscale ............................ 69
4.3 Simulation of the Development of Dislocation Structures .......... 71
4.4 Stochastic Constitutive Model ............................. 73
4.5 Material-Parameter Identification ........................... 77
4.5.1 Characteristics of the inverse problem ....................... 77
4.5.2 Multiple-shooting methods ............................... 77
4.5.3 Hybrid optimization of costfunction ......................... 77
4.5.4 Statistical analysis of estimates and experimental design .......... 79
4.5.5 Parallelization and coupling with Finite-Element analysis .......... 79
4.5.6 Comparison of experiments and simulations ................... 81
4.5.7 Consideration of experimental scattering ...................... 82
4.6 Finite-Element Simulation ................................ 83
4.6.1 Implementation and numerical treatment of the model equations .... 83
4.6.1.1 Transformation of the tensor-valued equations .................. 84
4.6.1.2 Numerical integration of the differential equations ............... 85
4.6.1.3 Approximation of the tangent modulus ....................... 86
4.6.2 Deformation behaviour of a notched specimen ................. 86
4.7 Conclusions .......................................... 88
References ........................................... 88
5 On the Physical Parameters Governing the Flow Stress
of Solid Solutions in a Wide Range of Temperatures ........... 90
Christoph Schwink, Ansgar Nortmann
Abstract ............................................. 90
5.1 Introduction .......................................... 90
5.2 Solid Solution Strengthening .............................. 92
5.2.1 The critical resolved shear stress, s o ......................... 92
5.2.2 The hardening shear stress, s d ............................. 92
5.3 Dynamic Strain Ageing (DSA) ............................ 93
5.3.1 Basic concepts ........................................ 93
5.3.2 Complete maps of stability boundaries ....................... 94
5.3.3 Analysis of the processes inducing DSA ...................... 97
5.3.4 Discussion ........................................... 99
5.4 Summary and Relevance for the Collaborative Research Centre ..... 102
References ........................................... 102
VII

Contents
6 Inhomogeneity and Instability of Plastic Flow
in Cu-Based Alloys .................................... 104
Hartmut Neuhäuser
6.1 Introduction .......................................... 104
6.2 Some Experimental Details ............................... 105
6.3 Deformation Processes around Room Temperature .............. 106
6.3.1 Development of single slip bands ........................... 106
6.3.2 Development of slip band bundles and Lüders band propagation .... 112
6.3.3 Comparison of single crystals and polycrystals ................. 116
6.3.4 Conclusion ........................................... 117
6.4 Deformation Processes at Intermediate Temperatures ............. 118
6.4.1 Analysis of single stress serrations .......................... 118
6.4.2 Analysis of stress-time series .............................. 121
6.4.3 Conclusion ........................................... 124
6.5 Deformation Processes at Elevated Temperatures ................ 124
6.5.1 Dynamical testing and stress relaxation ....................... 124
6.5.2 Creep experiments ..................................... 126
6.5.3 Conclusion ........................................... 128
Acknowledgements ..................................... 128
References ........................................... 129
7 The Influence of Large Torsional Prestrain on the Texture
Development and Yield Surfaces of Polycrystals .............. 131
Dieter Besdo, Norbert Wellerdick-Wojtasik
7.1 Introduction .......................................... 131
7.2 The Model of Microscopic Structures ........................ 131
7.2.1 The scale of observation ................................. 131
7.2.2 Basic slip mechanism in single crystals ...................... 132
7.2.3 Treatment of polycrystals ................................. 133
7.2.4 The Taylor theory in an appropriate version ................... 133
7.3 Initial Orientation Distributions ............................ 135
7.3.1 Criteria of isotropy ..................................... 135
7.3.2 Strategies for isotropic distributions ......................... 136
7.4 Numerical Calculation of Yield Surfaces ...................... 137
7.5 Experimental Investigations ............................... 140
7.5.1 Prestraining of the specimens .............................. 140
7.5.2 Yield-surface measurement ............................... 141
7.5.3 Tensile test of a prestrained specimen ........................ 142
7.5.4 Measured yield surfaces ................................. 143
7.5.5 Discussion of the results ................................. 146
7.6 Conclusion ........................................... 146
References ........................................... 147
VIII

Contents
8 Parameter Identification of Inelastic Deformation Laws Analysing
Inhomogeneous Stress-Strain States ....................... 149
Reiner Kreißig, Jochen Naumann, Ulrich Benedix, Petra Bormann,
Gerald Grewolls, Sven Kretzschmar
8.1 Introduction .......................................... 149
8.2 General Procedure ...................................... 149
8.3 The Deformation Law of Inelastic Solids ..................... 150
8.4 Bending of Rectangular Beams ............................ 152
8.4.1 Principle ............................................. 152
8.4.2 Experimental technique .................................. 152
8.4.3 Evaluation ........................................... 155
8.4.3.1 Determination of the yield curves ........................... 155
8.4.3.2 Determination of the initial yield-locus curve .................. 158
8.5 Bending of Notched Beams ............................... 160
8.5.1 Principle ............................................. 160
8.5.2 Experimental technique .................................. 161
8.5.3 Approximation of displacement fields ........................ 163
8.6 Identification of Material Parameters ........................ 165
8.6.1 Integration of the deformation law .......................... 165
8.6.2 Objective function, sensitivity analysis and optimization .......... 167
8.6.3 Results of parameter identification .......................... 169
8.7 Conclusions .......................................... 170
Acknowledgements ..................................... 172
References ........................................... 173
9 Development and Improvement of Unified Models
and Applications to Structural Analysis .................... 174
Hermann Ahrens, Heinz Duddeck, Ursula Kowalsky,
Harald Pensky, Thomas Streilein
9.1 Introduction .......................................... 174
9.2 On Unified Models for Metallic Materials .................... 174
9.2.1 The overstress model by Chaboche and Rousselier .............. 175
9.2.2 Other unified models .................................... 177
9.3 Time-Integration Methods ................................ 178
9.4 Adaptation of Model Parameters to Experimental Results ......... 181
9.5 Systematic Approach to Improve Material Models ............... 186
9.6 Models Employing Distorted Yield Surfaces ................... 190
9.7 Approach to Cover Stochastic Test Results .................... 197
9.8 Structural Analyses ..................................... 201
9.8.1 Consistent formulation of the coupled boundary
and initial value problem ................................. 202
9.8.2 Analysis of stress-strain fields in welded joints ................. 203
9.8.3 Thick-walled rotational vessel under inner pressure .............. 205
IX

Contents
9.8.4 Application of distorted yield functions ...................... 206
9.8.5 Application of the statistical approach of Section 9.7 ............. 209
9.8.6 Numerical analysis for a recipient of a profile extrusion press ...... 212
Acknowledgements ..................................... 214
References ........................................... 215
10 On the Behaviour of Mild Steel Fe 510
under Complex Cyclic Loading ........................... 218
Udo Peil, Joachim Scheer, Hans-Joachim Scheibe,
Matthias Reininghaus, Detlef Kuck, Sven Dannemeyer
10.1 Introduction .......................................... 218
10.2 Material Behaviour ..................................... 219
10.2.1 Material, experimental set-ups, and techniques ................. 219
10.2.2 Material behaviour under uniaxial cyclic loading ................ 219
10.2.2.1 Parameters ........................................... 219
10.2.2.2 Results of the uniaxial experiments ......................... 220
10.2.3 Material behaviour under biaxial cyclic loading ................ 225
10.2.3.1 Parameters ........................................... 225
10.2.3.2 Relations of tensile and torsional stresses ..................... 226
10.2.3.3 Yield-surface investigations ............................... 229
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510 ........ 236
10.3.1 Extended-two-surface model .............................. 236
10.3.1.1 General description ..................................... 236
10.3.1.2 Loading and bounding surface ............................. 237
10.3.1.3 Strain-memory surfaces .................................. 238
10.3.1.4 Internal variables for the description on non-proportional loading .... 241
10.3.1.5 Size of the yield surface under uniaxial cyclic plastic loading ...... 242
10.3.1.6 Size of the bounding surface under uniaxial cyclic plastic loading . . . 242
10.3.1.7 Overshooting ......................................... 242
10.3.1.8 Additional update of d in in the case of biaxial loading ............ 243
10.3.1.9 Memory surface F' .................................... 243
10.3.1.10 Additional isotropic deformation on the loading surface
due to non-proportional loading ............................ 244
10.3.1.11 Additional isotropic deformation of the bounding surface
due to non-proportional loading ............................ 244
10.3.2 Comparison between theory and experiments .................. 248
10.4 Experiments on Structural Components ...................... 248
10.4.1 Experimental set-ups and computational method ................ 248
10.4.2 Correlation between experimental and theoretical results .......... 248
10.5 Summary ............................................ 251
References ........................................... 252
X

Contents
11 Theoretical and Computational Shakedown Analysis
of Non-Linear Kinematic Hardening Material
and Transition to Ductile Fracture ........................ 253
Erwin Stein, Genbao Zhang, Yuejun Huang,
Rolf Mahnken, Karin Wiechmann
Abstract ............................................. 253
11.1 Introduction .......................................... 253
11.1.1 General research topics .................................. 253
11.1.2 State of the art at the beginning of project B6 .................. 254
11.1.3 Aims and scope of project B6 ............................. 254
11.2 Review of the 3-D Overlay Model .......................... 256
11.3 Numerical Approach to Shakedown Problems .................. 259
11.3.1 General considerations .................................. 259
11.3.2 Perfectly plastic material ................................. 260
11.3.2.1 The special SQP-algorithm ............................... 260
11.3.2.2 A reduced basis technique ................................ 261
11.3.3 Unlimited kinematic hardening material ...................... 261
11.3.4 Limited kinematic hardening material ........................ 263
11.3.5 Numerical examples .................................... 264
11.3.5.1 Thin-walled cylindrical shell .............................. 264
11.3.5.2 Steel girder with a cope .................................. 265
11.3.5.3 Incremental computations of shakedown limits
of cyclic kinematic hardening material ....................... 267
11.4 Transition to Ductile Fracture ............................. 269
11.5 Summary of the Main Results of Project B6 ................... 272
References ........................................... 273
12 Parameter Identification for Inelastic Constitutive Equations Based
on Uniform and Non-Uniform Stress and Strain Distributions ... 275
Rolf Mahnken, Erwin Stein
Abstract ............................................. 275
12.1 Introduction .......................................... 275
12.1.1 State of the art at the beginning of project B8 .................. 275
12.1.2 Aims and scope of project B8 ............................. 276
12.2 Basic Terminology for Identification Problems ................. 277
12.2.1 The direct problem: the state equation ....................... 277
12.2.2 The inverse problem: the least-squares problem ................. 278
12.3 Parameter Identification for the Uniform Case .................. 280
12.3.1 Mathematical modelling of uniaxial visco-plastic problems ........ 280
12.3.2 Numerical solution of the direct problem ..................... 282
12.3.3 Numerical solution of the inverse problem .................... 282
12.4 Parameter Identification for the Non-Uniform Case .............. 283
12.4.1 Kinematics ........................................... 284
XI

Contents
12.4.2 The direct problem: Galerkin weak form ..................... 285
12.4.3 The inverse problem: constrained least-squares optimization problem . 286
12.5 Examples ............................................ 287
12.5.1 Cyclic loading for AlMg ................................. 287
12.5.2 Axisymmetric necking problem ............................ 290
12.6 Summary and Concluding Remarks ......................... 294
References ........................................... 296
13 Experimental Determination of Deformation- and Strain Fields
by Optical Measuring Methods ........................... 298
Reinhold Ritter, Harald Friebe
13.1 Introduction .......................................... 298
13.2 Requirements of the Measuring Methods ..................... 298
13.3 Characteristics of the Optical Field-Measuring Methods ........... 299
13.4 Object-Grating Method .................................. 300
13.4.1 Principle ............................................. 300
13.4.2 Marking ............................................. 301
13.4.3 Deformation analysis at high temperatures .................... 302
13.4.4 Compensation of virtual deformation ........................ 303
13.4.5 3-D deformation measuring ............................... 305
13.4.6 Specifications of the object-grating method .................... 305
13.5 Speckle Interferometry .................................. 305
13.5.1 General ............................................. 305
13.5.2 Technology of the Speckle interferometry ..................... 307
13.5.3 Specifications of the developed 3-D Speckle interferometer ........ 308
13.6 Application Examples ................................... 309
13.6.1 2-D object-grating method in the high-temperature area ........... 309
13.6.2 3-D object-grating method in fracture mechanics ................ 309
13.6.3 Speckle interferometry in welding .......................... 310
13.7 Summary ............................................ 313
References ........................................... 317
14 Surface-Deformation Fields from Grating Pictures
Using Image Processing and Photogrammetry ................ 318
Klaus Andresen
14.1 Introduction .......................................... 318
14.2 Grating Coordinates .................................... 319
14.2.1 Cross-correlation method ................................. 319
14.2.2 Line-following filter .................................... 321
14.3 3-D Coordinates by Imaging Functions ....................... 324
14.4 3-D Coordinates by Close-Range Photogrammetry .............. 325
14.4.1 Experimental set-up .................................... 325
XII

Contents
14.4.2 Parameters of the camera orientation ........................ 326
14.4.3 3-D object coordinates .................................. 327
14.5 Displacement and Strain from an Object Grating: Plane Deformation . 328
14.6 Strain for Large Spatial Deformation ........................ 329
14.6.1 Theory .............................................. 329
14.6.2 Correcting the influence of curvature ........................ 332
14.6.3 Simulation and numerical errors ............................ 333
14.7 Conclusion ........................................... 335
References ........................................... 335
15 Experimental and Numerical Analysis of the Inelastic
Postbuckling Behaviour of Shear-Loaded Aluminium Panels ..... 337
Horst Kossira, Gunnar Arnst
15.1 Introduction .......................................... 337
15.2 Numerical Model ...................................... 339
15.2.1 Finite-Element method .................................. 339
15.2.1.1 Ambient temperature – rate-independent problem ............... 340
15.2.1.2 Elevated temperature – visco-plastic problem .................. 341
15.2.2 Material models ....................................... 341
15.2.2.1 Ambient temperature – rate-independent problem ............... 341
15.2.2.2 Elevated temperature – visco-plastic problem .................. 344
15.3 Experimental and Numerical Results ........................ 349
15.3.1 Test procedure ........................................ 349
15.3.2 Computational analysis .................................. 349
15.3.2.1 Monotonic loading – ambient temperature .................... 350
15.3.2.2 Cyclic loading – ambient temperature ........................ 351
15.3.2.3 Time-dependent behaviour ................................ 356
15.4 Conclusion ........................................... 358
List of Symbols ....................................... 359
References ........................................... 360
16 Consideration of Inhomogeneities in the Application
of Deformation Models, Describing the Inelastic Behaviour
of Welded Joints ...................................... 361
Helmut Wohlfahrt, Dirk Brinkmann
16.1 Introduction .......................................... 361
16.2 Materials and Numerical Methods .......................... 362
16.2.1 Materials and welded joints ............................... 362
16.2.2 Deformation models and numerical methods ................... 365
16.2.2.1 Deformation model of Gerdes ............................. 365
16.2.2.2 Fitting calculations ..................................... 365
16.3 Investigations with Homogeneous Structures ................... 365
XIII

Contents
16.3.1 Experimental and numerical investigations .................... 366
16.3.1.1 Tensile tests .......................................... 366
16.3.1.2 Creep tests ........................................... 369
16.3.1.3 Cyclic tension-compression tests ........................... 370
16.3.2 Discussion ........................................... 372
16.4 Investigations with Welded Joints ........................... 374
16.4.1 Deformation behaviour of welded joints ...................... 375
16.4.1.1 Experimental investigations ............................... 375
16.4.1.2 Numerical investigations ................................. 375
16.4.1.3 Finite-Element models of welded joints ...................... 375
16.4.1.4 Calculation of the deformation behaviour of welded joints ......... 375
16.4.2 Strain distributions of welded joints with broad weld seams ........ 376
16.4.3 Strain distributions of welded joints with small weld seams ........ 380
16.4.4 Discussion ........................................... 380
16.5 Application Possibilities and Further Investigations .............. 382
References ........................................... 383
Bibliography .................................................. 384
XIV

Preface
The Collaborative Research Centre (Sonderforschungsbereich, SFB 319), “Material
Models for the Inelastic Behaviour of Metallic Materials – Development and Technical
Application”, was supported by the Deutsche Forschungsgemeinschaft (DFG) from July
1985 until the end of the year 1996. During this period of nearly 12 years, scientists
from the disciplines of metal physics, materials sciences, mechanics and applied engineering
sciences cooperated with the aim to develop models for metallic materials on a
physically secured basis. The cooperation has resulted in a considerable improvement
of the understanding between the different disciplines, in many new theoretical and experimental
methods and results, and in technically applicable constitutive models as
well as new knowledge concerning their application to practical engineering problems.
The cooperation within the SFB was supported by many contacts to scientists and
engineers at other universities and research institutes in Germany as well as abroad.
The authors of this report about the results of the SFB 319 wish to express their thanks
to the Deutsche Forschungsgemeinschaft for the financial support and the very constructive
cooperation, and to all the colleagues who have contributed by their interest
and their function as reviewers and advisors to the results of our research work.
Introduction
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
The development of mathematical models for the behaviour of technical materials is of
course directed towards their application in the practical engineering work. Besides the
projects, which have the technical application as their main goal, in all projects, which
were involved in experiments with homogeneous or inhomogeneous test specimens –
where partly also the numerical methods were further investigated and the implementation
of the material models in the programs was performed –, experiences concerning
the application of the models for practical problems could be gained. The whole-field
methods for measuring displacement and strain fields, which were developed in connection
with these experiments, have given valuable support concerning the application
of the developed constitutive models to practical engineering.
The research concerning the identification of the parameters of the models has
proven to be very actual. The investigations for most efficient methods for the parameter
identification will in the future still find considerable attention, where the cooperation
of scientists from engineering as well as applied mathematics, which was started in
the SFB, will continue. As is shown in a later chapter, it is of increasing importance to
XV

use not only homogeneously, uniaxially loaded test specimen, but also to analyze stress
and deformation fields in complexly loaded components. In connection with these investigations,
methods for the design of experiments should be developed, which can be
used for the assessment of the structure of the material models and the physical meaning
of the model parameters. The results obtained up to now have shown, also by comparisons
in cooperation with institutions outside the SFB, that the predictive properties
of the developed material models are of equal quality as those of other models used in
the engineering practice. They have however the advantage that they are based on results
of material physics and therefore can use further developments of the knowledge
about the mechanisms of inelastic deformations on the microscale.
During the work in the different projects, a surprising number of similar problems
have been found. Due to the close contacts between the working groups, they could be
investigated with much higher quality than without this cooperation.
The exchange of thought between metal physics, materials sciences, mechanics
and applied engineering sciences was very stimulating and has resulted in the fact that
the groups oriented towards application could be supported by the projects working
theoretically, and on the other hand, the scientists working in theoretical fields could
observe the application of their results in practical engineering.
Research Program
Preface
The main results of the activities of the SFB have been models for the load-deformation
behaviour as well as for damage development and the development of deformation
anisotropies. These models make it possible to use results from the investigations from
metal physics and materials sciences in the SFB in the continuum mechanics models.
The research work in metal physics and materials sciences has considerably contributed
to a qualitative understanding of the processes, which have to be described by constitutive
models. The structure of the developed models and of the formulations found in
literature, which have been considered for comparisons and supplementation of our
own development, have strongly influenced the work concerning the implementation of
the material models in numerical computing methods and the treatment of technical
problems. The models could be developed to a status, where the results of experimental
investigations can be used to determine the model parameters quantitatively.
This has resulted in an increasing activity on the experimental side of the work
and also in an increase of the cooperation within the SFB and with institutions outside
of Braunschweig (BAM Berlin, TU Hamburg-Harburg, TH Darmstadt, RWTH Aachen,
KFA Jülich, KFZ Karlsruhe, École Polytechnique Lausanne). In the SFB, joint research
was undertaken in the fields of high-temperature experiments for the investigation of
creep, cyclic loading and non-homogeneous stress and displacement fields for technical
important metallic materials, and their comparison with theoretical predictions. The developed
whole-field methods for measuring deformations have shown to be an impor-
XVI

tant experimental method. The increasing necessity to obtain experimental results of
high quality for testing and extending the material models has resulted in the development
of experimental equipment, which also allows to investigate the material behaviour
under multiaxial loadings in the high- and low-temperature range.
The determination of model parameters and process quantities from experiments
has put the question for reliable methods for the parameter identification in the foreground.
The earlier used methods of least-squares and probabilistic methods, such as
the evolution strategy, have given satisfying results. In the SFB, however, the knowledge
has developed that methods for the parameter identification, which consider the
structure of the material models and the design of optimal experiments and discriminating
experiments, deserve special consideration.
If numerical values for the model parameters are given, the possibility exists to
examine these values concerning their physical meaning, and in cooperation with the
scientists from metal physics and materials sciences to investigate the connection between
the knowledge about the processes on the microscale and the macroscopic constitutive
equations.
The SFB was during its activities organized essentially in three project areas:
A: Materials behaviour
• Phenomena
• Material models
• Parameter identification
B: Development of computational methods
• General computational methods under consideration of the developed material models
• Special computational methods (e.g. shells structures, structural optimization, shakedown)
C: Experimental verification
• Whole-field methods
• Examination of the transfer of results
• Mock-up experiments.
Project area A: materials behaviour
Research Program
The research in the project area A was mainly concerned with theoretical and experimental
investigations concerning the basis for the development of material models and
damage development from metal physics and materials sciences. In the following, a
short description of the activities within the research projects is given. Methods and results
are in detail given in later chapters.
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Correlation between energetic and mechanical quantities of face-centred cubic
metals, cold-worked and softened to different states (Kaps, Haeßner)
One of these basic investigations is concerned with calorimetric measurements in connection
with the description of recovery. After measurements based on the sheet rolling
process, final investigations were performed concerning higher deformation temperatures
and more complex deformation processes. Here, torsion experiments were examined
due to the fact that this process allows the investigation of very high deformations
as well as a simple reversal of the deformation direction and cyclic experiments.
Recovery and recrystallization are in direct competition with strain hardening. If a
material is cold-worked, its yield stress increases. This process, denoted strain hardening,
leads to a gain in internal energy. Recovery and recrystallization act to oppose
strain hardening. Already upon deformation or during subsequent annealing, these
forces transform the material back into a state of lower energy. Although this reciprocity
has been known for some time, the exact dependence of the process upon the type
and extent of deformation, upon the temperatures during deformation and softening anneal
as well as upon the chemical composition of the material is as yet only qualitatively
known. Consequently, the predictability of the processes is as poor as it has always
been so that, even today, one is still obliged to refer to experience and explicit
experiments for help.
Material state after uni- and biaxial cyclic deformation (Gieseke, Hillert, Lange)
The investigations concerning the material behaviour at multiaxial plastic deformation
were performed using the material AlMg 3, copper and the austenitic stainless steel AISI
316L. To find the connection between damage development and microstructure, the dislocations
structure at the tip of small cracks and at surface grains with differently pronounced
slip-band development was investigated. With the aim to check the main assumptions
of the two-surface models explicitly, measurements of the development of
the yield surface of the material from the initial to the saturation state and within a saturation
cycle were considerably extended. Consecutive yield surfaces along different
loading histories were measured. The two-surface models of Ellyin and McDowell
were implemented in the computations.
Technical components and structures today are increasingly being designed and
displayed by computer-aided methods. High speed computers permit the use of mathematical
models able to numerically reconstruct material behaviour, even in the course
of complex loading procedures.
In phenomenological continuum mechanics, the cyclic hardening and softening
behaviour as well as the Bauschinger effect are described by yield-surface models. If a
physical formulation is chosen as a basis for these models, then it is vitally important
to have exact knowledge of the processes occurring in the metal lattice during deformation.
Two-surface models, going back to a development by Dafalias and Popov, describe
the displacement of the elastic deformation zone in a dual axis stress area. The
yield surfaces are assumed to be v. Mises shaped ellipses. However, from experiments
with uniaxial loading, it is known that the yield surfaces of small offset strains under
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Research Program
load become characteristically deformed. In the present subproject, the effect of cyclic
deformation on the shape and position of the yield surfaces is studied, and their relation
to the dislocation structure is determined. To this end, the yield surfaces of three
materials with different slip behaviour were measured after prior uni- or biaxial deformation.
The influence of the dislocation structures produced and the effect of internal
stresses are discussed.
Plasticity of metals and life prediction in the range of low-cycle fatigue: description of
deformation behaviour and creep-fatigue interaction (Rie, Wittke, Olfe)
In the field of investigations about the connection between creep and low-cycle fatigue,
the development of models for predicting the componente lifetime at creep fatigue was
the main aim of the work. Measuring the change of the physical magnitudes in the
model during an experiment results in an investigation and eventually a modification of
the model assumptions. The model was also examined for its usability for experiments
with holding-times at the maximum pressure loading during a loading cycle.
For hot working tools, chemical plants, power plants, pressure vessels and turbines,
one has to consider local plastic deformation at critical locations of structural
components. Due to cyclic changes of temperature and load, the components are subjected
to cyclic deformation, and the components are limited in their use by fatigue.
After a quite small number of cycles with cyclic hardening or softening, a state of cyclic
saturation is reached, which can be characterized by a stress-strain hysteresis-loop.
Cyclic deformation in the regime of low-cycle fatigue (LCF) leads to the formation of
cracks, which can subsequently grow until failure of a component part takes place.
In the field of fatigue fracture mechanics, crack growth is correlated with parameters,
which take into account information especially about the steady-state stress-strain
hysteresis-loops. Therefore, it can be expected that a more exact life prediction is possible
by a detailed investigation of the cyclic deformation behaviour and by the description
of the cyclic plasticity, e.g. with constitutive equations.
At high temperatures, creep deformation and creep damage are often superimposed
on the fatigue process. Therefore, in many cases, not one type of damage prevails,
but the interaction of both fatigue and creep occurs, leading to failure of components.
The typical damage in the low-cycle fatigue regime is the development and
growth of cracks. In the case of creep fatigue, grain boundary cavities may be formed,
which interact with the propagating cracks, this leading to creep-fatigue interaction. A
reliable life prediction model must consider this interaction.
The knowledge and description of the cavity formation and growth by means of
constitutive equations are the basis for reliable life prediction. In the case of diffusion-controlled
cavity growth, the distance between the voids has an important influence on their
growth. This occurs especially in the case of low-cycle fatigue, where the cavity formation
plays an important role. Thus, the stochastic process of void nucleation on grain boundaries
and the cyclic dependence of this process has to be taken into consideration as a
theoretical description. The experimental analysis has to detect the cavity-size distribution,
which is a consequence of the complex interaction between the cavities.
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Preface
Up to now, only macroscopic parameters such as the total stress and strain have
been used for the calculation of the creep-fatigue damage. But crack growth is a local
phenomenon, and the local conditions near the crack tip have to be taken into consideration.
Therefore, the determination of the strain fields in front of cracks is an important
step for modelling.
Development and application of constitutive models for the plasticity of metals (Steck,
Thielecke, Lewerenz)
The inelastic material behaviour in the low- and high-temperature ranges is caused by
slip processes in the crystal lattice, which are supported by the movement of lattice defects
like dislocations and dislocation packages. The dislocation movements are opposed
by internal barriers, which have to be overcome by activation. This is performed
by stresses or thermal energy. During the inelastic deformation, the dislocations interact
and arrange in a hierarchy of structures such as walls, adders and cells. This forming of
internal material structures influences strongly the macroscopic responses on mechanical
and thermal loading.
A combination of models on the basis of molecular dynamics and cellular automata
is used to study numerically the forming of dislocation patterns and the evolution
of internal stresses during the deformation processes. For a realistic simulation, several
glide planes are considered, and for the calculation of the forces acting on a dislocation,
a special extended neighbourhood is necessary. The study of the self-organization
processes with the developed simulation tool can result in valuable information for the
choice of formulations for the modelling of processes on the microscale.
The investigations concerning the development of material models based on
mechanisms on the microscale have resulted in a unified stochastic model, which is
able to represent essential and typical features of the low- and high-temperature plasticity.
For the modelling of the dislocation movements in crystalline materials and their
temperature and stress activation, a discrete Markov chain is considered. In order to describe
cyclic material behaviour, the widely accepted concept is used that the dislocation-gliding
processes are driven by the effective stress as the difference between the
applied stress and the internal back stress. The influence of effective stress and temperature
on the inelastic deformations is considered by a metalphysically motivated
evolution equation. A mean value formulation of this stochastic model leads to a
macroscopic model consisting of non-linear ordinary differential equations. The results
show that the stochastic theory is helpful to deduce the properties of the macroscopic
constitutive equations from findings on the microscale.
Since the general form of the stochastic model must be adapted to the special
material characteristics and the considered temperature regime, the identification of the
unknown material parameters plays an important role for the application on numerical
calculations. The determination of the unknown material parameters is based on a Maximum-Likelihood
output-error method comparing experimental data to the numerical simulations.
For the minimization of the costfunction, a hybrid optimization concept parallelized
with PVM is considered. It couples stochastic search procedures and several
Newton-type methods. A relative new approach for material parameter identification is
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the multiple shooting approach, which allows to make efficient use of additional measurement-
and apriori-information about the states. This reduces the influence of bad initial
parameters. Since replicated experiments for the same laboratory conditions show a significant
scattering, these uncertainties must be taken into account for the parameter identification.
The reliability of the results can be tested with a statistical analysis.
Several different materials, like aluminium, copper, stainless steel AISI 304 and
AISI 316, have been studied. For the analysis of structures, like a notched flat bar, the
Finite-Element program ABAQUS is used in combination with the user material subroutine
UMAT. The simulations are compared with experimental data from grating
methods.
On the physical parameters governing the flow stress of solid solutions in a wide range
of temperatures (Schwink, Nortmann)
In the area of the metal-physical foundations, investigations on poly- and single-crystalline
material have been performed. The superposition of solution hardening and ordinary
hardening has found special consideration. Along the stress-strain curves, the limits
between stable and unstable regions of deformation were investigated, and their dependencies
on temperature, strain rate and solute concentration were determined. In regions
of stable deformation, a quantitative analysis of the processes of dynamic strain
ageing (“Reckalterung”) was performed. The transition between regions of stable and
unstable deformation was investigated and characterized.
At sufficiently low temperatures, host and solute atoms remain on their lattice
sites. The critical flow stress is governed by thermally activated dislocations glide (Arrhenius
equation), which depends on an average activation enthalpy DG0, and an effective
obstacle concentration cb. The total flow stress is composed of the critical flow
stress and a hardening stress, which increases with the dislocation density in the cell
walls.
Detailed investigations on single crystals yielded expressions for the critical
resolved shear stress, s0 ˆ s0…DG0; cb; T; _e†, and the hardening shear stress,
sd ˆ wGbq1=2 w . Here, w is a constant, w ˆ 0:25 0:03, G the shear modulus, and
qw the dislocation density inside the cell walls. The total shear stress results as
s ˆ s0 ‡ sd.
At higher temperatures, the solutes become mobile in the lattice and cause an additional
anchoring of the glide dislocations. This is described by an additional enthalpy
Dg…tw; Eam† in the Arrhenius equation. In the main, it depends on the activation energy
Eam of the diffusing solutes and the waiting time tw of the glide dislocations arrested at
obstacles. Three different diffusion processes characterized by EaI; EaII; EaIII were found
for the two f.c.c.-model systems investigated, CuMn and CuAl, respectively. In both,
Dg reaches values up to about 0.1 DG0. Under certain conditions, the solute diffusion
causes instabilities in the flow stress, the well-known jerky flow phenomena (Portevin-
Le Châtelier effect). Finally, above around 800 K in copper-based alloys, the solutes become
freely mobile, and the critical flow stress as well as the additional enthalpy vanish.
In any temperature region, only a small total number of physical parameters is sufficient
for modelling plastic deformation processes.
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Preface
Inhomogeneity and instability of plastic flow in Cu-based alloys (Neuhäuser)
In a second project, the main goal of the research is to clarify the physical mechanisms,
which control the kinetics of the deformation, especially in such parameter regions,
which are characterized by inhomogeneity and instability of the deformation process.
It is looked for a realistic interpretation of the magnitudes, which will be used
with empirical material equations as it is necessary for a sensible application and extrapolation
to extended parameter regions. Especially, reasons and effects of deformationinhomogeneities
and -instabilities in the systems Cu-Al and Cu-Mn, which show tendencies
to short-range order, were investigated. Determining dislocation-generation
rates and dislocation velocities in the case of gradients of the effective stress were as
well aim of the investigations as the influence of diffusion processes on the generation
(blocking, break-away) and motion (obstacle destruction and regeneration) of dislocations.
Investigations were also performed concerning the use of the results for single
crystals for the description of the practically more important case of the behaviour of
polycrystals. In this case, especially the influence of the grain-boundaries on generation
and movement of dislocations or dislocation groups has to be considered.
The special technique used in this project is a microcinematographic method,
which permits to measure the local strain and strain rate in slip bands, which are the
active regions of the crystal. Cu-based alloys with several percent of Al and Mn solutes
are considered in order to separate the effects of stacking-fault energy from those of solute
hardening and short-range ordering, which are comparable for both alloy systems,
while the stacking-fault energy decreases rapidly with solute concentration for CuAl
contrary to CuMn alloys. Both systems show different degrees of inhomogeneous slip
in the length scales from nm to mm (slip bands, Lüders bands), and, in a certain range
of deformation conditions, macroscopic deformation instabilities (Portevin-Le Châtelier
effect). These effects have been studied in particular.
The influence of large torsional prestrain on the texture development and yield surface
of polycrystals – experimental and theoretical investigations (Besdo, Wellerdick-Wojtasik)
This research project consists of a theoretical and an experimental part. The topic of
the theoretical part was the simulation of texture development and methods of calculating
yield surfaces. The calculations started from an initially isotropic grain distribution.
Therefore, it was necessary to set up such a distribution. Different possibilities were
compared with an isotropy test considering the elastic and plastic properties. With some
final distributions, numerical calculations were carried out. The Taylor theory in an appropriate
version and a simple formulation based on the Sachs assumption were used.
Calculation of yield surfaces from texture data can be done in many different ways.
Some examples are the yield surfaces calculated with the Taylor theory, averaging methods
or formulations, which take the elastic behaviour into account. Several possibilities
are presented, and the numerical calculations are compared with the experimental results.
In order to measure yield surfaces after large torsional prestrain, thin-walled tubular
specimens of AlMg 3 were loaded up to a shear strain of c ˆ 1:5, while torsional
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Research Program
buckling was prevented by inserting a greased mandrel inside the specimens. Further
investigations of the prestrained specimens were done with the testing machine of the
project area B.
At least one yield surface, represented by 16 yield points, was measured with
each specimen. The yield point is defined by the offset-strain definition, where generally
the von Mises equivalent offset strain is used. Three different loading paths were
realized with the extension-controlled testing machine. Thus, the results were yield surfaces
measured with different offsets and loading paths.
The offset-strain definition is based on the elastic tensile and shear modulus.
These constants were calculated at the beginning of each loading path, and since they
strongly effect the yield surfaces, this must be done with the highest amount of care.
The isotropic specimens are insensitive to different loading paths, and the measured
yield surfaces seem to be of the von Mises type. By contrast, the prestrained specimens
are very sensitive to different loading paths. Especially the shape and the distorsion of
the measured surfaces changes as a result of the small plastic strain during the measurement.
Therefore, it seems that the shape and the distortion of the yield surface were not
strongly effected by the texture of the material.
Parameter identification of inelastic deformation laws analysing inhomogeneous stressstrain
states (Kreißig, Naumann, Benedix, Borman, Grewolls, Kretzschmar)
In the last years, the necessity of solutions of non-linear solid mechanics problems has
permanently increased. Although powerful hard- and software exist for such problems,
often more or less large differences between numerical and experimental results are observed.
The dominant reason for these defects must be seen in the material-dependent
part of the used computer programs. Either suitable deformation laws are not implemented
or the required parameters are missing.
Experiments on the material behaviour are commonly realized for homogeneous
stress-strain states, as for example the uniaxial tensile and compression test or the thinwalled
tube under combined torsion, tensile and internal pressure loading. In addition
to these well-known methods, experimental studies of inhomogeneous strain and stress
fields are an interesting alternative to identify material parameters.
Two types of specimens have been investigated. Unnotched bending specimens
have been used to determine the elastic constants, the initial yield locus curve and the
uniaxial tension and compression yield curves. Notched bending specimens allow experiments
on the hardening behaviour due to inhomogeneous stress-strain states.
The numerical analysis has been carried out by the integration of the deformation
law at a certain number of comparative points of the ligament with strain increments,
determined from Moiré fringe patterns, as loads. The identification of material parameters
has been performed by the minimization of a least-squares functional using deterministic
gradient-type methods. As comparative quantities have been taken into account
the bending moment, the normal force and the stresses at the notch grooves.
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Preface
Project area B: development of computational methods
The essential goals of the project area B were the transfer of experimental results in
material models, which describe the essential characteristics of the complex non-linear
behaviour of metallic materials in a technically satisfactory manner. For this reason,
known formulations of material models, developments of the SFB and new formulations
had to be examined with respect to their validity and the limits of their efficiency.
To be able to describe processes on the microscale of the materials, the material models
contain internal variables, which can either be purely phenomenological or be based on
microstructural considerations. In the frame of the SFB, the goal was the microstructural
substantiation of these internal variables.
For the adjustment of the model parameters on the experimental results, optimization
strategies are necessary, which allow judging the power of the models. The obtained
results showed that this question is of high importance, also for further research.
Extensions for multiaxial loading cases have been developed and validated. For the investigated
loadings of metals at high temperatures and alternating and cyclic loading
histories as well as for significantly time-dependent material behaviour, the literature
shows only a first beginning in the research concerning such extensions.
The material models had firstly to be examined concerning the materials. For the
practical application, however, their suitability for their implementation in numerical algorithms
(e.g. Finite-Element methods) and the influence on the efficiency of numerical
computations had to be examined.
Especially for the computation of time-dependent processes, numerically stable
and – because of the expensive numerical calculations – efficient computational algorithms
had to be developed (e.g. fast converging time-integration methods for strongly
non-linear problems).
The developed (or chosen) material models and algorithms had to be applied for
larger structures, not only to test the computational models, but simultaneously also –
by reflection to the assumptions in the material models – to find out which parameters
are of essential meaning for the practical application, and which are rather unimportant
and can be neglected. This results in the necessity to perform on all levels sensitivity
investigations for the relevancy of the variants of the assumptions and their parameters.
At loading histories, which describe alternating or cyclic processes due to the alternating
plastification, the question of saturation of the stress-strain histories and
shakedown are of special importance. The projects in the project area B were investigating
these problems in a complementary manner. They were important, central questions
conceived so that related problems were investigated to accelerate the progress of
the work and to allow mutual support and critical exchange of thought.
Development and improvement of unified models and applications to structural analysis
(Ahrens, Duddeck, Kowalsky, Pensky, Streilein)
Especially for structures of large damage potentials, the design has to simulate failure
conditions as realistic as possible. Therefore, inelastic and time-dependent behaviour
such as temperature-induced creep have to be considered. Besides adequate numerical
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Research Program
methods of analyses (as non-linear Finite-Element methods), mathematically correct
models are needed for the thermal-mechanical material behaviour under complex loadings.
Unified models for metallic materials cover time-independent as well as time-dependent
reactions by a unified concept of elasto-viscoplasticity.
Research results are presented, which demonstrate further developments for unified
models in three different aspects. The methodical approach is shown firstly on the
level of the material model. Then, verifications of their applicability are given by utilizing
them in the analyses of structures. The three aspects are the following problems:
1. Discrepancies between results of experimental and numerical material behaviour
may be caused by
• insufficient or inaccurate parameters of the material model,
• inadequate material functions of the unified models,
• insufficient basic formulations for the physical properties covered by the model.
It is shown that more consistent formulations can be achieved for all these three
sources of deficits by systematic numerical investigations.
2. Most of the models for metallic materials assume yield functions of the v. Mises
type. For hardening, isotropic and/or kinematic evolutions are developed, that correspond
to affine expansions or simple shifting of the original yield surface, whereas
experimental results show a distinctive change of the shape of the yield surfaces
(rotated or dented) depending on the load path. To cover this material behaviour of
distorted yield surfaces, a hierarchical expansion of the hardening rule is proposed.
The evolutionary equations of the hardening (expressed in tensors) are extended by
including higher order terms of the tensorial expressions.
3. Even very accurately repeated tests of the same charge of a metallic material show a
certain scattering distribution of the experimental results. The investigation of test
series (provided by other projects of the SFB) proved that a normal Gaussian distribution
can be assumed. A systematic approach is proposed to deal with such experimental
deviations in evaluating the parameters of the material model.
The concepts in all of the three items are valid in general although the overstress
model by Chaboche and Rousselier is chosen here for convenience.
In verifying the conceptual improvements, it is necessary to provide accurate and
efficient procedures for time-integration processes and for the evaluation of the model
parameters via optimization. In both cases, different procedures are elaborately compared
with each other.
Results of the numerical analyses of different structures are given. They demonstrate
the efficiency of the proposed further developments by applying Finite-Element
methods for non-linear stress-displacement problems. This includes:
• investigations of welded joints with modifications of the layers of different microstructures,
• thick-walled vessels in order to demonstrate the effects of different formulations of
the material model on the stress-deformation fields of larger structures,
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Preface
• distorted yield functions to a plate with an opening,
• effects of stochastic distribution of material behaviour to a plate with openings,
• the application of material models based on microphysical mechanisms to a larger
vessel, the recipient of hot aluminium blocks for a profile extrusion press.
On the behaviour of mild steel Fe 510 under complex cyclic loading (Peil, Scheer,
Scheibe, Reininghaus, Kuck, Dannemeyer)
The employment of the plastic bearing capacity of structures has been recently allowed
in both national and international steel constructions standards. The ductile material behaviour
of mild steel allows a load-increase well over the elastic limit. To make use of
this effect, efficient algorithms, taking account of the plastic behaviour under cyclic or
random loads in particular, are an important prerequisite for a precise calculation of the
structure.
The basic elements of a time-independent material model, which allows to take
into account the biaxial or random load history for a mild steel under room temperature,
are presented. In a first step, the material response under cyclic or random loads
has to be determined. The fundamentals of an extended-two-surface model based on
the two-surface model of Dafalias and Popov are presented. The adaptations have been
made in accordance with the results of experiments under multiaxial cyclic loadings.
Finally, tests on structural components are performed to verify the results obtained
from the calculations with the described model.
Theoretical and computational shakedown analysis of non-linear kinematic hardening
material and transition to ductile fracture (Stein, Zhang, Huang, Mahnken, Wiechmann)
The response of an elastic-plastic system subjected to variable loadings can be very
complicated. If the applied loads are small enough, the system will remain elastic for
all possible loads. Whereas if the ultimate load of the system is attained, a collapse
mechanism will develop and the system will fail due to infinitely growing displacements.
Besides this, there are three different steady states, that can be reached while the
loading proceeds:
1. Incremental failure occurs if at some points or parts of the system, the remaining
displacements and strains accumulate during a change of loading. The system will
fail due to the fact that the initial geometry is lost.
2. Alternating plasticity occurs, this means that the sign of the increment of the plastic
deformation during one load cycle is changing alternately. Though the remaining
displacements are bounded, plastification will not cease, and the system fails locally.
3. Elastic shakedown occurs if after initial yielding plastification subsides, and the system
behaves elastically due to the fact that a stationary residual stress field is
formed, and the total dissipated energy becomes stationary. Elastic shakedown (or
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Research Program
simply shakedown) of a system is regarded as a safe state. It is important to know
whether a system under given variable loadings shakes down or not.
The research work is based on Melan’s static shakedown theorems for perfectly plastic
and linear kinematic hardening materials, and is extended to generally non-linear limited
hardening by a so-called overlay model, being the 3-D generalization of Neal’s 1-D model,
for which a theorem and a corollary are derived. Finite-Element method and adequate
optimization algorithms are used for numerical approach of 2-D problems. A new lemma
allows for the distinction between local and global failure. Some numerical examples illustrate
the theoretical results. The shakedown behaviour of a cracked ductile body is investigated,
where a crack is treated as a sharp notch. Thresholds for no crack propagation
are formulated based on shakedown theory.
Parameter identification for inelastic constitutive equations based on uniform and nonuniform
stress and strain distributions (Mahnken, Stein)
In this project, various aspects for identification of parameters are discussed. Firstly, as
in classical strategies, a least-squares functional is minimized using data of specimen
with stresses and strains assumed to be uniform within the whole volume of the sample.
Furthermore, in order to account for possible non-uniformness of stress and strain
distributions, identification is performed with the Finite-Element method, where also
the geometrically non-linear case is taken into account. In both approaches, gradientbased
optimization strategies are applied, where the associated sensitivity analysis is
performed in a systematic manner. Numerical examples for the uniform case are presented
with a material model due to Chaboche with cyclic loading. For the non-uniform
case, material parameters are obtained for a multiplicative plasticity model, where
experimental data are determined with a grating method for an axisymmetric necking
problem. In both examples, the results are discussed when different starting values are
used and stochastic perturbations of the experimental data are applied.
Project area C: experimental verification
Material parameters, which describe the inelastic behaviour of metallic materials, can
be determined experimentally from the deformation of a test specimen by suitable chosen
basic experiments. One-dimensional load-displacement measurements, however, are
not providing sufficient informations to identify parameters of three-dimensional
material laws. For this purpose, the complete whole-field deformation respectively
strain state of the considered object surface is needed. It can be measured by optical
methods. They yield the displacement distribution in three dimensions and the strain
components in two dimensions. So, these methods make possible an extensive comparison
of the results of a related Finite-Element computation.
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Preface
Experimental determination of deformation- and strain fields by optical measuring
methods (Ritter, Friebe)
Mainly, two methods were developed and adapted for solving the mentioned problems:
the object-grating method and the electronic Speckle interferometry.
As known, the object-grating method leads to the local vector of each point of the
considered object surface marked by an attached grating, consisting of a deterministic
or stochastic grey value distribution, and recorded by the photogrammetric principle.
Then, the strain follows from the difference of the displacement vectors of two neighbouring
points related to two different deformation states of the object and related to
their initial distance.
The electronic Speckle interferometry is based on the Speckle effect. It comes
into existence if an optical rough object surface is illuminated by coherent light, and
the scattered waves interfere. By superposing of the interference effects of an object
and reference wave related to two different object states, the difference of the arising
Speckle patterns leads to correlation fringes, which describe the displacement field of
the considered object.
Regarding the object-grating method, grating structures and their attachment have
been developed, which can be analysed automatically and which are practicable also at
high temperatures up to 1000 8C, as often inelastic processes take place under this condition.
Furthermore, the optical set-up, based on the photogrammetric principle, was
adapted to the short-range field with testing fields of only a few square millimeters.
The object-grating method is applicable if the strain values are greater than 0.1%.
For measurement of smaller strain values down to 10 –5 , the Speckle interferometric
principle was applied. A 3-D electronic Speckle interferometer has been developed,
which is so small that it can be adapted directly at a testing machine. It is based
on the well-known path of rays of the Speckle interferometry including modern optoelectronic
components as laser diodes, piezo crystals and CCD-cameras.
Furthermore, both methods are suitable for high resolution of a large change of
material behaviour. Finally, the measurement can be conducted at the original and takes
place without contact and interaction.
Surface deformation fields from grating pictures using image processing and photogrammetry
(Andresen)
The before-mentioned grating techniques are optical whole-field methods applied to derive
the shape or the displacement and strain on the surface of an object. A regular
grating fixed or projected on the surface is moved or deformed together with the object.
In different states, pictures are taken by film cameras or by electronic cameras.
For plane surfaces parallel to the image plane, one camera supplies the necessary information
for displacement and strain. To get the spatial coordinates of curved surfaces,
two or more stereocameras must be used. In early times, the grating patterns were evaluated
manually by projecting the images to large screens or by use of microscope techniques.
Today, the pictures are usually digitized, yielding resolutions from 200 ×200 to
2000 ×2000 picture elements (pixels or pels) with generally 256 grey levels (8 bit). By
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Research Program
suitable image-processing methods, the grating coordinates in the images are determined
to a large extent automatically. The corresponding coordinates on plane objects
are derived from the image coordinates by a perspective transformation. Considering
spatial surfaces, first, the orientation of the cameras in space must be determined by a
calibration procedure. Then, the spatial coordinates are given by intersection of the rays
of adjoined grating points in the images.
The sequence of the grating coordinates in different states describes displacement
and strain of the considered object surface. Applying suitable interpolation gives continuous
fields for the geometrical and physical quantities on the surface. These experimentally
determined fields are used for
• getting insight into two-dimensional deformation processes and effects,
• supplying experimental data to the theoretically working scientist,
• providing experimental data to be compared with Finite-Element methods,
• deriving parameters in standard constitutive laws,
• developing constitutive laws with new dependencies and parameters.
Experimental and numerical analysis of the inelastic postbuckling behaviour of shearloaded
aluminium panels (Kossira, Arnst)
As a practical problem of aircraft engineering, the case of shear-loaded thin panels out
of the material AlCuMg 2 under cyclic, quasistatic loading was investigated by experimental
and numerical methods. Beyond the up-to-now used classical theory of plasticity,
the theoretical research was based on the “unified” models, which were developed
and adjusted to numerical computational methods in other areas of the research project.
Shear-loaded panels are in general substructures of aerospace constructions since
there are always load cases during a flight mission, in which shear loads are predominant
in the thin-walled structures of subsonic as well as in supersonic and hypersonic
aircrafts. The good-natured postcritical load-carrying behaviour of shear-loaded panels
at moderate plastic deformations can be exploited in emergency (fail safe) cases since
they exhibit no dramatic loss of stiffness even in the high plastic postbuckling regime.
The temperature at the surface of hypersonic vehicles may reach very high values, but
with a thermal protection shield, the temperatures of the load-carrying structure can be
reduced to moderate values, which allow the application of aluminium alloys. Therefore,
the properties of the mostly used aluminium alloy 2024-T3 are taken as a basis
for the experimental and theoretical studies of the behaviour of shear-loaded panels at
room temperature and at 200 8C.
The primary aim of these studies is the understanding of the occurring phenomena,
respectively the examination of the load-carrying behaviour of the considered
structures under different load-time histories, and to provide suitable data for the design.
Besides experimental investigations, which are achieved by a specially designed
test set-up, the development of numerical methods, which describe the phenomena, was
necessary to accomplish this intention. The used numerical model is based on a Finite-
Element method, which is capable of calculating the geometric and physical non-linear
– in case of visco-plastic material behaviour time-dependent – postbuckling behaviour.
XXIX

Preface
A substantial problem within the numerical method was the simulation of the non-linear
material properties. Using a rate-independent two-surface material model and a
modified visco-plastic material model of the Chaboche type, the non-linear properties
of the aluminium alloy 2024-T3 are approximated with sufficient accuracy at both considered
temperatures.
Some results of the theoretical and experimental studies on the monotonic and
cyclic postbuckling behaviour of thin-walled aluminium panels under shear load at ambient
and elevated temperatures are presented. The applied loads exceed the theoretical
buckling loads by factors up to 40, accompanied by the occurrence of moderate inelastic
deformations. Apart from the numerical model, the monotonic loading, subsequent
creep rates, the snap-through behaviour at cyclic loading, the inelastic processes during
loading, and the influence of the aspect ratio are major topics in the presented discussion
of the results for shear-loaded panels at room temperature and at 200 8C.
Consideration of inhomogeneities in the application of deformation models, describing
the inelastic behaviour of welded joints (Wohlfahrt, Brinkmann)
A second practical problem was the investigation of the influence of welded joints on
the mechanical behaviour of components, which is due to the high degree of “Werkstoffnutzung”
in modern welded structures of high importance. Special consideration
was given here to the important question of the material behaviour at cyclic loading as
well from the point of view of numerical computation of these processes and the connected
effects as from the point of view of the problems connected with aspects of
materials sciences.
The local loads and deformations in welded joints have rarely been investigated
under the aspect that the mechanical behaviour is influenced by different kinds of microstructure.
These different kinds of microstructure lead to multiaxial states of stresses
and strains, and some investigations have shown that for the determination of the total
state of deformation of a welded joint, the locally different deformation behaviour has
to be taken into account. It is also published that different mechanical properties in the
heat-affected zone as well as a weld metal with a lower strength than the base metal
can be the reason or the starting point of a fracture in welded joints. A new investigation
demonstrates that in TIG-welded joints of the high strength steel StE690, a finegrained
area in the heat-affected zone with a lower strength than that of the base metal
is exclusively the starting zone of fracture under cyclic loading in the fully compressive
range. These investigations support the approach described here that the mechanical behaviour
of the different kinds of microstructure in the heat-affected zone of welded
joints has to be taken into account in the deformation analysis. The influences of these
inhomogeneities on the local deformation behaviour of welded joints were determined
by experiments and numerical calculations over a wide range of temperature and loading.
The numerical deformation analysis was performed with ttformat
he method of Finite-Elements, in which recently developed deformation models
simulate the mechanical behaviour of materials over the tested range of temperature
and loading conditions.
XXX

1 Correlation between Energetic
and Mechanical Quantities of Face-Centred Cubic
Metals, Cold-Worked and Softened to Different States
Lothar Kaps and Frank Haeßner*
1.1 Introduction
Cold-worked metals soften at higher temperatures. The details of this process depend
on the material as well as on the type and degree of deformation. The kinetic parameters
can in principle be determined by calorimetric methods. By combining calorimetrically
determined values with characteristics measured mechanically and with microstructural
data, information can be gained about the strain-hardened state and the mechanism
of the softening process.
This materials information can support critical assessment of the structure of
material models and hence be utilized for the appropriate adjustment of constitutive
models to material properties.
1.2 Experiments
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
One objective of the work in this particular area of research was to investigate the dependence
of the softening kinetics of face-centred cubic metals on the deformation. The
chosen types of deformation were torsion, tension and rolling. In the cases of torsion
and tension, additional cyclic experiments with plastic amplitudes of 0.01 to 0.1 were
carried out. The materials studied were aluminium, lead, nickel, copper and silver.
Thus, in this order, metals of very high to very low stacking fault energy were investigated.
In the following presentation of the results, the emphasis will be on copper.
To determine the mechanical data, the first step was to characterize the deformation
with the aid of the crystallographic slip a, the shear stress s N normalized to the
* Technische Universität Braunschweig, Institut für Werkstoffe, Langer Kamp 8,
D-38106 Braunschweig, Germany
1

1 Correlation between Energetic and Mechanical Quantities
shearing modulus G: sN ˆ sc=G, and the strain-hardening rate H ˆ dsN=da. The conversion
to crystallographic quantities was effected using calculated Taylor factors [1, 2].
This procedure permits direct comparison between different types of deformation.
Figure 1.1 shows the family of curves that are obtained when copper is subjected
to torsion at various temperatures. The characterization is clear because for increasing
deformation temperature, a decreasing yield stress results.
Figure 1.2 shows the strain-hardening rate versus the normalized shear stress of
copper to extreme deformation. The strain-hardening rate can be subdivided into three
regions, which, following the literature, may be denoted strain-hardening regions III to
V [3]. Regions III and V show a linearly decreasing strain-hardening rate with shear
stress. Region IV, as region II, is characterized by constant strain hardening.
The occurrence of these different regions depends strongly on the type of deformation.
Thus, for tensile deformation, in consequence of instability, only deformation to region
III can be realized. Rolling permits greater deformation, but brings with it the problem
of defining a specific measurement to categorize the strain-hardening regions. The temperature
effect of the deformation fits well into the scheme proposed by Gil Sevillano [4].
According to this scheme, all flow curves in region III may be described by a fixed initial
strain-hardening rate H III
0 and a variable limiting stress sIII S . This latter is affected by dynamic
recovery and is therefore dependent on deformation temperature and velocity. It
decreases for increasing deformation temperature and increases for higher deformation
velocities.
This statement is also true for the other characteristic stresses sIV ; sV ; sV S : The logarithm
of the characteristic stress decreases linearly with the normalized deformation
temperature, TN ˆ kT=Gb3 : The normalization was proposed by Mecking et al. [5]. It
Figure 1.1: Flow curves of copper at temperatures of –20 8C to 120 8C.
2

1.2 Experiments
Figure 1.2: Strain-hardening rate of copper under torsion at room temperature versus normalized
shear stress.
has been successfully applied to our own measurements. However, it may be seen that
the dependence on temperature is different for the individual stresses (Figure 1.3).
Careful evaluation of the experiments taking account of the effects of texture and
sample shows similarities as well as differences between the two deformation types tension
and torsion. Up to a slip value of a =0.4, the flow curve shows little difference between
tension and torsion. Above that value, the hardening is greater for the tension experiment
(Figure 1.4).
The differences are more pronounced when the hardening rate is studied rather
than the flow curve. From the start, the former lies higher for tension than for torsion.
The different procedures may be followed microstructurally using a transmission elec-
Figure 1.3: Characteristic stresses for the strain hardening of copper versus the normalized temperature.
3

tron microscope. Other authors have described this influence of the load path on the
microstructure [6–9]. The reason for this may be that different average numbers of slip
systems are necessary for deformation [10]. This also affects the development of activation
energies DG0 and activation volumes V. To determine these quantities, velocities
are varied in tension and torsion experiments, i.e. during a unidirectional experiment,
the extension rate is momentarily increased. In those sections with an increased extension
rate, the material shows a higher flow stress. For the evaluation, the following ansatz
was chosen for the relationship between the extension rate _e and the flow stress r:
_e ˆ _e0 exp
1 Correlation between Energetic and Mechanical Quantities
Figure 1.4: Comparison of flow curves from tension and torsion experiments.
…DG0 Vr†
kT
: …1†
The activation volume and energy are the important quantities for the constitutive equations
developed in the subproject A6 [11, 12]. The comparison of the deformation
types tension and torsion shows a definite difference in the development of activation
volumes with sN. This is manifest by the tension (strain) deformation, which exhibits a
constant velocity sensitivity even for significantly smaller degrees of deformation (Figure
1.5). The activation volumes are a particularly indicative measurement for the
velocity sensitivity. In region III for torsion, they show a continuous decrease, which
becomes less only upon reaching region IV. For tension, on the other hand, there are
also two sections with decreasing or nearly constant activation volumes. However, the
transition in the curve of the activation volume versus the stress already lies in the
strain-hardening region III.
The activation energies DG0 for torsion were determined from the characteristic
stresses for different temperatures (cf. Figure 1.3). The resultant values for the stresses
sIII S ; sIVand sV S are 3.15, 2.79 and 2.79 eV/atom, respectively.
To obtain the energy data, the stored energy ES of the plastic deformation was determined
using a calorimeter. As expected, the stored energy shows a monotonic in-
4

1.2 Experiments
Figure 1.5: Activation volume of copper deformed in tension and torsion at room temperature.
crease with deformation. Moreover, dynamic recovery counteracts energy storage as it
does hardening. Hence, there is an unequivocal correlation between the deformation
temperature and stored energy such that an increasing deformation temperature leads to
less stored energy (Figure 1.6).
Figure 1.6 demonstrates the great influence of the stacking fault energy. The value
of the reduced stacking fault energy for silver lies at 2.4·10 –3 compared with the value
of 4.7·10 –3 for copper. Lower stacking fault energies lead to a greater separation of par-
Figure 1.6: Stored energy versus shear strain for distorted copper and silver deformed at different
temperatures.
5

tial dislocations. This hinders dynamic recovery because the mechanism of cross slip is
impaired.
The connection between stored energy and shearing stress was studied for deformation
by torsion, tension and push-pull. There is a clear tendency to store more energy
with increasing deformation temperature at constant shearing stress. It would appear
that energy storage by more fully condensed states is more effective. The measured
values for tension and push-pull in this sequence lie above those for the greatest
torsional deformation. For the same shearing stress, silver also clearly stores more energy
than copper.
This relationship is represented in the Figures 1.7 and 1.8. Figure 1.7 comprises
torsion experiments up to extreme deformation. Figure 1.8 shows a comparison of various
types of deformation. For better resolution, the abscissa here is confined to small
and intermediate values of stress. The variable behaviour of the materials and the effect
of the types of deformation may also be demonstrated in measurements of the softening
kinetics to be discussed. In analogy to the strain-hardening rate, an energy storage
rate HE ˆ dES=dsN has been defined. This quantity represents independent information.
The development of the energy storage rate is clearly correlated with the strainhardening
stages (Figure 1.9). The combination of energetic and mechanical measurements
permits a statement on the change in dislocation density q, to a first approximation
proportional to the stored energy, with increasing flow stress. A linearly increasing
energy storage rate with stress leads to a law of the type:
qES
qsN
1 Correlation between Energetic and Mechanical Quantities
asN ) sc
G ˆ k1
p
ES : …2†
This kind of behaviour is found only up to the middle of region III. After that, the energy
storage rate increases overproportionally until region IV is reached. In region IV, it
decreases slightly and then increases linearly again in region V. This time, however,
Figure 1.7: Stored energy versus normalized shear stress for copper and silver deformed at different
temperatures.
6

1.2 Experiments
~ Cu 235 K s Cu 293 K * Cu 373 K + Cu 293 K Tension × Cu 293 K Push-Pull
Figure 1.8: Stored energy versus normalized shear stress for copper deformed in torsion, tension
and push-pull.
Figure 1.9: Stored energy (upper curve) and rate of energy storage of distorted copper versus normalized
shear stress.
with a different proportionality factor a of value rather below the one pertaining to region
III. The factor a may only be analytically assessed for deformation in the region
of the strain-hardening stage II. For greater plastic deformation, which would then be
deformation in region III of the strain-hardening stage, this factor is of a qualitative nature.
The evolution of a for various materials, deformation temperatures and types of
deformation is collated in Table 1.1. The stress in the second column indicates the end
of the linear storage rate in the strain-hardening region III.
7

1 Correlation between Energetic and Mechanical Quantities
Table 1.1: The constant k 1 according to Equation (2) for various temperatures. The constant k 2
applies to extreme deformation in the region V.
Cu 253 K 6.4·10 –4
Cu 293 K 5.7·10 –4
Cu 373 K 5.5·10 –4
Cu 293 K tension 5.0·10 –4
Ag 253 K 6.0·10 –4
Ag 293 K 5.5·10 –4
If X denotes the softened fraction of the material, one may attempt to describe the
softening kinetics _X by a product of functions, which combines the thermal activation
and the nature of the reaction in one appropriate multiplier:
_X
Q
ˆ f …X†g…T† ˆf …X† exp : …3†
RT
Equation (3) is easily handled numerically. The activation energy of the softening Q
and the form function f may be determined separately. Equation (3) offers the added advantage
that, as a rate equation, it may be directly incorporated into a constitutive equation
if the quantities Q and f …X† are known. The simpler analysis considers the product
and in its place the reaction temperature. This temperature is a direct measure of the
stability of the deformed state.
The thermal results show that for increasing stored energy, the softening process
takes place at lower temperatures. An influence of the deformation temperature becomes
apparent. Higher deformation temperatures promote easier reaction for the same
stored energy. Exact analysis of these facts shows that the form function makes only a
negligible contribution here. The effect is induced by a reduced activation energy.
Different types of deformation show a stronger influence on the reaction temperature
than the deformation temperature. At lower energies, distorted samples soften faster
than extended or rolled ones. At higher energies, the reverse is true: Rolled samples
react faster. It is noticeable that cyclically deformed samples, for torsion as well as for
push-pull, do not diverge from the unidirectionally deformed samples of the same deformation
mode. This is remarkable because, particularly for tension and push-pull deformation,
there are substantial differences in the activation energy.
The activation energy describes the purely temperature dependence of the reaction.
For small deformation and stored energies of distorted copper at a value of
170 kJ/mol, it lies below the activation energy of volume self diffusion (200 kJ/mol).
Unidirectionally extended samples show a higher activation energy (190 kJ/mol); pushpull
deformed samples, on the other hand, show significantly lower activation energies
(130 kJ/mol). With increasing energy, the activation energies of all deformation types
fall. Figure 1.10 demonstrates these relationships.
With the aid of torsional deformation, it is unequivocally proved that only upon
reaching the strain-hardening stage V, one may presume constant activation energy. At
8
k1 sc/G k2
1.5·10 –3
1.25·10 –3
1.12·10 –3
1.5·10 –3
1.5·10 –3
1.25·10 –3
4.8·10 –4
4.5·10 –4
4.5·10 –4
–
4.5·10 –4
4.4·10 –4

1.2 Experiments
Figure 1.10: Activation energy of differently deformed copper versus the stored energy.
values of 80 to 90 kJ/mol, here for all deformation temperatures, the activation energy
lies in the region of grain boundary self diffusion or diffusion in dislocation cores. Tension
and push-pull samples do not achieve these high stored energies; for these deformation
modes, there is therefore no region of constant activation energy. Elevated deformation
temperatures result in a lower softening activation energy. One may interpret
this as strain hardening at higher temperature producing a microstructure that softens
faster. This effect should be accounted for when setting up constitutive equations.
There is a theory for the softening of deformed metals through the mechanism of
primary recrystallization by Johnson and Mehl [13], Avrami [14–16] and Kolmogorov
[17]. In the following, this will be denoted the JMAK theory. Comparison of the measured
activation energies with those predicted by the JMAK theory allow conclusions
to be drawn regarding the basic mechanisms of primary recrystallization.
Accordingly, for high deformation continuous nucleation must be assumed,
whereas for low deformation site, saturated nucleation is more probable. Table 1.2
shows the comparison in detail. For high deformation, this interpretation complies with
studies according to the microstructural-path method [18]. The grain spectra of weakly
deformed and recrystallized material show agreement with calculated spectra after sitesaturated
cluster nucleation.
Table 1.2: Effective activation energies from the JMAK theory compared with measured values
for low/high deformation.
Site-saturated Continuous Measured values [kJ/mol]
nucleation nucleation
[kJ/mol] [kJ/mol] low deformation high deformation
Copper 166–120 125–86 170 ±8 85 ±5
Silver 143–115 107–86 120 ±8 85 ±5
9

1 Correlation between Energetic and Mechanical Quantities
The second component of the kinetics, the pure reaction form, is described by the
function f …X†. For all nucleation-nucleation growth reactions, this function, by way of
the transformed fraction, is parabolic with zero points at the beginning and end of the
reaction. A more significant picture results when this function is compared with the
JMAK theory. For ideal nucleation-nucleation growth reactions, the theory demands for
f …X†=…1 X† a higher order function of ln…1 X† with an exponent …n 1†=n independent
of X. The Avrami exponent n takes the value 4 or 3, respectively.
In reality, however, independent of the measurement method, one finds Avrami
exponents that decrease with X. The thermal data show this particularly clearly. As an
example, Figure 1.11 shows the curve of the Avrami exponent as a function of the
transformed fraction for distorted copper. The horizontal reference line outlines the
curve for low degrees of deformation …c ˆ 0:8or1:4†, the central reference line applies
to intermediate degrees of deformation …c ˆ 2:4or3:0†:
Rolling and cyclic torsion act in the same way as unidirectional torsion if the
stored energy is taken as the comparative measure instead of the strain-hardening regions.
Complementary studies using the transmission electron microscope show that the
microstructural details are similar for these deformations (cf. Nix et al. [9]). The deformation
types unidirectional tension and push-pull are very different from torsion. The
Avrami exponents are very large for unidirectional tension.
In summary, the combination of stored energy, softening temperature and activation
energy as well as the softening form function is unequivocal for the material states
studied here. The degree and type of deformation of a sample may thus be identified
with no knowledge of its prior mechanical history.
Figure 1.11: Avrami exponent versus the transformed fraction for distorted copper with shear
strains 3.4 ≤c ≤7.0.
10

1.3 Simulation
1.3 Simulation
Primary recrystallization as one of the main processes of thermal softening was simulated
by a cellular automaton (CA). These latter are networks of computational units, which
develop their properties through the interaction of numerous similar particles [19, 20].
They are comprehensively described by the four properties geometry, environment, states
and rules of evolution. Cellular automatons were first applied to primary recrystallization
for the two-dimensional case by Hesselbarth et al. [21, 22]. For the extension to three
dimensions, a cubic lattice of identical cubes is defined. Each of these small cubes represents
a real sample volume of about 0.6 lm 3 . This value is obtained by comparison with
real grain sizes. The whole field is then equivalent to a mass of 0.007 mg. Compared with
the mass of thermal samples at 150 mg, this is very little. The geometrically closest cells
are counted as the nearest neighbours. It turns out that an alternating sequence of 7 and 19
nearest neighbours yields the best results. Stochastically changing environments influence
the kinetics in consequence of the resultant rough surface of the growing grains.
Figure 1.12 shows the 7 nearest neighbours on the left and the 19 on the right,
starting with a nucleus in the second time-step. The change of environment with each
time-step causes all grains in odd time-steps to be identical. The resultant grain shape
looks like a flattened octahedron.
Figure 1.12: Sequence of the recrystallization in the three-dimensional space.
11

1 Correlation between Energetic and Mechanical Quantities
The possible states of the cells are recrystallized and non-recrystallized. For the
extension to different grain boundary velocities, the non-recrystallized state was subdivided
further. The fourth descriptive characteristic after the geometry, environment and
possible states are the rules of evolution. These stipulate, which states the cells will
adopt in the next time-step. If a cell already has a recrystallized environment, the rules
predict that in the next time-step, this cell will also adopt the recrystallized state. Using
this simple cellular automaton, it is possible to solve the differential equation of the
JMAK theory. The quality of the solution improves with the field size.
Alternatively, several calculations may be combined. The deviation of simulated from
theoretical kinetics is of the order of 1%. A great advantage of cellular automatons is that
boundary conditions are automatically taken into account. They do not have to be stated
explicitly. This advantage should not be underestimated because the problem of collision of
growing grains for arbitrary site-dependent nucleation is non-trivial. In this way, it is possible
to calculate even complicated geometries not amenable to analytical solution.
The objective of simulations is to support the discussion on the various possible
causes for the deviation of real recrystallization kinetics from the theoretically predicted
processes. In so doing, one differentiates between topological and energetic causes.
Namely, the classical JMAK theory leans on two hypotheses, which strongly limit its universal
applicability. The first in the assumption that all processes are statistically distributed
in space; this applies to nucleation in the first instance and thus subsequent grain
growth. Any kind of nucleation concentration on chosen structural inhomogeneities alters
the collision course of growing nuclei and hence the correction factors of the extendedvolume
model. The second restrictive assumption concerns the process rates. Nucleation
and nuclear growth are assumed to be site-independent and constant in time. However,
comparison of various strongly deformed samples shows at once that for different stored
energies, even if they are mean values, recrystallization occurs at different rates. If, therefore,
we have structural components with different energies side by side in the same sample,
one must be aware that a uniform process rate does not exist.
Non-statistical nucleation was intensively studied for point clustering. The model
postulates stochastically placed centres, which show an increased nucleation rate. The
nucleation density follows a Normal distribution around the chosen centres. On a line
between two concentration centres, one obtains the distribution for the nucleation rates
shown in Figure 1.13.
This yields two boundary cases, which are also being discussed in the literature
[23–25]. First, we have very broad scatter of nuclei and, secondly, a high concentration
on the chosen sites. In a narrow parameter range between these boundary cases, the kinetics
are very sensitive to change (Figure 1.14).
It is possible to simulate the continuously decreasing Avrami exponents of the
strongly deformed samples as well as the low Avrami exponents at the beginning of
the transformation found for weakly deformed samples.
Another structural characteristic, the contiguity, describes the cohesion of recrystallized
areas. This quantity may also be calculated using the cellular automaton for
various site functions of nucleation. Comparison with experimentally determined contiguity
curves indicates that nucleation clustering can also be found in real materials.
The evaluation of grain-size distributions also points to clustering.
12

1.3 Simulation
Figure 1.13: Model of point clustering (left); plot of the nucleation rate between two concentration
centres (right).
Figure 1.14: Avrami exponent due to the restriction of nucleation to point clustering.
Introduction of site-dependent process rates is effected through an extension of possible
non-recrystallized states. One differentiates between mobility and driving force. With
reference to the literature [26, 27], a value around the factor 3 is taken. The result is 9
different velocities. Two degrees of recrystallization are defined, one of which refers to
the energy, the other to the volume. If the kinetics of the JMAK theory are appropriately
evaluated, there is hardly any difference between these two definitions.
The introduction of different velocities causes the reaction rate to decline towards
the end of the transformation. If the proportionality of the areas of equal velocity and
the resulting grain size is changed, the kinetics may be influenced to a degree. The kinetics
of strongly deformed samples may be simulated if the areas of equal velocity are
larger than the resultant grain size. Smaller initial areas do not give the desired effect;
the decline of the effective rates is too late and too weak. The kinetics of weakly de-
13

formed samples with low Avrami exponents cannot be calculated using this ansatz. An
experimental indication of rate retardation is obtained from studies on strongly rolled
copper by the microstructural-path method [18].
Finally, it may be said that there are indicators for each ansatz in real recrystallization
processes. Considering the experimental results, a weighted mixture of both
would appear to be a realistic course, which can doubtless be applied in the model.
Coupling to a constitutive equation is directly possible, for example by introducing the
stored energy as a function of deformation. The cellular automaton, on the other hand,
is able to calculate partially softened material structures. The strength of the composite
may then be determined from this using a parallel or series network. In future, this type
of model coupling will become more important in those areas, where modelling with
constitutive equations on the basis of discontinuous phenomena only such as dynamic
recrystallization do not produce the desired results.
1.4 Summary
1 Correlation between Energetic and Mechanical Quantities
Shortly summarizing this report, we can make the following basic statements:
• The diverse strain-hardening stages of face-centred cubic metals, identifiable from
mechanical data, which correspond to different structures of the strain-hardened
material, may also be determined from the thermally measured stored energy and
from the rate of energy storage. One finds that the energy storage of more fully
condensed states is particularly effective.
• The softening kinetics investigated via the stored energy are strongly influenced by
the details of the type of deformation (for example, unidirectional deformation-alternate
deformation). In the case of the primary recrystallization as the cause of the
softening, the process may be described well by quoting the activation energy and
the Avrami exponent. Knowledge of these two parameters for a strain-hardened
state allows the degree of softening to be numerically calculated for a freely chosen
temperature-time programme. Qualitatively, the activation energy and the Avrami
exponent are a measure of the thermal stability, that is, for the ease of reaction of
the deformed material.
• Utilizing a suitably fitted cellular automaton, it is possible to simulate the microstructural
processes underlying the softening and hence to control the topological as well as
the energetic model hypotheses. An important result of this simulation is the proof that
the Avrami theory, which is based on stereological elements, may be applied to calorimetrically
determined softening data. The kinetics in both cases are very similar.
The results presented here are the compilation of numerous data; a comprehensive publication
is given in [28].
14

References
References
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[2] U.F. Kocks, M.G. Stout, A.D. Rollett: The influence of texture on strain hardening. In:
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[3] J. Diehl: Zugverformung von Kupfer Einkristallen. Z. f. Metallkunde 47 (1956) 331.
[4] J. Gil Sevillano: The cold-worked state. Materials Science Forum 113–115 (1993) 19.
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[6] M.N. Bassim, C. D. Liu: Dislocation cell structures in copper in torsion and tension. Mater.
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of hc copper rods deformed by torsion and tension. Acta metall. 34 (1986)
1257.
[11] R. Gerdes: Ein stochastisches Werkstoffmodell für das inelastische Materialverhalten metallischer
Werkstoffe im Hoch- und Tieftemperaturbereich. Mechanik-Zentrum der TU Braunschweig
(Dissertation), Braunschweig, 1995.
[12] H. Schlums, E.A. Steck: Description of cyclic deformation processes with a stochastic
model for inelastic creep. Int. J. Plast. 8 (1992) 147.
[13] W. A. Johnson, R.F. Mehl: Reaction kinetics in process of nucleation and growth. Trans.
Am. Inst. Min. Engrs. 135 (1939) 416.
[14] M. Avrami: Kinetics in phase change: I. General theory. J. Chem. Phys. 7 (1939) 1103.
[15] M. Avrami: Kinetics in phase change: II. Transformation-time relations for random distribution
of nuclei. J. Chem. Phys. 8 (1940) 212.
[16] M. Avrami: Kinetics in phase change: III. Granulation, phase change and microstructure.
J. Chem. Phys. 9 (1941) 177.
[17] A.E. Kolmogorov: Zur Statistik der Kristallvorgänge in Metallen (russ. mit deutscher Zusammenfassung).
Akad. Nauk. SSSR Ser. Mat. 1 (1937) 335.
[18] R. A. Vandermeer, D. Juul Jensen: Quantifying recrystallization nucleation and growth kinetics
of cold-worked copper by microstructural analysis. Metall. Mater. Trans. 26A (1995)
2227.
[19] S. Wolfram: Statistical mechanics of cellular automata. Reviews of modern physics 55
(1983) 601.
[20] S. Wolfram: Cellular automata as models of complexity. Nature 311 (1984) 419.
[21] H.W. Hesselbarth, I.R. Göbel: Simulation of recrystallization by cellular automata. Acta
metall. mater. 39 (1991) 2135.
[22] H.W. Hesselbarth, L. Kaps, F. Haeßner: Two dimensional simulation of the recrystallization
kinetics in the case of inhomogeneously stored energy. Materials Science Forum 113–115
(1993) 317.
[23] J.W. Cahn: The kinetics of grain boundary nucleated reactions. Acta metall. 27 (1979) 449.
[24] J.W. Cahn, W. Hagel: Decomposition of austenite by diffusional processes. In: Z.D. Zackay,
H.I. Aarosons (Eds.), Interscience Publ., New York, 1960.
[25] R. A. Vandermeer, R. A. Masumura: The microstructural path of grain-boundary-nucleated
phase transformations. Acta metall. mater. 40 (1992) 877.
15

1 Correlation between Energetic and Mechanical Quantities
[26] J.S. Kallend, Y. C. Huang: Orientation dependence of stored energy of cold work in 50%
cold rolled copper. Metal Science 18 (1984) 381.
[27] F. Haeßner, G. Hoschek, G. Tölg: Stored energy and recrystallization temperature of rolled
copper and silver single crystals with defined solute contents. Acta metall. 27 (1979) 1539.
[28] L. Kaps: Einfluss der mechanischen Vorgeschichte auf die primäre Rekristallisation. Shaker
Verlag, Aachen, 1997.
16

2 Material State after Uni- and Biaxial Cyclic
Deformation
Walter Gieseke, K. Roger Hillert and Günter Lange*
2.1 Introduction
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
Technical components and structures today are increasingly being designed and displayed
by computer-aided methods. High speed computers permit the use of mathematical
models able to numerically reconstruct material behaviour even in the course of
complex loading procedures.
In phenomenological continuum mechanics, the cyclic hardening and softening
behaviour as well as the Bauschinger effect are described by yield surface models. If a
physical microstructural formulation is chosen as a basis for these models, then it is vitally
important to have exact knowledge of the processes occurring in the metal lattice
during deformation. Two surface models, going back to a development by Dafalias and
Popov [1–4], describe the displacement of the elastic deformation zone in a dual axis
stress area. The yield surfaces are assumed to be v. Mises shaped ellipses. However,
from experiments with uniaxial loading [5, 6], it is known that the yield surfaces of
small offset strains under load become characteristically deformed. In the present subproject,
the effect of cyclic deformation on the shape and position of the yield surfaces
is studied, and their relation to the dislocation structure. To this end, the yield surfaces
of three materials with different slip behaviour were measured after prior uni- or biaxial
deformation. The influence of the dislocation structures produced and the effect of inner
stresses are discussed.
* Technische Universität Braunschweig, Institut für Werkstoffe, Langer Kamp 8,
D-38106 Braunschweig, Germany
17

2 Material State after Uni- and Biaxial Cyclic Deformation
2.2 Experiments and Measurement Methods
Copper of 99.99% purity was chosen as a material exhibiting typical wavy slip behaviour.
Most of the experiments were performed using the technically important material
AlMg 3 of 99.88% purity. Its behaviour may be described as being somewhere intermediate
between planar and wavy slip 1 . Commercial austenitic steel 1.4404 (AISI
316L) was used as a material with typical planar slip behaviour. The total strain amplitude
was varied from 0.25% to 0.75% for AlMg 3 and steel, and between 0.05%
and 0.5% for copper. All materials were previously solution annealed or recrystallized.
AlMg 3 and the austenitic steel were quenched in water, the copper samples
cooled in the oven. After the thermal treatment, a h100i slightly fibrous texture was
identified, which did not change during the subsequent cyclic deformation. The copper
showed almost no texture. The yield surfaces of the initial materials were isotropic, independent
of the offset used [7, 8].
Tubular samples were used in the experiments. Their outer diameter and wall
thickness were 28 mm and 2 mm for AlMg 3 and copper, 29 mm and 1.5 mm for steel,
respectively. The measuring distance was 54 mm long for all samples. The following
cyclic experiments were carried out using a servo-hydraulic Schenck testing machine,
which had been augmented by a laboratory-made torsional drive [9]: uniaxial tension/
compression, alternating torsion; biaxial equal phase superposition of tension/compression
and alternate torsion; a 90 8 antiphase combination of tension/compression and alternate
torsion. The dislocation structures were subsequently investigated using a Philips
120 kV transmission electron microscope. For the strain-controlled experiments, a
triangular nominal value signal with constant strain rate of 2 · 10 –3 s –1 was chosen. The
equivalent strains were calculated after v. Mises according to:
eeq ˆ e 2 ‡ 1
3 c2
1=2
with e ˆ 1
p c : …1†
3
Two methods were applied to determine the yield surfaces. Using the definition via an
offset strain of 2 ·10 –4 %, the load was increased in steps of 6 N/mm 2 in the r-direction
or of 2.5 N/mm 2 in the s-direction until the given yield limit was reached. There was a
10 s intermission at each level. Before the next point on the curve was measured, several
load cycles were run through again to set the material to the same initial state.
The second measurement method was the recording of directionally dependent
stress-strain diagrams. Here, a new sample was used for each point measured. It was
stressed under predetermined load paths immediately following the cyclic treatment far
into the plastic region. In this way, static strain ageing effects were avoided. Further, it
was possible to determine yield surfaces of higher offset strain and areas of equal tangent
modules. For the evaluation of the yield surfaces and the tangent module areas,
besides the yield conditions after v. Mises and Tresca, a formulation developed within
the scope of this project was used:
1 The results for copper and AlMg3 presented in this report and their interpretation are taken
from the thesis by Walter Gieseke [9].
18

0 eq ˆ …r rA† 2 ‡ E
2
…s sA†
G
e 0 eq ˆ …e eA† 2 ‡ G
E …c c A† 2
1=2
1=2
; …2†
: …3†
The advantage of these equations lies in the fact that all equivalent stress-strain diagrams
show a Young’s modulus appropriate increase in the elastic region. In the case
of AlMg3, the s, c-hysteresis can be converted into the equivalent req,eeq-hysteresis,
which are in almost complete agreement with the measured r,e-hysteresis values. Figure
2.1 a shows the strain paths for the measurement of a family of yield surfaces of
varying offset strain and tangent modules after prior tension/compression loading.
The starting point for the measurement was set here in the centre of the elastic region
after load reversal in the load maximum. Figure 2.1b shows the appropriate load
paths, Figure 2.1 c the relevant equivalent stress-strain diagrams. The yield points of
various offset strains were determined by parallel shift of the elastic straight line. For
areas with the same tangent modules, the equivalent stress-strain curves were differentiated;
for a given tangential gradient, one obtains the pertinent r,s-points.
The yield surfaces in Figure 2.2 show that the yield conditions according to Equations
(2) and (3) produce the same results as the evaluation after v. Mises or Tresca
(AlMg 3, tension/compression loading, starting from the stress zero crossover, offset
strain 0.2% or 0.01%, respectively).
2.3 Results
2.3.1 Cyclic stress-strain behaviour
2.3 Results
Figure 2.3 a shows a plot for AlMg 3 of the stress amplitudes as a function of number
of cycles for the appropriate given equivalent total strain amplitude of Deeq ˆ 0.5%.
The three proportional loads are compared and that for the 90 8 anti-phase combination
of tension/compression and alternating torsion.
For all four load types, the saturation state is reached after about 500 cycles. The
curves for proportional loading almost coincide. Larger torsional fractions cause a
slight increase in the stress amplitudes. The curve for disproportional loading systematically
assumes higher values. This additional hardening effect is much more pronounced
at the beginning of the fatigue at about 25% than in the saturation stage, where it is
only about 5%.
Figure 2.3b shows the appropriate curves for the lower total strain amplitude of
Deeq ˆ 0.3%.
19

20
2 Material State after Uni- and Biaxial Cyclic Deformation
Figure 2.1: a) Strain paths for measurement of yield surfaces and areas of equal tangent modules; b) load paths for Figure 2.1 a; c)
equivalent stress-strain diagrams for the stress and strain values of Figures 2.1 a and b. Calculated according to Equations (2) and
(3).

