Section 6: Material modelling in solid mechanics - GAMM 2012

Section 6: Material modelling in solid mechanics 1
Section 6: Material modelling in solid mechanics
Organizers: Daniel Balzani (Universität Duisburg-Essen), Wolfgang Dreyer (WIAS Berlin)
S6.1: Microstructures in Elasto-Plasticity I Tue, 13:30–15:30
Chair: Thorsten Bartel S1|01–A03
A model for martensitic microstructure, its geometry and interface effects
Mehdi Goodarzi, Klaus Hackl (Universität Bochum)
Martensitic materials demonstrate a characteristic microstructure in the form of parallel layers of
compatible phases. This is the consequence of a symmetry-breaking phase transformation which
can be well understood within a continuum mechanical framework by investigating energetically
favorable configurations of phase mixtures. We present a micromechanical model built upon this
approach that reflects numerous attributes of the microstructure.
In this model, a specific class of laminate geometries is constructed based on coherence condition
allowing for curved twin interfaces as well as three-dimensional aggregates. Proper forms of
surface energies are then proposed based on scaling arguments and crystallographic considerations.
We afterwards look into energy minimizing geometries within the class of morphologies that
are considered. The results are notably compliant to the observed scale properties, size effects
and accommodation patterns in the microstructure of martensite.
A Formulation of Finite Gradient Crystal Plasticity with Systematic Separation of
Long- and Short-Range States
S. Mauthe, F. Hildebrand, C. Miehe (Universität Stuttgart)
With the ongoing trend of miniaturization and nanotechnology, the predictive modeling of size
effects play an increasingly important role in metal plasticity. These size effects mainly stem
from geometrically necessary dislocations whose description requires gradient-extended theories
of crystal plasticity. However, a key challenge of the formulation and numerical implementation of
gradient crystal plasticity is the complexity within full multislip scenarios, in particular in context
of rate-independent settings.
In order to partially overcome this difficulty, we suggest a new viscous regularized formulation
of rate-independent crystal plasticity, that exploits in a systematic manner the long- and
short-range nature of the involved variables. To this end, we outline a multifield scenario, where
the macro-deformation and the plastic slips on crystallographic systems are the primary fields.
Related to these primary fields, we define as the long-range state the deformation gradient, the
plastic slips and their gradients. We then introduce as the short-range plastic state the plastic
deformation map, the dislocation density tensor and scalar hardening parameters associated with
the slip systems. It is then shown that the evolution of the short range state is fully determined
by the evolution of the long-range state. This separation into long- and short-range states
is systematically exploited in the algorithmic treatment by a new update structure, where the
short-range variables play the role of a local history base.
The model problem under consideration accounts in a canonical format for basic effects related
to statistically stored and geometrically necessary dislocation flow, yielding micro-force balances
including non-convex cross-hardening, kinematic hardening and size effects. Further key ingredients
of the proposed algorithmic formulation are geometrically exact updates of the short-range
state and a destinct regularization of the rate-independent dissipation function that preserves the
range of the elastic domain.
The formulation is shown to be fully variational in nature, goverend by rate-type continuous

2 Section 6: Material modelling in solid mechanics
and incremental algorithmic variational principles. We demonstrate the modeling capabilities and
algorithmic performance by means of representative numerical examples for multislip scenarios
in fcc single crystals.
Numerical Implementation of a Three-Dimensional Continuum Dislocation Microplasticity
Theory
Stephan Wulfinghoff, Thomas Böhlke (KIT)
The modelling of the mechanical behaviour of micro-devices can not be established by classical
continuum mechanical plasticity models without internal length scale. Depending on the system
size, ab-initio simulations or Discrete Dislocation Dynamics serve as mature tools to predict the
mechanical response of very small systems. Anyhow, the gap between these discrete and classical
continuum mechanical methods is not yet filled. Besides phenomenological gradient plasticity
models, dislocation density based approaches have emerged which account explicitly for dislocation
transport and line length increase. The presentation is based on the kinematical continuum
mechanical dislocation framework of Hochrainer et al. [1] which can be considered as a generalization
of Nye’s theory [3]. The theory transfers the kinematics of three-dimensional systems of
discrete dislocations to a continuum mechanical representation in terms of continuously distributed
dislocations. An averaged version of the theory by Hochrainer et al. [2] (see also Sandfeld et
al. [4]) leads to a significant reduction of the computational simulation effords and will mainly
be addressed. The kinematical dislocation theory is coupled to a crystal plasticity framework.