2.3 Results
Figure 2.2: 0.2% and 0.01% offset saturation yield surfaces measured in the stress zero crossover.
Evaluation using the v. Mises and Tresca conditions and Equations (2) and (3).
Figure 2.3 a: Cyclic strain hardening behaviour for Deeq ˆ 0.5%, material: AlMg 3.
The saturation state is reached after about 900 cycles. Here too, the curves of proportional
loading approximately coincide. For disproportional loading, a weak additional
hardening effect appears at the beginning of the fatique stage, yet this reverses in
saturation.
The additional hardening effect may usually be explained by the fact that for an
appropriately large plastic strain amplitude, the anti-phase loading leads to an additional
hardening because more slip systems are activated than for proportional loading.
This is particularly the case for the high strain amplitude of Deeq ˆ 0.5% at the beginning
of the fatigue. For strain amplitudes of Deeq ˆ 0.3%, the plastic fraction of
21

2 Material State after Uni- and Biaxial Cyclic Deformation
Figure 2.3 b: Cyclic strain hardening behaviour for Deeq ˆ 0.3%, material: AlMg 3.
saturation is so small that the additional hardening in consequence of anti-phase loading
is not enough to compensate the overall smaller stress values of the sum of the individual
components. At strain amplitudes of Deeq 0.4%, AlMg3 shows Masing behaviour
for all proportional loads. Deviations occur at smaller amplitudes: The length
of the elastic regions increases with decreasing strain amplitude. Similar behaviour is
found for planar flowing a-brass [10].
For copper, a total strain amplitude of Deeq ˆ 0.5% under phase-shifted loading
produces a pronounced additional hardening effect throughout the whole fatigue region
(Figure 2.4 a).
As for AlMg3, the curves for proportional loading approximately coincide, though
the pure torsional load yields the lowest values. The stress values of the phase-shifted
Figure 2.4 a: Cyclic strain hardening behaviour at Deeq ˆ 0.5%, material: copper.
22

2.3 Results
Figure 2.4 b: Cyclic strain hardening behaviour at Deeq ˆ 0.1%, material: copper.
loading reach saturation after about 30 cycles. Under proportional loading, on the other
hand, constant stress values are only measured after about 50 cycles. For an amplitude
of Deeq ˆ 0.1%, the effect occurs only at the onset of fatigue (Figure 2.4b).
For a further reduction to Deeq ˆ 0.05%, the stress values for phase-shifted
loading in the saturation region lie below those for synchronous loading (Figure 2.4 c).
The considerations regarding the additional hardening effect in AlMg 3 are equally
applicable here.
The austenitic steel 1.4404 for proportional loading at Deeq ˆ 0.75% shows a relatively
short strain hardening region already reaching saturation after about 20 cycles. But
the 90 8 phase-shifted loading produces a strong additional hardening effect. The appropri-
Figure 2.4 c: Cyclic strain hardening behaviour at Deeq ˆ 0.05%, material: copper.
23

2 Material State after Uni- and Biaxial Cyclic Deformation
Figure 2.5 a: Cyclic strain hardening behaviour at Deeq ˆ 0.75%, material: steel 1.4404.
Figure 2.5 b: Cyclic strain hardening behaviour at Deeq ˆ 0.5%, material: steel 1.4404.
ate stress values compared with proportional loading are increased by more than 60%. The
saturation plateau is only reached after about 30 cycles (Figure 2.5 a).
For a strain amplitude of Deeq ˆ 0.5% too, the material reaches saturation for
proportional loading after about 20 cycles. The increase of the stress amplitudes is less
here, however. Phase-shifted loading (Figure 2.5 b) also yields a distinct additional
hardening effect. The appropriate stress amplitudes as for Deeq ˆ 0.75% are greatly
increased. The additional hardening effect may be regarded here as a consequence of
the planar flow behaviour.
2.3.2 Dislocation structures
The dislocation structure of AlMg 3 is characterized by walls of prismatic edge dipoles.
Mobile screw dislocations lie between them. For all strain amplitudes and loading types
studied, the dipolar walls lie in (111) planes at the onset of fatigue. The value and type
24

2.3 Results
of loading determine the resultant saturation structure. Using a model by Dickson et al.
[11, 12], all wall orientations that differ from (111) planes can be indexed.
For low strain amplitudes …Deeq 0.3%) after proportional and disproportional
loading, the (110) walls and the initial (111) walls dominate. In almost all cases, formation
of the (110) walls was the work of a single slip system. Hereby, the walls were compressed
perpendicular to the Burgers vector. Similar structures are also found in brass with
15 at% zinc [13]. At high amplitudes …Deeq > 0.3%) after proportional loading, the
(100) besides the (311), (210), (211) and (110) walls are predominant. By contrast, for
phase-shifted loading, the initial orientation of the (111) walls is conserved.
During proportional loading, a maximum of three slip systems are set in motion.
Phase-shifted loading on the other hand, because of the rotating stress vector, usually
activates more than four slip systems. Figure 2.6 a shows the typical example of a dislocation
structure after proportional loading. The arrangement can be designated anisotropic
since the dipolar walls in almost all grains are oriented in only one or two crystallographic
directions. The anisotropy essentially results from the small number of active
slip systems in the proportional loading case, expressing a certain planarity in the
slip behaviour.
Disproportional loading at high total strain amplitudes, however, results in generally
more isotropic structures (Figure 2.6 b). Here, the dipolar wall structure is quite often destroyed
along favourably oriented (111) planes (Figure 2.6 c). Parallel arrays of elongated
screw dislocations are often observed in these bands, which infers high local slip activity.
Depending upon loading amplitude, for copper, characteristic dislocation structures
evolve, which differ much more strongly from each other than for AlMg 3. In saturation,
copper does not show Masing behaviour. The saturation state, depending on amplitude,
is reached following various amounts of accumulated plastic strain. On the basis
of the experimental results, it appears meaningful to classify into small
(Deeq 0.2%), medium ( 0.2% Deeq 1%) and high (Deeq 1%) amplitudes.
After Hancock and Grosskreutz [14], in the medium amplitude region
(Deeq ˆ 0.375%) at the onset of fatigue, bundles of multipoles initially appear separated
by dislocation poor regions. The majority of dislocations in the bundles are primary
edge dislocations in parallel slip planes, which mutually interact in some sections
to form dipoles and multipoles. Further, as for AlMg 3, prismatic loops are formed
through jog-dragging processes. Screw dislocations on the other hand are hardly found
in this fatigue stage; it is assumed that they are largely annihilated through cross-slip.
In the continued course of fatigue, the density of primary and particularly secondary
dislocations increases in the bundles. The dipoles are divided into small pieces through
cutting processes with dislocations of other slip systems. This causes additional hardening:
The dipole ends now present in higher concentrations are less mobile. A similar
process is also presumed for AlMg 3. The bundles gradually combine to cell-like structures.
Finally, elongated dislocation cells are produced, the walls of which are sharply
outlined against the dislocation poor interstices. The walls comprise short dipoles of
high density. In the dislocation poor regions, screw dislocations stretch from one wall
to the next (Figure 2.7 a, proportional loading with Deeq ˆ 0.5%). According to Laird
et al. [15], one may expect the spatial arrangement of the structure in Figure 2.7 a to
yield approximately cylindrical dislocation cells, the cross-sectional areas of which are
shown here.
25

2 Material State after Uni- and Biaxial Cyclic Deformation
a) b)
After 90 8 phase-shifted overlap of tension/compression and alternate torsion, in copper
with an equivalent strain amplitude of Deeq ˆ 0.5%, isotropic cells dominate. Their
walls are composed of elongated, regularly ordered single dislocations (Figure 2.7 b) as
found by Feltner and Laird for the high plastic strain amplitude Depl ˆ 0.5% [16].
26
c)
Figure 2.6: a) Equal phase overlap of tension/compression and alternate torsion, Deeq ˆ 0.5%,
saturation, Z=[100], multibeam case; b) 90 8 phase-shifted overlap of tension/compression and alternate
torsion, Deeq ˆ 0.5%, saturation, Z=[01-1], g =[-111]; c) deformation band parallel to
the (11-1) plane, tension/compression, Deeq ˆ 0.5%, saturation, Z=[001], g =[200].

2.3 Results
a) b)
Figure 2.7: a) Elongated cells with dipolar walls for copper, tension/compression, Deeq ˆ 0.5%,
saturation, Z=[011], g =[1-11]; b) isotropic, non-dipolar cell structure after phase-shifted loading
for copper, Deeq ˆ 0.5%, saturation, Z=[011], multibeam case.
The lack of dipolar structures is explained by Feltner and Laird as being due to
unhindered cross-slip. The rotating stress vector activates the slip systems required to
create isotropic cell structures at even smaller stress amplitudes than in the proportional
case. Annihilation of screw dislocations is facilitated, thus producing the dislocation
poor inner cell regions. In addition, the enhanced cross slip ability of the screw dislocations
suppresses the creation of prismatic loops.
Thus copper, for proportional and disproportional loading at Deeq ˆ 0.5%, always
exhibits different slip mechanisms. For proportional loading, the screw dislocations
glide to and fro parallel to the walls in the dislocation poor areas. At the same
time, new screw dislocations are continually being pressed out of the walls until they
reach the opposite wall. In between the walls too, new screw dislocations are formed.
The walls themselves take part in the slip by flip-flop movement.
For disproportional loading, only slip dislocations participate in the deformation;
these are pressed out of the walls and after crossing the cell interior are reincorporated
into the opposite cell wall. It follows that copper shows an additional hardening effect,
which is retained in saturation (cf. Figure 2.4a). For austenitic steel 1.4404, the additional
hardening effect predominates at 90 8 phase-shifted loading with equivalent total
strain amplitude of Deeq ˆ 0.75% and 0.5%. Study using the transmission electron
microscope shows for disproportional loading that although a large number of stacking
faults are produced, there is no deformation-induced martensite. For steel 1.4306, this
transformation already occurs at strain amplitudes of Depl ˆ 0.3% under uniaxial
loading [17]. Figure 2.8 a shows a typical dislocation structure after proportional loading
with Deeq ˆ 0.75%. The walls of the elongated cells comprise dislocation bun-
27

dles with a preferential orientation parallel to the {111} planes. On the other hand, the
stronger tendency to multiple slip produces a labyrinthine structure after phase-shifted
loading (Figure 2.8b). The walls are sharply defined against the cell interior.
2.3.3 Yield surfaces
2 Material State after Uni- and Biaxial Cyclic Deformation
a) b)
Figure 2.8: a) Dislocation structure after proportional loading, Deeq ˆ 0.75%, saturation; b) dislocation
structure after 90 8 phase-shifted loading, Deeq ˆ 0.75%, saturation.
The discussion of yield surface measurements may be exemplified by experiments with
equivalent strain amplitude of Deeq ˆ 0.5%. The materials were in the cyclic saturation
state.
2.3.3.1 Yield surfaces on AlMg 3
Figure 2.9 collates the dynamically measured 0.01% offset yield surfaces for AlMg3 for
the four chosen loading types. The starting point each time was the reversal point of
the stress hysteresis.
The yield surfaces for proportional loading (in the following denoted proportional
yield surfaces) are flattened in each relief direction compared with an elliptical shape.
The 0.01% surfaces are in general agreement with those presented in [9]: the 2 ·10 –4 %
offset yield surfaces determined by method 1.
The yield surfaces determined after disproportional loading (hereafter denoted disproportional
yield surfaces) come closest to an isotropic shape (v. Mises ellipse). The
28

2.3 Results
Figure 2.9: 0.01% offset yield surfaces measured in the load reversal points, saturation, AlMg3,
Deeq ˆ 0.5%.
proportional yield surfaces, by contrast, show definitely anisotropic shapes. Compared
with the axial ratio of the v. Mises ellipse, the values measured perpendicular to the
loading direction (transverse yield surface values) are clearly larger than the cross sections
found in the loading direction (longitudinal yield surface values). As the comparison
of yield surfaces measured at the upper reversal point (Figure 2.9) and at the stress
zero crossover (cf. Figure 2.2) shows, both the transverse values and the contracted
longitudinal values within a cycle remain constant. The shape of the yield surface,
however, changes from the flattened form at the load reversal point to an essentially
symmetrical ellipse in the stress zero crossover. During the further course of the negative
half-cycle, this then changes into a flattened shape once more (flattening again on
the origin side). This deformation may also be observed on yield surfaces with the
small offset strain of 2 ·10 –4 % and on tangent module areas with high tangential gradients.
Figure 2.10 shows the proportional and disproportional yield surfaces measured at
the load reversal points for the relatively large offset strain of 0.2%.
In consequence of the high plastic fraction, during deformation, all four yield surfaces
practically coincide and are almost elliptical in shape. Referring to the v. Mises
condition or Equation (2), the longitudinal values are slightly less than the transverse
ones. The surfaces thus show, in weaker form, the same anisotropy as those measured
with small offsets. The torsional yield surface is slightly flattened in the relief direction.
29

2 Material State after Uni- and Biaxial Cyclic Deformation
Figure 2.10: 0.2% offset yield surfaces measured at the load reversal points, saturation, AlMg3,
Deeq ˆ 0.5%.
2.3.3.2 Yield surfaces on copper
Copper behaves in many aspects like AlMg 3. In Figure 2.11, the three proportional
yield surfaces measured at the load reversal points are shown in contrast with the disportional
surface.
The proportional surfaces are again flattened in the relief direction. The shortened
axis too remains the same throughout the whole hysteresis cycle. The disproportional
yield surface approaches the elliptical shape, which is significantly larger. The distinct
additional hardening effect of copper thus causes an additional isotropic hardening.
2.3.3.3 Yield surfaces on steel
Figure 2.12 shows 0.02% offset yield surfaces measured after equal phase superposition
at the upper and lower reversal points of the saturation hysteresis. As for AlMg 3
and copper, the displacement of the yield surface in the loading direction and the flattening
on the origin side are clearly seen.
Figure 2.13 represents the 0.02% offset yield surfaces measured after disproportional
loading at the reversal points of tension and compression (c =0). For this load path, the
yield surface follows the rotating stress vector. Both yield surfaces are symmetrical to
the tensile stress axis and again show the typical flattening on the origin side.
30

2.4 Sequence Effects
2.4 Sequence Effects
Figure 2.11: 0.01% offset yield surfaces measured at the load reversal points, saturation, copper,
Deeq ˆ 0.5%.
Figure 2.12: 0.02% offset yield surfaces measured at the load reversal points, proportional loading
(tension/compression and alternate torsion), saturation, steel 1.4404, Deeq ˆ 0.5%.
On AlMg 3 and copper in the saturation state, the variation of the loading direction
from tension/compression to alternating torsion, and the reverse, was investigated. The
experiments were meant to show how inner stresses affect the shape of the yield surfaces.
For the offset strain of 0.01%, the points of yield onset were taken from the as-
31

2 Material State after Uni- and Biaxial Cyclic Deformation
Figure 2.13: 0.02% offset yield surfaces measured at the load reversal points of the tension/compression
hysteresis, disproportional loading, saturation, steel 1.4404, Deeq ˆ 0.5%.
cending and descending branches of the hysteresis curves and from these, the yield surfaces’
diameters (cross sections) determined. In similar fashion, the diameters of the
surfaces with equal tangent modules were also determined [9, 18]. In addition, the transition
of the maximum stress amplitude to the new saturation state was observed. Figure
2.14 shows the change of the 0.01% yield surface diameter of AlMg3 and copper
(equivalent total strain amplitude Deeq ˆ 0.5%) for the transition from pure tension/
compression to pure alternating torsion.
The broken lines show the transverse values of the tension/compression saturation
yield surface, respectively (state before the change). The continuous lines represent the
longitudinal values of the saturation yield surfaces, which would have appeared following
pure torsion. The yield surfaces’ diameters of torsion hysteresis in the case of
AlMg3 already decrease drastically in the first cycle and quickly reach a new saturation
state yet without recurring to the saturation longitudinal value following pure torsion.
The new loading state must therefore differ from the initial state with regard to the inner
stress, or else, in consequence of isotropic hardening, the saturation yield surfaces
are larger after prior tension/compression than after pure alternating torsion.
The second option is confirmed by the dislocation structure. As already demonstrated,
for AlMg 3, an anisotropic dislocation structure evolves after proportional loading.
In extensive grain areas, only few slip systems are activated; the dipolar walls generally
take up only one or two crystallographic directions. Since different slip systems
are involved in tension/compression loading than in alternating torsion, the dipolar
walls orient themselves in different crystallographic directions. The screw dislocations
move in dislocation poor channels parallel to the dipolar walls, adjacent to the respective
slip systems. If a tension/compression experiment is immediately followed by one
with alternating torsion, the dislocation structure is initially unfavourable for torsion.
With changing loading direction, the sources of torsional slip dislocations are activated
first and then later on, the dipolar walls change their orientation to one more favourable
for torsional loading. As is seen from Figure 2.14, the greater fraction of the torsional
slip dislocations is already activated in the first three cycles after the change of
32

2.4 Sequence Effects
Figure 2.14: AlMg3 and copper, 0.01% yield surface diameter after changing the loading direction
from tension/compression to alternating torsion.
the loading direction. The result is an immediate drastic reduction of the yield surface
diameter. However, further restructuring of the dislocation arrangement appears to be
impeded; the diameter remains constant during subsequent cycles. The dislocation
structure anticipated for pure torsional loading is clearly unable to evolve following
previous tension/compression loading. The large number of activated slip systems after
the change in loading direction may offer some explanation.
For copper, the yield surface diameter from torsional hysteresis also seriously decreases
in the first torsional half-cycle after changing the loading type. Moreover, in
contrast to AlMg 3, it falls continuously until the saturation longitudinal value for pure
torsion is reached. The longitudinal values for saturation yield surfaces thus come
about independent of previous history. The same is true for the diameters of the tangent
module areas. The dislocations in copper arrange themselves in a similarly isotropic
way as in AlMg 3 (elongated cells, dipolar walls: see Figure 2.7 a). Yet after the change
in loading direction, they reorientate themselves completely. This property characterizes
materials with wavy slip behaviour [15].
In a further experiment to assess the effect of inner stress, samples of AlMg 3 and
copper were relieved from various points in the torsional hysteresis branch. The yield
surface diameters were taken from these partial cycles and plotted in Figure 2.15 as a
function of the offset strain and of the strain values (initial stress relieving points).
33

For a given offset strain, both materials showed the same yield surface longitudinal
values at every point in the hysteresis. The influence of inner stress, assumed load
dependent, upon the shape of the yield surfaces would therefore not appear to be significant.
It seems that the dislocation structure exerts the critical influence.
2.5 Summary
2 Material State after Uni- and Biaxial Cyclic Deformation
Figure 2.15: Yield surface longitudinal values for various offsets determined after stress relief
from various points on the torsional hysteresis.
We have presented yield surfaces on AlMg 3, copper and austenitic steel 1.4404 (AISI
316L) after tension/compression and alternating torsional loading as well as proportional
and phase-shifted superposition of both loads. The materials were first cycled to
saturation with maximum deformation amplitudes of 0.75%, whereby substantial additional
hardening effects occurred. The development of the appropriate dislocation
structures was studied using a transmission electron microscope.
34

Yield surfaces measured in all three materials at the reversal points of the stress
deformation hysteresis, for small offset strains (0.01% or 2 · 10 –4 %), after proportional
alternate loading, show a flattened shape in the off-load direction compared with the
v. Mises ellipse. At the stress zero crossover points of the hysteresis, the yield surfaces
assume a symmetrical shape. Transverse and longitudinal values of the yield surfaces
remain constant independent of the starting point in the hysteresis. This behaviour and
the sequence effects confirm that the anisotropy of the yield surfaces is caused by the
appropriately anisotropic dislocation structure of the materials. Inner stresses obviously
play a minor role.
After disproportional loading, generally isotropic yield surfaces result. This may
be explained quite simply by the relevant isotropic dislocation structures. Yield surfaces
of higher offset strains and areas of equal tangent modules for small tangential gradients
also evolve essentially isotropically since sufficient slip systems are activated during
the measurement procedure and the dislocation walls participate in the slip process.
Acknowledgements
The authors thank Mr. Horst Gasse for his decisive contribution to the development of the
experimental apparatus, the measuring technique and the performance of the experiments.
References
References
[1] Y. F. Dafalias, E.P. Popov: Plastic Internal Variables Formalism of Cyclic Plasticity. Journal
of Applied Mechanics 63 (1976) 645–651.
[2] Y. F. Dafalias: Bounding Surface Plasticity, I Mathematical Foundation and Hypoplasticity.
Journal of Engineering Mechanics 12 (9) (1986).
[3] D.L. McDowell: A Two Surface Model for Transient Nonproportional Cyclic Plasticity,
Part 1: Development of Appropriate Equations, Part 2: Comparison of Theory with Experiments.
Journal of Applied Mechanics 85 (1986) 298–308.
[4] F. Ellyin: An isotropic hardening rule for elastoplastic solids based on experimental observations.
Journal of Applied Mechanics 56 (1969) 499.
[5] N.K. Gupta, H.A. Lauert: A study of yield surface upon reversal of loading under biaxial
stress. Zeitschrift für angewandte Mathematik und Mechanik 63(10) (1983) 497–504.
[6] J.F. Williams, N.L. Svensson: Effect of torsional prestrain on the yield locus of 1100-F aluminium.
Journal of Strain Analysis 6(4) (1971) 263.
[7] R. Hillert: Austenitische Stähle bei ein- und bei zweiachsiger, plastischer Wechselbeanspruchung.
Dissertation TU Braunschweig, 2000.
[8] W. Gieseke, G. Lange: Veränderung des Werkstoffzustandes bei mehrachsiger plastischer
Wechselbeanspruchung. In SFB Nr. 319 Arbeitsbericht 1991–1993, TU Braunschweig.
35

2 Material State after Uni- and Biaxial Cyclic Deformation
[9] W. Gieseke: Fließflächen und Versetzungsstrukturen metallischer Werkstoffe nach plastischer
Wechselbeanspruchung. Dissertation TU Braunschweig, 1995.
[10] H.J. Christ: Wechselverformung der Metalle. In: B. Ilschner (Ed.): WFT Werkstoff-Forschung
und Technik 9, Springer Verlag Berlin, 1991.
[11] J.I. Dickson, J. Boutin, G.L. ’Espérance: An explanation of labyrinth walls in fatigued
f.c.c. metals. Acta Metallurgica 34(8) (1986) 1505–1514.
[12] J.L. Dickson, L. Handfield, G.L. ’Espérance: Geometrical factors influencing the orientations
of dipolar dislocation structures produced by cyclic deformation of FCC metals.
Materials Science and Engineering 81 (1986) 477–492.
[13] P. Lukás, M. Klesnil: Physics Status solidi 37 (1970) 833.
[14] J.R. Hancock, J.C. Grosskreutz: Mechanisms of fatigue hardening in copper single crystals.
Acta Metallurgica 17 (1969) 77–97.
[15] C. Laird, P. Charlsey, H. Mughrabi: Low energy dislocation structures produced by cyclic
deformation. Materials Science and Engineering 81 (1986) 433–450.
[16] C. E. Feltner, C. Laird: Cyclic stress-strain response of FCC metals and alloys II. Dislocation
structures and mechanism. Acta Metallurgica 15 (1967) 1633–1653.
[17] M. Bayerlein, H.-J. Christ, H. Mughrabi: Plasticity-induced martensitic transformation during
cyclic deformation of AISI 304L stainless steel. Materials Science and Engineering A
114 (1989) L11–L16.
[18] W. Gieseke, G. Lange: Yield surfaces and dislocation structures of Al-3Mg and copper
after biaxial cyclic loadings. In: A. Pineau, G. Cailletaud, T. C. Lindley (Eds.): Multiaxial
fatigue and design, ESIS 21, Mechanical Engineering Publications, London, 1996, pp. 61–
74.
36

3 Plasticity of Metals and Life Prediction in the Range
of Low-Cycle Fatigue: Description of Deformation
Behaviour and Creep-Fatigue Interaction
Abstract
Kyong-Tschong Rie, Henrik Wittke and Jürgen Olfe*
Results of low-cycle fatigue tests are presented and discussed, which were performed at
the Institut für Oberflächentechnik und plasmatechnische Werkstoffentwicklung of the
Technische Universität Braunschweig, Germany. The cyclic deformation behaviour was
investigated at room temperature and high temperatures. The investigated materials are
copper, 2.25Cr-1Mo steel, 304L and 12%Cr-Mo-V steel. (Report of the projects A5
and B4 within the Collaborative Research Centre (SFB 319) of the Deutsche Forschungsgemeinschaft.)
3.1 Introduction
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
Low-cycle fatigue (LCF) and elasto-plastic cyclic behaviour of metals represent a considerable
interest in the field of engineering since repeated cyclic loading with high amplitude
limit the useful life of many components such as hot working tools, chemical
plants, power plants and turbines. During loading in many cases after a quite small
number of cycles with cyclic hardening or softening, a state of cyclic saturation is
reached. This saturation state can be characterized by a closed stress-strain hysteresisloop.
Cyclic deformation in the regime of low-cycle fatigue (LCF) leads to the formation
of cracks, which can subsequently grow until failure of a component part takes
place.
The crack growth is correlated with parameters of fracture mechanics, which take
into account informations especially about teh steady-state stress-strain hysteresis-loops.
* Technische Universität Braunschweig, Institut für Oberflächentechnik und plasmatische Werkstoffentwicklung,
Bienroder Weg 53, D-38106 Braunschweig, Germany
37

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Therefore, a more exact life prediction is possible by investigating the cyclic deformation
behaviour in detail and describing the cyclic plasticity, e.g. with constitutive equations.
In this paper (see Section 3.3), the investigated cyclic deformation behaviour was
described by analytical relations and, moreover, by relations, which take into account
physical processes as the development of dislocation structures.
When components are loaded at high temperature, additional processes are superimposed
on the fatigue. Besides corrosion, which is not discussed here, creep deformation
and creep damage are the most important. Therefore in many cases, not one type
of damage prevails, but the interaction of both fatigue and creep occurs leading to failure
of components.
A reliable life prediction model for creep-fatigue must consider this interaction as
proposed by the authors (see Section 3.4.1). In this model, the propagating crack,
which is the typical damage in the low-cycle fatigue regime, interacts with grain
boundary cavities. Cavities are for many steels and some other metals the typical creep
damage and also play an important role in the case of creep-fatigue. The possibility of
unstable crack advance, which is the criterium for failure, is given if a critical configuration
of the nucleated and grown cavities is reached.
Therefore, the basis for reliable life prediction is the knowledge and description
of the cavity formation and growth by means of constitutive equations. In the case of
diffusion-controlled cavity growth, the distance between the voids has an important influence
on their growth. This occurs especially in the case of low-cycle fatigue, where
the cavity formation plays an important role. Thus, the stochastic process of pore nucleation
on grain boundaries and the cyclic dependence of this process have to be taken
into consideration as a theoretical description. The experimental analysis has to detect
the cavity size distribution, which is a consequence of the complex interactions between
the cavities (see Section 3.4.2).
Formerly, the total stress and strain have been used for the calculation of the
creep-fatigue damage. However, these are macroscopic parameters, whereas the crack
growth is a local phenomenon. Therefore, the local conditions near the crack tip have
to be taken into consideration. The determination of the strain fields in front of cracks
is an important first step for modelling (see Section 3.4.3).
3.2 Experimental Details
3.2.1 Experimental details for room-temperature tests
The materials used for the uniaxial fatigue tests at room temperature were polycrystalline
copper and the steel 2.25Cr-1Mo (10 CrMo910). Specimens of 2.25Cr-1Mo were
investigated in as-received conditions, in the case of copper, the material was annealed
at 650 8C for 1/2 h.
38

3.2 Experimental Details
The tests were controlled by total strain and carried out at room temperature in
air. The strain rates were _e=10 –3 s –1 (or, for a small number of tests, _e=2·10 –3 s –1 ) for
steel, and _e =10 –4 s –1 and _e=10 –3 s –1 for copper. Most of the tests were single-step tests
(SSTs) with a constant strain amplitude De/2, some tests were performed as two-step
tests (2STs) and other as incremental-step test (ISTs). In the case of the two-step test,
the specimens had been cycled to a steady-state regime before the strain amplitude was
changed in the next step. The strain amplitudes were in general in the low-cycle fatigue
range and a few amplitudes in the range of high-cycle fatigue (HCF) and in the transition
regime between low-cycle fatigue and extremely low-cycle fatigue (ELCF): The
tests with copper were performed with strain amplitudes between 0.1 and 1.7%, the
tests with steel with amplitudes between 0.185 and 1.2%.
The incremental-step tests were carried out with constant strain rate and with given
values for the lowest and the highest strain amplitude, (De/2)min and (De/2)max. The
factor of subsequent amplitudes q a in the ascending part of the IST-block or, alternatively,
the difference of amplitudes d a is constant.
For most of the tests, smooth cylindrical specimens were used. Usually, the diameter
and the length of the gauge were 14 mm and 20 mm, for the tests with very high
strain amplitudes (near the ELCF-regime), the diameter was 14.7 mm and the length
10 mm. For some tests, flat specimens were used with the values 8.7 ×5mm 2 for the
rectangular cross-section.
The steady-state microstructure of tested specimens was investigated with transmission
electron microscope at the Institut für Schweißtechnik (Prof. Wohlfahrt [1]),
the Institut für Metallphysik und Nukleare Festkörperphysik (Prof. Neuhäuser [2]) and
the Institut für Werkstoffe (Prof. Lange [3]). They are all at the Technische Universität
Braunschweig and involved in the Collaborative Research Centre (SFB 319).
3.2.2 Experimental details for high-temperature tests
The creep-fatigue tests were carried out on 304L austenitic stainless steel and on 12%
Cr-Mo-V ferritic steel. The tests were total strain-controlled low-cycle fatigue tests with
a tension hold time up to 1 h at 600 8 and 650 8C for the 304L, and 5508C for the ferritic
steel. For the tests for the lifetime determination and the tests for analysing the cavity
configuration, we used round and polished specimen. After low-cycle fatigue testing,
the specimens were metallographically prepared for stereological analysis of the density
and cavity size distribution (see Section 3.4.2.1).
A furnace with a window and special optics allow high magnification observation
of the specimen surface continuously during the test with a video system and a subsequent
measurement of the crack growth, the crack tip opening and the crack contour on
flat and polished specimens in an inert atmosphere. In-situ measurement of the strain
field in front of the crack was performed by means of the grating method [4–9].
The surface of the specimen was prepared with a grating of TiO 2 with a line
distance of 200 lm, which was photographed at the beginning of the test and at given
loads after cycling. By means of digital image analysis, the local strain at every cross
of the grating was calculated by the group of Prof. Ritter [10] and Dr. Andresen [11]
39

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.1: Deformed grid and corresponding strain in the direction of the load (in-situ, 5508C).
(TU Braunschweig, Collaborative Research Centre (SFB319)). The following picture
(Figure 3.1) shows a photograph of the grid and the digitized picture with the regions,
where the local strain is higher than 4% and 5%. The position of the crack is illustrated
by means of a straight line, and the line, which surrounds the 5% deformed zone at the
crack tip, is shown in the figure. From the figure, the size of the 5% deformed zone in
direction of its maximum expansion was taken. In the following, this distance was
designed as R0.05 in analogy to Iino [12]. It has been used to describe the development
of the highly deformed zone in dependence on the crack length and the tension hold
time.
3.3 Tests at Room Temperature:
Description of the Deformation Behaviour
3.3.1 Macroscopic test results
In single-step tests, annealed copper shows cyclic hardening in nearly the whole range
of lifetime. After a quite small number of cycles, the end of a rapid hardening regime
is reached. Due to the effect of secondary hardening, in some ranges of amplitudes, no
saturation was observed, but, as first approximation, the effect of secondary hardening
can be neglected [13]. Examples for cyclic hardening curves up to saturation are shown
in Figure 3.2 a.
In the case of single-step tests with 2.25Cr-1Mo, there is cyclic softening in
nearly the whole range of strain amplitudes. In the first cycles, rapid hardening can be
found before cyclic softening takes place. After this, a steady-state regime can be
40

3.3 Tests at Room Temperature: Description of the Deformation Behaviour
Figure 3.2: Copper; a) cyclic hardening curves, _e =10 –4 s –1 ; b) cyclic stress-strain curves: amplitudes
of applied and internal stress vs. amplitude of plastic strain. &: data of SSTs with _e =10 –3 s –1 ; *:
data of SSTs and 2STs with _e =10 –4 s –1 ; n, ~: data of stress relaxation tests after SSTs with
_e =10 –3 s –1 or _e =10 –4 s –1 , respectively.
found, which continues until a failure takes place. While in the case of copper, there is
a very clear effect of rapid hardening, in the case of 2.25Cr-1Mo, the effects of cyclic
hardening and softening are less pronounced.
For both materials, after a certain number of cycles, a state of saturation is
achieved. The stress-strain behaviour is represented by a hysteresis-loop. (To avoid confusion,
it may be useful to mark characteristic values of the steady-state hysteresis-loop
with an index. For example, the amplitude of stress Dr/2 can be written in the case of
saturation as (Dr/2) s. Nevertheless, no index is used in this paper because it is usually
clear from the context whether the instantaneous or the steady-state values are referred.)
In Figure 3.2 b, an example for cyclic stress-strain curves, Dr/2 or Dr i/2 vs. De p/2,
are shown, which are constructed with the aid of steady-state hysteresis-loops. The values
of the plastic strain e p are given in dependence on total strain e and stress r by:
ep ˆ e r=E ; …1†
where E is the Young’s modulus. This equation is used to describe also the relation between
the amplitudes of plastic strain De p/2, total strain De/2 and stress Dr/2. The amplitudes
of the internal stress, Dr i/2, are found with the aid of stress relaxation tests
(see [14]). Most of the experimental points shown in Figure 3.2 b were found from 24
tests with amplitudes in the range of LCF (single-step tests and two-step tests with
low-high amplitude-sequences; 0:16% De=2 1:0%†. Additionally, one test in the
high-cycle fatigue (HCF) regime and three tests in the transition regime between LCF
and extremely low-cycle fatigue (ELCF: compare Komotori and Shimizu [15]) are
taken into consideration. In the case of copper, the 24 tests are used to study various
41

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.3: Steady-state stress-strain hysteresis-loops; a) 2.25Cr-1Mo, _e=10 –3 s –1 ; b) copper, hysteresis-loops
in relative coordinates.
parameters of the material, in the case of 2.25Cr-1Mo, data are used from 13 singlestep
tests (12 LCF-tests, one HCF-test).
Examples for steady-state hysteresis-loops are shown in the Figure 3.3 a and b. In
Figure 3.3 a, stress-strain hysteresis-loops of 2.25Cr-1Mo are shown, in Figure 3.3 b, hysteresis-loops
in relative coordinates, r r and e r, are shown in the case of copper. The relative
coordinate system is defined by an origin, which is set at the point of minimum stress and
strain of the hysteresis-loop. The material exhibits Masing behaviour when the upper
branches of different hysteresis-loops follow a common curve in the relative coordinate
system. In contrast, copper exhibits non-Masing behaviour in single-step tests as can be
seen in Figure 3.3 b. Also for 2.25Cr-1Mo, non-Masing behaviour was found. Only in a
small range of the tested amplitudes, in the range of 0.185%< De/2

3.3 Tests at Room Temperature: Description of the Deformation Behaviour
Figure 3.4: Copper (_e =10 –4 s –1 ) shifted; hysteresis-loops and master curve.
3.3.2 Microstructural results and interpretation
For both materials, dislocation cell structures were found. For 2.25Cr-1Mo, cell structure
was found in single-step tests in the range of amplitudes, in which the materials exhibit
non-Masing behaviour. In the case of single-step tests with copper, the cell structure is
well developed for high amplitudes, for low amplitudes, other dislocation structures are
dominating as e.g. vein structure. Often, the shape of the cells is not cuboidal but elongated.
With increasing strain amplitude, the cell size is decreasing (compare Feltner and
Laird [22]). Schubert [18] proposed a microstructure-dependent cyclic proportional limit
rprop ˆ rL ‡ 2 MS Gb=dm ; …2†
where rL is the lattice friction stress, MS is the Sachs factor, G is the shear modulus
and b is the absolute value of the Burgers vector. The decrease of the mean cell size
d m and the increase of r prop with increasing strain amplitude is in agreement with the
non-Masing behaviour of the materials [13, 14]. For 2.25Cr-1Mo, the value of d m in
Equation (2) corresponds to the mean distance of precipitates for low amplitudes and to
the mean cell size for high amplitudes (De/2 > 0.4%). Therefore, in the case of low amplitudes,
Masing behaviour was found [18].
A typical steady-state dislocation structure of the second step of a two-step test
with an amplitude-sequence high-low is shown in Figure 3.5. A dislocation cell structure
can be seen although the dominating structure of the low amplitude in the case of
a single-step test is vein structure (see [18]). While in two-step tests with amplitude-sequences
low-high the microstructure is history-independent, it is obviously not independent
in the case of a test with an amplitude-sequence high-low (compare [21]). Nevertheless,
the dependence of the macroscopic behaviour on this history-dependent microstructural
behaviour is almost negligible.
In incremental-step tests with sufficiently high values of (De/2)max, dislocation
cell structure can be found in cyclic saturation. The dislocation structure is assumed as
43

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.5: Copper, dislocation cell structure of a two-step test; _e=10 –4 s –1 , strain amplitude sequence
0.4–0.2%: steady-state dislocation structure of the second step.
quasi-stable: In cyclic saturation, the dislocation structure does not change within one
IST-block. The quasi-stable dislocation structure correlates well with the Masing behaviour
of the incremental-step test (for details: see Schubert [18]).
Experimental values of the dislocation cell size or cell wall distance, respectively,
are:
dm ˆ 0:85 l for De=2 ˆ 0:2% ;
dm ˆ 0:76 l for De=2 ˆ 0:4% ;
dm ˆ 0:58 l for De=2 ˆ 0:7% ;
in the case of copper and SSTs for _e ˆ 10 4 s 1 . In the case of 2.25Cr-1Mo, SSTs,
_e ˆ 10 3 s 1 , the experimental values are:
dm ˆ 0:85 l for De=2 ˆ 0:6% ; and
dm ˆ 0:65 l for De=2 ˆ 1:2% ‰13Š :
These values were used to calculate the cyclic proportional limit rprop, and a good
agreement with the macroscopic cyclic proportional limit defined by an offset of 0.01%
was found [18]. Moreover, the values of [18] are used for further evaluation (Sections
3.3.4.1 and 3.3.4.2).
44

3.3 Tests at Room Temperature: Description of the Deformation Behaviour
3.3.3 Phenomenological description of the deformation behaviour
3.3.3.1 Description of cyclic hardening curve, cyclic stress-strain curve
and hysteresis-loop
It is shown by Wittke [13] that the first part of a cyclic hardening curve of a single-step
test, the rapid hardening regime, can be described excellently with a stretched exponential
function for stress amplitude Dr/2 vs. cycle number N (or more exact: N – 0.25):
Dr=2 ˆ A0 ‡…As A0† 1 exp ‰ …N 0:25†=N0Š
kH
h i
: …3†
The constants A 0 and A s are closely related to the monotonous and cyclic stress-strain
curve, respectively, the constants N 0 and k H are found by trial and error. A simple dependence
of the parameters on the steady-state value of the plastic portion of the total
strain amplitude can be found [8]. Moreover, the stretched exponential function, Dr/2
vs. N, is applicable also for two-step tests in the case of hardening and softening in
good approximation. The comparison between experimental and calculated cyclic hardening
curves is given in Figure 3.2 a.
It is usual to describe the cyclic stress-strain curve (css-curve) by a power law.
As can be seen in Figure 3.2b, in the case of copper, the description of the cyclic
stress-strain curves by the solid line and the dotted line is quite good. The double-logarithmic
cyclic stress-strain curves, Dr/2 vs. De p/2, for different strain rates are nearly
parallel. Also in the case of 2.25Cr-1Mo, the description of the cyclic stress-strain
curve by a power law is good. In the case of 2.25Cr-1Mo, we get with
Dr=2 ˆ k 0 …Dep=2† n0
and by using the constants k 0 ˆ 803 MPa and n 0 ˆ 0:138 good agreement between experimental
and calculated values …_e ˆ 10 3 s 1 ; E ˆ 208 GPa†.
For copper, the values of the constants for the different css-curves in Figure 3.2b are:
k 0 ˆ 554:6 MPa; n 0 ˆ 0:228 for _e ˆ 10 3 s 1 ;
k 0 ˆ 565:9 MPa; n 0 ˆ 0:238 for _e ˆ 10 4 s 1 ; and
k 0 ˆ 441:3 MPa; n 0 ˆ 0:220 for internal stress measurements tests :
With regard to fatigue fracture mechanics and lifetime estimation, the description of the
steady-state hysteresis-loop is the most important point in this Section 3.3. In first approximation,
also in the case of the hysteresis-loop, a power law between relative stress
and relative plastic strain, r r and e pr, can be assumed (see Morrow [23]):
rr ˆ kH e b pr : …5†
It should be mentioned that the parameters kH and b are dependent on the plastic strain
amplitude.
…4†
45

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Although the description of the hysteresis-loop with a power law is quite rough, it
may be useful to apply such a law for fracture mechanical estimation (see Rie and
Wittke [24]). To get a better description of the hysteresis-loop shape, other relations are
necessary. The hysteresis-loop can be described, e.g. by a two-tangents method (compare
[19]), as follows:
epr ˆ…rr=k0† 1=b 0 ‡…rr=kE† 1=b E : …6†
For each hysteresis-loop, four constants, k 0, b 0, k E and b E, have to be determined. An
example for the applicability of this relation in the case of the mild steel Fe510 (St-52),
which was tested at the Institut für Stahlbau of the TU Braunschweig (Prof. Peil [25]),
is shown in Figure 3.6.
We have developed other very exact relations with only three constants. They are
expressed by:
rr ˆ AG ‰1 exp … …epr=dG† G †Š ; …7†
or alternatively by:
rr ˆ Cq exp … jq ‰ln …epr=dE†Š 2 † : …8†
The three constants of the stretched exponential function (Equation (7)) are A G, d G and
G, the constants of the exponential parabola function (Equation (8)) are C q, j q and d E.
Examples for the excellent applicability of both equations are shown by Rie and Wittke
[14] and Wittke [13]. In contrast to other relations, in the case of Equations (7) and (8),
a good agreement between experiment and calculation can be found even for the second
derivative of the hysteresis-loop branch, d 2 r r/de 2 r vs. e r (see Section 3.3.4.1).
Figure 3.6: Mild steel Fe510; hysteresis-loop in relative coordinates; De/2=0.5%, _e =10 –4 s –1 ;
comparison between experiment and calculation; calculation according to Equation (6), k 0 =45418
MPa, b 0 =0.553, k E=1238 MPa, b E=0.095; E=210 GPa.
46

3.3 Tests at Room Temperature: Description of the Deformation Behaviour
3.3.3.2 Description of various hysteresis-loops with few constants
A very exact description of the shape of various hysteresis-loops with few constants
can be obtained when the parameters of the power law (Equation (5)), kH and b, are
given as simple functions of the plastic strain range Dep. Such functional relations are
developed in [13, 26]. Furthermore, a method to calculate the parameters of the exponential
parabola function (Equation (8)), C q, j q, and d E, in dependence on the plastic
strain amplitude is described in [13].
As shown above, in the case of non-Masing behaviour, it is possible to get a master
curve. This master curve together with the cyclic stress-strain curve can be used to
construct each hysteresis-loop (see [17]). In contrast to the power-law master-curve proposed
by Lefebvre and Ellyin [17], better results were achieved, e.g. by a stretched exponential
function or an exponential parabola function (see [13]). In the latter case, the
master curve can be described by:
r ˆ Cq exp … jq ‰ln …epr=dE †Š 2 † ; …9†
where Cq , jq and dE are constants. For copper, almost independent on strain rate, the
values of the parameters of the master curve (compare Figure 3.4) are:
Cq ˆ 246:1 MPa; jq ˆ 0:03576 ; dE ˆ 2:3194% :
All these methods, which were used to describe various steady-state hysteresis-loops of
copper with few constants, are also applicable in the case of 2.25Cr-1Mo.
3.3.4 Physically based description of deformation behaviour
3.3.4.1 Internal stress measurement and cyclic proportional limit
For a physically based description of the cyclic deformation behaviour, it is necessary
to take into consideration that the applied stress r can be separated into the internal
and the effective stress, r i and r eff. The effective stress is that fraction of the total
stress causing dislocations to move at a specific velocity, the internal stress can be defined
as the stress needed to balance the dislocation configuration at a net zero value of
the plastic strain rate (see Tsou and Quesnel [27]).
At room temperature, internal stress can be easily obtained experimentally by
stress relaxation tests. For this purpose, test specimens were cycled to approximated
saturation in uniaxial push-pull tests in the range of LCF prior to the relaxation tests.
Figure 3.7 a shows hysteresis-loops for copper with both the total stress r and the internal
stress r i plotted vs. the plastic strain e p. In agreement with the method of Tsou and
Quesnel [27], the stress value after 30 min of relaxation is adopted as the internal
stress value. Figure 3.7 b shows for a stress relaxation test performed after a monotonic
strain-controlled tension test (_e 10 4 s 1 † that this is a good approximation: After
47

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.7: Copper; a) hysteresis-loops: total and internal stress vs. plastic strain; De/2=0.4%,
_e =10 –3 s –1 ; *: experimental data of stress relaxation tests, ------:calculation analogous to
Equation (7), AG=260 MPa, dG=0.039%, G=0.412; b) data of strain-controlled tension test (interrupted
at e =9.3%) and relaxation test.
less than 30 min=1800 s, the stress value is almost constant. (In Figure 3.7 b, t=0 is
defined by the start of the stress relaxation procedure.)
The twelve experimental points of Figure 3.7 a are expressed by the dotted-line fit
curve. The fit curve can be described by a relation, which is analogous to Equations
(7) or (8), respectively. By considering different r i-e p-hysteresis-loops, the cyclic stressstrain
curve Dr i/2 vs. De p/2 can be determined. This cyclic stress-strain curve has been
shown already in Figure 3.2b. With the same amplitude of the plastic strain, the shape
of the r i-e p-hysteresis-loop is assumed to be independent of the strain rate of the prior
cyclic test (compare Tsou and Quesnel [27], Hatanaka and Ishimoto [28]).
The proposed Equations (7) and (8) are well appropriate to fit the experimental
points. Therefore, one of them (here Equation (8)) is used in the following for checking
whether the ri-ep-hysteresis-loops exhibit Masing or non-Masing behaviour. To investigate
the Masing or non-Masing behaviour of the r i-e p-hysteresis-loops, several hysteresis-loops
are presented in Figure 3.8 a in relative coordinates. In this figure, r ir-e pr-hysteresis-loops
for a small, a medium and a quite large strain amplitude of the LCF range
are shown. Non-Masing behaviour can be seen clearly.
In the following, the dependence of the non-Masing behaviour on microstructure
will be quantified with the model of Schubert [18]. As usual, a macroscopic cyclic proportional
limit can be defined by a strain offset, e.g. 0.01%. Nevertheless, the value of
the strain offset is arbitrary and has no physical meaning. Therefore, in the case of the
r ir-e pr-hysteresis-loops, a better way is chosen: At first, a hypothetic hysteresis-loop r ir
vs. e r is constructed with the given values of r ir, e pr and the analogous relation to Equation
(1):
48
er ˆ epr ‡ rir=E : …10†

3.3 Tests at Room Temperature: Description of the Deformation Behaviour
In the next step, the second derivative of a half branch of this hysteresis-loop, d 2 r ir/de 2 r
vs. e r, is constructed. The e r-value of the extreme point (minimum) of this second derivative
is called here e r, ex. Now, we define a macroscopic cyclic yield stress:
ryc ˆ er; ex E=2 …11†
in agreement with a statistical approach based on the distribution of elementary volumes
with different yield stresses (compare Polák et al. [29]): r yc is interpreted as the
yield stress with the highest probability density within the material. By this definition,
an uniquely applicable and physically better justified macroscopic cyclic proportional
limit is found. With the values of d m and r prop (see Equation (2)) for copper given by
Schubert [18] or Rie et al. [19], respectively, the good agreement between the two cyclic
proportional limits, r prop and r yc, can be seen in Figure 3.8b.
The effective stresses contribute also to the non-Masing behaviour of materials, but
in agreement with the above mentioned model, the main reason of the non-Masing behaviour
is thought to be governed by the non-Masing behaviour of the r i-e p-hysteresis-loops.
In the case of 2.25Cr-1Mo, the described model is also applicable. Evaluation of
a stress relaxation test for another charge of the material give a value of Dr i/Dr =0.907
for De/2=0.6%. This value is quite similar to the tests with copper.
Figure 3.8: Copper; a) hysteresis-loops of relative internal stress vs. relative plastic strain (without
experimental values; dotted line hysteresis-loops: calculation by Equation (8)); parameters of the
former performed single-step tests: strain rate _e =10 –4 s –1 ; strain amplitudes De/2: 0.16%, 0.4%
and 0.7%; b) cyclic proportional limits, r prop and r yc, in dependence on plastic strain range. The
values of rprop are calculated in dependence on experimental values of dm; the values of ryc are
determined with the aid of the r i-e p-hysteresis-loops and described by the dotted-line fit function.
49

3.3.4.2 Description of cyclic plasticity with the models of Steck and Hatanaka
By Schlums and Steck [30], a model was proposed, which allows to describe high-temperature
cyclic deformation behaviour in terms of metal physics and thermodynamics.
A modification was given by Gerdes [31] to use the model for low temperatures (e.g.
room temperature). The model was applied in the case of copper. According to this
model, the plastic strain rate can be calculated as follows:
_ep ˆ CG exp
QG
RT sinh DVG …r ri†
RT
: …12†
The evolution equations of the internal parameters, r i and DV G (internal stress and activation
volume), are:
and
3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
_ri ˆ HG E exp
b GDVGri sign …r ri†
RT
_ep
…13†
D _VG ˆ K1DV 2 G j_epj‡K2DVGj_epj ; …14†
where R=8.3147·10 –3 kJ mol –1 K –1 , T=293 K, Q G=49.0 kJ mol –1 . The Young’s modulus
is dependent on temperature (room temperature: E=116 GPa), the other constants,
C G, Q G, H G, b G, K 1, K 2, and the initial value of the activation volume DV G0, have to
be determined, e.g. by a parameter identification procedure. The original model is
three-dimensional, but here it is used only in the uniaxial case.
In cooperation with the Institut für Allgemeine Mechanik und Festigkeitslehre
(Prof. Steck) at the Technische Universität Braunschweig, a set of parameters was
found. This set of parameters takes the results of internal stress measurements and the
dependence of the deformation behaviour on strain rate into account and is given by:
CG ˆ 0:3670 10 5 s 1 ; HG ˆ 1:784 ; b G ˆ 0:3676 ;
K1 ˆ 47:20 MPa mol kJ 1 ; K2 ˆ 10:328 ; and
DVG0 ˆ 1:182 kJ mol 1 MPa 1 :
With these parameters, a good description of rapid hardening and cyclic saturation is possible
[13]. Results in the case of saturated hysteresis-loops are shown in Figure 3.9 a.
It can be shown that the model describes the non-Masing behaviour of the
material in single-step tests. Furthermore, a relatively exact description of the hysteresis-loop
shape is possible. Some modifications seem to be necessary because the parameters
are valid only in a limited range of amplitudes and strain rates. More modifications
are needed to describe the stress-strain behaviour also in the case of incremental-step
tests and two-step tests with sufficient accuracy (for details: see Wittke [13]).
50

3.3 Tests at Room Temperature: Description of the Deformation Behaviour
Figure 3.9: Application of physical based models: comparison of experiment and calculation; a)
copper, _e =10 –4 s –1 , calculation with the Steck model [30]; b) 2.25Cr-1Mo, _e=10 –3 s –1 , calculation
with the modified model of Hatanaka [28] (calculation: solid line; experiment: dotted line).
By Hatanaka and Ishimoto [28], another physically based model was proposed to
describe cyclic plasticity. In this model, assumptions are made concerning the evolution
of dislocation density and concerning the mean dislocation velocity. We have modified
the original model by taking into account also the evolution of dislocation structure
(for details: see [13]). It is shown that the modified model can be applied for copper
and also for steady-state hysteresis-loops for 2.25Cr-1Mo [13]. An example for the latter
case is shown in Figure 3.9 b.
3.3.5 Application in the field of fatigue-fracture mechanics
Usually, crack growth data are correlated with a fracture mechanical parameter such as
e.g. DJ or DJ eff. According to the proposals of Dowling [32] and with the results of
Shih and Hutchinson [33], it is possible to estimate DJ in the case of various specimen
and crack geometries. Schubert [18] measured the growth of cracks, which were approximated
as half circular surface cracks in circular specimens. Crack growth of
2.25Cr-1Mo was measured with the ACPD method. In crack closure measurements, a
crack closure parameter U was found, which is nearly constant: U=0.9 (compare Rie
and Schubert [34] and Schubert [18]). Crack growth was successfully correlated with
DJ [18] and DJ eff [34], respectively.
The value U=0.9 was used also in the case of crack growth measurements of
edge cracks in flat specimens [13].
For the calculation of DJ, characteristic values of the deformation behaviour are
needed. According to Dowling [32], the cyclic integral DJ is calculated, e.g. in dependence
on the cyclic hardening exponent n'. As proposed by Rie and Wittke [24], n' is
51

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
replaced by the exponent b of Equation (5). By this replacement, non-Masing behaviour
is taken into consideration. The values of b are calculated according to the method
proposed in [26] (for details: see Wittke [13]). With this method, the effective part of
DJ for edge cracks is estimated as:
2 Dr2
DJeff ˆ 7:88 U
2E
4:84
‡ p
b
DrDep
1 ‡ b
!
a ; …15†
where a is the crack depth.
The growth of edge cracks in flat specimens for the steel 2.25Cr-1Mo was measured
with optical method. Tests with three different strain amplitudes (strain rate:
_e ˆ 2 10 3 s 1 ) were performed. The relation between crack growth per cycle da/dN
and DJeff is described by:
da=dN ˆ CJ…DJeff† J ; …16†
where C J and J are constants. With an assumed initial crack depth and with a crack
depth, which defines failure, lifetime can easily predicted by integrating Equation (16)
(compare Schubert [18]). With values for the characteristic parameters of hysteresisloops,
Dr, De p and b, the constants
CJ ˆ 3:89 10 5 ; J ˆ 1:16
were found (with da/dN in mm and DJ eff in Nmm/mm 2 ). The correlation between da/
dN and DJeff is quite satisfactory as can be seen in Figure 3.10.
Figure 3.10: 2.25Cr-1Mo, _e=2·10 –3 s –1 , correlation between crack growth per cycle da/dN and
DJ eff.
52

3.4 Creep-Fatigue Interaction
3.4.1 A physically based model for predicting LCF-life
under creep-fatigue interaction
In this section, the original model of the author proposed in 1985 [35] was described to
illustrate in the following the modifications and the experimental verifications made
successively in the last years.
3.4.1.1 The original model
Unstable crack advance occurs if the crack progress per cycle, da/dN, becomes approximately
equal to the spacing of the nucleated intergranular cavities [35, 36]. The crack
tip opening displacement d/2 may be seen as the upper bound to crack growth [36] and
the relation can be written as:
da
dN
3.4 Creep-Fatigue Interaction
d
ˆ …k 2r† ; …17†
2
where k is the cavity spacing, r is the radius of the r-type cavity and is a constant.
The crack tip opening displacement may be represented in analogy to the total
strain by an elastic term De el plus a contribution due to plastic deformation De p and by
thermally activated, time-dependent processes ec [37]:
d ˆ a…K1Deel ‡ K2Dep ‡ K3ec† ˆaCcal ; …18†
where K 1, K 2, K 3 are constants [35], and a is the crack length.
Under repeated loading, there will be a dependence of the number of created cavities
on the number of cycles. In analogy to the Manson-Coffin relationship, we postulate
a constitutive equation for the cycle-dependent cavity nucleation under cyclic creep
and low-cycle fatigue condition with superimposed hold time. Assuming that only the
plastic strain imposed is responsible for cavity nucleation and disregarding stress dependence,
the maximum number of cavities n max is given by:
nmax ˆ pN j Dep ; …19†
where De p is the plastic strain range, N is the number of cycles, p is the cavity nucleation
factor, and j is the cyclic cavity nucleation exponent. It was proposed that p was
identical with the density of grain boundary precipitates. Since it was found that not
every precipitation necessarily produces a cavity, experimental constant has been
used to adapt the observation in our first model. was used as a fit factor to have best
results in life prediction.
53

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Cavity nucleation under creep-fatigue condition is favoured on grain boundaries
perpendicular to the load axis and the cavity spacing k can be written as:
k ˆ 1
p : …20†
n
Nucleation of cavities is governed by a deformation of the matrix, and the cavity
growth is controlled by diffusion. In the first model, the cavity growth model of Hull
and Rimmer [38] was taken for describing the cavity growth rate during creep-fatigue.
This was done because the Hull-Rimmer model could be integrated analytically for
every cycle, and the result could be expressed in one compact equation for the life prediction.
In this case, the lifetime is reached if:
a
2 …K1Deel ‡ K2Dep ‡ K3ec†
ˆ
1
pNj 8
><
p
>: Dep
v
u p
u4XdgbDgb
pDep
2t
…j ‡ 2†kT
Z t
0
r…t† dtN j 2 ‡1
9
>=
: …21†
>;
By integrating, it was assumed that the kinetics of the cavity growth in tension are the
same as the kinetics of cavity shrinking in compression.
3.4.1.2 Modifications of the model
The empirical constant could reduce the versatile character of the basic concept on
unzipping of cavitated material as the failure criterion. Therefore, in a first step of modification,
we use:
d
ˆ k 2r : …22†
2
p was taken from direct experimental observation of cavities as will be shown in the
following. Therefore, it is not necessary to consider the influence of precipitation on
the nucleation of cavities, and n max in Equation (19) could be replaced by the real density
of cavities on grain boundaries n.
The cavities nucleated by tensile stresses can be healed during periods of compressive
stress if the compressive stress is applied for a long enough time. It has been
observed that the time required to heal the cavity by compressive stress is up to six
times longer than the time to nucleate the cavity by tensile stress [39]. In a second
step, the incomplete healing response has been modelled in dividing the rate of the radius
changes dr/dt in the growth models by a factor of 6 if the stress is negative.
In a third step, a numerical procedure for integrating the cavity growth models was
introduced. With this, it is possible to use any model depending on the physical parameters,
which may prevail. The models of Hull-Rimmer [38], Speight-Harris [40] and Riedel [41]
54

were compared, and it could be shown that in case of the calculated lifetimes, the influence
of the model on the result is negligible [42]. The model of Riedel [41] is used in the following
because it is successfully checked directly by experiments (see Section 3.4.2.3).
3.4.1.3 Experimental verification of the physical assumptions
Both the cavity nucleation factor p and the cavity growth were determined experimentally
by means of stereometric metallography as will be shown in Section 3.4.2. These
values have been used for the life prediction.
A furnace with window and special optics allows high magnification observation
of the specimen surface continuously during the test with a video system and a subsequent
measurement of the crack growth and the crack tip opening. The value for crack
tip opening was determined in a distance of 250 lm [43].
With this method, the fundamental assumption of the life prediction model about
the dependence of the crack tip opening displacement on the crack length and the
strain range expressed in Equation (18) could be experimentally checked. An example
of the crack tip opening displacement in dependence on the crack length is shown in
Figure 3.11. The slope of the straight line C exp for the experiments ranges between
0.043 and 0.058. The calculated slope determined with Equation (18) for the same experiment
is C cal=0.044. From that, it can be concluded that the calculation of the crack
tip opening displacement in the original life prediction model leads to values, which
are in the right order of magnitude.
3.4.1.4 Life prediction
3.4 Creep-Fatigue Interaction
The fatigue life of high-temperature low-cyclic fatigue under arbitrary cyclic loading
situations including wave shapes and hold time can be estimated using the unstable
crack advance criterion of the critical cavity configuration expressed in Equation (22).
Figure 3.11: Crack tip opening d 250 lm vs. crack length.
55

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.12: Variation of cavity growth with time, respectively number of cycles (De p/2=1%).
Figure 3.13: Comparison of experimental life and predicted life.
The effective cavity spacing k can be estimated from Equations (19) and (20), the
crack tip opening displacement d from Equation (18). And the cavity radius r can be
obtained by integrating the history-dependent cavity growth equation expressed in
Equations (24) to (27).
Because the cavity spacing k influences the cavity growth rate (Equations (24) to
(27)), the number of created cavities and their growth have to be calculated for every
cycle separately (Figure 3.12). From the radius of every nucleated and subsequently
grown cavity, we calculate the mean value r m and compare k–2r m with d/2 (Equation
(22)) to get the critical life. Figure 3.13 shows the good agreement between experimental
data and predicted life using the pore growth model of Riedel [41].
56

3.4.2 Computer simulation and experimental verification of cavity formation
and growth during creep-fatigue
The fundamental physically based assumptions in the life prediction model about the
development of the cavity density (Equation (19)) and the cavity growth have been experimentally
verified. The results of this gave rise to the development of a new 2-dimensional
cavity growth model, which describes the complex interaction between the
cavities, and thus leading to constitutive equations of the damage development, which
could be directly measured.
3.4.2.1 Stereometric metallography
After low-cycle fatigue testing, the specimens were metallographically prepared for
stereometric analysing for the density and cavity size distribution. For this purpose, the
cavities were photographed by a Scanning Electron Microscope, and the cavity density
and the distribution of the radii on the polished surface were detected. For every test,
nearly 100 cavities were measured. The measured values of size and density on the metallographic
section are much different from the real cavity configuration in the volume.
For calculating the real cavity size distribution and density on the grain boundaries,
the following assumptions are made: All cavities are on boundaries oriented perpendicular
to the load axis with a maximum deviation of 30 8. All grains are of identical
size, which is the mean value (in this case 62 lm), and all cavities are spherically
shaped.
The principal procedure is divided into two steps: First, the cavity size distribution
and the cavity density in the volume of the specimen were calculated. This was
done from the corresponding values in the metallographic section by means of a numerical
procedure. Spheres were placed in a given volume by means of the Monte Carlo
method. The spheres are randomly distributed. The size distribution of the spheres
was set as a logarithmic Gaussian distribution. The resulting size distribution was calculated
in a section of the volume, which is designed as the imaginary metallographic
surface. The determined values of this section were compared with the experiment, and
this procedure was repeated by varying the density and the parameters of the Gaussian
distribution. This was done until the resulting density and the size distribution were
identical to the values of the metallographic section.
Second, the real density on the grain boundaries n gb from the density in the volume
n v was calculated by means of a formula, which was provided by Needham and
Gladman [44]:
ngb ˆ nv
3.4 Creep-Fatigue Interaction
i
: …23†
2q
i is the size of the grains determined by means of the intercepted-segment method, and
the constant q (q =0.134) depends on the angle between the cavitated grain boundaries
and the load axis.
57

3.4.2.2 Computer simulation
For the simulation, the cavities were placed one after another on a given area of the
grain boundary by the Monte Carlo method. After the nucleation of one additional cavity,
the growth of all cavities on the grain boundary was calculated until the next was
formed. The growth of every cavity in our calculations depends on the spacing to the 6
neighbouring pores in plane and is described by so extended diffusion-controlled cavity
growth model. The extension is illustrated in Figure 3.14.
The advantage of the proposed model in comparison to the existing cavity growth
models is the inhomogeneous distribution in plane. To calculate the cavity growth, it
can be assumed that the total vacancy flow to the cavity considered is the sum of the
flows from all 6 segments as illustrated in Figure 3.14. We propose that the flow from
every segment depends on the distance only to the nearest cavity within the segment
considered. This is analogous to existing 1-dimensional cavity growth models [45–48],
but overestimates the vacancy flow because the contribution of the far distant cavities
within this segment is supposed to be the same as the nearest. The same considerations
will be applied for other segments. A fit factor is introduced, which takes this into consideration.
This factor is set to the value of 0.2 to have the best fit of the experiments.
To calculate the growth rate _r from the cavity distance k under the actual stress r b, the
cavity growth model proposed by Riedel [41] was chosen:
58
3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
_r ˆ 2XdDb‰rb r0…1 x†Š
kTh…w†q…x†r 2 ; …24†
r0 ˆ 2c s
r
x ˆ 2r
k
sin w ; …25†
2
; …26†
q…x† ˆ 2lnx …3 x†…1 x† : …27†
Figure 3.14: Statistically distributed cavities in plane.

The meaning and the values of the constants for 304L are: h (W)=0.61 the relation between
the cavity volume and the volume of a sphere with radius r, X =1.21·10 –29 m 3
the atomic volume, dD b =2·10 –13 exp (–Q/RT)m 3 /s with Q=167 kJ/mol the grain boundary
diffusion coefficient times the grain boundary width, 2W =708 the void tip angle,
cs=2 kJ/m 2 the specific surface energy, k Boltzmann constant and T the temperature.
This differential equation (Equation (24)) was solved numerically. A possible coalescence
has been taken into account. In this case, the two cavities were replaced by a
new cavity, with the volume of both at the centre of the connecting line. As is shown
below, this coalescence of cavities plays an important role in the case of fatigue because
the accumulated strain, which controls the cavity formation, is relatively high
compared to unidirectional tests. The cavity development during creep has also been
successfully simulated, but will not be the subject of this paper.
In the case of low-cycle fatigue, the cavity density nGb depends on the number of
cycles N and is calculated by a power law function between n Gb and N (Equation
(19)). This is one of the basic assumptions of our life prediction model and is verified
by the experiments as will be shown below.
Note that during the creep-fatigue, the stress is not constant, whereas it is constant
in the case of pure creep. Therefore, in the calculation, the changing stress was
taken into consideration.
The cavities are formed continuously during the tension period of the cycle until
the strain maximum is reached. During the hold period, stress relaxation occurs and no
cavities are formed. The actual stress for calculating the cavity growth after the formation
of every single pore was taken directly from the experiment. During the compression
period, no further cavity formation occurs. Due to the negative stress, the cavities
are shrinking. However, the influence of shrinkage is negligible for this kind of test
without compressive hold time and therefore will not be further discussed in this paper.
3.4.2.3 Results
3.4 Creep-Fatigue Interaction
In Figure 3.15, the experimentally detected cavity density on the grain boundary is
plotted versus the number of cycles. The cavity density during creep-fatigue depends
on the number of cycles N by a power law as suggested before. With p =12·10 –2 1/
lm 2 and j =0.4 in Equation (19), a good fit of the experimental data is possible (not
plotted in the figure), and the fundamental idea about the cavity formation in the life
prediction model is verified.
With this basic assumption about the cavity density in dependence on the number
of cycles, the simulation of the cavity growth proceeds as follows. After a few cycles,
the distribution is cut off at the right-hand side of the curve as proposed by Riedel
[41]. When cycling continues, more and more cavities coalesce, and therefore, large
cavities are formed. At the end of the simulation, the distribution is nearly Gaussian.
In Figure 3.16, the cumulative frequency of the cavity radii for the experiment
and the simulation, which is the solid line, is given for different numbers of cycles. In
the case of the experiment, the size distribution in the metallographic section is plotted.
The size distribution for the simulation is transformed to the resulting distribution in
the imaginary section by means of the method described in Section 3.4.2.1.
59

60
3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.15: Experimentally detected cavity density and cavity density of the simulation vs. number
of cycles.
Figure 3.16: Comparison of computer simulation with experimentally detected cavity size distribution
for creep-fatigue tests (304L, e a=1%, T=650 8C, 1 h tension hold).