The presentation will focus on the numerical implementation of the theory, especially on promising
schemes for the numerical coupling of the set of PDEs. Finite Element simulation results
demonstrate the performance of the implementation in three-dimensional applications.
[1] T. Hochrainer, M. Zaiser and P. Gumbsch, A three-dimensional continuum theory of dislocations:
kinematics and mean field formulation. Philosophical Magazine 87 (2007), 1261–1282.
[2] T. Hochrainer, M. Zaiser and P. Gumbsch, Dislocation transport and line length increase in
averaged descriptions of dislocations. arXiv:1010.2884v1
[3] J. F. Nye, Some geometrical relations in dislocated crystals. Acta Metallurgica 1 (1953),
153–162.
[4] S. Sandfeld, T. Hochrainer, M. Zaiser and P. Gumbsch, Continuum modeling of dislocation
plasticity: Theory, numerical implementation, and validation by discrete dislocation
simulations. J. Mater. Res. 26 (2011), 623–632.
Rigorous derivation of a dissipation for laminate microstructures
Sebastian Heinz (WIAS Berlin)
We study models for finite plasticity in the framework of rate-independent evolutionary systems.
The corresponding incremental minimization problems, in general, admit no solutions due to
the creation of microstructure, see [1]. We focus on the case where the microstructure is made of
simple laminates only. We show in a mathematical rigorous way how the incremental minimization
problem can be relaxed and identify the relaxed dissipation as the one introduced in [2].
[1] C. Carstensen, K. Hackl, A. Mielke. Non-convex potentials and microstructures in finite-strain
plasticity, Proc. Royal Soc. London, Ser. A 458 (2002), 299 – 317.

Section 6: Material modelling in solid mechanics 3
[2] K. Hackl, D. M. Kochmann. Relaxed potentials and evolution equations for inelastic microstructures.
In B. Daya Reddy, editor, IUTAM Symposium on Theoretical, Computational
and Modelling Aspects of Inelastic Media, pages 27 – 39. Springer-Verlag, 2008.
Thermodynamically and variationally consistent modeling of distortional hardening:
application to magnesium
Baodong Shi (Helmholtz-Zentrum Geesthacht), Jörn Mosler (Helmholtz-Zentrum Geesthacht, TU
Dortmund)
To capture the complex elastoplastic response of many materials, classical isotropic and kinematic
hardening alone are often not sufficient. Typical phenomena which cannot be predicted by the
aforementioned hardening models include, among others, cross hardening or more generally, the
distortion of the yield function. However, such phenomena do play an important role in several
applications in particular, for non-radial loading paths. Thus, they usually cannot be ignored. In
the present contribution, a novel macroscopic model capturing all such effects is proposed. In contrast
to most of the existing models in the literature, it is strictly derived from thermodynamical
arguments. Furthermore, it is the first macroscopic model including distortional hardening which
is also variationally consistent. More explicitly, all state variables follow naturally from energy
minimization within advocated framework.
A viscosity-limit approach to the evolution of microstructures in finite plasticity
Christina Günther, Klaus Hackl (Universität Bochum)
Material microstructures in finite single-slip crystal plasticity occur and evolve due to deformation.
Their formation is not arbitrary, they tend to form structured spatial patterns. This hints at a
universal underlying mechanism, in the same manner as the minimization of the global energy governs
the behavior of purely elastic materials. For non-quasiconvex potentials, the minimizers are
small scale fluctuations, related to probability distributions of the deformation gradients, which
can be found by the relaxation of the potential.
As in the approach of D.Kochmann and K.Hackl, we use a variational framework, focusing on the
Lagrange functional. For the treatment of the laminates, the potentials have to be adapted. The
new approach rests on the introduction of a small smooth transition zone between the laminates
in order to avoid a global minimization. This makes the evolution equations more handable for
numerical calculations. We present explicit time-evolution equations for the volume fractions and
the internal variables. We outline a numerical scheme and show some examples.
S6.2: Polymers and Elastomers I Tue, 13:30–15:30
Chair: Mikhail Itskov, Vladimir Kolupaev S1|01–A1
Constitutive modelling of chemical ageing
Alexander Lion, Michael Johlitz (Universität der Bundeswehr München)
In order to model chemical ageing of rubber a three-dimensional theory is proposed. The fundamentals
of this approach are different decompositions of the deformation gradient in combination
with an additive split of the Helmholtz free energy into three parts. Its first part belongs to the
volumetric material behaviour. The second part is a temperature-dependent hyperelasticity model
which depends on an additional internal variable to consider the long-term degradation of the
primary rubber network. The third contribution is a functional of the deformation history and a
further internal variable; it describes the creation of a new network which remains free of stress

4 Section 6: Material modelling in solid mechanics
when the deformation is constant in time. The constitutive relations for the stress tensor and the
internal variables are deduced using the Clausius-Duhem inequality. In order to sketch the main
properties of the model, expressions in closed form are derived with respect to continuous and
intermittent relaxation tests as well as for the compression set test. Under the assumption of near
incompressible material behaviour, the theory can also represent ageing-induced changes in volume
and their effect on the stress relaxation. The simulations are in accordance with experimental
data from literature.
[1] A. Lion, M. Johlitz, On the representation of chemical ageing of rubber in continuum mechanics,
International Journal of Solids and Structures, under review.
Multi-phase modeling of shape memory polymers
Nico Hempel, Markus Böl (TU Braunschweig)
Shape memory polymers are a highly versatile class of so-called smart materials. They are able to
store a certain state of deformation and remember another one by means of an external stimulus
such as light or temperature. Usually, the physical mechanism responsible for this behavior is a
temperature-dependent phase transition between an “active”, entropy-elastic phase and a “frozen”,
energy-elastic phase. As polymers are usually capable of experiencing large deformations, models
are required which are able to represent both the phase transitions and states of large deformation.
In the present work, we propose a model based on the idea of the multiplicative decomposition
of the deformation gradient. Evolution equations for the several deformation components are
presented that provide for the storage of the entropy-elastic strain and its recovery during the
transition between the frozen phase and the active phase. First characteristic shape memory cycles
will be presented as a last point.
On response functions in linear thermoelastic models of shape memory polymers
Aycan Özlem Aydin, Rasa Kazakevičiute-Makovska , Holger Steeb (Universität Bochum)
There are two basic classes of constitutive models for thermoresponsive Shape Memory Polymers
(SMPs), rheological and thermoelastic models. In this work, we present a detailed analysis of
linear thermoelastic models. The first model within this class has been proposed by Liu et al. [1]
and in the following years numerous modifications of that model have been presented (cf. e.g. [2]).
All these models have a common mathematical structure with three basic response functions.
Different models developed within this concept follow from the general theory by specifications
of the relevant response functions. The general mathematical form of the response functions
is discussed and an experimental methodology determining of response functions directly from
experimental data is presented. Representative examples illustrate the quality and the efficiency
of the proposed methodology. It is shown that a reliable evaluation of thermoelastic models for
SMPs requires a comparison with data for all four steps of the termomechanical cycle, both under
strain- and stress controlled conditions.
[1] Y. Liu, K. Gall, M. L. Dunn, A. R. Greenberg, J. Diani, Thermomechanics of shape memory
polymers: Uniaxial experiments and constitutive modeling, Int. J. Plasticity, vol. 22, pp.
279-313, 2006.