3.4 Creep-Fatigue Interaction
The proposed model and the constitutive equations for simulating the pore configuration
in the case of creep-fatigue leads to a good agreement between calculation and
experimentally detected cavity size distribution, and also the cavity density as shown in
Figure 3.15. From Figure 3.15, one can draw the conclusion that the cavity coalescence
plays an important role in the case of creep-fatigue. The values of the given density,
that is the density, which would exist without coalescence, is much higher than the resulting
density on the grain boundaries by coalescence.
3.4.3 In-situ measurement of local strain at the crack tip during creep-fatigue
In the previous sections, the total strain and stress were used for calculating the damage
development and predicting the fatigue life. But in the LCF-regime, failure is a local
phenomenon, which takes place in front of the crack. Therefore, the strain has been
measured in front of the crack for giving the basis of a local application of material
laws and a local damage model. The method provided by the group of Prof. Ritter [10]
is usable for long time creep-fatigue tests at high temperature. The stability of the grating
is sufficient for high accuracy measurement in argon for more than two weeks [49].
3.4.3.1 Influence of the crack length and the strain amplitude
on the local strain distribution
The size of the highly deformed zone in front of the crack depends on the crack
length. This effect can be measured with this method. For both steels, the size of the
highly deformed zone increases with the crack length, which is shown in Figure 3.17
by plotting R 0.05 vs. crack length. The size of the highly deformed zone also depends
on the amplitude e a of the total strain. This is also demonstrated in Figure 3.17.
The increase of the plastic zone size with both the crack length and the total
strain amplitude will be explained by means of the theory of Shih and Hutchinson [33]
and by observations of Iino [12]. Finite-Element calculations by Shih and Hutchinson
[33] showed that both the crack length a and the strain amplitude ea are directly proportional
to the crack tip opening displacement d, Iino [12] observed the linear dependence
of the highly deformed zone size R 0.05 on the crack tip opening in the case of
low-cycle fatigue:
a d ; ea d Shih and Hutchinson
R0:05 d Iino : …28†
From both theory and experiment, the measured relationships between R 0.05 and e a as
well as between R 0.05 and a will be expected as shown in Figure 3.17. In the case of
high-cycle fatigue, these effects are well known and can be explained in terms of linear-elastic
fracture mechanics [50].
61

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
Figure 3.17: Increase of the highly deformed zone size in dependence on the total strain range.
3.4.3.2 Comparison of the strain field in tension and compression
The strain in front of the crack tip measured at the maximum stress in tension and compression
of the same cycle is shown in Figure 3.18. It can be seen that also in compression,
the local strain just in front of the crack tip is positive. This has been found in all tests and
for all crack lengths in both steels. Three different explanations are possible:
• A small amount of oxygen remains in the inert gas atmosphere, which leads to oxidation
of the crack surfaces. As a consequence, the crack surfaces can not return to
their original position of the previous cycle [51]. Therefore, the high deformation
developed at the tensile strain maximum cannot be completely reversed.
Figure 3.18: Local strain in direction of the load for the tension and compression maximum of
the same cycle vs. distance from crack tip in direction of the maximum expansion of the 5% deformed
zone.
62

3.4 Creep-Fatigue Interaction
• A small shifting of the crack surfaces during opening of the crack may lead to an
incomplete crack closure, and therefore to positive strain at the crack tip in compression.
• Due to the notch effect, stress and strain concentration occur at the crack tip during
crack opening. However, when the crack closes, no stress concentration appears
and, as a consequence, the maximum stress at the crack tip in compression is equal
to the total stress. Therefore, the stress at the crack tip in tension is higher than the
stress in compression. The mean value of the stress in front of the crack is positive,
and the consequence is the measured positive strain.
The fact that a positive strain appears in compression supports the high importance of
the local strain measurement for crack growth calculation and life prediction. For the
demonstrated test, the crack advance is 4 lm per cycle. The size of the zone of positive
mean strain in front of the crack is estimated at 1 mm. This means that the propagating
crack advances for more than 250 cycles through a material, which has been cycled under
positive mean stress.
3.4.3.3 Influence of the hold time in tension on the strain field
The values of the strain in front of the crack are lower in the case of tests with tension
hold times compared to tests without hold. Figure 3.19 shows the development of R 0.05
(size of the 5%-deformed zone from crack tip) as a function of crack length for 304L
and different hold times. The same results are given for the ferritic steel in a paper of
the authors [49].
The strain field depends on the hold time of the test, but remains the same during
the hold period of each cycle within the accuracy of the measurement. In-situ monitoring
of the crack advance and crack path indicates that the increase of crack growth rate
Figure 3.19: Plastically deformed zone size (size of the 5%-deformed zone from crack tip R0.05)
vs. crack length a in 304L.
63

3 Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue
with tension hold time is related to a transition from a trans- to an intercrystalline crack
path. Metallographic observation of the microstructure shows that during tension, hold
grain boundary cavitation and microcrack formation occur.
We conclude that the microscopic changes of the material during the creep-fatigue
such as the grain boundary damage lead to the change of the stress-strain behaviour in
front of the crack. This phenomenon has to be emphasized particularly because the macroscopic
stress-strain behaviour is not influenced by the grain boundary damage [37].
To explain the strain behaviour in front of the crack, we propose a model, which is
based on the reduction of strain to rupture in front of the crack if cavities are formed [49].
Comparable results for creep-fatigue cannot be found in literature, but Hasegawa
and Ilschner [52] have detected a reduction of the strains in front of cracks in the case
of high temperature tension tests if cavities are formed.
3.5 Summary and Conclusions
The cyclic deformation behaviour at room temperature was investigated for copper and
steel 2.25Cr-1Mo. It can be concluded that for both materials, non-Masing behaviour
has to be taken into consideration. The investigation of the microstructure shows that
for both copper and 2.25Cr-1Mo, dislocation cell structures were found for sufficient
high strain amplitudes.
The deformation behaviour can be described by analytical relations. Especially for
the steady-state stress-strain hysteresis-loops, very exact relations are proposed.
With the aid of stress relaxation experiments, a cyclic yield stress r yc can be defined
and correlated with a microstructure-dependent proportional limit r prop. Calculations
with the physically based models of Steck and Hatanaka, respectively, show good
agreement with experimental results. The model of Hatanaka was modified by taking
results concerning the dislocation structure into account.
An application of test results in the field of fatigue fracture mechanics is shown
by correlating da/dN and DJ eff.
The generalized life prediction model of the authors has the capability to predict
lifetime of high temperature low-cycle fatigue under various wave shapes and hold
times. Physically based constitutive equations for cavity nucleation and subsequent
growth under variable loading histories are considered, and the unzipping of the cavitated
grain boundary is taken as criterion for catastrophic failure. The crack tip opening
displacement is seen as the upper bound to crack growth. These physically based assumptions
in the model are verified by corresponding experiments.
The development of intergranular cavitation in austenitic steels can be simulated by
the proposed 2-dimensional cavity growth model with good agreement to the experiment.
It is important that not only the cavity size distribution but also the resulting cavity density
on grain boundaries are in accordance with the experiment. From this, it can be concluded
that the coalescence of neighbouring voids is very important for the cavity growth during
low-cycle fatigue and is the main reason for the existence of relatively large cavities.
64

The grating method is a very useful tool for determining the local strain in front
of cracks during creep-fatigue. The high accuracy of this method for measuring the
plastic deformation remains even for long time tests. It can be shown that the magnitude
of the local strain at the crack tip during high temperature, low-cycle fatigue testing
depends on the crack length and on the total strain range. During cycling, the local
strain in front of the crack tip is positive even in compression maximum. By means of
the grating method, it can be shown that the high crack growth rate of creep-fatigue is
associated with a relatively small size of the plastically deformed zone.
References
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[8] J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtemperatur-Low-Cycle-Fatigue.
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[9] J. Olfe: Wechselwirkung zwischen Kriechschädigung und Low Cycle Fatigue und ihre Berücksichtigung
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Clausthal-Zellerfeld, 1996.
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Measuring Methods. This book (Chapter 13).
[11] K. Andresen: Surface-Deformation Fields from Grating Pictures Using Image Processing
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Fatigue ’96, Proceedings of the Sixth International Fatigue Congress, Pergamon, 1996, pp.
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haviour of Materials – 3, Elsevier Applied Science, London, New York, 1992, pp. 136–
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[18] R. Schubert: Verformungsverhalten und Rißwachstum bei Low Cycle Fatigue. Fortschrittsber.
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Behaviour and Microstructure in the LCF-Range. In: K.-T. Rie (Ed.): Low Cycle Fatigue
and Elasto-Plastic Behaviour of Materials – 3, Elsevier Applied Science, London, New
York, 1992, pp. 514–520.
[20] C. E. Feltner, C. Laird: Cyclic Stress-Strain Response of f.c.c. Metals and Alloys – I. Phenomenological
Experiments. Acta Metallurgica 15 (1967) 1621–1632.
[21] G. Hoffmann, O. Öttinger, H.-J. Christ: The Influence of Mechanical Prehistory on the Cyclic
Stress-Strain Response and Microstructure of Single-Phase Metallic Materials. In: K.-T.
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[22] C. E. Feltner, C. Laird: Cyclic Stress-Strain Response of F.C.C. Metals and Alloys – II: Dislocations
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Damping, and Cyclic Plasticity. ASTM STP 378 (1965) 45–87.
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(1996).)
[25] U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behaviour
of Mild Steel Fe 510 under Complex Cyclic Loading. This book (Chapter 10).
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Approach. (To be published in: Int. J. Fatigue (1996/97).)
[27] J.C. Tsou, D.J. Quesnel: Internal Stress Measurements during the Saturation Fatigue of
Polycrystalline Aluminium. Mat. Sci. and Engin. 56 (1982) 289–299.
[28] K. Hatanaka, Y. Ishimoto: A Numerical Analysis of Cyclic Stress-Strain-Response in Terms
of Dislocation Motion in Copper: In: H. Fujiwara, T. Abe, K. Tanaka (Eds.): Residual
Stresses – III, Elsevier Applied Science, 1991, pp. 549–554.
[29] J. Polák, M. Klesnil, J. Heles˘ic: The Hysteresis Loop: 2. An Analysis of the Loop Shape.
Fatigue of Engineering Materials and Structures 5(1) (1982) 33–44.
[30] H. Schlums, E.A. Steck: Description of Cyclic Deformation Process with a Stochastic Model
for Inelastic Behaviour of Metals. Int. J. of Plasticity 8 (1992) 147–159.
[31] R. Gerdes: Ein stochastisches Werkstoffmodell für das inelastische Materialverhalten metallischer
Werkstoffe im Hoch- und Tieftemperaturbereich. Braunschweiger Schriften zur Mechanik
20 (1995).
[32] N.E. Dowling: Crack Growth During Low-Cycle Fatigue of Smooth Axial Specimens; Cyclic
Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Growth. ASTM STP
637 (1977) 97–121.
[33] C. F. Shih, J.W. Hutchinson: Fully Plastic Solutions and Large Scale Yielding Estimates for
Plane Stress Crack Problems. Journal of Engin. Mat. and Technol., Oct. 1976, Transactions
of ASME, pp. 289–295.
[34] K.-T. Rie, R. Schubert: Note on the crack closure phenomenon in low-cycle fatigue. Int.
Conf. Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, Munich 1987, Elsevier
Applied Science, pp. 575–580.
[35] K.-T. Rie, R.-M. Schmidt, B. Ilschner, S.W. Nam: A Model for Predicting Low-Cycle Fatigue
Life under Creep-Fatigue Interaction. In: H.D. Solomon, G.R. Halford, L.R. Kai-
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[36] G.J. Lloyd: High Temperature Fatigue and Creep Fatigue Crack Propagations Mechanics,
Mechanisms and Observed Behaviour in Structural Materials. In: R. P. Skelton (Ed.): Fatigue
at High Temperatures, Applied Science Publishers, London, New York, 1983, pp.
187–258.
[37] R.-M. Schmidt: Lebensdauer bei Kriechermüdung im Low-Cycle Fatigue Bereich. Dissertation
TU Braunschweig, VDI Fortschritt-Berichte Nr. 47 (1988).
[38] D. Hull, D.E. Rimmer: The Growth of Grain-Boundary Voids Under Stress. Philosophical
Magazine 4 (1959) 673–687.
[39] B. K. Min, R. Raj: Hold Time Effects in High Temperature Fatigue. Acta Metall. 26 (1978)
1007–1022.
[40] H.E. Evans: Mechanisms of Creep Fracture. Elsevier Applied Science Pub. LTD., 1984,
pp. 251–263.
[41] H. Riedel: Fracture at High Temperatures. Springer-Verlag, Berlin Heidelberg, 1987.
[42] K.-T. Rie, J. Olfe: A physically based model for predicting LCF life under creep fatigue interaction.
In: K.-T. Rie (Ed.): Proc. 3rd Int. Conf. on Low Cycle Fatigue and Elasto-Plastic
Behaviour of Materials, Elsevier Applied Science, London, New York, 1992, pp. 222–228.
[43] K. Tanaka, T. Hoshide, N. Sakai: Mechanics of Fatigue Crack-Tip plastic blunting. Engineering
Fracture Mechanics 19 (1984) 805–825.
[44] N.G. Needham, T. Gladman: Nucleation and growth of creep cavities in a Type 347 steel.
Mat. science 14 (1980) 64–66.
[45] S.J. Fariborz: The effect of nonperiodic void spacing upon intergranular creep cavitation.
Acta metall. 33 (1985) 1–9.
[46] S.J. Fariborz: Intergranular creep cavitation with time-discrete stochastic nucleation. Acta
metall. 34 (1986) 1433–1441.
[47] J. Yu, J.O. Chung: Creep rupture by diffusive growth of randomly distributed cavities – I.
Instantaneous cavity nucleation. Acta metall. 38 (1990) 1423–1434.
[48] J. Yu, J.O. Chung: Creep rupture by diffusive growth of randomly distributed cavities – II.
Continual cavity nucleation. Acta metall. 38 (1990) 1435–1443.
[49] K.-T. Rie, J. Olfe: In-situ measurement of local strain at the crack tip during creep-fatigue.
In: Proceedings of the International Symposium on Local Strain and Temperature Measurements
in Non-Uniform Fields at Elevated Temperatures, March 14–15, Berlin, 1996.
[50] K.-H. Schwalbe: Bruchmechanik metallischer Werkstoffe. Carl Hanser Verlag, München,
Wien, 1980.
[51] T. Ericsson: Review of oxidation effects on cyclic life at elevated temperature. Canadian metallurgical
quarterly 18 (1979) 177–195.
[52] T. Hasegawa, B. Ilschner: Characteristics of crack tip deformation during high temperature
straining of austenitic steels. Acta metall. 33(6) (1985) 1151–1159.
67

4 Development and Application of Constitutive Models
for the Plasticity of Metals
Abstract
Elmar Steck, Frank Thielecke and Malte Lewerenz*
The macroscopic behaviour of crystalline materials under mechanical or thermal loadings
is determined by processes in the microregion of the material. By a combination
of models on the basis of molecular dynamics and cellular automata, it seems possible
to simulate numerically the formation of internal structures during the deformation processes.
The stochastical character of these mechanisms can be considered by modelling
them as stochastic processes, which result in Markov chains. By a mean value formulation,
this leads to a macroscopic model consisting of non-linear ordinary differential
equations. The determination of the unknown material parameters is based on a Maximum-Likelihood
output-error method comparing experimental data to the numerical simulations.
With Finite-Element methods, it is possible to use the material models for
the design of components and structures in all fields of technical application and for
the numerical simulation of their behaviour under complex loading situations.
4.1 Introduction
Metallic materials show, like other crystalline substances, typical macroscopic responses
on mechanical loading, which are caused by processes on the microscale. Figure
4.1 shows a typical cyclic stress-strain diagram with constant strain amplitude.
Cyclic hardening can be observed as well as the Bauschinger effect, which can be
recognized by the fact that plastic flow occurs after load reversal at significantly lower
stresses than those, from which the load reversal was done. For the technical use of
metallic materials, the description of this kind of processes in material models is of
high importance.
* Technische Universität Braunschweig, Institut für Allgemeine Mechanik und Festigkeitslehre,
Gaußstraße 14, D-38106 Braunschweig, Germany
68
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

4.2 Mechanisms on the Microscale
Figure 4.1: Cyclic stress-strain diagram for 304 stainless steel.
The moving of dislocations is the main microscopic mechanism responsible for
the plastic deformations in metallic materials. In the following, a stochastic model is
presented, which is able to consider hardening and recovery processes by means of
Markov chains.
During the deformation process, the dislocations arrange in a hierarchy of structures
such as walls, adders or cells. This forming of structures influences the macroscopic
behaviour of the materials considerably. The principle of cellular automata in
combination with the method of molecular dynamics is used for the numerical simulation
of these processes.
For the material parameter identification, the minimization of the Maximum-Likelihood
costfunction by hybrid optimization methods parallelized with PVM is considered.
With a multiple shooting method, additional information about the states can be
taken into account, and thus the influence of bad initial parameters will be reduced. For
the analysis of structures like a notched flat bar, the Finite-Element Program ABAQUS
is used in combination with the user material subroutine UMAT. The results are compared
with experimental data from grating methods.
4.2 Mechanisms on the Microscale
The movement of dislocations and the connected plastic deformations caused by external
loading is determined by two important activation mechanisms. The stress activation
is caused by the external loads. The thermal activation supports at elevated temperatures
the dislocation movements and therefore the plastic deformations.
69

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.2: Stress and thermal activation of dislocation motion.
Figure 4.2 shows schematically the obstacles, which resist the dislocation movements
on the microscale in the form of barrier potentials U*, the possible position – determined
by temperature – of the dislocations relative to these barrier potentials, and
the effect of an external load and a temperature increase on the energetic situation of
the dislocations. It is visible that the potential U r of the external forces by superposition
changes the potentials of the actual obstacles so that the dislocation movement in
the direction of the applied stress is more probable than in the opposite direction, and
that the thermal activation supports this process [1–4].
The barriers, which oppose the dislocation movements, are on the one side given
by the crystalline structure of the material itself, on the other hand, foreign atoms and
grain boundaries can form obstacles. One of the most important reasons for the hindering
of the dislocations, however, are the dislocations themselves. During plastic deformation,
continuously new dislocations are produced. In the beginning, the ability of the
material for deforming plastically is increased. With increasing dislocation density, a
mutual influence of the lattice disturbances occurs, which results in isotropic hardening.
Due to the lattice distortions connected with the plastic deformation, elastic energy
is stored in the material, which also hinders the movements of the dislocations,
which are generating it. This process is called kinematic hardening. The internal stresses,
however, support the dislocation movements in the opposite direction and result in
e.g. the Bauschinger effect. At elevated temperatures – above half of the melting temperature
of the material –, thermally activated reorganization processes in the crystals
occur, which reduce the mutual hindering of the dislocations and result macroscopically
in recovery.
Significant magnitudes for these processes are given in Figure 4.3, which shows a
dislocation, which is influenced by other dislocations. The shaded area is a measure for
the activation volume DV ˆ bA, which decreases in size with increasing isotropic hardening.
The Burgers vector b determines with his orientation relative to the dislocation
line the character of the dislocation. q w is the density of the so-called forest dislocations,
i.e. the dislocations, which hinder the movement of the others [4].
Table 4.1 shows the connection between the activation volume and the most important
dislocation mechanisms for different regions of the homologous temperature T=Tm.
70

4.3 Simulation of the Development of Dislocation Structures
Figure 4.3: Activation volume and forest dislocations.
Table 4.1: Activation volume depending on deformation mechanism and temperature.
Mechanism Temperature Activation volume
Climbing >0.5 b 3 remains constant during deformation
Movement of
dislocation jumps
>0.5 10–1000 b 3 , the value of the activation volume decreases
during deformation
Cross slip 0.2–0.4 10–100 b 3 , the value remains approximately constant
during deformation
Cutting of >0.3 1000 b
dislocations
3 , the activation volume decreases due to increase
of the density of forest dislocations with increasing
deformation
4.3 Simulation of the Development of Dislocation Structures
For unidirectional as well as for cyclic plastic deformation, it is observed that dislocation
structures are developed in the shape of e.g. adders or dislocation cells, which in a
typical manner depend on the loading history and the loading magnitude (Figure 4.4).
Due to the fact that this forming of dislocation patterns influences the macroscopic
behaviour of the materials considerably, the simulation of these self-organization
processes can result in valuable information for the choice of formulations for the modelling
of processes on the microscale.
The interaction of a large number of identical particles is the basic idea for the
definition of cellular automata. It is an idealization of real physical systems, where
space as well as time are discrete. A cellular automaton is completely characterized by
the following four properties [5]: geometry of the cell arrangement, definition of a
neighbourhood, definition of the possible states of a cell, and evolution rules. Each cell
can during the evolution in time only assume values (states) out of a finite set. For all
71

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.4: Characteristic dislocation structures.
cells, the same evolution rules are valid. The change in state of a cell depends on its
own state and those of the neighbouring cells.
Opposite to the usual assumptions for cellular automata, where the state of a cell
only depends on the states of the next neighbours, for the simulation of dislocation
movements, it has to be taken into account that the dislocations possess long-range acting
stress fields. With this model, it is possible to compute the dynamics of some thousand
edge- or screw-dislocations on parallel slip planes in areas of arbitrary magnitude.
A basic model, for which only one slip system in horizontal direction was chosen,
assumes a grid of rectangular cells, which can be occupied by edge- or screw-dislocations
with positive or negative sign [5]. The transition rules are: A positive or negative
occupied cell becomes an empty cell if the dislocation in the cell will move due to
the acting forces to a neighbouring cell or if an annihilation with a dislocation in a
neighbouring cell occurs. The step width of a dislocation is always one cell size per
time step. Reachable cells are the cells left, right, up and down from the actual cell.
This characterizes a so called v. Neumann neighbourhood. For the calculation of the
forces acting on a dislocation, a larger neighbourhood is necessary due to the longrange
acting stress of the dislocations. The balance of forces decides, if and in which
direction a dislocation will move. It is computed for each time step and each dislocation
for both degrees of freedom.
A much more realistic simulation for the development of dislocation structures is
obtained from models, which consider several glide planes [6]. Figure 4.5 shows a twodimensional
projection for the glide system for a cubic face-centred lattice, and modelling
of the glide processes on this system with three glide directions under angles of respectively
608.
The simulation results in wall- and labyrinth-structures of the dislocations (Figure
4.6). An extension of the model with consideration of vacancies and a suitable velocity
law is under progress.
72

4.4 Stochastic Constitutive Model
Figure 4.5: Cell arrangement and neighbourhood of simulation model.
Figure 4.6: Simulation of dislocation structures.
4.4 Stochastic Constitutive Model
The description of the processes responsible for plastic deformations shows that they
are strongly stochastic. Figure 4.7 shows for a simplified case for processes at high
temperatures, under consideration of kinematic hardening only, the used stochastic
model [1, 2].
Over the state axis, which represents the value of the kinematic hardening rkin,
and therefore the strength of the obstacles resisting the dislocation movements, the distribution
of the “flow units” (dislocations, dislocation packages or grain boundaries) is
given. The effect of the external stress is reduced by the hardening stress, therefore
only the effective stress reff ˆ r rkin is responsible for the dislocation movements.
Depending on reff, a hardening probability
73

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.7: Stochastic model for high temperatures.
V ˆ c1 exp
dDVrkin
sign reff _eie …1†
RT
is formulated. This transition probability is based on the condition that thermal activation
of the dislocations can be taken as an empirical Arrhenius function. R is the gas
constant and c1; d; DV are constants, which have to be determined by experiment. It
can be seen that the transition probability from a certain hardening state to the next
higher decreases with increasing hardening.
Hardening is opposed by a recovering process according to:
E ˆ c2 exp
F0
RT
jrkinj
r0
m
sinh DVrkin
RT
; …2†
which is thermally activated and not dependent on the external stress. The constants c 2
and m have also to be determined by experiment. The strength of the lattice distortions
increases with increasing hardening. It supports the recovery process. Therefore, the
transition probabilities for recovery increase with increasing hardening.
The model simulates hardening and recovery by transitions of dislocations at a
barrier strength rkin;i to higher barriers rkin;i‡1 and lower barriers rkin;i 1: The probability
that a flow unit remains in the actual position is given by:
B ˆ 1 V E : …3†
The transition probabilities of the model can be arranged in a stochastic matrix:
74

0
B
S ˆ B
@
1 V1 E2
V1
B2
V2
0
4.4 Stochastic Constitutive Model
. . .
. .
.
.
. .
Ei
Bi
Vi
0
.
. .
. .
.
. .
.
Ek 1
Bk 1 Ek
Vk 1 1 Ek
1
C : …4†
C
A
The change of the structure, which is described by the state vector z, during one
time step Dt is given by the Markov chain:
z…t ‡ Dt† ˆSz…t† : …5†
For constant stress and temperature (homogeneous process), the state vector after
n time steps is given by z…t0 ‡ nDt† ˆS n z…t0†: The stochastic matrix given by Equation
(4) can be transformed to principal axes and yields then:
S ˆ M 1 0
1
B 0
SM ˆB
@ 0
0
k2
0
0
0
.
. .
1
0
0 C ;
0 A
…6†
0 0 0 kn
where M is the modal matrix, i.e. the matrix of the columnwise arranged eigenvectors
of the matrix S. Due to the fact that the maximal principal value of stochastic matrices
is 1 and all other eigenvalues have magnitudes

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.8: Distribution function for r kin and DV=bA.
_eie ˆ C exp
_rkin ˆ H d exp
F0
RT
jreffj
r0
n
sinh DVreff
RT
dDVrkin
signreff _eie R exp
RT
; …7†
F0
RT
jrkinj
r0
m
sinh DVrkin
RT
D _V ˆ K1DV 2 j_eiej‡K2DVj_eiej : …9†
The material behaviour is described by a relation for the inelastic strain rate, where the
actual values for isotropic and kinematic hardening occur as internal variables. This
general form of the constitutive equations is also the basis for the development of a
hierarchical model classification [7]. A concrete model must be chosen with respect to
the intended application purpose. The values C; n; H; d; R ; m; K1; K2 and DV0 are
material parameters, which have to be determined by comparison with experimental results.
The parameter identification, which consists in integrating the non-linear, ordinary
differential equations for varying parameter sets and by appropriate optimization
methods to search for the optimal parameter sets, deserves special recognition in aspect
of the used mathematical methods [7, 8]. An additional scaling of the functions like
F0 1 1
exp
is necessary to improve the parameter identifiability and the
R T T0
macroscopical interpretations.
76
;
…8†

4.5 Material-Parameter Identification
4.5.1 Characteristics of the inverse problem
Under the assumption of normal distributed measurement errors with zero mean and
known measurement-covariance matrix C…ti†; the costfunction is:
L2…x; p† ˆ 1
2 jjrjj2 ˆ 1 X
2
n
‰z…ti† x…ti†Š
iˆ1
T C 1 ‰z…ti† x…ti†Š?min : …10†
The minimization of this weighted least squares function yields a Maximum-Likelihood
estimate of the parameters, which reproduces the observed behaviour z of the real process
with maximum probability [9].
Typical features of the identification are that the constitutive model is not only
highly non-linear in states x, but also in parameters p. Due to incomplete measurement
information, the problem is ill-conditioned, parameters are highly correlated. Because
of unbalanced parameters, the model may change its characteristics and becomes stiff
or even pathological. Since replicated tests for the same laboratory conditions show a significant
scattering and thus incompatibility of the data, this uncertainty must be taken into
account for the development and identification of the constitutive models [7, 10].
4.5.2 Multiple-shooting methods
4.5 Material-Parameter Identification
The measurements of the kinematic back stress, e.g. by relaxation test, yield very important
informations about the deformation process and thus can be used to get more
reliable parameters. In general, there are no (complete) measurements for the internal
states. However, engineers have a lot of additional apriori-information, which should be
used to improve the model prediction capacity. Although it is possible to formulate additional
weighted least-squares terms for the Maximum-Likelihood function, a much
more efficient method is to use multiple shooting (Figure 4.9) [11, 12].
The basic idea is to subdivide the integration interval by a suitable chosen grid
and to treat the discretized model equations as non-linear constraints of the optimization
problem. The initial state estimates at the nodes of the grid allow to make efficient
use of measurement- and apriori-information about the solution [13, 14].
4.5.3 Hybrid optimization of costfunction
For the identification of the material parameters, a hybrid optimization concept is used.
Starting with evolution strategies as a pre-optimization to get reliable initial parameters,
the main-optimization is done with a damped Gauß-Newton method [15].
77

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.9: Principle of multiple shooting.
Adaptive Evolution Strategies attempt to imitate the organic evolution process,
e.g. a collective learning of a population with the principles mutation, recombination
and selection [16]. The self-learning process of strategy parameters adapts this optimization
procedure to the local topological requirements. Since it is possible to overcome
local minima with a special destabilization method, evolution strategies even work with
bad initial model parameters [8].
Damped Gauß-Newton methods are most widely used for the minimization of
non-linear least-squares functions [7, 11]. Starting from initial parameters p 0, improved
parameters are iteratively obtained by the solution of a linear least-squares problem linearized
about p k . The steplength parameter k k is chosen to enforce the convergence
properties:
1
2 jj r…pk †‡J k Dp k jj 2 ! min with p k‡1 ˆ p k ‡ kkD p k ; …11†
solution with pseudo inverse: D p k ˆ J ‡ …p k †r…p k † :
A study of different search and gradient-based methods like the algorithms of Powell,
special subspace simplex methods or sequential quadratic programming are given in [7].
The numerical sensitivity analysis is a very important and most time consuming
part of the identification. Since the calculation is very closely related to the numerical
integration of the differential equations and the available accuracy, the sensitivity analysis
may be a critical point. Three different concepts are used to generate the sensitivity
matrix. The commonly used finite difference approximation:
qxi
qpj
xi…pj ‡ dpj† xi…pj†
dpj
is easy to implement, but the efficiency and reliability are low. Better concepts are
based on the integration of the sensitivity equations:
qx
qp
ˆ qf
qx
qx
qp
‡ qf
qp
…12†
: …13†
It is obvious that the solution of the model and the sensitivity equations should be
coupled. A very powerful coupling is available by Internal Numerical Differentiation
78

4.5 Material-Parameter Identification
(IND) [11]. This means that the internally generated discretization scheme of the integrator
is differentiated with respect to the parameters.
4.5.4 Statistical analysis of estimates and experimental design
The parameter estimates are only useful if also a statistical analysis of their reliability
is computed. Using the pseudo inverse J + at the solution of the Gauß-Newton method,
the calculation of standard deviations and correlations for the parameters is quite easy.
Very important for further work is to improve the calculation by better experimental
designs. Based on design criteria like the minimization of det…J T J† 1 ; different
methods have been considered and tested for typical growth function and a fundamental
constitutive model. These studies also show that the bad identifiability of the inverse
problems can be overcome with a special scaling of the states [7].
4.5.5 Parallelization and coupling with Finite-Element analysis
The separable multiexperiment structure leads to a coarse-grained parallelism of the parameter
identification problem. In addition, evolution strategies and multiple shooting
provide inherent parallelism on a high level. Thus, efficient parallel computation of
model functions and derivatives can be easily performed on a workstation-cluster with
PVM (Figure 4.10).
Figure 4.10: Parallel simulation concept.
79

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.11: Creep tests for aluminium.
Since the development and identification of anisotropic damage models became
more and more important, a three-dimensional Finite-Element program was coupled
with the estimation software by a special interface. The flexibility and modular structure
of this approach may be very useful for a lot of other applications, e.g. structure
optimization.
For the application of the damped Gauß-Newton method, the Internal Numerical
Differentiation was adapted to the Finite-Element analysis. Thus, not only the simulation
results but also the sensitivities have to be transferred. The results of the simulations
are compared with experimental strain fields obtained by grating methods [17].
80

Figure 4.12: Cyclic tests for copper.
4.5 Material-Parameter Identification
4.5.6 Comparison of experiments and simulations
A lot of different materials like pure aluminium, pure copper or the austenitic steels
AISI 304 and AISI 316 have been extensively studied.
Figure 4.11 shows some results of parameter identifications for aluminium
Al 99.999. The temperature regime was between 500 8C and 700 8C. Since only monotonic
tests were evaluated, a constitutive model with only one structure variable for the
internal stress is used. The parameters were identified for the given stresses simultaneously
so that the calculated curves were obtained by a single parameter set [7, 8, 15].
Figure 4.12 gives two examples on copper. The experimental database consists of
seven strain-controlled cyclic tests at room temperature [18]. Two strain rates _e =10 –4 ,
10 –3 , and a multitude of strain amplitudes De=2 =0.2–0.7% are examined.
For this application of the stochastic constitutive model, the special characteristics
of the material and the measurements have to be considered. In the low-temperature regime,
hardening is the most important phenomenon, while the recovery influence is
negligible. In contrast to high temperatures, metal physical results also indicate that the
81

4 Development and Application of Constitutive Models for the Plasticity of Metals
Figure 4.13: Scattering of creep and tension-relaxation tests for AISI 316.
influence of the effective stress can be modelled only by a sinushyperbolicus function.
Thus, the low-temperature model has only five parameters.
4.5.7 Consideration of experimental scattering
The experimental data to determine the parameters of constitutive equations usually
consist of only one observed trajectory for each temperature and loading condition.
Nevertheless, replicated tests for the same laboratory conditions show a significant scattering
and thus incompatibility of the measurements (Figure 4.13). Based on a statistical
analysis, this uncertainty can be taken into account for more reliable modelling and
parameter identification.
The modelling of the experimental uncertainties is based on the scattering of the
parameters or the initial values (Figure 4.14) [10].
Based on these concepts, realistic simulations of the uncertainties in experimental
data due to measurement errors and scattering are possible [7].
82

4.6 Finite-Element Simulation
4.6 Finite-Element Simulation
Figure 4.14: Probability density function and correlation of scattered parameters.
The aim of using constitutive models is to predict the behaviour of metallic structures
under mechanical and thermal loading. This requires the solution of a coupled initialboundary
value problem, given by the momentum equilibrium and the constitutive
equations. Since the boundary value problem is usually solved by the Finite-Element
Method (FEM), the constitutive model has to be implemented in an appropriate way.
Since the code ABAQUS/STANDARD is used, the theoretical aspects of the model implementation
are discussed for application of the user subroutine UMAT.
The developed method of implementation is described in Section 4.6.1. The main
characteristics of this method are its applicability to any unified constitutive model of
the class described above and to small as well as to large deformations theory. In Section
4.6.2, some numerical and experimental results are given, which show that the
model presented here works well.
4.6.1 Implementation and numerical treatment of the model equations
The considered constitutive model can be mathematically classified as a coupled system
of non-linear ordinary differential equations (CSNODE), which builds an initial
value problem. Its solution to a time increment can be embedded in an incremental Finite-Element
formulation with displacement approach, leading to the well known impli-
83

cit FEM-problem for non-linear material equations, which has to be solved iteratively
(see e.g. [19]).
Since this iteration requires a repeated solution of the initial value problem, the
computational cost of the FEM-simulation can be minimized by optimizing the numerical
solution of the CSNODE. This can be reached by:
• simplifying the model equations with some appropriate transformation,
• the use of an efficient numerical integration scheme, and
• an efficient algorithm to approximate the so-called tangent modulus.
The proposals worked out to this, three aspects are summarized in the following subsections
(for further details see [7, 20]).
4.6.1.1 Transformation of the tensor-valued equations
Using the v. Mises hypothesis, the multiaxial formulation of the model equations takes
the form:
_rij ˆ fij…_ekl; rkl; r kin
kl ; DV† ; …14†
_r kin
ij ˆ fij…rkl; r kin
kl ; DV† ; …15†
_DV ˆ f …rkl; r kin
kl ; DV† ; …16†
where rij is the Cauchy stress tensor, r kin
ij is the back stress tensor and _eij is the deformation
rate tensor. Each of these symmetric tensors is defined by six independent components,
so that the whole CSNODE contains thirteen scalar equations.
Since the v. Mises equivalent stress
rv ˆ 3
2 r0ijr0 r
ij
just depends on the deviatoric stresses, the inelastic part of the tensor equations are
also purely deviatoric. Therefore, the deviatoric rates _rij and _r kin
ij can be described in
some interval ‰t0; t0 ‡ DtŠ as a linear combination of the three deviatoric tensors
rij…t0†; r kin
ij …t0† and _eij…t0†, as long as _eij is constant in Dt. Using a suitable transformation,
the deviatoric rates can be expressed in a subspace by only three independent
components. After this transformation, the initial value problem (Equations (14) to
(16)) can be written as:
84
4 Development and Application of Constitutive Models for the Plasticity of Metals
…17†
_y i ˆ fi…yj† ; yi…t0† ˆyi;0 ; i; j ˆ 1 ...n ; …18†

4.6 Finite-Element Simulation
and contains only seven scalar equations. Hornberger [21] shows that the subspace dimension
can be reduced to two if a special integration scheme is used. Nevertheless,
this idea is neglected here in order to obtain a free integrator choice.
This transformation concept can be applied to plane strain, plane stress and uniaxial
states as well. Although the number of scalar equations cannot be reduced in these
cases, the main advantage is that the transformed model equations in the subspace are
of identical form for each of these cases. Based on this fact, the model implementation
for one-, two- and three-dimensional states can be performed very easily.
Using special large deformation formulations (see e.g. ABAQUS Theory Manual
[22]), this form of implementation can be used with small or large deformation theory
as well.
4.6.1.2 Numerical integration of the differential equations
Due to its complexity and non-linearities, the CSNODE (see Section 4.6.2) has to be
integrated numerically. In oder to choose an appropriate integration algorithm, the integration
task is classified as follows:
• medium required integration tolerances (corresponding to usual FEM-tolerances),
• a small integration interval (given by the incremental FEM-solution),
• an associated efficient method for error estimation, and
• a stable solution (to guarantee a stable FEM-solution).
Numerical integration methods on the other hand can be classified by their integration
order p, which describes the discretization error R in dependency of the step size h by
R h p (for an overview see [23, 24]). There are:
• methods with fixed integration order like multi-step methods, Runge-Kutta methods,
and Taylor series methods, and
• methods with variable integration order like extrapolation methods.
Extrapolation methods are efficient only for high integration tolerances, while multi-step
methods loose efficiency for small integration intervals. The use of Taylor series methods
is not practicable since it requires higher derivatives of the CSNODE, which are usually
not given directly. So, explicit and implicit, Runge-Kutta methods are widely used for the
integration of constitutive equations in FEM-analysis (see e.g. [19, 21, 25]).
Butcher [26] pointed out that the mentioned methods with fixed integration order
can be combined to get new classes of integration methods. For example, so-called Rosenbrock
methods result from the combination of Runge-Kutta methods and Taylor series
methods based on the first derivative of the CSNODE (also called the Jacobean of
the system). The main advantage of these methods is their unconditional stability – as
in implicit Runge-Kutta methods – that is reached with an explicit algorithm without
any iteration process. Rosenbrock methods as well as Runge-Kutta methods can be designed
as embedded integration formulae, which lead directly to a method of internal
error estimation without additional numerical cost.
85

4 Development and Application of Constitutive Models for the Plasticity of Metals
Verner [27] proposed families of embedded explicit Runge-Kutta processes, which
allow to rise integration order p without dismissing the results of an integration with a
lower starting order p0. If this concept is used with Rosenbrock methods, the resulting
integration process is kind of an optimal numerical integration method for constitutive
rate equations in FEM-analysis because
• it is an efficient algorithm especially for medium error tolerances,
• it is unconditionally stable,
• it is designed for computing the solution for the whole (but small) integration interval
in one step, and
• an internal error estimation nearly without additional cost is possible.
For details and comparison to classical methods see [7, 20].
4.6.1.3 Approximation of the tangent modulus
In non-linear implicit FEM-analysis, the tangent modulus qDrij
is used to compute the
qDekl
element stiffness matrix, which is the tangent operator for the applied Newton iteration
method. Due to the necessity of numerical integration, the stress increment Drij is a
discrete value and so, the partial derivative cannot be built analytically. Therefore, it
has to be approximated numerically too. This can be done by an Internal Numerical
Differentiation (IND), which was proposed by Bock [11]. Illustratively, IND means to
compute the derivative of the numerical integration algorithm, which leads to the discrete
stress increment. The IND computes an approximation of the partial derivative
that is of similar relative exactness as the solution of the integration itself.
4.6.2 Deformation behaviour of a notched specimen
Some results of the simulated relaxation behaviour of a notched flat bar are shown in
Figure 4.15. Since the main advantage of the proposed method of model implementation
is its easy applicability to three-dimensional as well as to plane state or even onedimensional
(uniaxial) FEM-problems, the numerical results of two simulations using
three-dimensional and plane stress theory are compared. Additionally, experimental results
of Ritter and Friebe [17] show that the model is able to predict the material response
correctly.
86

4.6 Finite-Element Simulation
Figure 4.15: Normal strain in load direction after two hours relaxation time. Comparison between
experimental and numerical results. Material: SS 304 L, temperature: 923 K. ESZ means plane stress.
87

4 Development and Application of Constitutive Models for the Plasticity of Metals
4.7 Conclusions
The mechanisms on the microscale of crystalline materials can be examined on different
scales of magnitude. Starting from a scale, where the processes are described by
help of activation energies and activation volumes as mechanically and thermally activated,
it is possible to consider their stochastical nature by stochastic processes, from
which by mean value considerations, a transition to macroscopic material equations is
possible.
To support the formulation of these models, simulations can be useful, which consider
the multi-particle properties of the processes, and use the methods of cellular
automata or molecular dynamics.
For the numerical simulation and the parameter identification, a variety of sophisticated
methods have been considered. The results show that it is possible to use the
material models for the analysis of structures even under complex loading situations.
References
[1] E. Steck: A Stochastic Model for the High-Temperature Plasticity of Metals. Int. J. Plast.
(1985) 243–258.
[2] E. Steck: The Description of the High-Temperature Plasticity of Metals by Stochastic Processes.
Res. Mechanica 25 (1990) 1–19.
[3] H. Schlums: Ein stochastisches Werkstoffmodell zur Beschreibung von Kriechen und zyklischem
Verhalten metallischer Werkstoffe. Dissertation TU Braunschweig, Braunschweiger
Schriften zur Mechanik 5 (1992).
[4] R. Gerdes: Ein stochastisches Werkstoffmodell für das inelastische Materialverhalten metallischer
Werkstoffe im Hoch- und Tieftemperaturbereich. Dissertation TU Braunschweig,
Braunschweiger Schriften zur Mechanik 20 (1995).
[5] H. Hesselbarth: Simulation von Versetzungsstrukturbildung, Rekristallisation und Kriechschädigung
mit dem Prinzip der zellulären Automaten. Dissertation TU Braunschweig, Braunschweiger
Schriften zur Mechanik 4 (1992).
[6] D. Sangi: Versetzungssimulation in Metallen. Dissertation TU Braunschweig, 1996.
[7] F. Thielecke: Parameteridentifizierung von Simulationsmodellen für das viskoplastische Verhalten
von Metallen – Theorie, Numerik, Anwendung. Dissertation TU Braunschweig, 1997.
[8] F. Thielecke: Gradientenverfahren contra stochastische Suchstrategien bei der Identifizierung
von Werkstoffparametern. ZAMM Z. angew. Math. Mech. 75 (1995).
[9] E. Steck, M. Lewerenz, M. Erbe, F. Thielecke: Berechnungsverfahren für metallische Bauteile
bei Beanspruchungen im Hochtemperaturbereich, Arbeits- und Ergebnisbericht 1991–
1993. Subproject B1, Collaborative Research Centre (SFB 319), 1993.
[10] F. Thielecke: New Concepts for Material Parameter Identification Considering the Scattering
of Experimental Data. ZAMM Z. angew. Math. Mech. 76 (1996).
[11] H.G. Bock: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer
Differentialgleichungen. Bonner Mathematische Schriften, Bonn, Vol. 183 (1985).
[12] J. Schlöder: Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Parameteridentifizierung.
Dissertation Universität Bonn, 1987.
88

References
[13] F. Thielecke: Ein Mehrzielansatz zur Parameteridentifizierung von viskoplastischen Werkstoffmodellen.
ZAMM Z. angew. Math. Mech. 76 (1996).
[14] R. Jategaonkar, F. Thielecke: Evaluation of Parameter Estimation Methods for Unstable
Aircraft. AIAA Journal of Aircraft 31(3) (1994).
[15] R. Gerdes, F. Thielecke: Micromechanical development and identification of a stochastic
constitutive model. ZAMM Z. angew. Math. Mech. (1996).
[16] I. Rechenberg: Evolutionsstrategie ’94, Werkstatt Bionik und Evolutionstechnik, Band 1.
Frommann-Holzboog, Stuttgart, 1994.
[17] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical
Measuring Methods. This book (Chapter 13).
[18] K.-T. Rie, H. Wittke: Inelastisches Stoffgesetz und zyklisches Werkstoffverhalten im LCF-Bereich,
Arbeits- und Ergebnisbericht 1991–1993. Subproject B4, Collaborative Research
Centre (SFB 319), 1993.
[19] E. Hinton, D.R. J. Owen: Finite Elements in Plasticity: Theory and Practice. Pineridge
Press, Swansea, 1980.
[20] M. Lewerenz: Zur numerischen Behandlung von Werkstoffmodellen für zeitabhängig plastisches
Materialverhalten. Dissertation TU Braunschweig, 1996.
[21] K. Hornberger: Anwendung viskoplastischer Stoffgesetze in Finite-Element-Programmen.
Dissertation Universität Karlsruhe, 1988.
[22] Hibbitt, Karlsson, Sørensen, Inc.: ABAQUS THEORY MANUAL, Version 5.4. Pawtucket,
RI, United States, 1994.
[23] E. Hairer, S.P. Nørsett, G. Wanner: Solving Ordinary Differential Equations I (Nonstiff
Problems). Springer, Berlin, 1987.
[24] E. Hairer, G. Wanner: Solving Ordinary Differential Equations II (Stiff Problems). Springer,
Berlin, 1991.
[25] S.W. Sloan: Substepping Schemes for the Numerical Integration of Elastoplastic Stress-
Strain Relations. Int. J. Numer. Meth. Eng. 24 (1987) 893–911.
[26] J.C. Butcher: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and
General Linear Methods. John Wiley & Sons Ltd, Chichester, 1986.
[27] J.H. Verner: Families of Imbedded Runge-Kutta-Methods. SIAM J. Numer. Anal. 16 (1979)
857–875.
89

5 On the Physical Parameters Governing
the Flow Stress of Solid Solutions
in a Wide Range of Temperatures
Abstract
Christoph Schwink and Ansgar Nortmann*
At sufficiently low temperatures, host and solute atoms remain on their lattice sites. The
critical flow stress r0 is governed by a thermally activated dislocation glide (Arrhenius
equation), which depends on an average activation enthalpy, DG0, and an effective obstacle
concentration, cb. The total flow stress r is composed of r0 and a hardening stress rd,
which increases with the dislocation density q w in the cell walls according to rd /…q w† 1=2 .
At higher temperatures, the solutes become mobile in the lattice and cause an additional
anchoring of the glide dislocations. This is described by an additional enthalpy Dg in the
Arrhenius equation. In the main, Dg depends on the activation energy Ea of the diffusing
solutes and the waiting time tw of the glide dislocations arrested at obstacles. Three different
diffusion processes were found for the two f.c.c.-model systems investigated,
CuMn and CuAl, respectively. Under certain conditions, the solute diffusion causes instabilities
in the flow stress, the well-known jerky flow phenomena (Portevin-Le Châtelier
effect). Finally, above around 800 K in copper based alloys, the solutes become freely
mobile and r0 as well as Dg vanish. In any temperature region, only a small total number
of physical parameters is sufficient for modelling plastic deformation processes.
5.1 Introduction
The intention of the present project was to find out the physically relevant parameters,
which determine the stable flow stress r in metallic systems of model character over a
given wide range of temperatures and strain rates.
* Technische Universität Braunschweig, Institut für Metallphysik und Nukleare Festkörperphysik,
Mendelssohnstraße 3, D-38106 Braunschweig, Germany
90
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

5.1 Introduction
Any theory describing plastic deformation modes of such systems will have to
make use of these – and only these – parameters. As model systems we choose single
phase binary f.c.c. solid solutions. They are on the one hand simple, macroscopically
homogeneous materials, on the other hand exhibit all basic processes, which occur also
in more complex alloys of technical interest. To cover a wide range of different characteristics
existing in various binary alloys, we studied the two systems CuMn and CuAl,
which differ appreciably in some salient properties (Table 5.1).
We point to the misfit parameter, the variation of the stacking fault energy with
solute concentration and the tendency for short range ordering. The systems have in
common a metallurgical simplicity and a large range of solubility. The samples used
were rods of polycrystals, for CuMn also of single crystals oriented either for single or
for multiple ([100], [111]) glide.
For low enough temperatures, i.e. roughly below room temperature, host and solute
atoms remain on their lattice sites in our systems. Then, the flow stress is recognized to
consist of two additive parts, which are in single crystals the critical resolved shear stress
s0, and the shear stress sd produced by strain hardening. s0 is best examined on crystals
oriented for single glide, while results on sd originated from studies on [100] and [111]
crystals. The parameters governing s0 and sd are discussed in Section 5.2.
At higher temperatures, the solute atoms become increasingly mobile and start to
diffuse to sinks, e.g. dislocations. As a consequence, an additional anchoring of glide
dislocations occurs, known as dynamic strain ageing (DSA), which results in an additional
contribution to flow stress, DrDSA, and in a decrease of the strain rate sensitivity
(SRS) with increasing deformation. If the SRS reaches a critical negative value, jerky
flow sets in, the so-called Portevin-Le Châtelier (PLC) effect. The mechanisms inducing
DSA and the relevant parameters represent the main part of project A8 and are reported
in Section 5.3.
We restrict the report on the own main results. For details and further literature,
the reader is referred to the publications cited.
Table 5.1: Metallurgical and physical properties of CuAl and CuMn.
Misfit d
d ˆ…Da†=…aDc†
CuAl CuMn
d ˆ0.067
(weak hardening)
Bulk diffusion QD ˆ 1:86 ...2:01 eV
D0 ˆ…0:8 ...5:6† 10 5 m 2 s 1
Stacking fault energy strongly decreasing with
increasing cAl
d ˆ0.11
(strong hardening)
QD ˆ 2:03 ...2:12 eV
D0 ˆ…7:4 ...14:2† 10 5 m 2 s 1
independent of cMn
Slip character 5 ... 10 at% Al planar 0.5 ...5:5 at% Mn homogeneous
Short range order marked and increasing with cAl negligible up to 5 at%
91

5 On the Physical Parameters Governing the Flow Stress of Solid Solutions
5.2 Solid Solution Strengthening
In the frame of the project, invited overviews on “Hardening mechanisms in metals with
foreign atoms” [1], “Solid solution strengthening” [2] (in collaboration with project A9),
and “Flow stress dependence on cell geometry in single crystals” [3] have been published.
5.2.1 The critical resolved shear stress, s 0
A detailed investigation on CuMn [4] showed that the thermally activated process governing
s0 is phenomenologically completely characterized by two parameters, the average
activation enthalpy, DG0, of the effective glide barriers, and the concentration of
the latter, cb. This concentration resulted about 20 times smaller than the solute concentration,
cMn. For DG0, values around 1.3 eV were found.
The magnitude of cb and DG0 suggest the effective glide barriers to consist of
complexes of at least two solute atoms. A dislocation segment after having surpassed
an effective barrier sweeps in the subsequent elementary glide step an area containing
cMn=cb solute atoms on the average.
Altogether, we arrive at s0 ˆ s0…DG0; cb; T; _e†.
5.2.2 The hardening shear stress, s d
Detailed mechanical and TEM-studies have been performed on CuMn-crystals oriented
along [100] and [111] [5]. The hardening shear stress resulted as equal to the reduced
stress, sd ˆ…s s0†, and obeying the known relation [6]:
sd ˆ t Gbq 1=2
t : …1†
Here, q t is the average total dislocation density, G the shear modulus. A surprising result
was that t depends on the solute concentration, it decreases with increasing cMn.
This means that for a given value of the reduced stress, the dislocation density q t is
higher in an alloy than in the pure host. A further analysis showed that q t is stored
nearly completely in the cell walls, which are fully developed already at small stresses
and strains. The next result of relevance was the increase of the wall area fraction fw
with solute concentration. Defining a mean dislocation density inside the cell walls, q w,
by q w ˆ q t=fw, we can rewrite Equation (1) as:
sd ˆ t f 1=2
w Gbq1=2 w ˆ w Gbq 1=2
w : …2†
The prefactor w now turns out as independent of cMn and practically constant for a
fully developed cell structure. w ˆ 0:25 0:03 from the experiments favourably com-
92

pares with the lowest values for calculated by theory [7] (cf. also [8]). This suggests
the view that the most favourably oriented dislocation segments will cross the obstacle
field and will be followed via the unzipping effect by all others at nearly the same
stress, which is the lowest possible one.
A TEM-investigation on Cu1.3 at% Mn crystals oriented for single glide [9], the
first systematic TEM-study on a solid solution, yielded for the extended stage I a prevailing
primary dislocation density, q prim, and a continuous decrease of the strain hardening
rate with increasing strain. In stage II and above, the reduced flow stress was
found as completely governed by the density of all secondary dislocations taken together,
q sec. It is:
…s s0† ˆ Gbq 1=2
sec
; with ˆ 0:32 0:04 : …3†
The total shear stress, s, results as a linear superposition of a “solid solution stress”, s0,
and a strain hardening stress, sd, as found also for the multiple glide crystals. It is completely
described by four parameters, apart from the obvious ones, T and _e:
s ˆ s0…DG0; cb; T; _e†‡sd…q t; fw; T; _e† : …4†
The generalization for polycrystals adds the problem of compatibility of neighbouring
grains. It is of importance mainly for small stresses and strains and introduces essentially
the average grain diameter as an additional parameter in the case of a random assembly of
grains (cf. [3]). At higher stresses, the relevant parameters are the same as in Equation (4).
5.3 Dynamic Strain Ageing (DSA)
5.3.1 Basic concepts
5.3 Dynamic Strain Ageing (DSA)
In the commonly applied models [10–12], the contribution of DSA to flow stress, DrDSA,
increases proportional to the increase in the line concentration C of glide obstacles on
arrested, “waiting” dislocations by DC during the waiting times, tw [11–13]. It is generally
assumed that DC is a function of …D…T†tw†, where D…T† is the diffusion coefficient
of the underlying process with the activation energy Ea. The waiting time tw is connected
with the strain rate _e via tw ˆ X=_e, where X represents the “elementary strain” [14].
Phenomenologically, DSA can be described by an additional free activation enthalpy
Dg ˆ Dg…Ea; tw† entering besides DG0 the well-known Arrhenius equation [15].
Ample DSA leads to flow stress instabilities (PLC-effect). Details of the processes
inducing jerky flow can be studied in the region of stable flow preceding a PLC-region
by measuring with high accuracy i) stress-strain curves, r…e† (Figure 5.1), ii) strain rate
sensitivities (SRS) of flow stress, …Dr=D ln _e†, along whole stress-strain curves and over
a wide range of temperature. The results are presented in the following.
93

5 On the Physical Parameters Governing the Flow Stress of Solid Solutions
Figure 5.1: Schematic stress-strain curve showing the definition of the critical stresses …ri† and
strains …ei†. r0 is the critical flow stress (see [16]).
5.3.2 Complete maps of stability boundaries
We succeeded in establishing the first complete maps of boundaries of stable flow.
From copper-based solid solutions, polycrystalline samples of six different Mn- and
three Al-concentrations have been studied, furthermore CuMn-crystals oriented for single
and multiple ([100]) glide [15–20].
Figure 5.2 shows a survey of occurrence and types of instabilities in
Cu 2.1 at% Mn. The critical reduced stresses …ri r0† (gained from about 40 r…e†curves
(see Figure 5.1) running parallel to the …ri r0† axis) are plotted as function of
temperature T. There are three transitional temperature intervals, labelled a, b and c,
where several regions of stable and unstable deformation alternate along r…e†-curves.
Outside these intervals, the stress-strain curves are either stable or jerky throughout. In
the small interval c, 290 8C9T9305 8C, an irregular sequence of bursts of type C
Figure 5.2: Mode of deformation map: dependence of reduced critical stresses …ri r0† on temperature
T for Cu 2.1 at% Mn. Basic strain rate _e1 ˆ 2:45 10 6 s 1 . The hatched areas represent
domains of unstable deformation with the predominant types of instabilities indicated (see [18]).
94

5.3 Dynamic Strain Ageing (DSA)
Figure 5.3: Reduced critical flow stresses for the beginning and end of jerky flow as functions of
T. (a) polycrystals; (b) [100]-crystals, Cu 2 at% Mn (see [19]).
prevents the existence of a unique dependence of ri on T (or _e) as could be established
for intervals a and b. For further details see [16, 18].
The strain hardening coefficient, which roughly remains constant for temperatures
up to about 300 K, decreases strongly with further increasing temperature owing to recovery
processes. At about T ˆ 600 8C, it becomes nearly zero [18], the critical flow
stress simultaneously vanishes as well as the additional enthalpy Dg and a steady state
of deformation exists across the whole deformation curve. The solutes are now moving
freely through the lattice [21].
Figure 5.3 a gives stability maps for several Mn-concentrations over the intervals
a and b for polycrystals, Figure 5.3 b the same for [100]-crystals of 2 at% Mn [19].
The similarity of both is closest if the boundaries for the [100]-crystal are compared
with those for a polycrystal of about 1.2 at% Mn. Contrarily, the boundary map for a
single glide ([sg]-) crystal of 2 at% Mn (Figure 5.4) looks quite different [19]. Only a
single boundary occurs over the whole range of temperatures. However, the curve can
be divided into two parts, which for good reasons are noted as intervals a and b, too
(see Section 5.3.3).
Finally, boundary maps for CuMn-polycrystals have been compared with those
measured for CuAl [20]. Figure 5.5 presents characteristic examples. The complete correspondence
of Cu 0.63 at% Mn with Cu 5 at% Al is obvious and indicates the existence
of two different PLC-domains. They are labelled as domains I and III. With in-
95

96
5 On the Physical Parameters Governing the Flow Stress of Solid Solutions
Figure 5.4: Reduced critical flow stresses s a;b
x for Cu 2 at% Mn single glide crystals as functions
of T. In contrast with multiple glide crystals (poly-, [100]-, see Figure 5.3) at any temperature,
only one boundary of stability occurs (see [19]).
Figure 5.5: Deformation-mechanism maps of CuAl (a–c) (see [20]) and CuMn (d–f) as obtained
at different temperatures but constant basic strain rate _e1 ˆ 2:5 10 6 s 1 . The reduced critical
stresses …ri r0† indicate the transition between stable and unstable deformation (stress-strain
curves running parallel to the ordinate). The hatched areas indicate the PLC-regions. a)
Cu 5 at% Al, b) Cu 7.5 at% Al, c) Cu 10 at% Al (values for CuAl from [20]), d) Cu 0.63 at% Mn
(*) and Cu 0.95 at% Mn (n), e) Cu 1.1 at% Mn, f) Cu 2.1 at% Mn (values for CuMn from [18]).

5.3 Dynamic Strain Ageing (DSA)
creasing concentration, a “bulge” develops on the boundary r2…T; _e†. It is clearly visible
already for 0.95 at% Mn (Figure 5.5 d), and is extended to a peak for 1.1 at% Mn
(Figure 5.5 e) [18]. As a consequence, an additional PLC-domain II develops bounded
by the anomalous boundary r2 at the lower temperature side (Figure 5.5 b and e). The
island of stability, which appears in Cu 1.1 at% Mn and Cu 2.1 at% Mn above about
400 K is covered in CuAl by the domain II (Figure 5.5 b,c,e and f) [20].
5.3.3 Analysis of the processes inducing DSA
Precise measurements of the critical stresses ri for the onset of jerky flow [16] on the
one hand, and of changes in the flow stress, Dr, after variations in strain rate [17, 18]
on the other, are the basis of an analysis of DSA. Figure 5.6 shows as an example variations
in shear stress measured in stages I and II of a crystal oriented for single glide
[22]. Generally, one has to distinguish between the instantaneous variations, Dsi, occurring
immediately after a change in _e, and the stationary ones, Dss. (Remark: For single
crystals, the flow stress r is always replaced by the resolved shear stress, s.)
It is the difference, …Dss Dsi† ˆD…DsDSA†; which reflects the effect of DSA
and causes a decrease of the SRSˆ …Ds=D ln _e† T [19]. Analogously, for polycrystals is
SRSˆ …Dr=D ln _e† T [18].
The boundaries r2, r3 of the “island of stability” in temperature interval b (see
Figure 5.3) are governed by a thermally activated process as demonstrated in Figure
5.7: A decrease in T is qualitatively equivalent to an increase in _e. We can take ri as
indicating the onset of the thermally activated process and derive from
…ri r0† ˆA ‡ B
‡ C ln _e ! ln _e ˆ A0
T
B=C
T
values for the activation energies Qm ˆ B=C …m ˆ 2; 3† [18].
Figure 5.6: Change in resolved shear stress, Ds, after a change in external strain rate of
_e2=_e1 ˆ 2 : 1, against incremental true strain, Dc, taken in stage II at s ˆ 32:4 MPa and
c ˆ 52:6%. The plot is corrected for the average strain hardening rate (see [22]).
…5†
97

5 On the Physical Parameters Governing the Flow Stress of Solid Solutions
Figure 5.7: Dependence of the reduced critical stresses on (a) 1=T at _e1 ˆ 2:45 10 6 s 1 , and on
(b) ln _e1 at T ˆ 460 K; both plots for Cu 2.1 at% Mn in interval b (see [18]).
Another way of determining these energies starts from a consideration of the normalized
instantaneous and stationary SRS, denoted by Si and Ss, respectively [18, 19].
Figure 5.8 gives their course with increasing stress along the stress-strain curve of a
[sg]-crystal [22], Figure 5.9 shows Ss…r† alone for a polycrystal at various temperatures.
Following the above mentioned models [11, 12], the marked dependence of
SDSA ˆ…Ss Si† on T (and also _e) is for short enough tw described by the relation [18]:
SDSA / c D…T†
_e
n
/ c_e n exp
nEa
kT
: …6†
Here, Ea is the activation energy of pipe diffusion entering D…T† ˆD0 exp… Ea=kT†.
Strain rate exponent n and Ea are best obtained from Equation (6) in regions, where
SDSA varies linearly with stress yielding constant slopes M ˆ …qSDSA=qr† T;_e . One easily
finds [18]:
n ˆ
q ln M
; and nEa ˆ
q ln _e T
q ln M
q…1=kT† _e
: …7†
The activation energies Qm and Ea;m found for the diffusion processes generating the
PLC-domains I, II and III are compiled in Table 5.2 [20]. Where Qm and Ea;m can both
98

e measured for the same process, they are found equal, Qm ˆ Ea;m, within scatter.
Average values for the three DSA-processes are denoted by EI, EII, EIII.
An important further quantitative result is that the strain rate exponent resulted as
n ˆ 1=3 (within scatter) in all cases.
5.3.4 Discussion
5.3 Dynamic Strain Ageing (DSA)
Figure 5.8: Dimensionless instantaneous …Si† and stationary …Ss† SRS against reduced stress
…s s0†; s0 ˆ critical resolved shear stress; T ˆ 263 K (see [22]).
Figure 5.9: The dependence of the stationary, normalized SRS on reduced stress for
Cu 3.5 at% Mn at temperatures of interval a. All data points refer to states of stable deformation.
The critical stresses are indicated for 73.4 8C. The Mm denote the linear slopes of the S-…r r0†curves
(see [18]).
The strain rate exponent n has for a long time been commonly assumed to equal 2/3
according to Cottrell and Bilby’s theory of lattice diffusion [23, 24]. Already the first
experimental determination of n yielded, however, n ˆ 1=3 and has been explained by
a pipe-diffusion mechanism governing the DSA-process concerned [25].
99

5 On the Physical Parameters Governing the Flow Stress of Solid Solutions
Table 5.2: Activation energies of DSA-processes in CuAl and CuMn.
Average CuxAl CuxMn 1
value
5 10 0.63 1.3 2.1 3.5
Ea1 [eV] 0.74 ±0.15 0.79 ±0.12 – – 0.88 ±0.05 0.86 ±0.10 2
Q1 [eV] × 0.75 ±0.10 × × × ×
Q2 [eV] × 0.76 ±0.10 – 0.88 ±0.10 0.91 ±0.10 0.87 ±0.10
EI [eV] 0.74 0.77 – 0.88 0.89 0.86
E …a†
a3
Shortly after, Schlipf [26] pointed out that a more general relation than Equation
(6) for SDSA is conceivable, viz. SDSA / DC q , which by use of DC /…D…T†=_e† r
yields SDSA / c q _e qr (see also [27]). Now q ˆ 1=2 and r ˆ 2=3 would give an exponent
n ˆ qr ˆ 1=3 also in the case of lattice diffusion. On the other side, it became
more and more clear that n ˆ 1=3 holds quite generally for any DSA-process [20, 28,
29].
To clarify the puzzling situation, a more extensive experimental analysis of SDSA
has been undertaken by studying and simultaneously evaluating the dependence of
SDSA on flow stress as well as on solute concentration. We found [30] that
• the Mulford-Kocks model of DSA [12] describes the experiments clearly better than
the van den Beukel model [11], and
• the data – taking the most reliable ones – are in favour of a simple proportionality
to solute concentration, i.e. q ˆ 1.
This would exclude a lattice diffusion and is suggesting an own pipe diffusion mechanism
for each DSA-process. Recent theoretical work [31] points to an even probable
existence of several modes of pipe diffusion [20] along dissociated dislocations.
A recently found method to measure immediately the average waiting time tw of
dislocations [32] showed that the elementary strain X continuously increases with the
flow stress, the total increase never exceeding a factor of only 10. In principle, X is deducible
from a knowledge of the dislocation arrangement …q t; q f; fw† and of the density
of glide barriers …cb† [19]. However, a general theory is still missing. Therefore, X…r†
and with it tw, which governs stress transients, are still to be considered as parameters.
100
[eV] × × × 0.81 ±0.10 0.86 ±0.10 0.87 ±0.10
Q2 [eV] × 1.1 ±0.30 × × × ×
EII [eV] × 1.1 × 0.81 0.86 0.87
E …b†
a3
[eV] 1.42 ±0.25 × 1.9 1.53 ±0.10 1.27 ±0.10 1.15 ±0.10
Q …b†
3 [eV] 1.46 ±0.30 × – 1.-59±0.08 1.25 ±0.05 1.16 ±0.10
EIII [eV] 1.44 × 1.9 1.56 1.26 1.15
1 2
Values for CuMn from [18]; Cu 4.1 at% Mn; –: not measured; ×: not defined or not measurable.

101
Table 5.3: Overview of the parameters investigated quantitatively in project A8. They characterize the flow stress and its strain rate sensitivity
in single phase random f.c.c. solid solutions along stress-strain curves taken over a wide range of temperatures. The DSA parameters come into
play only at higher temperatures (about room temperature!). In the future, some of the parameters will prove derivable from more complete
theories.
Elementary process Characteristic magnitude Parameter (quanitatively measured) Literature
Solid solution hardening critical flow stress, r0
(single crystal: crss, s0)
Dislocation hardening reduced flow stress, rd
…rd ˆ r r0; sd ˆ s s0†
Dynamic strain ageing (DSA) flow stress contribution, DrDSA,
or additional enthalpy,
Dg ˆ VDrDSA,
Dg ˆ Dg…Ea; …tw† n †
Exhaustion of DSA limiting Dg-value, Dgmax
0:1DG0
Transitions of DSA owing to
variations in _e
Variations in mobile dislocation
density, owing to variations in _e
transition from instantaneous
to stationary flow stress,
Dri ! Drs
i) average activation enthalpy,
DG0 (eV)
ii) effective barrier concentr.
cb ˆ f …c†
i) total dislocation density, q t…r†
[m –2 ]
ii) volume fraction of disloc.
walls, fw…r†
i) activation energy Ea;m (eV),
m ˆ I, II, III, of the diffusion
inducing DSA
ii) tw…s† ˆX=_e ˆ waiting time
of arrested dislocations
iii) strain rate exponent, n
Wille, Gieseke and Schwink [4];
Neuhäuser and Schwink [2]
Neuhaus and Schwink [6];
Neuhaus, Buchhagen and Schwink
[33];
Heinrich, Neuhaus and Schwink [9]
Springer and Schwink [25];
Kalk, Schwink and Springer [17];
Kalk and Schwink [18];
Nortmann and Schwink [20]
relaxation constant, B Springer, Nortmann and Schwink
[30]
relaxation time H waiting time
tw
Schwarz [13]; McCormick [34];
Springer and Schwink [32]
active slip volume, Va b a ˆ 1 dlnVa=dln_c Schwink and Neuhäuser [35];
Neuhäuser [36];
Traub, Neuhäuser and Schwink
[37];
Nortmann and Schwink [22]
5.3 Dynamic Strain Ageing (DSA)

5 On the Physical Parameters Governing the Flow Stress of Solid Solutions
The detailed analysis of the SRS also allows to evaluate quantitatively the variations
of the additional enthalpy by varying the strain rate, D…Dg†, along the whole
stress-strain curves, and to determine the value of the quantity Dg…_e† itself [30]. It
amounts up to about 10% of DG0 ˆ 1:3 eV (see Section 5.3.1) [15, 30].
At higher flow stresses (0 100 MPa), Dg is observed to approach a limit with decreasing
strain rate. The course of Dg…_e† can be best discussed for data on CuAl. At
the time, the approach of the limit is described by a kind of relaxation parameter B. It
is concluded that an exhaustion of solute atoms available for the diffusion process in
question limits the increase of DrDSA, and not a saturation of glide dislocations by the
diffusion-induced glide obstacles [30].
Finally, the question has been addressed, whether the SRS might be influenced
besides of DSA-processes also by variations of the mobile dislocation density q m when
varying the strain rate [22]. In fact, a superposition of both effects could be demonstrated.
There exists a very small interval around a transition temperature, above which
DSA-effects are dominating the SRS, while q m-effects dominate below.
5.4 Summary and Relevance
for the Collaborative Research Centre
Shortly summarizing this report, we can say that any mechanism contributing to flow
stress can be accounted for by a few measurable parameters in a model description.
Whether a mechanism and parameter is relevant or negligible depends on the experimental
conditions, e.g. on temperature. In any case, the total number of relevant parameters
is defined and quite limited. Table 5.3 is to give a concise overview of all results
obtained as far as they concern the parameters investigated.
The methods developed in this project to determine these parameters (cf. Table
5.3) can be applied to any material. The parameters will enter any final constitutive
material equations developed, e.g. those of project A6 of the Collaborative Research
Centre (SFB). Results and experiences of our project have been also exchanged with
project A1. Throughout the work, there was an intimate contact to project A9.
References
[1] Ch. Schwink: Rev. Phys. Appl. 23 (1988) 395.
[2] H. Neuhäuser, Ch. Schwink: In: H. Mughrabi (Ed.): Materials Science and Technology,
Vol. 6. VCH Weinheim, 1993, p. 191.
[3] Ch. Schwink: Scripta metall. mater. 27 (1992) 963 (Viewpoint Set No 20).
[4] Th. Wille, W. Gieseke, Ch. Schwink: Acta metall. 35 (1987) 2679.
102

References
[5] R. Neuhaus, Ch. Schwink: Phil. Mag. A 65 (1992) 1463.
[6] For a review referring mainly to pure copper, see: S. J. Basinski, Z.S. Basinski: In: F. R. N.
Nabarro (Ed.): Dislocations in Solids, Vol. 4. North-Holland, Amsterdam, 1979, p. 261.
[7] W. Püschl, R. Frydman, G. Schöck: phys. stat. sol. (a) 74 (1982) 211.
[8] G. Saada: In: G. Thomas, J. Washburn (Eds.): Electron Microscopy and Strength of Crystals.
Interscience, New York, 1963, p. 651.
[9] H. Heinrich, R. Neuhaus, Ch. Schwink: phys. stat. sol. (a) 131 (1992) 299.
[10] For reviews see:
a) Y. Estrin, L.P. Kubin: Acta metall. 34 (1986) 2455.
b) Y. Estrin, L.P. Kubin: Mat. Sci. Eng. A 137 (1991) 125.
c) P. G. McCormick: Trans. Indian Inst. Metals 39 (1986) 98.
d) L.P. Kubin, Y. Estrin: Rev. Phys. Appl. 23 (1988) 573.
e) H. Neuhäuser: In: D. Walgraef, E.M. Ghoniem (Eds.): Patterns, Defects and Materials
Instabilities, Kluwer Ac. Publ., Dordrecht, 1990, p. 241.
[11] A. van den Beukel: phys. stat. sol. (a) 30 (1975) 197.
[12] R. A. Mulford, U.F. Kocks: Acta metall. 27 (1979) 1125.
[13] R. B. Schwarz: Scripta metall. 16 (1982) 385.
[14] L.P. Kubin, Y. Estrin: Acta metall. mater. 38 (1990) 697.
[15] Th. Wutzke, Ch. Schwink: phys. stat. sol. (a) 137 (1993) 337.
[16] A. Klak, Ch. Schwink: phys. stat. sol (b) 172 (1992) 133.
[17] A. Kalk, Ch. Schwink, F. Springer: Mater. Sci. Eng. A 164 (1993) 230.
[18] A. Kalk, Ch. Schwink: Phil. Mag. A 72 (1995) 315.
[19] A. Kalk, A. Nortmann, Ch. Schwink: Phil. Mag. A 72 (1995) 1229.
[20] A. Nortmann, Ch. Schwink: Acta metall. mater. 45 (1997) 2043-2050, 2051–2058.
[21] H. Neuhäuser: This book (Chapter 6).
[22] A. Nortmann, Ch. Schwink: Scripta metall. mater. 33 (1995) 369.
[23] A.H. Cottrell, B. A. Bilby: Proc. Phys. Soc. Lond. A 62 (1949) 49.
[24] J. Friedel: In: Dislocations, 368 Pergamon, Oxford, 1964, p. 405.
[25] F. Springer, Ch. Schwink: Scripta metall. mater. 25 (1991) 2739.
[26] J. Schlipf: Scripta metall. mater. 29 (1993) 287; Scripta metall. mater. 31 (1994) 909.
[27] H. Flor, H. Neuhäuser: Acta metall. 28 (1980) 939.
[28] C. P. Ling, P. G. McCormick: Acta metall. mater. 41 (1993) 3127.
[29] S.-Y. Lee: Thesis, Aachen, 1993.
[30] F. Springer, A. Nortmann, Ch. Schwink: phys. stat. sol. (a) 170 (1998) 63–81.
[31] J. Huang, M. Meyer, V. Pontikis: Phil. Mag. A 63 (1991) 1149; J. Phys. III 1 (1991) 867.
[32] F. Springer, Ch. Schwink: Scripta metall. mater. 32 (1995) 1771.
[33] R. Neuhaus, P. Buchhagen, Ch. Schwink: Scripta metall. 23 (1989) 779.
[34] P. G. McCormick: Acta metall. 36 (1988) 3061.
[35] Ch. Schwink, H. Neuhäuser: phys. stat. sol. 6 (1964) 679.
[36] H. Neuhäuser: In: F. R. N. Nabarro (Ed.): Dislocations in Solids, Vol. 6, North-Holland, Amsterdam,
1983, p. 319.
[37] H. Traub, H. Neuhäuser, Ch. Schwink: Acta metall. 25 (1977) 437.
The publications [1–5, 9, 15–20, 22, 25, 30, 32, 33] resulted from work performed in the present
project of the Collaborative Research Centre (SFB).
103

6 Inhomogeneity and Instability of Plastic Flow
in Cu-Based Alloys
Hartmut Neuhäuser*
6.1 Introduction
Slip deformation in crystal is inhomogeneous by nature as it is accomplished by the
production and movement of dislocations on single crystallographic planes. Usually,
only few dislocation sources are activated and produce slip on few planes, where, in
particular in fcc and hcp crystals and even more pronounced in alloys with low stacking-fault
energy, many dislocations move on the same plane. This is provoked in particular
if the dislocations on their path through the crystal change the obstacle structure in
the slip plane, e.g. in short-range ordered or in short-range segregated alloys (i.e. in
nearly all alloys) so that the first dislocation feels a stronger “friction” than the succeeding
ones. As the macroscopic elongation of the sample is distributed in general heterogeneously
among the crystallographic planes, the quantities of resolved strain a and
strain rate _a, defined as:
a ˆ l=l 0l0 and _a ˆ _l=l 0l0
(with _l=macroscopic deformation rate, l0 =specimen length, l 0 =Schmid orientation factor),
and commonly used in the formulation of constitutive equations, cannot be directly
connected with realistic dislocation behaviour.
Therefore, in this work, the local strain and strain rate in slip bands, which are
the active regions of the crystal [1], have been measured by a micro-cinematographic
method [2]. Cu-based alloys turned out to be a convenient model system for experimental
reasons: Single crystals can be grown easily in reasonable perfection and the
stacking-fault energy c can be varied by changing the alloy composition. In Cu-
2...16 at% Al, c varies from 35 to 5 mJ/m 2 with increasing Al content, while it remains
(nearly) constant (c & 40 mJ/m 2 ) for Cu-2...17 at% Mn. Thus, the effects of
stacking-fault energy can be separated from those of solute hardening and short-range
ordering, which are comparable for both alloy systems.
104
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
* Technische Universität Braunschweig, Institut für Metallphysik und Nukleare Festkörperphysik,
Mendelssohnstraße 3, D-38106 Braunschweig, Germany
…1†

While solid solution hardening has been extensively studied and is well documented
and appears well understood in the temperature range below room temperature
[3–5], several open questions remain, which are particularly connected with inhomogeneity
of slip above ambient temperature. In a certain range of deformation conditions,
even macroscopic deformation instabilities occur like the Portevin-Le Châtelier
(PLC) effect. This effect appears to be a consequence of the mobility of solute atoms
in the strain field of dislocations (“strain ageing”) and are extensively studied in [6].
In the following, we briefly review our local slip line observations performed during
deformation and accompanied by EM and AFM (atomic force microscope) investigations
of the slip line fine structure and of dislocation structure by TEM. The conclusions
reached so far as well as the still open questions are summarized. According to
the changes of principal mechanisms, the chapter will be divided into the ranges
around room temperature, at intermediate temperatures, and at elevated temperatures.
6.2 Some Experimental Details
6.2 Some Experimental Details
Observations with video records of slip line development during deformation are performed
in two special set-ups with tensile deformation machines equipped with light microscopes.
The slip steps are visualized in dark field illumination as bright lines, where the
scattered light intensity is a measure of slip step height. The minimum step height resolved
is around Smin ˆ 5 to 10 nm (depending on the quality of the electropolished crystal surface),
changes of larger step heights down to dS 5 nm can be resolved.
One apparatus is designed for very high resolution in time (down to 3 ls) [7, 8],
using photo diodes and a storage oscilloscope with pretrigger parallel to video recording.
From the rate of intensity increase and by comparison with interference microscopy
of the same slip band after full development, the local rate of step height increase
and thus the local shear rate can be determined. The time shift of curves of development
recorded by two neighbouring photo diodes immediately yields the velocity
of growth in length, corresponding to the velocity of screw dislocations if the observations
are performed on the “front” surface, where the plane with Burgers vector and
crystal axis cuts the crystal surface (cf. [9]). By using a second microscope and video
system observing the opposite front surface of the plate-shaped crystal, the time, which
slip needs to traverse the crystal thickness, can be determined.
The second apparatus is designed for observations at various temperatures (up to
500 8C) [10] and with a wide field of view between 0.3 and 4.2 mm in order to check
spatial correlations between activated slip bands. The video system usually records with
a frame rate of 50 s –1 and can be increased up to 500 s –1 .
For investigation of the fine structure of slip lines, which is not resolved by light
microscopy, after deformation EM replica and AFM observations are performed. In addition,
in some cases, the dislocation structure developed during deformation steps has
been studied by TEM.
105

For creep tests at elevated temperatures, a special creep set-up was designed,
using the controlling system of the drive of the Instron tensile machine to keep arbitrary
constant stress values and recording strain and strain rate versus time. In particular,
the system permits rapid changes between deformation conditions, e.g. between
testing at constant deformation rate and at constant stress. The specimen is inside a vacuum
tube (p

a)
6.3 Deformation Processes around Room Temperature
b)
Figure 6.1: a) Temperature dependence of the critical resolved shear stress (crss s0)ofCu-2...15 at%
Al single crystals oriented for single slip at a deformation rate of _l=2 ·10 –3 mm/s (_a =3.6 · 10 –5 s –1 ).
In the range of macroscopic slip instabilities (“PLC effect”, dotted line), the stress intervals of serrations
are plotted. b) Temperature dependence of the (normalized) strain rate sensitivity S ˆ ds=d ln _a
(determined from stationary back extrapolated stress jumps during strain rate changes) for one selected
Al concentration (c=15 at%). Interval with arrows indicates PLC effect (jerky flow). The
plots a) and b) contain data from literature (` cf. refs. in [5]) in addition to our own measurements
(*, • and I, indicating the interval between stress maxima and minima in serrated flow).
for the velocity of screw dislocation groups at the edge of an expanding new slip band
on the front surface. For this example of Cu-15 at% Al, the velocity of edge dislocations
can be estimated from the measured growth rate in height Sb if a reasonable distance
de of the (edge) dislocations moving in groups is assumed. As we consider here
the very first dislocation group produced by the source, we use an average distance between
edge dislocations in the group as determined on TEM micrographs for single,
slightly piled-up groups, i.e. d e =0.2 lm [13, 14]. Then
ve ˆ Ssb de=b 3m=s …3†
107

esults, where S sb is the slope of the step height versus time curve in the very first few
ms. The ratio v s/v e&8 (at least 3) appears reasonable in view of the interaction
strength of solute obstacles with different dislocation characters and with estimates of
friction stresses on dislocations from the shape of dislocations on TEM micrographs
[13].
Figure 6.2 a shows the typical slip band development recorded by video and the
photo diode; in Figure 6.2 b, the local shear rate during the development is shown in a
double log plot. After the very first rapid growth of step height, the rate slows down
gradually when more and more slip lines are added to the slip band. While the very
first dislocation group appears to move under overstress, resulting in a local slip instability
with the shear rate exceeding that imposed by the deformation machine (cf.
dotted line in Figure 6.2 b), the successive groups feel opposing internal stresses. This
behaviour can be modelled by a (local) work hardening [11, 15]. It shows that the local
strain rate
108
6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
_aloc ˆ R mbv …4†
a)
b)
Figure 6.2: a) Records of slip step growth (step height S sb versus time t) of a single slip band,
evaluated from photodiode and digital storage oscilloscope (note ms time scale and high level of
noise). b) Variation of the growth rate in step height Ssb (=local slopes of a)) for several slip
bands in Cu-15 at% Al (compared with earlier results in Cu-30 at% Zn), plotted in double log
scales versus time t. The dotted line indicates the growth rate, which would be necessary to accommodate
the imposed deformation rate by one single slip band.