[2] R. Kazakevičiute-Makovska, H. Steeb, A. Özlem Aydın, On the evolution law for the frozen
fraction in the linear theories of shape memory polymers, Arch. App. Mech., in press, 2011.

Section 6: Material modelling in solid mechanics 5
Modeling the finite strain deformation and initial anisotropy of amorphous thermoplastic
polymers
Philipp Hempel, Thomas Seelig (KIT)
The present work deals with modeling the temperature dependent finite strain deformation behavior
of amorphous thermoplastic polymers incorporating the effect of an initial anisotropy. The
anisotropy prevails in form of a frozen-in pre-orientation of molecular chains and results from preceding
manufacturing processes (e.g. injection molding) at elevated temperatures and subsequent
rapid cooling. The initial molecular orientation affects the mechanical response in terms of flow
strength and hardening.
The standard finite strain kinematics with a multiplicative split of the deformation gradient into
an elastic and plastic part is modified by introducing a network deformation gradient which comprises
the actual plastic deformation and the initial (processing induced) pre-deformation [1],[2].
As a computational example, an injection molded plate is investigated which displays nonuniform
shrinkage and buckling during heating due to the action of the frozen-in network stress. The
spatial distribution of molecular pre-orientation, and hence initial anisotropy, in the FE model is
estimated from optical birefringence.
[1] E.M. Arruda, M.C. Boyce, Evolution of plastic anisotropy in amorphous polymers during
finite straining, International Journal of Plasticity (1993), 6 –697.
[2] M.C. Boyce, D.M. Parks, A.S. Argon, Plastic flow in oriented glassy polymers, International
Journal of Plasticity (1998), 6 – 593.
Modeling of induced anisotropy at large deformations for polymers
Ismail Caylak, Rolf Mahnken (Universität Paderborn)
In this presentation we develop a model to describe the induced plasticity of polymers at large
deformations. Polymers such as stretch films exhibit a pronounced strength in the loading direction.
The undeformed state of the films is isotropic, whereas after the uni-axial loading the
material becomes anisotropic. In order to consider this induced ansiotropy during the stretch
process, a spectral decomposition of the inelastic Cauchy-Green tensor is done. Therefore, the
yield function can be formulated as a function of the anisotropic tensor, where again the anisotropic
tensor is a function of the maximum eigenvalue. A backward Euler scheme is used for
updating the evolution equations, and the algorithmic tangent operator is derived. The numerical
implementation of the resulting set of constitutive equations is used in a finite element program
and for parameter identification.
S6.3: Microstructures in Elasto-Plasticity II Tue, 16:00–18:00
Chair: Dennis Kochmann, Sebastian Heinz S1|01–A03
Microstructure development in a Cosserat continuum as a consequence of energy
relaxation
Muhammad Sabeel Khan, Klaus Hackl (Universität Bochum)
A rate-independent inelastic material model for a Cosserat continuum is presented. The free energy
of the material is enriched with an interaction potential taking into account the intergranular

6 Section 6: Material modelling in solid mechanics
kinematics at the continuum scale. As a result the total energy becomes non-convex, thus giving
rise to the development of microstructure. To guarantee the existence of minimizers an exact
quasi-convex envelope of the corresponding energy functional is derived. As a result microstructure
occurs in both the displacement and micro-rotation field. The corresponding relaxed energy
is then used for finding the minimizers of the two field minimization problem corresponding to a
Cosserat continuum. Finite element formulation and numerical simulations are presented. Analytical
and numerical results are discussed.
About the Microstructural Effects of Polycrystalline Materials and their Macroscopic
Representation at Finite Deformation
Eva Lehmann, Stefan Loehnert, Peter Wriggers (Universität Hannover)
In the sheet bulk metal forming field, the strict geometrical requirements of the workpieces result
in a need of a precise prediction of the material behaviour. The simulation of such forming processes
requires a valid material model, performing well for a huge variety of different geometrical
characteristics and finite deformation.
Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically
observable plastic deformation is traced back to dislocations within considered slip
systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level.