(R m =mobile dislocation density) varies with time in the activated slip zones by many
orders of magnitudes and that the assumption of an average strain rate according to
Orowan’s equation (Equation (4)), which is frequently used in constitutive modelling,
is not realistic and somewhat arbitrary. The local strain rate _aloc can be connected with
the external deformation rate only by using the “active crystal length” la instead of the
total crystal length l 0 in Equation (1):
where
_aloc ˆ _l=l 0la ; …5†
la ˆ nabBsb
6.3 Deformation Processes around Room Temperature
(n ab =number of simultaneously active slip bands, B sb=active width of a slip band measured
along the crystal axis [2]) is a function of deformation rate, stress, strain, temperature,
and time in general. Instead, the nucleation rate formulation of Orowan can
be used to express the average strain rate
_a ˆ _NbF ; …7†
where the rate _N of successive source activations is required (i.e. in our case, the rate
of slip band activations), and the details of slip band development do not matter because
only the total area F swept by all active dislocations during the event enters.
The slip instability at the onset of each slip band evolution can be detected as a
slight stress drop in special experiments (using very thin, short specimens with the load
cell directly connected to one crystal grip [16]) and in acoustic emission records (e.g.
[17]); they are too small to be resolved in case of common specimens (y 4 mm, length
120 mm) in usual tensile machines with their large inertia.
The firstly activated dislocation source of a new slip band is always on that crystal
surface, which due to the bending and lattice rotation by local shear feels a slight
overstress (surface “high”, see below). The average times t HL for the edge dislocation
group to traverse the crystal from this front surface to the opposite one are found, for
plate-shaped Cu-15 at% Al crystals of D=170 and 220 lm thickness, to be 11.1 and
0.7 s, respectively, corresponding to average velocities of the edge dislocations of 22
and 440 lm/s (slip plane inclined by 458 to the crystal axis). The large difference to
Equation (3) is due to the opposing stress gradient along this dislocation path and reflects
the high strain rate sensitivity around room temperature (cf. Figure 6.1 b).
An important process in the formation of dislocation groups from each activated
source is the partial destruction of obstacles by the dislocations cutting across obstacles
in the slip plane. In case of the present alloys Cu-Al and Cu-Mn, their well-known tendency
to short-range ordering suggests that the effective obstacles in the yield region
are groups of solute atoms in an at least partially ordered configuration. This will be
destroyed by a cutting dislocation so that the next dislocation will be able to move at a
lower stress. Although some energetically favourable solute configurations will be “repaired”
by the following dislocations, the net effect is a destruction of “friction” to a
lower value in the activated slip plane. This process was modelled [14] using realistic
…6†
109

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
next-neighbour pair potentials determined from diffuse X-ray scattering measurements
on Cu-15 at% Al crystals [18] in Monte Carlo simulations of a model crystal with the
measured short-range order, and the resulting dislocation configurations of the group
compared with those observed in TEM [14] (Figure 6.3). This indicates quite high intrinsic
“friction” stresses of the original alloy. The dislocation group is able to move at
a distinctly lower stress because the first dislocation feels, in addition to the external
stress, the internal stress from the following piled-up dislocations.
The resulting fluctuations in local stress are especially pronounced in the case of
Cu-Al alloys, where the dislocation groups on single slip planes are much more extended
than in Cu-Mn alloys as a consequence of the low stacking-fault energy in the
former case, which prevents dislocations from easy cross-slip. This tendency is clearly
observed in TEM micrographs of the dislocation structure after deformation in stage I
(Figure 6.4 a,b [14]) and in the slip band fine structure imaged by EM replica in Figure
6.4 c,d, and by AFM in Figure 6.4 e,f [19]. In particular, the high resolution of the
Figure 6.3: a) Variation of the diffuse antiphase boundary energy in the slip plane by passage of a
number n of dislocations crossing the slip plane and changing near neighbour short-range-ordered
configurations. b) Interaction stresses between dislocations in single dislocation groups (sww
dotted lines) observed by TEM for annealed and quenched Cu-10.7 at% Al crystals. Full lines
sSRO give the difference between these curves (*) and the simulation result (^) from a), assuming
sSRO ˆ c SRO=b [14, 23].
110
a)
b)

6.3 Deformation Processes around Room Temperature
a) b)
c) d)
e)
0.5 lm
Figure 6.4: Comparison of dislocation structures: TEM micrographs after deformation in stage I
at room temperature: a) Cu-14.4 at% Al; b) Cu-12 at% Mn), and slip line structures, EM replica:
c) Cu-10.7 at% Al; d) Cu-8 at% Mn; AFM micrographs: e) Cu-15 at% Al; f) Cu-17 at% Mn.
last method permits to decide that in Cu-Al, the activated slip line is indeed on a single
crystallographic plane according to the measured step angle (cf. for Cu-30 at% Zn [20],
for Cu-7.5 at% Al [21]). In Cu-Mn alloys, on the other hand, the high probability for
cross-slip (c c Cu) appears to be the reason for the Cu-like slip line arrangement with
clusters of activated slip planes during the work-hardening stages II and III, while in
Cu-Al alloys with its low c value, very strong local variations of slip behaviour occur
[22]. This again indicates that the average stress and strain usually given in stress-strain
111
f)

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
diagrams may differ considerably from the local values relevant at the active dislocation
sources and for the moving dislocations.
6.3.2 Development of slip band bundles and Lüders band propagation
The process of successive activation of slip lines in the slip band and of slip bands in the
slip band bundle or at the front of a propagating Lüders band has been investigated in
detail by observations on thin flat crystals [16] and by FEM calculations of the stress
around slip steps as well as calculations of the stress field resulting from excess dislocation
groups below the surface necessary to shield the notch stress of the slip step (Figure
6.5). These calculations show that in the surface region, maxima of shear stress occur in
characteristic distances ahead of a previously activated slip plane (irrespective of the details
of dislocation arrangement in the group), i.e. in a distance of 200 nm and in a distance
of 30 lm. The former corresponds to the observed distances of slip lines d se, the latter to
those of slip bands d sb; the numbers depend on the positions of the front and last dislocation
of the excess group. Thus, the activation of a new source occurs under a certain overstress,
which explains the above-mentioned slip instability in the first stage of slip band
growth. It also indicates that the externally measured crss or yield stress in stage I has
to be considered with some caution, although, owing to the high strain rate sensitivity
(Figure 6.1 b), the local stress will exceed the average value by only a few percent.
Figure 6.5: Resolved shear stresses in the slip planes near the upper surface of the crystal (cf. sketch
below) around a slip step and from the stresses of dislocations (B) below the surface, which are necessary
to shield the notch stress of about 50 MPa (A). Note the maxima of the resulting stress
around distances of 200 nm and 30 lm, which are prefered locations for next source activation. Calculation
for S =100 nm, a =200 nm, n =50 dislocations, distance to the front dislocation=33.5 lm.
112

6.3 Deformation Processes around Room Temperature
The above-mentioned differences between Cu-Al and Cu-Mn disappear in the mesoand
macroscopic level: The appearance of slip bands, slip band bundles and the Lüders
band is quite similar (Figure 6.6 a–d). In observations specially designed for examining
the long-range correlations of slip by applying low magnification in the light microscope,
it was found [23, 24] that the neat and simple Lüders band configuration (Figure
6.6 c,d) usually observed in thin flat crystals [16] can be produced also in thick cylindrical
crystals (4 mm y) if the external load (i.e. applied deformation rate) is selected low
enough. Such a deformation front, which is shown schematically in cross section in Figure
6.7 (left side), propagates with a certain velocity v LB from one crystal grip to the other
during tensile deformation in stage I (yield region) in a nearly stable configuration (solitary
wave [25]). The first source of a new slip band ahead of the front is activated at (or near)
surface “high” (Figure 6.7), and slip gradually crosses the crystal towards the opposite
surface “low” against a gradient of bending stress (Figure 6.5). The average plastic front
is normal to the crystal axis and the propagation velocity along the crystal, as determined
from the measured distances and times of front slip bands (Figure 6.8), is found to be
proportional to the external deformation rate _l if this remains below the critical value.
Above that, the deformation mode changes to the formation of slip band bundles (Figure
6.6 a,b) whose trace across the crystal follows the crystallographic slip planes (Figure
6.7, right side). Now, the stress due to the increased deformation rate appears to be high
enough to activate sources more or less at random along the crystal length. They grow to
slip band bundles by adding neighbouring slip bands according to the mechanism shown in
Figure 6.5 (cf. [26, 27]). From such a slip band bundle, the Lüders band starts when the
bundle has reached a certain sufficiently high integrated shear, implying enough stress
concentration due to the bending moment, the thickness reduction and the lattice rota-
a) b)
c) d)
Figure 6.6: Light micrographs of slip band structure on the crystal front surface for the two deformation
modes in stage I of Cu-Al and Cu-Mn: Formation and growth of slip band bundles: a) Cu-
10.7 at% Al; b) Cu-17 at% Mn, and formation and propagation of a Lüders band front: c) Cu-
15 at% Al; d) Cu-17 at% Mn.
113

114
6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
Figure 6.7: Schematic representation (crystal cross section along its axis) indicating the shear distribution
in the Lüders band front which propagates with velocity vLB, and in slip band bundles
(cf. Figure 6.6 a,b). The slip bands are initiated in the Lüders band at surface “high”, at the edge
of the slip band bundle at its right on surface “high”, at its left on surface “low”, according to the
bending stresses and the stress patterns of Figure 6.5.
a)
b)
Figure 6.8: Determination of Lüders band propagation rates v LB from plots of cumulated distances
x F and times t F of the front slip bands of Lüders bands at various external deformation
rates of _ l=2, 4, and 10 lm/s (selected below the critical value) for Cu-10.7 at% Al (a) and Cu-
12 at% Al (b).

a)
6.3 Deformation Processes around Room Temperature
b)
Figure 6.9: a) FEM analysis of the stress pattern around the Lüders band front; b) Plot of the resolved
shear stress near the surface across the Lüders band front from the sheared (left) to the virgin
part (right), for different radii of curvature (R) in the Lüders band region (cf. [23]). The increased
stress at the left, mainly due to the reduced cross section, is compensated by work hardening
(kinematic stress).
tion, which accompany the local shear of the single crystal (with slip planes inclined by 458
to the crystal axis). This stress concentration then helps to propagate the Lüders band constriction
along the crystal. The Finite-Element Method (FEM) analysis of stresses (Figure
6.9) shows a stress maximum at the tail and a minimum at the front of the Lüders band; the
latter explains the large gaps between the front slip bands and indicate the necessity of local
stress concentrations from neighbouring slip bands (Figure 6.5) to initiate the next new one
ahead of the Lüders band front. In a recent approximate treatment, Brechet et al. [28] have
described such transitions between homogeneous slip, bundled slip and propagating deformation
fronts in quite general terms reflecting many of the above observations.
Macroscopically, the existence of stress concentrations is realized in the yield
points observed during first loading of the specimen. In fact, calculating the propagation
stress from the external load by using the specimen cross section at the most active
part in the Lüders band region, we arrive at the same stress as that is observed at
the yield point calculated from load and original cross section (cf. [23, 24]). This indicates
that for these alloys the initial yield point is of purely geometrical origin (cf. [29];
the yield points due to strain ageing are smaller and will be discussed below).
115

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
6.3.3 Comparison of single crystals and polycrystals
An important further stage of the investigations concerns the possibility to transfer the
single crystal results to the case of polycrystals. As a first step in thin flat specimens of
Cu-5, 10 and 15 at% Al with grain sizes around 200 lm, the slip bands have been observed
during several steps of tensile deformation [30] recorded by video and examined
in detail after the deformation steps in the light and electron microscope. In exceedingly
large grains, often fronts of slip bands propagate similar to slip band bundles or
Lüders bands in single crystals. In exceedingly small grains, slip activity is often retarded
due to stresses from neighbouring grains. In the average sized grains, several
(mostly 2 to 3) slip planes are activated, often one after the other and different ones in
different parts of the grain (Figure 6.10). This reflects the local influence of compatibility
stresses exerted by the neighbouring grain. It also explains why not all, but most
slip systems are activated according to the magnitude of the Schmid factor. In the Cu-
Al alloys, the plastic relaxation near grain boundaries occurs frequently, in spite of the
low stacking-fault energy, by cross-slipping of primary dislocations [12, 30]. This appears
to be easier than to activate new sources on secondary systems. It is important in
particular that the kinetics of single slip bands in polycrystals appear to be quite similar
Figure 6.10: Video records of slip line formation in single grains of a polycrystalline thin flat Cu-
10 at% Al specimen shown at three stages of deformation (e=0.5, 1.5 and 8%) at room temperature.
The numbers in the scheme indicate the succession of activated slip planes.
116

to that in single crystals as shown in Table 6.1 for the average total times of activity of
single slip bands. Contrary to single crystals, in the investigated polycrystals, no Lüders
bands were observed to propagate; according to experience in the literature [31],
the grain sizes for that have to be chosen much smaller.
A pilot experiment was performed in cooperation with Harder [32, 33] and Bergmann
[34] on a thin flat specimen of Cu-5 at% Al containing 3 grains of different known
orientations. The observations of slip band activity correspond well with the measurements
of local strains by the multigrid method and with the FEM calculations [35].
6.3.4 Conclusion
6.3 Deformation Processes around Room Temperature
Table 6.1: Comparison of average times of formation of slip bands on single crystals (plate
shaped, thickness 0.18 mm) and in grains of polycrystals (plate shaped, thickness 0.4 mm, grain
size 0.2 mm) for the Cu-Al alloys with 5, 10 and 15 at% Al. For the single crystals, the range of
observed values is given in parenthesis.
t B (s) Single Crystals
(thickness 0.18 mm)
Cu-5 at% Al 7.8
(3...15)
Cu-10 at% Al 0.15
(0.1...0.2)
Cu-15 at% Al 0.04
(0.02...0.06)
Polycrystals
(grain size 0.2 mm)
In single and polycrystals of the considered Cu-Al and Cu-Mn alloys, deformation proceeds
by production and movement of groups of strongly correlated dislocations across
slip zones. This strong correlation and the destruction of short-range order lead to localized
deformation and micro-instabilities of slip. Owing to the variation of the slip kinetics
during the activity of each slip band, a description of the overall kinetics by the
nucleation rates of slip bands (Equation (7)) including local work hardening (i.e. kinematic
stress) appears appropriate. Thus, the flow units used in [36] consist of such spatially
and temporarily correlated dislocations in groups. Their local stress concentrations
are important in the propagation of slip along the crystal. Details of the mechanisms
and kinetics of dislocation multiplication inside the slip bands still remain to be
explored. The first steps done to study the influence of surrounding grains on the activity
of a considered grain in a polycrystal should be extended, in particular by combining
them with FEM analyses of the local stresses.
12.7
0.73
0.05
117

6.4 Deformation Processes at Intermediate Temperatures
The range of “intermediate” temperatures is characterized by an increasing mobility of the
solute atoms in the alloy, in particular in the neighbourhood of dislocations. Although first
atomic site changes seem to occur in the dislocation core region already at temperatures
well below room temperature, as evidenced by strain ageing effects during and after stress
relaxation experiments [37], well pronounced influences of solute mobility are observed at
temperatures exceeding room temperature (the lower, the higher the solute concentration,
cf. [6]). In a certain range of temperature and external strain rate, dynamic strain ageing
leads to repeated rapid local slip events even observable as serrations in the load-time
curve in ordinary deformation experiments, the well-documented Portevin-Le Châtelier
(PLC) effect (e.g. [31]). Supplementing the research in [6], where most investigations
are performed in the range preceding this instability region, the present study concentrates
on the evolution of such plastic instabilities. Their temperature region for Cu-Al
single crystals oriented for single glide and deformed in stage I is indicated in Figure
6.1 a by the dotted lines. Figure 6.1 b shows that it nearly coincides with the range
of negative strain rate sensitivity if this is determined from the back-extrapolated stress
course during strain rate changes [38, 39]. For the more general behaviour and ranges
of existence of the PLC effect during work hardening for various crystal orientations
and polycrystals, cf. [6, 40].
6.4.1 Analysis of single stress serrations
Applying an especially rapid data acquisition system to record the load (or stress) simultaneously
with slip line recording by video, the course of PLC load drops has been
directly correlated to the formation of new slip bands at the crystal surface [38, 41].
Figure 6.11 a shows a series of several selected frames taken during the stress serration
given in Figure 6.11 b. Thus, in this range of temperature, one macroscopic instability
event involves the rapid formation of a whole cluster of new slip bands. Obviously,
after breakaway of a first source dislocation from its solute cloud, rapid dislocation
multiplication occurs, where dislocations move fast enough to develop only minor solute
clouds implying high dislocation mobility. The slip transfer mechanism of Section
6.3.2 (Figure 6.5) with local stresses in the surface region rapidly produces a series
of neighbouring slip bands (i.e. a slip band bundle) at a rate higher than that necessary
to comply with the deformation rate imposed by the tensile machine. Therefore, the
load decreases and thus the production rate and dislocation velocity, too. This in turn
permits the solute cloud to grow further and to slow down the dislocation until it stops
suddenly and the specimen is again elasticly reloaded up to the next breakaway event.
The quantitative formulation of this behaviour [38] permits to estimate the change in
the effective enthalpy dDG due to ageing:
118
6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
DG ˆ DG0 ‡ dDG ˆ DG0 ‡ Kf …tw† …8†

a)
6.4 Deformation Processes at Intermediate Temperatures
b)
Figure 6.11: Sequence of frames: a) with slip bands originating during a stress drop (b)), in the PLC
regime (T=500 K) of a Cu-10 at% Al crystal deformed in stage I with a rate of _l ˆ 2 10 3 mm/s.
119

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
in the waiting times for thermal activations
tw ˆ tw0 exp …DG=kT† ; …9†
where K and t w0 are constants and the function f describes the ageing kinetics. We find
dDG 0.16 eV during the stress drop and &0.14 eV during reloading, i.e. & 0.3 eV
in total for Cu-10 at% Al at 580 K [39], which compares quite well with the values determined
from different experiments and different arguments [42, 43]. This change
amounts to roughly 10% of the total effective activation enthalpy DG0 in this temperature
range of stress serrations.
The breakaway stress rises with temperature due to an increasing solute cloud, up to
a stress maximum s0…TM† ˆs0M, which occurs at lower T M for higher solute concentrations
c (Figure 6.1 a, in more detail Figure 6.12a). Beyond the crss maximum, no serrations
occur and slip bands can no longer be detected: Slip becomes virtually homogeneous
for T > TM (cf. Section 6.5). The correlation of s0M with solute concentration (Figure
6.12b) agrees quite well with the classical formula proposed by Friedel [44]:
s0M ˆ A…W 2 m c=kTMb 3 † …10†
for the boundary between dislocation breakaway from the (unsaturated!) solute cloud
(T TM†, which by
rapid diffusion reforms fast enough to be “dragged along” with the moving dislocation.
This relation permits to estimate the mean binding enthalpy W m of solute atoms to the
dislocation, i.e. for c=2...15 at% Al: Wm 0.12 eV taking the structure factor A=0.1
as determined for Cu-Mn alloys by Endo et al. [45]. These Wm values compare well
with earlier results from internal friction [46] and from theoretical estimates [47].
A summary of the temperature dependence of the correlation of serrations (load fluctuations)
with slip activity is given in Figure 6.13, where, on the right, the mean stress
drop amplitude Ds is plotted, while, on the left, the magnitude of simultaneously active
slip band bundles, n aB, is given as determined from the video records according to:
naB ˆ _l= _NbSsb ; …11†
where _l=external deformation rate, _Nb =formation rate of new slip bands in one recorded
active slip band bundle, S sb=average slip step height (normal to the crystal surface),
which does not change noticeably with temperature from room temperature (cf. Section
6.3) up to the PLC range. It is evident that at low T, where n aB is high, the fluctuations
in this number average out well so that a smooth load trace results. However, when
naB becomes small (1 to 10), fluctuations in the load trace are resolved, and they turn into
serrations when n aB formally falls below 1, i.e. when only one slip band bundle is active
for a short time with intervals of elastic reloading until breakaway of the next event.
120

6.4 Deformation Processes at Intermediate Temperatures
a)
b)
Figure 6.12: a) High temperature part of the temperature dependence of the crss s0) (cf. Figure
6.1 a) around its maximum, measured for various Al concentrations (2...15 at%) for Cu-Al single
crystals oriented for single slip, at a deformation rate of _l=1.7 · 10 –3 mm/s (crystal length
l 0 =100 mm) (* Cu-15 at% Al, * Cu-10 at% Al, & Cu-7.5 at% Al, n Cu-5 at% Al, ~ Cu-
3.5 at% Al, s Cu-2 at% Al); b) plot of the maximum stress s 0M=s 0 (T=T M) at the temperatures
TM of the crss maxima versus alloy concentration to check Equation (10) by Friedel [44] for the
transition between dislocation breakaway from and dragging along of the solute cloud.
6.4.2 Analysis of stress-time series
The recorded time series of load (or stress) in the range of plastic instabilities were analysed
by several methods with respect to deterministic chaos or randomness and under
the influence of measurement noise. Different methods proposed in literature for such
dynamic time series analyses have been compared [48] such as reconstruction in phase
space and correlation integral [49, 50], determination of Eigenvalues [51], and determination
of Lyapunov exponents and K 2 entropy [52, 53]. The problems with finding optimum
embedding parameters have been studied relativating first attempts to detect the
existence of chaos in jerky flow [54]. More successful appears a space-time analysis
121

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
Figure 6.13: Correlation of the temperature dependence of the number of active slip band bundles
n aB (Equation (11)) and the average height of stress serrations Ds for Cu-10 at% Al crystals deformed
with a rate of _ l=1.7 · 10 –3 mm/s. Below the shape of the load-time curves in indicated.
Note the abrupt disappearance of serrations at T M.
[48], which permits to take into account temporal correlations of the correlation integral,
such as in case of quasi-periodic behaviour, after checking the autocorrelation
function (for determination of a proper cut-off) and the power spectrum (for detecting
periodicities).
In evaluations of stress-time series, special care must be taken in case of changes of
the specimen structure during deformation as common in deformation due to work hardening.
This is shown in the examples of Figure 6.14 [48] for polycrystals deformed in the
PLC regime at different temperatures and for a single crystal oriented for single glide, both
for Cu-10 at% Al. For single crystals of Cu-5...15 at%Alandforpolycrystals (Cu-15 at%
Al), the PLC instabilities are of statistic rather than chaotic (deterministic) nature supporting
the recent theoretical treatment by Hähner [55]. For polycrystals, in certain ranges of
deformation conditions at least some deterministic contributions are identified, which are
periodic and seem to correspond to the propagation of the various types of PLC bands. The
long period “type A” serrations (at T=1008C in Figure 6.14) is superposed by a short
period at higher temperature (“type B” at T=150 8C in Figure 6.14), while the single crystal
does not show any periodicity, but indicates a change of structure from stage I to stage II.
While, according to McCormick [56], the type A serrations are associated with a continuous
propagation of plastic PLC deformation bands, type B corresponds to discontinuous
propagation of bands, and during type C, serrations at still higher temperature with spatially
122

6.4 Deformation Processes at Intermediate Temperatures
Figure 6.14: Sequences of stress-time series measured in the PLC region of polycrystals (T= 100 8C,
T=150 8C) and a single crystal (single glide, transition from stage I to stage II, T=300 8C).
uncorrelated local deformation bands occur. Accordingly, for types A and B from an analysis
of the time series characteristic parameters of the deformation bands (band width, local
plastic shear and shear rate in the band) and of their propagation rate v B can be evaluated
[48]. Figure 6.15 gives examples of the latter quantity for type A and B bands in Cu-15 at%
Al polycrystrals, which show different dependences on total strain e (Figure 6.15a) and _e
(Figure 6.15 b). The model of Jeanclaude and Fressengeas [57] would predict a decrease of
v B with increasing e if spatial coupling of local deformations occurs by cross-slip, while an
increase would indicate spatial coupling by internal stresses [58] (cf. Figure 6.5). The observed
dependence in Figure 6.15 a then would mean a change of cross-slip transfer to
internal stress transfer with increasing temperature, which does not seem quite reasonable.
Further investigations appear necessary and are under way for clarification.
a) b)
Figure 6.15: a) Propagation rates vB of PLC deformation bands evaluated from time series like
those in Figure 6.14 for the serrations of type A (T=100 8C) and type B (T=150 8C) for various
total strains e at a strain rate of _e =1 ·10 –4 s –1 ; b) strain dependence of type B propagation rates
(T=150 8C) for variations of external strain rates _e ˆ _l=l.
123

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
6.4.3 Conclusion
The strain ageing process forming solute clouds around the dislocations leads to macroscopically
pronounced plastic instabilities in a certain range of deformation conditions,
which are again intimately connected with strain localization. Here, the reason is the
breakaway of a dislocation from its solute cloud and subsequent rapid multiplication of
less aged dislocation groups. Thus, the overall kinetics (neglecting the serrations) can
be again described in the nucleation rate approach for aged dislocations, where the kinetics
of ageing enters the rate equations [40, 55, 56, 58]. The evolution of each single
stress instability event can be described in such an approach, too, while the details of
the dislocation multiplication and in particular the role of cross-slip processes in the
slip transfer from the active into the bordering region still remains to be clarified.
6.5 Deformation Processes at Elevated Temperatures
6.5.1 Dynamical testing and stress relaxation
As indicated above in connection with Figure 6.12, for T >TM, the deformation occurs
in a nearly ideal homogeneous manner. This was checked by EM slip line replica and
TEM: No traces of slip could be detected on the crystal surface, and TEM does not
show dislocation groups, but randomly distributed heavily jogged dislocations indicating
easy cross-slip of screw and climb of edge dislocations. Therefore, no slip line observations
are possible. In this range, viscous glide behaviour of dislocations can be assumed,
and the classical Orowan equation (Equation (4)) _a ˆ R mbv with a definite dislocation
velocity v and mobile dislocation density R m appears realistic. According to
the analysis from Figure 6.12 b, these dislocations carry along their (unsaturated) solute
cloud, which now decreases with increasing temperature for entropy reasons. Thus, the
alloying effect diminishes with increasing temperature as seen in Figure 6.1 a and Figure
6.12 a. The observation of a smaller yield stress for higher alloy concentrations (for
c>5 at%) at T >T M, which looks surprising at first sight, can be explained by the wellknown
increase of the diffusion constant D(c) with solute concentration c [59] in the
treatment of Friedel [44]: The relation between strain rate _a and applied stress s is
_a ˆ 2R m…b=k†D sinh …sb 2 k=kT† 2R mb 3 Ds=kT ; …12†
where kˆbcM ˆbc exp …Wm=kT† is the distance of pinning solute atoms along the dislocation.
This relation also describes well the observed strain-rate sensitivity in stage I
for T >TM (Figure 6.16c, where different c values are plotted), which agrees well if determined
from either stress relaxations (Figure 6.16a) or from strain-rate changes (Figure
6.16 b), where the initial stress jumps (constant structure) are evaluated.
The course of stress relaxations in this temperature regime can be well described
by a viscous dislocation velocity [60]:
124

c)
a)
b)
6.5 Deformation Processes at Elevated Temperatures
Figure 6.16: Strain-rate sensitivities (cf. Figure 6.1 b) in the range of elevated temperatures for
single crystals of Cu-Al with different Al concentrations measured from stress relaxations (a) and
from strain rate changes (b) taking the “initial” stress changes of the transients (i.e. without structural
changes) (_e2 ˆ 5_e1; _e1 =1.7 · 10 –5 s –1 ; symbols as in Figure 6.12 a); c) plot of the strain-rate
sensitivity for T>T M (T M=temperature at the crss maxima in Figure 6.12 a) versus 1/T to check
Equation (12), for initial and stationary (i.e. back extrapolated) stress changes, using all data with
different alloy concentrations ≥5 at%.
125

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
v s ; …13†
and either by a stress-dependent mobile dislocation density:
R m s n ; …14†
or by a Gaussian spectrum of free activation enthalpies. For the first approach, the observed
n values correspond well with the m=n+1 values (=3.6 ...3.8±0.5 for Cu-
3.5...10 at% Al) usually found from creep experiments for this type of alloys [61]. For
the latter approach, the temperature dependence of the average characteristic relaxation
times changes abruptly at the temperature of the crss maximum, indicating again the
change of rate-controlling mechanism, i.e. breakaway of dislocations from their solute
cloud for TT M.
6.5.2 Creep experiments
In order to check by more direct measurements and evaluations creep data for T>T M,
additional creep tests have been performed in the special creep set-up described in Section
6.2. Figure 6.17 shows some typical creep curves in the plot of strain rate versus
strain: a) at a fixed stress for various temperatures, and b) at a fixed temperature for various
applied stresses for polycrystalline Cu-10 at% Al. After a rapid decrease, the strainrate
approaches stationarity (the following rapid increase of _e ˆ _l=l is due to specimen
constriction and should be disregarded). In Figure 6.17b for sufficiently low stresses
a) b)
Figure 6.17: Creep tests on Cu-10 at% Al polycrystals, in plots of strain rate _e versus strain e: a)
performed at constant stress r/G (G=shear modulus) for various temperatures, and b) at constant
temperature T/T m (T m=melting temperature) for various stresses. Note the oscillating strain rate at
low stresses in b).
126

a)
6.5 Deformation Processes at Elevated Temperatures
Figure 6.18: a) Critical strains for the onset of dynamic recrystallization (DRX), determined from
its first appearance (*) and from the distance of strain rate maxima (*); b) examples for initiating
dynamic recrystallization by a change of stress to lower values during creep tests.
(or strain rates), creep occurs with an oscillating strain rate showing the characteristics
known for dynamic recrystallization (e.g. [62]). For instance, the critical strain for the onset
of dynamic recrystallization increases in proportion to the applied stress (Figure
6.18 a). The dynamic recrystallization can be induced by a rapid change to a lower
stress in the critical range (Figure 6.18b). This seems to be accompanied by changes of
the dislocation structure, which are to be studied in more detail to obtain more information
on the nature of the recovery processes in this temperature range T>T M.
The stationary creep rate, approximated by the minimum rate _emin in Figure 6.17,
varies with stress and temperature (Figure 6.19) according to
a) b)
Figure 6.19: Plots of the stationary creep rate (cf. Figure 6.16) versus stress (a) and temperature
(b) to determine the parameters in Equation (15).
127
b)

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys
_emin r 5 exp … Q=kT† …15†
with Q&2 eV, which approximates the activation for diffusion of solutes in the alloy or
for self diffusion. The stress exponent (m&5) is slightly higher than that quoted above
from stress relaxations, which has been clarified in [63, 64].
The first rapidly decreasing part of the creep curve contains information on dislocation
multiplication. At very low stresses, this part of primary creep may even show
increasing strain rate for some time. Observed differences to creep tests in conventional
creep machines [65] can be traced back to the different kinetics of loading. These processes
will be explored further by rapid changes from strain rate to stress-controlled
conditions at different levels of stress (or strain) (cf. [63, 64]).
6.5.3 Conclusion
In the temperature region T >T M, diffusion processes are dominant. The deformation kinetics
can be well described by the viscous glide approach with the dislocation velocity
governed by dragging of solute clouds and a stress-dependent mobile dislocation density.
This is the result of dislocation multiplication and simultaneous intensive recovery
processes, where dislocation climb and cross-slip are important similar to pure metals
[66, 67]. The details of these processes in solid solutions have to be further clarified.
Acknowledgements
This work was possible through the engagement and essential contributions of my coworkers,
C. Engelke, A. Hampel, A. Nortmann, J. Plessing, in their dissertation works,
and Ch. Achmus, U. Hoffmann, T. Kammler, M. Kügler, H. Rehfeld, S. Riedig, M.
Schülke, H. Voss, G. Wenzel, A. Ziegenbein, in their diploma works.
In addition, I acknowledge gratefully the continuous discussions and cooperation
with Prof. Dr. Ch. Schwink, and the cooperation in SRO measurements with Prof. Dr.
O. Schärpf (ILL Grenoble) and Dr. R. Caudron (LLB Saclay) by neutron scattering,
and with Prof. Dr. G. Kostorz and Dr. B. Schönfeld (ETH Zürich) by X-ray scattering
(with financial support of the Volkswagenstiftung). In particular, the financial support
of our work by the Deutsche Forschungsgemeinschaft in the Collaborative Research
Centre (Sonderforschungsbereich, SFB 319-A9) is gratefully acknowledged.
128

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130

7 The Influence of Large Torsional Prestrain
on the Texture Development
and Yield Surfaces of Polycrystals
Dieter Besdo and Norbert Wellerdick-Wojtasik*
7.1 Introduction
The simulation of forming processes applying the Finite-Element method is more and
more in use today. If the results are close to reality, the simulation can save costs involved
in the forming of testing tools and shorten the development stage of new products.
But this aim can only be achieved if the model of the forming process is physically
plausible. The treatment of contact problems and the modelling of the material behaviour,
e.g., present many problems. The material properties of the anisotropy caused
or at least modified by the forming process in particular are problematic.
In classical continuum mechanics, the material behaviour is described by phenomenological
laws; the inner structure of the material is not considered in detail. Today,
the available CPU’s have reached a performance level that allows us to take the microscopic
behaviour into account as in texture analysis (see Figure 7.1). Thus, it seems
possible to develop constitutive laws based on an improved physical basis and to use
them in Finite-Element calculations.
7.2 The Model of Microscopic Structures
7.2.1 The scale of observation
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
In papers on texture analysis and on theories of polycrystals, the expressions ‘microscopic’
and ‘macroscopic’are often used. It is thus necessary to define the scale of observation.
The resolving power of the microscopic observer is usefully described by the
* Universität Hannover, Institut für Mechanik, Appelstraße 11, D-30167 Hannover, Germany
131

7 The Influence of Large Torsional Prestrain on the Texture Development
Figure 7.1: View of material structure in continuum mechanics and in texture analysis.
following definition. The observer knows the physical phenomenon and mechanism of
slipping, but he is not able to locate the area of slipping in the grain. Thus, if slipping
occurs, he is forced to treat it as a homogeneous in the grain-distributed action. This
also means that all points of the grain are describable by only one stress tensor or velocity
gradient.
The expressions ‘macroscopic’, ‘global’ or ‘polycrystal’ are not related to a deformed
body of a special form; they refer to a volume of many crystals. This volume is
often called a control volume or representative volume, which is large compared with
the microscopic scale. Although it consists of many crystals, it is small in contrast to
any deformed body. Thus, a deformed specimen consists of many representative volumes.
To start calculations in the interior of the representative volume, one is forced to
have some state quantities of the macroscopic scale as well as of the microscopic scale.
The macroscopic information could be a velocity gradient, for example.
7.2.2 Basic slip mechanism in single crystals
The plastic deformation of a single crystal is assumed to be caused only by slipping in
certain slip systems. A slip system consists of a slip direction and a slip plane. The
planes and directions are determined by the structure of the crystal. In face-centred cubic
crystals, e.g., the primary slip systems are formed by the {111} planes and the
h110i directions. Plastic deformation by slipping of a system is only possible if the
shear stress s on the slip system exceeds a critical value s c. The deformation gradient,
F and the velocity gradient L, relative to a lattice fixed frame, are then given by:
F ˆ I ‡ c …s m T † and L ˆ _c …s m T † ; …1†
where s and m are the orthogonal lattice vectors of the slip direction and the slip
plane. The magnitude of shear in the active slip system is called c . The equations
132

above are only valid if single slip occurs, but generally more than one slip system will
be operating simultaneously. The appropriate equations for multislip follow for the velocity
gradient by superposition of single slips:
L ˆ X _c …s m T † : …2†
Nevertheless an analogue treatment of the deformation gradient F is not valid. Furthermore,
when the elastic distortion of the lattice is considered as well, the expressions become
more complicated because the distorted lattice vectors must be used for an appropriate
formulation (see e.g. Havner [1]).
7.2.3 Treatment of polycrystals
7.2 The Model of Microscopic Structures
The main problem of modelling crystal structures is not the formulation for the single
crystal. It is more difficult to find a suitable averaging method to obtain the properties
of the polycrystal. The interactions of the crystals at their grain boundaries during their
deformation are so complex that there are still some simplifications necessary to make
the problem mathematically treatable.
Several texture models have been developed to deal with this problem. Some of
them ignore the grain interactions, while others try to consider them in different ways.
The first and basic models are those of Sachs [2] and Taylor [3, 4]. Simulations based
on the Taylor model show better results compared with textures measured in experiments.
It is therefore till now often the basis of texture simulations. Generally, most
methods differ from each other in terms of whether the homogeneity of deformations
or the homogeneity of stresses are partly or completely satisfied. A comprehensive
overview of modelling plastic deformation of polycrystals is given in [5].
All texture models require some basic data of the microscopic scale. Usually, at
least the following specifications are considered:
• The polycrystal consists of N j crystallites with equal volume. No restrictions about
the grain form are made.
• The orientation of each crystal is given by the Eulerian angles u 1, y and u 2. No information
about the arrangement of the crystals in the polycrystal is available.
• The elastic constants of the single crystals are given as well as the slip systems including
their critical shear stresses. The assumption of the known critical shear
stresses presents a problem in practice.
7.2.4 The Taylor theory in an appropriate version
The Taylor model, often called Full-Constrained model, is the most often used texture
model. The fundamental assumption of the model is that in each crystal, five slip sys-
133

7 The Influence of Large Torsional Prestrain on the Texture Development
tems are activated in a way such that the microscopic velocity gradient for the incompressible
flow is identical with the global one:
grad V ˆ L ˆ A T j
� )
X 5
_c …s m T †
ˆ1
|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}
microscopic
Aj ‡ Xj with Xj ˆ A _ T j Aj : …3†
The expression Xj ˆ A _ T j Aj given by the transformation tensor A and its time derivative
A _ is most important. This is the lattice spin of the crystal. To solve the problem,
the equation is decomposed in symmetric and antimetric parts, D and W, respectively.
Thus, if the global velocity gradient is given, the symmetric part can be solved as a set
of linear equations. This leads to 384 possible solutions in the case of 12 slip systems.
The correct solution is the one, which minimizes the internal power of the grain. Especially,
if the critical shear stresses are equal on all slip systems, this selection criterion
is not unique, and all combinations of five slip systems that lead to internal power inside
a tolerance limit are supposed to be active. As more than five slip systems operate
simultaneously, this is an extension of the Taylor theory. Another way to solve the ambiguous
problem is to vary the initial critical shear stresses with a random generator.
This solution is not a restriction of the model; in some cases, it might improve the
quality of the texture models.
If the magnitude of shear is known for all slip systems, the lattice spin is given
by:
Xj ˆ W
X12
T
A _c
ˆ1
ges …s m T m s T � )
† Aj ; …4†
hence, A _ j can be calculated. Integration of Equation (3) leads to the new orientation of
the lattice. Finally, the microscopic stresses can be calculated with the known slip systems.
A macroscopic stress tensor and also a mean spin tensor are obtained by averaging
the crystal data.
Normally the hardening of the crystal is considered by a law of the form:
_s c ˆ f X12
lˆ1
c l; c b; _u m
!
; b; l 2‰1; ...; 12Š ; …5†
which must be evaluated after each step of calculation. Some examples for suitable
hardening laws can be found in [6] and [1]. The calculations documented in [6] also
show that the hardening law strongly effects the stress response and has hardly any influence
on the texture development of the polycrystal.
134
|‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚}
macroscopic

7.3 Initial Orientation Distributions
For a practical comparison of measured and calculated textures, the initial orientations
of the crystals should be measured as single orientations of single grains or as non-discretized
orientation distribution function (ODF). In theoretically based works and research
projects, it is quite normal to start the calculation with an isotropic state. Therefore,
it is necessary to generate a distribution with initial global isotropic properties.
7.3.1 Criteria of isotropy
Before initial orientations can be used for numerical simulations, it is necessary to
check whether an initial isotropy is actually guaranteed and not only orthotropy. Many
criteria can be used to check this although not all of them are sufficient. In [7], e.g.,
the components of the average elastic stiffness tensor were regarded. But for small deviations
from the isotropy configuration, there can be remarkable deviations of the elastic
modulus for different directions of loading. In [6], the plastic isotropy is proved by
calculating the yield surfaces of the single crystals. If all these yield surfaces are regularly
distributed in the stress plane, the distribution is thought to be isotropic. This
approach considers only the first possible slip system, and if multislip occurs, the isotropy
might not be satisfied. It therefore seems to be best to introduce an isotropy test,
which checks the elastic properties as well as the plastic properties under consideration
of multislip.
A suitable test of the elastic isotropy is to calculate the average elastic stiffness tensor.
The method introduced by Hill [8], which leads to good estimations in the case of
randomly distributed crystals, seems to be the simplest and best method of approximation.
A quantity denoting the elastic anisotropy of an orientation distribution may be:
AE ˆ Emax Emin
Eh111i Eh100i 7.3 Initial Orientation Distributions
AE ˆ 0 ) elastic isotropy ;
AE ˆ 1 ) single crystal isotropy ;
where the maximum difference of the calculated average elastic modulus is related to
the corresponding data of the single crystal. Thus, the value is independent of the
constants of the single crystal and only the quality of the distribution is assessed. In the
case of ideal isotropy, the quantity A E will vanish. A helpful visualization is to draw
the elastic properties of different directions as a body of elastic moduli. Here, distributions
with lower quality, concerning isotropy, show remarkable deviations from the
ideal spherical form.
The Taylor model is an ideal tool to check the plastic properties under consideration
of multislip because at least five slip systems are active. The function
fAP…e† ˆ1 AP…e† with AP…e† ˆ jr1?…e† r2?…e†j
s c
…6†
…7†
135

7 The Influence of Large Torsional Prestrain on the Texture Development
can be used to judge the plastic properties. A P is the ratio of the differences of stresses
orthogonal to the tension direction to a mean value of the critical shear stress. In the
best case of isotropy, the function will reach fAP…e† ˆ18e, and the accompanying plot
will show a sphere.
7.3.2 Strategies for isotropic distributions
A special strategy is only required if the distribution should consist of as few crystals
as possible. But given a later implementation of such a texture-based constitutive law
in a Finite-Element program, this should always be the aim.
Several authors use random distributions generated with the help of a random
generator. Unfortunately, these distributions are only usuable, considering the isotropy,
if they consist of many orientations (>1000). Distributions created by a proper strategy
are generally better than distributions generated randomly when the number of orientations
is equal.
Several strategies are based on a discretization of the Euler space. Here, the space
built from the possible combinations of Euler angles is discretized. On account of the
crystal symmetry, it is not necessary to consider the entire Euler space; a small portion
is sufficient. For cubic crystal symmetry and orthorhombic symmetry of the specimen,
the relevant Euler space was given by Pospiech [9]. Unfortunately, this field has a nonlinear
boundary and therefore it is not easy to discretize it. Müller [6] and Harren [7]
use a corresponding discretization and obtain distributions of 32 ...128 and 385 orientations,
respectively. Isotropy is not satisfied in every case, but the distributions of Müller
[6] are better although they consist of fewer orientations.
Asaro and Needlemann [10] and Harren and Asaro [11] use a combination of specific
method and random distribution. The unit triangle of the stereographic projection
is used to fix one of the global axes. The attachment of the base in space is done with
an angle given by a random generator. Figure 7.2 shows the elastic properties calculated
with data given in [11].
The distributions show a noticeable anisotropy, which is assumed to be caused by
the random generator. The method was used again to check the plastic isotropy with
the result that all distributions obtained had better properties than the original ones.
Another method is given by Müller [6], who discretized the surface of a sphere to
obtain the positions of local basis vectors. This method leads to distributions of good
quality (see e.g. Figure 7.3), but it is always combined with the problem of the spherical
geometry.
This problem can be avoided if one takes the area of a circle for discretization and
obtains the points on the sphere by an equal area projection. A detailed description of the
method is given in [12]. The quality of the distributions naturally depends on the division
of the area and on the number of orientations, but for the same number of orientations, the
isotropy is better or at least comparable to that of the so-called Kugel distributions. Figure
7.4 shows the isotropy test of a distribution generated with this method. Although it
consists of only roughly one hundred orientations, the isotropy is nearly guaranteed.
136

7.4 Numerical Calculation of Yield Surfaces
Figure 7.2: Global elastic modulus body of some distributions given in [11].
Figure 7.3: Test of isotropy of the distribution kugel192 given in [6].
Figure 7.4: Test of isotropy of the distribution kr104 given in [12].
7.4 Numerical Calculation of Yield Surfaces
The numerical calculation of yield surfaces with data from orientation distributions can
be carried out in many different ways. But with regard to a comparison with experimental
data, research methods, which allow the consideration of the sequence of an experiment,
should be preferred. Generally, all methods are averaging methods, but the
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7 The Influence of Large Torsional Prestrain on the Texture Development
procedure of averaging and the basic assumptions vary. The methods can be categorized
as follows:
Static methods of averaging are based only on the Schmid law and no strain is
considered. Methods of this type are not suitable for a comparison with experimental
data, as the later are usually measured with an offset strain. In [6], a method is proposed
based on averaging the single crystal yield surfaces. This method can also consider
kinematic hardening when the surface lies outside the origin. Figure 7.5 shows
two yield surfaces on an initial distribution. The values of the stresses r XX, r YY and
s XY are related to a mean value of the critical shear stresses. This normalization is also
done in the figures below. In [13], this method is combined with some offset simulations.
When the offset is large, the resulting yield surfaces are similar. Another method,
called MHSSS (Most Highly Stressed Slip Systems), is proposed by Toth and Kovács
[14]. This method uses a double averaging, first in the grain and second for the polycrystal.
It is shown in [12] that the calculated yield stress is the harmonic mean of the
five lowest possible stresses causing yielding in different slip systems. The arithmetric
or geometric mean may be used in the same way.
The classical Taylor yield surfaces are based on statics as well. The yield surfaces
shown in Figure 7.6 have been calculated by applying 80 loading paths, which are
marked by the arrows. The best and fastest method for calculating the yield surface in
this manner was introduced by Bunge [15].
Stress-controlled methods are based on a global given stress tensor. The stress is
increased incrementally until the shear stress in one slip system exceeds the critical value.
It is then possible to calculate the amount of shearing that is needed for a static equilibrium
with the hardening law. If the critical shear stress is not reached during a step, the
deformation is assumed to be purely elastic. After all deformations of the crystals have
been obtained, the mean value of strain is calculated. The method continues until the offset
strain is reached. Unfortunately, only the stress is given and no information about the
global velocity gradient is supplied. Therefore, antimetric parts are hardly considered. But
for small deformations (e.g. for a simulation of the elastic-plastic transition), this method
may be suitable. Figure 7.7 shows initial yield surfaces of an initial kugel distribution. The
graph is due to an ideal calculation, and the symbols correspond to a calculation under
consideration of strain hardening, loading path effects and orientation alterations. The offset
strain used is 0.2%, which is a standard value in material testing. One may notice that
Figure 7.5: Initial yield surfaces calculated with the radial averaging method.
138

7.4 Numerical Calculation of Yield Surfaces
Figure 7.6: Initial yield surfaces calculated with the Taylor model.
Figure 7.7: Initial yield surfaces calculated with a stress-controlled method.
although a large offset is used, the resultant yield surfaces are smaller than the ones calculated
with the Taylor model. Therefore, the later ones are only valid for comparison with
yield surfaces measured when a large offset strain is used.
More examples and a detailed description of this method can be found in [12].
Strain-controlled methods are based on a given deformation or velocity gradient.
In a simple manner, the yield stress of the Taylor model is calculable with a strain path
like an experiment. When the components of the global velocity gradient are given for
the stress plane considered, e.g. in the form
Lij ˆ fXL X ij ‡ fYL Y ij with fX ˆ cos b and fY ˆ sin b ; …8†
it is possible to apply the loading path desired by choosing suitable values for the angle
b. LX ij and LYij are the tensor components of pure loading in X and Y direction, respectively.
When b changes, the equivalent strain rate changes, too; it is therefore necessary
to vary the time increment of the integration to achieve a constant step of
equivalent strain increment during each loading step. Thus, this method is suitable for
calculating yield surfaces as shown in Figure 7.8. Exactly 80 loading paths starting
with pure tension and then continuing counter clockwise round the stress plane are applied.
For comparison, the ideal Taylor yield surfaces are shown too. This model shows
the expected effect as the expanding of the yield surfaces caused by the loading path
under consideration of hardening. The hardening law used is the isotropic PAN law
with parameters proposed in [6] and the offset strain is 0.1%.
139

7 The Influence of Large Torsional Prestrain on the Texture Development
Figure 7.8: Initial yield surfaces calculated with a strain-controlled Taylor simulation.
When kinematic hardening is considered, the yield surfaces become distorted and
may not be closed.
The only disadvantage is that the classical Taylor theory starts with the full plastic
material state. Thus, the elastic-plastic transition is not taken into consideration. A better
method might be the Lin model [16], which similar to the Taylor model assumes
that all crystals have the same strain. Furthermore, nearly the same hardening law can
be used. This method is used in [6] for the calculation of offset-strain dependent yield
surfaces. The disadvantage is the small deformation area of application. Thus, it is not
useful for texture simulations. The problem is discussed further in [12], and it is shown
that the numerical evaluation can be simplified without restrictions.
7.5 Experimental Investigations
The aim of the experimental investigations was to measure yield surfaces of large prestrained
materials. The large deformation was achieved with a torsion-testing machine
at the Institut für Mechanik of the Universität Hannover. The measurement of the yield
surfaces was done with a testing machine at the Institut für Stahlbau of the Technische
Universität Braunschweig. The material of the specimens was always the aluminium alloy
AlMg 3.
7.5.1 Prestraining of the specimens
The prestraining of the specimens was achieved with a torsion-testing machine, further
described in [12]. In order to measure yield surfaces after the deformation, it was necessary
to twist thin walled tubular specimens. The final nominal length, inside diameter
and wall thickness of each specimen, were 60 mm, 24 mm and 2 mm, respectively. If
the accuracy of the manufactured specimen is high (e.g. by using a CNC-controlled
lathe), large deformations without buckling can be achieved. To prevent buckling and
140

to ensure that the cylindrical form of the specimens was maintained, a lubricated mandrel
was inserted inside the specimens. A maximal amount of shear of
c a ˆ tan w ˆ 1:5 could be reached with that configuration, where w describes the angle
of an axial direction on the surface of the specimen after the deformation. That means
a twist of about 360 degrees for the specimens. The elongation of the specimen was
not suppressed with the result that an elongation always occurred, which nearly depended
linearly from the twisting angle in agreement with the research done by Pöhlandt
[17] with specimens of aluminium. The maximal elongation was D` ˆ 1:25 mm.
Since the measurement of the yield surfaces was done in another apparatus, the specimens
were fully unloaded after the torsional deformation.
7.5.2 Yield-surface measurement
Four different material states have been investigated: specimens without any prestrain and
ones with c a=0.5, c a=1.0 and c a=1.5 magnitudes of shear. The testing apparatus was a
strain-controlled machine with the capability of combined tension-torsion loadings. The
yield point of the material was detected with the offset strain definition. In all tests, one
specimen was used for 16 loading paths, starting with pure tension and then continuing
counter clockwise round the r-s-plane. This was done for three reasons. First, this reduced
the costs of specimens. Second, it was not guaranteed that the prestrain is reproducible
and last, the multipath measurement data are needed for the comparison with the
theoretical models. These data are ideal to check whether the texture model including
the hardening law is able to describe the material behaviour during such a loading history.
The interpretation of the data measured is based on an additive decomposition of
the total strain increment in an elastic and a plastic part. If the total strain increment is
given by the testing machine, the plastic parts of it are given by:
Depl ˆ Deges
Dr
E
and Dc pl ˆ Dc ges
when the constants E and G are known. In determining the yield surface, the offset von
Mises equivalent plastic strain was computed using the equation:
De vM
pl ˆ De2 1
pl ‡
3 Dc2 r
pl
7.5 Experimental Investigations
Ds
G
…9†
: …10†
The yielding point was reached when the calculated plastic strain exceeded the given
offset strain:
X
vM
Depl eoff ; …11†
where offsets between 0.0015% and 0.1% have been used. It is essential for the offset
definition that the values of E and G are known with high accuracy. Otherwise, large
141

7 The Influence of Large Torsional Prestrain on the Texture Development
Figure 7.9: Calculated value of E in dependence on the number of used measuring points.
errors may be the result. If E is measured too large, the resultant yield stress will be
smaller than the real one. In an extreme case, yielding is supposed although the
material is still in the elastic state. On the other hand, if E is measured too small, the
resultant yield stress will be larger than the real one. The shear modulus G has an appropriate
influence. This possible error in determining the yield stress increases with
decreasing offset. Data obtained by using very small offsets should therefore be treated
with caution.
The determination of the elastic constants E and G is normally done with the first
measured points of a new loading path. The best way is to calculate the regression
coefficients. Although the regression coefficient in the elastic range should always be
constant, that is practically not the case. It always varies in a small range depending on
the number of considered measuring points as shown in Figure 7.9.
Thus, if another number of measuring points is selected for calculating the modulus
E, the resultant value E and as a consequence also the resultant yield stress will be
changed. This is an especially critical case for the modulus E; the shear modulus G
shows better relations.
7.5.3 Tensile test of a prestrained specimen
This test was done to investigate the appearance of the cross-effect. A cross-effect is
given when the maximum yield stress in the tensile component of stress is altered by
the strain hardening in torsion and vice versa. Normally, the cross-effect and related issues
are investigated by the measurement of yield surfaces when the plastic deformation
at most reaches the usually small offset strain. This tensile test was realized to investigate
the cross-effect on a larger scale.
Two tensile test specimens DIN 50125-B 10 ×50 have therefore been produced,
one of nearly isotropy material and the other of prestrained material. Hence, a cylindrical
specimen was twisted up to fracture, which occurs at a shear rate of c a=1.65. The
142

tensile specimen was produced from the broken rest as shown in Figure 7.10. An estimate
led to an amount of shear of c a=0.55 at the radius of the final test specimen, but
as a result of the processing, the two specimens were not distinguishable. The result of
the tensile test is shown as a diagram of force and elongation in Figure 7.10.
Additionally, some mechanical properties are given in Figure 7.10. As expected,
the prestrained material is more brittle compared with the other one. Furthermore, the
mechanical strength properties are greater than those of the unstrained specimen. Values
never reachable for the unstrained specimen were obtained. This shows that there is
a remarkable cross-effect.
7.5.4 Measured yield surfaces
7.5 Experimental Investigations
Figure 7.10: Tensile test of pre- and unstrained material.
Some measured yield surfaces are presented below. Detailed discussions and further investigations
about the cross-effect and the loading path are given in [12].
First, it is remarkable that 0.0015% was the smallest practicable offset-strain for
the unstrained specimens, while this value was too small for the prestrained specimens.
There were several runaways among the data measured and therefore the smallest offset
was chosen to 0.005%. A larger one of 0.05% was also chosen for comparison.
Other offsets were only used for unique specimens.
Figure 7.11 shows the measured yield surfaces of unstrained specimens with increasing
offset.
As expected, the yield surfaces have an elliptical form and for small values, the
axial ratio r/s is closely to the von Mises yield surface. This ratio, however, increases
with increasing offset as well. The data obtained from the largest offset used show an
expansion of the surface, which is surely caused by the specific loading path and the
large offset of 0.1% inducing significant plastic deformation.
143

7 The Influence of Large Torsional Prestrain on the Texture Development
Figure 7.11: Yield surfaces of unstrained specimens.
A comparison of yield surfaces of prestrained specimens is given in Figure 7.12.
There are remarkable concave areas, which seem to disappear for the more prestrained
specimens. This is causually connected with the loading path because in the
second and third quadrant, no such areas occur. On the other hand, this concave area is
due to the first measured point and, therefore, it is the first loading differing from the
torsional preloading. This might be an important fact. Furthermore, the surfaces show a
hardening with increasing degree of prestrain. There is a significant distortion, a flattening
of the portion of the surface opposite the loading direction, and a kinematic hardening
or a so-called Bauschinger effect occurs.
When the large offset is applied, the most remarkable characteristics disappear.
the yield surfaces shown in Figure 7.13 are ellipses slightly shifted in the loading direction.
Furthermore, a hardening with increasing degree of prestraining is noticeable. Exceptionally,
the yield surface of the unstrained specimen is measured with an alternate
loading path, which does not affect the shape strongly.
Figure 7.12: Yield surfaces of prestrained specimens (small offset strain).
144

7.5 Experimental Investigations
Figure 7.13: Yield surfaces of prestrained specimens (large offset strain).
Figures 7.12 and 7.13 show that it is often difficult to assign the characteristics
observed. One may ask if the effects are due to the prestraining or to the parameters of
the measurement. Especially, the loading path for each specimen could be the cause of
some effects. In order to investigate the influence of these parameters, some specimens
were applied to the measure procedure three times. First, the small offset was used;
then the larger one and finally again the small offset. The resultant properties of the unstrained
material are shown in Figure 7.14, where only the surfaces measured with the
small offset are presented. The data measured characterized by the * is of the third
measurement of this specimen. Thus, an influence of the loading path can be seen because
this yield surface is slightly shifted to the direction of the last loading of the previous
path with the large offset.
In fact, there is an influence of the loading path, which does not seem to be too
large because the form of the surface is not affected.
Surprisingly, the prestrained material shows a very different behaviour. In Figure
7.15, correspondent measurement of a specimen prestrained up to c a=0.5 is shown.
Figure 7.14: Influence of the loading path (unstrained material).
145

7 The Influence of Large Torsional Prestrain on the Texture Development
Figure 7.15: Influence of the loading path (prestrained material).
The first surface measured with the small offset shows the properties already detected
in Figure 7.12. The second has nearly an elliptical form. The third surface, measured
with the small offset, is a slightly shifted ellipse. Considering the previous yield
surface, the third one shows properties as expected, but compared with the first one,
there are hardly any common characteristics. Form, size and position have changed remarkably.
Thus, the prestrained specimens are very sensitive to further deformations
compared with the unstrained ones.
7.5.5 Discussion of the results
The measurement of yield surfaces is not problematic for unstrained specimens even
when small offset strains are used. The data are reproducible and if the offset is small,
there is only a small influence caused by the loading path.
On the other hand, the prestrained specimens were very sensitive. When the form
of the yield surface is not known, it is difficult to identify runaway data and to assign
the effects to parameters of the measure procedure. An amplification of these effects is
due to the problem of determination of the correct elastic modulus.
In comparison with similar investigations (e.g. in [18–24]) agreement as well as
different results can be found.
7.6 Conclusion
In several investigations, the models of polycrystals are based on the motivation that
these models lead to better results in simulating the distortion of yield surfaces. The
146

distortion and the resultant anisotropy are often assumed to be caused by the orientation
distribution of the single crystals in the material.
The results presented prove that especially prestrained material is very sensitive to
small deformation. This means that the principle form of the yield surface is strongly
sensitive to small plastic deformations. Since this small deformation hardly affects the
texture of the material, it must be assumed that the texture is not the real cause for the
distortion of the yield surface. Additional events and mechanisms must occur in the
material during any plastic deformation.
Further investigations on the numerical calculation of yield surfaces will be undertaken.
Especially the question concerning, which method leads to results similar to the
surfaces measured and what kind of microscopic hardening law is needed, will be considered.
The fact that almost all parameters of the hardening law must be identified by
the mechanical properties of the polycrystal is problematic. At least, one should be able
to identify all these parameters with standard methods in material testing. Otherwise, it
does not make sense to use microscopic-based material laws. The final aim is to do the
calculation first and then proceed in manufacturing. The other way of doing an experiment
first and then trying to reach the same results in simulation may be practicable
for research projects, but this is surely not senseful for practical applications.
Finally, a search for a texture model to describe small deformations as well as
large deformations and all this in an acceptable calculation time will be undertaken.
Then, an implementation in a Finite-Element program may be useful.
References
References
[1] K.S. Havner: Finite Plastic Deformation of Crystalline Solids. University Press, Cambridge,
1992.
[2] G. Sachs: Zur Ableitung einer Fließbedingung. Zeitschrift des Vereins deutscher Ingenieure
72 (1928) 734–736.
[3] G.I. Taylor: Plastic Strain in Metals. J. Inst. Metals 62 (1938) 307–323.
[4] G.I. Taylor: Analysis of Plastic Strain in Cubic Crystals. In: J.M. Lessels (Ed.): Stephen
Timoshenko 60th Anniversary Volume, 1938, pp. 307–323.
[5] E. Aernoudt, P. van Houtte, T. Leffers: Deformation and Textures of Metals at Large
Strains. In: H. Mughrabi (Ed.): Plastic Deformation and Fracture of Materials, Vol. 6 of
Materials Science and Technology: A Comprehensive Treatment (Vol.-Eds.: R. W. Cahn, P.
Haasen, E.J. Kramer), VCH, Weinheim, 1993, pp. 89–136.
[6] M. Müller: Plastische Anisotropie polykristalliner Materialien als Folge der Texturentwicklung.
VDI Fortschrittsberichte Reihe 11: Mechanik/Bruchmechanik, VDI-Verlag, Düsseldorf,
1993.
[7] S.V. Harren: The Finite Deformation of Rate-Dependent Polycrystals: I. A Self-Consistent
Framework. J. Mech. Phys. Solids 39 (1991) 345–360.
[8] R. Hill: The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc. London A 65
(1952) 349–354.
[9] J. Pospiech: Symmetry Analysis in the Space of Euler Angles. In: H.J. Bunge, C. Esling
(Eds.): Quantitative Texture Analysis, 1982.
147

7 The Influence of Large Torsional Prestrain on the Texture Development
[10] R. J. Asaro, A. Needlemann: Texture Development and Strain Hardening in Rate Dependent
Polycrystals. Acta. metall. 33 (1985) 923–953.
[11] S.V. Harren, R. J. Asaro: Nonuniform Deformations in Polycrystals and the Aspects of the
Validity of the Taylor Model. J. Mech. Phys. Solids 37 (1989) 191–232.
[12] N. Wellerdick-Wojtasik: Theoretische und experimentelle Untersuchungen zur Fließflächenentwicklung
bei großen Scherdeformationen. Dissertation Universität Hannover, 1997.
[13] D. Besdo, M. Müller: The Influence of Texture Development on the Plastic Behaviour of
Polycrystals. In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications.
IUTAM Symposium Hannover/Germany 1991, Springer-Verlag, Berlin, Heidelberg,
1992, pp. 135–144.
[14] L.S. Toth, I. Kovács: A New Method for Calculation of the Plastic Properties of Fibre Textures
Materials for the Case of Simultaneous Torsion and Extension. In: J.S. Kallend, G.
Gottstein (Eds.): Proc. 8th Int. Conf. on Textures of Materials ICOTOM, 1988.
[15] H.J. Bunge: Texture Analysis in Materials Science. Cuvillier, Göttingen, 1993.
[16] T. H. Lin: Analysis of Elastic and Plastic Strains of a Face-Centered Cubic Crystal. J.
Mech. Phys. Solids 5 (1957) 143–149.
[17] K. Pöhlandt: Beitrag zur Optimierung der Probengestalt und zur Auswertung des Torsionsversuches.
Dissertation TU Braunschweig, 1977.
[18] P. M. Nagdhi, F. Essenburg, W. Koff: An Experimental Study of Initial and Subsequent
Yield Surfaces in Plasticity. J. Appl. Mech. 25 (1958) 201–209.
[19] H.J. Ivey: Plastic Stress-Strain Relations and Yield Surfaces for Aluminium Alloys. J.
Mech. Engng. Sci. 3 (1961) 15–31.
[20] W. M. Mair, H.L.D. Pugh: Effect of Prestrain on Yield Surfaces in Copper. J. Mech.
Engng. Sci. 6 (1964) 150–163.
[21] J.F. Williams, N.L. Svensson: Effect of Torsional Prestrain of the Yield Locus of 1100-F
Aluminium. Journal of Strain Analysis 6 (1971) 263–272.
[22] A. Phillips, C. S. Liu, J.W. Justusson: An Experimental Investigation of Yield Surfaces at
Elevated Temperatures. Acta Mechanica 14 (1972) 119–146.
[23] P. Cayla, J.P. Cordebois: Experimental Studies of Yield Surfaces of Aluminium Alloy and
Low Carbon Steel under Complex Biaxial Loadings. Preprints of MECAMAT 92, International
Seminar on Multiaxial Plasticity, 1992, pp. 1–17.
[24] A.S. Khan, X. Wang: An Experimental Study on Subsequent Yield Surface after Finite
Shear Prestraining. Int. J. of Plasticity 9 (1993) 889–905.
148

8 Parameter Identification of Inelastic Deformation
Laws Analysing Inhomogeneous Stress-Strain States
Reiner Kreißig, Jochen Naumann, Ulrich Benedix, Petra Bormann,
Gerald Grewolls and Sven Kretzschmar*
8.1 Introduction
The rapid development of numerical mechanics has resulted in
• an increased need for the identification of material parameters,
• new procedures, developed to solve these problems.
A common property of material parameters consists in the fact that they could not be
measured directly.
The classical method of the determination of material parameters is to demand a
quite good agreement between measured data from properly chosen experiments and
comparative data taken from numerical analysis. This will be carried out by the optimization
of a least-squares functional.
Furtherly, the parameter identification based on experiments with inhomogeneous
stress-strain fields, the usage of global and local comparative quantities in the objective
function and optimization by deterministic methods will be described.
8.2 General Procedure
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
In addition to classical material-testing methods, current research is done to identify
material parameters of inelastic deformation laws by the experimental and theoretical
analysis of inhomogeneous strain and stress fields. A new method is the parameter
identification using the comparison of numerical results obtained by the Finite-Element
* Technische Universität Chemnitz, Institut für Mechanik, Straße der Nationen 62,
D-09009 Chemnitz, Germany
149

method with experimental data, for instance, with displacement fields measured by optical
techniques [1–7]. The papers [1–4] were realized within the Collaborative Research
Centre (Sonderforschungsbereich 319).
Unlike this Finite-Element algorithm based method, in this paper, another procedure
is presented to identify material parameters of inelastic deformation laws. The
principle consists in the experimental determination of the strain distributions in the ligament
of a notched bending specimen at several load steps and the numerical integration
of the deformation law at a certain number of points along the ligament with measured
strain increments as load. The actual material parameters can be found using the
global equilibrium of the stresses integrated along the ligament with the known external
loads. Besides also local quantities, for instance, the stresses in the grooves of the
notch could be compared. A detailed scheme of this procedure is shown in Figure 8.1.
Below constitutive equations, in the framework of the classical plasticity and
materials as sheet metals or metal plates are studied. The elastic properties should be
isotropic. Viscoplastic effects are neglected. An initial anisotropy, especially a planar
orthotropy is taken into account.
8.3 The Deformation Law of Inelastic Solids
As an example, the deformation law of classical plasticity theory with small strains as
used in the material subroutines of the integration algorithm (cf. Section 8.6.1) will be
considered. At the yield limit holds the yield condition:
F…r; h; p† ˆ0 : …1†
The linear elasticity law
_r ˆ E_e …2†
is valid for loads in the elastic domain
T qF
F < 0 or F ˆ 0 and _r
qr ˆ _eT T qF
E < 0 : …3†
qr
For loads into the plastic domain
T qF
F ˆ 0 and _r
qr ˆ _eT T qF
C > 0 ; …4†
qr
the deformation law becomes:
150
8 Parameter Identification of Inelastic Deformation Laws
_e ˆ _eel ‡ _epl ˆ E 1 _r ‡ _ k qF
qr
: …5†

8.2 The Deformation Law of Inelastic Solids
Figure 8.1: Scheme for the identification of material parameters by bending tests.
In this case, the inner variables are assumed to develop in accordance with:
_h ˆ _ kq…r; h; p† : …6†
The material stiffness matrix C…r; h; p† arises from Equation (5) by elimination of _ k
with the help of the consistency condition.
151

8.4 Bending of Rectangular Beams
8.4.1 Principle
8 Parameter Identification of Inelastic Deformation Laws
The simultaneous determination of the uniaxial stress-strain curves for tension and compression
by bending test is known since 1910 [8]. On the other hand, examples for application
were relatively rare publicated [9–11]. Here, this technique is applied to analyse
the material properties in the delivery state. Besides, it was developed to calculate
the initial yield surface (Figure 8.2).
8.4.2 Experimental technique
Test material for the own investigation was the stainless steel X6CrNiTi 18.10, which
exhibits a gradual transition from the elastic to the plastic behaviour. All specimens
were made from one and the same metal plate with a thickness of 6 mm.
It is assumed that the sheet metal has a planar orthotropy coming from the production
process. To determine these initially anisotropic material properties, two specimens
are prepared, the axes of which are parallel to the orthotropy directions (Figure 8.3).
Bending tests, being able to provide yield curves and the initial yield surface,
were carried out in a specially constructed four-point bending device positioned in a
conventional 100 kN material testing machine. The longitudinal forces in the specimens
can be neglected because the four loads are easily moveable in horizontal direction
with the increasing deformation (Figure 8.4).
Figure 8.2: General procedure for the investigation of initial material properties.
152

8.4 Bending of Rectangular Beams
Figure 8.3: Specimen geometry for bending beams.
Figure 8.4: Device for four-point-bending.
All tests were made at a velocity of 0.108 mm/min and at a maximum strain rate
of about 10 –5 s –1 .
The axial and lateral strains at the outer fibres of the bending beams were measured
with high-elongation strain gages up to maximum strains of 5%. In separate extensive
investigations, the measuring accuracy of strain gages was analysed at higher
strains using the Moiré technique as an experimental reference method. As result, equa-
153

8 Parameter Identification of Inelastic Deformation Laws
Figure 8.5: Experimental results in pure bending of a rectangular beam: a) bending moment; b)
strains at the outer fibres (C compression, T tension).
tions for the determination of strain in the elastic-plastic region were obtained, whereas
in the elastic region, the producer’s strain gage factor can be used [12, 13].
As an example for the primary experimental results, the measured curves for the
bending moment and the strains at the outer fibres are shown in Figure 8.5.
The high density of data and the smoothness of the curves are a good basis for
evaluation. A maximum strain level of about 0.5% is sufficient to determine initial
material properties and to study the elastic-plastic transition.
154

8.4.3 Evaluation
8.4.3.1 Determination of the yield curves
The following fundamental assumptions are made for the evaluation of elastic-plastic
pure bending tests:
• The cross-sectional areas remain plane.
• The stress state is uniaxial.
The trapezoid-shaped distortion of the cross-section is taken into account, but the deformations
should be only moderate so that the stress states can be assumed as uniaxial.
In pure bending, the distribution of the usual technical strain e ˆ…l l0†=l0 is a
linear function of the coordinate y at the deformed beam (Figure 8.6):
ex…y† ˆ jy j ˆ 1
h …eTx e C x † …7†
with j as the curvature of the neutral axis. Then, the bending stress curve is an image
of the uniaxial stress-strain relation of the material (Figure 8.6).
The equivalence between the bending stresses rx and the bending moment Mb
and the longitudinal force N ˆ 0, respectively, requires:
Mb…j† ˆ
N…j† ˆ
ZhC…j†
hT…j†
ZhC…j†
hT…j†
8.4 Bending of Rectangular Beams
rx…y†b…y†ydy ˆ 1
j 2
rx…y†b…y†dy ˆ 1
j
Ze T x …j†
e C x …j†
Ze T x …j†
e C x …j†
Figure 8.6: Strain and stress in a rectangular beam under pure bending.
rx…ex†b…ex†exdex ; …8†
rx…ex†b…ex†dex ˆ 0 : …9†
155

In Equations (8) and (9), rx…ex† represents the uniaxial yield curves for tension and
compression. The actual width b of the beam at the coordinate y can be transformed in
b…ex† ˆb0‰1 ‡ ez…ex†Š with b0 as the initial width of the rectangular cross-section.
Differentiation of Equations (8) and (9) with respect to the parameter j gives:
…1 ‡ e T z †eT x
de T x
dj rT x
…1 ‡ e C z †eC x
de C x
dj rC x
ˆ j
b0
2M ‡ j dM
dj
…1 ‡ e T z †deT x
dj rTx …1 ‡ e C z † deCx dj rCx and from there, the stresses rT x …eTx † and rC x …eCx † in the outer fibres:
r T;C
x
ˆ 1
b0h0
2M ‡ j dM
dj
de T;C
x
dj
h
h0
1
…1 ‡ e T;C
z †
; …10†
ˆ 0 ; …11†
depending on the curvature j. In Equation (12), eT x …j† and eCx …j† represent the corresponding
strains in the outer fibres of the bending beam.
In contrast to the former works [8–11], the last term in Equation (12) reflects the
trapezoid-shaped distortion of the cross-section and takes into account anisotropic, especially
orthotropic material behaviour. The quantity h=h0 in this term can be determined by:
h
h0
ˆ
j
e T x
Z hC
h T
e C x
dy
1 ‡ ey…y†
ˆ
Z eT
e C
e T x
e C x
…1 ‡ ex†‰1 ‡ ez…ex†Šdex
assuming incompressibility for the total strains.
In Equation (12), the denominator causes different yield curves for tension and
compression, respectively, whereas the numerator defines the general level of the stress.
These two influences are already seen in the primary experimental results of Figure 8.5.
As a first step in evaluation, the limit of the linear part in the curves of Figure 8.5
was estimated by a multiphase regression method. From these data, the elastic
constants in form of the Young’s modulus E and the Poisson’s ratio m can be easily determined
using linear elasticity. In a second step, the data curves were approximated by
cubic spline functions. After differentiating, the stresses can be calculated numerically
point by point from Equation (12).
In Figure 8.7, stress-strain relations following from the data of Figure 8.5 are shown.
In addition to Figure 8.5, the yield curves for the whole strain range up to 5% are
presented in Figure 8.8.
The initial yield locus curve is characterized by the uniaxial yield stresses and the
directions of de pl in these stress points (cf. Figure 8.2). The yield stresses in tension
and compression, respectively, are defined from the stress-strain relations assuming a
relatively small offset strain of 0.1‰. The plastic strains can be calculated from the
156
8 Parameter Identification of Inelastic Deformation Laws
…12†
…13†

8.4 Bending of Rectangular Beams
Figure 8.7: Yield curves determined from the data of Figure 8.5.
Figure 8.8: Yield curves up to a maximum strain of 5%.
157

Figure 8.9: Plastic strain e pl
measured total strains at the outer fibres in longitudinal and transverse direction using
the incompressibility of plastic deformation and the elastic deformation law. Then, the
direction of depl can be found from differentiating the curves epl y …epl x † at approximately
ˆ 0:1‰ (Figure 8.9).
e pl
x
8 Parameter Identification of Inelastic Deformation Laws
8.4.3.2 Determination of the initial yield-locus curve
From bending tests of straight specimens, the yield stresses and the directions of plastic
flow, i.e. the normal directions of the yield surface at measured yield stresses, for pure
tension and compression in x- and y-direction were determined.
If one assumes a quadratic yield function, the most general expression for principal
stress states is the quadratic form:
f …rx; ry† ˆh1r 2 x ‡ h2rxry ‡ h3r 2 y ‡ h4rx ‡ h5ry ‡ h6 ˆ 0 : …14†
One constant is free, hence h2 was set to be (–1) and h6 will be positive for:
f …rx; ry† ˆh1r 2 x rxry ‡ h3r 2 y ‡ h4rx ‡ h5ry h6 ˆ 0 : …15†
The first part of the objective function should minimize the squares of the yield function
at the measured yield stresses ^r:
158
y …epl x
† from the data of Figure 8.5.

y1 ˆ 1 Xh
i2 f …^rx ; ^ry † ! min : …16†
…q† …q† 2
q
Then, the optimality condition
8.4 Bending of Rectangular Beams
grad y 1 ˆ 0 …17†
gives a system of linear equations for the constants h1, h3, h4, h5 and h6:
‰^r 4 xŠ ‰^r2x^r2 yŠ ‰^r3xŠ ‰^r2x^ryŠ ‰ ^r 2 xŠ ‰^r 2 x^r2 yŠ ‰^r4yŠ ‰^rx^r2 yŠ ‰^r3yŠ ‰ ^r2 yŠ ‰^r 3 xŠ ‰^rx^r2 yŠ ‰^r2xŠ ‰^rx^ryŠ ‰ ^rxŠ
‰^r 2 x^ryŠ ‰^r3 yŠ ‰^rx^ryŠ ‰^r2 yŠ ‰ ^ryŠ
‰ ^r 2 xŠ ‰ ^r2 yŠ ‰ ^rxŠ
0
10
1
h1
B
C
B
CB
h3 C
B
CB
C
B
CB
h4 C
B
CB
C
@
A@
h5 A
‰ ^ryŠ ‰1Š h6
ˆ
‰^r 3 x^ryŠ ‰^rx^r 3 yŠ ‰^r 2 x^ryŠ ‰^rx^r 2 yŠ 0 1
B C
B C
B C
B C …18†
@ A
‰ ^rx^ryŠ
with ‰...Šˆ P
…...†. All stresses are normalized.
q
For yield stresses, all representing only pure tension and compression, the right
side will be zero. To overcome this, the normal directions n ˆ…nx; ny† were taken into
a second part of the objective function:
y2 ˆ 1 X
2
qF qF
^n x…q† ^n y…q† ! min : …19†
2 qr q
y…q† qrx…q† The optimality condition gives another system of linear equations for h1, h3, h4 and h5:
‰4^r 2 x^n2 yŠ ‰ 4^rx^ry^nx^nyŠ ‰2^rx^n 2 ‰ 4^rx^ry^nx^nyŠ ‰4^r
yŠ ‰ 2^rx^nz^nyŠ
2 y^n2 xŠ ‰ 2^ry^nx^nyŠ ‰2^ry^n 2
xŠ ‰2^rx^n 2
yŠ ‰ 2^ry^nx^nyŠ ‰^n 2
yŠ ‰ ^nx^nyŠ
‰ 2^rx^nx^nyŠ ‰2^ry^n 2
xŠ ‰ ^nx^nyŠ ‰^n 2
xŠ 0
B
@
1
0 1
h1 C
CB
C
CB
h3 C
CB
C
C@
h4 A
A
h5
‰2^rx^ry^nyŠ
ˆ
‡ ‰ 2^r 2 y^nx^nyŠ ‰ 2^r 2 y^nx^nyŠ ‡ ‰2^rx^ry^n 2 xŠ ‰^ry^n 2
yŠ ‡ ‰ ^rx^nx^nyŠ
‰ ^ry^nx^nyŠ ‡ ‰^rx^n 2
xŠ 0
B
@
1
C :
A
…20†
Since h6 not occurs, this system gives only information about the shape of the ellipse
but not of its extension. Now, we superpose the two parts of the objective function under
consideration of weighting factors. The weighting factors were chosen to be reciprocal
to the variance of measured stresses and normal directions, respectively:
y ˆ 1
s 2 …r† y 1 ‡ 1
s 2 …n† y 2 : …21†
159

The approximated initial yield-locus curve is shown in Figure 8.10.
It could be described well by a shifted von Mises yield ellipse. The initial values
of the deviatoric back stresses could be derived from the centre coordinates of the ellipse
to be x0 ˆ 8:3 MPa and y 0 ˆ 3:2 MPa.
8.5 Bending of Notched Beams
8.5.1 Principle
8 Parameter Identification of Inelastic Deformation Laws
Figure 8.10: Initial yield-locus curve: 0.01% offset-strain, half-axes ration a=b ˆ 3
p and angle
c ˆ 45 8 given; rFo ˆ 218:3 MPa, rM x ˆ 19:8 MPa, rMy ˆ 14:7 MPa.
Pure bending of a deeply notched specimen is a suitable test to investigate material
properties also under conditions of different plane stress states. The geometry of the
specimen is shown in Figure 8.11.
The shape of specimen was optimized to get large ratios ry=rx of the principal
stresses in the material particles lying in the ligament. Bending tests have the advan-
160

tages that with only one specimen, tension as well as compression can be analysed and
that a reversal loading can be realized easily in the experiment.
In this chapter, the determination of the strain distributions in the ligament of the
specimens for each load step is described.
8.5.2 Experimental technique
8.5 Bending of Notched Beams
Figure 8.11: Specimen geometry for notched bending beams.
The in-plane Moiré technique, about which was reported earlier [14], was used to measure
the deformation of the specimens. The geometric Moiré is based on the superposition
of two gratings – the deformed object grating and the reference grating. Each
Moiré fringe, the so-called isothetic, describes the geometric location of all points with
the same Cartesian displacement component. This simple and clear fringe parameter is
reflected in the following fundamental equation:
ux…x; y† ˆpmx…x; y† ; …22†
where ux is the in-plane component of the displacement vector, mx fringe order, p pitch
(Figure 8.12), x; y Eulerian coordinates. The preparation of the specimen (left) and a
microphotograph (right) of the object grating were presented in Figure 8.12.
Parallel to the Moiré technique, strain gage measurements were carried out in the
grooves of notches and in the outer parts of the ligament.
As an example, the steps of loading (1–6), unloading (7) and reversal loading (8–
14) of a notched bending specimen are stated in Table 8.1.
As typical Moiré fringe fields in Figure 8.13, the isothetics are presented for the
load step 4 from Table 8.1.
The pictures show high concentrations of the strain ex close to the grooves of the
notches and only small strains ey in the ligament. Because of that, an intense plane
stress state can be expected along the ligament.
Finally, Figure 8.14 demonstrates the remarkable effect that the deformations in the
whole specimen are removed almost completely at a defined reversal bending moment.
161

Figure 8.12: Moiré technique.
162
8 Parameter Identification of Inelastic Deformation Laws
Table 8.1: Load steps in bending of a notched specimen.
Loading Moment [Nm] Unloading Moment [Nm] Reversal loading Moment [Nm]
1 477 7 2 8 –93
2 552 9 –187
3 609 10 –273
4 655 11 –430
5 689 12 –503
6 736 13 –763
14 –817
Figure 8.13: Isothetic fields for load step 4 from Table 8.1 (left ux, right uy).

8.5.3 Approximation of displacement fields
By means of Moiré measurements, we have got some measured values ^ux…^xs; ^y s† and
^uy…^xt;^yt† of the displacements but at different points …^xs; ^y s† and …^xt; ^y t† for ^ux and ^uy,
respectively.
To determine the values at given points like needed in our case, an approximation
of the displacement fields will be done. The displacements ux and uy are independent
from each other so that they could be handled separately. The used method is only described
for ux, a detailed discourse is given in [15]. The general idea for approximations
is to use a local approach valid in a distinct area
~ux…x; y† ˆ X
amum…x; y† …23†
m
8.5 Bending of Notched Beams
Figure 8.14: ux-Isothetic field for load step 12 in Table 8.1.
with scalar factors am and properly chosen functions u m. Polynomial functions are well
established, but if one uses such with higher degree, the solution shows undesired
wave-like effects. On the other hand, it is useful to take functions, which are at least
C 1 -continuously to get also a good approximation of the strains.
The problem was solved by a Finite Element-like approximation. Good results
were obtained using functions of the Serendipity-class of isoparametric 8-nodes-rectangular
elements.
The mesh generation could be done with the preprocessor of an arbitrary Finite-
Element program. If one takes the same mesh to approach both ux and uy, this procedure
has the additional advantage that transformation of the mesh including the approximations
into the initial configuration is possible. Hence, the description in material coordinates
and the tracking of material points will be possible.
The functional to determine the scalar factors am was chosen to be:
163

U ˆ Xne 1 Xmi …~ux…^xij; ^y ij† ^uxij† 2
" #
‡ c1 iˆ1
‡ c 2
8 Parameter Identification of Inelastic Deformation Laws
mi
jˆ1
X nr
rˆ1
2
6
4
lr
Z
lr
… ~uxI;n
~uxII;n† 2 dlr
X ne
iˆ1
2
6
4
Gi
Z
Gi
3
…~u 2 x;xx ‡ 2 ~u2 x;xy ‡ ~u2 x;yy †dGi
7
5
3
7
5 : …24†
The first term consists of the least-squares sum between approximated and measured
displacements of all ne elements. It will be weighted elementwise by the number mi of
measured points per element to exclude the influence of a possibly non-regular distribution
of measured points. The second term serves the smoothness of the approximation
inside the elements. The third term reduces discontinuities in the first derivatives normal
to the boundaries of neighbouring elements.
In Figure 8.15, the approximated isothetics for the load step 4 are shown (cf. Figure
8.13).
For bending specimens, small strains are coupled with finite displacements.
Hence, the parameter identification demands the tracking of material points, the deformation
fields were transformed into the initial configuration. The points in Figure 8.15
are the originally measured positions of Moiré fringes, whereas the isolines were numerically
calculated from the Finite-Element approximation. One can observe a really
good agreement between measured and approximated values.
Figure 8.15: Approximation of the isothetic fields from Figure 8.13.
164

8.6 Identification of Material Parameters
Figure 8.16: Deformations ex and ey for load step 4.
In Figure 8.16, the deformations ex and ey in the ligament for load step 4 (cf. Figures
8.13 and 8.15) are shown.
8.6 Identification of Material Parameters
8.6.1 Integration of the deformation law
The equations of elastic-plastic material behaviour, i.e. the deformation law (Equation
(5)), the hardening rule (Equation (6)) and the flow condition (Equation (1)), could be
summarized as:
E 1 _r ‡ _ k qF
qr
_e ˆ 0 ; …25†
_h kq…r; _ h; p† ˆ0 ; …26†
F…r; h; p† ˆ0 : …27†
The integration of these equations will be carried out for measured load steps De at
each reference point of the ligament. Equation (25) represents ordinary differential
equations for the stresses r, Equation (26) such one for the internal variables h. Both
systems are coupled and include the additional unknown scalar _ k. The yield condition
(Equation (27)) is needed as an additional algebraic condition.
165

Using implicit Euler time stepping, one finds the following system of non-linear
equations:
E 1 …rn rn 1†‡ _ kn
hn hn 1
qF
Dt Den ˆ 0 ; …28†
qrn
_knq nDt ˆ 0 ; …29†
F…rn; hn† ˆ0 ; …30†
to compute the values rn, hn and _ k for the n-th load step.
To solve the non-linear Equations (28) to (30), a Newton method like in [16] was
used. With k as suffix for the number of iterates and d for the increments of the variables,
we get:
Jj k
n
with the matrix
dr
dh
d_ 0
0 1 Den B
@ A B
ˆ B
k @
Jj k
n ˆ
h k‡1
n
_k k‡1
n
8 Parameter Identification of Inelastic Deformation Laws
qF
qr jk
n
_k k qF
n
qr jkn
Dt E 1 …r k n rn 1†
_k k
nqjk 1
C
nDt …hkn
hn 1† C
A
Fj k
n
qF
qh jk
n
…31†
E 1 ‡ _ k k q
n
2 F
jk
qr2 nDt k_ k q
n
2 F
qrqh jkn
Dt
qF
qr jkn
Dt
_k k qq
n
qr jkn
Dt I k_ k qq
n
qh jkn
Dt qkn Dt
0
B
@
1
C ;
C
A
0
…32†
and the …k ‡ 1†th iterate will be:
rk‡1 0 1
n r
@ A ˆ
k n
h k
0 1
dr
@ A ‡ dh
d_ 0 1
@ A : …33†
k
n
_k k
n
To improve convergence for considerably large load steps Den, it is possible to subdivide
the load steps into a certain number of linear sub-load steps.
166

8.6.2 Objective function, sensitivity analysis and optimization
To determine the material parameters, the objective function
U ˆ 1
2
X
n
‰c 1…~r M n …p1† ^r M n †2 ‡ c 2…~r N n …p1†† 2 ‡ c 3…~r lK
n …p1† ^r lK
n †2
‡ c 4…~r uK
n …p1† ^r uK
n †2 Š!min
p
will be minimized. The calculated and measured values of bending stresses rM n and
stresses rN n caused by the normal force, and the longitudinal stresses rlK n and ruK n at the
lower and upper notch grooves were compared. Moment M and normal force N were
computed by numerical integration of stresses at the reference points along the ligament.
To get the stresses ~r M n and ^rM n , the moment will be divided by the resistance moment
W ˆ 1
4 bh2 for ideal plasticity, ~r N n is got by dividing the normal force by the
cross-sectional area. ^r N n will be zero for the four-point-bending specimen.
The usage of deterministic optimization algorithms requires the knowledge of the
gradient dU
of the objective function with sufficient accuracy. In our case caused by
dp
the non-linear equations solved by implicit methods, the derivatives are not available
analytically.
One way to compute the gradient is the numerical sensitivity analysis performed
by the variation of the material parameters. For m parameters, the value of the objective
function has to be determined additionally m times for every iteration of the
material parameters. The choice of parameter increments is unsure, and the control of
the accuracy of the derivatives is difficult and also unsure.
Therefore, a semi-analytical sensitivity analysis will be preferred. The objective
function,
…34†
U ˆ U…z…p†† ; …35†
depends indirectly on the parameters p. The total differential is:
dU ˆ qU
qz
8.6 Identification of Material Parameters
T qz
qp dp ˆ aT dp ; …36†
a is called the sensitivity vector and gives a measure for the sensitivty of the objective
function at arbitrarily parameter changes [17].
The derivatives qU
could be determined easily from the objective function. The
qz
only difficulty is the determination of qz
. This will be done by implicit differentiation
qp
similar to [15, 18].
The system G of Equations (28) to (30) depends on the variables r, h and _ k.
These will be combined to the enlarged state vector:
167

z…p† ˆ…r…p†; h…p†; _ k† T ; …37†
which yet depends on the vector of material parameters. The iterated system (Equations
(28) to (30)) to compute the n-th load step could be summarized as:
Gn…zn…p†; zn 1…p†; p† ˆ0 : …38†
Implicit differentiation gives:
so that
dGn
dp
ˆ qGn
qzn
qzn qGn qzn 1 qGn
‡ ‡ ˆ 0 …39†
qp qzn 1 qp qp
qGn qzn
ˆ
qzn qp
qGn qzn 1
qzn 1 qp
qGn
qp
…40†
is a system of linear equations to determine qzn
qGn
. The matrix represents the iterated
qp qzn
Jacobean Jjn from Equation (32), which is known in its decomposed form.
Furthermore is:
and
qGn
qzn 1
0
ˆ @
qGn
qp ˆ
q
_kn
2 0 1
F
B
qrqp
C
B C
B C
B qq C
_kn
C
B qp
C
B C
@ qF A
qp
E 1 0
0
I
1
0
0 A …41†
0 0 0
have to be supplied by the material subroutine. Under consideration that qz0
ˆ 0, it is
qp
possible to compute the derivatives fo the next load step from that of the previous one
with the accuracy of the integration of the deformation law.
To solve the optimization problem of parameter identification (Equation (34)),
several gradient-based optimization methods like steepest descent, BFGS, Gauß-Newton
and Levenberg-Marquardt’s methods were tested. The Levenberg-Marquardt algorithm,
which combines steepest descent and Gauß-Newton method [19, 20], was preferred
as a very robust and suitable procedure for the non-linear least-squares optimization.
Hereby, the parameter vector will be changed according to
168
8 Parameter Identification of Inelastic Deformation Laws
…42†

p w‡1 ˆ p w ‡ s w
with the search direction
s w ˆ R w GN ‡ lI
where R w GN
…43†
1 rU w ; …44†
means the actual Gauß-Newton matrix.
Far from the solution, l will be taken large so that the procedure is nearly a steepest
descent, and in the neighbourhood of the solution, l will be taken small so that it
is nearly a Gauß-Newton algorithm. The control of the choice of the parameter l will
be done in such a manner that the next iterate will be searched in a “model-trust region”
with kswk < d.
8.6.3 Results of parameter identification
The implemented generalized material model consists of the following equations:
• quadratic yield function (Baltov and Sawczuk [21])
F ˆ Nijkl r D ij ij r D kl kl r 2 F…epl v †ˆ0 ; …45†
• evolutional relations
• yield stress – isotropic hardening
rF ˆ A…e pl
v † ; …46†
• kinematic hardening by Backhaus [22]
_ ij ˆ B…e pl
v †_epl ij ; …47†
• distorsional hardening by Danilov [23]
v †
_Nijkl ˆ C…epl
_e pl
v
_e pl
ij _epl
kl ; …48†
where A, B and C could be chosen as arbitrary functions of the equivalent plastic strain
e pl
v .
8.6 Identification of Material Parameters
Some special formulations will be at disposal:
• yield function (shifted von Mises)
F ˆ 3
2 …rDij ij†…r D ij ij† r 2 F ˆ 0 ; …49†
169

• yield stress
– modified power law
rF ˆ rFo ‡ a1‰…e pl
– arctan law
v ‡ a2† a3 a a3Š
; …50†
2
rF ˆ rFo ‡ a1 arctan …a2e pl pl
v †‡a3ev ; …51†
• kinematic hardening
– by Prager (cf. [22])
_ ij ˆ b1_e pl
ij
; …52†
– by Armstrong-Frederick (cf. [24])
_ ij ˆ b1_e pl
ij
b2_e pl
v ij : …53†
All computations have been carried out with the initial values for rFo, 11o and 22o
taken from the approximation of the initial yield-locus curve (cf. Figure 8.10).
The power law (Equation (50)) gave no satisfying results because the parameters
depend on each other and it shows no asymptotic behaviour of the uniaxial flow
curves.
Therefore, the arctan law (Equation (51)) for isotropic hardening combined with
kinematic hardening and initial anisotropy has been used for further computations. Results
are shown in Figures 8.17 and 8.18.
The consideration of kinematic hardening by Armstrong-Frederick [24] and of
distortional hardening by Danilov [23] did not reveal any quantitatively better approximation.
8.7 Conclusions
8 Parameter Identification of Inelastic Deformation Laws
In the paper, some possibilities of bending tests to identify material parameters of inelastic
materials were described. For that reason, the co-operation between theoretically
and experimentally working academics has to be much closer, as this is usual at german
technical universities. The foundation of the Deutsche Forschungsgemeinschaft –
Collaborative Research Centre (Sonderforschungsbereich 319) – has been proved as a
very effective measure to overcome reservations and to realize such a close co-operation.
An important result of this project is that bending tests are suitable very well to
study elastic-plastic material properties. Bending tests have the fundamental advantages
170

8.7 Conclusions
Figure 8.17: Parameter identification for the bending test of the notched specimen, arctan law for
isotropic hardening, kinematic hardening by Prager (cf. [22]), rFo ˆ 218:3 MPa, 11o ˆ
8:3 MPa, 22o ˆ 3:2 MPa; optimized parameters: a1 ˆ 94, a2 ˆ 1626, a3 ˆ 380, b1 ˆ 1600;
mean quadratic deviation: 20.7 MPa.
that in one experiment, tension as well as compression exist and that a reversal loading
to identify kinematic strain hardening can easily be realized. In the experiments, some
difficulties result from the fact that strains in a very large extension must be detected.
So, for definition of yield stresses, the offset strain is about 10 –4 , and on the other side,
a maximum strain of 5 ·10 –2 should be measured. In both cases, the stress level is approximately
the same. This taking into account, stable numerical results for the elastic
constants, the yield stresses as well as the yield curves and the anisotropic yield-locus
curve were obtained from bending tests of specimens.
The identification of hardening parameters was carried out by analysing the displacement
fields of notched specimens. The suitability of such experiments has been
proved. Further investigations should use all informations about the whole experimentally
investigated area. Therefore, the use of the Finite-Element method to calculate the
numerical comparative solution is absolutely necessary.
Shortened calculations as described here require low computational times and
may give a deep insight into the effects of several material models in combination with
171

a special analysis of measured data. Such calculations could be used also furthermore
to get reliable starting values for the Finite-Element-based parameter identification.
The authors’ hope that they can continue the research about the relatively new
idea to identify inelastic material properties and parameters by means of the evaluation
of inhomogeneous strain and stress fields. For instance some possibilities are given in
the DFG project [6].
Acknowledgements
8 Parameter Identification of Inelastic Deformation Laws
Figure 8.18: Related uniaxial flow curves to Figure 8.17.
The authors would like to thank Prof. Dr. Dr. E.h.E. Steck and Prof. Dr. R. Ritter from
Technical University of Braunschweig for their extensive support during the preparation
and realization of this project.
172

References
References
[1] E. Stein, D. Bischoff, R. Mahnken: Identifikation mit Finite-Element Methoden. Arbeitsund
Ergebnisbericht 1991–1993. Subproject B8, Collaborative Research Centre (SFB 319).
[2] E. Stein: Parameteridentifikation mit Finite-Element Methoden. Förderungsantrag 1994–
1996. Subproject B8, Collaborative Research Centre (SFB 319).
[3] R. Mahnken, E. Stein: Parameter Identification for Inelastic Constitutive Equations Based
on Uniform and Non-Uniform Stress and Strain Distributions. This book (Chapter 12).
[4] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifikation
für ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Bauingenieur
71 (1996) 21–31.
[5] R. Kreissig, J. Naumann: Weiterentwicklung der Theorie der plastischen Verfestigung und
ihre experimentelle Verifikation mit Hilfe des Moiréverfahrens. DFG-Projekt 1994–1996/
Zwischenbericht 1996.
[6] R. Kreissig, A. Meyer: Effiziente parallele Algorithmen zur Simulation des Deformationsverhaltens
von Bauteilen aus elastisch-plastischen Materialien. Förderungsantrag 1996–
1998. Subproject D1, Collaborative Research Centre (SFB 393).
[7] R. Kreissig: Parameteridentifikation inelastischer Deformationsgesetze. Technische Mechanik
16(1) (1996) 97–106.
[8] H. Herbert: Über den Zusammenhang der Biegungselastizität des Gußeisens mit seiner
Zug- und Druckelastizität. Mitt. und Forschungsarbeit. VDI 89 (1910) 39–81.
[9] A. Nadai: Plasticity. McGraw-Hill, New York, London, 1931.
[10] V. Laws: Derivation of the tensile stress-strain curve from a bending data. J. Materials Sci.
16 (1981) 1299–1304.
[11] R. A. Mayville, I. Finnie: Uniaxial stress-strain curves from a bending test. Exp. Mech.
22(6) (1982) 197–201.
[12] M. Stockmann, J. Naumann, P. Bormann, F. Pelz: Zur Widerstandsänderung von Dehnungsmeßstreifen
bei großen Deformationen. Materialprüfung 38(4) (1996) 134–138, 38(5)
(1996) 216–219.
[13] P. Bormann, J. Naumann, M. Stockmann: Biegeversuche zur Ermittlung einachsiger
Fließkurven für Zug und Druck. In: O.T. Bruhns (Ed.): Große plastische Formänderungen.
Bad Honnef, 1994. Mitteilungen des Inst. für Mechanik, Nr. 93, Ruhr-Universität Bochum.
[14] J. Naumann: Grundlagen und Anwendung des In-plane-Moiréverfahrens in der experimentellen
Festkörpermechanik. VDI-Fortschrittsberichte, Reihe 18, Nr. 110, Düsseldorf, 1992.
[15] E. Bohnsack: Continuous field approximation of experimentally given data by finite elements.
Computer & Structures 63(6) (1997) 1195–1204.
[16] D. Michael, A. Meyer: Some remarks on the simulation of elasto-plastic problems on parallel
computers. Preprint-Reihe Chemnitzer DFG-Forschungsgruppe “Scientific Parallel Computing”,
Technische Universität Chemnitz-Zwickau, March 1995.
[17] H. Eschenauer, W. Schnell: Elastizitätstheorie. BI-Wissenschaftsverlag, Mannheim,1993.
[18] R. Mahncken, E. Stein: Identification of Parameters for Visco-plastic Models via Finite-Element
Methods and Gradient Methods. IBNM-Bericht 93/5, Institut für Baumechanik und
Numerische Mechanik der Universität Hannover, 1995.
[19] J.E. Dennis, R. B. Schnabel: Numerical Methods for Unconstrained Optimization and Nonlinear
Equations. Prentice Hall, Englewood Cliffs, 1983.
[20] S.S. Rao: Engineering Optimization. Wiley & Sons, New York, 1996.
[21] A. Baltov, A. Sawczuk: A Rule of Anisotropic Hardening. Acta Mechanica 1 (1965) 81–92.
[22] G. Backhaus: Deformationsgesetze. Akademie-Verlag, Berlin, 1983.
[23] V. L. Danilov: K formulirovke zakona deformacionnogo upročnenija. Mechanika tverdogo
tella, Moskva 6 (1971) 146–150.
[24] V. Dorsch: Zur Anwendung und Numerik elastisch-plastischer Stoffgesetze. In: Prozeßsimulation
in der Umformtechnik, Band 9, Springer, Berlin Heidelberg, 1996.
173

10 On the Behaviour of Mild Steel Fe 510
under Complex Cyclic Loading
Udo Peil, Joachim Scheer, Hans-Joachim Scheibe, Matthias Reininghaus,
Detlef Kuck and Sven Dannemeyer*
10.1 Introduction
The aim of this project is to develop a material model for the prediction of the material
behaviour of mild steel Fe 510 under multiaxial cyclic plastic loading.
First of all, detailed information about the material response under cyclic plastic
loading are necessary. Therefore, extensive experimental investigations are made including
uniaxial single- and multiple-step tests and biaxial tension-torsion tests with
various prestrains, increasing or decreasing strain amplitudes, and several proportional
and non-proportional biaxial loading paths, respectively. Cyclic hardening or softening
in the uniaxial case, or additional hardening under non-proportional loading are some
of the observed effects.
To describe the material peculiarities, the two-surface model of Dafalias-Popov
has been modified. The improvements result in a rate-independent, isothermal two-surface
model, which is presented here.
Experimental data from the uniaxial and biaxial tests, and, in addition, from tests
on structural components are compared with corresponding calculations made with the
new model.
The results demonstrate the capabilities of the extended-two-surface model to predict
the behaviour of mild steel and steel constructions under multiaxial cyclic plastic
loading.
* Technische Universität Braunschweig, Institut für Stahlbau, Beethovenstraße 51,
D-38106 Braunschweig, Germany
218
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

10.2 Material Behaviour
10.2.1 Material, experimental set-ups, and techniques
The investigated material was mild steel Fe 510. The chemical and mechanical characteristics
of the different heats used in the investigations can be found in the corresponding
papers (Scheibe [1], Reininghaus [2]).
Two types of specimens were used in the investigations. Figures 10.1 and 10.2
show sketches of both the cylindrical specimens used in the experiments with uniaxial
loading, and the tubular specimens for the biaxial investigations.
Differentservohydraulic testingmachines wereused in the uniaxialinvestigations.The
biaxial investigations were performed on a 160 kN-tension-compression “SCHENK” universal
testing machine, extended with a 1000 Nm-torsion drive and an extensometer.
10.2.2 Material behaviour under uniaxial cyclic loading
10.2.2.1 Parameters
10.2 Material Behaviour
In the strain-controlled experiments, the strain rate _e was chosen for values between 3.5
and 24‰/min. For the force-controlled experiments, the loading rate was 3 kN/s. All
Figure 10.1: Cylindrical specimen for uniaxial experiments.
Figure 10.2: Tubular-type specimen for biaxial experiments.
219

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.3: Varied parameters of the uniaxial experiments.
tests were performed at room temperature. The varied parameters in the uniaxial loaded
tests were the strain amplitude e a (2, 3, 5, 8, and 12‰), the mean strain e m (–40, –25,
0, 10, 25, and 40‰), and the strain history (see Figure 10.3 for explanation).
The uniaxial tests were performed paying particular attention to effects of the sequence
of loadings, the evolution of cyclic hardening and softening, the relaxation of
mean stresses, and the size of the elastic region.
To investigate the ratchetting effects, stress-controlled experiments were performed
varying the initial strain e m, the stress amplitude r a, the mean stress r m, and
the stress ratio Rˆr u/r o.
10.2.2.2 Results of the uniaxial experiments
Strain-controlled experiments
Typical results of the different multiple-step tests without mean strain are shown in Figures
10.4 and 10.5.
In these figures, the courses of the half-range of stresses vs. the number of halfcycles
are shown. In addition, the strain vs. time is plotted for explanation in the upper
left part of the diagram. The resulting stress vs. strain is given in the upper right part.
Figure 10.4: Multiple-step test (MST), ea ˆ 12-8-5-3-2‰.
220

10.2 Material Behaviour
Figure 10.5: Multiple-step test (RFE), ea ˆ 2-12-2‰.
After passing the linear-elastic region, the distinct yield point and the yield plateau,
mild steel Fe 510 shows the well known Bauschinger effect if the specimen is unloaded.
During the cyclic loading, the maximum stresses of a hysteresis-loop are not
constant: For amplitudes ea smaller than 5‰, the maximum stresses decrease from cycle
to cycle, and a saturated state is reached after 500 or more cycles (cyclic softening).
Amplitudes e a higher than 5‰ cause an increase of maximum stresses during the first
40 cycles (cyclic hardening).
Note that cyclic softening or hardening is understood as a decrease or an increase of
the stress amplitude in comparison with the stress level of the monotonic stress-strain
curve at this strain level. For e a 12‰, the stress level of the monotonic stress-strain
curve is almost constant (with r ˆ r F). Therefore, cyclic hardening or softening can be
observed easily. Stress levels Dr/2 (for symmetric amplitudes) greater than rF are seen
by definition as cyclic hardening, and stress levels Dr/2 lower than rF as cyclic softening.
A cyclic stress-strain curve is achieved by plotting the corresponding stress ranges
Dr/2 at the stabilized states vs. the corresponding strain amplitudes e a. Figure 10.6
shows a typical monotonic stress-strain curve along with two cyclic stress-strain curves.
The intersection of the curves at a strain of 5‰ characterizes the border between cyclic
softening and cyclic hardening.
Figure 10.6: Cyclic stress-strain curves.
221

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
The cyclic stress-strain curves obtained after 40 cycles and the one after 500 cycles
differ if smaller amplitudes are used. This shows that a saturated state is not reached after 40
cycles at all. With higher amplitudes, the two curves tend to become identical: The steady
state is reached during the first 40 cycles. However, after prestraining with a higher amplitude
(e.g. path MST, Figure 10.4), the stabilized state is already reached after a few cycles.
Figure 10.7 shows a multiple-step test (MST) with a mean strain of e mˆ+40‰.
The maximum stress at 52‰ (or the maximum stress level Dr/2) decreases within the
first cycles. Note that this transient behaviour is not a cyclic softening according to the
above definition because, during the first loops of the amplitude e a ˆ 12‰, only the
mean stress caused by the mean strain returns to zero.
Parallel to this so-called mean-stress relaxation, a cyclic hardening of the
eaˆ12‰ amplitude takes place, but due to the dominance of the mean-stress relaxation,
only a softening of the stress level Dr/2 can be observed.
A comparison of the stabilized loops for amplitudes e aˆ12‰ with different mean
strains shows that the different mean strains do not significantly influence the shape or
the maximal stress amplitudes of the hysteresis-loops.
Figure 10.8 shows the cyclic stress-strain curves for different mean strains. It is
seen that the maximum stress amplitude Dr/2 depends only on the strain amplitude e a.
This result is the basis for the above definition of cyclic hardening or cyclic softening
under a given strain amplitude.
Another important point besides the effect of cyclic hardening is the influence of
the cyclic loading on the size of the elastic region. The elastic regions were determined
using an offset-proof-strain method (see Scheibe [1] for further explanation) with a
proof strain of 0.03‰ and a constant elastic modulus E 0 of 206 000 MPa.
Figure 10.9 deals with the evolution of the elastic domain during uniaxial cyclic
loading. The diagram shows that the size of the elastic domain k is a function of the
maximum strain amplitude emax. If the former is greater than 5‰, the size of the elastic
region is reduced to the value ks&100 MPa. For strain amplitudes lower than 5‰, the
elastic region decreases very slowly (e aˆ2‰ in Figure 10.9). The dashed line of the
amplitude e aˆ2‰ shows that the size of the elastic region tends to reach the same value
as that of the greater amplitudes within some hundred cycles.
Figure 10.7: Multiple-step test (MST) 12-8-5-3-2‰, with a mean strain of 40‰.
222

10.2 Material Behaviour
Figure 10.8: Cyclic stress-strain curve for different mean strains.
Figure 10.9: The evolution of the elastic region under cyclic loading.
Stress-controlled experiments
As a result of the stress-controlled experiments concerning the ratchetting effect (Kuck
[3]), different variations of the ratchetting effect were found (Figure 10.10):
a) quick saturation, especially with small r a and r m,
b) positive increase of strain without saturation,
c) negative increase of strain with gradual saturation (at a large number of cycles),
d) increase of strain with gradual saturation, reversal, and then constant increase of
strain.
In addition, the influence of the mentioned cross-section of the specimen and with it the
kind of stress (initial cross-section: technical stress, and actual cross-section: effective
stress) on the number of cycles needed to reach the saturated state, was investigated.
The tests show an increase of ratchetting with decreasing minimum stress at constant
maximum stress (Figure 10.11), and, in addition, a distinct mean stress (34.5 MPa), where
no ratchetting is found (Figure 10.12). The amount of the occurred ratchetting depends on
the difference between the actual mean stress and the mean stress, which leads to zero
ratchetting (Kuck [3]).
223

224
10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.10: Different appearances of the ratchetting effect.
Figure 10.11: Influence of R ˆ ru=ro:

Figure 10.12: Influence of the mean stress.
10.2.3 Material behaviour under biaxial cyclic loading
10.2.3.1 Parameters
For the biaxial tests, the loading path (Figure 10.13), the sequence of different loading
paths, the loading intensity eB (defined by Equation (1)), and the ratio D between tensile
and torsional loads (defined by Equation (2)) were varied. Note that eB is not a mechanical
derived equation, it is used only to compare biaxial and uniaxial loads here.
For the experimental investigations, different loading intensities e B between 1.90‰ and
7.10‰ were chosen.
r
eB ˆ e 2 a
‡ 1
3 c2 a
10.2 Material Behaviour
; …1†
D ˆ 3
p ea
: …2†
ca The achieved stress curves and the calculated uniaxial equivalent stresses (based on the
theory of v. Mises) show the influences of various sequences of loading paths or the
additional-hardening effect.
Moreover, the yield-surface investigations give additional insight into the evolution
of the elastic region under complex loadings. Here, the influence of the intensity
eB and the loading path on the location, shape, and size of the measured yield surface
were investigated.
Additional experiments were made to determine the influence of several parameters
coming out of the process of yield point probing itself, i.e. technique and sequence
of yield point probing, choice of the starting point, or fixing of the elastic modulus.
225

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.13: Biaxial loading paths.
10.2.3.2 Relations of tensile and torsional stresses
Figure 10.14 gives some characteristic results of the evolution of the tensile and torsional
stress in the transient state of biaxial cyclic-loaded mild steel.
The evolution of the stresses of the biaxial proportional and non-proportional tests
differs with increasing intensities. For a proportional loading, only the maximum stresses
increase and the stress curve grows without changing its shape (Figure 10.14 a).
"
Figure 10.14 a)–h): Tensile and torsional stresses under biaxial proportional and non-proportional
loadings.
226

10.2 Material Behaviour
a) Path 03; D ˆ 1:0 b) Path 07; D ˆ 1:0
c) Path 10; D ˆ 1:0 d) Path 09; D ˆ 1:0
e) Path 09; D ˆ 1:0 f) Path 09; D ˆ 2:4
g) Path 08; D ˆ 1:0 h) Path 08; D ˆ 1:0
227

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
On the other hand, for some of the non-proportional loadings, a significant change in
the shape of the stress course in addition to the increase of the maximum stresses can be
found (Figure 10.14 b,g–i). Note that the loading path and the relation D between ea and ca
does not change. The results of a varying D can be seen in Figure 10.14 d–f. In Figure
10.15, the uniaxial equivalent stresses r v are plotted vs. the maximum equivalent
strain or intensity e B. In these experiments, the cyclic hardening in the uniaxial cyclic
stress-strain curve starts at about 3‰. The difference between the cyclic stress-strain
curve of the uniaxial experiments described in Section 10.2, and the curve seen in Figure
10.15, depends on the different heats of Fe 510 used in the two investigations.
Comparing the cyclic stress-strain curve with the equivalent stresses of the proportional
loadings, no significant difference in the maximum stresses between the uniaxial
and the biaxial proportional loadings can be found. A significant difference between
the cyclic stress-strain curve and the equivalent stresses is found however for
non-proportional loading paths. This effect is named additional hardening here.
The additional hardening appears to be strongly dependent on the type of nonproportional
loading. In general, the additional hardening increases by an increasing
phase angle u (Figure 10.13). For higehr values of u (between 60 and 90 degrees, path
06 and 07), the additional hardening is almost constant. The amount of additional hardening
of path 10 corresponds to that of path 07. The highest amount of additional hard-
228
i) Path 08; D ˆ 1:0 j) Step 1: Path 03
k) Step 2: Path 07 l) Step 3: Path 03
Figure 10.14 i)–l): Tensile and torsional stresses under biaxial proportional and non-proportional
loadings.

ening is found in experiments with the strain path 09 (“Butterfly”). Here, the equivalent
stress in the saturated state reaches the level of the uniaxial tensile strength. The
additional hardening fades when the non-proportionality of the loading has decreased.
Figure 10.14 j–l shows the results of an experiment, where a specimen was first
uniaxially loaded (e aˆ6‰) up to a saturated state, then underwent a non-proportional
loading path 07 (e B of 8‰), and finally was uniaxially loaded again.
The cyclic hardening during the first loading stage is followed by an additional
hardening in the second stage. During the third loading, the additional hardening fades
so that the saturated stress-strain curves of the first and the third loading level are
nearly identical.
10.2.3.3 Yield-surface investigations
10.2 Material Behaviour
Figure 10.15: Uniaxial equivalent stresses of biaxial loading paths.
Yield surfaces were investigated in the transient and in the saturated state of the
material behaviour. To reduce time-consumptional manual controlling, a computer program
was developed, which allows yield surfaces at any point of any complex loadings
to be established automatically. This program and the experimental set-up is described
comprehensively by Dannemeyer [4].
An offset-proof-strain method was used to define the onset of plastic flow. For an
increasing offset-proof strain, the yield surfaces of mild steel Fe 510 in saturation show
the same effects as described for other materials (e.g. Michno and Findley [5]). The distinct
corner in preloading direction and the flattening opposite to it tend to blur, and the size of
the surface increases, mainly against the preloading direction (Figure 10.16). The
Bauschinger effect is responsible for this uneven expansion when the offset value increases.
An offset-proof strain of 0.05‰ was used for the yield-surface investigations in
the saturated state. To determine yield surfaces in the transient state of the material behaviour,
the offset-proof strain was reduced to 0.03‰. A further decrease of the offsetproof
strain was restricted by the precision of the technical set-up.
In this investigation, several yield-surface determinations were performed on a
single specimen. 16 yield probings with different ratios of sizes of tensile and torsional
strain increments are combined to build up a yield surface. To investigate the influence
229

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.16: Effect of various offset-proof strains on the yield surface of mild steel Fe 510.
of the sequence of the different yield-probing directions, yield surfaces were determined
using two different sequences (Figure 10.17).
For materials, which reach a cyclically stable state during cyclic loading, the influence
of the probing sequence can be excluded if the specimen is loaded with several
cycles between two yield probings. The small hardening effects drawn by the previous
yield probing disappear completely, and a similar loading state can be reached at the
beginning of every new yield probing (Figure 10.18). This allows yield surfaces to be
investigated with negligible dependence on the probing path (Figure 10.19). (The turnover
points and the first yield points are marked with arrows here. The loading path for
each of the yield surfaces is given by symbols.)
This method is called the single-point technique (SPT) here, in contrast to the multiple-point
technique (MPT), where several yield points are investigated one after another,
each time unloading to a starting point located somewhere within the yield surface.
Figure 10.17: Sequences of yield point probings.
230
p
3 s

Figure 10.18: Single-point technique.
p
3 s
10.2 Material Behaviour
Figure 10.19: Yield surfaces determined with the single-point technique and current elastic moduli
under proportional and non-proportional loading paths.
231

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
The influence of the probing sequence on a yield surface determined with the
multiple-probing technique is systematic by nature (Figure 10.20). So, the measured
yield surface can be corrected, at least in quality, if a surface investigated with the single-probing
technique is used as a reference.
Another effect, which influences a yield-point determination on mild steel Fe 510,
should be described here. In Figure 10.21, the stress-strain curve of the initial loading
and the elastic region after the turnover of the 20th cycle of a specimen under uniaxial
tension-compression load is plotted.
In order to compare the two gradients, the turnover point of the 20th cycle is
moved to the origin. It is seen that the slope of the initial loading curve is nearly constant
over the plotted range of 1.3‰. After being loaded 20 times into the plastic
range, a distinct proportional region is missing.
This non-linearity of the stress-strain curve under preloading affects the yield-surface
determination in different ways. In general, for the determination of a yield locus,
it is necessary to start the yield-point determinations somewhere in the elastic region, if
possible in the centre of the elastic region to secure an almost rectangular touch of the
elastic-plastic border.
Due to the non-linear area after a turnover point, the unloading strain e s becomes
a parameter of the size of the established yield surface.
The results of an experiment, shown in Figure 10.22, demonstrate the effect of
various unloading strains on the yield surface of a uniaxial cyclic tension-compressionloaded
specimen. It is obviously that the expansion of the yield surface against the preloading
direction depends on the amount of the unloading strain. The diameter of the
yield surface rectangular to the preloading direction is not affected, however.
After preloading, the lack of a distinct area of proportionality influences the yieldsurface
determination not only in the setting of a starting point but also in the calculation
of plastic strains. Using an offset-proof-strain definition in combination with an automa-
Figure 10.20: Yield surfaces determined with the multiple-point technique and current elastic
moduli under proportional and non-proportional loading paths.
232
p
3 s

10.2 Material Behaviour
Figure 10.21: Sections of the stress-strain curve of mild steel Fe 510 under uniaxial cyclic tension-compression
loading.
p
3 s
Figure 10.22: The effect of various unloading strains es on yield loci under cyclic uniaxial tension-compression
loading.
tized experimental procedure, a continuous separation of elastic and plastic strains has to be
carried out on-line during a yield probing. The elastic moduli (E and G) can be set either to
constant values for the whole experiment, or they can be determined from the gradient of a
defined number of data considered to be linear-elastic. If the current elastic moduli change
during a cyclic loading and constant moduli are considered in the on-line calculation of
plastic strains, the resulting yield point does not correspond to the existing yield point.
Figure 10.23 shows the results of yield-surface determinations with fixed and current
elastic characteristics. In comparison to the Figures 10.19 and 10.20, it can be seen
that the size of the yield surface depends strongly on how the elastic moduli are given.
233

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.23: Yield surfaces investigated with constant elastic moduli under proportional and
non-proportional loading paths.
The SPT-loci with constant elastic moduli show a corner and a distinct flattening,
whereas the MPT-loci of the same type are smoother and have increased in size. If
non-proportional loading paths are used, the difference in size of the SPT- and MPTloci
is at its greatest (Figure 10.23, upper left yield surface).
In the transient state of material behaviour, yield-surface investigations are more
sensitive than in the saturated state. Even in the area of the yield plateau, a yield-point
determination with its incursion into the plastic region influences strongly the later
yield-point probes. In general, the intensive hardening during the first few cycles excludes
the use of the single-point technique in the transient state.
The multiple-point technique and a further decrease of the offset-proof strain to smallest
values seem to help getting realistic results for mild steel at this stage of a cyclic loading.
When a new specimen is first loaded, the yield surface contracts and starts to move
in the stress space immediately after the elastic limit is passed. In addition, a distortion of
the initial round yield surface (r- 3
p s-space) occurs. The pronounced isotropic softening
is strongest during the first cycle, and in the case of a proportional load, it is usually completed
after a few cycles depending on the load intensity (see also Figure 10.9).
If the load is of a non-proportional type, the shrinkage of the yield surface is almost
completed at the beginning of the second cycle (Figure 10.24). This isotropic softening is
always connected with a kinematic hardening. In further states of the cyclic loading, the
yield surface continues to move to higher stresses after the shrinkage is already completed.
In Figure 10.24, the results of an experiment in the transient state are presented.
During the first cycle, the yield surface decreases at constant uniaxial equivalent stresses
rv, and from the second to the fifth cycle, a distinct increase of the maximum stresses
obtained by a movement of the yield surface takes place. It can be stated that the additional
hardening of a non-proportional load is also a type of kinematic hardening.
234
p
3 s

p
3 s
10.2 Material Behaviour
Figure 10.24: Yield surfaces of the first, second and fifth cycle of a non-proportional cyclic
loaded specimen in the transient state.
This distortion of the yield surface (formative hardening), including the deviation
of the initial round shape by forming a corner in preloading direction and a flat side at
the opposite, is the third mechanism of hardening besides the isotropic and kinematic
hardening found on cyclic loaded mild steel Fe 510.
In general, the shape of a preloaded yield surface depends on the type of load.
All investigated proportional loading paths of the same intensity show yield surfaces,
which are nearly identical in size and shape. They show a distinct corner in preloading
direction and a flattening opposite to it. In addition, the two diameters of the yield surface
change differently during a proportional load. In the saturated state, the diameter
in direction of the preloading (d 1, Figure 10.25) has shrunk to 40% of the initial value,
whereas the second diameter rectangular to the preloading direction (d 2) decreases only
to 70% as seen in Figure 10.19, for example. A non-proportional load forms a more
Figure 10.25: Definition of diameters of a yield surface.
235

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
rounded and smoother shape, and the diameters decrease more homogeneously because
of the changes in the direction of the load increments.
An approximate alignment of the yield surface with the corner and the flattening
towards the direction of the stress increment can be found only at proportional loading
paths. At some of the non-proportional loading paths, the direction of the stress increment
at the turnover point differs distinctly from the direction of the suggested axis of
symmetry of the yield surface (see Figure 10.19).
Additional yield-surface investigations were performed for the subproject A10
(Prof. Besdo [6]) on torsional preloaded AlMg 3.
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
10.3.1 Extended-two-surface model
10.3.1.1 General description
Extensive investigations were made into the suitability of several models to describe
the characteristical effects of the material behaviour of cyclic loaded mild steel Fe 510
(Heuer [7], Scheibe [1]). Based on these investigations, it can be concluded that the
two-surface model of Dafalias and Popov [8] represents a suitable basis for further developments.
The extensions of the original two-surface model are presented here. The
new extended-two-surface model (ETS-model) is described completely by Scheibe [1],
Reininghaus [2], Scheer et al. [9], Peil and Kuck [10], Peil and Reininghaus [11], and
Reininghaus [12]. Here, only the fundamental description of the model is given.
The fundamental parts of this model are:
• three memory surfaces in the strain space,
• the consideration of the additional-hardening effect based on experimental findings
to describe non-proportional loadings and the consideration of
• a softening of the loading surface, and
• an isotropic hardening of the bounding surface.
One important aspect of the presented model is that the material or model parameters,
which are determined once for the mild steel Fe 510, are fixed for this material. All calculations
shown in this paper were carried out with the same set of parameters. There
was no extra fitting necessary for any special kind of loading path or calculation.
236

10.3.1.2 Loading and bounding surface
The original two-surface model of Dafalias and Popov assumes that the yield surface
or loading surface and the memory or bounding surface (Figure 10.26) harden kinematically
and isotropically, while the two surfaces are in contact. If there is no contact between
the two surfaces, the memory surface remains unchanged, while the loading surface
moves according to the Mróz rule [13]. This rule secures a tangential contact without
any intersection if the two surfaces come into contact.
Both the loading surface and the bounding surface are assumed as hyperspheres
in the deviatoric stress space and are represented as:
with
F ˆ 1
2 … ~ rD
^F ˆ 1
2 … ~ rD
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
~ †… ~ rD
^
~ †… ~ rD
~ †
^
~ †
1
3 k2 ˆ 0 ; …3†
1
3 ^k 2 ˆ 0 …4†
k
^k
radius of the yield or loading surface,
radius of the bounding surface,
~ ^
~
~
centre of the loading surface (kinematic hardening),
centre of the bounding surface,
rD deviatoric stress tensor.
In this basic formulation, the model has some disadvantages:
• the yield plateau of mild steel cannot be predicted,
• there is no possibility to distinguish between monotonic and cyclic loadings, and
• an update problem of the variable d in (overshooting problem) occurs.
In the ETS-model, which is described here, the loading surface remains unchanged:
The surface can contract or expand, move, but not be distorted. In contrast to the origi-
Figure 10.26: Loading and bounding surface.
237

nal model of Dafalias and Popov, the bounding surface in the ETS-model is only able
to harden or soften isotropically. Furthermore, two bounding surfaces instead of one are
implemented. The inner one corresponds to the original bounding surface in the model
of Dafalias and Popov. The outer one is used as a control surface and is activated only
under non-proportional loadings.
The plastic modulus in the ETS-model is calculated from Equation (5):
with
P ˆ ^P0 ‡ h
d
din d
d distance between the actual stress point and the bounding surface (see Figure 10.26),
din distance between the yield point and the bounding surface (see Figure 10.26),
^P0 plastic modulus of the strain hardening region,
h shape parameter.
The plastic modulus P is split up into a modulus for the kinematic hardening Pa and
the isotropic hardening Pk:
P ˆ Pa ‡ Pk : …6†
Pk is calculated from the formulation of the isotropic hardening or softening of the
yield surface. Pk is a function of the size of the new implemented memory surface in
the strain space. The plastic modulus Pa for the kinematic hardening is determined
from the difference between the total plastic modulus P and the plastic modulus for the
isotropic hardening Pk.
During the development of the model, different rules for the kinematic hardening
of the surface were tested (Reininghaus [2]). The best results are obtained with the
original kinematic hardening rule of Mróz [13], therefore all results presented here are
calculated with this rule.
10.3.1.3 Strain-memory surfaces
The strain-memory surfaces in the strain space allow the monotonic and cyclic material
behaviour to be taken into account.
• Strain memory Mm for monotonic behaviour
The size qm of this memory surface can be understood as the maximum plastic strain
amplitude during the whole cyclic loading:
238
10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Mm ˆ 1
2 … ~ ep
~ bm†… ~ e p
~ bm†
…5†
3
4 q2m ˆ 0 ; …7†

with
Z
qm ˆ
dqm ; …8†
dqm ˆ 1
2 Hm n nmdevp ;
~ ~
…9†
~ bm
Z
ˆ d bm ;
~
…10†
d ~ bm ˆ 1
2 Hm d ~ e p
Hm ˆ 0 for
�
Mm 0
Mm ˆ 0 and
~ n ~ nm < 0 ;
Hm ˆ 1 for Mm ˆ 0 and
~ n ~ nm 0 ; …13†
~ n normal to the loading surface,
~ nm
~
normal to the monotonic strain-memory surface,
ep ~
plastic strain tensor,
bm
devp
centre of the monotonic strain-memory surface,
equivalent plastic strain increment,
size of the monotonic strain-memory surface.
qm
• Strain memory Ms for cyclic behaviour
The increase of size qs of the memory surface Ms during a cycle only depends on an
additional factor (cs in Equations (15) and (16)). The change in the size of this memory
surface describes cyclic hardening or softening:
Ms ˆ 1
2 … ~ ep
~ bs†… ~ e p
~ bs†
…11†
…12†
3
4 q2s ˆ 0 ; …14†
dqs ˆ…1 Hm†Hscs ~ n ~ ns devp ; …15†
d ~ bs ˆ‰…1 Hm†Hs…1 cs†‡HmŠd ~ e p ; …16†
qs ˆ
Z
~ bs
Z
ˆ d bs
~
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
dqs ; …17†
…18†
239

with
Hs ˆ 0 for
�
Ms 0
Ms ˆ 0 and
Hs ˆ 1 for Ms ˆ 0 and
~ n ~ ns < 0 ;
~ ns normal to the saturated strain-memory surface,
~ bs centre of the saturated strain-memory surface,
qs size of the saturated strain-memory surface,
cs factor for the isotropic hardening per cycle.
• Strain memory Ma for the actual loading direction
…19†
~ n ~ ns 0 ; …20†
The memory surface Ma for the actual loading direction will decrease to zero when the
angle between the normal vector of the loading surface and the normal vector of the
memory surface Ma exceed 908:
with
Ma ˆ 1
2 … ~ ep
~ ba†… ~ e p
~ ba†
3
4 q2a ˆ 0 ; …21†
dqa ˆ 1
2 ~ n ~ na devp ; …22†
d ~ ba ˆ 1
2 d ~ ep ; …23†
qa ˆ
Z
dqa ; …24†
~ ba ˆ…1 Ha† e
~ p Z
‡
Ha ˆ
�
1 for
0 for
d ~ ba
~ n ~ na < 0
~ n ~ na 0
~ na normal to the actual strain-memory surface,
~ ba centre of the actual strain-memory surface,
qa size of the actual strain-memory surface.
240
10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
…25†
; …26†

10.3.1.4 Internal variables for the description of non-proportional loading
The variable Z is used to distinguish between a proportional and a non-proportional
loading within the current loading increment. Z is defined as:
Z ˆ 1
d D
jd r
~ D D
j j r
~ D ~
j
r
~ r
for jd ~ r D j > 0 ; …27†
Z ˆ 0 for jd ~ r D jˆ0 : …28†
The internal variables FS and FL are functions of Z and devp. In the case of proportional
loading, FS and FL are assigned to zero. During non-proportional loading, both
variables rise to the value one. During the process, FL has a temporal delay to FS. Effects,
which occur immediately with the set-in of a non-proportional loading, are controlled
by the variable FS, and those processes, which occur slowly during a non-proportional
loading, are controlled by FL. If the non-proportional loading is followed by a
proportional one, both internal variables decrease to zero again to simulate the erasure
of the additional hardening found in the experiments (see Section 10.2.3.2):
FS ˆ
Z
dFS ; …29†
dFS ˆ W2…Z FS†devp …30†
with W2 ˆ 0:1 tanh…qm=q † for Z FS ;
W2 ˆ 0:01 for Z < FS :
FL ˆ
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
Z
dFL ; …31†
dFL ˆ W3…Z …1 cos 308††devp …32†
with W3 ˆ…1 FL† 0:1
for Z 1 cos 308 ;
W3 ˆ FL for Z < 1 cos 308 :
241

10.3.1.5 Size of the yield surface under uniaxial cyclic plastic loding
The size of the yield surface k is defined to be:
k ˆ ks ‡…k0 km†2 10qm
q …ks km†2 10qs
q …33†
with
km ˆ 0:6 k0 size of the yield surface after the first plastic loading,
ks ˆ 0:4 k0 size of the yield surface for the saturated state.
10.3.1.6 Size of the bounding surface under uniaxial cyclic plastic loading
The mathematical formulation of the bounding surface has to describe a curve, which
is formed from the parts of the monotonic or the cyclic stress-strain curve, which
delimit the current maximum stress for a given strain (see Figure 10.15).
• Section I for qm < q ; qs < q :
^k ˆ const ˆ k0 ; …34†
• Section II for qm q ; qs < q :
^k ˆ k0 ‡ 0:73 …qm q † 0:0162 …qm q † 2 ; …35†
• Section III for qm q ; qs q :
^k ˆ k0 ‡ 0:73 …qm q † 0:0162 …qm q † 2 "
‡ D^k 1
qs
q
#
5
qs
qm
q
q
with
k0 initial size of the yield surface,
D^k cyclic hardening due to uniaxial loading.
10.3.1.7 Overshooting
The overshooting problem is connected with the update of the initial value din. To prevent
the overshooting effect, the initial value din is limited to:
242
10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
…36†
din;min ˆ 0:35 dmax with dmax ˆ 2
r
2…^k k† : …37†
3

10.3.1.8 Additional update of din in the case of biaxial loading
In the case of uniaxial loading, a new din has to be determined as soon as there is an
angle higher than 90 8 between the normal vector of the strain-memory surface Ma and
the normal vector of the yield surface. In the case of biaxial loading, an additional update
is made if the angle between the normal vector of the yield surface and the current
deviatoric stress vector exceeds 30 8.
10.3.1.9 Memory surface F 0
The memory surface F 0 is defined as a surface in the stress space, which allows the
model to remember a previous or current non-proportional loading. The surface is connected
with the internal variables FS and FL. Its formulation is similar to the v. Mises
yield rule:
F 0 ˆ… ~ r 0
0
†…~ r
~
0
0
†
~
2
3 k02 ˆ 0 : …38†
In the case of uniaxial loading, this surface corresponds to the loading surface in size
and location. During a non-proportional loading, the memory surface shows an additional
isotropic hardening:
k 0 Z
ˆ 2:0 P 0 devp ‡ k …39†
with
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
P 0 ˆ 5:0 …k 0 max k 0 †FS …k 0
k†…1 FS† 2 d dD
din
2
; …40†
k 0 max ˆ k0 PAR FS ‡ k ; …41†
P0 plastic modulus for the isotropic hardening of the memory surface,
k0 actual size of the memory surface,
k0 max maximal size of F0 k
,
0 PAR maximal difference between the size of the loading surface and this memory surface.
If the current stress point lies within the memory surface, the location of F 0 remains unchanged.
For a contact of the stress point with the memory surface, the surface moves
in a way that the normal vector of the memory surface corresponds to the normal vector
to the yield surface. Meanwhile, the stress point remains on the memory surface.
The displacement of this surface is defined as:
243

0
ˆ r
~ ~ D
… ~ r D
~ † k0
k
: …42†
An additional considerable influence on the predicted material behaviour is obtained by
a modification of the update behaviour of din, in combination with the definition of F 0 .
For a stress point lying within the memory surface, the value din is constantly updated
so that the material behaves in a quasi-elastic manner. However, in regions with
din < din;min, the constant value din;min (see Equation (37)) instead of the updated din is
taken into consideration.
10.3.1.10 Additional isotropic deformation of the loading surface
due to non-proportional loading
The isotropic deformation of the memory surface F 0 and the loading surface F differs
by the amount of 2:0 R P 0 devp (see Equation (39)). Parallel to the isotropic deformation
of the memory surface F 0 , the loading surface changes in a way that a hardening of the
memory surface causes an additional softening of the loading surface. The amount of
the change of the loading surface is R P 0 devp.
The incremental deformation of the loading surface is given by:
dk ˆ …k0km† b
q abqm
q ln a dqm …ks km† b
q abqs
q ln a dqs
Z
P 0 devp :
The additional deformation of the loading surface can only be expressed by an incremental
form. The incremental equations are formulated in a way that the integrated increments
converge to limits. These limits are the known maximal values of the parameters.
During a cyclic multiaxial loading, where the stress point is located mostly on the
loading surface, this additional deformation of the loading surface generates a deformation
of the calculated stress path.
10.3.1.11 Additional isotropic deformation of the bounding surface
due to non-proportional loading
To describe effects of biaxial loading, an additional isotropic deformation of the bounding
surface is necessary. Therefore, the additional parameters FD and dlim have to be defined.
The definition of the parameter FD depends on the memory surface F 0 . For a
stress path going through the memory surface, this parameter converges to its maximal
value. For a stress point on the memory surface, the parameter is reduced to zero during
cyclic loading:
244
10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
FD ˆ
Z
…43†
PDdevp ; …44†

• stress point within the memory surface and rv < k0:
PD ˆ P …FD;max FD†
FL
FD;max
with FD;max fitparameter,
• stress point on the memory surface and rv < k0:
PD ˆ 0:01P FD
FD;max
• stress point on the memory surface and rv k0:
…45†
; …46†
PD ˆ 0:0 : …47†
The parameter dlim is defined as a limitation of the distance between the loading and
the bounding surface. For values d < dlim, an additional isotropic hardening of the
bounding surface occurs. For d > dlim, this additional hardening of the bounding surface
decreases. The plastic modulus of the additional isotropic deformation of the
bounding surface is given by:
with
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
…FB;max FB†
^PZ ˆ P FS
FB;max
^PZ ˆ P FB
FB;max
0:2
for d dlim ; …48†
…0:001 ‡ FS† for d > dlim …49†
FB;max ˆ…D^k1FS ‡ FD 2:0†FL ; …50†
Figure 10.27: Multiple-step test.
245

246
10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.28: Path 03, eB ˆ 7.1‰.
Figure 10.29: Path 07, eB ˆ 2.8‰.
Figure 10.30: Path 07, eB ˆ 4.9‰.

10.3 Modelling of the Material Behaviour of Mild Steel Fe 510
Figure 10.31: Path 08, eB ˆ 4.9‰.
Figure 10.32: Path 09, eB ˆ 2.8‰.
Figure 10.33: Path 09, eB ˆ 4.9‰.
247

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Z
FB ˆ
^PZdevp ; …51†
dlim ˆ 1:0 …D^k1F 0:2
S ‡ FD 2:0† ; …52†
^PZ plastic modulus for the additional isotropic hardening of the bounding surface due
to non-proportional loading,
D^k material parameter,
FB additional isotropic hardening of the bounding surface due to non-proportional
loading.
The additional hardening resulting from a non-proportional loading FB (Equation (51))
is added to the size of the bounding surface ^k (Equations (34) to (36)) regardless of the
amount of qm and qs.
10.3.2 Comparison between theory and experiments
Figures 10.27 to 10.33 show the experimental results in the left column and the results
of the calculations with the ETS-model in the right column. Note that all calculations
(uniaxial, proportional and non-proportional) are made with the same set of parameters.
It can be seen from these figures that the response of mild steel Fe 510 under uniaxial,
proportional and non-proportional loading histories is well predicted by the model.
10.4 Experiments on Structural Components
10.4.1 Experimental set-ups and computational method
Calculations using several models were made on typical components of steel constructions
like necked girders, girders with holes, or plates with holes. Figures 10.34 to
10.36 show specimens used in these investigations.
All experiments were performed force-controlled. The longitudinal strains in the
interesting sections were measured with strain gauges.
For the description of the structural behaviour, the Finite-Element method was
used. Precise informations concerning details of the computational methods can be
found in the corresponding papers [1, 2, 9, 14].
10.4.2 Correlation between experimental and theoretical results
First, the results of an experiment with a necked girder and the corresponding calculations
are presented. The cyclically loaded girder (Figure 10.35) shows repeating plastic
deformations in the area of the neck.
248

Figure 10.34: Plate with a hole.
Figure 10.35: Necked girder.
10.4 Experiments on Structural Components
249

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
Figure 10.36: Girder with holes.
Figure 10.37: Force-strain diagrams of a necked girder (experiment and calculation).
In the right diagram of Figure 10.37, the force-strain relation measured directly at
the edge of the neck is presented. The force-strain relation predicted by the ETS-model
is plotted in the left diagram of this figure.
The calculated force-strain relation shows a good correspondence with the results
of the experiment. The amount of the increase of the plastic strains per cycle is well
predicted for both amplitudes. The strains of the upper turnover points are almost 10%
smaller than the strains in the experiment. The higher difference of the measured and
calculated strains in the area of the lower turnover points is caused by the more intensive
bulge of the calculated hysteresis-loops.
250

[½]
As a second example, Figure 10.38 shows a comparison between measured and calculated
strains exx of a plate with a hole (X ˆ 0:0 in Figure 10.34) for the first 10 load
steps. It is seen that the differences between the results of the calculations with the models
of Reininghaus (biaxial) [2] and Scheibe (uniaxial, Z ˆ 0) [1] are small. As an additional
result, Figure 10.38 shows that the “biaxial” extensions of the ETS-model to include nonproportional
effects do not influence the results of the calculations of the uniaxial loaded
plate with a hole. If a non-proportional load occurs, distinct differences are obtained
merely because this kind of load is not mentioned in the model of Scheibe.
10.5 Summary
10.5 Summary
Figure 10.38: Plate with a hole, LK2, cycle 1–10, experiment and calculations.
Both the exact knowledge of the material behaviour and a model to simulate this behaviour
are necessary for a precise calculation of the response of structures under plastic
cyclic loads.
Extensive investigations were carried out into the material behaviour of structural
mild steel Fe 510 under uniaxial, load- and strain-controlled loads as well as under different
biaxial proportional and non-proportional loads combined with yield-surface investigations.
251

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading
The two-surface model of Dafalias-Popov was extended to fit the individual characteristics
of mild steel behaviour under cyclic loads. This extended-two-surface model
is able to simulate precisely the behaviour of Fe 510 under various proportional and
non-proportional multiaxial cyclic loads.
Linking up with a Finite-Element program, structures and structural elements undergoing
cyclic or random plastic deformations are calculated.
References
[1] H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Berücksichtigung
in Konstruktionsberechnungen. Tech. Univ. Braunschweig, Institut für Stahlbau, Bericht
Nr. 6314, 1990.
[2] M. Reininghaus: Baustahl Fe 510 unter zweiachsiger Wechselbeanspruchung. Tech. Univ.
Braunschweig, Institut für Stahlbau, Bericht Nr. 6326, 1994.
[3] D. Kuck: Experimentelle Untersuchungen zum Ratchetting-Verhalten bei Baustahl ST52-3.
Dissertation TU Braunschweig, 1996.
[4] S. Dannemeyer: Zur Veränderung der Fließfläche von Baustahl bei mehrachsiger plastischer
Wechselbeanspruchung. Dissertation TU Braunschweig, 1999.
[5] M.J. Michno, W. N. Findley: A Historical Perspective of Yield Surface Investigations for
Metals. Int. J. Non-Linear Mechanics 11 (1976) 59–82.
[6] D. Besdo, N. Wellerdick-Wojtasik: The Influence of Large Torsional Prestrain on the Texture
Development and Yield Surfaces of Polycrystals. This book (Chapter 7).
[7] H. Heuer: Untersuchung zur Anwendbarkeit des Einfließflächen-Modells auf das zyklische
Materialverhalten von Baustahl. Diplomarbeit, Tech. Univ. Braunschweig, 1988.
[8] Y. F. Dafalias, E.P. Popov: A Model of Nonlinearly Hardening Materials for Complex Loadings.
Acta Mechanica 21 (1975) 173–192.
[9] J. Scheer, H.-J. Scheibe, D. Kuck, M. Reininghaus: Stahlkonstruktionen unter zyklischer Belastung.
Arbeits- und Ergebnisbericht 1987–1990. Subproject B5, Collaborative Research
Centre (SFB 319): Stoffgesetze für das inelastische Verhalten metallischer Werkstoffe – Entwicklung
und Technische Anwendung, Tech. Univ. Braunschweig, Institut für Stahlbau,
1990.
[10] U. Peil, D. Kuck: Stahlkonstruktionen unter zyklischer Belastung. Arbeits- und Ergebnisbericht
1991–1993. Subproject B5, Collaborative Research Centre (SFB 319): Stoffgesetze für
das inelastische Verhalten metallischer Werkstoffe – Entwicklung und Technische Anwendung,
Tech. Univ. Braunschweig, Institut für Stahlbau, 1993.
[11] U. Peil, M. Reininghaus: Baustahl unter mehrachsiger zyklischer Belastung. Arbeits- und
Ergebnisbericht 1991–1993. Subproject B5, Collaborative Research Centre (SFB 319):
Stoffgesetze für das inelastische Verhalten metallischer Werkstoffe – Entwicklung und Technische
Anwendung, Tech. Univ. Braunschweig, Institut für Stahlbau, 1993.
[12] M. Reininghaus: Baustahl ST52 unter plastischer Wechselbeanspruchung. Dissertation TU
Braunschweig, 1994.
[13] Z. Mróz: An Attempt to Describe the Behavior of Metals under Cyclic Loads Using a More
General Workhardening Model. Acta Mechanica 7 (1968) 199–212.
[14] J. Scheer, H.-J. Scheibe, D. Kuck: Zum Verhalten ausgeklinkter Träger unter zyklischer
Beanspruchung. Bauingenieur 65 (1990) 463–468.
252

11 Theoretical and Computational Shakedown Analysis
of Non-Linear Kinematic Hardening Material
and Transition to Ductile Fracture
Abstract
Erwin Stein, Genbao Zhang, Yuejun Huang, Rolf Mahnken
and Karin Wiechmann*
The research work of this project is based on Melan’s static shakedown theorems for
perfectly plastic and linear kinematic hardening materials. Using a 3-D generalization
of Neal’s 1-D model for limited hardening, a so-called overlay model, a new theorem
and a corollary are derived for general non-linear kinematic hardening materials. For
the numerical treatment of the structural analysis of 2-D problems, the Finite-Element
method (FEM) is used, while enhanced optimization algorithms are used to perform the
shakedown analysis effectively. This will be demonstrated with some numerical examples.
Treating a crack as a sharp notch, the shakedown behaviour of a cracked ductile
body is investigated and thresholds for no crack propagation are formulated.
11.1 Introduction
11.1.1 General research topics
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
The response of an elastic-plastic system subjected to variable loadings can be very
complicated. If the applied loads are small enough, the system will remain elastic for
all possible loads. Whereas if the ultimate load of the system is attained, a collapse
mechanism will develop and the system will fail due to infinitely growing displacements.
Besides this, there are three different steady states that can be reached, while the
loading proceeds: 1. Incremental failure occurs if at some points or parts of the system,
* Universität Hannover, Institut für Baumechanik und Numerische Mechanik, Appelstraße 9 a,
D-30167 Hannover, Germany
253

11 Theoretical and Computational Shakedown Analysis
the remaining displacements and strains accumulate during a change of loading. The
system will fail due to the fact that the initial geometry is lost. 2. Alternating plasticity
occurs, this means that the sign of the increment of the plastic deformation during one
load cycle is changing alternately. Though the remaining displacements are bounded,
plastification will not cease and the system fails locally. 3. Elastic shakedown occurs if
after initial yielding plastification subsides and the system behaves elastically due to
the fact that a stationary residual stress field is formed and the total dissipated energy
becomes stationary. Elastic shakedown (or simply shakedown) of a system is regarded
as a safe state. It is important to know if a system under given variable loadings shakes
down or not.
11.1.2 State of the art at the beginning of project B6
In 1932, Bleich [1] was the first to formulate a shakedown theorem for simple hyperstatic
systems consisting of elastic, perfectly plastic materials. This theorem was then
generalized by Melan [2, 3] in 1938 to continua with elastic, perfectly plastic and linear
unlimited kinematic hardening behaviour. Koiter [4] introduced a kinematic shakedown
theorem for an elastic, perfectly plastic material in 1956, that was dual to Melan’s
static shakedown theorem. Since then, extensions of these theorems for applications
of thermoloadings, dynamic loadings, geometrically non-linear effects and internal
variables have been carried out by different authors (Corradi and Maier [5], König [6],
Maier [7], Prager [8], Weichert [9], Polizzotto et al. [10]). However, little progress has
been made in the formulation of a corresponding shakedown theorem for materials
with non-linear kinematic hardening. The only attempt was made by Neal [11], who
formulated a static shakedown theorem for materials with non-linear kinematic hardening
in a 1-D stress state by using the Masing overlay model [12]. Several papers were
published concerning especially 2-D and 3-D problems (Gokhfeld and Cherniavsky
[13], König [14], Sawczuk [15, 16], Leckie [17]). The shakedown investigation of
those problems leads to grave mathematical problems. Thus, in most of these papers,
approximate solutions based on the kinematic shakedown theorem of Koiter [4] or on
the assumption of a special failure form were derived. But these solutions often lost
their bounding character due to the fact that simplifying flow rules or wrong failure
forms were estimated. Until the beginning of project B6, only a few papers were published,
in which the Finite-Element method was used for the numerical treatment of
shakedown problems (Belytschko [18], Corradi and Zavelani [19], Gross-Weege [20],
Nguyen Dang and Morelle [21], Shen [22]).
11.1.3 Aims and scope of project B6
In the framework of the geometrically linearized theory, the shakedown behaviour of
linear elastic, perfectly plastic, of linear elastic, linear unlimited kinematic hardening
and of linear elastic, non-linear limited kinematic hardening materials were taken under
254

1.1 Introduction
consideration. The theoretical and numerical treatment of the shakedown behaviour of
these material models was one major scope of project B6. Based on static shakedown
theorems, the numerical treatment of 2-D and 3-D field problems for arbitrary non-linear
kinematic hardening materials by Finite-Element method should be realized. One
special task was the formulation and the proof of a static shakedown theorem for limited
non-linear kinematic hardening materials. In Section 11.2, a 3-D overlay model is
presented, that was developed to describe non-linear, limited kinematic hardening
material behaviour. This model is an extension of the 1-D overlay model of Neal [11].
A static shakedown theorem and a corollary, that were formulated and proved for the
proposed material model, are extensions of Melan’s static shakedown theorems for perfectly
plastic and linear kinematic hardening materials [2].
While analytical solutions of shakedown problems can only be derived for very
simple systems, Finite-Element methods based on displacement methods should be
used for the numerical treatment and solution of 2-D and 3-D shakedown problems.
After discretizing the system and accounting for the shakedown conditions, usually a
non-linear mathematical optimization problem is derived, that is very large scaled.
Solving optimization problems like these is normally very difficult. Thus, effective optimization
algorithms should be formulated and implemented, that were designed especially
to take account of the special structure of the problems. Section 11.3 is concerned
with the numerical approach based on static shakedown theorems. The discretized optimization
problems for the proposed material models are discussed briefly.
In Section 11.3.5, numerical examples show the effectivity of the proposed formulation.
Solutions for perfectly plastic and kinematic hardening materials are compared.
One important scope of project B6 was the examination of hardening and softening
materials. While classic shakedown theorems imply implicitly that a material under cyclic
loading behaves stable after only one or two loading cycles, experimental investigations
show that stable cycles can be reached only after several loading cycles and
sometimes only asymptotically. Thus, the influence of cyclic hardening and softening
on the shakedown behaviour of materials had to be taken under consideration. The examination
of cyclic hardening material with the Chaboche constitutive equation [23],
will be considered in Section 11.3.5.3. An incremental-failure analysis for this material
is carried out, and the results are compared with those of the 3-D overlay model described
in Section 11.2.
Stress singularities occur if macroscopic cracks develop in a solid material. Under
these circumstances, classical shakedown theorems cannot be used. Thus, one major
aim of project B6 was to apply shakedown theory directly to fatigue fracture to include
stress singularities into shakedown investigations. In Section 11.4, we will apply shakedown
theory to fatigue fracture to derive thresholds for no crack propagation. Classic
shakedown theorems predict a zero shakedown limit load for a cracked body because
of singular stresses at the crack tip. But experiments for ductile materials show that
limits exist, for which no crack propagation occurs. We will consider the crack as a
sharp notch, the notch root of it being a material constant at threshold level (Neuber
[24]). The threshold of a fatigue crack follows then from the stationarity of the plastic
energy dissipated in the cracked body.
255

11.2 Review of the 3-D Overlay Model
There exist many mathematical models to describe the kinematic hardening behaviour
of materials, for example the Prager linear kinematic hardening model [25], its modification
by Ziegler [26], Mróz multisurface model [27], Dafalias and Popov’s two-surface
model [28] and so on. In Stein et al. [29], a so-called 3-D overlay model was developed
to describe the non-linear kinematic hardening material behaviour. We will
give here a brief review of the proposed model.
A macroscopic material point x 2 X IR 3 is assumed to be composed of a spectrum
of microscopic elements (or microelements). Each microelement is numbered with
a scalar variable n 2‰0; 1Š. Stresses and strains are separately defined for the macroscopic
material point (macrostress and macrostrain) and the microelements (microstresses
and microstrains). They are denoted by r…x†; e…x† and w…x; n†; g…x; n†, respectively.
The macrostress r has to fulfil the equilibrium condition:
div r…x† ˆb…x†; 8 x 2 X : …1†
In the framework of geometrically linear continuum mechanics, the kinematic relation
e…x† ˆ 1
2 …ru…x†‡…ru…x††T †; 8 x 2 X ; …2†
holds between the displacement u and the strain e.
By assuming that the stress r of a macroscopic material point (macrostress) is the
weighted sum of the stresses w of all microelements (microstresses), the macroscopic
material point and the corresponding microelements deform in the same way, and we
get the following static and kinematic relations:
r…x† ˆ
Z 1
0
w…x; n†dn ; …3†
g…x; n† ˆe…x†; 8 n 2‰0; 1Š : …4†
Furthermore, we suppose that all microelements are linear elastic, perfectly plastic and
have the same temperature, the same elastic moduli, but different yield stresses k…n†.
For convenience, k…n† can be considered as a monotone growing function of n. Additionally,
the validity of the additive decomposition of the microstrain g in an elastic
and a plastic part is supposed. Thus, the following relations can be derived for the microelements:
256
11 Theoretical and Computational Shakedown Analysis
g…n† ˆg E …n†‡g P …n† ; …5†

g E …n† ˆE 1 w…n† ; …6†
U…w† k 2 …n† ; …7†
_g P …n† ˆ _ k…n† qU
qw ; _ k…n† 0 ; …8†
_k…n†‰U…w† k 2 …n†Š ˆ 0 ; …9†
where E stands for the symmetrical elasticity tensor and U… † is the yield function. For
simplicity, the argument x of the fields will be omitted partly.
Assume that at a certain macrostress, the microelement number n begins to yield,
the function k…x; n† is then uniquely determined by the macroscopic r; e-function in the
1-D case:
r…n† ˆ
Z n
0
k…n†dn ‡…1 n†k…n† : …10†
It is easy to show that, similar to k…n†; r is also a monotone growing function of the
variable n (see Figure 11.1). For n ˆ 0, we have r ˆ r…n ˆ 0† ˆk…n ˆ 0† ˆk0. The
maximum value of r (denoted by rY or K) is derived by setting n ˆ 1 in Equation
(10):
rY ˆ r…n ˆ 1† ˆK ˆ
11.2 Review of the 3-D Overlay Model
Z 1
0
k…n†dn : …11†
Thus, k0 and K can be identified with the initial yield stress ro and the ultimate stress
of the macroelement rY, respectively. For the 3-D case, there is an analogous relation
between r and k…n† as Equation (10), namely:
Figure 11.1: Kinematic hardening of a macroscopic material point and yield stresses of the microelements
in 1-D case.
257

U…r† ˆ
Z n
nˆ0
k…n†dn ‡…1 n†k…n† : …12†
Now, we define the difference between the microstress w…n† and the macrostress r as
the residual microstress and denote it by p…n†:
p…n† ˆw…n† r : …13†
For the residual microstress defined in this way, we have:
and
p…n† ˆ0; n 2‰0; 1Š; for krk ro ; …14†
p…n† 6ˆ 0; n 2‰0; 1Š; for krk > ro : …15†
It is necessary to notice that Equation (15) holds for all n 2‰0; 1Š even if U…r† is smaller
than k…n†.
Integration of Equation (13) yields:
Z 1
0
p…x; n†dn ˆ 0; 8 x 2 X ; …16†
where Equation (3) has been used. From Equation (13) can be concluded that the resultant
of p…n† does not contribute to the macrostress r. All microstress fields, which fulfil
Equation (16), can be regarded as residual microstress fields.
For a kinematic hardening material, it is possible to represent the residual microstress
p…x; n† by a backstress …x†, which describes the translation of the initial yield
surface:
p…x; n† ˆ
11 Theoretical and Computational Shakedown Analysis
U…r† k…n†
U…r† k…0†
…x† : …17†
It is easy to show that the microstress in Equation (17) satisfies Equation (16).
For the 3-D overlay model described above, the following static shakedown theorem
has been formulated and proved (e.g. Stein et al. [30], Zhang [31]):
Theorem 1: If there exist a time-independent residual macrostress field q…x† and a
time-independent residual microstress field p…x; n†, satisfying
U‰mp…x; 0†Š ‰K…x† k0…x†Š 2 ; 8 x 2 X ; …18†
such that for all possible loads within the load domain, the condition
258

Ufm‰r E …x; t†‡q…x†‡p…x; 0†Šg k 2 0…x† …19†
is fulfilled 8 x 2 X and 8 t > 0, where m > 1 is a safety factor against inadaptation, then
the total plastic energy dissipated within an arbitrary load path contained within the load
domain is bounded, and the system consisting of the proposed material will shake down.
The static shakedown theorem 1 is formulated by using the residual microstress p. Considering
the relation between the backstress and the residual microstress p in Equation
(17), it is also possible to formulate the following corollary directly in terms of the backstress
, i.e.:
Corollary 1: If there exist a time-independent residual macrostress field q…x† and a
time-independent field …x†, satisfying
U‰m …x†Š ‰K…x† k0…x†Š 2 ; 8 x 2 X ; …20†
such that for all possible loads within the load domain, the condition
Ufm‰r E …x; t†‡q…x† …x†Šg k 2 0…x† …21†
is fulfilled 8 x 2 X and 8 t > 0, then the system will shake down.
For the formulation of the static shakedown theorem 1 and the corresponding corollary
1, only the values of k0 and K have been used. That means that the shakedown limits
for systems of the considered material do not depend on the particular shape of the
function k…n†, and therefore do not depend on the particular r; e-curve but, solely, on
the magnitudes of k0 and K.
For K…x† ˆk0…x† (an elastic, perfectly plastic material), we have ˆ 0 due to
Equation (20), and theorem 1 reduces to Melan’s theorem [2, 3] for an elastic, perfectly
plastic material. For K…x† !‡1 (materials with unlimited kinematic hardening), the
constraint (Equation (20)) imposed on the backstresses …x†, can never become active
and therefore may be dropped. In this case, we get the static shakedown theorem of
Melan for a linear, unlimited kinematic hardening material.
11.3 Numerical Approach to Shakedown Problems
11.3.1 General considerations
11.3 Numerical Approach to Shakedown Problems
For the numerical approach of shakedown problems, both the static and the kinematic
theorems can be employed. However, the use of the static theorems has the advantage
that the discretized optimization problem is regular. Therefore, only static theorems
were used for the following considerations. Furthermore, for the Finite-Element discre-
259

tization, only elements based upon the displacement method were used. In the following
sections, we will limit our attention to plane-stress problems (for more details and
other stress situations, see e.g. Zhang [31], Stein et al. [32], Mahnken [33]).
11.3.2 Perfectly plastic material
We are interested in finding the maximal possible enlargement of the load domain allowing
still for shakedown. Thus, for systems consisting of linear elastic, perfectly plastic
material, we get the following optimization problem in matrix notation:
b ! max ; …22†
X NG
iˆ1
11 Theoretical and Computational Shakedown Analysis
Ciq i ˆ Cq ˆ 0 ; …23†
U‰br E i …j†‡qiŠ r 2 o ; 8…i; j† 2I J ; …24†
for load domains of the form of a convex polyhedron with M load vertices. Here, b is
the maximal possible enlargement of the load domain, NG is the number of Gaussian
points. Iˆ‰1; ...; NGŠ and Jˆ‰1; ...; MŠ are sets containing all Gaussian points and
load vertices, respectively. C is a system-dependent matrix and rE i …j† is the elastic
stress vector at the i-th Gaussian point, which corresponds to the j-th load vertex P…j†
of the load domain. The Equations (23) and (24) represent the discretized static equilibrium
conditions for the residual stresses and the shakedown conditions controlled at the
Gaussian points, respectively. qi is the residual stress vector in the i-th Gaussian point.
In general, the discretized shakedown problem (Equations (22) to (24)) is a large
scaled optimization problem. Direct use of standard optimization algorithms such as the
sequential quadratic programming method (SQP-method) is not effective. Thus, two optimization
algorithms were formulated and implemented to take account of the special
structure of the problem (Equations (22) to (24)).
11.3.2.1 The special SQP-algorithm
Dual methods do not attack the original constrained problem directly, but instead attack
an alternative problem, the dual problem, whose unknowns are the Lagrange multipliers
of the primal problem.
In order to solve the dual problem, a projection method is used. Each subproblem
is then solved iteratively. The iteration matrix needed to do so was implemented due to
Bertsekas’s algorithm. To make the algorithm even more effective for large sized problems,
a Quasi-Newton method was used and a BFGS-update formula was implemented
(for details of the proposed methodology, see Stein et al. [32], Mahnken [33]).
260

11.3.2.2 A reduced basis technique
The main idea is to solve Equations (22) to (24) in a sequence of reduced residual
stress spaces of very low dimensions. Starting from the known state b …k 1† and q …k 1† ,
a few …r† basis vectors b p;k ; p ˆ 1; ...r, are selected from the residual stress space B d .
They form a subspace B r;k (or reduced residual stress space) of B d . The improved state
…k† is determined by solving the reduced optimization problem. Due to its low dimension,
the reduced problem can be solved very efficiently by using a SQP- or a penalty
method. The k-th state is obtained by an update for q k and Db k . Selecting new reduced
basis vectors, the process is repeated until a convergence criterion for Db k is fulfilled.
One way for generating the basis vectors is to carry out an equilibrium iteration. The
intermediate stresses during the iteration are in equilibrium with the same external load,
and their differences are thus residual stresses. Details can be found in [29–31].
11.3.3 Unlimited kinematic hardening material
For the investigation of the shakedown behaviour of systems consisting of unlimited kinematic
hardening material, we restrict the load domain in the same way as in Section
11.3.1, and the following optimization problem can be formulated [31]:
b ! max ; …25†
X NG
iˆ1
11.3 Numerical Approach to Shakedown Problems
Ciq i ˆ 0 ; …26†
U‰br E i …j†‡qi iŠ r 2 o ; 8…i; j† 2I J ; …27†
where i is the affine backstress vector at the i-th Gaussian point.
Defining now vectors y i in such a way that
y i ˆ q i i ; …28†
where the y i are not restricted. Thus, the equality constraints (Equation (26)) of Equations
(25) to (27) can be dropped:
b ! max ; …29†
U‰br E i …j†‡yiŠ r 2 o ; 8…i; j† 2I J : …30†
A modified optimization problem is derived, that has a very simple structure. Zhang
[31] formulated and proved a lemma to solve this problem:
261

11 Theoretical and Computational Shakedown Analysis
Lemma 1: The maximal shakedown load factor b s of Equations (29) and (30) can be
derived through
b s ˆb; with b ˆ min
i2I b i ; …31†
where b i, …i 2I†, is the solution of the subproblem
b i ! max ; …32†
U‰bir E i …j†‡yiŠ r 2 o : …33†
The dimension of Equations (32) and (33) is very low. Thus, it can be solved effectively
with a usual SQP-method.
Relationship between perfectly plastic and kinematic hardening material
The shakedown load of a system consisting of unlimited kinematic hardening material
is determined by that point xip , where the maximum possible enlargement of the elastic
stress domain S E
is the least in comparison to other points. Thus, the failure is of local
ip
character. This reflects the fact that a system, that consists of unlimited kinematic hardening
material and is subjected to cyclic loading, can fail only locally in form of alternating
plasticity [34]. Incremental collapse cannot occur since it is connected with a
non-trivial, kinematic compatible plastic strain field, which has global character.
The shakedown load of a system consisting of perfectly plastic material cannot be
larger than the shakedown load of the same system consisting of unlimited kinematic
hardening material with the same initial stress ro. However, it is possible that the
shakedown loads for perfectly plastic and kinematic hardening material are identical.
This is the case only if alternating plasticity is dominant in both cases. It is easy to
show that
q ip ˆ y ip …34†
holds.
Concluding, the following conclusions can be drawn from lemma 1:
1. A system consisting of unlimited kinematic hardening material and subjected to
variable loading can fail only locally in form of alternating plasticity.
2. The kinematic hardening does not influence the shakedown load if the same system
with perfectly plastic material, subjected to the same loading, fails in form of
alternating plasticity in such a manner that there exists at least one point xip , for
which the enlarged elastic stress domain bsS E
is just contained in the yield surface
ip
shifted to qip . A further shift of the yield surface at this point would cause that a
portion of the enlarged elastic domain b sS E
leaves the yield surface. In the se-
ip
quel, the alternating plasticity with the special character mentioned before will be
denoted by APSC. In all other cases, an increase of the shakedown load due to kinematic
hardening is expected.
262

3. If the shakedown loads for perfectly plastic and unlimited hardening material are
identical, then alternating plasticity is the dominant failure form in both cases.
4. The shakedown load determined for perfectly plastic material is exact if it is identical
with that determined for unlimited kinematic hardening material, provided the
latter is determined exactly.
11.3.4 Limited kinematic hardening material
As mentioned in Section 11.2, the shakedown limit of a system consisting of limited kinematic
hardening material depends only on the values k0 and K. For this reason, the given
function k…n† may be replaced by a step function of n 2‰0; 1Š. The step function has to be
chosen such that its minimum is equal to k0 and its area is equal to K. That is,
K ˆ Xm
lˆ1
Dn l kl ; m 2 ; …35†
where m is the number of the intervals of the step function and kl is the value of the
step function of the l-th interval with the length Dn l.
For plane stress problems, the microelements may be incorporated in a natural
way. The thickness t of the structure is divided into several (m 2) layers with the
thicknesses tl, l ˆ 1; 2; ...; m. Each layer behaves elastic, perfectly plastic and has a
corresponding yield stress kl (one value of the step function). The thicknesses of the
layers have to be chosen such that
tl
t ˆ Dn l ; 8 l 2‰1; ...; mŠ : …36†
By doing so, a unique relation between the layers of the structure and the intervals of
the step function is established. The microelements of the l-th interval are replaced by
the l-th layer of the structure. The parallel connection of the microelements is realized
by discretizing all layers in the same way. That is, elements that lie on top of each
other have the same nodes. Thus, it is guaranteed that elements of different layers have
the same kinematics.
Dividing the structure into Ne elements with NG Gaussian points and m layers,
we get the discretized shakedown problem (Stein et al. [32]):
b ! max ; …37†
XNG X
Ci
iˆ1
m
sˆ1
11.3 Numerical Approach to Shakedown Problems
ts
t qi;s ˆ XNG
iˆ1
Ciq i ˆ Cq ˆ 0 ; …38†
U…br E i …j†‡qi;1† k 2 1 ˆ k2 0 ; 8…i; j† 2I J ; …39†
263

11 Theoretical and Computational Shakedown Analysis
U…pi;1† …K k1† 2 ˆ…K k0† 2 ; 8 i 2I: …40†
The Equations (38), (39) and (40) represent the discretized static equilibrium conditions
for the residual macrostresses and the shakedown conditions controlled at the Gaussian
points, respectively. q i;s and pi;s are the residual stress vector and the residual microstress
vector in the s-th layer of the i-th Gaussian point. Between q i;s; pi;s and the residual
macrostress vector q i, the following relation holds:
q i;s ˆ q i ‡ pi;s : …41†
For m ˆ 1 and K ˆ k1, Equations (37) to (40) reduce to the discretized shakedown
problem for a perfectly plastic material. In this case, we have pi;1 ˆ 0; q i ˆ q i;1 and the
constraint (Equation (40)) can be dropped. We come to the other extreme case by assuming
K !1, which corresponds to an unlimited kinematic hardening material. Due
to K !1, the constraint (Equation (38)) can never become active, and thus pi;1 is not
constrained. Correspondingly, the constraint (Equation (40)) may be dropped as well.
To solve the optimization problem (Equations (37) to (40)) effectively, the reduced
basis technique presented in Section 11.3.2.2 was extended (for details, see e.g.
Zhang [31], Stein et al. [32]).
11.3.5 Numerical examples
In this section, numerical examples of different structures are considered. The influence
of kinematic hardening on the shakedown limit is demonstrated.
11.3.5.1 Thin-walled cylindrical shell
A cylindrical shell with wall thickness d and radius R is subjected to an internal pressure
p and an internal temperature Ti (Figure 11.2). The external temperature Te is
equal to zero for all times t. For Ti ˆ Te and p ˆ 0, the system is assumed to be stressfree.
The pressure and the temperature can vary between zero and their maximum values
p and Ti. The corresponding load domain is defined by:
0 p bc 1po ˆ p ; 0 c 1 1 ; 0 Ti bc 2T o i ˆ Ti ; 0 c 2 1 : …42†
To visualize the influence of kinematic hardening on the shakedown limits, the following
three constitutive laws are considered:
1. Elastic, perfectly plastic material with initial yield stress ko
11.2 b).
(curve 1 in Figure
2. Limited kinematic hardening material with K ˆ 1:35 ko (curve 2 in Figure 11.2 b).
3. Unlimited kinematic hardening with K ˆ‡1(curve 3 in Figure 11.2 b).
264

Curves 1, 2 and 3 in Figure 11.2b show that the kinematic hardening does not always
increase shakedown limits. The common part of the curves represents load domains,
which lead exclusively to APSC (Section 11.3.3.1) in all three cases, whereas both alternating
plasticity and incremental failure can occur for the remaining load domains.
11.3.5.2 Steel girder with a cope
11.3 Numerical Approach to Shakedown Problems
Figure 11.2: Thin-walled cylindrical shell: a) system and loads; b) shakedown diagram.
A steel girder with a length of 4 m will be investigated. It consists of an IPB-500 profile
with a cope on either side. The girder is simply supported and, in the middle, it is
subjected to a concentrated single load P. Both corners of the copes are provided with
a round drill hole of radius r ˆ 8 mm (see Figure 11.3a). The material is St 52-3 with
an initial yield stress ro ˆ 37:5 kN/cm 2 and a maximal hardening rY ˆ 52:0 kN/cm 2 .
The hardening can be regarded as kinematic.
Experimental investigations
At the Institute for Steel Construction of the University of Braunschweig, the girder
was investigated experimentally [35]. First of all, the behaviour of the system subjected
to cyclic loading was of interest. Additionally, for comparison, the ultimate load was
determined experimentally as well.
Firstly, the girder was subjected to different cyclic load programs. The load program
of the first 15 cycles is shown in Table 11.1. Then, the girder was subjected to a
load program, where the load varied between 0 and 600 kN with a velocity of 600 kN/
min. The number of load cycles, that led to a crack (with a length of 1 mm) at a drill
hole, was 145. The number of load cycles, that led to a collapse of the girder, was 372.
After collapse due to cyclic loading, the girder was shortened on either side by
50 cm, and it was recoped as before. Then, for this system, the ultimate load was determined
as 887 kN. Note that this value can also be regarded as the ultimate load of the
initial system since only those cross-sections near the cope are responsible for failure
of the system.
265

11 Theoretical and Computational Shakedown Analysis
Figure 11.3: a) Discretization of the steel girder; b) load-strain diagram at one of both drill holes
of the girder.
Table 11.1: Loading program of the first 15 cycles for a steel girder.
Cycles 1–5 Cycles 6–10 Cycles 11–15
P: 0 > 540 kN P: 0 > 320 kN P: 0 > 540 kN
Numerical investigations
Due to the symmetry of the system and the loading, only one quarter of the system was
considered for numerical investigation. For the Finite-Element discretization, two different
types of elements were employed. The web of the girder was discretized by using 8128
isoparametric elements each with 4 nodes (see Figure 11.3 a). The upper and the lower
flanges were discretized with 768 and 640 DKT (discrete Kirchhoff triangle)-elements,
respectively. Apart from bending forces, the DKT-elements can also be stressed in plane
in order to take account of the fact that the flanges are not subjected to pure bending.
For the numerical investigation, both the ultimate load and the shakedown load
were calculated. The solutions were obtained by the reduced basis technique. In order
to demonstrate the influence of kinematic hardening on the ultimate load and on the
shakedown load, respectively, the calculations were performed both for a perfectly plastic
and a kinematic hardening material. The results are shown in Table 11.2.
Note that the value for the ultimate load determined numerically (877.7 kN) was
1% lower than the value determined by experiment (887 kN).
From Table 11.2, it can be seen that, while the ultimate load increases by a factor of
rY=ro due to kinematic hardening, the shakedown load remains unaltered. In this case, the
girder fails due to alternating plasticity near the drill holes (see Section 11.3.2.1).
266