A finite crystal plasticity model is used to model polycrystalline materials in representative
volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation
gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the
elastic contribution. In order to circumvent singularities stemming from the linear dependency
of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the
plastic slips and slip resistances.
The model is validated with experimental microstructural data under deformation. The validation
on the macroscopic scale is performed through the reproduction of the experimentally
calculated initial yield surface. Additionally, homogenised stress-strain curves from the microstructure
build the outcome for a suitable effective material model. Through optimisation techniques,
effective material parameters can be determined and compared to results from real forming processes.
A rheological model for arbitrary symmetric distortion of the yield surface
A.V. Shutov, J. Ihlemann (TU Chemnitz)
A new rheological model is presented, which provides insight into phenomenological modelling of
combined nonlinear kinematic and distortional hardening. The model is constructed by coupling
idealized two-dimensional rheological elements like Hooke-body, Newton-body, and modified St.
Venant element [1,2]. The symmetric distortion of the yield surface and its orientation depending
on the recent loading path are captured by the rheological model in a vivid way. We emphasize the
flexibility of the proposed approach since it can be used to capture any smooth convex saturated
form of the yield surface observed experimentally, if the yield surface is symmetric with respect to
the recent loading direction. In particular, an arbitrary sharpening of the saturated yield locus in
the loading direction combined with a flattening on the opposite side can be taken into account
[2]. Moreover, the yield locus evolves smoothly and its convexity is ensured at each hardening
stage.
The rheological model serves as a guideline for construction of new constitutive relations. The
kinematic assumptions, the ansatz for the free energy and for the yield function are motivated by
the rheological model. Additionally to kinematic and distortional hardening, a nonlinear isotropic
hardening is introduced as well. Normality flow rule is considered, and a rigorous proof of ther-

Section 6: Material modelling in solid mechanics 7
modynamic consistency is provided. Finally, the predictive capabilities of the resulting material
model are verified using the experimental data for a very high work hardening annealed aluminum
alloy 1100 Al.
[1] A.V. Shutov, S. Panhans, R. Kreißig, A phenomenological model of finite strain viscoplasticity
with distortional hardening, ZAMM, 8, 653 - 680.
[2] A.V. Shutov, J. Ihlemann, A viscoplasticity model with an enhanced control of the yield
surface distortion, Submitted to International Journal of Plasticity.
Towards the simulation of Internal Traverse Grinding – from mesoscale modelling to
process simulations
Raphael Holtermann (TU Dortmund), Andreas Menzel (TU Dortmund / Lund University)
The present work aims at the modelling and simulation of Internal Traverse Grinding of hardened
100Cr6/AISI 52100 using electro plated cBN grinding wheels. We focus on the thermomechanical
behaviour resulting from the interaction of tool and workpiece in the process zone on a mesoscale.
Based on topology analyses of the grinding wheel surface, two-dimensional single- and multigrain
representative numerical experiments are performed to investigate the resulting loaddisplacement-behaviour
as well as the specific heat generation due to friction and plastic dissipation.
A thermoelastic-viscoplastic constitutive model is used to capture thermal softening of the
material taken into account. Based on previous work [1,2], an adaptive remeshing scheme which
uses a combination of error estimation and indicator methods, is applied to overcome mesh dependence.
In consequence, the formulation allows to resolve the complex deformation patterns
and to predict a realistic thermomechanical state of the resulting workpiece surface [3].
As a future goal, we aim at coupling the above cutting zone model to a process scale simulation
to model the thermomechanical behaviour of the entire three-dimensional workpiece.
[1] C. Hortig and B. Svendsen, Simulation of chip formation during high-speed cutting, J. Mat.
Processing Technology 186, 66–76 (2007)
[2] C. Hortig and B. Svendsen, Adaptive modeling and simulation of shear banding and high
speed cutting, Proc. 10th ESAFORM Conference on Material Forming CP907, 721–726
(2007)
[3] D. Biermann, A. Menzel, T. Bartel, F. Höhne, R. Holtermann, R. Ostwald, B. Sieben, M.