11.3 Numerical Approach to Shakedown Problems
Table 11.2: Numerical ultimate and shakedown load for a steel girder.
Material type Ultimate load in kN Shakedown load in kN
1. Perfectly plastic 633.2 164.2
2. Kinematic hardening 877.7 164.2
It should be pointed out that, originally, the experiment was not intended to analyse
shakedown behaviour. During the experiment, the amplitudes of the cyclic loads
were higher than the theoretical shakedown load. Thus, no shakedown behaviour could
be observed. However, some valuable information can be drawn from the load-strain
diagram for the first 5 load cycles shown in Figure 11.3 b. The strains were measured
directly at one of the drill holes.
In Figure 11.3 b, a region can be observed, where the load P is linearly proportional
to the strains, i.e., where the system behaves purely elastic. The amplitude of the
region is between 155.2 kN and 179.5 kN. A comparison shows that the numerical results
are in good agreement with the experiment.
11.3.5.3 Incremental computations of shakedown limits of cyclic kinematic hardening
material
To describe the cyclic kinematic hardening behaviour of materials, many models have
been developed. Mróz’s multisurface model [27], the two-surface model of Dafalias
and Popov [28] and Chaboche’s model [23] are three of the best known examples.
Here, we use Chaboche’s model as an example and investigate its shakedown behaviour.
Due to the fact that no shakedown theorems for cyclic hardening materials have
been formulated yet, an incremental method will be used to calculate the shakedown limit.
Examples and comparison
As our first example, we consider the square plate with circular hole illustrated in Figure
11.4. The length of the plate is L and the ratio between the diameter D of the hole
and the length of the plate is 0.01. The thickness of the plate is t ˆ 1 cm. The system
is subjected to the biaxial loading p1 and p2. Both can vary independently between
zero and their maximum values p1 and p2. The corresponding load domain is defined
by:
0 p1 bc 1ro ˆ p1 ; 0 c 1 1 ; …43†
0 p2 bc 2ro ˆ p2 ; 0 c 2 1 : …44†
The results are shown in Table 11.3, where b pih denotes the shakedown limit for the
path-independent hardening material and b cyh the shakedown limit for the cyclic kinematic
hardening material.
267

11 Theoretical and Computational Shakedown Analysis
Figure 11.4.: Geometry and loading conditions of a square plate with a centric circular hole.
Table 11.3: Shakedown limits for a plate with centric hole.
p1=p2 b pih b cyh N
0.0/1.0 0.69633 0.69629 300
0.2/1.0 0.65458 0.65417 20
0.4/1.0 0.61758 0.61750 20
0.6/1.0 0.58433 0.58417 20
0.8/1.0 0.55471 0.55417 20
1.0/1.0 0.52775 0.52707 20
It is shown that the results from optimization methods with cyclic independent hardening
properties are upper bounds for those from incremental computation with cyclic
kinematic hardening materials. It turned out that for about 20 cycles, the values b cyh
approach b pih with an error less than 1%. It should be pointed out that the computation
efforts for the cyclic processes are much higher in comparison with optimization methods.
The second example to be considered is a CT-specimen with a notch as shown in
Figure 11.5. It is subjected to a uniaxial loading p, which may vary between 0 and p.
Five different values of notch root radius r are used to obtain a wide range of
shakedown limits. Table 11.4 shows the results of the incremental computations.
For this example too, the shakedown limits are the same as those of numerical
optimization apart from the fact that the number of load paths, which we have taken
for the incremental hardening, has no influence on the shakedown limit of the system.
Remark 1: Elastic shakedown does not implement damage and creep phenomena. Engineers
are interested in the admissible number of cycles for low-cycle fatigue,
which cannot be derived from classical shakedown theorems. An approximated
reduced load factor b < b for scalar-valued damage can be achieved in a postprocess
assuming conservatively that the loads always alternate between their
maximum and minimum values in all load cycles.
268

11.4 Transition to Ductile Fracture
Figure 11.5: Geometry and loading conditions of a compact tension specimen.
Table 11.4: Shakedown limits for a CT-specimen.
Remark 2: There is an important connection between shakedown theory and structural
optimization admitting inelastic deformations, which is a demanding task in the
frame of the design under modern safety considerations related to the failure of a
structure as given in the new EURO-CODES.
11.4 Transition to Ductile Fracture
b pih b cyh N
r ˆ 0:1 0.069325 0.069292 20
r ˆ 0:2 0.085133 0.085083 20
r ˆ 0:3 0.092504 0.092083 20
r ˆ 0:4 0.101117 0.101042 20
r ˆ 0:5 0.107392 0.107292 20
To solve the optimization problem (Equations (37) to (40)) for a system with complicated
geometry and load domain, the numerical methods like Finite-Element method should be
usually used (see [32, 36]). For a problem with no more than two loading parameters (load
domain with four vertices), Stein and Huang [37] developed an analytical method for determination
of the shakedown-load factor b. The advantage of this method is that only the
maximum effective stress in the system must be calculated. The shakedown-load factor b
follows directly from a closed form. For a load domain with only one parameter (load
domain with two vertices), the result for b is especially simple, it reads:
269

ˆ 2ro
reff
; …45†
where reff is von Mises effective stress and ro is the elastic limit. From Equation (45)
can be concluded that the shakedown limit load of the system is as large as twice as its
elastic limit.
Making use of this result, Huang and Stein [38] calculated the shakedown limit
load of a notched body under variable tension r as illustrated in Figure 11.6 a. It reads:
b ˆ 2ro
ˆ
reff
11 Theoretical and Computational Shakedown Analysis
Figure 11.6: a) A compact tension specimen under cyclic loading; b) modified notch; c) modified
crack.
p
ro pr
K 1 m ‡ m2 p ; …46†
where K is the stress-intensity factor (SIF), m is Poisson’s ratio. Thus, the maximal
stress-intensity factor Ksh, under which the system will still shake down, reads:
Ksh ˆ bK ˆ ro
p
pr
1 ‡ m m2 p : …47†
If the applied SIF K does not exceed the shakedown limit SIF Ksh, the notched body
shakes down. Otherwise, alternating plasticity occurs at the notch root.
Shakedown limit SIF for a cracked body
In [38], Huang and Stein applied Neuber’s material block concept [24] to the shakedown
investigation of a cracked body. Accordingly, the continuum ahead of a sharp
notch is considered as a material block with finite linear dimension e (Figure 11.6 b).
Across this block, no stress gradient can develop. The original notch should be replaced
by an effective notch with radius r …> r†. The stress-concentration factor is re-
270

duced due to the enlarged notch radius. The classic strength theory is then still usable.
The effective notch radius r is obtained in such a way that the average stress over the
block e of the original notch is equal to the maximum stress of the modified notch, i.e.:
rymax j r
Z
1
ˆ
e
e
ryjrdr1 : …48†
0
Using Craeger’s relation for the stress distribution [39] at the notch root, the following
relation can be derived:
2K
p ˆ
pr
1
e
For r , one gets:
Z e
0
K
r
p 1 ‡
2p…r=2 ‡ r1† 2…r=2 ‡ r1† dr1
2K
ˆ p : …49†
p…r ‡ 2e†
r ˆ r ‡ 2e : …50†
Thus, the effective notch radius is equal to the original one plus two times Neuber’s
material block size e.
In the case of a crack (Figure 11.6 c), the effective crack-tip radius, denoted by q, can
be obtained immediately by setting r ) r ‡ 2e and r ˆ 0 in Equation (47), yielding:
q ˆ 2e : …51†
Equation (51) indicates implicitly that a crack can be treated as a notch with radius q.
In fact, experiments with different materials done by Frost [40], Jack and Price [41],
and Swanson et al. [42] show that a cracked body under cyclic loadings behaves in an
identical manner as a notched body does if the latter has the same geometry as the
cracked body and the notch root radius r is small enough (see also [43, 44]). The
shakedown limit stress-intensity factor reads:
1
p
m ‡ m2 p : …52†
Ksh ˆ ro pq
11.4 Transition to Ductile Fracture
Thus, the shakedown limit stress-intensity factor of a cracked body is proportional to
the initial yield stress ro times the square root of the effective crack-tip radius q. Fora
material, ro is usually given. The problem now is to establish the effective crack-tip radius
q. A direct measurement of this parameter is difficult. Using an indirect method,
Kuhn and Hardrath [45] calculated the effective crack-tip radii for metallic materials
and proposed a relationship between q and the ultimate strength rY of a material depicted
in a diagram.
For a given material, the initial yield stress ro and ultimate stress rY can be measured
by a simple tension experiment. The effective crack-tip radius q is taken directly
from the diagram given in [45]. Knowing these quantities, the shakedown limit SIF Ksh
271

11 Theoretical and Computational Shakedown Analysis
Table 11.5: Shakedown limit SIFs Ksh and fatigue thresholds Kth for various materials.
Material ro [MPa] rY [MPa] q
p m 1/2
Ksh [Mnm –3/2 ] Kth [Mnm –3/2 ] Ref.
2 1/4 Cr-1Mo 345 528 13.549 · 10 –3
SA 387-2.22 390 550 12.433 · 10 –3
SA 387-2.22 340 520 13.708 · 10 –3
SA 387-2.22 290 500 14.346 · 10 –3
Docol 350 260 360 19.128 · 10 –3
SS 141147 185 322 22.316 · 10 –3
HP Steel 210 304 25.504 · 10 –3
HP Steel 160 279 28.692 · 10 –3
HP Steel 120 242 36.662 · 10 –3
follows immediately from Equation (52). In Table 11.5, the shakedown limit SIFs Ksh
for some materials are selected. At the same time, fatigue thresholds for the same
materials are listed there. They were obtained by other authors with experimental methods
of other theoretical approaches [46, 47].
It can be seen that the shakedown limit SIFs Ksh of cracked bodies agree quite
well with their fatigue thresholds Kth. This agreement indicates that the reason for
crack arrest in these materials is the shakedown of the cracked bodies. In these cases,
the fatigue threshold of a cracked body can be predicted by using shakedown theory.
11.5 Summary of the Main Results of Project B6
One major issue of project B6 was the formulation of a 3-D overlay model for non-linear
hardening materials. For this class of materials, a static shakedown theorem and a
corresponding corollary were formulated and proved, which are generalizations of Melan’s
static shakedown theorems for perfectly plastic and linear kinematic hardening
materials. A systematic investigation of the numerical treatment of shakedown problems
using Finite-Element method was carried out. The findings were used to employ
efficient optimization strategies and algorithms developed to take advantage of the special
structure of the arising optimization problems. As an important result of these investigations,
explicit conclusions about the failure forms of cyclically loaded systems
could be drawn, i.e. incremental collapse or alternating plasticity. The influence of cyclic
hardening and softening on the shakedown behaviour of structures was studied incrementally.
The results were compared with those derived for the 3-D overlay model.
A new methodology was proposed to include stress singularities into shakedown
investigations allowing for the prediction of fatigue thresholds of ductile cracked
bodies. Thus, a transition from shakedown theory to cyclic fracture mechanics was
achieved.
272
8.25 8.3 [45]
8.6 8.3 [45]
8.14 8.6 [45]
7.3 9.6 [45]
8.9 5.4 [46]
7.6 6.0 [46]
9.5 6.2 [46]
8.1 6.7 [46]
7.8 8.2 [46]

References
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des elastisch-plastischen Verhaltens der Baustoffe. Bauingenieur 13 (1932) 261–267.
[2] E. Melan: Der Spannungszustand eines Mises-Henckyschen Kontinuums bei veränderlicher
Belastung. Sitzber. Akad. Wiss. Wien IIa 147 (1938) 73–78.
[3] E. Melan: Zur Plastizität des räumlichen Kontinuums. Ing.-Arch. 8 (1938) 116–126.
[4] W. T. Koiter: A New General Theorem on Shakedown of Elastic-Plastic Structures. Proc.
Koninkl. Acad. Wet. B 59 (1956) 24–34.
[5] L. Corradi, G. Maier: Inadaptation Theorems in the Dynamics of Elastic-Workhardening
Structures. Ing.-Arch. 43 (1973) 44–57.
[6] J.A. König: A Shakedown Theorem for Temperature Dependent Elastic Moduli. Bull. Acad.
Polon. Sci. Ser. Sci. Tech. 17 (1969) 161–165.
[7] G. Maier: A Shakedown Matrix Theory Allowing for Workhardening and Second-Order
Geometric Effects. In: Proc. Symp. Foundations of Plasticity, Warsaw, 1972.
[8] W. Prager: Shakedown in Elastic-Plastic Media Subjected to Cycles of Load and Temperature.
In: Proc. Symp. Plasticita nella Scienza delle Costruzioni, Bologna, 1956.
[9] D. Weichert: On the Influence of Geometrical Nonlinearities on the Shakedown of Elastic-
Plastic Structures. Int. J. Plasticity 2 (1986) 135–148.
[10] C. Polizzotto, G. Borino, S. Caddemi, P. Fuschi: Shakedown Problems for Material Models
with Internal Variables. Eur. J. Mech. A/Solids 10 (1991) 787–801.
[11] B. G. Neal: Plastic Collapse and Shakedown Theorems for Structures of Strain-Hardening
Material. J. Aero Sci. 17 (1950) 297–306.
[12] G. Masing: Zur Heyn’schen Theorie der Verfestigung der Metalle durch verborgen elastische
Spannungen. Technical Report 3, Wissenschaftliche Veröffentlichungen aus dem Siemens
Konzern, 1924.
[13] D.A. Gokhfeld, O.F. Cherniavsky: Limit Analysis of Structures at Thermal Cycling. Sijthoff
& Noordhoff, 1980.
[14] J.A. König: Theory of Shakedown of Elastic-Plastic Structures. Arch. Mech. Stos. 18
(1966) 227–238.
[15] A. Sawczuk: Evaluation of Upper Bounds to Shakedown Loads of Shells. J. Mech. Phys.
Solids 17 (1969) 291–301.
[16] A. Sawczuk: On Incremental Collapse of Shells under Cyclic Loading. In: Second IUTAM
Symp. on Theory of Thin Shells, Kopenhagen, Springer Verlag, Berlin, 1969.
[17] F. A. Leckie: Shakedown Pressure for Flush Cylinder-Sphere Shell Interaction. J. Mech.
Eng. Sci. 7 (1965) 367–371.
[18] T. Belytschko: Plane Stress Shakedown Analysis by Finite Elements. Int. J. Mech. Sci. 14
(1972) 619–625.
[19] L. Corradi, I. Zavelani: A Linear Programming Approach to Shakedown Analysis of Structures.
Comp. Math. Appl. Mech. Eng. 3 (1974) 37–53.
[20] J. Gross-Weege: Zum Einspielverhalten von Flächentragwerken. PhD Thesis, Inst. für
Mech., Ruhr-Universität Bochum, 1988.
[21] H. Nguyen Dang, P. Morelle: Numerical Shakedown Analysis of Plates and Shells of Revolution.
In: Proceedings of 3rd World Congress and Exhibition on FEMs, Beverley Hills,
1981.
[22] W. P. Shen: Traglast- und Anpassungsanalyse von Konstruktionen aus elastisch, ideal plastischem
Material. PhD thesis, Inst. für Computeranwendungen, Universität Stuttgart, 1986.
[23] J.L. Chaboche: Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity. Int.
J. Plast. 3 (1989) 247–302.
[24] H. Neuber: Kerbspannungslehre. Springer Verlag, 1958.
273

11 Theoretical and Computational Shakedown Analysis
[25] W. Prager: A New Method of Analyzing Stresses and Strains in Workhardening Plastic Solids.
J. Appl. Mech., 1956, pp. 493–496.
[26] H. Ziegler: A Modification of Prager’s Hardening Rule. Quart. Appl. Math. 17 (1955) 55–
65.
[27] Z. Mróz: On the Description of Anisotropic Workhardening. J. Mech. Phys. Solids 15
(1967) 163–175.
[28] Y. F. Dafalias, E.P. Popov: A Model of Nonlinearly Hardening Materials for Complex Loading.
Acta Mechanica 43 (1975) 173–192.
[29] E. Stein, G. Zhang, R. Mahnken, J.A. König: Micromechanical Modeling and Computation
of Shakedown with Nonlinear Kinematic Hardening Including Examples for 2-D Problems.
In: Proc. CSME Mechanical Engineering Forum, Toronto, 1990, pp. 425–430.
[30] E. Stein, G. Zhang, J.A. König: Shakedown with Nonlinear Hardening Including Structural
Computation Using Finite Element Method. Int. J. Plasticity 8 (1992) 1–31.
[31] G. Zhang: Einspielen und dessen numerische Anwendung von Flächentragwerken aus ideal
plastischem bzw. kinematisch verfestigendem Material. PhD thesis, Institut für Baumechanik
und Numerische Mechanik, Universität Hannover, 1992.
[32] E. Stein, G. Zhang, R. Mahnken: Shakedown Analysis for Perfectly Plastic and Kinematic
Hardening Materials. In: Progress in Computational Analysis of Inelastic Structures,
Springer Verlag, 1993, pp. 175–244.
[33] R. Mahnken: Duale Methoden in der Strukturmechanik für nichtlineare Optimierungsprobleme.
PhD thesis, Institut für Baumechanik und Numerische Mechanik, Universität Hannover,
1992.
[34] J.A. König: Shakedown of Elastic-Plastic Structures. PWN-Polish Scientific Publishers,
1987.
[35] J. Scheer, H.J. Scheibe, D. Kuck: Untersuchung von Trägerschwächungen unter wiederholter
Belastung bis in den plastischen Bereich. Bericht Nr. 6099, Institut für Stahlbau, TU
Braunschweig, 1990.
[36] E. Stein, G. Zhang, Y. Huang: Modeling and Computation of Shakedown Problems for
Nonlinear Hardening Materials. Computer Methods in Mechanics and Engineering 321
(1993) 247–272.
[37] E. Stein, Y. Huang: An Analytical Method to Solve Shakedown Problems with Linear Kinematic
Hardening Materials. Int. J. of Solids and Structures 18 (1994) 2433–2444.
[38] Y. Huang, E. Stein: Shakedown of a Cracked Body Consisting of Kinematic Hardening
Material. Engineering Fracture Mechanics 54 (1996) 107–112.
[39] M. Craeger: Master Thesis, Lehigh University, 1966.
[40] N.E. Frost: Notch Effects and the Critical Alternating Stress Required to Propagate a
Crack in an Aluminium Alloy Subject to Fatigue Loading. J. Mech. Engng. Sci. 2 (1960)
109–119.
[41] A.R. Jack, A.T. Price: The Initiation of Fatigue Cracks from Notches in Mild Steel Plates.
International Journal of Fracture Mechanics 6 (1970) 401–409.
[42] R. E. Swanson, A.W. Thompson, I.M. Bernstein: Effect of Notch Root Radius on Stress Intensity
in Mode I and Mode III Loading. Metallurgical Transactions A 17A (1986) 1633–
1637.
[43] N.E. Dowling: Fatigue at Notches and the Local Strain and Fracture Mechanics Approaches.
In: Fracture Mechanics, 1979.
[44] D. Taylor: Fatigue Thresholds. Butterworth, 1989.
[45] P. Kuhn, H.F. Hardrath: An Engineering Method for Estimating Notch-Size Effect in Fatigue
Test on Steel. Technical report, NACA technical note, 1952.
[46] R. O. Ritchie: Near-Threshold Fatigue Crack Growth in 2 1/4 Cr-1Mo Pressure Vessel Steel
in Air and Hydrogen. J. of Eng. Materials and Technology 102 (1980) 293–299.
[47] J. Wasen, K. Hamberg, B. Karlsson: The Influence of Grain Size and Fracture Surface Geometry
on the Near-Threshold Fatigue Crack Growth in Ferritic Steels. Mat. Sci. Engng.
102 (1998) 217–226.
274

12 Parameter Identification for Inelastic Constitutive
Equations Based on Uniform and Non-Uniform
Stress and Strain Distributions
Abstract
Rolf Mahnken and Erwin Stein*
In this contribution, various aspects for identification of material parameters are discussed.
The underlying experimental data are obtained from specimen, where stresses and strains
can be either uniform or non-uniform within the volume. In the second case, the associated
simulated data are obtained from Finite-Element calculations. A gradient-based optimization
strategy is applied for minimization of a least-squares functional, where the corresponding
sensitivity analysis if performed in a systematic manner. Numerical examples
for the uniform case are presented with a material model due to Chaboche with cyclic
loading. For the non-uniform case, material parameters are obtained for a multiplicative
plasticity model, where experimental data are determined with a grating method for an
axisymmetric necking problem. In both examples, the effect of different starting values
and stochastic perturbations of the experimental data are discussed.
12.1 Introduction
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
12.1.1 State of the art at the beginning of project B8
The project B8 of the Collaborative Research Centre (SFB 319) has been started in
1991 with the intention to identify material parameters of constitutive models for inelastic
material behaviour. Though the interest for reliable modelling has always been
very high in the engineering community, up to that time, the concepts for parameter
identification concerning experimental and numerical issues were fairly limited. In particular
the state of the art was as follows:
* Universität Hannover, Institut für Baumechanik und Numerische Mechanik, Appelstraße 9 a,
D-30167 Hannover, Germany
275

12 Parameter Identification for Inelastic Constitutive Equations
• The experiments producing the experimental data were mostly conducted in simple
tension, compression or torsion, respectively. In this way, the sample, e.g. a cylindrical
hollow specimen, is subjected to an axial load (force or displacement), which produces
strains and stresses assumed to be uniform within the whole volume of the specimen.
• The identification process for the underlying material models was performed in the
framework of a geometric linear theory.
• For optimization of the resulting least-squares functional (at least within the SFB
319) evolutionary strategies were preferred.
• In some situations it may occur that more than one set of parameters can give reasonable
fits. This issue of instability (or even non-uniqueness in the case of identical
fits) has not been considered.
12.1.2 Aims and scope of project B8
As a main consequence of the first item in the above overview it was observed that parameters
derived from an optimal least-squares fit of uniaxial experiments do not necessarily
predict non-uniform deformations. This is due to the facts that (i) a uniaxial experiment
does not provide enough information to obtain an accurate simulation of the
non-uniform case, and (ii) the ideal test conditions of uniformness often cannot be realized
in the laboratory. In nearly all mechanical tests deformations eventually cease to
be uniform due to localization, fracture and other failure mechanisms. E.g., non-uniformness
is unavoidable in the case of necking of the sample in tension tests or barreling
due to friction of the sample in compression tests. Therefore a main object of project
B8 was to develop a more general approach, which accounts for this inhomogeneity
by performing parameter identification using Finite-Element simulations.
The next issue is concerned with the geometric setting. It is quite obvious that
material parameters obtained from a fit within a geometric linear setting in general do
not carry over to the finite-deformation regime. This in particular holds if extreme
loads are subjected to the specimen thus yielding large deformations. To this end parameter
identification within a geometric non-linear theory has been performed.
In a common – classical – approach, parameter identification is formulated as an
optimization problem, where a least-squares functional is minimized in order to provide
the best agreement between experimental data and simulated data in a specific norm.
Algorithms for solution of this problem, basically, may be classified into two classes,
i.e. methods, which only need the value of the least-squares function (zero-order methods)
and descent methods, which require also the gradient of the least-squares function
(first-order methods). Very often an evolutionary method is preferred in practice because
of its versatility (see e.g. Müller and Hartmann [1] and Kublik and Steck [2]).
However, in general, these methods are very time-consuming due to many function
evaluations (several hundred thousand). Thus for reasons of efficiency an optimization
strategy based on gradient evaluations has been developed.
For determination of the gradient of the associated least-squares functional, basically
two variants are known from the literature: (1) The finite-difference method: This tech-
276

12.2 Basic Terminology for Identification Problems
nique, though conceptionally very simple in general, is regarded as inefficient due to many
function evaluations and accuracy problems. (2) The sensitivity analysis: In this concept
the gradient is determined analytically consistent with the formulation of the underlying
direct problem. As part of the work of project B8 the latter concept has been developed
firstly to the uniform case, and then it was carried over to the non-uniform case.
Another object of project B8 was to discuss and investigate the stability of the results
for the identification process since instability is a typical feature of inverse problems
(see Baumeister [3], Banks and Kunisch [4]). To this end two indicators are investigated:
We examine the eigenvalues of the Hessian of the least-squares function, and
we study the effect of perturbations of the experimental data on the parameters.
Furthermore, for the case of numerically instable results, we introduce a regularization
due to Tikhonov, which can be interpreted as an enhancement of the basic least-squares
functional by adequate model information.
Parameter identification essentially relies on experimental data obtained in the laboratory.
In this respect it is obvious that for the non-uniform case spatially distributed
data give more information as data evaluated only at certain points, e.g. using strain
gauges. Therefore optical methods turned out to be the ideal tools in order to obtain
the underlying data sets, and in our examples the experimental data were obtained with
a grating method in collaboration with the projects C1 (Dr. Andresen [5]), C2 (Prof.
Ritter [6]) and B5 (Prof. Peil [7]).
In this contribution we will describe our approach for parameter identification,
firstly, to the conventional uniform case, and secondly, to the non-uniform case, where
the Finite-Element method is applied. To specify, this work is structured as follows: In
the next Section 12.2 the basic terminology for identification problems pertaining to the
direct problem and the inverse problem is introduced. In Section 12.3 a systematic concept
for parameter identification is briefly described for the uniform case, and in Section
12.4, it is extended to the non-uniform case. In Section 12.5 two examples for parameter
identification based on experimental data obtained within the Collaborative Research Centre
(SFB 319) are presented. In the first example the Chaboche model [8] is considered
with a sample in cyclic loading, and in the second example we investigate an axisymmetric
necking problem of mild steel. Section 12.6 gives a summary of the main results
of project B8, and furthermore, we discuss issues of future research work.
12.2 Basic Terminology for Identification Problems
12.2.1 The direct problem: the state equation
In the sequel, we denote by K a (vector-)space with elements j of admissible material
parameters, and g…j; ^u…j†† is the state equation, which may represent e.g. the (non-linear)
state of the discretized form of an initial value problem or the variational form of
an initial boundary value problem. The state equation g may be dependent on the parameters
j, both explicitly and implicitly, where the implicit dependency is defined via
277

12 Parameter Identification for Inelastic Constitutive Equations
the state variable ^u…j† 2U, and where U is a (function-)space of admissible state variables
^u…j†. With this notation, we formulate the direct problem:
Find u…j† 2U such that g…j; u…j†† ˆ 0 for given j 2K: …1†
In what follows, we assume existence of the solution u…j†:ˆ Arg fg…j; ^u…j†† ˆ 0g for
all j 2K.
12.2.2 The inverse problem: the least-squares problem
Let ~D denote an observation space, and let ~d 2 ~D denote given data from experiments.
In general experimental data are not complete, e.g. for cyclic loading tests very often
they are available only for a part of the cycles. To account for this possible incompleteness,
we introduce an observation operator M mapping the trajectory u…j† to points
M u…j† in the observation space P (Banks and Kunisch [4], p. 54). With this notation
we formulate the inverse problem:
Find j 2K such that M u…j† ˆ ~d for given ~d 2 ~D : …2†
An identification process based on experimental data is typically influenced by two
types of errors: Using the notations ^u for the true state and j for the correct parameter
vector, then the following situations may arise (Banks and Kunisch [4]):
• M ^u 6ˆ ~d due to measurement errors,
• ^u 6ˆ u…j † due to model errors.
In general, the first error type is taken into account by statistical investigations of the
data. The second type is reduced by increasing the complexity of the model, which in
general is accompanied by an increase of material parameters np.
Referring to the classical definitions of Hadamard [9], a problem is well-posed if the
conditions of (i) existence, (ii) uniqueness and (iii) continuous dependence on the data for
its solution are satisfied simultaneously. If one of these conditions is violated, then the
problem is termed ill-posed. Since in practice the number of experimental data is larger
than the number of unknown parameters, problem (2) in general is overdetermined, thus
excluding the existence of a solution due to measurement and/or model errors. The classical
strategy therefore uses an optimal approach of simulated data u…j† and experimental
data ~d, thus replacing problem (2) by the least-squares optimization problem:
Find j 2K such that for given ~d 2 ~D: f …j†:ˆ 1
2 kM u…j† ~dk 2 ~D ! min : …3†
j2K
• Remarks
1. In practice, experimental data are given at discrete time- or load-steps. Therefore,
for the following discussions it is natural to set ~D IRndat for the observation space,
278

12.2 Basic Terminology for Identification Problems
with ndat as the number of experimental data. Furthermore, very often parameters are
independent of each other such that K IR np can be separated, where np is the number
of material parameters, Next, we use the short hand notation M u…j† ˆ: d…j† 2IR ndat ,
thus indicating the transformation of the simulated data to the observation space (e.g.
by an interpolation procedure). Then, the resulting least-squares problem reads:
f …j† ˆ 1
2 kd…j† ~dk 2
2 ! min ; KˆYnp
Ki ; Ki :ˆ fai ji big : …4†
j2K
iˆ1
Here, ai; bi are lower and upper bounds for the material parameters, respectively.
2. In many situations, the problems (3) or (4), though well-posed, may lead to numerically
instable solutions, i.e. small variations of ~d then lead to large variations of
the parameters j. These difficulties are caused if
(a) the material model has (too many) parameters, which yield (almost) linearly dependencies
within the model, or if
(b) the experiment is inadequate in the sense that some effects intended by the model
are not properly “activated”.
It has already been mentioned that a typical step for reducing the model error is to increase
the complexity of the model, which generally is accompanied by an increase of
the material parameters. A typical example is the modification of the standard J2-flow
theory with the linear Prager rule in order to account for anisotropic hardening effects.
A further extension is possible with the non-linear Chaboche model [8] (see also Equation
(8) in the forthcoming Section 12.3) in order to account for non-linear kinematic
hardening effects. In doing so, it should be realized that the introduction of additional
material parameters may also result into the aforementioned numerical instability for
the identification process if appropriate steps are not performed when planning the experiment.
To summarize, the contradictory requirements for numerical stable results
and reducing the model error have to be carefully balanced.
3. In some cases, even non-uniqueness for the parameter set may occur: This was observed
by Mahnken and Stein [10, 11] for two material models under certain loading
conditions. The main consequences are that at least cyclic loading becomes necessary
in case of the Chaboche model [8], and for identification of the Steck model [12] experiments
have to be performed at different temperatures.
4. As a consequence of the above Remark 2, it is strongly recommended to study the
effect of perturbations of the experimental data on the parameters. This may indicate
possible instabilities of the identification process. Furthermore, the eigenvalue structure
of the Hessian of f …j† gives further information about the stability. However, a systematic
strategy to detect possible instabilities so far is not available.
5. A mathematical tool, suitable to overcome possible numerical instabilities, is a regularization
of the functional in Equation (4), and this leads to the more general problem:
fc…j†:ˆ 1
2 Wd…d…j† ~d†
2
2
‡ c
2 Wl…j ~j†
2
2
! min : …5†
np j2K IR
279

Here, the matrices Wd 2 IRndat IRndat and Wl 2 IRnp IRnp ‡ , the scalar c 2 IR and the
a priori parameters j 2 IRm are regularization parameters (see Baumeister [3]). Note
that the first part of the functional in Equation (5) is also obtained when considering
parameter identification based on statistical investigations in the context of a Maximum-Likelihood
method in order to account for measurement errors. It is noteworthy
that the r.h.s. of the functional is also related to the Bayesian estimation (see Bard [13],
Pugachew [14]). The above functional provides the opportunity to include physical interpretation
of some parameters, obtained e.g. by “hand fitting”, into the optimization
process if numerical instabilities occur. However, a systematic concept for determination
of the regularization parameters in the context of parameter identification for viscoplastic
material models so far is not available.
6. Problems of the above kind like Equations (4) or (5) with separable constraints
may be solved with the projection algorithm due to Bertsekas [15]:
j …j‡1† ˆPfj …j† …j† H …j† rf …j …j† †g ; …Pfjg† i :ˆ min…bi; …max…ji; ai†† ;
i ˆ 1; ...; np : …6†
Note that the above iteration scheme requires the gradient of the associated leastsquares
functional. This task is generally performed in the sensitivity analysis, where
the gradient is determined consistent with the formulation of the underlying direct problem.
Note also that the iteration matrix H has to be “diagonalized” in order to insure
descent properties of the search directions for algorithm (see Bertsekas [15] and Mahnken
[16] for an explanation of this terminology and further details).
12.3 Parameter Identification for the Uniform Case
In this section, we briefly describe a systematic strategy for parameter identification for
the uniform case within a geometric linear setting. A detailed description is given in
Mahnken and Stein [11, 17].
12.3.1 Mathematical modelling of uniaxial visco-plastic problems
Let Iˆ‰0; TŠ be the time interval of interest. The uniaxial stress is designated by
r ˆ r11 : I!IR, while eel and ein : I!IR, are the elastic and inelastic parts of the
small strain tensor components ein ij and eel ij , respectively. The model equations representing
one-dimensional visco-plasticity with small strains are summarized as follows:
280
12 Parameter Identification for Inelastic Constitutive Equations
e ˆ e el ‡ e in
additive split of total strains ; …7a†
e el ˆ 1
r elastic strains ; …7b†
E

12.3 Parameter Identification for the Uniform Case
_e in ˆ _ ^e in …r; q; h; e in ; ...;j† evolution for inelastic strains ; …7c†
_q ˆ _ ^q…r; q; h; e in ; ...;j† evolution for internal variables . …7d†
Here, additionally, we defined the temperature h, the elastic modulus E, and j 2 IR np is
a vector of np material parameters characterizing the inelastic material behaviour.
There exists a great variety of constitutive relations in the literature according to
the above skeletal structure (Equations (7a) to (7d)) (see e.g. Miller [18], Lemaitre and
Chaboche [19] and references therein). Many approaches intend to provide for a number
of different characteristic effects such as strain rate-dependent plastic flow, creep or
stress relaxation. In doing so, a yield criterion with the inherent specification of loading
and unloading conditions as in time-independent classical plasticity is not needed. The
resulting equations are currently referred to as “unified models”. Concerning the internal
variables, in principle they are argued for macroscopic or microscopic reasons depending
on the basic conception.
Three representative examples for the evolution equations _e in and _q in Equation
(7) were treated by Mahnken and Stein in [11, 17] within project B8, i.e. the models of
Chaboche [8], Bodner and Partom [20] and Steck [12] (see also Kublik and Steck [2]).
In this contribution only the Chaboche model with the evolution equations
_e in F
ˆ K0 n0 8
><
sign …r † if F > 0
>:
0 else
…8a†
_R ˆ b…q R†_e in sign …_e in † isotropic hardening …8b†
_ ˆ c…c sign …_e in ††_e in
F ˆ…r † sign …r † R k 0
kinematic hardening …8c†
overstress …8d†
shall be considered, where j :ˆ ‰n 0 ; K 0 ; k 0 ; b; q; c; cŠ T is the vector of material parameters
related to the inelastic material behaviour.
In addition to the Equations (7a) to (7d), we assume that initial conditions
r…t ˆ 0† ˆr0 ; e in …t ˆ 0† ˆe in
0 ; q…t ˆ 0† ˆq 0
are given, which complete the formulation of the initial value problem.
The representation above – and in the forthcoming two subsections – is based on
stress-controlled experiments. Of course, analogous arguments hold for the complementary
tests, i.e. strain-controlled experiments, where experimental data are given for a
stress distribution ~r…t†; t 2I.
…9†
281

12 Parameter Identification for Inelastic Constitutive Equations
12.3.2 Numerical solution of the direct problem
We define N as the number of time steps Dtk ˆ tk tk 1; k ˆ 1; ...; N; t0 ˆ 0; tN ˆ T.
Using the second order midpoint-rule at each time step, from Equations (7a) to (7d),
we obtain the update relations
ek ˆ ek 1 ‡ Dtk _e in
k 1=2 ‡ Deel k 1 ; …10†
q k ˆ q k 1 ‡ Dtk _q k 1=2 ; …11†
where we applied the notation:
_e in
k 1=2 ˆ _ ^e in …1=2…rk 1 ‡ rk† ; 1=2…q k 1 ‡ q k†; ...;j† ; …12†
_q k 1=2 ˆ _ ^q…1=2…rk 1 ‡ rk† ; 1=2…q k 1 ‡ q k†; ...;j† ; …13†
De el 1
k 1 ˆ
E …rk rk 1† : …14†
Since the state variables ek and q k are not known in advance, the following non-linear
system of equations has to be solved at each time step:
gk;1…ek; qk†:ˆ ek ek 1 Dtk _e in
k 1=2 Deel k 1 ˆ 0 ; …15†
g k;2…ek; q k†:ˆ q k q k 1 Dtk _q k 1=2 ˆ 0 : …16†
Defining Gk :ˆ ‰gk;1; g T k;2 ŠT and a vector of state variables Yk :ˆ ‰ek; q T k ŠT , Equations
(15) and (16) may be summarized as:
Gk…Yk† ˆ0 : …17†
Referring to the notation of Section 12.2.1, Equation (17) will be termed as the state
equation, describing the state of the variables Yk :ˆ ‰ek; q T k ŠT at the k-th time step.
Furthermore, using the notation of Section 12.2.1, we have the direct problem:
Find Yk…j† such that Gk…Yk…j†† ˆ 0; k ˆ 1; ...; N for given j 2K: …18†
The iterative solution of Equation (17) is obtained with a Newton method. Details of
this strategy with applications to the material models of Bodner-Partom, Chaboche and
Steck are described in Mahnken and Stein [11, 17].
12.3.3 Numerical solution of the inverse problem
For reasons of simplicity, in the sequel we will assume that the discrete values for time
integration ftkg N
kˆ1 Iand for the experimental data ft exp
k gndat kˆ1 Ido coincide for both
282

the numerical values ek…j† ˆe…tk; j† and the observations ~ek ˆ ~e…tk†; k ˆ 1; ...; N. The
following considerations can be extended to more complex situations in a straightforward
manner. Theresultinginverseproblemthenbecomestheleast-squaresoptimizationproblem:
Find j 2Ksuch that for given ~d 2 ~D:
f …j† ˆ 1 X
2
N
…ek…j† ~ek†
kˆ1
2 ! min ; np j2K IR
Kˆ Ynp
12.4 Parameter Identification for the Non-Uniform Case
Figure 12.1: Schematic flow chart of the optimization strategy for the uniform case with outer
and inner iteration loops.
iˆ1
Ki ; Ki :ˆ fai ji big : …19†
As before, ai; bi are lower and upper bounds for the material parameters.
A schematic flow chart for solution of problem (19) with a simplified description is
shown in Figure 12.1. It can be seen that basically an outer loop for iteration of the material
parameters and an inner loop for iteration of the state variables ^Yk…j† are performed. In the
outer loop the Bertsekas algorithm(Equation (6))is applied,wherethe gradientis determined
in a sensitivityanalysis. For detailspertaining to this strategy with applicationsto the material
models of Bodner-Partom,Chaboche and Steck, we also referto Mahnken and Stein [11, 17].
12.4 Parameter Identification for the Non-Uniform Case
As already mentioned in Section 12.1.2, very often the assumption of uniform stress
and strain distributions during the experiment cannot be guaranteed due to the experi-
283

mental conditions or failure mechanisms. Therefore, we will consider parameter identification
with Finite-Element simulations in order to take into account inhomogeneities
within the sample. In what follows, we give a brief review for a geometrically non-linear
continuum-based formulation of the direct problem as a variational problem and
furthermore of the associate least-squares problem. For solution of these problems, a
standard linearization procedure is applied for the direct problem and a sensitivity analysis
is performed for the inverse problem. In the forthcoming sections, we will comment
on the similarities of these associated concepts. More details of our approach are
documented in Mahnken and Stein [10, 21] for the geometric linear case, and in Mahnken
and Stein [22], this concept has been extended to the geometric non-linear case.
12.4.1 Kinematics
12 Parameter Identification for Inelastic Constitutive Equations
Let B IR ndim be the reference configuration of a continuous body B with smooth
boundary @B, and let X 2B IR ndim be the position vector in the Euclidian space IR ndim
with spatial dimension ndim ˆ 1; 2; 3. We shall denote by @uB and @rB those parts of
the boundary @B, where configurations are prescribed as u and boundary tractions are
prescribed as t, respectively. As usual, we assume @uB [@rB ˆ@B and
@uB \@rB ˆ;. In addition, b denotes the body force per unit volume. As before, we
define I:ˆ ‰t0; TŠ 2IR‡ as a time interval of interest, and K IR np designates the
(vector-)space of material parameters.
Following Barthold [23] the fundamental mapping for describing the current configuration
of the body for varying time t 2I and varying parameter j 2Kis given as:
^u:ˆ B I K !IRndim ,
…X; t; j† 7! u ˆ ^u…X; t; j† :
As usual, we restrict ourselves to configurations u satisfying J :ˆ det…F† > 0 and
u ˆ u on @uB, where we use the shorthand notation F:ˆ ^F…X; t; j†:ˆ @Xu for the deformation
gradient at …X; t; j†. The exposition that follows crucially depends on the basic
assumption:
…20†
X ˆ ^u…X; t ˆ t0; j†8 …X; j† 2 …B K† ; …21†
i.e. the initial configuration B at time t ˆ t0 is independent of the parameter set j. Itis
noteworthy that this restriction, e.g., does not hold for more complex situations in shape
optimization, and would necessitate the introduction of a reference configuration invariant
of the design variables j (Barthold [23], Haber [24]). Thus, we will regard the set
…X; t; j† 2B I Kas the independent variables in the ensuing considerations.
An illustration of the mapping (Equation (20)) for the body B at fixed time t for
two different parameter sets j1; j2 2Kis shown in Figure 12.2.
284

12.4 Parameter Identification for the Non-Uniform Case
Figure 12.2: Illustration of two-parameter-dependent configurations at fixed time.
12.4.2 The direct problem: Galerkin weak form
Let j 2Kbe given, and let us assume a partition of the time interval Iˆ SN
‰tk 1; tkŠ
kˆ1
into N subintervals (for problems of elasticity and plasticity tk refers to the load step).
Denoting by uk ˆ ^u… ; tk; j† the configuration at time tk for the parameter set j, the
balance equation of linear momentum and the set of Dirichlet and Neumann boundary
conditions at the (k)-th step read:
divrk ‡ qb ˆ 0 in B ;
uk ˆ uk on @uB ;
rkn ˆ tk on @rB : …22†
Here, rk designates the Cauchy stress tensor. Using the notation h : i for the L2 dual
pairing on B of functions, vectors or tensor fields, an equivalent formulation is the classical
weak form (principal of virtual work) of the momentum equations at time tk. Ina
spatial description this results into the direct problem:
Find uk such that g…uk†ˆhs : ddij gj ˆ 0 8 du and for given j 2K; …23†
tˆtk tk
where the Kirchhoff stress tensor s ˆ Jr is introduced. Furthermore, a spatial rate of deformation
tensor induced by the virtual displacement du is defined as dd :ˆ sym…qudu†,
and g:ˆ hb dui‡ht dui@rB designates the external part of the weak form. For the case
of inelastic problems the above set of equations has to be supplemented by initial conditions
Z…X; t0; j† ˆZ0, where Z denotes a set of history variables.
The iterative solution of the non-linear problem (Equation (23)) is based on a standard
Newton method, in which a sequence of linearizations of the weak form (Equation
(23)) is performed. To this end the Gâteaux derivative qDg…uk† of problem (Equation (23))
as shown in Table 12.1 is determined. Here, lD :ˆ quDu and dD :ˆ sym…lD† are a velocity
gradient and a spatial rate of deformation tensor, respectively, induced by the linearization
increment Du. Additionally, c is the fourth order spatial material operator.
285

12 Parameter Identification for Inelastic Constitutive Equations
Table 12.1 : Weak form, Gâteaux derivative for linearization and linear equation for parameter
sensitivity in a spatial formulation.
• Weak form (principle of virtual work)
g…uk†ˆhs : ddij gj ˆ 0
tˆtk tk
• Gâteaux derivative for linearization
qDg…uk†ˆh…c : dD† : dd ‡ lDs : ddijtˆtk • Linear equation for parameter sensitivity
qjg…uk†ˆh…c : dj† : dd ‡ ljs : dd ‡ q p
js : ddij ˆ 0
tˆtk
12.4.3 The inverse problem: constrained least-squares optimization problem
As in Section 12.3, we assume identical time (load) steps ftkg N
kˆ1 and observation
states ftjg ntdat jˆ1 , where experimental data ~dj 2 ~D are available, and where ~D denotes the
observation space. In particular, ~dj may contain stresses, strains, displacements, reaction
force fields etc. Since in general only incomplete data are available from the experiment,
we introduce an observation operator M mapping the configuration trajectory
uk ˆ ^u… ; tk; j†; k ˆ 1; ...; N to points of the observation space ~D. Note that this definition
of M also accounts for quantities such as stresses, strains, reaction force fields etc.
since these quantities can be written in terms of the basic dependent variable uk. Then
we consider the least-squares optimization problem:
Find j 2Ksuch that for given ~d 2 ~D: f …j†:ˆ 1 X
2
N
kMuk kˆ1
~dkjj 2 ~D
! min ; …24†
j2K
where uk satisfies the weak form of the direct problem (Equation (23)).
The solution procedure for problem (Equation (24)) is schematically illustrated in
Figure 12.3. As for the uniform case in Figure 12.1, an outer loop is performed with
the Bertsekas algorithm (Equation (6)). Then determination of Muk, needed for evaluation
of the least-squares functional (Equation (24)), is performed only at the converged
state of the direct problem, i.e. when problem (Equation (23)) is satisfied at the k-th
time (load) step.
Next we will briefly resort to the parameter sensitivity qjuk. To this end firstly,
the parameter sensitivity qjg…uk† of the weak form (Equation (23)) is determined, at
equilibrium, analogously as in the linearization procedure. The resulting expression for
qjg…uk† is given in Table 12.1 in a spatial setting, where now a spatial rate of deformation
tensor induced by the parameter sensitivity qjuk ˆ vj is defined as
dj :ˆ sym …quvj†. Note that qjg…uk† also requires the spatial material operator c of the
linearization procedure. Furthermore we defined the sensitivity-load term hq p
js : ddijtˆtk ,
which excludes dependencies of s via the configuration u. The determination of this
term becomes a major task of the sensitivity analysis, and we refer to Barthold [23]
and Mahnken and Stein [22] for further details.
286

In the practical implementation, firstly, the sensitivity load term is determined in a
preprocessing procedure consistent with the underlying integration algorithm. Having
obtained qju k by solution of the associate linear equation, the total derivative qjwk…u†
of any quantity w…u k† is performed in a postprocessing procedure. Further details of
the procedure are described in Mahnken and Stein [22].
We close this section with the remark that the above results can be easily extended
to the enhanced element formulation described by Simo and Armero [25].
12.5 Examples
12.5.1 Cyclic loading for AlMg
12.5 Examples
Figure 12.3: Schematic flow chart of the identification process for the non-uniform case with
outer and inner iteration loops.
In this example parameters for the Chaboche model (Equations (7) and (8)) are determined
in the case of an aluminium/magnesium alloy. The underlying experimental data
were obtained from project A2 (Prof. Lange [26]).
The experiments were performed at room temperature for a cylindrical hollow
specimen with an outer radius of 28 mm and a thickness of 2 mm. The specimen were
subjected to a periodic strain of an amplitude of emax ˆ 0:3% at a strain rate of
_e ˆ 0:2% s –1 . 110 cycles were generated during the test, however, experimental data
are available only for 20 cycles out of these. Youngs modulus has been predetermined
as E ˆ 1:09 10 5 MPa.
As an objective function for the inverse problem the simple least-squares function
287

12 Parameter Identification for Inelastic Constitutive Equations
f …j† ˆ 1
kr…j† ~rk2 2
2
is minimized, which is the analogue of Equation (19) for the case of strain-controlled experiments.
Note that due to the incompleteness of the data set ~r contains data only for 20
cycles out of the total of 110. The minimization was performed with an evolutionary strategy
as described in Schwefel [27] and with the Bertsekas algorithm (Equation (6)). For the
first method we used three “parents” and “twenty descendants”, whilst for the latter the
BFGS-matrix was used as an iteration matrix and a Gauss-Newton matrix for preconditioning.
The computer runs were performed on an IBM-250T.
The starting vector and the solution vectors are given in Table 12.2. Concerning the
Bertsekas algorithm, three different runs were made. Run 1 and Run 2 were started with
the vector in the second column, however, for Run 2, a regularization was performed
using the extended functional Equation (5). Here, for Bd and for Bl, the unity matrix is
chosen, and we set c ˆ 10 5 . It can be seen that no convergence was attained for Run
1 after 2000 iteration steps, whilst minimization with the regularized functional attained
convergence after 201 steps. The corresponding minimal eigenvalue of the Hessian at
the solution point is 7.62 · 10 –2 , thus indicating stable results. This also is confirmed by
Run 3, where each data was perturbed stochasticly with a maximal value of 10%, and
where the effect of this perturbation is negligible. In the last two columns of Table 12.2
results for the evolutionary strategy are shown. After 897 iterations the results are still
poor (Run 4), and after 10256 iterations and 168 h, the value for the objective function
is still above that obtained by the Bertsekas algorithm (Run 2) in 24 min.
Table 12.2: Cyclic loading for AlMg: starting and obtained values for the material parameters of
the Chaboche model for AlMg in case of different optimization strategies and least-squares functions.
Concerning Run 3 see Section 12.5.1. ITE and NFUNC denote the number of iterations and
function evaluations, respectively.
n0 [–] 5.0 · 10 0
K0 [MPa] 1.0 · 10 2
b0 [–] 1.0 · 10 2
q [MPa] 1.0 · 10 2
c [–] 1.0 · 10 2
c [MPa] 1.0 · 10 2
k 0 [MPa] 1.0 · 10 1
288
Bertsekas algorithm Evolutionary strategy
Start Run 1 Run 2 Run 3 Run 4 Run 5
4.582 · 10 0
2.344 · 10 2
5.195 · 10 0
6.233 · 10 1
0.206 · 10 –1
9.840 · 10 4
0.000 · 10 0
1.360 · 10 1
4.242 · 10 1
4.824 · 10 0
6.827 · 10 1
1.542 · 10 3
4.719 · 10 1
0.000 · 10 0
1.360 · 10 1
4.242 · 10 1
4.824 · 10 0
6.827 · 10 1
1.542 · 10 3
4.719 · 10 1
0.000 · 10 0
4.971 · 10 0
1.972 · 10 2
5.115 · 10 0
6.488 · 10 1
1.173 · 10 2
1.792 · 10 2
8.543 · 10 –1
…25†
1.250 · 10 1
3.896 · 10 1
5.068 · 10 0
6.697 · 10 1
1.546 · 10 3
4.736 · 10 1
2.238 · 10 0
f …j† 3.491 · 10 5
9.620 · 10 3
2.135 · 10 3
3.335 · 10 3
8.936 · 10 3
2.135 · 10 3
fl…j† – – 3.582 · 10 –5
– – –
CPU [min] – 192 24 – 1440 605 ·10 3
ITE – 2000 201 –
897 10 256
NFUNC
2072 250 –
17752 20 493
Remark – no convergence
regularized perturbed – –

12.5 Examples
t [s]
Figure 12.4: Cyclic loading for AlMg. Up: Stress versus time for the solution parameter set for
the first 18 out of 110 cycles. Note the incompleteness of the experimental data set. Down: Stresses
versus strains for three different cycles. The numbers 1, 30, 110 correspond to the specific cycles.
289

12 Parameter Identification for Inelastic Constitutive Equations
Figure 12.4 depicts the stresses versus strains for three different cycles for the solution
vector of the material parameters. It can be seen that very substantial agreement
of experimental and simulated data is obtained, except for the first cycle, where the
model is not able to simulate the horizontal plateau. This explains the relatively high
model error for the values of the objective function at the solution point in Table 12.2.
12.5.2 Axisymmetric necking problem
In this section numerical results for the necking of a circular bar are presented. The material
of the specimen is a mild steel, Baustahl St52, due to the german industrial codes
for construction steel. The experimental data were obtained with a grating method. For
this purpose, firstly, grid marks were positioned on the surface of the sample, and these
were recorded by a digital CCD-camera at different observation states ti; i ˆ 1; ...ntdat
in the displacement-controlled experiment. In Figure 12.5, the sample with the grating
is shown at four observation states 5, 7, 10, 13 as introduced in Figure 12.7. More details
concerning the grating method are given in the contribution of project C2 (Prof.
Ritter [6]). The next step concerns the image processing by use of numerical methods
in order to obtain the final data for the identification process. This task is described in
more detail in the contribution of project C1 (Dr. Andresen [5]).
The elastic constants are E ˆ 20 600 kN/cm 2 for Youngs modulus and m ˆ 0:3 for
Poissons ratio. The material is assumed to be elasto-plastic, modelled by large strain multiplicative
von Mises elasto-plasticity with non-linear isotropic hardening summarized in
Table 12.3 (see Simo and Miehe [28] for further details). Solution of the direct problem
is done with a product-formula algorithm according to Simo [29]. The solution of the inverse
problem is based on the general setting described in the previous section. Details
Figure 12.5: Axisymmetric necking problem: photographs with a CCD-camera of the sample and
the grating at four different observation states NLST as introduced in Figure 12.7.
290

12.5 Examples
Table 12.3: Large strain multiplicative von Mises elasto-plasticity.
s ˆ ldev…ln b el †‡K ln Jg 1 r
2
U…s; e; j† ˆkdev…s†k W
3
Kirchhoff stress
0 …e; j† yield function
W 0…e; j† ˆr0 ‡ q…1 exp… be†† flow stress
1
2 Lt…b el 1 el
†b ˆ c dev…s†
r kdev…s†k
2
_e ˆ c
3
flow rule
variable evolution
c 0; U 0; cU ˆ 0 loading and unloading conditions
j :ˆ ‰r0; b; qŠ T ; np ˆ dim…j† ˆ3 vector of material parameters
Figure 12.6: Axisymmetric necking problem: different levels for spatial discretization.
Table 12.4: Axisymmetric necking problem: starting and obtained values for the material parameters
of a mild steel, Baustahl St52, for three different optimization runs. nite denotes the number
of iterations.
Run 1 (Q1/E4) Run 2 (Q1/E4) Run 3 (Q1/E4)
Starting Solution Starting Solution Starting Solution
r0 [MPa] 300.0 360.26 400.0 360.26 400.0 346.09
b [–] 10.0 3.949 20.0 3.949 20.0 3.951
q [MPa] 800.0 436.72 2000.0 436.72 2000.0 419.73
f …j† [–] 1.738 · 10 4
4.210 · 10 2
3.012 · 10 4
4.210 · 10 2
2.685 · 10 4
3.245 · 10 2
Perturbed no no yes
nite 34 34 39
291

12 Parameter Identification for Inelastic Constitutive Equations
Figure 12.7: Axisymmetric necking problem: comparison of simulation and experiment. Up: Load
versus elongation. Down: Necking displacement versus total elongation.
concerning the sensitivity analysis consistent with the product-formula algorithm of Simo
[29] for determination of the load term q p
j sk can be found in Mahnken and Stein [22].
An axisymmetric enhanced strain element (Q1/E4) described by Simo and Armero
[25] is used in the element formulation, and only a quarter of the bar is considered
for discretization using the appropriate symmetry boundary conditions.
The object is to identify the 3 parameters b; q; r0 of Table 12.3, which characterize
the inelastic behaviour of the material. To this end the following least-squares functional
is minimized
292

f …j† ˆ Xnt dat
iˆ1
1
2 ku j …j† ~u j k2 ‡ Xnt dat
12.5 Examples
Figure 12.8: Axisymmetric necking problem: comparison of experiment and Finite-Element simulation
for the configurations at four different observation states NLST as introduced in Figure
12.7. The circles represent the experimental data.
iˆ1
1
2 …w…Fi…j† ~Fi†† 2 ; …26†
where ~u and ~Fi…j†, ~Fi, i ˆ 1; ...; ntdat denote data for configurations and the total loads,
j
respectively. The number of load steps is N ˆ 40, the number of observation states is
ntdat ˆ 9, and we have nmp ˆ 12 for the number of observation points. A multilevel
strategy is applied in order to accelerate the optimization process by using solutions on
coarser grids as starting values on finer grid. In this example the total number of levels
is five (Figure 12.6).
In Table 12.4 results for the parameters j of three different runs are listed. Whilst
Run 1 and Run 2 differ in their starting vectors in order to take into account possible
local minima, the purpose of Run 3 is to investigate the effect of a perturbation of the
experimental data on the final results [11]. The perturbation was performed stochasticly,
whereby each data was varied by a maximum value of 5%. It can be observed that
Run 1 and Run 2 give identical results, thus indicating no further local minima. The results
for Run 3 differ only slightly from the results of Run 1 and Run 2, thus indicating
a stable solution with respect to measurement errors.
In Figure 12.7 the results for the total load versus total elongation and maximal
necking displacement versus total elongation are compared for simulation and experiment.
Figure 12.8 depicts a 2-D illustration of the final Finite-Element grid at different
observation states and the corresponding experimental data along the side of the sample.
It can be observed that for both quantities of different type, displacements and
forces, excellent agreement is obtained after optimization.
293

12.6 Summary and Concluding Remarks
In this contribution the main issues of project B8 of the research period 1991–1996 for
identification of parameters for inelastic constitutive equations were presented. The
main purpose of the project was to develop a general concept of gradient-based optimization
strategies for the uniform case and to extend this strategy to the non-uniform,
geometrically non-linear case in order to take into account inhomogeneities of stresses
and strains within the sample during the experiment. The main results of the project
and the cooperation with other projects from the Collaborative Research Centre (SFB
319) are listed as follows:
Theoretical results
• A gradient-based optimization algorithm has been developed for minimization of a
least-squares functional. In particular a projection algorithm due to Bertsekas is applied,
which accounts for possible upper and lower bounds for the parameters. Quasi-Newton
methods with Gauss-Newton preconditioning are used as iteration matrices.
• A unified strategy for an analytical sensitivity analysis in the case of uniform stress
and strain distributions is obtained, valid for a certain class of constitutive equations
with internal variables. The resulting scheme is consistent with the corresponding
time integration scheme, and as a main result a recursion formula is obtained. It
has been applied to the material models of Chaboche, Steck and Bodner-Partom.
• A unified strategy for an analytical sensitivity analysis of the variational (Galerkin)
form in the case of non-uniform stress and finite-strain distributions is obtained. As
for the uniform case it is consistent with the time-integration scheme.
• Investigations of uniqueness (or stability, respectively) for the inverse problem for
certain material models were performed (see Remark 3 of Section 12.2.2). The
main consequences are that at least cyclic loading becomes necessary in case of the
Chaboche model [11], and for identification of the Steck model experiments have to
be performed at different temperatures [10].
• A regularization technique for stabilization of the least-squares problem based on a
priori information has been introduced. Applications were done for the Bodner-Partom
model [11].
Numerical results
12 Parameter Identification for Inelastic Constitutive Equations
• Parameter identification for the methods of Chaboche, Steck and Bodner-Partom
based on experimental data for experiment with uniform stresses and strains was
performed.
294

• Comparative results with the evolutionary strategy showed great advantage of gradient-based
schemes with respect to execution time.
• Parameter identification with the Finite-Element method was performed in the
frame of non-linear multiplicative plasticity using experimental data obtained with a
grating method.
Co-work with other projects of the Collaborative Research Centre (SFB 319)
• Experimental work was done for a compact tension specimen and a necking problem
in cooperation with the projects C1 (Dr. Andresen [5]), C2 (Prof. Ritter [6]),
B5 (Prof Peil, Prof. Scheer [7]).
• Results for the compact tension specimen were published in a joint publication in
Der Bauingenieur (see Andresen et al. [30]).
• For parameter identification for an aluminium/magnesium alloy subjected to cyclic
loads, data of project A2 (Prof. Lange [26]) were used.
Concluding remarks
12.6 Summary and Concluding Remarks
The concepts proposed in this paper provide a flexible approach for identification of inelastic
material models. This opens the possibility to obtain more insight into the nonuniformness
of the samples during the experiment and on the reliability of the numerical
results. However, some open questions remain to be considered for future work:
• Further development in this area should take into account phenomena such as damage,
localization and temperature-dependent effects, which very often are highly
non-uniform during the experiment.
• In practice we know that measurement techniques possess limited accuracy. A probabilistic
investigation of these dispersion phenomena can be done with the Maximum-Likelihood
method (see Bard [13], Pugachew [14] and project B1 (Prof. Steck
[31]). Furthermore we know that repetition of the same experiment with different
samples in general yield different values. Reasons for this scattering of the data
may be due to inhomogeneities, load uncertainty, production, nature of the phenomenon.
Therefore this randomness also necessitates a probabilistic approach, where
e.g. the Bayesian estimation is a common strategy (see Bard [13] and Pugachew
[14]). The consideration of the above uncertainties seems to be a major task when
performing parameter identification in the future.
• In the work done so far the effect of discretization errors both in space and in time
is not considered, especially in the frame of a proper error control. Therefore it
would be of interest to take into account adaptive strategies in the context of the
optimization process.
295

• The shape of the least-squares functional, which is the objective function of the resulting
optimization problem, may not be convex, and thus different local minima
may occur. In this respect, a more systematic approach could be a hybrid method,
that is to combine a stochastic method with our deterministic strategy.
• The material model at hand might be insufficient for specific physical effects such
that an error-controlled adaptive modelling might be necessary.
References
12 Parameter Identification for Inelastic Constitutive Equations
[1] D. Müller, G. Hartmann: Identification of material parameters for inelastic constitutive
models using principles of biologic evolution. J. Eng. Mat. Tech. (ASME) 111 (1989) 299–
305.
[2] F. Kublik, E.A. Steck: Comparison of Two Constitutive Models with One- and Multiaxial
Experiments. In: D. Besdo, E. Stein (Eds.): IUTAM Symposium Hannover 1991, Finite Inelastic
Deformations – Theory and Applications, Springer-Verlag, Berlin, 1992.
[3] J. Baumeister: Stable solution of inverse problems. Vieweg, Braunschweig, 1987.
[4] H.T. Banks, K. Kunisch: Estimation Techniques for Distributed Parameter Systems. Birkhäuser,
Boston, 1989.
[5] K. Andresen: Surface-Deformation Fields from Grating Pictures Using Image Processing
and Photogrammetry. This book (Chapter 14).
[6] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical
Measuring Methods. This book (Chapter 13).
[7] U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behaviour
of Mild Steel Fe 510 under Complex Cyclic Loading. This book (Chapter 10).
[8] J.-L. Chaboche: Viscoplastic Constitutive Equations for the Description of Cyclic and Anisotropic
Behavior of Metals. Bull. Acad. Pol. Sci. Ser. Sci. Tech. 25 (1977) 33.
[9] J. Hadamard: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale
University Press, New Haven, 1923.
[10] R. Mahnken, E. Stein: The Parameter-Identification for Visco-Plastic Models via Finite-Element-Methods
and Gradient-Methods. Modelling Simul. Mater. Sci. Eng. 2 (1994) 597–
616.
[11] R. Mahnken, E. Stein: Parameter Identification for Viscoplastic Models Based on Analytical
Derivatives of a Least-Squares Functional and Stability Investigations. Int. J. Plast.
12(4) (1996) 451–479.
[12] E.A. Steck: A Stochastic Model for the High-Temperature Plasticity of Metals. Int. J. of
Plast. 1 (1985) 243–258.
[13] Y. Bard: Nonlinear Parameter Estimation. Academic Press, New York, 1974.
[14] V. S. Pugachew: Probability Theory and Mathematical Statistics for Engineers. Pergamon
Press, Oxford, New York, 1984.
[15] D.P. Bertsekas: Projected Newton methods for optimization problems with simple constraints.
SIAM J. Con. Opt. 20(2) (1982) 221–246.
[16] R. Mahnken: Duale Verfahren für nichtlineare Optimierungsprobleme in der Strukturmechanik.
Dissertation, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität
Hannover, F 92/3, 1992.
296

References
[17] R. Mahnken, E. Stein: Gradient-Based Methods for Parameter Identification of Viscoplastic
Materials. In: H.D. Bui, M. Tanaka (Eds.): Inverse Problems in Engineering Mechanics,
A.A. Balkama, Rotterdam, 1994.
[18] A.K. Miller: Unified Constitutive Equations for Creep and Plasticity. Elsevier Applied
Science, London New York, 1987.
[19] J. Lemaitre, J.L. Chaboche: Mechanics of solid Materials. Cambridge University Press,
Cambridge, 1990.
[20] S.R. Bodner, Y. Partom: Constitutive equations for elastic-viscoplastic strain-hardening materials.
Trans. ASME, J. Appl. Mech. 42 (1975) 385–389.
[21] R. Mahnken, E. Stein: A Unified Approach for Parameter Identification of Inelastic
Material Models in the Frame of the Finite Element Method. Comp. Meths. Appl. Mech.
Eng. 136 (1996) 225–258.
[22] R. Mahnken, E. Stein: Parameter Identification for Finite Deformation Elasto-Plasticity in
Prinicpal Directions. Comp. Meths. Appl. Mech. Eng. 147 (1997) 17–39.
[23] F. J.B. Barthold: Theorie und Numerik zur Berechnung und Optimierung von Strukturen aus
isotropen, hyperelastischen Materialien. Dissertation, Forschungs- und Seminarberichte aus
dem Bereich der Mechanik der Universität Hannover, F 93/2, 1993.
[24] R. B. Haber: Application of the Eulerian Lagrangian Kinematic Description to Structural
Shape Optimization. Proc. of NATO Advanced Study Institute on Computer-Aided Optimal
Design, 1986, pp. 297–307.
[25] J.C. Simo, F. Armero: Geometrically Nonlinear Enhanced Strain Mixed Method and the
Method of Compatible Modes. Int. J. Num. Meth. Eng. 33 (1992) 1413–1449.
[26] W. Gieseke, K.R. Hillert, G. Lange: Material State after Uni- and Biaxial Cyclic Deformation.
This book (Chapter 2).
[27] K.P. Schwefel: Numerische Optimierung von Computer-Modellen mittels der Evolutionsstrategie.
Birkhäuser Verlag, Basel, 1977.
[28] J.C. Simo, C. Miehe: Associative Coupled Thermoplasticity at Finite Strains: Formulation,
Numerical Analysis and Implementation. Comp. Meths. Appl. Mech. Eng. 98 (1992) 41–
104.
[29] J.C. Simo: Algorithms for Static and Dynamic Multiplicative Plasticity that Preserve the
Classical Return Mapping Schemes of the Infinitesimal Theory. Comp. Meths. Appl. Mech.
Eng. 99 (1992) 61–112.
[30] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifikation
für ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Der Bauingenieur
71 (1996) 21–31.
[31] E. Steck, F. Thielecke, M. Lewerenz: Development and Application of Constitutive Models
for the Plasticity of Metals. This book (Chapter 4).
297

13 Experimental Determination of Deformationand
Strain Fields by Optical Measuring Methods
Reinhold Ritter and Harald Friebe*
13.1 Introduction
The experiment is an essential basis for the development of material laws, which describe
the inelastic behaviour of metallic materials. It is needed first of all to observe
such a behaviour in order to get knowledge of the process of the corresponding methods.
Furthermore, the parameters for such material laws must be measured. They can
be achieved from experimentally determined, multi-dimensional load-deformation distributions.
The experiment is finally necessary for the comparison with the calculation in
order to verify the implanted material laws. The requirements of the measuring methods
result from the tasks of the experiment.
13.2 Requirements of the Measuring Methods
Since the Finite-Element programs, set up with the developed material laws, lead to
two- or three-dimensional distributions of the searched values, also such measuring
methods are needed, which allow larger object areas to be analysed connected and twodimensional.
As one must furthermore plan on a large local change of the material behaviour,
a high local resolution of the measuring method is also required. In addition,
measuring systems must be developed, with which deformation- and strain fields can
be measured even in the transitional areas from inelastic to elastic material behaviour.
The measurements should preferably be able to be executed on the original because
the model laws for the transfer of results from the model to the original are in
general very complicated, especially for inelastic material behaviour. Inelastic material
behaviour is often observed at high temperatures up to approx. 1000 8C. The measuring
methods should also be usable in such temperature areas. A measurement directly on
the testing machine in the testing field during an ongoing test is practical.
* Technische Universität Braunschweig, Institut für Messtechnik und Experimentelle Mechanik,
Schleinitzstraße 20, D-38106 Braunschweig, Germany
298
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

13.3 Characteristics of the Optical Field-Measuring Methods
Finally, measuring methods without contact and interaction are advantageous. All
of these requirements lead to the development and the use of the optical field-measuring
techniques. The corresponding methods are distinguished by the following characteristics.
13.3 Characteristics of the Optical Field-Measuring Methods
First, two-dimensional structured patterns are generated in the form of intensity distributions,
which are related to the searched values of the considered object surface. The
Figures 13.1 and 13.2 show two such patterns. In Figure 13.1, there are two groups of
lines, which consist of straight lines of constant width. They include an angle of 90 8
and form the so-called cross grating. If this is e.g. firmly attached to the considered
surface, then both will be deformed in the same way when by a loading.
The pattern in Figure 13.2 is produced e. g. if laser light illuminates a rough surface.
The remitted beams interfere. As a result of the distribution of different amplitudes
and phases, a granular-like intensity distribution comes into existence, which is
called the Speckle effect [1].
Figure 13.1: Cross grid.
Figure 13.2: Speckle pattern.
299

In a second step, such patterns are recorded in the image plane by recording cameras,
and their image points are determined from the digital image processing. From
this, the searched object values can be determined by a calibrated test set-up [2, 3].
This process is described in the following by the example of the object-grating method,
which is suited mostly for the deformation analysis of inelastic material behaviour.
13.4 Object-Grating Method
13.4.1 Principle
13 Experimental Determination of Deformation- and Strain Fields
The precondition is a grating structure, which is firmly attached to the considered object
surface. This is recorded from two or more different positions and orientations in
reference to the object by cameras [4] (Figure 13.3). By retransforming the digitally determined
image coordinates, e.g. the grating intersection points, into the object space,
the local vectors of the corresponding object points can be determined.
The difference of the local vectors of an object point as a result of a deformation
of the object leads to the deformation vector. The change of the distance between two
neighbouring object points related to their original distance describes the strain [5].
The stereophotographic set-up of Figure 13.3 represents the simplest arrangement
according to the photogrammetric principle [6]. There in general, only the local vector
of an object point is of interest.
In case of plane deformation of the object, only one recording camera of the optical
set-up is needed. With regard to the previously described 3-D object-grating
method, then this is called a 2-D object-grating method.
The following results have been achieved for the development of this measuring
method for the deformation analysis of objects with inelastic material behaviour.
Figure 13.3: Principle of the object-grating method.
300

13.4.2 Marking
13.4 Object-Grating Method
First, the technology for marking the object had to be improved as marks are needed,
which not only remain attached to the object caused by greater deformations of the object
surface, but which also can be recognized at high temperatures. The well-known
screen print principle was the basis for the further development [7].
If a mixture of TiO2-particles of approx. 0.3 lm diameter and ethanol is sprayed
through a screen-like mask on the polished object surface, then a grating-like structure
comes into existence after the ethanol is vaporized and the mask is removed. This is
composed of the blank polished object surface and local limited fields, which each consist
of a great number of such TiO2-particles. Figure 13.4 shows a REM-picture of
such a grating structure.
By dark field illumination, the object surface appears dark, and each grating field
is light due to the diffuse remission of the individual TiO2-particles. Figure 13.5 is one
example for this type of illumination and also the recognizability of the grating at high
temperatures [8].
Figure 13.4: TiO 2-grating in REM.
Figure 13.5: Cross grating at 8508C.
301

13 Experimental Determination of Deformation- and Strain Fields
a) b)
Figure 13.6: Cross grating on a curved object surface; a) not deformed; b) strongly deformed.
This type of grating structure can also follow a very large deformation of the object
surface without being destroyed (Figure 13.6).
13.4.3 Deformation analysis at high temperatures
The object-grating method is also suitable for deformation analyses at high temperatures.
The specimen with the attached TiO2-grating is enclosed by a radiation heater in
the testing machine. It is heated up by infra-red radiation. The visible radiation part
serves as a dark field illumination of the test surface with the grating. This is recorded
through a glass window built in the wall of the heater by a camera (Figure 13.7).
So far, the method has been developed for only in-plane deformation analysis. In
the case of the 3-D object-grating method, the heater would have to be furnished with
a second window. However, for this, another heater or illumination concept would be
Figure 13.7: Principle of the optical deformation analysis at high temperatures (top view).
302

13.4 Object-Grating Method
necessary in order to produce the necessary dark field illumination of the grating for
both cameras also at high temperatures. In addition, the test set-up cannot be calibrated
at high temperatures in the existing heater.
13.4.4 Compensation of virtual deformation
As a result of complicated initial conditions, the recorded grating images for determining
the searched deformation are perhaps superimposed by unwanted rigid body movements
of the considered specimen. These can lead to large errors in determining the deformation
according to the 2-D grating method. In Figure 13.8, the possible grating distortions
are listed.
With the aid of an eight parameter pseudo-affine transformation [9] by Equations
(1) and (2), it is possible by knowing the parameters a1 to a8 to transform the image
coordinates distorted by rigid body movements to undistorted ones:
Figure 13.8: Grating distortions as a result of translatory and rotatory rigid body movements in relation
to the camera coordinate system.
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13 Experimental Determination of Deformation- and Strain Fields
xt ˆ a1 ‡ a2x ‡ a3y ‡ a4xy ; …1†
yt ˆ a5 ‡ a6x ‡ a7y ‡ a8xy : …2†
The following steps are necessary in order to compensate possible virtual deformations
with an otherwise in-plane deformation of the object:
A reference object with an attached grating is fastened to the specimen so that the
rigid body movements of both are equal, but such that the reference object is not deformed.
After orienting the plane specimen surface and reference object parallel to the image
plane of the recording camera, the simultaneous recording of the specimen and reference
grating takes place.
From the coordinates of the reference grating referring to the non-deformed and deformed
specimen state, the parameters for the retransformation i.e. its virtual deformation
can be determined from Equations (1) and (2). With the aid of the now known retransformation
instruction, the true deformation of the specimen can be obtained from the image
coordinates referring to both loading states observed in the specimen. Figure 13.9 shows
two fields of lines of the same strain, one with (a) and the other without (b) virtual strains.
Figure 13.9: Lines of the same strain a) with and b) without virtual strains.
304

13.4.5 3-D deformation measuring
A strategy has been developed for the exact calibration of the system for a 3-D deformation
measurement [10, 11]. For this purpose, formulations and algorithms have been
developed, which are based on the photogrammetric principle. Afterwards, the inner
and outer orientation of the cameras used are determined with an appropriate calibration
object and the well-known bundle adjustment.
13.4.6 Specifications of the object-grating method
In Table 13.1, the exemplary data for the accuracy of the object-grating method when
using a camera with 1024 ×1024 pixels has been put together. These refer in one case
to the camera and as an example to an object measuring area of 20 ×20 mm 2 .
A camera with a higher resolution leads either to a higher resolution of the measuring
area or at the same resolution to recording a larger object area.
This method can currently be used for temperatures up to 1000 8C and measuring
areas from 0.1 ×0.1 mm 2 to any size. It primarily provides the field of the local vectors
of the observed object points and the displacement- and strain fields derived from them.
13.5 Speckle Interferometry
13.5.1 General
13.5 Speckle Interferometry
Table 13.1: Example for the accuracy of the object-grating method.
Camera Object
Measuring area: 1024 ×1024 Pixel 20 ×20 mm 2
Number of measuring points approx.: 75 ×75 75 ×75
Accuracy of the displacement approx.: 0.02 Pixel 0.4 lm
Reference length of the strain: 13 Pixel 0.25 mm
Accuracy of the strain approx.: 0.2% 0.2%
When developing material laws for the inelastic behaviour of metallic materials, the
transition to elastic procedures must be included. Therefore, optical field-measuring
methods are needed, with which both areas can be analysed.
Since the object-grating method despite of all of its advantages is not sensitive
enough for determining strains, which are smaller than 0.3%, a supplementary measuring
method had to be developed, with which lower scales are also attainable.
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13 Experimental Determination of Deformation- and Strain Fields
Figure 13.10: Correlation fringes.
The measuring method developed for this is based on well-known optical paths of
rays of the Speckle interferometry for measuring the in- and out-of-plane displacement
of an object surface [12, 13].
With this, one can e.g. make each one of the both in-plane components of the
displacement visible directly in the form of correlation fringes without the out-of-plane
component being included (Figure 13.10). The visibility and thus also the possibility of
obtaining qualitative values on-line can be done e.g. on a video monitor. This offers
the possibility of selectively controlling the deformation processes.
The quantitative evaluation of the Speckle interferometric measuring is carried out
according to the well-known phase-shift principle [14]. The primary results consist in
the phase differences (Figure 13.11), from which the field distributions of the single
displacement components can be derived (Figure 13.12). The strain distributions are obtained
from the displacement field by numerical differentiation.
An optical differentiation can be realized by the shearographic principle. Due to
the relative shift of the interfering paths of rays, only a small number of correlation
fringes can be obtained in comparison to the previously mentioned displacement mea-
Figure 13.11: Phase pictures of x-, y- and z-displacement.
306

13.5 Speckle Interferometry
Figure 13.12: Lines of constant displacement for x-, y- and z-direction.
suring. This strain information is overlapped by large geometric influences and slope
influences. Therefore, no on-line observation of the strain is possible. Studies have
shown that shearography is better suited for a qualitative proof of deformations.
13.5.2 Technology of the Speckle interferometry
The use of the electronic Speckle interferometry (ESPI) in the material- and construction
testing requires a compact and transportable measuring head, which can be directly
adapted on a testing machine.
The developed and practically tested measuring instrument is based on the application
of modern optoelectronic elements such as laser diodes as a light source, piezo crystals
for a nanometer exact phase shift of the light and a CCD-camera [15]. For every displacement
component, a path of rays of illumination with a laser diode, a phase shift device
and a shutter is built in the measuring head. The three displacement directions are
recorded nearly simultaneously by rapidly switching between the illumination directions.
Switching to the individual sensitivity directions takes only a few milliseconds.
In this way, it is possible to record slow running deformation processes in 3-D.
The development of the measuring head includes the construction of a control device
as well as the programming of the software to control and evaluate the measuring
data saved in an adapted computer.
Figure 13.13 shows the measuring head. It has the dimensions 250 ×250
×350 mm 3 and is adapted on a testing machine (Figure 13.14).
For a quantitative evaluation of correlation fringes by the phase-shift principle, an
initial value for the phase order is needed. This is given in the easiest case manually by
on-line observation. In principle, the heterodyne method can be used for the automation
of the order determination. It is based on using two light sources of different wave-
307

13 Experimental Determination of Deformation- and Strain Fields
Figure 13.13: 3-D-ESPI measuring head.
Figure 13.14: 3-D-ESPI on a testing machine.
lengths. For this technique, the measuring head has been expanded to two illumination
sources for each path of rays. However, this procedure requires, in addition to a very
high degree of accuracy for the phase determination, also a stronger protection against
disturbing surrounding influences than with the measuring set-up introduced here.
13.5.3 Specifications of the developed 3-D Speckle interferometer
The essential specifications of the developed 3-D-ESPI are represented in Table 13.2.
This method primarily leads to the field of displacements and by numeric differentiation
the strains of the observed object surface.
308

Table 13.2: Specifications of the 3-D-ESPI.
13.6 Application Examples
13.6 Application Examples
Measuring surface: 10 ×7 mm 2 to 600 ×450 mm 2
Measuring area out-of-plane: 0.4 ...20 lm
in-plane: 1 ...50 lm
Accuracy out-of-plane: 0.04 lm
in-plane: 0.1 lm
Strain resolution: ca. 10 –6
Local resolution: 768 ×580 Pixel or 1024 ×1024 Pixel
Object distance: 100 mm to 2000 mm
Measuring head dimensions: 250 ×250 ×350 mm 3
Displacement measurements qualitative: on-line
quantitative: 3-D by phase evaluation
The application possibilities of the object-grating method and the electronic Speckle interferometry
are introduced by three examples. They are related to their use in deformation-
and strain analysis at high temperatures, in fracture mechanics as well as welding.
13.6.1 2-D object-grating method in the high-temperature area
To examine the inelastic material behaviour at high temperatures, a tensile test was
done with a notched tensile specimen (Figure 13.15) according to the set-up in Figure 13.7
at 6508C. The task was to determine its plane deformation by using the 2-D objectgrating
method. To correct possible virtual deformations, the specimen was furnished
with a reference object (Section 13.4.4).
The essential test data are listed in Table 13.3.
In Figure 13.16, a grating section of the non-deformed and the deformed state of
the specimen are shown.
Figure 13.17 shows the determined strain fields for the in-plane directions.
13.6.2 3-D object-grating method in fracture mechanics
This example refers to the deformation- and strain analysis in the area of the crack tip
of a fracture mechanic CT-specimen (Figure 13.18). Since in addition to the in-plane
strain, also the out-of-plane displacement was searched, the 3-D object-grating method
was used.
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13 Experimental Determination of Deformation- and Strain Fields
Figure 13.15: Tensile specimen; a) incl. reference object; b) geometry.
Table 13.3: Data from the tensile test at high temperature.
Temperature: 650 8C
Material: Steel: X2CrNi18 9
Measuring method: 2-D object-grating method (p =0.2 mm)
Dimensions: H=100 mm, W=13 mm
Testing field: 19 mm×12.2 mm
Material behaviour: elastic/inelastic
Displacement measurement: in-plane
Figure 13.19 shows the calibrated testing set-up with two CCD-cameras directed
at the specimen. In Table 13.4, the essential testing data are listed.
From the recorded local vectors describing the different form states, the displacement-
and strain fields were derived. Figure 13.20 shows the searched out-of-plane displacement
and Figure 13.21 the strain distribution in the tensile direction.
13.6.3 Speckle interferometry in welding
For the experimental testing of the elastic and inelastic behaviour of a cold pressure
butt welding Copper-Aluminium specimen (Figure 13.22), the electronic Speckle interferometry
was applied. The ESPI shown in Figure 13.14, which was adapted to the tensile
machine, was used.
310

13.6 Application Examples
Figure 13.16: Section of the tensile specimen; a) non-deformed; b) deformed.
Since the measurement of all three displacement directions is possible for small
load intervals, the deformation of the elastic state could be observed on-line and quantitatively
recorded as well as the change between two purely inelastic states. In Figure 13.23,
the phase images of the in-plane displacements (corresponding to lines of constant displacement)
of an elastic, and in Figure 13.24 of an inelastic deformation are shown.
Finally, the displacement fields were determined by evaluating the phase images
according to the mentioned phase-shift principle, and the 2-D strain distributions were
Figure 13.17: Lines of constant strain a) in x- and b) in y-direction.
311

312
13 Experimental Determination of Deformation- and Strain Fields
Figure 13.18: Geometry of the CT-specimen.
Figure 13.19: Testing arrangement with two CCD-cameras.
Table 13.4: Data of the fracture mechanic test.
Material: AlMg3
Measuring method: 3-D object-grating method (p =77 lm)
Dimensions: W=20 mm, B=2 mm
Test field: 6 ×4.5 mm 2
Material behaviour: inelastic
Displacement measurement: in-plane, out-of-plane

Figure 13.20: Out-of-plane displacement field.
derived from these. Figure 13.25 shows the isolinear representation of these strains
from the inelastic deformation.
13.7 Summary
13.7 Summary
With the optical field-measuring methods, object values can be determined two- or
three-dimensionally. Therefore, they are used especially when contours, deformations
and strains of a larger area of the observed object surface should be measured together.
Although these methods are based on optical principles, which have been wellknown
for a long time, they were first able to be used when compact lasers for the generation
of coherent light and efficient PCs including adapted software for digital image
processing of a large quantity of optical measuring data were developed.
313

314
13 Experimental Determination of Deformation- and Strain Fields
Figure 13.21: e x-strain field.
Figure 13.22: Geometry of the Cu-Al specimen.