Tiffe, A. Zabel, Experimental and computational investigation of machining processes for
functionally graded materials, Procedia Engineering, accepted for publication, 2011
Multiscale crystal plasticity based on continuum theory of dislocation mechanics: an
extension to ductile fracture.
Mubeen Shahid, Klaus Hackl (Universität Bochum)
The presentation focuses on the modelling of phenomena associated with plastic deformations
in crystalline materials. The plastic deformation in crystalline materials strongly depends on
aggregate behaviour of dislocations. However there is no universal constitutive frame-work which
directly relates all the attributes of dislocations at microscale to the macroscale deformations,
both qualitatively and quantitatively.

8 Section 6: Material modelling in solid mechanics
First the macroscale constitutive model based on the continuum theory of dislocations [1,2]
is discussed. The plastic deformation and crack growth during ductile fracture mutually effect
each other, where dislocations’ movement shape the crack growth, and the latter effects the
dislocations density [3]. We discuss the numerical implementation of the proposed crystal viscoelastoplasticity
model coupled with modern dislocation measures [1,2] and present an example
where the mechanisms of crystal plasticity are linked to the ductile fracture. The results focus
the effect of severe plastic deformation on dislocations evolution and crack growth behaviour,
following the approach of Cherepanov [3].
[1] T. Hochrainer, M. Zaiser, P. Gumbsch, A three-dimensional continuum theory of dislocation
systems: kinematics and mean-field formulation, Philosophical Magazine, 87:8-9 (2007),
1261-1282.
[2] S. Sandfeld, T. Hochrainer, P. Gumbsch, M. Zaiser, Numerical implementation of a 3D
continuum theory of dislocation: dynamics and application to micro-bending, Philosophical
Magazine, 90:27-28 (2010), 3697-3728.
[3] G.P. Cherepanov et al., Dislocation generation and crack growth under monotonic loading,
J. Appl. Phys., 78(10) (1995), 6249-6264.
Multiscale modelling and simulation of micro machining of titan
Richard Lohkamp, Ralf Müller (TU Kaiserslautern)
The topology of micro machined surfaces depends strongly on the underlying heterogeneous microstructure
of the material. The crystal structure influences the deformation and separation
characteristics. In the case of α-titanium the deformation is dictated by the hcp crystal structure
with its specific slip systems. In the crystal plastic deformation it is essential to take self and latent
hardening into account. Furthermore to capture the rate dependent behavior a visco-plastic
evolution law is used. This setting serves as a framework for more complex constitutive laws, such
as the one given in [1].
As a first attempt to model the cutting process, the fracture mechanisms in a crystalline αtitanium
are analysed within the concept of configurational forces. To this end the theory of
configurational forces is presented for a standard dissipative medium and is specialized to the
crystal plasticity setting. The numerical implementation of the material law and the configurational
forces is done in a consistent way within the finite element method. The application of
configurational forces in the crystal plasticity setting is discussed and demonstrated by illustrative
examples.
S6.4: Polymers and Elastomers II Tue, 16:00–18:00
Chair: Alexander Lion, Joachim Schmitt S1|01–A1
How to approximate the inverse Langevin function?
Mikhail Itskov, Roozbeh Dargazany, Karl Hörnes (RWTH Aachen)
The inverse Langevin function directly results from the non-Gaussian theory of rubber elasticity
as the chain force and represents an indispensable ingredient of full-network rubber elasticity models.
However, the inverse Langevin function cannot be represented in a closed-form and requires
an approximation as for example a Padé approximation. The Padé approximations can be given

Section 6: Material modelling in solid mechanics 9
in a relatively simple form and are able to describe the asymptotic behavior of the inverse Langevin
function in the vicinity of the maximum chain extensibility. Far away from the asymptotic
area the approximation error can be, however, relatively large. In the present contribution we
compare various Padé approximants with the Taylor power series representation of the inverse
Langevin function. To this end, a simple recursive procedure calculating power series coefficients
of the inverse function is proposed. The procedure can be applied to any function which can be
expanded into Taylor series. Within the convergence radius the resulting series of the inversed
Langevin function demonstrates better agreement with the analytical solution than the Padé approximations.