13.7 Summary
a) b)
Figure 13.23: Phase image in the elastic area a) in x- and b) in y-direction.
a) b)
Figure 13.24: Phase image in the inelastic area a) in x- and b) in y-direction.
Optical field-measuring methods are primarily used today in the manufacturingand
quality control as well as in the material- and construction testing. In the manufacturing-
and quality control, they are used among other things for the automatic recording
of form dimensions and to recognize global or locally limited defects. In the material-
and construction testing, these methods are preferably used for determining displacement-
and strain fields when testing objects with complicated structures with respect to
their dimensioning.
In connection with the development of material laws for the description of the inelastic
behaviour of metallic materials, especially the object-grating method and the
electronic Speckle interferometry have been further developed. Here, the main goal was
their adaptation for the solution of three essential tasks: Firstly, they should be used for
the observation of inelastic processes in order to get knowledge for the development of
such material laws. Secondly, parameters had to be measured for these laws. Finally,
experimentally determined displacement- and strain fields were required as a comparison
to the corresponding achieved data obtained by Finite-Element calculations in order
to verify the material laws included in them.
315

13 Experimental Determination of Deformation- and Strain Fields
Figure 13.25: Lines of constant strain a) in x- and b) in y-direction from the inelastic deformation.
The essential result of the further development of the object-grating method and
the Speckle interferometry for material- and construction testing consists in realizing
compact measuring instruments, which can be adapted directly on a testing machine
thereby making measurements directly in the testing field possible. This was achieved
by applying modern optoelectronic elements such as fibre optics, laser diodes and
CCD-cameras. In addition, both elastic and inelastic processes can be recorded from
room temperature up to high temperatures (approx. 10008C). The measurement is
made without contact and interaction. The methods yield primarily the field of the 3-D
local vectors (object-grating method) or the field of the 3-D displacement vectors
(Speckle interferometry). During the further development of the object-grating method,
principles of the near-field photogrammetry were used; the Speckle interferometric measuring
method, which is now available, is based on well-known interferometric paths of
rays, which have been integrated in a compact 3-D system here.
The further developed field-measuring methods have, in the meantime, been used
many times in various ways in material testing with respect to their reliability.
The results obtained and experiences made have encouraged further tests. It
should be tested in this way whether these field methods are also suitable for an online
measurement with the goal of process control. Furthermore, it is planned to modify
the methods so that larger object surfaces can be recorded in order to e. g. carry out an
automated construction or building supervision.
316

References
References
[1] R. K. Erf: Speckle Metrology. Academic Press, INC., London, 1978.
[2] R. Ritter: Messung von Weg und Dehnung mit Feldmeßmethoden. Materialprüfung 36(4)
(1994) 130–133.
[3] R. Ritter: Optische Feldmeßmethoden. In: W. Schwarz (Ed.): Vermessungsverfahren im
Maschinen- und Anlagenbau. Verlag Konrad Wittwer GmbH, Stuttgart, 1995, pp. 217–234.
[4] R. Ritter: Moireverfahren. In: C. Rohrbach (Ed.): Handbuch für experimentelle Spannungsanalyse.
VDI-Verlag, Düsseldorf, 1989, pp. 299–322.
[5] M. Erbe, K. Galanulis, R. Ritter, E. Steck: Theoretical and experimental investigations of
fracture by finite element and grating methods. Engineering Fracture Mechanics 48(1)
(1994) 103–118.
[6] K. Kraus: Photogrammetrie, Band 1: Grundlagen und Standardverfahren. Dümmler-Verlag,
Bonn, 1986.
[7] V. Cornelius, C. Forno, J. Hilbig, R. Ritter, W. Wilke: Zur Formanalyse mit Hilfe hochtemperaturbeständiger
Raster. VDI-Berichte Nr. 731, 1989, pp. 285–302.
[8] J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messung von Dehnungsfeldern bei Hochtemperatur-Low-Cycle-Fatigue.
Zeitschrift für Metallkunde 81(11) (1990) 783–789.
[9] D. Winter: Optische Verschiebungsmessung nach dem Objektrasterprinzip mit Hilfe eines
flächenorientierten Ansatzes. Dissertation TU Braunschweig, 1993.
[10] D. Bergmann, R. Ritter: 3D Deformation Measurement in Small Areas Based on Grating
Method and Photogrammetry. SPIE’s Proceedings Vol. 2782, Besançon, 1996, pp. 212–223.
[11] J. Thesing: Entwicklung eines Versuchsstandes und eine Auswertestrategie zur dreidimensionalen
Verformungsmessung nach dem Objektrasterprinzip. Studienarbeit am Institut für
Technische Mechanik, Abteilung Experimentelle Mechanik, TU Braunschweig 1995 (unpublished).
[12] A. Felske: Speckle-Verfahren. In: C. Rohrbach (Ed.): Handbuch für experimentelle Spannungsanalyse.
VDI-Verlag, Düsseldorf, 1989, pp. 372–397.
[13] R. Jones, C. Wykes: Holographic and Speckle Interferometry. Cambridge University Press,
1983.
[14] D. Bergmann, B.-W. Lührig, R. Ritter, D. Winter: Evaluation of ESPI-phase-images with
regional discontinuities: Area based unwrapping, SPIE’s Proceedings Vol. 2003, Interferometry
VI, San Diego, 1993, pp. 301–311.
[15] J. Hilbig, K: Galanulis, R. Ritter: Zur 3D-Verformungsmessung mit einem Elektronischen
Speckle Pattern Interferometer (ESPI). VDI-Berichte Nr. 882, 1991, pp. 233–242.
317

14 Surface-Deformation Fields from Grating Pictures
Using Image Processing and Photogrammetry
Klaus Andresen*
14.1 Introduction
Grating methods provide a well-known technique for deriving the shape, the displacement
or the deformation of the surface of an object [1]. A regular periodic grating may
be projected or fixed on the surface of an observed object. If the surface is flat, a single
image is sufficient to derive the physical grating coordinates on the object. For a
curved surface, two or more images taken from different locations are needed to calculate
spatial coordinates by photogrammetric methods. In the images, the coordinates of
suitable marks will be determined. Depending on the experimental set-up, the physical
coordinates of a plane surface or the 3-D coordinates of a curved surface will be derived
from these data [2].
According to the application, a different type of grating is applied, e.g. a line grating,
a point grating, a cross grating or a circle grating. Here, cross gratings generated
by two mainly orthogonal bands of lines will be investigated because the related image-processing
methods proved to be most stable when analysing largely deformed
grating patterns by line-following algorithms. Moreover, the cross point coordinates
could be determined in most cases automatically with subpixel accuracy [3, 4].
Projected gratings provide a simple and cheap means to deliver cross points of
the surface if only the shape of the object is asked for. However, if the deformation is
needed, only a fixed grating is applicable because the displacement of material points
from an undeformed and a deformed state must be given. The experimental set-ups and
different techniques of fixing cross gratings on a surface will be explained in the next
section.
The accuracy of the grating coordinates in the images of roughly 0.1 pixel limits
the possible applications. Depending on the resolution of the digitized image and on
the scale between image and object, one has approximately a relative error of “0.1/
number of pixels”. When a CCD-video camera with 512 pixel is used, an accuracy of
20 lm is obtainable in an object region of 100 100 mm 2 . The local accuracy of the
* Technische Universität Braunschweig, Rechenanlage des Mechanik-Zentrums,
Schleinitzstraße 20, D-38106 Braunschweig, Germany
318
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

strain within a mesh, however, is almost independent of the scale and is about
0:002 ...0:004 if the pitch of the grating lines is about 15–20 pixels and the line
width is about 5–7 pixel, which is an optimal assumption. This means that only inelastic
deformation of metals in a range larger than 0.01 or 1 percent strain delivers a suitable
accuracy.
Whole-field methods for elastic strains are based on interference or Speckle techniques.
The related optical patterns need quite different image-processing methods as
e.g. Moiré or phase-shift algorithms [5], which will not be treated in this text.
14.2 Grating Coordinates
For deformation analysis, initially periodic gratings will be applied in practice. Their
basic patterns are points given by filled circles, crosses given by two intersecting bands
of lines or overlapping circles for very large deformation of sheet metal. Each pattern
may be distorted to a certain extent during the deformation process. The coordinates
are principally defined as the centre of the circle or by the intersection of the two arms
of a cross. The geometrical structure of the grating points is assumed to be matrix-like,
i.e. a single point is characterized by its row index i and its column index j.
Also the digital image of a grating taken by a CCD-camera is stored in a rectangular
matrix of e.g. 512 rows (index x) and columns (index y), respectively. Each element
contains a grey or intensity value (0 ...256; 8 bit), which is proportional to the
light intensity of a related small region of the object surface.
The grating coordinates in the images are expressed in pixel. By suitable filtering
techniques, subpixel accuracy is reachable even in noisy images with low contrast. For
a simulated image in Figure 14.1 with a relatively large deformation of the grating, the
frequency distribution of the errors [pixel] is given in Figure 14.2. The results are derived
with a line-following filter as described in the next section. Obviously, about
85% of the deviations from the theoretical coordinates are less than 0.1 pixel. The
larger deviations will be observed in regions with a big curvature.
14.2.1 Cross-correlation method
14.2 Grating Coordinates
For less deformed cross pattern, a correlation-filter method has proved to supply cross
coordinates with subpixel accuracy [4]. Such a filter will be constructed according to
an idealized intensity distribution of a cross, where the filter constants cij are proportional
to that distribution. Then, a filtered value ~fkl is calculated by a convolution sum:
319

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
Figure 14.1: Simulated cross grating.
Figure 14.2: Frequency distribution of deviation [pixel].
~fkl ˆ XK=2
X L=2
iˆ K=2 jˆ L=2
cijfk i;l j : …1†
If this filter is applied to a picture, a smooth correlation function is resulting, which
mainly amplifies the cross region and which shows its maximum values in the centre
of each cross.
The pixel indices (xm; ym) of the maximum points could be taken as cross coordinates,
however, with a limited accuracy of one pixel. This can be improved if the maximum
…~xmax; ~y max† of a local 2-D polynomial of the form
320
f …~x; ~y† ˆa0 ‡ a1~x ‡ a2~y ‡ a3~x 2 ‡ a4~x~y ‡ a5~y 2
…2†

is calculated, which approximates the grey values of the central maximum point and its
8 neighbouring points using local coordinates …~x ˆ x xm; ~y ˆ y ym†. This technique
generally provides an accuracy of 0:1 pixel, and it was applied to different deformation
processes showing relatively small changes of the original rectangular cross grating
structure.
14.2.2 Line-following filter
14.2 Grating Coordinates
For highly deformed cross gratings, the above described technique failed because of
the big change of the width and the inclination of the crosses and its arms. For these
images, a line-following filter was developed [6]. It is used to determine both bands of
cross lines separately with subpixel accuracy. The intersection of these two bands then
delivers the wanted cross coordinates with high accuracy. This technique proved to be
stable even for strongly deformed and curved grating lines.
For initializing the line-search procedure, first, one point on the line and the related
line direction must be given. This is performed manually by two click points perpendicular
to the line or automatically with a rotational invariant filter [7], which determines the
centre of the line and its direction. But this filter is not stable when passing through cross
points. Hence, a new elliptic correlation filter of the following form was developed:
G…x; y† ˆGx…x†Gy…y†Wy…y† ˆcos p 3p p
x cos y cos y : …3†
2A 2B 2B
A and B in Equation (3) may be regarded as the principal semi-axes of an ellipse and
hence as half the filter length and the filter width, respectively (Figure 14.3). This filter
is rotated locally into grating-line direction (Figure 14.4). The hat-like filter form Gx in
x-direction amplifies all values on the line and the relatively large extension in that direction
guarantees a stable line-following quality. The cosine-like filter function Gy
with two negative side lobes is known to detect lines with an intensity distribution similar
to the central lobe. The negative side lobes provide a zero filter response if applied
to a constant grey level region in the image. The weight function Wy decreases the
function Gx steady to zero at y ˆ B, which provides a smooth filter response.
Figure 14.3: Filter function of an elliptical filter.
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14 Surface-Deformation Fields from Grating Pictures Using Image Processing
Figure 14.4: The rotatable elliptic line searching filter.
This filter is moved perpendicular through a line in 5 7 discrete steps. The resulting
filter response takes on a maximum value in the centre of the line. To provide
subpixel accuracy, the filter responses are approximated by a second order polynomial
in the maximum point. The maximum value of the polynomial then delivers coordinates
with subpixel accuracy. A similar method determines the centroid of an area between
the filter responses and a suitable threshold to define the centre coordinates of a
line. In practice, the filter is moved in column or line direction in the image according
to which direction is closer to the perpendicular direction because a shift in an arbitrary
angle through a grating line needs complex interpolation procedures.
For optimal results, the filter width 2B should be about 2 2:5 times the gratingline
width W and a filter ratio A=B ˆ 2 2:5 with the larger values for noisy data
guarantees a stable and robust line-following characteristic even through cross points
and small gaps in the line. The needed line direction for the filter rotation is derived
from the foregoing points by extrapolation.
To demonstrate the power of the described technique, a low quality image of the
surface of a metal block, deformed by forging, is evaluated (Figure 14.5). The originally
rectangular grating pattern of equal line width, etched into the material, becomes
strongly curved in some regions. Also, the line width and the intensity distribution of
the lines are quite different in horizontal and vertical direction according to the compression
and extension of the material. Hence, different filters must be used for each
line. Figure 14.6 directly shows the grey values of a small subsection marked in Figure
14.5 by a box, and Figure 14.7 supplies the same information in a 3-D representation.
In both Figures, the resulting lines of the filter process are drawn into the images. Obviously,
the centre lines are smooth even when the image is noisy and when the line
pattern has a very low contrast.
322

14.2 Grating Coordinates
Figure 14.5: Deformed grating on the surface of a metal block deformed by forging.
Figure 14.6: Grey distribution of a cross.
Figure 14.7: 3-D intensity distribution.
323

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
14.3 3-D Coordinates by Imaging Functions
When looking at the 3-D displacement of small flat deformation fields – e.g. crack tips
in a volume of 10 10 2mm 3 –, a simplified numerical method can be used instead
of stereo photogrammetry to derive the spatial grating coordinates [8]. A calibrated
grating on a glass plate is moved exactly parallel and perpendicular to its plane in precise
3 ...5 steps DZ in Z-direction. The coordinates …Xijk; Yijk; Zijk† form a dense rectangular
grid in space, and they are known exactly. Since from each step, an image is recorded,
also the related image coordinates are known. Hence, a pair of 3-D polynomial-imaging
functions n ˆ f …X; Y; Z†; g ˆ g…X; Y; Z† for each camera can be calculated,
which transforms each grid point …X; Y; Z† into an image point …n; g†, i.e. it approximates
the real stereo-imaging function. Suitable approximating functions are polynomials
with free parameters since that provides a linear-equation system when using a
Gaussian least-squares fit. Hence one has:
n…a† ˆ X aijkX i Y j Z k ; g…b† ˆ X bijkX i Y j Z k ; …4†
where the vectors a ˆ…aijk†; b ˆ…bijk† describe free parameters. Each vector is determined
separately by minimization. For a, one has:
X
…nijk;measured nijk…a†† 2 ˆ min : …5†
A similar equation holds for g…b†. For each camera, this supplies a set of two functions,
which transform any point within the grid volume into an image point. Vice versa,
also a point in space …XP; YP; ZP† can be calculated if its image coordinates
…n 1m; g 1m†, …n 2m; g 2m† are known in at least 2 images. Then, one has:
n 1…XP; YP; ZP† ˆn 1m ; n 2…XP; YP; ZP† ˆn 2m ; …6†
g 1…XP; YP; ZP† ˆg 1m ; g 2…XP; YP; ZP† ˆg 2m : …7†
These are 4 equations for the 3 unknown spatial coordinates …XP; YP; ZP†, which easily
are solved by numerical iteration.
This technique also works if the image plane in the camera is tilted (Scheimpflug
condition), which provides a larger, well-focussed area in space when using stereo cameras.
Moreover, the method is easy to program and there proved to be no convergence
problems. It was applied to the propagation of a crack tip [9, 10].
324

14.4 3-D Coordinates by Close-Range Photogrammetry
14.4.1 Experimental set-up
A general method for measuring spatial coordinates of an object grating is adopted
from close-range photogrammetry. A measuring device was developed consisting of 2
or 3 stereo cameras in a stiff framework. A movable support, which first holds a calibrating
glass plate later on holds the considered objects [11, 12]. Before any measurement
can take place, the exterior and the intrinsic orientation of the cameras must be
known. The related calibration procedure is based on a high quality cross grating,
which is fixed on a plane glass plate. The orthogonal grating lines define a global coordinate
system …X; Y; Z†; …X; Y† in the plane and …Z† perpendicular to it. With respect to
this system, the exterior orientation – the projection centre …X0; Y0; Z0† and a rotation
matrix R describing the rotation of the local camera system …x; y; z† into the global system
– must be determined. The constants of the intrinsic orientation are the focal
length c, called camera constant, and lens distortion factors A; B; R0, described later on.
The glass plate is moved in 3 parallel steps in DZ 0 -direction, which might be inclined
by small angles x; y against the Z-direction; in Figure 14.8, only a plane configuration
is shown. Given the pitch DX; DY of the cross grating and an arbitrary shift
of the origin …X0; Y0† in that plane, the coordinates of the spatial grid coordinates are:
Xijk ˆ XS ‡ iDX ‡ axZk ; …8†
Yijk ˆ YS ‡ jDY ‡ ayZk ; …9†
Zijk ˆ Zk
14.4 3-D Coordinates by Close-Range Photogrammetry
Figure 14.8: Set-up for camera calibration, plane configuration.
…10†
325

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
with i ˆ 1; ...; M; j ˆ 1; ...; N in the plane and k ˆ 1; ...; L in shift direction, where
ax ˆ tan x, ay ˆ tan y and Zk ˆ Z0 k = 1 ‡ a2x ‡ a2 q
y;
which is the related distance on
the Z-axis due to a parallel shift of Z0 k . XS; YS; DX; DY are given parameters of the cross
grating, while x; y, and Zk…k ˆ 1; ...; L 1† are unknown quantities, which will be
determined together with the parameters of the camera orientation in a modified bundle-block
adjustment. Instead of moving the cross grating, it is also possible to shift the
cameras and fasten the grating if this results in a simpler set-up.
In each shifted position, the cross grating is recorded. This means that each camera
takes the images of a spatial grid, which approximately coincides with the measuring
volume of the device.
14.4.2 Parameters of the camera orientation
Here, only a short summary will be given for the theory of the bundle-block adjustment
because it is well-known in the relevant publications [2]. The intrinsic and exterior
orientation of a camera in space is described by its projection centre …X0; Y0; Z0†,
the focal length c, the rotation matrix R ˆ…rij† usually given by 3 Euler angles, and
the distance …n 0; g 0† of the origin in the image plane to the optical axis. Then the transformation
from the space coordinates …X; Y; Z† to the image coordinates …n; g† is for
each point (i; j; k) within the grid:
n ˆ n0 ‡ c …X X0†r11 ‡…Y Y0†r21 ‡…Z Z0†r31
‡ u…n; g† ; …11†
…X X0†r13 ‡…Y Y0†r23 ‡…Z Z0†r33
g ˆ g0 ‡ c …X X0†r12 ‡…Y Y0†r22 ‡…Z Z0†r32
‡ v…n; g† : …12†
…X X0†r13 ‡…Y Y0†r23 ‡…Z Z0†r33
u…n; g†; v…n; g† describe the lens distortion, which are assumed to be radial symmetric:
u…n; g† ˆ…A1…R 2 0 R 2 †‡A2…R 4 0 R 4 ††…n n 0† ; …13†
v…n; g† ˆ…A1…R 2 0 R 2 †‡A2…R 4 0 R 4 ††…g g 0† ; …14†
where the radius R is given by:
R ˆ …n n0† 2 ‡…g g0† 2
q
: …15†
R0 is a constant of the objective, usually about 70% of the maximum width of the image
plane, and A1; A2 are the required distortion parameters.
Now p will be defined as the vector of the unknown parameters:
326

p ˆ…fX0; Y0; Z0; n 0; g 0; rij; c; A1; A2g l ; ax; ay; Z1; ...; ZL 1; fXijk; Yijk; Zijkg m † ; …16†
where subscript l means that this set of parameters is repeated for each camera. Further
on, a certain number m of spatial coordinates are not used directly, but they are taken
to be unknown parameters. This provides a higher accuracy because it couples the pa-
rameters of the cameras in a global least-squares fit. …n 0
ijkl ; g0 ijkl
† are measured image co-
ordinates of the camera l. By a bundle-block adjustment, the unknown parameters in p
are determined altogether by minimizing the sum of the squared differences between
the measured and the calculated image coordinates. Here, the expression for n is given,
a similar expression holds for g:
F…p† ˆ X
14.4 3-D Coordinates by Close-Range Photogrammetry
camera Xgrid l
ijk
…n ijkl…p† n 0
ijkl †2 ˆ min : …17†
To solve Equation (17), it is linearized with respect to an initial parameter vector p 0:
F…p0†‡ qF…p0† D p ˆ 0 for every …i; j; k; l† …18†
qp
yielding an overdetermined system of linear equations for an increment D p, which is
solved by the least-squares method. Starting from suitable initial values p 0, a global
iteration is performed until D p is less than a given threshold. p and its standard deviation
are the final result. Numerical experience has proved that about half the number of
grating points should be dealt with as unknown points to get an optimal convergence
and accuracy of the non-linear iteration process.
Programming of the bundle-block adjustment and also its application requires a
lot of experience, especially choosing suitable initial values becomes a very sensitive
task. Meanwhile, commercial products are available [13].
14.4.3 3-D object coordinates
If the parameters of the orientation are known, there are well-known algorithms [2] to
determine spatial object points by ray intersection, provided the coordinates of the adjoined
points in the stereo images are given. This is generally true for grating images.
However, it is also possible to determine whole elements in space without knowing
adjoined points if the elements are to be described analytically by a set of free parameters
[14, 15]. Practical examples are circles, straight lines and curved lines. Also
cylinders and spheres in space can be treated if a sequence of contour points in the
images are detected.
327

14.5 Displacement and Strain from an Object Grating:
Plane Deformation
The strain tensor of a local grating point in a plane (x; y) can be calculated from the
displacement of the vectors dr 1
i …i ˆ 1; ...; 4† in the deformed state and dr0i
in the undeformed
state, where dri means the connection to the neighbouring 4 grating points.
Generally, the displacement is due to a rigid body motion and a deformation of
the object. For the calculation of strain in a grating point, first, the centre of the deformed
element will be shifted parallel into the related undeformed centre. Then, the
variation of the four dr-vectors describes a rotation and the desired plastic deformation
or strain. The rotation will be separated from the strain in a theory of large deformation
as follows. Assuming, the vectors …dr 0
1 ; dr02
† will be deformed into …dr11
; dr12
†, then a deformation
gradient F is calculated from the linear relations:
dr 1
1 ˆ Fdr01
; dr12
ˆ Fdr02
: …19†
F will be split into the left rotation tensor R and a right deformation tensor U:
F ˆ RU : …20†
From the right Cauchy-Green tensor G:
G ˆ F T F ˆ U T R T RU ˆ U T U ; …21†
a deformation tensor [16] is given:
U ˆ G
p ˆ c01 ‡ c1G …22†
with
1
c1 ˆ p p ; c0 ˆ c1det G : …23†
trG ‡ 2 det G
trG ˆ g11 ‡ g22 is the trace of G and det G is the related determinant. The elements of
the deformation tensor U
U11 U12
U21 U22
ˆ 1 ‡ ex exy
exy 1 ‡ ey
include the well-known plane-strain components ex; ey; exy. In a cross point, 4 strain values
according to the four meshes surrounding the point are calculated. The average of
these values is taken to define the local strain tensor in the central point.
A similar technique can be applied to spatial surfaces if the curvature is relatively
small within the considered area. Then, 4 pairs of vectors as in the plane case are taken
328
14 Surface-Deformation Fields from Grating Pictures Using Image Processing
…24†

to calculate the strain in its centre point. After moving the deformed centre point and
the related vectors into the undeformed state, each of the pairs of spatial vectors, forming
a triangle, are rotated into an arbitrary reference plane, in which a plain-strain tensor
can be calculated as described in the foregoing section.
14.6 Strain for Large Spatial Deformation
14.6.1 Theory
14.6 Strain for Large Spatial Deformation
The described procedure for calculating the spatial deformation fails if the curvature is
large. Then, virtual strains arise already from the rotation of an element into a reference
plane.
Geometrically based methods for evaluating large strain are published in [17, 18].
Here, an improved method based on a deformation function is proposed. It delivers a
deformation gradient for the central point using the 8 neighbouring points in the grating.
To derive the deformation function, 4 meshes of a plane grating are considered
with basic coordinates …x; y; z† as shown in Figure 14.9. The coordinates of the undeformed
grating are …xij; yij; zij ˆ 0† with indices (i; j ˆ 1; 0; 1) or written as a vector
xij ˆ…xij; yij; 0† T . The coordinates of the deformed element are xij ˆ…xij;yij;zij† T .
Figure 14.9: Undeformed grating element and spatially deformed one.
329

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
The total displacement of each point again is given by a rigid body translation, a
rotation and a plastic deformation of the element. To eliminate the rigid body motion,
first, the central vector x00 will be subtracted:
~xij ˆ xij x00 : …25†
Then, a rotation matrix will be determined, which moves the normal vector n of the
surface element into the z-axis. Approximately, the normal vector is derived from the
cross product of the difference vectors:
dx ˆ ~x10 ~x 10 ; dy ˆ ~x01 ~x 0 1 ; …26†
n ˆ dx dy : …27†
The unit vector n0 delivers the 3. row of the rotation matrix R. The first row is taken
from the unit vector d 0
x , which means that the deformed x-direction nearly coincides
with the related undeformed direction after rotating the element. The 2. row is given by
the cross product n0 d 0
x .
Now, the rotated coordinates are:
^xij ˆ Rxij …i; j ˆ 1; 0; 1† ; …28†
and the related displacement vectors:
q ij ˆ ^xij xij : …29†
Regarding these vectors, a deformation function for each coordinate can be assumed,
which describes the displacement from the undeformed to the deformed state:
^x ˆ x ‡ f …x; y† ; ^y ˆ y ‡ g…x; y† ; ^z ˆ h…x; y† : …30†
The functions f ; g will be approximated by polynomials of second order, e.g.:
f …x; y† ˆf0 ‡ f1x ‡ f2y ‡ f3x 2 ‡ f4xy ‡ f5y 2
with f0 ˆ 0, since ^x00 ˆ 0. Forg, a similar function with coefficients gk is taken. The
deformation function h for ^z is less relevant since only the deformation in the tangential
plane is considered.
From these functions, the deformation gradients in the central point x00 are derived.
Hence, one has:
…31†
q^x=qx ˆ 1 ‡ f1 ; q^x=qy ˆ f2 ; …32†
q^y=qx ˆ g1 ; q^y=qy ˆ 1 ‡ g2 : …33†
To determine the coefficients fk…k ˆ 1 ...5†, a least-squares method applied to the xcomponent
of displacement vector qxij requires
330

X
fqxij f …xij; yij†g 2 ˆ ! min : …34†
ij
In the case of rectangular meshes with pitches Dx; Dy in the basic plane, the matrix of
the normal equations can be calculated analytically. With respect to symmetry, the coefficients
f1; f2 are totally uncorrelated and one has:
f1 ˆ X
xijqxij= X
x 2 ij ; f2 ˆ X
yijqxij= X
; …35†
ij
and similar equations for g1; g2:
g1 ˆ X
xijqyij= X
x 2 ij ; g2 ˆ X
ij
ij
ij
ij
ij
ij
y 2 ij
yijqyij= X
y
ij
2 ij
: …36†
Generally, weight coefficients wij may be introduced according to the significance of
the displacement values qij, leading e.g. to f1 ˆ P
xijqxijwij=
ij
P
wijx
ij
2 ij . If the totally 8
edge points are existing, the following simple expressions are resulting (wij ˆ 1):
f1 ˆf…qx11 ‡ qx10 ‡ qx1 1† …qx 11 ‡ qx 10 ‡ qx 1 1†g=…6Dx† ; …37†
f2 ˆf…qx11 ‡ qx01 ‡ qx 11† …qx1 1 ‡ qx0 1 ‡ qx 1 1†g=…6Dy† : …38†
Similar equations hold for g1; g2 with qxij replaced by qyij. Taking weight coefficients
wij ˆ 0 in the four corner points, the simple central differences, e.g. f1 ˆ
…qx10 qx 10†=2Dx, are resulting. This proved to yield the least errors for noise-free
grating coordinates as shown in the next section.
Now, the deformation gradient F
F ˆ 1 ‡ f1 f2
g1 1 ‡ g2
14.6 Strain for Large Spatial Deformation
is given in point (0,0), and the strain can be calculated according to Equations (19) to
(24).
Generally, already the undeformed element may be spatially curved and no longer
rectangular. Then, both elements, undeformed 1 and deformed 2, will be shifted into
the origin. There, they are rotated as described in the foregoing section, meaning that
the normal vector n1; n2 coincide with the z-axis and that the deformed x-directions
show into the basis x-direction. Then, a displacement vector is calculated from the differences
of the rotated coordinates:
qij ˆ ^x 2 ij ^x 1 ij ; …40†
and according to Equations (30) to (34), a system of normal equations can be determined.
In this case, Equation (34) must be built up and solved numerically because no
symmetry of the meshes is provided.
…39†
331

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
Figure 14.10: Correcting the displacement vector by arc length.
Figure 14.11: Hat-like deformed metal sheet.
Figure 14.12: Principal strain eI.
14.6.2 Correcting the influence of curvature
Numerical simulation, as described in the next section, shows a systematic error for the
strain. In the average, it is always too small because only the tangential projection of
the curved surface is used. A noticeable improvement is derived when the arc length of
332

14.6 Strain for Large Spatial Deformation
Figure 14.13: Difference of eIth eI without compensation of curvature.
Figure 14.14: Difference of eIth eI with 70% compensation of curvature.
deformed surface is considered in the x-z- ory-z-plane, respectively. This can be performed
approximately by calculating a second-order polynomial of displacement in zdirection
through the deformed but translated and rotated points ^x 10, ^x00, ^x10 in Figure
14.10. Using numerical integration, the arc length of this curve is determined.
Then, it proved to give optimal deviations with almost zero average when adding 50%
to 70% of the difference between the length of the curve and its projection to the displacement
in x-direction. A similar procedure holds for the y-direction. The errors of
the strain could be decreased by 20% to 50% depending on the pitch of the grating.
14.6.3 Simulation and numerical errors
The spatial strain procedure was tested on a hat-like deformed metal sheet. The deformation
functions u…r† and w…r† in z-direction were assumed to be radial symmetric:
u…r† ˆcu sin …2pr=R0† ; w…r† ˆcw cos …pr=R0† ; …41†
333

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
Figure 14.15: Relative maximum error of strain eIth eI with respect to grating pitch.
r ˆ …x2 ‡ y2 p
† ; u ˆ arctan …y=x† : …42†
Then, the deformed hat is given by:
^x ˆ x ‡ u…r† cos …u† ; ^y ˆ y ‡ u…r† sin …u† ; ^z ˆ w…r† : …43†
The exact principal strains in r-direction are:
q
eIth ˆ …1 ‡ du=dr† 2 ‡…dw=dr† 2
1 ; eIIth ˆ u=r : …44†
Regarding the following figures, the parameters were cu ˆ 0:5, cw ˆ 4;
R0 ˆ 5; Dx ˆ Dy ˆ 0:33. In Figure 14.11, the hat-like deformed sheet is given for
30 30 lines. Figure 14.12 shows the principal strain eI calculated according to Sections
14.5 and 14.6.1. Figure 14.13 demonstrates the related error eIth eI if no compensation
of the curvature is taken into account. Obviously, the difference is always
positive with an average value of 0.006.
Figure 14.14 shows the same deviation with 70% curvature compensation due to
Section 14.6.2. The average value is reduced to 0.002 and the maximum error from
0.011 to 0.007. Hence, in this example, the maximum deviation from the theoretical
values is always less than 0.7% of the maximum strain eI ˆ 0:96. A 100% compensation
does not reduce the maximum values but increases the minimum values to –0.007.
Figure 14.15 shows the influence of the grating pitch on the relative maximum error
for the above chosen example. The error increases with the pitch in the interesting
334

interval approximately like a second-order polynomial. For a 10 10 grating, the related
maximum error becomes already 6%.
14.7 Conclusion
The grating techniques, as described in this chapter, were applied to a large variety of
deformation processes, e.g. crack-tip propagation, low-cycle fatigue, high-temperature
creep, cold-welded zones of Cu-Al-specimen, necking of cylindrical tension specimen,
holes in thin-sheet metal, crystal deformation and many examples more. This was performed
mainly for research purposes to get insight into deformation processes, to test
Finite-Element results or to determine the parameters in new constitutive laws for inelastic
deformation [19, 20].
With some modifications, the photogrammetric equipment and the software also
were applied to measure the dimension of industrial parts [8] to determine the radius of
cutting tools and to derive the contour of a fossil fish.
A new field of practical applications comes from metal forming. Especially for
sheet metal forming of the body works of cars, the grating methods are used for failure
analysis. Often, it must be decided whether the characteristics of the material or the
shape of the pressing tool cause the defects on the surface. Also, the influence of oil
films on the flow of the material and on the friction between tool and sheet metal shall
be investigated to optimize the forming process and the mechanical equipment.
For these future applications, the image-processing hardware and software must
be improved further with respect to the following topics:
• developing fully automatic programs for incrementally deformed pattern series,
• improving the portability of the software as to support PC’s and workstations,
• looking for low-cost hardware equipment for technology transfer into industry, and
• implementing state of the art graphical user interfaces and plotting software.
These goals seem to be in the reach within the next years because the computing
power still increases every year, thus allowing more sophisticated algorithms for automatic
image processing.
References
References
[1] P. J. Sevenhuijsen, J.S. Sirkis, F. Bremand: Current trends in obtaining data from grids. Exper.
Techn. 27 (1993) 22–26.
[2] K. Kraus: Photogrammetrie, Vol. 2. Dümmler, Bonn, 1984.
[3] J.S. Sirkis, T. J. Lim: Displacement and strain measurement with automated grid methods.
Exp. Mech. 31 (1991) 382–388.
335

14 Surface-Deformation Fields from Grating Pictures Using Image Processing
[4] K. Andresen, B. Hübner: Calculation of Strain from an Object Grating on a Reseau Film
by a Correlation Method. Exp. Mech. 32 (1992) 96–101.
[5] K. Andresen, Q. Yu: Robust phase unwrapping by spin filtering combined with a phase direction
map. Optik 94 (1993) 145–149.
[6] Z. Lei, K. Andresen: Subpixel grid coordinates using line following filtering. Optik 100
(1995) 125–128.
[7] P.-E. Danielsson, Q.-Z. Ye: A new procedure for line enhancement applied to fingerprints.
Report of Linköping University, Dept. of Electrical Engineering, Linköpin, Sweden, 1983,
p. 581.
[8] K. Andresen: 3D-Vermessungen im Nahbereich mit Abbildungsfunktionen. Mustererkennung
92, 14. DAGM Symposion, Dresden, 1992, pp. 304–309.
[9] K. Andresen, B. Kamp, R. Ritter: 3D-Contour of Crack Tips Using a Grating Method. Second
International Conference on Photomechanics and Speckle Metrology, San Diego 1991.
SPIE Proceedings Vol. 1554A (1991) 93–100.
[10] K. Andresen, B. Kamp, R. Ritter: Three-dimensional surface deformation measurement by
a grating method applied to crack tips. Opt. Eng. 31 (1992) 1499–1504.
[11] K. Andresen, Z. Lei, K. Hentrich: Close range dimensional measurement using grating
techniques and natural edges. SPIE 2248 (1994) 460–467.
[12] K. Andresen, K. Hentrich, B. Hübner: Camera Orientation and 3D-Deformation Measurement
by Use of Cross Gratings. Optics and Lasers in Engineering 22 (1995) 215–226.
[13] CAP – Combined Adjustment Program, Users Manual. Fa. Rollei Braunschweig, FRG,
1989.
[14] K. Andresen: Ermittlung von Raumelementen aus Kanten im Bild. Zeitschrift für Photogrammetrie
und Fernerkundung 59 (1991) 212–220.
[15] F. Neugebauer: Calculation of curved lines in space from non-homologues edgepoints. Optical
3-D Measurement Techniques III, Eds. Gruen/Kahmen, Vienna 1995, pp. 506–515.
[16] J. Stickforth: The Square Root of a Three-Dimensional Positive Tensor. Acta Mechanica 67
(1987) 233–235.
[17] F. Bredendick: Methoden der Deformationsermittlung an verzerrten Gittern. Wiss. Zeitschrift
der Techn. Univ. Dresden 18 (1969) 531–538.
[18] L. Eberlein, P. Feldmann, R.V. Thi: Visioplastische Deformations- und Spannungsanalyse
beim Fliesspressen. Umformtechnik 26 (1992) 113–118.
[19] K. Andresen, R. Ritter, E. Steck: Theoretical and Experimental Investigations of Fracture
by FEM and Grating Methods. Defect Assessment in Components – Fundamentals and Applications.
Mechanical Engineering Publications, London, 1991, pp. 345–361.
[20] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifikation
für ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Bauing. 71
(1996) 21–31.
336

15 Experimental and Numerical Analysis
of the Inelastic Postbuckling Behaviour
of Shear-Loaded Aluminium Panels
Horst Kossira and Gunnar Arnst*
15.1 Introduction
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
The engineering problem of the presented research project is based on the design and
loading of the aerospace structure shown in Figure 15.1. Although the example in Figure
15.1 depicts a possible structure of a hypersonic vehicle, the construction is typical
for supersonic and common subsonic transport aircrafts. To examine the basic phenomena
of the load-carrying behaviour in the postbuckling range, the analysis of such structures
can be reduced to a simplified mechanical model of an initially flat, shear-loaded
panel. For practical reasons, our investigations were limited to aluminium (Al2024-T3)
panels at ambient temperature and 200 8C. Within the high postbuckling regime or at
high load levels and elevated temperatures in supersonic vehicles, moderate inelastic
strains occur and the behaviour of the considered structure becomes geometric and
physically non-linear.
Due to the high complexity of the problem, analytical investigations must be accompanied
by tests in order to validate the numerical model. It is based on the Finite-
Element method and therefore can be easily applied to different geometries and boundary
conditions. The main problem of the numerical model is the choice of suitable
material models since no universal material model for the description of the inelastic
behaviour of arbitrary metallic materials exists. To simplify the adaption of the presented
numerical model to different materials, the used solution algorithm is designed
to allow a very easy implementation of different material models. In case of the considered
aluminium alloy, the performance of several material models is examined for the
rate-independent plasticity at ambient temperature and visco-plastic behaviour at elevated
temperature. The identification of their parameters from suited material test results
is demonstrated.
All shear tests, including quasistatic monotonic, cyclic and creep- and relaxation
tests at elevated temperatures, are conducted with a specially designed test set-up
* Technische Universität Braunschweig, Institut für Flugzeugbau und Leichtbau,
Hermann Blenk Straße 35, D-38108 Braunschweig, Germany
337

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.1: Typical structure.
(“PApS”, Figure 15.2) generating pure shear load with clamped boundary conditions.
The rigid edges of the picture frame are pin-jointed, where the pins are located exactly
at the corners of the square specimen. The pins are parted so that the tested area of the
specimen forms a square field with no cutouts with a dimension of 500 ×500 mm 2 .
Therefore, the comparison of numerical and theoretical results achieved with the mechanical
model shown in Figure 15.3 are not influenced by uncertainties in the assumption
of the geometry or the boundary conditions.
The test set-up is equipped with nine infrared radiators in front of the shear panel
and eight heating elements, which are placed directly on the edges of the shear frame.
In combination with five separate temperature controllers, a nearly constant temperature
distribution can be achieved in the tested area of the panel up to 200 8C. Until now, 78
monotonic and cyclic tests at ambient and elevated temperature and different panel
thicknesses have been performed on this test set-up.
Figure 15.2: Test set-up PApS.
338

Figure 15.3: Mechanical model.
15.2 Numerical Model
15.2.1 Finite-Element method
15.2 Numerical Model
Two partly different formulations of the fundamental equations and affiliated solution
methods are used. The behaviour of the considered material at room temperature can
be described by means of rate-independent material models for spontaneous plasticity.
This type of non-linear material models are implemented in the framework of the geometrically
non-linear static equations of the plate theory, and the problem is solved by incremental-iterative
methods. At the other hand, the visco-plastic problem introduces real
time-dependency with higher demands regarding the time-integration accuracy and stability.
Therefore, a more closed formulation of the fundamental equations of the continuum
theory and the constitutive equations is used to apply a suitable time-integration method.
Both methods base on the following assumptions of the continuum theory: Adopting a
total Lagrange formulation with an additive decomposition of the strain tensor, large deformations
but only small strains are admissible. The mixed variational principle, which is
used to derive the finite elements, bases on the Kirchhoff-Love plate theory. This theory
yields reasonably good results since the considered panels are sufficiently slender. All material
models are implemented in the numerical model by means of the normality rule of
the classical theory of plasticity, applying a v. Mises type of inelastic potential. A mixed
principle is chosen since such a formulation produces displacements and stresses with the
same degree of approximation as primary unknowns, and therefore provides some advantages
concerning the computational effort in the treatment of the non-linearities. All primary
unknowns of the described variational formulations are approximated with bilinear
polynoms, yielding a four-noded plate element.
339

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
15.2.1.1 Ambient temperature – rate-independent problem
The used variational functional is derived from the principle of virtual displacements
with the strain-displacement relation as a restriction and reads for a certain state t:
dA t Z
ˆ d
F
r t c
1
2 rt C 1 r dF
Z
F
dv t p dF ˆ 0 : …1†
This principle is transformed into an incremental form, yielding the linear stiffness matrix,
the tangent and the secant matrix. The solution for each increment is achieved by
an arc length-controlled modified Newton-Raphson iteration. In this form, the material
law in the incremental form De ˆ DDr is comprised in the tangent matrix, and the
non-linear constitutive equations of the rate-independent material models can be easily
included. The in-plane, coupling and bending stiffnesses of the plate material are determined
by integration of the actual elastic-plastic tangent moduli D given by the used
material model over the plate thickness. This integration assumes discrete layers with
constant properties in thickness direction, and the description of the material behaviour
is reduced to the plain stress constitutive equation in each layer. The elastic-plastic tangent
modulus is updated only once for each load increment using the results of the previous
load step, yielding an Euler-Cauchy type of integration of the non-linear constitutive
equations. This method considerably reduces the numerical effort. Furthermore, no
differentiation of the constitutive equations is needed as it would be the case in implicit
integration methods, and therefore, the material model can be changed very easily. Stability
problems in the integration of the used material models were never observed, and
it can be shown that the error in the solution for the used constitutive equations remains
sufficiently small since the magnitude of the load increments, which is restricted
by the geometric non-linearities, is small enough. Details can be found in [1] and [2].
The described solution method is in principle capable of calculating snap-through effects.
However, due to the quasistatic basis of the method, a so-called instable equilibrium
path connects the starting- and the end point of the snap-through. This path represents
a fair approximation only for moderate snap-throughs. Simulations of severe
snap-throughs, which occur in the range of unloading during cyclic shear tests after
very high load amplitudes, lead – in connection with the development of plastic strains
during the snap-through – to obviously wrong solutions and numerical problems. To
improve the capabilities of the used method, the described Finite-Element formulation
was extended to calculate dynamic effects by solving the complete equation of movement.
For simplicity, the damping matrix is formed by a linear combination of the used
consistent mass matrix and the system-stiffness matrix. The magnitude of damping is
fitted to experimental results. The accelerations and the velocities are derived from the
displacements using the Newmark scheme. To reduce the numerical effort, the dynamic
method is only used if the quasistatic method detects a limit point in the loading path
and a snap-through is starting. In this point of loading, the determinant of the system
matrix changes its sign. When after the snap-through, the velocities of the structure are
small enough, the quasistatic method is used again.
340

15.2.1.2 Elevated temperature – visco-plastic problem
The temporal derivative of the basic variational principle (Equation (1)) is used to derive
the Finite-Element formulation. To achieve a closed formulation of the problem,
the equations related to the material model are included in the principle by means of
Lagrange multipliers. In case of the Chaboche model, those equations are: the consistency
condition, the overstress function and the evolution equations for isotropic and
kinematic hardening. The corresponding Lagrange multipliers can be determined by the
requirement that each term of the variational principle has to form an energy. Like in
the rate-independent problem, a layered model is adopted. However, the plate stiffness
is formulated directly in terms of the primary unknowns. In the sense of a “rate
approach” [3], the primary unknowns are the velocities, the temporal derivatives of the
stresses, respectively the membrane forces, and the bending moments plus the temporal
derivative of the visco-plastic potential and the effective strain rate of each layer. The
complete visco-plastic problem is solved by a predictor-corrector method similar to a
midpoint-type time-integration algorithm. One predictor and one corrector step is used
for solving the equations in one time-increment. A comprehensive discussion of this
method is given in [4]. This time-integration scheme was chosen in order to avoid the
analytical or numerical determination of the tangent-stiffness matrix needed in the
framework of implicit time-integration schemes to establish the Newton-Raphson iteration.
Therefore, this method reduces the difficulties of the implementation of new constitutive
equations. An automatic time-step control is established by limiting the magnitude
of the increment of equivalent total strain within each time-step. The critical value
for this increment of equivalent total strain is determined by numerical experiments.
15.2.2 Material models
15.2 Numerical Model
All investigations are made for the aluminium alloy 2024-T3 (AlCuMg2/3.1354T3),
where T3 denotes the rolling and prestraining pretreatment. The results of tension tests
at room temperature and elevated temperature are given in Figure 15.4. There is a distinct
plastic orthotropy at room temperature, which must be taken into consideration in
the material model. This orthotropy vanishes at higher temperatures. The results of the
tests at 200 8C with different loading rates give an impression of the changed mechanical
properties of the material at elevated temperatures.
15.2.2.1 Ambient temperature – rate-independent problem
The cyclic elasto-plastic behaviour of the considered aluminium alloy 2024-T3 at ambient
temperature can be described by two or more surface rate-independent material
models based on the classical theory of plasticity. In the conducted investigations, the
performance of twelve different material models, respectively combinations of their basic
characteristics, were examined.
341

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.4: Tensile tests at room temperature and at 200 8C.
Those basic characteristics are:
• the number (Mroz) and shape of the yield surfaces, i.e. v. Mises, Hill, Rees,
Doong-Socie,
• the rule for the translation of the yield surface, respectively kinematic hardening,
i.e. Mroz, Phillips-Weng, Tseng-Lee,
• the rule for the isotropic hardening, i.e. Ellyin, and
• the way, the plastic tangent modulus is determined, i.e. Dafalias or McDowell.
A more detailed description of the used material models is given in [1, 5, 6]. The influence
of the translational rule on the performance material models shall be discussed in
more detail. All of the described models use v. Mises-type yield surfaces. Deformable
yield surfaces shall not be discussed here in more detail.
The Mroz model for the material used is sketched in Figure 15.5. Four surfaces
are located in the stress space. The inner one surrounds the elastic region and is of the
Hill-type. This type of yield surface is based on the v. Mises surface, but can be deformed
by adjusting additional shape parameters. All surfaces can move in the stress
space in the sense of kinematic hardening expressed in terms of the backstress tensor.
This effect is substantial for the representation of the Bauschinger effect occurring in
cyclic loading of metallic materials. Furthermore, the description of the plastic anisotropy
is made possible by adjusting the starting values of the backstress tensor. Therefore,
kinematic hardening is necessary even for the simulation of tests with monotonic
loading. Plastic loading takes place when the stress point is located on the yield surface
and moves in an outward direction. Stress states beyond the yield surface are not admissible.
This is controlled in all types of material models of this category by the socalled
consistency condition. Therefore, the movement of the yield surface during plastic
loading with kinematic hardening is restricted by this condition. The direction of the
movement of the yield surface has to be established separately. For the Mroz model,
342

Figure 15.5: Mroz model.
15.2 Numerical Model
the translation of the yield surfaces is chosen in a way that subsequent yield surfaces
get into contact in points with equal directed normals. In other words, the surfaces
approach each other tangentially.
An isotropic hardening during plastic loading, which means an expansion of the
surfaces, is possible. This type of hardening describes the stabilization of the materials
hysteresis and is controlled by means of the accumulated effective plastic strain. Its
contribution to the entire hardening is small since the considered alloy shows a rapid
stabilization under cyclic loading. Each surface of the Mroz model is connected with a
constant plastic tangent modulus. This causes a piecewise linear approximation of the
inelastic stress-strain relation, which is a central advantage of this model since the initial
position of the yield surfaces, their diameter, the shape parameters of the yield surfaces
and the plastic tangent moduli can easily be determined as the graphic in Figure
15.5 demonstrates. Unfortunately, the storage capacity needed for this model in numerical
analyses is comparatively high since three backstress tensors have to be stored.
The second described material model is the Tseng-Lee model. It consists of only
one yield surface and a so-called memory surface in the stress space. The yield surface
shows kinematic and isotropic hardening, whereas the memory surface can only expand
in the sense of isotropic hardening. The translational rule for the kinematic hardening is
chosen in an approximation of the test results of Phillips [7]. These results indicate that
the translation on the yield surface is directed along the actual stress increment. To
avoid an intersection of the yield- and the memory surface, Tseng and Lee [8] formulated
their translational rule as a combination of the “Phillips direction” and a direction
of movement, which is determined by the distance measure between the actual stress
point on the yield surface and a point on the memory surface with the same outer nor-
343

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
mal. Unfortunately, the Mroz direction predominates the kinematic hardening in some
constellations of the position of the yield surface and loading direction. To improve this
behaviour, an own modified formulation for a combined translational rule was developed,
which enforces the movement along the “Phillips direction”. In both models, the
tangent modulus is determined as a function of a distance measure between the actual
stress point and a point on the memory surface. Considering non-proportional hardening
effects, the distance measure used in the equation of Dafalias for the tangent modulus
is directed along the stress increment. If the yield surface is in contact with the
memory surface and the distance measure becomes zero, the tangent modulus is determined
by a simple Ramberg-Osgood power law, which is fitted to the properties of the
material under monotonic loading. The usage of two different equations for the tangent
modulus – for repeated loading and for monotonic loading – is affirmed by tests conducted
by Phillips [7].
The fourth model in this comparison is formed by a two-surface model with a
pure Mroz-type translational rule and the Dafalias equation for the tangent modulus.
All parameter identifications for the considered models are performed by means
of stochastic, respectively evolutional optimizing methods, using a least-square formulation
of the object function. To use the material models for the calculation of the behaviour
of the shear-loaded panels, their parameters are identified simultaneously by the results
of four uniaxial tension-compression tests with specimen made of the considered
aluminium alloy in the typical pretreatment state. The results of the tension-compression
tests are fairly good approximated by all four described material models. An example
is given for the Tseng-Lee model in Figure 15.4, where only the first tensile
loadings are depicted.
Since wide areas of cyclic shear-loaded and buckled panels exhibit non-proportional
paths for one component of normal strain and the shear strain, special emphasis
is laid on the capabilities of the material models in reproducing non-proportional hardening
effects. Results of strain-controlled tension-torsion test (provided by G. Lange et
al. [9], Institute of Material Science, Techn. Univ. Braunschweig) are used to examine
the performance of the material models considering the effect of non-proportional hardening.
The results of the simulation of a typical non-proportional strain path are shown
in Figure 15.6. The first loading of the tubular specimen leads to pure shear. Then, the
specimen is unloaded and a combined tension-torsion loading starts. After total unloading,
this load cycle starts again. Obviously, the models using a pure Mroz-type translational
rule show a poorer correlation with the test data, especially in the development
of the tensile stress. In contrast, the models using the combined translational rules are
able to give a good simulation of the behaviour even up to high cycle numbers.
With one indicated exception, all results given in this paper, concerning the behaviour
of shear-loaded panels at ambient temperature, are calculated with the described
Tseng-Lee material model.
15.2.2.2 Elevated temperature – visco-plastic problem
The visco-plastic problem is treated with unified, respectively overstress material models.
The models of Steck [10] (in an isothermal formulation, see Equations (2) and (3))
344

and Chaboche [11] with several modifications are in examination. Several creep, stressand
strain-controlled uniaxial tests were performed (supported by K.-T. Rie et al. [12],
Institute of Surface Engineering and Plasmatechnology, Techn. Univ. Braunschweig) to
provide results for the process of parameter identification. This is necessary because of
the strongly underdetermined character of such a problem, and since it is known that a
parameter identification with only one distinct type of test result usually cannot represent
the properties of the material sufficiently.
• Steck model:
e_ in ˆ A1e
V2r iso
‰sinh …V1r eff †Š N sign …r eff † with r_ iso ˆ h1e
r_ kin ˆ h2e …V1r iso V5r kin sign …r eff †† e_ in
• Chaboche model:
e_ in ˆ f
K
15.2 Numerical Model
Figure 15.6: Results of tension-torsion tests compared to different material models.
iso V3r
je_ in iso V4r
j A2e ; …2†
A3 sinh …V6 r kin † and r eff ˆ r r kin : …3†
N
; for f > 0 with f_ ˆ D…T X† R_ ; X ˆ C 2
3 aein
X_e in
and R_ ˆ c …Q R†e_ in : …4†
Figure 15.7 depicts the results for the simulation of three creep tests with the Steck model
and the basic Chaboche model (Equation (4)). The parameters of both models are simul-
345

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.7: Simulation of creep tests with the Steck and the basic Chaboche material model.
taneously identified from the three creep tests. The creep rates of the uniaxial tests are of
the same magnitude as the typical rates occurring in certain spots of the corresponding
shear-panel tests. However, the results of Finite-Element calculations performed with
these sets of material parameters are not in good correlation with the shear tests. It is assumed
that better results – regarding the calculation of the behaviour of the shear panels –
can be achieved if a fair approximation of a set of creep tests, tension tests at higher loading
rates, and a special transient, stress-controlled test are achieved.
The simultaneous identification of the parameters of the material models from
those tests is performed by optimizing methods using a combination of gradient and
stochastic algorithms. Unfortunately, parameter identifications with both models are not
successful. Numerical experiments show that the main problem is the reproduction of
the tension test at high loading rates (the result of a test at 1 MPa/s is depicted in Figure
15.4). To avoid this problem, the overstress function of the Chaboche model is
modified by adding a term accounting for additional, rate-independent inelastic strains.
This “overlaying method” is among others described in [13] and [14]. Within the engineering
approach, the interaction between both parts of inelastic strains is neglected.
Good results are achieved with a very simple approximation of those rate-independent
strains by a type of Ramberg-Osgood power law for isotropic hardening. It is the same
approximation as it is assumed for the monotonic hardening regime within the rate-independent
Tseng-Lee model described above. With this additional term, two new parameters
are introduced into the material model. A very good starting value (regarding
346

15.2 Numerical Model
Figure 15.8: Simulation of creep tests with the modified Chaboche model (combined identification).
further parameter identification of the complete material model) for those parameters
can be found manually within a few iterations if it is assumed that the major part of
the inelastic strain, occurring in a fast tension test (see Figure 15.4), is described only
by this Ramberg-Osgood term. The approach without kinematic hardening is possible
here since no inelastic orthotropy is present. As a matter of fact, no cyclic effects concerning
this part of inelastic strains can be simulated. The results of a simultaneous
identification of the parameters of this model are given in Figure 15.4 for the tension
tests, in Figure 15.8 for the creep tests, and in Figure 15.9 for a transient test.
Additional tests with notched specimen (measurement and processing of strain
distribution was provided by R. Ritter and H. Friebe [15], Institute of Measurement
Techniques and Experimental Mechanics, Techn. Univ. Braunschweig) were conducted
to characterize the multiaxial behaviour of the considered material and to examine the
accuracy of the material models in the multiaxial case. Figure 15.10 shows the measured
strain distribution (left-hand side) and the numerical results (right-hand side)
achieved with the modified Chaboche model for a creep test after 12 hours. The specimen
was loaded to a nominal value of tensile stress of 180 MPa in the smallest cross
section. Considering the resolution of the optical measuring method of 0.1–0.2% strain,
the numerical results are in very good correlation with the test results. All results given
in this paper concerning the behaviour of shear-loaded panels at elevated temperature
are calculated with the described modified Chaboche material model.
347

348
15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.9: Simulation of transient tests with the modified Chaboche model (combined identification).
Figure 15.10: Test and numerical results for a creep test with an inhomogeneous specimen.

15.3 Experimental and Numerical Results
15.3.1 Test procedure
All tests conducted on the test set-up PApS at ambient temperatures are incrementalstep
tests. The temporal course of loading of the specimen at elevated temperature is
controlled by a computer. In case of tests at elevated temperatures, the specimen is
mounted loosely in the shear frame at ambient temperature first. Then, the shear frame
and the panel are heated. After that, the 100 screws, which clamp the specimen between
the halves of the shear frame, are tightened. This procedure avoids a preloading
of the panel and the occurrence of significant thermal buckles due to the different thermal
strains of the aluminium panel and the steel shear frame. Nevertheless, the imperfections
of the panel at the beginning of the mechanical loading are slightly higher than
in tests at room temperature. The angle of shear (see Figure 15.3) and the central deflection
of the buckled panel are determined by inductive displacement transducers. Additionally,
strain gauge rosettes are positioned at different points of the shear panels,
and in some cases, the entire displacement field has been measured by means of engineering
photogrammetry.
15.3.2 Computational analysis
15.3 Experimental and Numerical Results
The shear panels are discretized with regular meshes as it is shown in Figure 15.11. Simulations
of shear-angle-controlled tests are conducted with prescribed deformations of
the edges of the panels. The resulting load is obtained by integrating the stress resultants
along the edges. In case of load-controlled tests like creep tests, the rigid clamping
of the panels is simulated by introducing constraints for the nodal deformations on
the edges into the Finite-Element equation system. The load is then applied as a single
force on one corner of the panel. An examination of the convergency of the spatial discretization
of the panels showed that regular meshes with 20 ×20 elements are sufficient
since a further refinement gives no significant improvement of the results for the
central deflection (Figure 15.11) or the effective shear load. Since the computational effort
– especially for the solution of the visco-plastic problem – is very high even
16×16 element meshes are used. Meshes with refinements near the clamped boundaries
of the panel improve the solution only when the whole mesh is very coarse. It
has been shown by comparative analyses that the idealization of the panel with ten
layers in thickness direction is sufficient for the rate-independent problem. Typical
creep analyses are conducted with only seven layers without a significant loss of accuracy
in the global and local results since the distribution of the inelastic strains along
the cross section of the panels is more smooth than in the spontaneous plasticity problem.
The determination of the critical time-step sizes related to accuracy and stability
shall not be discussed here in detail. Numerical experiments show that the creep behaviour
of the shear panels is reproduced within acceptable accuracy for the current param-
349

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.11: First loading of shear panels with a thickness of 1.4 mm.
eters of the material model if a time-step of about 10 s is used at the very beginning of
the creep process. This comparatively small time-step can be increased very rapidly.
The critical time-step for stability can be determined also by numerical experiments
and is larger than 160 s.
15.3.2.1 Monotonic loading – ambient temperature
Figure 15.11 shows some experimental and numerical results for the first loading of
panels with 1.4 mm thickness at ambient temperature. The angle of shear is plotted versus
the central deflection. For an undamaged plate, the first symmetric buckling mode
always corresponds to the lowest eigenvalue. For this reason, the plate will buckle symmetrically.
Within the pre- and the lower postbuckling range, the behaviour of the panels
is strongly influenced by initial geometric imperfections. The influence of the geometric
imperfections vanishes at least when the angle of shear reaches values of about
0.12 8. As the first plastic deformation occurs at an angle of shear of about 0.2 8, the
geometric imperfections do not affect the plastic deformation. Numerous numerical analyses
show that the angle of shear at first yielding is approximately a constant for panel
thicknesses between 1.2 mm and 3.0 mm.
The first yielding takes place at a spot on the edges, where the main buckle is
constrained by the clamping. Further load increase leads to a propagation of plastic regions
at this spot and along the tension diagonal on the concave sides of the buckle.
350

15.3 Experimental and Numerical Results
The load-angle-of-shear diagram shows no direct influence of the first plastic deformations
on the overall stiffness of the panels until – depending on the panel thickness –
the angle of shear reaches values of 0.5 8 or more. Figure 15.12 a depicts theoretical results
for the deformation state and current distribution of the tangent modulus in three
planes of the plate for a 1.6 mm panel at different loading states. The tangent modulus
is a measure for the local stiffness and the development of plastic strains since it represents
the slope of the uniaxial reference-stress-strain curve of the material. Therefore, in
the regime of elastic straining, its value is equal to the elastic modulus and decreases
with increasing plastic loading. The distribution of the tangent modulus at an angle of
shear of 0.35 8 (superceding the value of the critical buckling load by a factor of 20),
which is shown on the left-hand side of Figure 15.12 a, shows still large areas of nearly
elastic states. Higher loads lead to a distribution of the tangent modulus, which is
shown on the right-hand side of Figure 15.12 a. In this case – at an angle of shear of
0.6 8 –, the plastic regions cover the whole plate, and in the tension field, the plastic or
tangent modulus decreases considerably due to large plastic deformations. From a comparison
of the deformation states follows that the magnitude of the buckling deflections
are not very much increased from the lower to the higher load as the load-carrying
mechanism of the panel is shifted from bending to membrane tension.
15.3.2.2 Cyclic loading – ambient temperature
The load reversal represents the most critical point of the behaviour of the cyclically
loaded panel since remaining deformations from prior plastic loading act like geometri-
Figure 15.12 a: Deformation states and distributions of the plastic tangent modulus at maximum
load.
351

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.12 b: Deformation states and distributions of the effective plastic strain before snapthrough
after different maximum loads.
cal imperfections each time when zero load is passed. The computed distribution of the
deflection and the corresponding distribution of the effective plastic strain at zero load
are shown in Figure 15.12 b. As a matter of fact, the larger remaining deformations
after higher loads lead to more intense snap-throughs from the current buckling form to
the buckling form after load reversal with buckles perpendicular to the former ones.
This can be seen in Figure 15.13, where the results of the first cycle of shear tests
(1.4 mm panel thickness) at increasing amplitudes of the applied load, respectively of
the angle of shear, are illustrated. Due to inevitable small disturbances in the tests, the
direction of the central deflection after passing zero load is not predictable. Therefore,
there are two possible paths of the central deflection after each snap-through.
With increasing load amplitude, the snap-through becomes more complicated
since it can run through different even unsymmetric buckling forms as intermediate
states. In cases when the central deflection has the same sign in both load extrema, the
load reversal can lead to a “double-snap-through” with an intermediate state with opposite
central deflection. For the theoretical computations, a change of the deformation
path after reaching the bifurcation point at load reversal – the so-called branch switching
– is obtained by using small geometric imperfections or disturbances corresponding
to the desired buckling mode. In cases of a more intense and complicated snap-through
behaviour, the described dynamic method is used. In this case, the branch switching is
managed by applying a suitable distribution of accelerations, which disturbs the system
and induces the dynamic snap-through procedure. This is illustrated in Figure 15.14,
where the subsequent deformation states and the distribution of the velocity normal to
the plate in a moment corresponding to the depicted intermediate deformation state are
shown.
352

15.3 Experimental and Numerical Results
Figure 15.13: Angle of shear vs. central deflection, results for 1.4 mm panels at different load amplitudes.
Figure 15.14: Dynamic snap-through, deformations and velocity distribution.
353

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.15: Angle of shear vs. central deflection, results for 1.6 mm panels at very high load
amplitude (left) and at higher numbers of cycles (right).
The snap-through in Figure 15.14 was computed in the first cycle of a cyclic
shear test (panel thickness 1.6 mm) at the very high amplitude of the angle of shear of
0.7 8. The results for the central deflection of this test are given in Figure 15.15. Additionally
to the numerical results achieved with the Tseng-Lee model, the starting- and
ending points of the first and second snap-through computed with the Mroz model are
shown in this figure. As can be easily seen, the accuracy of the used material model
has a large influence on the reproduction of the snap-through behaviour since it depends
strongly on the remaining plastic deformations at the bifurcation points. In most
cases, the repeated buckling behaviour remains the same after the second load cycle. In
this test, the cyclic buckling behaviour did not change. Only in one of 22 cyclic tests, a
random behaviour in changing the sign of the central deflection was observed. Furthermore,
Figure 15.15 shows that the global cyclic load-deformation course is obviously
stabilized after the second cycle, and the results for the 10th and 50th cycle are nearly
identical. Since the correlations between numerical results and test results are fairly
good, the numerical model is applied to different aspect ratios (a/b, see Figure 15.3). In
Figure 15.16, the development of the central deflection and the buckling pattern of two
panels with different aspect ratios are shown. Typical results, which can be used in the
design of shear panels for defining the distance of stringers and frames, are shown in
Figure 15.17.
354

15.3 Experimental and Numerical Results
Figure 15.16: Buckling modes and central deflection, cyclic loading, aspect ratios a/b = 1.5 and
2.0.
Figure 15.17: Load at angle of shear of 0.3 8 and 0.5 8 vs. thickness, different aspect ratios.
355

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
15.3.2.3 Time-dependent behaviour
The general buckling behaviour and the buckling mode of panels loaded at 200 8C are
very similar to those at room temperature. Results for the monotonic first loading of
panels at different temperatures are shown in Figure 15.18. The loading rate for the
tests at elevated temperature is 0.17 kN/s. It is known from the parameter identification
of the material models, that the tangent modulus at the beginning of loading at 200 8C
is only 10% smaller than at room temperature. Therefore, the “global stiffness”, which
means the ratio between the global angle of shear and the load, is nearly the same for
all examined temperatures in the prebuckling regime. When the panel starts to buckle,
the occurring bending stresses together with the shear stress lead to early inelastic
strains, which yield a stronger decrease of the global stiffness than at room temperature.
Further analyses show that in the range of panel thicknesses between 1.2 and
1.8 mm, an increase of the panel thickness of approximately 0.2 mm covers the loss of
global stiffness in the postbuckling regime due to the increase of the temperature from
208C to 2008C. The development of the central deflection during monotonic loading
of the shear panels at 200 8C is almost equal to the situation at room temperature if the
postbuckling regime is concerned. The theoretical buckling loads at 200 8C tend to be
slightly higher, but due to the undetermined imperfections, a proper measurement of
this effect is impossible. The influence of different loading rates on the monotonic be-
Figure 15.18: Load vs. angle of shear for 1.6 mm panels at different temperatures.
356

15.3 Experimental and Numerical Results
Figure 15.19: Angle of shear vs. time for different creep tests, thickness 1.6 mm.
haviour of the panels is up to now examined between 0.17 kN/s and 0.01 kN/s. All
loading rates lead to no significant changes in the monotonic behaviour regarding buckling
load and postbuckling stiffness within the normal scatter of the test data.
Figure 15.19 depicts results of creep tests. The theoretical creep rates in the global
stationary creep regime are in good correlation with the measurements. The creep rates
in the primary phase are underestimated by the numerical model, especially at lower
load levels since there is no smooth transition between loading and creep phase. This is
not only a problem of the description of the primary and secondary creep behaviour of
the material since there is a second more geometric non-linear effect. In this first creep
phase, a more rapid relaxation of the bending stresses due to the buckling must take
place. In the case of spontaneous plasticity, the increase of the load reduces the share
of bending within the whole load-carrying mechanism and the tension component
along the diagonal increases. During a creep process, this load-carrying state is reached
due to the permanent generation of inelastic strains. The used numerical model gives
better results for the first creep phase if this tension load-carrying state is already
reached during the loading phase by applying higher creep loads (Figure 15.19). The
development of the central deflection indicates only small changes in the deformation
of the panel during creep. The creep test at a load of 65 kN shown in Figure 15.19
yields to an increase of the central deflection of 0.7 mm within the first 360 min. Tests
at lower creep rates yield even lower increases of the central deflection.
357

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
Figure 15.20: Increase of the angle of shear after 360 min.
Figure 15.20 illustrates the influence of creep load and panel thickness on the
creep rates. In this figure, the creep-shear angle is defined as the difference between the
shear angle at the start of the creep process and the value after 360 min giving an integral
measure for the occurring creep rates.
Finally, Figure 15.21 shows the computed development of the effective inelastic
strain for a creep test at 35 kN. The distributions at the start of the creep process on the
left-hand side are scaled by factor 10 4 , and those at the end of the creep process on the
right-hand side of Figure 15.21 by factor 10 3 . In general, the distributions are again
similar to those of the rate-independent problem. During the creep process, the highest
increases of the inelastic strain can be found at the corners on the lower side and along
the edges on the upper side in the vicinity of the tension diagonal.
15.4 Conclusion
The combination of numerous experimental results and a well-established numerical model
give a good insight of the behaviour of shear-buckled aluminium panels as far as the behaviour
at monotonic, cyclic and creep loading is concerned. In case of the rate-independent
problem, the Tseng-Lee material model is best suited to simulate the inelastic behaviour
of the material under consideration since this model describes the effect of non-pro-
358

portional loading very accurately. The visco-plastic problem is treated with the Chaboche
material model. To describe the material behaviour at higher load levels, an additional rateindependent
term is added to the Chaboche model. Very high amplitudes of the external
shear loads in cyclic tests require a Finite-Element method with an algorithm accounting for
dynamic effects to describe the complex snap-through behaviour. The simulation of the
behaviour of panels with different geometries was performed successfully. The accuracy
of the developed method is proved by the fact that very good results are achieved for durability
analyses using the output of the numerical simulations as input.
List of Symbols
List of Symbols
Figure 15.21: Effective inelastic strain distributions for a creep test at 35 kN; left-hand-side: factor
10 4 , t=5 min; right-hand side: factor 10 3 , t=360 min.
C matrix of in-plane, coupling and bending stiffness
D yield tensor
F area of the plate midsurface
p discrete force vector
v vector of the midplane displacements
r vector of membrane forces and bending moments, properties
of the 2. Piola-Kirchhoff stress tensor
c * vector of the in-plane and bending strain with Green-Lagrange properties
359

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour
a, Ai,Vi,N, K, Q, C, c
parameters of the material models
e in
uniaxial inelastic strain
r uniaxial stress
[...] t
state t
References
[1] K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen
aus kohlenstoffaserverstärktem Kunststoff. Inst. f. Flugzeugbau u. Leichtbau, Technische
Universität Braunschweig, 1989.
[2] P. Horst: Zum Beulverhalten dünner, bis in den plastischen Bereich zyklisch durch Schub
belasteter Aluminiumplatten. ZLR-Forschungsbericht 91-01, ISBN 3-9802073-5-8, Inst. f.
Flugzeugbau u. Leichtbau, Technische Universität Braunschweig, 1991.
[3] J.L. Chaboche: A Review of Computational Methods for Cyclic Plasticity and Viscoplasticity.
Proc. of Int. Conference: Computational Plasticity – Models, Software and Applications,
Barcelona, 1987, pp. 379–411.
[4] J. Knippers: Eine gemischt-hybride FE Methode für viskoplastische Flächentragwerke unter
dynamischen Einwirkungen. Berichte aus dem Konstruktiven Ingenieurbau, Heft 18, ISBN
3-79831548-5, Technische Universität Berlin, 1993.
[5] H. Kossira, P. Horst: Cyclic Shear Loading of Aluminium Panels with Regard to Buckling
and Plasticity. Thin-Walled Structures 11 (1991) 65–84.
[6] P. Horst, H. Kossira, G. Arnst: On the Performance of Different Elasto-Plastic Material
Models Applied to Cyclic Shear-Buckling. Proc. of the Int. ECCS-Colloquim: On the Buckling
of Shell Structures on Land, in the Sea and in the Air, Lyon, France, 1991.
[7] A. Phillips: A Review of Quasistatic Experimental Plasticity and Viscoplasticity. Int. J. Plasticity
2 (1986) 315–328.
[8] N.T. Tseng, G.C. Lee: Simple Plasticity Model of Two-Surface-Type. J. Engg. Mech. 109
(1983) 795–810.
[9] W. Gieseke, K.R. Hillert, G. Lange: Material State after Uni- and Biaxial Cyclic Deformation.
This book (Chapter 2).
[10] H. Schlums, E. Steck: Description of Cyclic Deformation Processes with a Stochastic Model
for Inelastic Behaviour of Metals. Int. Jour. Plasticity 8 (1992) 147.
[11] J.L. Chaboche, G. Rousselier: On the Plastic and Viscoplastic Constitutive Equations –
Part I: Rules Developed with Internal Variable Concept. J. Pressure Vessel Technology
(ASME) 105 (1983) 153–158.
[12] K.-T. Rie, H. Wittke, J. Olfe: Plasticity of Metals and Life Prediction in the Range of Low-
Cycle Fatigue: Description of Deformation Behaviour and Creep-Fatigue Interaction. This
book (Chapter 3).
[13] E.-R. Tirpitz, M. Schwesig: A Unified Model Approach Combining Rate-Dependent and
Rate-Independent Plasticity. Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials –
3, Berlin, 1992, pp. 411–417.
[14] E.-R. Tirpitz: Elastoplastische Erweiterung von viskoplastischen Stoffmodellen für Metalle –
Theorie, Numerik und Anwendung. Report 92-70, Inst. of Structural Mechanics, Techn.
Univ. Braunschweig, 1992.
[15] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical
Measuring Methods. This book (Chapter 13).
360

16 Consideration of Inhomogeneities in the Application
of Deformation Models, Describing the Inelastic
Behaviour of Welded Joints
Helmut Wohlfahrt* and Dirk Brinkmann**
16.1 Introduction
Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)
The local loads and deformations in welded joints have rarely been investigated under
the aspect that the mechanical behaviour is influenced by different kinds of microstructure
[1]. These different kinds of microstructure lead to multiaxial states of stresses and
strains, and some investigations [2–4] have shown that for the determination of the total
state of deformation of a welded joint, the locally different deformation behaviour
has to be taken into account. It is also published that different mechanical properties in
the heat-affected zone (HAZ) [5] as well as a weld metal with a lower strength as the
base metal [6] can be the reason or the starting point of a fracture in welded joints. A
new investigation demonstrates [7] that in TIG-welded joints of the high strength steel
StE 690, a fine-grained area in the heat-affected zone with a lower strength than that of
the base metal is exclusively the starting zone of fracture under cyclic loading in the
fully compressive range. These investigations support the approach described here that
the mechanical behaviour of the different kinds of microstructure in the heat-affected
zone of welded joints has to be taken into account in the deformation analysis. The influences
of these inhomogeneities on the local deformation behaviour of welded joints
were determined by experiments and numerical calculations over a wide range of temperature
and loading. The numerical deformation analysis was performed with the
method of Finite Elements, in which recently developed deformation models simulate
the mechanical behaviour of materials over the tested range of temperature and loading
conditions.
The starting point of these investigations was the question if such deformation
models are able to describe the deformation behaviour of welded joints sufficiently.
* Technische Universität Braunschweig, Institut für Schweißtechnik und Werkstofftechnologie,
Langer Kamp 8, D-38106 Braunschweig, Germany
** Volkswagen AG, D-38436 Wolfsburg, Germany
361

16 Consideration of Inhomogeneities in the Application of Deformation Models
16.2 Materials and Numerical Methods
16.2.1 Materials and welded joints
The investigations were carried out with the microalloyed steel StE 460, of which the
microstructure in the normalized state consists of ferrite and minor amounts of bainite
and pearlite. The hardness has a value of 220 HV. The chemical composition is listed
in Table 16.1.
For the deformation analysis, manual arc weldings were manufactured with two
different widths of the weld seam (24 mm and 16 mm) using the same welding parameters
– with the exception of the number of layers – and the same welding electrodes.
The different joints were welded by varying the distance between the two welded
plates. The chemical composition of the electrodes is also given in Table 16.1 and the
welding parameters (Uw=22.5 V, Iw=170 A, vw=0.2 cm/s) lead to a heat input per
unit length of 20 kJ/cm, which is on the upper limit for the use of these electrodes.
Microsections and hardness distributions of the welded joints show clearly the
three different sections of the joints base metal, heat-affected zone and weld metal
(Figures 16.1 and 16.2).
The microsections and the hardness distributions were not only used to identify these
different zones, but also to establish Finite-Element models for the calculations. Detailed
experimental and numerical investigations indicated that the heat-affected zone must also
be divided into zones because the mechanical properties are not constant over its width.
On the basis of the microsections, four significantly different kinds of microstructure
could be identified. The differences between these kinds of microstructure are caused
by the peak temperature and the number of weld cycles. To gain the mechanical properties
of each microstructure, it must be identified and then prepared in specimens with a
large diameter and a large measurement length by using the so-called weld simulation.
During the weld simulation, various specimens of the base metal were conductively
heated up to different peak temperatures and then cooled under nitrogen with different
t 8,5-cooling times. The simulation parameters for each structure can be determined
first of all numerically by using the thermal conduction equation and subsequently
optimized experimentally by comparison with microsections. The specimens
with homogeneously simulated microstructures over a measurement length of 25 up to
30 mm were used in tensile tests, creep tests and tension-compression tests. Micrographs
of all four kinds of microstructure are shown in Figure 16.3. The various microstructures
are listed in Table 16.2 together with their hardness values.
Table 16.1: Chemical composition of the base metal StE 460 and the weld electrodes.
362
C Si Mn P N Cu Ni
StE 460 0.14% 0.45% 1.62% 0.012% 0.006% 0.021% 0.56% Nb, V, S
Tenacito 70 0.06% 0.5% 1.6% – – – 0.9%

16.2 Materials and Numerical Methods
Figure 16.1: Microsections of manual arc-welded joints of the steel StE 460; up: width of the
weld metal: 24 mm; down: width of the weld metal: 16 mm.
Figure 16.2: Areas of the welded joints with hardness values below 230 HV (grey: base metal,
weld metal) and above 230 HV (black: HAZ).
In addition to the investigations with the four kinds of microstructure of the heat-affected
zone, the same mechanical tests were carried out with the base metal and the
weld metal. The specimens containing the weld metal were taken from welded joints
vertical to the weld seam. They were machined in that way that in the mechanical tests,
the deformation is concentrated in the weld metal.
363

364
16 Consideration of Inhomogeneities in the Application of Deformation Models
Figure 16.3: Microstructure of the base metal and microstructures N, F, C and M in the heat-affected
zone (from left to right).
Table 16.2: Kinds of investigated microstructures and Vickers hardness.
State of material Microstructure Hardness Notice
Base metal Ferrite (bainite, pearlite) 220 HV
Microstructure N Ferrite (bainite, pearlite) 230 HV fine-grained as base metal
Microstructure F Ferrite (bainite, pearlite) 270 HV fine-grained as base metal
Microstructure C Bainite, martensite 280 HV
Microstructure M Martensite (bainite) 375 HV
Weld metal – 220 HV

16.2.2 Deformation models and numerical methods
16.2.2.1 Deformation model of Gerdes
In these investigations, the high-temperature formulation of the deformation model of
Gerdes [8] was used:
e_in ˆ C1 exp
r_kin ˆ H1E exp
R1 exp
Fh
RT
jr rkinj
r0
n1
sinh
d1 bA h rkin sign …reff†
RT
Fh
RT sinh
bA h rkin
RT
e_in
bA h …r rkin†
RT
; …1†
; …2†
A_ h ˆ dV1je_inj‡dV2jrj‡dV3 : …3†
The model parameter r 0, which is used for the stress standardization, was substituted
by the Young’s modulus. The time-dependency of the activation volume is described
by a three-parametric function and is only used for the simulation of cyclic tests.
16.2.2.2 Fitting calculations
16.3 Investigations with Homogeneous Structures
The fitting calculations were carried out in cooperation with project number B1 with an
evolution algorithm to gain the model parameters for the calculations. The parameter
calculations were performed here only phenomenologically for each temperature and
each kind of microstructure. Additionally, the different types of tests were simulated
separately because the qualities of the model parameters became much better then, and
the fitting calculations needed even less time than the parameter estimations made for
common tensile and creep tests.
16.3 Investigations with Homogeneous Structures
All investigations were carried out to prove whether deformation models are able to describe
the mechanical behaviour of the steel StE 460, of four significant kinds of microstructure
of this steel and of the weld metal of manual arc weldings. Tensile, creep and
tension-compression tests were performed over a wide range of temperatures (room
temperature up to 700 8C) and loading conditions in order to characterize the mechanical
behaviour of each state of the base metal.
365

16 Consideration of Inhomogeneities in the Application of Deformation Models
16.3.1 Experimental and numerical investigations
16.3.1.1 Tensile tests
The results of the tensile tests made at room temperature are registered in Figure 16.4. The
base metal and the weld metal show a clearly visible yield strength, which is highly pronounced
in the base-metal deformation, whereas a proof strength has to be attributed to the
various kinds of microstructure produced through weld simulation. The arrangement of
the stress-strain curves in Figure 16.4 corresponds with the hardness values. The microstructure
M with the highest hardness values has the highest flow stresses. For the parameter
estimation, the stress-strain curve of the base metal has to be filtered so that the yield
strength is transformed into a proof strength. The fitting calculations with the deformation
model of Gerdes indicate that the mechanical behaviour of the various kinds of microstructure
and the weld metal can be described sufficiently well, but differences arise in
the simulation of the stress-strain curve of the base metal and its yield strength cannot
be simulated by the deformation model. The largest differences occur in the yieldstrength
range, where the stress values are underestimated. At large strains (≥3%), the
experimental and the calculated values differ less. The calculated stress-strain curve of
the high strength microstructure M shows a stress state of saturation, whereas a steadystrain
hardening is observed in the experiments with this kind of microstructure.
The tensile tests carried out at 300 8C show very similar results as the tests at
room temperature (Figure 16.5). The arrangement of the stress-strain curves has the
same order and corresponds also to hardness values. The mechanical behaviour of the
base metal differs from that at room temperature because the yield strength of the base
Figure 16.4: Stress-strain curves of the investigated microstructures at room temperature; symbols:
experimental curve; lines: fitted curve.
366

16.3 Investigations with Homogeneous Structures
Figure 16.5: Stress-strain curves of the investigated microstructures at 3008C; symbols: experimental
curve; lines: fitted curve.
metal is not as pronounced as at room temperature. Comparing Figure 16.4 and Figure
16.5, one sees that at strains above 3%, the strength values of all microstructures are
higher at 300 8C than at room temperature. The Young’s modulus of all microstructures
decreased to a value of 170 000 MPa. The mechanical behaviour of all kinds of microstructure
can be described again sufficiently by the fitting calculations and here, the
yield strength of the base metal is also adequately simulated. Only the calculated
stress-strain curve of the microstructure M includes a stress state of saturation at large
strains although the real stress-strain curve exhibits a strain-hardening behaviour.
The stress-strain curves at 500 8C (Figure 16.6) are again arranged in the same order
as at room temperature. At this temperature, the base metal shows no yield-strength effects
and its stress-strain curve is nearly the same as the curve of the microstructure N. The
stress-strain curve of the microstructure M reveals a softening behaviour at large
strains, and the differences between the curves of the structures M and C decreased.
The fitting calculations carried out by using a Young’s modulus of 150000 MPa simulate
the mechanical behaviour of all kinds of microstructure very well, only the softening of the
microstructure M is unsuitably modelled with a stress state of saturation.
It can be seen in Figure 16.7 that the stress-strain curves taken at 700 8C are not
arranged in the same order as at room temperature. At 700 8C, the base metal has a
higher strength than the microstructures N and F because the especially low grain size
of these two microstructures favours plastic deformation at high temperatures. The two
high strength microstructures M and C show a softening behaviour caused by a transformation
of the microstructure. The fitting calculations (Young’s modulus=130
000 MPa) correspond relatively well with the experimental results although the
softening behaviour is unsuitably modelled by horizontal lines.
367

368
16 Consideration of Inhomogeneities in the Application of Deformation Models
Figure 16.6: Stress-strain curves of the investigated microstructures at 5008C; symbols: experimental
curve; lines: fitted curve.
Figure 16.7: Stress-strain curves of the investigated microstructures at 7008C; symbols: experimental
curve; lines: fitted curve.

16.3.1.2 Creep tests
16.3 Investigations with Homogeneous Structures
The creep tests were carried out at 500 8C and 700 8C. Already at 500 8C (loading:
275 MPa), all kinds of microstructure show a remarkable creep (Figure 16.8). The order
of all creep curves agrees within the range of reproducibility with the results of the
tensile tests. The differences between the base metal and the microstructure N and the
differences between the microstructures M and C are very small. The fitting calculations
show a very good applicability of the deformation model of Gerdes to the creep
behaviour of all microstructures. Differences between the experiments and the calculations
are not noticeable in Figure 16.8.
The analysis of the creep behaviour at 700 8C (loading: 50 MPa) reveals results
analogous to those of the tensile tests. The highest creep strains occur in the microstructures
F and N (Figure 16.9), whereas the smallest creep strains occur in the microstructures
C and M. The base metal takes a middle position of all creep curves. The behaviour
of the microstructures F and N is highly influenced by the fine-grained microstructure,
which favours the plastic deformation. The fitting calculations simulate the
creep behaviour sufficiently well. The softening behaviour of the two microstructures
M and C caused by a transformation of the microstructure is modelled as steady-state
creep. It must be stated that it cannot be anticipated to find the consequences of the
transformation in the results because this behaviour has not been implemented in the
model equations.
Figure 16.8: Creep curves of the investigated microstructures at 5008C, loading: 275 MPa; symbols:
experimental curve; lines: fitted curve.
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16 Consideration of Inhomogeneities in the Application of Deformation Models
Figure 16.9: Creep curves of the investigated microstructures at 700 8C, loading: 50 MPa; symbols:
experimental curve; lines: fitted curve.
16.3.1.3 Cyclic tension-compression tests
Strain-controlled cyclic tension-compression tests were performed with varying strain
amplitudes at room temperature (±0.6%) and at 5008C (±0.4%). The strain rates of the
tests lie between 1 · 10 –4 1/s and 5 ·10 –4 1/s. The fitting calculations were made in two
steps because of the hardening or softening behaviour found in the different kinds of
microstructure. In the first step, the first five cycles of the tension-compression tests
were used to estimate the first model parameters. If necessary, these model parameters
were utilized in the second step as the starting set of parameters for new parameter estimations
to simulate the behaviour during the whole tension-compression tests.
Figure 16.10 shows the results of the first five cycles of all microstructures at
room temperature (strain rate: e_ ˆ 5 10 4 l/s). The order of all curves is the same as
that of the stress-strain curves achieved from tensile tests. The base metal exhibits a
pronounced yield strength, which has to be filtered for the fitting calculations. The
mathematical simulations describe the mechanical behaviour of all microstructures sufficiently
although the yield strength cannot be modelled by the deformation model. The
cyclic hardening of all microstructures can be described by the model.
Figure 16.11 contains the stress-time curves of all microstructures over the complete
testing time. The printed symbols are the points of return in the fully tension
range. It can be seen that all microstructures with the exception of the base metal soften
after the first hardening cycles. The hardening period lasts about 5 cycles and the
softening leads very fast (10 to 20 cycles) to a state of cyclic saturation. Only the base
metal already softens from the beginning of the test, and the cyclic state of saturation
is reached after less cycles.
370

16.3 Investigations with Homogeneous Structures
Figure 16.10: Stress-time curves of the investigated microstructures at room temperature, first five
cycles, strain amplitude: ±0.6%; symbols: experimental curve; lines: fitted curve.
Figure 16.11: Stress-time curves of the investigated microstructures at room temperature, 200 cycles,
strain amplitude: ±0.6%; symbols: experimental curve; lines: fitted curve.
371

16 Consideration of Inhomogeneities in the Application of Deformation Models
In Figure 16.12, a photo taken with the transmission electron microscope shows
the saturated state of the base metal. The state of saturation can be identified through
the array of dislocations in a cell structure. The fitting calculations made in the second
step indicate that the deformation model of Gerdes is not able to simulate the cyclic
softening behaviour of all microstructures. Figure 16.11 demonstrates that the stresses
in the initial hardening range were underestimated and in the following softening range
overestimated. The model describes the state of saturation by a horizontal line for all
microstructures.
The results of cyclic tension-compression tests carried out at 5008C are represented
in the following figures (strain rate: e_ ˆ 5 10 4 l/s). Figure 16.13 contains the
first five cycles of the tension-compression tests of all microstructures. The curves are
arranged in the same order as the curves of the tensile tests.
The cyclic behaviour agrees with the results found at room temperature. It can be
seen that all microstructures with exception of the base metal firstly harden and then
soften during the cyclic loading (Figure 16.14). The amounts of hardening and softening
are not as high as at room temperature. The fitting calculations simulated the first
five cycles of the stress-strain behaviour sufficiently (Figure 16.13). Because of the relatively
weak softening behaviour, the model parameters achieved from the first five
cycles could also be used to describe the behaviour during all following cycles. A second
fitting improves the quality of the parameters only insignificantly.
It can be seen from Figure 16.14 that all curves lead after few cycles to states of
cyclic saturation. The cyclic stress-time curves of all microstructures can be simulated
sufficiently well by the deformation model. But, as to be seen, the cyclic softening
after the cyclic hardening cannot be described by the model. The stresses are underestimated
in the range of hardening and overestimated in the range of softening, but the
differences between the experiments and calculations remain relatively small.
16.3.2 Discussion
All results indicate that recently developed deformation models can be successfully applied
to describe the mechanical behaviour of a microalloyed steel and of the different
microstructures identified in the heat-affected zone of its weldings. But, in view of accuracy
and calculation time, it is useful for all fitting calculations to estimate the parameters
separately for each microstructure, each temperature and each testing sequence.
The fitting calculations are carried out phenomenologically as it was not possible
to find relations between model parameters and microstructural parameters. Some
special aspects of the mechanical behaviour of the microalloyed steel and its various
kinds of microstructure cannot be modelled by the deformation model of Gerdes.
Firstly, the simulation of the yield strength is not successful because the stresses in this
range are underestimated and the model calculates only a proof strength. Secondly, the
softening behaviour of some kinds of microstructure at high temperatures cannot be described
with the deformation model. The model simulates the softening behaviour
through a steady-state behaviour. This behaviour is also detectable in the simulation of
cyclic tension-compression tests. In this case, the hardening and softening behaviour is
372

16.3 Investigations with Homogeneous Structures
Figure 16.12: TEM-photo of the state of saturation due to cyclic loading, base metal at room temperature
after 200 cycles.
Figure 16.13: Stress-time curves of the investigated microstructures at 5008C, first five cycles,
strain amplitude: ±0.4%; symbols: experimental curve; lines: fitted curve.
373

16 Consideration of Inhomogeneities in the Application of Deformation Models
Figure 16.14: Stress-time curves of the investigated microstructures at 5008C, 200 cycles, strain
amplitude: ±0.4%; symbols: experimental curve; lines: fitted curve.
also modelled by a steady-state or saturation behaviour. The experimentally observed
saturation behaviour of all microstructures is modelled sufficiently well.
From these results, a simplification for the Finite-Element calculations of the deformation
behaviour of welded joints can be derived. In order to lower the calculation
time, the number of zones in the heat-affected zone can be reduced if the mechanical
behaviour of the microstructures is nearly the same.
16.4 Investigations with Welded Joints
The deformation behaviour of welded joints was investigated with the two kinds of
welded specimens, which differ in the width of the weld seam (see Section 16.2.1).
Tensile tests at room temperature were made to find the strain distributions during loading,
and calculations were performed with the Finite-Element code ABAQUS with the
same control of the strain rate as in the experiments.
374

16.4.1 Deformation behaviour of welded joints
16.4.1.1 Experimental investigations
The experiments to gain the strain distributions were performed in cooperation with the
project C2. In these experiments, flat specimens taken from the welded joints vertical
to the weld seam were tested in tensile tests with the same strain control as in the tests
with the homogeneous structures. The strain distributions were determined with the
grating method [9]. For reasons of symmetry and of the grating size, only less than one
half of the welded specimens was observed during the tests. The inspected regions
were the heat-affected zone, the weld metal and small parts of the base metal. The
welded specimens show relatively large rigid body motions during the tests so that reference
objects had to be fixed to the specimens. These reference objects are subject to
the same rigid body motions as the welded specimens, but they are not deformed during
the tests. With these object motions, the fictitious strains can be detected and thus,
the real strains can be determined.
16.4.1.2 Numerical investigations
16.4 Investigations with Welded Joints
For the numerical investigations, the Finite-Element code ABAQUS has been used in
cooperation with the project B1.
16.4.1.3 Finite-Element models of welded joints
The Finite-Element meshes of the welded joints have been derived from hardness distributions
and microsections. It turned out during the investigations that only those points
had to be determined describing the transition from the base metal to the heat-affected
zone and from the heat-affected zone to the weld seam (see Figure 16.2). Then, the
mesh for the heat-affected zone was modelled with four equidistant zones containing
the information of the mechanical behaviour of the affiliated microstructures. The Finite-Element
calculations were performed with the model parameters gained from the
room-temperature fittings. After the first experiments, it could be observed that the deformations
are concentrated in the weld metal so that it should be possible to model
the heat-affected zone with one microstructure only. In some calculations, the heat-affected
zone was modelled only with the microstructure C. This procedure reduces the
calculation times for the tensile test simulation with ABAQUS. Therefore, the following
sections include calculations for 6-material models (weld metal, four regions of the
HAZ, base metal) and 3-material models (weld metal, HAZ, base metal).
16.4.1.4 Calculation of the deformation behaviour of welded joints
For the Finite-Element calculations, the same control of the strain rate as in the real
tensile tests was used. The load was attached as a boundary condition on one side of
375

16 Consideration of Inhomogeneities in the Application of Deformation Models
the Finite-Element mesh. The first calculations revealed that the calculation times are
very high in comparison to the real tests. The cpu-time to calculate a tensile test of 900
seconds was about 20000 cpu-seconds with the 6-material model (2300 elements) and
about 4000 cpu-seconds with the 3-material model (900 elements). A serious problem
for the Finite-Element calculations is the time-stepping during the first part of the simulation
(0 to 40 seconds). During this period, the time-steps for the numerical integration
of the model equations are reduced from 5 seconds to 0.05 seconds because the differences
between the material properties of the weld seam and the microstructures M or C
are too large for greater time-steps in order to calculate the equilibrium state.
16.4.2 Strain distributions of welded joints with broad weld seams
The first investigations with welded structures were carried out with tensile specimens
with a weld metal length of about 24 mm. The analysis of the measured strain distributions
during loading shows the following results. The first remarkable strains occur in
the weld metal possessing the lowest flow stresses of all microstructures. The deformations
in the heat-affected zone and in the base metal are smaller. Figure 16.15 illustrates
the distributions of the longitudinal strains measured with the grating method at a
stage of 2.2% medium strain of the whole specimen. Only the strains in the weld-metal
zone and in the heat-affected zone are visible because the grating was fixed only on
these zones. The longitudinal strains have maximum values of about 4.8% in the weldmetal
and less than 1% in the heat-affected zone. The curvature of the strain isolines indicates
that the soft weld metal is backed up by the harder heat-affected zone. The hindered
vertical deformations in the transition zone between weld-metal and heat-affected
zone influence not only the vertical strains (Figure 16.15 right) but also the distribution
of the longitudinal strains (Figure 16.15 left).
The strain calculations made with a 6-material model show nearly the same results
(Figure 16.16) as the experiments, but there are also some differences. The first
remarkable strains occur in the heat-affected zone in the region of the microstructures
N and F because they have a proof strength lower than the yield strength of the base
metal. But at a medium strain stage of 2.2%, the longitudinal strains in the weld metal
are much higher than the strains in the heat-affected zone. The calculated values reach
3.4% in the weld-metal and less than 1% in the heat-affected zone. The strains in the
base metal are already higher than in the heat-affected zone but smaller than in the
weld metal. The backing up of the soft weld metal by the harder heat-affected zone is
also determined. At the points, where the contact faces between weld-metal and heat-affected
zone break through the free surfaces, singularities appear in the calculated stress
and strain distributions. These singularities are caused by the sudden change of the material
properties between two neighbouring elements of the weld-metal and the heat-affected
zone. These numerical effects have been stated before by other authors [1, 5]
and must not be taken into account in the comparison between numerical and experimental
results. Only the strain distributions in a wider range around the singularities
can be compared with experimental results.
376

16.4 Investigations with Welded Joints
Figure 16.15: Strain distributions of a welded specimen with a long weld-metal zone measured
with the grating method, medium strain: 2.2%; left: longitudinal strains; right: vertical strains.
Figure 16.16: Strain distributions of a welded specimen with a long weld-metal zone calculated
with the 6-material model, medium strain: 2.2%; left: longitudinal strains; right: vertical strains.
377

16 Consideration of Inhomogeneities in the Application of Deformation Models
The singularities also occur in the analysis of the strains calculated with a 3material
model. Here, the first remarkable strains do not occur in the heat-affected zone
but in the weld metal as observed in the experiments. The strain values at a medium
strain stage of 2.2% are about 3.5% in the weld-metal and less than 1% in the heat-affected
zone (Figure 16.17). The backing up of the soft weld metal is also determined in
these investigations.
The next examined load step at a medium strain of 4.3% was at the end of the
tensile test. Figure 16.18 illustrates the strain distributions of this state. The curvatures
of the strains are the same as in the load step discussed before. They also demonstrate
the backing up of the soft zone by a harder zone. Figure 16.18 shows the longitudinal
strains, which have values of more than 8% in the weld-metal and less than 2% in the
heat-affected zone. The vertical strain distributions illustrate also the backing up clearly.
The strains along a line parallel to the transition area are higher in the middle of the
specimen than at the free surface.
The calculations for a load step of 4.5% show qualitatively the same results as
the experiments. The strains calculated with the 6-material model and the 3-material
model are represented in Figures 16.19 and 16.20.
In both figures, the calculated strains in the weld metal are lower than the measured
strains. Both maximum values of the longitudinal strains in the weld metal are
slightly lower than 7%. The arrangement of the strain isolines in the figures corresponds
with the experimental results and confirms the backing up of the soft weld metal
by the harder microstructure of the heat-affected zone. The figures representing the
vertical strain distributions calculated with the 6-material model (Figure 16.19 right)
and the 3-material model (Figure 16.20 right) verify also the experimental results.
Figure 16.17: Strain distributions of a welded specimen with a long weld-metal zone calculated
with the 3-material model, medium strain: 2.2%; left: longitudinal strains; right: vertical strains.
378

16.4 Investigations with Welded Joints
Figure 16.18: Strain distributions of a welded specimen with a long weld-metal zone measured by
the grating method, medium strain: 4.3%; left: longitudinal strains; right: vertical strains.
Figure 16.19: Strain distributions of a welded specimen with a long weld-metal zone calculated
with the 6-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.
379

16 Consideration of Inhomogeneities in the Application of Deformation Models
Figure 16.20: Strain distributions of a welded specimen with a long weld-metal zone calculated
with the 3-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.
16.4.3 Strain distributions of welded joints with small weld seams
The strain distributions measured after the loading of welded joints with a narrow zone
of weld metal are shown in Figure 16.21. At the end of the tensile tests (medium strain
4.3%), the longitudinal strains have values of about 9% in the weld-metal and lower
than 2% in the heat-affected zone (Figure 16.21). The strains in the base metal are
somewhat higher than 2%. The vertical strains in Figure 16.21 indicate the backing up
of the soft weld metal and base metal through the hard heat-affected zone. The curvature
of the strain isolines demonstrates this effect clearly. It can be seen that the strains
measured in specimens of a welded joint with a narrow weld seam are slightly higher
than the strains measured in specimens of a welded joint with a wider weld-metal zone.
The numerical results achieved with the 6-material model (Figure 16.22) and the
3-material model (Figure 16.23) correspond with the experimental results. The backing
up of the weld metal is also confirmed by the longitudinal and vertical strain distributions.
But the calculated strain values in the weld metal are lower than the measured
ones. The maximum strains in the weld metal have values of 7%, and the strains in the
heat-affected zone are about 1% and are lower than the measured ones.
16.4.4 Discussion
The results of all calculated strain distributions show that modern deformation models
are able to describe the mechanical behaviour of welded joints sufficiently well. Problems
arise in the numerical investigations with the automatic time-stepping in the Fi-
380

16.4 Investigations with Welded Joints
Figure 16.21: Strain distributions of a welded specimen with a short weld-metal zone measured
with the grating method, medium strain: 4.3%; left: longitudinal strains; right: vertical strains.
Figure 16.22: Strain distributions of a welded specimen with a short weld-metal zone calculated
with the 6-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.
381

16 Consideration of Inhomogeneities in the Application of Deformation Models
Figure 16.23: Strain distributions of a welded specimen with a short weld-metal zone calculated
with the 3-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.
nite-Element calculations, which increases the needed cpu-times of the calculations.
The small time-steps are caused by the great differences in the properties of the neighbouring
materials in the transition range between the weld metal and the hard microstructure
in the heat-affected zone. Also, the calculated strain and stress singularities
are not found in the experiments, and they are too caused by the sudden change of the
material properties. The 3-material model and the 6-material model lead to nearly the
same calculated strains so that the number of different regions in the heat-affected zone
of a welded joint can be reduced if the locally highest strains do not occur there.
16.5 Application Possibilities and Further Investigations
All results show that modern deformation models can be applied successfully to
welded joints. The mechanical behaviour of the base metal and of various kinds of microstructure
is in a reasonable good accordance with the experimental results.
But the analysis demonstrates also that some special aspects of the mechanical behaviour
of microalloyed steels cannot be simulated by the deformation model. This
lack of simulation includes yield-strength effects and the softening behaviour, which occurs
in the high temperature and cyclic range. New deformation models have to describe
these effects if they should be applied in further investigations. A big problem in
the calculations was the handling of the equations in Finite-Element calculations and in
382

the parameter estimation. Both numerical methods need very high calculation times so
that numerical investigations can only be made if the hardware is big enough. Further
investigations may solve this problem if numerical methods or deformation equations
are developed shortening the calculation times by faster algorithms or if model equations
are established, which are easier to handle in modern calculation methods.
The efforts of modelling welded joints can also be reduced according to these investigations.
If the highest strains or fractures do not occur in the heat-affected zone, it
must not be modelled with more than one microstructure. In this case, the Finite-Element
meshes can directly be derived from hardness distributions of welded joints. In
other cases, it can be recommended from these investigations that only two different
microstructures have to be taken into account for the calculation of the mechanical behaviour
of welded joints.
References
References
[1] U.H. Clormann: Örtliche Beanspruchungen von Schweißverbindungen als Grundlage des
Schwingfestigkeitsnachweises. Dissertation, Technische Hochschule Darmstadt, 1986.
[2] W. Schieblich: Rechnerische und experimentelle Ermittlung des Zeitstandverhaltens einer
austenitischen Schweißverbindung. Dissertation, Technische Hochschule Darmstadt, 1992.
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Staatliche Materialprüfanstalt (MPA) Stuttgart, 1992.
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unter besonderer Berücksichtigung des lokalen Verformungsverhaltens. Dissertation,
Technische Hochschule Aachen, 1985.
[5] U. Pätzold: Verformungsanalyse von Schweißverbindungen. Dissertation, Technische Universität
Braunschweig, 1992.
[6] H. Hickel: Eigenspannungen und Festigkeitsverhalten von Schweißverbindungen. Dissertation,
Universität Karlsruhe, 1973.
[7] J. Pucelik, Th. Nitschke-Pagel, H. Wohlfahrt: Relationship of tensile residual stresses and fatigue
crack propagation under cyclic loading in the fully compressive range. Poster at the
Fourth European Conference on Residual Stresses, 4.–6. 06. 1996, Cluny (F), to be published.
[8] R. Gerdes: Ein stochastisches Werkstoffmodell für das inelastische Materialverhalten metallischer
Werkstoffe im Hoch- und Tieftemperaturbereich. Dissertation, Technische Universität
Braunschweig, 1995.
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H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Berücksichtigung in
Konstruktionsberechnungen.
E. Beißner: Zum Tragverhalten stählerner Augenstäbe im elastisch-plastischen Zustand.
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H. Hesselbarth: Simulation von Versetzungsstrukturbildung, Rekristallisation und Kriechschädigung
mit dem Prinzip der zellulären Automaten.
F. Kublik: Vergleich zweier Werkstoffmodelle bei ein- und mehrachsiger Versuchsführung im
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H. Schlums: Ein stochastisches Werkstoffmodell zur Beschreibung von Kriechen und zyklischem
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H. Gröhlich: Finite-Element-Formulierung für vereinheitlichte inelastische Werkstoffmodelle ohne
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Subproject A8
R. Neuhaus: Entwicklung der Versetzungsstruktur und Härtung bei [100] und [111] orientierten
Kupfer und Kupfer-Mangan Kristallen im einachsigen Zugversuch.
A. Kalk: Dynamische Reckalterung und die Grenzen der Stabilität plastischer Verformung. Systematische
Untersuchungen an Viel- und Einkristallen aus CuMn.
Subproject A9
A. Hampel: Struktur und Kinetik der lokalisierten Verformungen in kubisch flächenzentrierten Legierungseinkristallen
– Experimente und Modellrechnungen.
Subproject A10
M. Müller: Plastische Anisotropie polykristalliner Materialien als Folge der Texturentwicklung.
Subproject B2
G. Kracht: Erschließung viskoplastischer Stoffmodelle für thermomechanische Strukturanalyse.
E.-R. Tirpitz: Elasto-plastische Erweiterung von viskoplastischen Stoffmodellen für Metalle.
H. Braasch: Ein Konzept zur Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle.
Subproject B5/B7
M.M. El-Ghandour: Low-Cycle Fatigue Damage Accumulation of Structural Steel St52.
Subproject B6
G. Zhang: Einspielen und dessen numerische Behandlung von Flächentragwerken aus ideal plastischem
bzw. kinematisch verfestigendem Material.
Subproject B8
R. Mahnken: Duale Methoden für nichtlineare Optimierungsprobleme.
Subproject C2
W. Wilke: Photogrammetrische Verformungsmessung durch Überlagerungseffekte frequenzmodulierter
periodischer Bildstrukturen.
385

J.O. Hilbig: Zur Beeinflussung der Speckleauswertung durch eine Kombination der Speckle- und
der Objektrastermethode.
D. Winter: Optische Verschiebungsmessung nach dem Objektprinzip mit Hilfe eines flächenorientierten
Ansatzes.
Subproject C3
P. Horst: Zum Beulverhalten dünner, bis in den plastischen Bereich zyklisch durch Schub belasteter
Aluminiumplatten.
D. Hachenberg: Untersuchungen zum Nachbeulverhalten stringerverstärkter schubbeanspruchter
Platten aus kohlenstoffaserverstärktem Kunststoff.
Subproject C4
U. Pätzold: Verformungsanalyse von Schweißverbindungen.
Publications Resulted from the Collaborative Research Centre
1985–1987
Bibliography
Subproject A1
W. Witzel, F. Haeßner: Zur Vergleichbarkeit von Werkstoffzuständen nach Dehnen, Stauchen und
Tordieren. Z. Metallkunde 78 (1987) 316.
F. Haeßner, H. Müller: Einfluss von Korngröße und Beanspruchungsrichtung auf die Fließspannung
texturbehafteter Zink-Bleche. Z. Metallkunde 78 (1987) Juli-Heft.
Subproject A2
G. Lange, D. Rode: Ermüdungsverhalten anisotroper Aluminium-Legierungen unter mehrachsiger
Beanspruchung im Zeitfestigkeitsbereich. Hauptversammlung der Deutschen Gesellschaft für
Metallkunde, 20.–23. Mai 1986, Vortrag Nr. EBO7.
D. Rode: Ermüdungsverhalten anisotroper Aluminium-Legierungen unter mehrachsiger Beanspruchung
im Zeitfestigkeitsbereich. Dissertation TU Braunschweig, 1987.
Subproject A3
K. Andresen, R. Ritter: The phase shift method applied to reflection Moiré pattern. Proc. of the
VIII. Int. Conf. on Exp. Stress Analysis, Amsterdam, 1986, pp. 351–358.
K. Andresen, R. Ritter: Dehnungs- und Krümmungsermittlung mit Hilfe des Phasenshiftprinzips.
Technisches Messen 6 (1987).
S. Angerer, J.P. Löwenau, B. Morche, W. Wilke: 3D-Verformungsmessung an einem gummiartigen
Dämpfungselement mit Hilfe der Stereophotographie und der digitalen Bildverarbeitung.
Fachkolloquium Experimentelle Mechanik, Stuttgart, 1986, pp. 33–40.
H.C. Götting, R. Schütze, R. Ritter, W. Wilke: Dehnungsmessungen an Faserverbundwerkstoffen
mit Hilfe des Beugungsprinzips. VDI-Berichte 631, VDI Verlag Düsseldorf, 1987, pp. 275–285.
J. Hilbig, R. Ritter: Zur Bestimmung von Neigungsänderungen schalenförmiger Objekte mit Hilfe
der Laser-Speckle-Photographie. Contribution to Joint Conf. on DGaO, NOC, OG-IoP, SFO
“Optics 86”, Scheveningen, 1986.
R. Ritter, M. Hahne: Interpretation of Moiré-effect for curvature measurement of shells. Proc. of
the VIII. Int. Conf. on Exp. Stress Analysis, Amsterdam, 1986, pp. 331–340.
R. Ritter: Raster- und Moiré-Methoden. In: C. Rohrbock, N. Czaika (Eds.): Handbuch der experimentellen
Spannungsanalyse, Düsseldorf, 1987.
R. Ritter, W. Wilke: Dehnungsmessung nach dem Gitterprinzip. Lecture at the Colloquy of the
Collaborative Research Centre (SFB 319), Goslar, 4.–5. Dezember 1986.
386

Publications Resulted from the Collaborative Research Centre
C. Schulze: Optische Verformungsmessung an Schalen nach dem Rasterprinzip. Dissertation an
der Fakultät für Maschinenbau und Elektrotechnik der TU Braunschweig, 1986.
Subproject A5
K.-T. Rie, R.-M. Schmidt, B. Ilschner, S.W. Nam: A Model for Predicting Low-Cycle Fatigue
Life under Creep-Fatigue Interaction. In: H.D. Solomon, G.R. Halford, L.R. Kaisand, B. N.
Leis (Eds.): Low Cycle Fatigue, ASTM STP 942, American Society for Testing and Materials,
Philadelphia, 1988, pp. 313–328.
K.-T. Rie, R.-M. Schmidt: Life Prediction for Low-Cycle Fatigue under Creep-Fatigue Interaction.
Fifth International Conference on Mechanical Behaviour of Materials, Peking, 1987.
K.-T. Rie, R.-M. Schmidt: Lifetime Prediction under Creep-Fatigue Conditions. Proceedings of
the Second International Conference on Low-Cycle Fatigue and Elasto-Plastic Behaviour of
Materials, September 1987, München.
R.-M. Schmidt: Low-Cycle Fatigue under Special Aspects of Quality Control. Braunschweig Kolloquium
1985, DVS, pp. 107–119.
K.-T. Rie, R.-M. Schmidt: High Temperature Low-Cycle Fatigue of Austenitic and Ferritic Weldments.
ECF 6, Amsterdam, 1986, pp. 1096–1113.
K.-T. Rie, R.-M. Schmidt: Ermüdung von Schweißverbindungen im Bereich geringer Schwingspielzahlen.
Schweißen und Schneiden 38 (1986) 509–514.
K.-T. Rie, R.-M. Schmidt: Low-Cycle Fatigue of Welded Joints. Welding and Cutting 38 (1986)
E172–E175.
Subproject A6, B1
E. Steck: Entwicklung von Stoffgesetzen für die Hochtemperaturplastizität. Grundlagen der Umformtechnik.
Internationales Symposium, Stuttgart 1983, Springer Verlag, 1984, pp. 83–113.
E. Steck: A stochastic model for the high-temperature plasticity of metals. Intern. Journal of Plasticity
1 (1985) 243–258.
E. Steck: A stochastic model for the high-temperature plasticity of metals. Trans. 8th Intern.
SMiRT-Conf., North-Holland, 1985.
E. Steck: Ein stochastisches Modell für die Wechselwirkung von Plastizität und Kriechen. Workshop
Werkstoff und Umformung, Stuttgart 1986, Springer Verlag, 1986.
Subproject A8
Th. Wille, W. Gieseke, Ch. Schwink: Quantitative analysis of solution hardening in selected copper
alloys. Acta Met. 35 (1987) 2679–2693.
Th. Steffens, C.-P. Reip, Ch. Schwink: Anomalous dislocation densities in fcc solid solutions.
Scripta Met. 21 (1987) 335–339.
Subproject A9
H. Neuhäuser, O.B. Arkan, H.H. Potthoff: Dislocation multipoles and estimation of frictional
stress in fcc copper alloys. Mat. Sci. Eng. 81 (1986) 201–209.
H. Neuhäuser: Physical manifestation of instabilities in plastic flow. In: V. Balakrishnan, C.E.
Bottani (Eds.): Mechanical Properties and Behaviour of Solids: Plastic Instabilities World Scientific,
Singapore, 1986, pp. 209–252.
O.B. Arkan, H. Neuhäuser: Dislocation velocities in Cu-Ni alloys determined by the stress pulseetch
pit technique and by slip line cinematography. phys. stat. sol. (a) 99 (1987) 385–397.
H. Neuhäuser, O.B. Arkan: Dislocation motion and multiplication in Cu-Ni single crystals. phys.
stat. sol. (a) 100 (1987).
Subproject B2
H.K. Nipp: Temperatureinflüsse auf rheologische Spannungszustände im Salzgebirge. Report Nr.
82-36 from the Institut für Statik der TU Braunschweig.
A. Schmidt: Berechnung rheologischer Zustände im Salzgebirge mit vertikalen Abbauen in Anlehnung
an In-situ-Messungen. Report Nr. 84-43 from the Institut für Statik der TU Braunschweig.
387

Subproject B4
K.-T. Rie, R. Schubert: Einfluss einer Druckwasserstoffumgebung auf das Ermüdungs- und Risswachstumsverhalten
bei Low-Cycle Fatigue; Wasserstoff in Metallen. Results of the Schwerpunktprogramm,
Deutsche Forschungsgemeinschaft, Bonn, 1986.
K.-T. Rie, R. Schubert: Note on the Crack Closure Phenomenon in LCF. 2. Int. Conf. on LCF and
Elasto-Plastic Behaviour of Materials, München 1987, Elsevier Applied Science, pp. 575–580.
Subproject C1
K. Andresen, R. Ritter, R. Schuetze: Application of grating methods for testing of material and
quality control including digital image processing. SPIE-Potics in Engineering Measurement,
Cannes, 1985.
K. Andresen: Auswertung von Rasterbildern mit der digitalen Bildverarbeitung. Seminar of the
Collaborative Research Centre (SFB 319), Braunschweig.
K. Andresen, F. Hecker: Das Phasenshiftverfahren zur digitalen Auswertung von Moiré-Mustern
und spannungsoptischen Bildern. Nieders. Mechanik-Kolloquium, Braunschweig.
K. Andresen, R. Ritter: The Phase Shift Method Applied to Reflection Moiré Pattern. Int. Conf.
Exp. Stress Anal., Amsterdam, 1986.
S. Angerer, J.P. Löwenau, B. Morche, W. Wilke: 3D-Verformungsmessung an einem gummiartigen
Dämpfungselement mit Hilfe der Stereophotographie und der digitalen Bildverarbeitung.
Fachkolloquium Experimentelle Mechanik, Stuttgart, 1986, pp. 33–40.
K. Andresen: Das Phasenshiftverfahren zur Rasterbildauswertung. Colloquy of the Collaborative
Research Centre (SFB 319), Goslar.
K. Andresen: Verformungsmessung mit Rasterverfahren und der digitalen Bildverarbeitung. Oberseminar
Mechanik, Universität Hannover.
K. Andresen: Konturermittlung und Verformungsmessung mit der digitalen Bildverarbeitung. Mechanik/Colloquy
of the Collaborative Research Centre (SFB), TU Berlin.
Subproject D1
J. Ruge, R. Linnemann: Festigkeits- und Verformungsverhalten von Bau-, Beton- und Spannstählen
bei hohen Temperaturen. Arbeitsbericht 1984–1986. Subproject B4, Collaborative Research
Centre (SFB 148), TU Braunschweig.
R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Methoden.
Dissertation TU Braunschweig, 1987.
Subproject D2
G. Bahr: Kommunizierende Versuchstechnik und simultane theoretische und experimentelle Tragwerksuntersuchung.
Dissertation TU Braunschweig, 1984.
W. Maier, M. Klahold: Das Konzept der kommunizierenden Versuchstechnik für Stabtragwerke.
Nachdruck eines Vortrages auf dem Fach-Kolloquium Experimentelle Mechanik 1986, Eigenverlag,
Ingenieurwiss. Zentrum, Inst. für Modellstatik, Uni Stuttgart.
1988–1990
Bibliography
Subproject A1
F. Haeßner, K. Sztwiertnia: The Misorientation Distributions Associated with the Texture of Polycrystalline
Aggregates of Cubic Crystals. In: J.S. Callend, G. Gottstein (Eds.): ICOTOM 8,
The Metallurgical Society, 1988, pp. 163–168.
F. Haeßner, J. Schmidt: Recovery and recrystallization of different grades of high purity aluminium
determined with a low temperature calorimeter. Scripta Met. 22 (1988) 1917.
U. Meyer: Einfluss verschiedener Verformungsparameter auf die Rekristallisationskinetik von
Kupfer. Dissertation TU Braunschweig, 1989.
388

Publications Resulted from the Collaborative Research Centre
J. Schmidt: Untersuchung der Erholungs- und Rekristallisationsvorgänge in tieftemperaturverformten
Metallen mit Hilfe kalorischer Messungen. Dissertation TU Braunschweig, 1990.
J. Schmidt, F. Haeßner: Stage III-Recovery of Cold Worked High-Purity Aluminium Determined
with a Low-Temperature Calorimeter. Z. f. Phys. B-Condensed Matter 81 (1990) 215.
F. Haeßner: The Study of Recrystallization by Calorimetric Methods. In: T. Chandra (Ed.): Recrystallization
’90, The Minerals, Metals and Materials Society, 1990, pp. 511.
Subproject A2
D. Rode, G. Lange: Beitrag zum Ermüdungsverhalten mehrachsig beanspruchter Aluminiumlegierungen.
Metall 42 (1988) 582.
Subproject A5/B4
R. Schubert, K.-T. Rie: Verfestigung, Fließfläche und Versetzungsstruktur bei Low-Cycle-Fatigue.
Mat.-wiss. und Werkstofftech. 19 (1988) 376–383.
K.-T. Rie, R. Schubert, J. Olfe: Untersuchungen zur Oberflächenausbildung, Mikrostruktur und
Ermüdungsschädigung im LCF-Bereich. DFG-Kolloquium: Schadensfrüherkennung und Schadensablauf
bei metallischen Bauteilen, Darmstadt, Sept. 1989, Berichtsband DVM, Berlin,
1989, pp. 55–62.
R. Schubert: Verformungsverhalten und Risswachstum bei Low-Cycle-Fatigue. Dissertation TU
Braunschweig, 1989.
J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtemperatur-LCF.
Zeitschrift für Metallkunde 81(11) (1990) 783–789.
Subproject A6/B1
E. Steck: Constitutive Laws for Strain-, Strainrate- and Temperature Sensitive Materials. Keynote,
Proceedings 2. Intern. Conf. on Technology of Plasticity (ICTP), Stuttgart, Springer Verlag,
Berlin, 1987.
E. Steck: Development of Constitutive Equations for Metals at High Temperatures. Proceedings 2.
Intern. Conf. on Technology of Plasticity (ICTP), Stuttgart, Springer Verlag, Berlin, 1987.
E. Steck: On the Development of Material Laws for Metals. Festschrift Heinz Duddeck zu seinem
60. Geburtstag, Institut für Statik, TU Braunschweig, 1988.
E. Steck: A Stochastic Model for the Interaction of Plasticity and Creep in Metals. In: F. Zeigler,
G.I. Schueller (Eds.): Nonlinear Stochastic Dynamic Engineering Systems, IUTAM Symposium
Igls, 1987, Springer Verlag, Berlin, 1988.
E. Steck: Marcov-Chains as Models for the Inelastic Behaviour of Metals. In: D.R. Axelrad, W.
Muschik (Eds.): Constitutive Laws and Microstructure, Institute of Advanced Study, Berlin,
Springer Verlag, Berlin, 1988.
E. Steck: A Stochastic Model for the Interaction of Plasticity and Creep in Metals. Nuclear Engineering
and Design 114 (1989) 285–294.
E. Steck: The Description of the High-Temperature Plasticity of Metals by Stochastic Processes.
Res. Mechanica 25 (1990) 1–19.
E. Steck, H. Schlums: Discrete Models on the Microscale for the Plastic Behaviour of Metals.
Proceedings of Plasticity ’89, Second Intern. Symp. on Plasticity and its Current Applications,
Tsu, Japan, 1989, pp. 581–584.
Subproject A8
Ch. Schwink: Hardening mechanisms in metals with foreign atoms. Revue Phys. Appl. 23 (1988)
395–404.
R. Neuhaus, P. Buchhagen, Ch. Schwink: Dislocation densities as determined by TEM in h100i
and h111i CuMn crystals. Scripta Met. 23 (1989) 779–784.
L. Diehl, F. Springer, Ch. Schwink: Studies of hardening mechanisms of symmetrically oriented
single crystals of fcc solid solutions. In: P. O. Kettunen et al. (Eds.): Strength of Metals and Alloys
1, 1988, pp. 313–319.
389

Bibliography
Subproject A9
A. Hampel, H. Neuhäuser: Investigation of slip line growth in fcc Cu alloys with high resolution
in time. phys. stat. sol. (a) 104 (1987) 171–181.
H. Neuhäuser: The dynamics of slip band formation in single crystals. Res. Mechanica 23 (1988)
113–135.
H. Neuhäuser: On some problems in plastic instabilities and strain localization. Rev. Phys. Appl.
23 (1988) 571–572.
A. Hampel, O.B. Arkan, H. Neuhäuser: Local shear rate in slip bands of CuZn and CuNi single
crystals. Rev. Phys. Appl. 23 (1988) 695.
A. Hampel, H. Neuhäuser: Recording of slip line development with high resolution. In: O.Y.
Chiem, H.-D. Kunze, L.W. Meyer (Eds.): Proc. Int. Conf. on Impact Loading and Dynamic
Behaviour of Materials, DGM Verlag, 1988, pp. 845–851.
H. Neuhäuser: Slip propagation and fine structure. In: L.P. Kubin, O. Martin (Eds.): Proc. Coll.
Int. CNRS on Nonlinear Phenomena in Materials Science, Trans. Tech. Publ., Aedermannsdorf,
1988, pp. 407–415.
A. Hampel, M. Schülke, H. Neuhäuser: Dynamic studies of slip line formation on single crystals
of fcc solid solutions. In: P. O. Kettunen, T. K. Lepistö, M.E. Lehtonen (Eds.): Proc. 8. Int.
Conf. on Strength of Metals and Alloys, Pergamon Press, Oxford, 1988, pp. 349–354.
J. Olfe, H. Neuhäuser: Dislocation groups, multipoles, and friction stresses in -CuZn alloys.
phys. stat. sol. (a) 109 (1988) 149–160.
H. Neuhäuser: Plastic instabilities and the deformation of metals. In: D. Walgraef, N.M. Ghoniem
(Eds.): Proc. NATO Adv. Study Inst. on Patterns, Defects and Materials Instabilities,
Kluwer Acad. Publ., Dordrecht, 1990, pp. 241–276.
H. Neuhäuser, J. Plessing, M. Schülke: Portevin-LeChâtelier effect and observation of slip band
growth in CuAl single crystals. J. Mech. Beh. Metals 2 (1990) 231–254.
Subproject A10
D. Besdo: Finite-Element-Analyses of Strain-Space-Represented Elastic-Plastic Media Using Simplified
Stiffness Matrices. Proc. of the Intern. Conf. on Applied Mechanics, Beijing, China,
Pergamon Press, 1989, pp. 1403–1408.
D. Besdo, E. Doege, H.-W. Lange, M. Seydel: Zur numerischen Simulation des Tiefziehens. Lecture
at the 13. Umformtechnischen Kolloquium Hannover 14./15. März 1990, HFF-Bericht Nr.
11 (Ed.: E. Doege), HFF Hannoversches Forschungsinstitut für Fertigungsfragen E.V.
D. Besdo, L. Ostrowski: On the Creep-Ratchetting of AlMgSiO.5 at Elevated Temperature – Experimental
Investigations. Proc. of the Fourth IUTAM Symposium on Creep in Structures,
Krakow, 10.–14. Sept. 1990.
Subproject B2
M. Schwesig: Inelastisches Verhalten metallischer Werkstoffe bei höheren Temperaturen – Numerik
und Anwendung. Report Nr. 89-57 from the Institut für Statik der TU Braunschweig, 1989.
M. Schwesig, H. Ahrens, H. Duddeck: Erfahrungen aus der Anwendung des inelastischen Stoffgesetzes
nach Hart. Festschrift Richard Schardt, THD Schriftenreihe Wissenschaft und Technik
S1, Darmstadt, 1990.
M. Schwesig, H. Braasch, G. Kracht, H. Duddeck, H. Ahrens: Erfahrungen aus der Anwendung
inelastischer Stoffgesetze bei höheren Temperaturen. In: D. Besdo (Ed.): Numerische Methoden
der Plastomechanik, Tagungsband; Hannover, 1989.
D. Dinkler, M. Schwesig: Numerische Lösung von Anfangswertproblemen in der Statik und Dynamik.
Festschrift Heinz Duddeck, Braunschweig, 1988.
H. Duddeck, B. Kröplin, D. Dinkler, J. Hillmann, W. Wagenhuber: Berechnung des nichtlinearen
Tragverhaltens dünner Schalen im Vor- und Nachbeulbereich. Nichtlineare Berechnungen im
Konstruktiven Ingenieurbau, Hannover, DFG-Gz.: Du 25/28-7, 1989.
D. Dinkler: Stabilität elastischer Tragwerke mit nichtlinearem Verformungsverhalten bei instationären
Einwirkungen. Ingenieur-Archiv 60 (1989).
390

Publications Resulted from the Collaborative Research Centre
D. Dinkler: Stabilität dünner Flächentragwerke bei zeitabhängigen Einwirkungen. Report Nr. 88-
52 from the Institut für Statik der TU Braunschweig, 1988.
H. Duddeck, D. Winselmann, F. T. König: Constitutive laws including kinematic hardening for
clay with pore water pressure and for sand. Numerical Methods in Geomechanics, Innsbruck,
1988.
L. Pisarsky, H. Ahrens, H. Duddeck: FEM-Analysis for time-depending cyclic pore water cohesive
soil problems. Eurodyn 90, European Conference on Structural Dynamics, Bochum, DFG-
Gz.: Du 25/34-1-2, 1990.
R. Meyer, H. Ahrens: An elastoplastic model for concrete. Conference Proceedings of the Second
World Congress on Computational Mechanics, Stuttgart, 1990.
Subproject B5
H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Berücksichtigung in
Konstruktionsberechnungen. Dissertation TU Braunschweig, 1990.
J. Scheer, H.-J. Scheibe, M. Reininghaus: Wirtschaftliche Bemessung von Schraubenanschlüssen
bei Ausnutzung des duktilen Verhaltens von Stahl. Report Nr. 6018, Institut für Stahlbau, TU
Braunschweig, 1989.
J. Scheer, H.-J. Scheibe, D. Kuck: Untersuchungen von Trägerschwächungen unter wiederholter
Belastung bis in den plastischen Bereich. Report Nr. 6099, Institut für Stahlbau, TU Braunschweig,
1989.
E. Beißner: Zum Tragverhalten stählerner Augenstäbe im elastisch-plastischen Zustand. Dissertation
TU Braunschweig, 1989.
H.-J. Scheibe: Zur Berechnung zyklisch beanspruchter Stahlkonstruktionen im plastischen Bereich.
Lecture 8. Stahlbau-Seminar, Steinfurt, 1989.
Subproject B6
R. Mahnken, E. Stein, D. Bischoff: A globally convergence criterion for first order approximation
strategies in structural optimations. Int. J. Meths. Eng. 31 (1990).
E. Stein, G. Zhang, R. Mahnken, J.A. König: Micromechanical modelling and computation of
shake down with nonlinear kinematic hardening inducing examples for 2-D problems. Proc. of
CSME Mechanic Engineering Forum, Toronto, 1990.
E. Stein, G. Zhang, J.A. König: Micromechanic modelling of shake down with nonlinear kinematic
hardening inducing examples for 2-D problems. In: Axelrad, Muschik (Eds.): Recent
Developments in Micromechanics, Springer Verlag, 1990.
Subproject C1
K. Andresen, R. Helsch: Automatische Rasterkoordinatenermittlung mit Hilfe digitaler Filter. Informatik
Fachberichte 149, Mustererkennung, 1987, pp. 228.
K. Andresen, R. Helsch: Calculation of Grating Coordinates Using a Correlation Filter Technique.
Optik 80 (1988) 76–79.
K. Andresen, B. Kamp, R. Ritter: Verformungsmessungen an Rissspitzen nach dem Objekt-Raster-
Verfahren. VDI-Berichte 679 (1988) 393–403.
K. Andresen, B. Morche: Die Ermittlung von Rasterkoordinaten und deren Genauigkeit. Mustererkennung
198, pp. 277–283. 10. DAGM-Symposium Zürich, Berlin, New York, Tokio, 1988.
K. Andresen, H. Horstmann: Ermittlung der Verformungen und Spannungen in einer gelochten
Gummimembran mit Hilfe von Rasterverfahren. Forsch. Ing.-Wes. 55 (1989) 33–36.
K. Andresen, K. Hentrich: Vergleich von Frequenz- und Ortsfilterverfahren zur Moiré-Bildauswertung.
Optik 83 (1989) 113–121.
K. Andresen, P. Feng, W. Holst: Fringe Detection Using Mean and n-Rank Filters. Fringe 89,
Berlin, Physical Research 10 (1989) 45–49.
K. Andresen, R. Ritter: Digitale Bildverarbeitung in der Werkstoffprüfung und Qualitätskontrolle.
Tagungsband: Bildverarbeitung: Forschen, Entwickeln, Anwenden, Techn. Akad. Esslingen,
1989.
391

Bibliography
K. Andresen, R. Helsch: Calculation of Analytical Elements in Space Using a Contour Algorithm.
ISPRS-Commission V. Symp. on Close Range Photogrammetry and Machine Vision,
Zürich. SPIE 1395 (1990) 863–869.
K. Andresen: Evaluation of Moiré Fringes Using Space Filtering. Proc. 9th Int. Conf. Exp. Mechanics,
Copenhagen, 1990, pp. 1650–1659.
Subproject C2
J. Hilbig, R. Ritter, W. Wilke: Hochtemperaturdehnungsmessung nach dem Rasterprinzip am
Beispiel des LCF-Versuchs. Akademie der Wissenschaften der DDR/Institut für Mechanik, Report
Nr. 24, 8, Chemnitz, 1989, pp. 255–258.
R. Ritter, W. Wilke: Optische in-situ-Dehnungsfeldmessung unter Hochtemperatureinfluss mit dem
Rasterverfahren am Beispiel des LCF-Versuchs. Vortrags- und Diskussionstagung „Werkstoffprüfung
1990: Aussagefähigkeit von Prüfungsergebnissen für das Verarbeitungs- und Bauteilverhalten“
6./7. 12. 1990 Bad Nauheim, veranstaltet von DVM im Auftrag der Arbeitsgemeinschaft
Werkstoffe.
J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtemperatur
LCF. Zeitschrift für Metallkunde 81(11) (1990) 783–789.
R. Ritter, J. Strusch, W. Wilke: Formanalyse mit Hilfe des Reflexions-Rasterverfahrens und des
photogrammetrischen Prinzips. Österreich. Ingenieur- und Architektenzeitung 135(7/8) (1990)
346–348.
K. Galanulis, J. Hilbig, R. Ritter: Strain Measurement by the Diffraction Principle. Österreich. Ingenieur-
und Architektenzeitung 134(7/8) (1989) 392–394.
W. Cornelius, J. Hilbig, R. Ritter, W. Wilke, C. Forno: Zur Formanalyse mit Hilfe hochtemperaturbeständiger
Raster. VDI-Bericht 731, 5, Düsseldorf, 1989, pp. 295–302.
K. Galanulis, J. Hilbig, B. W. Lührig, R. Ritter: Strain Measurement by the Diffraction Principle
for Curved Surfaces. Proceedings of the 9th Int. Conference on Experimental Mechanics,
Technical University of Denmark, Lyngby/Dänemark, 20.–24. Aug. 1990.
Subproject C3
P. Horst, H. Kossira: Zum Beulverhalten dünner Aluminiumplatten bei wechselnder Schubbelastung.
In: Proceedings der Jahrestagung der Deutschen Gesellschaft für Luft- und Raumfahrt
(DGLR), Jahrbuch 1988 I der DGLR, Bonn, 1988.
K. Wolf: FIPPS – Ein Programm-Paket zur numerischen Analyse des linearen und nichtlinearen
Tragverhaltens von Leichtbaustrukturen. In: H. Kossira (Ed.): 50 Jahre IFL, ZLR-Bericht 89-
01, ISBN 3-9802073-0-7, Braunschweig, 1989.
K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus
kohlenstoffaserverstärktem Kunststoff. Dissertation, Inst. f. Flugzeugbau und Leichtbau, Technische
Universität Braunschweig, 1989.
P. Horst: Plastisches Beulen dünner Aluminiumplatten. In: H. Kossira (Ed.): 50 Jahre IFL, ZLR-
Forschungsbericht 89-01, ISBN 3-9802073-0-7, Braunschweig, 1989.
P. Horst, H. Kossira: Theoretical and experimental investigation of thin-walled aluminium-panels
under cyclic shearload. In: Proceedings of the International Conference on Spacecraft Structures
and Mechanical Testing of ESA, CNES and DFVLR, ESA-Special Report SP-289,
Noordwijk, 1989.
P. Horst, H. Kossira: Cyclic Shear Buckling of Thin-Walled Aluminium Panels. Proceedings of the
17th Congress of the International Council of the Aeronautical Sciences (ICAS) (Paper Nr. 90-
4.4.1), Stockholm, 1990.
392

Subproject D1 (C4)
J. Ruge, C.X. Hou, U. Pätzold: Bestimmung von Gefügeinhomogenitäten in der Wärmeeinflusszone
von Schweißverbindungen. Schweißen und Schneiden 41(3) (1989) 134–137.
R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Methoden.
Fortschr.-Ber. VPI-Reihe 18, Nr. 55, VPI-Verlag, Düsseldorf, 1988.
J. Ruge, S. Zhang, U. Pätzold: Spannungsberechnungen in Schweißnahtmodellen mit Hilfe neuer
Werkstoffgesetze. Mat.-wiss. und Werkstofftech., Heft 9, 1990.
1991–1993
Publications Resulted from the Collaborative Research Centre
Subproject A1
M. Zehetbauer, J. Schmidt, F. Haeßner: Calorimetric study of defect annihilation in low temperature
deformed pure Zn. Scripta Metallurgica et Materialia 25 (1991) 559.
F. Haeßner, K. Sztwiertnia, P. J. Wilbrandt: Quantitative analysis of the misorientation distribution
after the recrystallization of tensile deformed copper single crystals. Textures and Microstructures
13 (1991) 213.
J. Schmidt, F. Haeßner: Recovery and recrystallization of high purity lead determined within a
low temperature calorimeter. Scripta Metallurgica et Materialia 25 (1991) 969.
E. Woldt, F. Haeßner: Aspekte des Entfestigungsverhaltens von Kupfer. Z. f. Metallkunde 82
(1991) 329.
F. Haeßner: Calorimetric investigation of recovery and recrystallization phenomena in metals.
In: R.D. Shull, A. Joshi (Eds.): Thermal Analysis in Metallurgy, The Minerals, Metals and
Materials Society, 1992, pp. 233–257.
F. Haeßner, K. Sztwiertnia: Some microstructural aspects of the initial stage of recrystallization of
highly rolled pure copper. Scripta Metallurgica et Materialia 27 (1992) 2933.
P. Krüger, E. Woldt: The use of an activation energy distribution for the analysis of the recrystallization
kinetics of copper. Acta Metallurgica et Materialia 40 (1992) 2933.
E. Woldt: The relationship between isothermal and non-isothermal description of Johnson-Mehl-
Avrami-Kolmogorov kinetics. J. Phys. Chem. Solids 53 (1992) 521.
F. Haeßner, J. Schmidt: Investigation of the recrystallization of low temperature deformed highly
pure types of aluminium. Acta Metallurgica et Materialia 41 (1993) 1739.
K. Sztwiertnia, F. Haeßner: Orientation characteristics of the microstructure of highly pure copper
and phosphorus copper. Textures and Microstructures 20 (1993) 87.
H.W. Hesselbarth, L. Kaps, F. Haeßner: Two dimensional simulation of the recrystallization kinetics
in the case of inhomogeneously stored energy. Materials Science Forum 113–115 (1993) 317.
Subproject A2
G. Lange, W. Gieseke: Veränderung des Werkstoffzustandes von Aluminium-Legierungen durch
mehrachsige plastische Wechselbeanspruchungen. Festschrift zur Vollendung des 65. Lebensjahres
von Prof. Dr. rer. nat. Dr.-Ing. E. h. Eckhard Macherauch, DGM-Verlag Sept. 1991, pp. 49.
M. Heiser, G. Lange: Scherbruch in Aluminium-Legierungen infolge lokaler plastischer Instabilität.
Z. f. Metallkunde 83 (1992) 115.
G. Lange: Schadensfälle durch Schwingbrüche. Ingenieur-Werkstoffe 4(10) (1992) 62.
G. Lange: Schwingbrüche durch Steifigkeitssprünge. 15. Vortragsveranstaltung des Arbeitskreises
„Rastermikroskopie in der Materialprüfung“, Deutscher Verband für Materialforschung und
-prüfung, Berlin, 1992, pp. 285.
G. Lange: Schwingbrüche durch konstruktiv bedingte Spannungsspitzen (Teil 1: Sprunghafte
Querschnittsänderungen und Steifigkeitssprünge an Schweißverbindungen). Ingenieur-Werkstoffe
5(3) (1993) 58.
393

Bibliography
G. Lange: Schwingbrüche durch konstruktiv bedingte Spannungsspitzen (Teil 2: Verbindungs- und
Befestigungselemente). Ingenieur-Werkstoffe 5(4) (1993) 74.
G. Lange: Fractures in Aircraft Components. In: H.P. Rossmanith, K.J. Miller (Eds.): Mixed-
Mode Fatigue and Fracture, Mechanical Engineering Publications Limited, London, 1993, pp.
23.
Subproject A5/B4
K.-T. Rie, R. Schubert, H. Wittke: Cyclic Deformation Behaviour and Crack Growth in Low-Cycle
Fatigue Range. Mechanical Behaviour of Materials – VI, The Sixth International Conference,
Kyoto, Japan, Preprints of Additional Papers and Extended Abstracts, 1991, pp. 243–
244.
K.-T. Rie, H. Wittke, R. Schubert: The DJ-Integral and the Relation Between Deformation Behaviour
and Microstructure in the LCF-Range. In: K.-T. Rie (Ed.): Low Cycle Fatigue and Elastic-Plastic
Behaviour of Materials – 3, Elsevier Applied Science, London/New York, 1992, pp.
514–520.
K.-T. Rie, J. Olfe: Lokale Werkstoffbeanspruchung bei Hochtemperatur Low-Cycle Fatigue. IX.
Symposium Verformung und Bruch, Teil 1, August 1991, pp. 150–154.
K.-T. Rie, J. Olfe: Crack Growth and Crack Tip Deformation under Creep-Fatigue Conditions.
In: M. Jono, T. Inoue (Eds.): Mechanical Behaviour of Materials – VI, Volume 4, Pergamon
Press, 1991, pp. 367–372.
K.-T. Rie, J. Olfe: A Physically Based Model for Predicting LCF Life under Creep Fatigue Interaction.
In: K.-T. Rie (Ed.): Proc. 3rd Int. Conf. on Low Cycle Fatigue and Elastic-Plastic Behaviour
of Materials, Elsevier Applied Science, London/New York, 1992, pp. 222–228.
K.-T. Rie, J. Olfe: Dehnungsfelder vor Riss-Spitzen bei Kriechermüdung. Z. Metallkde. 84
(1993).
Subproject A6/B1
E. Steck: Stochastic Models for the Plasticity of Metals. In: O. Brüller, V. Mannl, J. Najar (Eds.):
Advances in Continuum Mechanics, Springer Verlag, 1991, pp. 77–87.
K. Rohwer, G. Malki, E. Steck: Influence of Bending-Twisting Coupling on the Buckling Loads of
Symmetrically Layered Curved Panels. Proceedings Intern. Colloquium on Buckling of Shell
Structures, on Land, in the Sea and in the Air, Lyon, France, Sept. 1991.
E. Steck, F. Kublik: Application of Constitutive Models for the Prediction of Multiaxial Inelastic
Behaviour. SMIRT 11, Transactions Vol. 1, Tokyo, Japan, 1991, pp. 557–567.
E. Steck, H.W. Hesselbarth: Simulation of Disclocation Pattern Formation by Cellular Automata.
In: J.-P. Boehler, A.S. Kahn (Eds.): Anisotropy and Localisation of Plastic Deformation, Proceedings
of Plasticity ’91, Elsevier, London, 1991, pp. 175–178.
E. Steck: Stochastic Modelling of Cyclic Deformation Process in Metals (Reprint). In: S.I. Andersen
et al. (Eds.): Proceedings of the 13th Riso International Symposium on Materials Science:
Modelling of Plastic Deformation and its Engineering Applications, Riso National Laboratory,
Roskilde, Denmark, 1992.
F. Kublik, E. Steck: Comparison of Two Constitutive Models with One- and Multiaxial Experiments.
In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications,
IUTAM Symposium Hannover, Germany 1991, Springer-Verlag, Berlin, Heidelberg, 1992.
H. Schlums, E. Steck: Description of Cyclic Deformation Process with Stochastic Models for Inelastic
Behaviour of Metals. Int. J. Plasticity 8 (1992) 147–169.
Subproject A8
F. Springer, Ch. Schwink: Quantitative investigations on dynamic strain ageing in polycrystalline
CuMn alloys. Scripta Metall. Mater. 25 (1991) 2739–2745.
R. Neuhaus, Ch. Schwink: The flow stress of [100]- and [111]-oriented Cu-Mn single crystals: a
transmission electron microscopy study. Phil. Mag. A65 (1992) 1463–1484.
H. Heinrich, R. Neuhaus, Ch. Schwink: Dislocation structure and densities in tensile deformed
CuMn crystals oriented for single glide. phys. stat. sol. (a) 131 (1992) 299–309.
394

Publications Resulted from the Collaborative Research Centre
A. Kalk, Ch. Schwink: On sequences of alternate stable and unstable regions along tensile deformation
curves. phys. stat. sol. (b) 172 (1992) 133–145.
Ch. Schwink: Flow stress dependence on cell geometry in single crystals. Scripta Metal. Mater.
27 (1992) 963–969 (Viewpoint Set No. 20).
H. Neuhäuser, Ch. Schwink: Solid solution strengthening. In: R. W. Cahn, P. Haasen, E.J. Kramer
(Eds.): Materials Science and Technology, Vol. 6 (Vol.-Ed.: H. Mughrabi), VCH, Weinheim,
1993, pp. 191–251.
A. Kalk, Ch. Schwink, F. Springer: On sequences of stable and unstable regions of flow along
stress-strain curves of solid solutions – experiments on Cu-Mn polycrystals. Mater. Sci. Eng.
A 164 (1993) 230–234.
Th. Wutzke, Ch. Schwink: Strain rate sensitivities and dynamic strain ageing in CuMn crystals
oriented for single glide. phys. stat. sol. (a) 137 (1993) 337–350.
Subproject A9
H. Neuhäuser: Collective Dislocation Behaviour and Plastic Instabilities – Micro and Macro Aspects.
In: J.-P. Boehler, A.S. Kahn (Eds.): Anisotropy and Localization of Plastic Deformation,
Elsevier Appl. Sci., London, 1991, pp. 77–80.
C. Engelke, P. Krüger, H. Neuhäuser: Stress Relaxation in Cu-Al Single Crystals at High Temperatures.
Scripta Metal. Mater. 27 (1992) 371–376.
J. Vergnol, F. Tranchant, A. Hampel, H. Neuhäuser: Mesoscopic Observations Related to Twinning
Instabilities in -CuAl Crystals. In: O. Martin, L. Kubin (Eds.): Non-Linear Phenomena
in Materials Science 11, Trans. Tech. Publ., Zürich, 1992, pp. 303–316.
H. Neuhäuser, Ch. Schwink: Solid Solution Hardening. In: R.W. Cahn, P. Hansen, E.J. Kramer
(Eds.): Materials Science and Technology – A Comprehensive Treatment, Vol. 6: Plastic Deformation
and Fracture of Materials (Vol.-Ed.: H. Mughrabi), VCH Verlagsgemeinschaft,
Weinheim, 1993, pp. 191–250.
A. Hampel, T. Kammler, H. Neuhäuser: Structure and Kinetics of Lüders Band Slip in Cu-5 to
15at%Al Single Crystals. phys. stat. sol. (a) 135 (1993) 405–416.
H. Neuhäuser: Collective Micro Shear Processes and Plastic Instabilities in Crystalline and
Amorphous Structures. Int. J. Plasticity 9 (1993) 421–435.
C. Engelke, J. Plessing, H. Neuhäuser: Plastic Deformation of Single Glide Oriented Cu-2 to
15at%Al Crystals at Elevated Temperatures. Mater. Sci. Eng. A 164 (1993) 235–239.
H. Neuhäuser: Problems in Solid Solution Hardening of Alloys. Physica Scripta T 49 (1993) 412–
419.
H. Neuhäuser, A. Hampel: Observation of Lüders Bands in Single Crystals. Scripta Metal. Mater.
29 (1993) 1151–1157.
Subproject A10
D. Besdo: Eine Erweiterung der Taylor-Theorie zur Erfassung der kinematischen Verfestigung.
ZAMM-Z. Angew. Math. Mech. 71 (1991) T264–T265.
D. Besdo, M. Müller: The Influence of Texture Development on the Plastic Behaviour of Polycrystals.
In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications,
IUTAM Symposium Hannover, Germany 1991, Springer Verlag, Berlin, Heidelberg, 1992.
M. Müller, D. Besdo: Simulation globaler Anisotropie mit Hilfe eines Vielkristallmodells. ZAMM-
Z. Angew. Math. Mech. 73 (1993) T658.
Subproject B2
G. Kracht: Erschließung viskoplastischer Stoffmodelle für thermomechanische Strukturanalyse.
Report Nr. 93-69 from the Institut für Statik der TU Braunschweig, 1993.
E.-R. Tirpitz: Elasto-plastische Erweiterung von viskoplastischen Stoffmodellen für Metalle. Report
Nr. 92-70 from the Institut für Statik der TU Braunschweig, 1992.
H. Braasch: Ein Konzept für Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle.
Report Nr. 92-71 from the Institut für Statik der TU Braunschweig, 1992.
395

Bibliography
L. Pisarsky: Zur Berechnung nichtmonoton beanspruchter wassergesättigter Tonböden. Report Nr.
91-65 from the Institut für Statik der TU Braunschweig, 1991.
Z. Huang: Beanspruchungen des Tunnelausbaus bei zeitabhängigem Materialverhalten von Beton
und Gebirge. Report Nr. 91-68 from the Institut für Statik der TU Braunschweig, 1991.
B. Hu: Berechnung des geometrisch und physikalisch nichtlinearen Verhaltens von Flächentragwerken
aus Stahl unter hohen Temperaturen. Report Nr. 93-72 from the Institut für Statik der
TU Braunschweig, 1993.
H. Braasch, H. Duddeck, H. Ahrens: A New Approach to Improve and Derive Materials Models.
J. Eng. Mat. Tech. (ASME) 117 (1995) 14–19.
E.-R. Tirpitz, M. Schwesig: A Unified Model Approach Combining Rate-Dependent and Rate-Independent
Plasticity. Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials-3, Berlin,
1992, pp. 411–417.
M. Schwesig, U. Kowalsky: Zur Formulierung und Anwendung eines elasto-plastischen Modells
für reibungsbehafteten Kontakt. Report Nr. 93-75 from the Institut für Statik der TU Braunschweig,
1993, pp. 75–94.
H. Braasch: Concept to Improve the Approximation of Material Functions in Unified Models.
Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials – 3, Berlin, 1992, pp. 405–410.
U. Kowalski: Verification of a Microstructure-Related Constitutive Model by Optimized Identification
of Material Parameters. Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials –
3, Berlin, 1992, pp. 405–410.
T. Streilein: Anwendung eines Überspannungsmodells zur Beschreibung ein- und mehraxialer zyklischer
Versuche. Report Nr. 93-75 from the Institut für Statik der TU Braunschweig, 1993,
pp. 29–46.
H. Braasch: Erfassung streuenden Materialverhaltens in Werkstoffmodellen. Report Nr. 93-75
from the Institut für Statik der TU Braunschweig, 1993, pp. 1–14.
Subproject B5/B7
J. Scheer, H.-J. Scheibe: Einachsige Zug-Druck-Versuche an Baustahl St52-3. Institut für Stahlbau,
TU Braunschweig, unpublished internal report.
J. Scheer, H.-J. Scheibe: Untersuchungen von zyklisch beanspruchten Lochscheiben aus Baustahl
St52-3. Institut für Stahlbau, TU Braunschweig, unpublished internal report.
S. Dannemeyer: Verhalten von thermomechanisch behandelten Baustählen unter zyklischer Beanspruchung
im elastisch-plastischen Bereich. Experimentelle Studienarbeit, Institut für Stahlbau,
TU Braunschweig, 1992.
M.M. El-Ghandour: Low-Cycle Fatigue Damage Accumulation of Structural Steel St52. Dissertation
TU Braunschweig, 1992.
L. Reifenstein: Verhalten modifizierter CT-Proben aus Baustahl St52-3 unter zyklischer Belastung
im elastisch-plastischen Bereich. Experimentelle Studienarbeit, Institut für Stahlbau, TU Braunschweig.
U. Peil: Dynamisches Verhalten abgespannter Maste. VDI Bericht Nr. 924, 1992.
Subproject B6/B8
E. Stein, G. Zhang, J.A. König: Shakedown with non-linear hardening including structural computation
using finite element methods. Int. J. Plasticity 8 (1992) 1–31.
E. Stein, G. Zhang: Theoretical and numerical shakedown analysis for kinematic hardening
materials. In: Proc. 3rd Conf. on Computational Plasticity, Barcelona, 1992, pp. 1–25.
E. Stein, G. Zhang, Y. Huang: Modelling and computation of shakedown problems for non-linear
hardening materials. Computer Methods in Mechanics and Engineering 103(1/2) (1993) 247–
272.
E. Stein, G. Zhang, R. Mahnken: Shake-down analysis for perfectly plastic and kinematic hardening
material. In: E. Stein (Ed.): Progress in computational analysis of inelastic structures,
CISM courses and lecture No. 321, Springer Verlag, Wien, New York, 1993, pp. 175–244.
E. Stein, Y. Huang: An analytical method to solve shakedown problems for materials with linear
kinematic hardening materials. Int. J. of Solids and Structures 18 (1994) 2433–2444.
396

Publications Resulted from the Collaborative Research Centre
G. Zhang: Einspielen und dessen numerische Behandlung von Flächentragwerken aus ideal plastischem
bzw. kinematisch verfestigendem Material. Ph. D. Thesis, Institut für Baumechanik
und Numerische Mechanik, Universität Hannover, 1992.
R. Mahnken: Duale Verfahren für nichtlineare Optimierungsprobleme in der Strukturmechanik.
Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover,
F92/3, 1992.
R. Mahnken, E. Stein: Parameter-Identification for Visco-Plastic Models via Finite-Element-Methods
and Gradient Methods. IUTAM-Symposium on Computational Mechanics of Materials,
Brown University, 1993.
E. Stein, R. Mahnken: On a solution strategy for parameter identification of visco-plastic models
in the context of finite elements methods. Proc. of Plasticity: Fourth Int. Symposium on Plasticity
and its Current Applications, 1993.
Subproject C1
K. Andresen, B. Hübner: Calculation of Strain from an Object Grating on a Reseau Film by a
Correlation Method. Exp. Mechanics 32 (1992) 96–101.
Q. Yu, K. Andresen, D. Zhang: Digital pure shear-strain moiré patterns. Applied Optics 31
(1992) 1813–1817.
K. Andresen, B. Kamp, R. Ritter: 3D-Contour of Crack Tips Using a Grating Method. Second International
Conference on Photomechanics and Speckle Metrology, San Diego 1991, SPIE Proceedings
1554 A (1991) 93–100.
K. Andresen, B. Kamp, R. Ritter: Three-dimensional surface deformation measurement by a grating
method applied to crack tips. Optical Engineering 31 (1992) 1499–1504.
K. Andresen: 3D-Vermessungen im Nahbereich mit Abbildungsfunktionen. Mustererkennung 92,
14. DAGM Symposion, Dresden, 1992, pp. 304–309.
K. Andresen, Q. Yu: Robust phase unwrapping by spin filtering combined with a phase direction
map. Optik 94 (1993) 145–149.
K. Andresen, Q. Yu: Robust Phase Unwrapping by Spin Filtering Using a Phase Direction Map.
Fringe 93-Bremen.
K. Andresen: Ermittlung von Raumelementen aus Kanten im Bild. Zeitschrift für Photogrammetrie
und Fernerkundung 59 (1991) 212–220.
K. Andresen, R. Ritter, E. Steck: Theoretical and experimental investigations of crack extension
by FEM- and grating methods. Defect assessment in components. Fundamentals and application.
ESIS/EGF9, Mechanical Engineering Publication, London, 1991, pp. 345–361.
Subproject C2
R. Ritter, W. Wilke: Gliederung der Moiréverfahren. Österreich. Ingenieur- und Architekten-Zeitung
136 (1991) 218–222.
R. Ritter, W. Wilke: Slope and Contour Measurement by the Reflection Grating Method and the
Photogrammetric Principle. Optics and Lasers in Engineering 15 (1991) 103–113.
K. Galanulis, J.O. Hilbig, R. Ritter: Zur 3D-Verformungsmessung mit einem Elektronik Speckle-
Pattern Interferometer (ESPI). VDI-Berichte Nr. 882, 1991, pp. 233–242.
J.O. Hilbig, R. Ritter: Speckle measurement for 3D surface movement. Proceedings of the Second
International Conference on Photomechanics and Speckle Metrology, San Diego 1991, SPIE
Proceedings 1554 A (1991) 588–592.
J.O. Hilbig, K. Galanulis, R. Ritter: Zur 3D-Verformungsmessung mit einem Elektronischen
Speckle-Interferometer. DVM-Tagungsband „Werkstoffprüfung 1991“, Bad Nauheim, 1991, pp.
103–110.
D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, D. Winter, H. Wohlfahrt: Zur Anwendung der
Speckle-Meßtechnik bei der Verformungsmessung in der Verbindungszone von Kaltpress-
Schweißverbindungen verschiedener Werkstoffe. DVM-Tagungsband „Werkstoffprüfung 1992“,
Bad Nauheim, 1992, pp. 261–271.
397

Bibliography
D. Brinkmann, K. Galanulis, M. Kassner, Ritter, D. Winter, H. Wohlfahrt: Deformation analysis
in the joining zone of cold pressure butt welding of different materials. International Symposium
on Mis-Matching of Welds, GKSS-Research Centre, Geesthacht, 1993.
D. Bergmann, B.-W. Lührig, R. Ritter, D. Winter: Evaluation of ESPI-phase-images with regional
discontinuities. SPIE Proceedings 2003, 1993.
K. Galanulis, R. Ritter: Speckle interferometry in material testing and dimensioning of structures.
SPIE Proceedings 2004, 1993.
R. Ritter, H. Sadewasser, D. Winter: Evaluation of wrapped phase images with regional discontinuities.
International Workshop “Fringe”, Akademie Verlag, Berlin, 1993.
Subproject C3
H. Kossira, P. Horst: Cyclic Shear Loading of Aluminium Panels With Regard to Buckling and
Plasticity. Thin-walled Structures 11 (1991) 65–84.
P. Horst: Zum Beulverhalten dünner, bis in den plastischen Bereich zyklisch durch Schub belasteter
Aluminiumplatten. Dissertation, ZLR Forschungsbericht 91-01, ISBN 3-9802073-5-8, Inst.
f. Flugzeugbau und Leichtbau, Technische Universität Braunschweig, 1991.
P. Horst, H. Kossira, G. Arnst: On the Performance of Different Elastic-Plastic Material Models
Applied to Cyclic Shear Buckling. Proc. of the Int. ECCS-Colloquium: On the Buckling of
Shell Structures on Land, in the Sea and in the Air, Lyon, France, 1991.
H. Kossira, M. Haupt: Buckling of Laminated Plates and Cylindrical Shells Subjected to Combined
Thermal and Mechanical Loads. Proc. of the Int. ECCS-Colloqium: On the Buckling of
Shell Structures on Land, in the Sea and in the Air, Lyon, France, 1991.
M. Haupt, H. Kossira, M. Kracht, J. Pleitner: A Very Efficient Tool for the Structural Analysis of
Hypersonic Vehicles under High Temperature Aspects. Proc. of the 18th ICAS-Congress,
Peking, China, 1992.
K. Wolf, H. Kossira: An efficient test method for the experimental investigation of the post buckling
behaviour of curved composite shear panels. Proceedings of the European Conferences on
Composite Materials (ECCM), Amsterdam, 1992.
M. Haupt, H. Kossira: Integrated Thermal and Mechanical Structural Analysis of Hypersonic Vehicles
by Using Adaptive Finite Element Methods. Proc. of the Third Aerospace Symposium
1991, Braunschweig. In: Orbital Transport – Technical, Metereological and Chemical Aspects,
Springer, 1993, pp. 165–178.
Subproject C4
J. Ruge, U. Pätzold: Einsatz einer vollautomatischen Härteprüfstation zur Prüfung von Schweißverbindungen
und Optimierung von Schweißverfahren. Tagungsband der DVM-Tagung Werkstoffprüfung,
1991.
U. Pätzold: Verformungsanalyse von Schweißverbindungen. Dissertation, Institut für Schweißtechnik,
Technische Universität Braunschweig, 1992.
D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, H. Wohlfahrt: Zur Anwendung der Speckle-
Meßtechnik bei der Verformungsmessung in der Verbindungszone von Kaltpress-Schweißverbindungen
verschiedenartiger Werkstoffe. Tagungsband der DVM-Tagung Werkstoffprüfung,
1992, pp. 261–271.
D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, D. Winter, H. Wohlfahrt: Deformation Analysis
in the Joining Zone of Cold Pressure Butt Welds of Different Materials. Proceedings of the
Conference on Mis-Matching of Welds, MEP, London, 1993.
398