Phenomenological modeling of a polymeric composite
Sebastian Borsch, Albrecht Bertram (Universität Magdeburg)
A composite material consisting of a polymeric matrix material and metallic filler particles is
modeled in a phenomenological way. The isotropic viscoplastic constitutive model is formulated
within the theory of finite deformations. The flow rule is a superposition of two terms. This
approach enables us to simulate the strong backflow behavior during unloading, which can be
observed in various polymeric materials. A quasi-static finite-element simulation has been performed
to compare the model with cyclic tension tests as well as a relaxation test.
Modelling of nanoindentation of polymers with effects of surface roughness and parameters
identification
Zhaoyu Chen, Stefan Diebels (Universität des Saarlandes)
Since the nanoindentation testing technique can measure the properties of extremely small volumes
with sub-m and sub-N resolution from the continuously sensed force-displacement curves,
it also became one of the primary testing techniques for polymeric materials and biological tissues.
The analysis of individual indentation tests using the conventionally applied Oliver & Pharr
method has limitations to capture the hyperelastic and rate-dependent properties of polymers.
Therefore, an inverse method with respect to the experimental testing, based on finite element
simulation and numerical optimisation has been used and evolved. However, nanoindentation is
composed of various error contributions, e. g. friction, adhesion, surface roughness and indentation
process associated factors. These contributions forming the systematic errors between the
numerical model and the experiments will often lead to large errors in the parameters identification.
Therefore, basic investigations and quantification of these influences are indispensable to
characterise the materials accurately from nanoindentation based on the inverse method.
In the present contribution, the characterisation of polymers through nanoindentation with effects
of the surface roughness based on inverse method will be investigated numerically. The boundary
value problems of nanoindentation of polymers are modelled with the FE code ABAQUS R○ .
In contrast to the traditional inverse method, virtual experimental data calculated by numerical
simulations with chosen parameters replace the real experimental measurements. Such a procedure
is called parameter re-identification. In this sense, the finite element code ABAQUS R○ is used as
our virtual laboratory. The model parameters are identified using an evolution strategy based on
the concept of numerical optimisation. The surface roughness effects are investigated numerically
based on the approach utilizing the phenomenological concepts. The surface roughness is chosen
to have a simple representation considering only one-level of roughness profile described by a
sine function. The influence of the surface roughness is quantified associated to the sine curve
parameters as well as to the indentation parameters. Moreover, the real surface topography is

10 Section 6: Material modelling in solid mechanics
characterised using multi-level of protuberance-on-protuberance profiles. The effects of the surface
roughness are investigated with respect to the identified model parameters using a surface
topography with one-level or multi-level profiles approximated by a sine function. The results are
expected to offer a deep insight into characterising the real surface roughness numerically.
Material force computation for the thermo-mechanical response of dynamically loaded
elastomers
Ronny Behnke, Michael Kaliske (TU Dresden)
Elastomers are widely used in our today’s life. The material is characterized by large deformability
upon failure, elastic and time dependent as well as non-time dependent effects which can
be also a function of temperature. In addition, cyclically loaded components show heat built-up
which is due to dissipation. As a result, the temperature evolution of an elastomeric component
can strongly influence the material properties and durability characteristics. Representing best
the real thermo-mechanical behaviour of an elastomeric component in its design process is one
motivation for the use of sophisticated, coupled material approaches within numerical simulations.
In order to assess the durability characteristics, for example regarding crack propagation, the
material forces (configurational forces) are one possible approach to be applied. In the present
contribution, the implementation of material forces for a thermo-mechanically coupled material
model including a continuum mechanical damage (CMD) approach is demonstrated in the context
of the Finite Element Method (FEM). Special emphasis is given to material forces resulting
from internal variables (viscosity and damage variables), temperature field evolution and dynamic
loading. Using an example of an elastomeric component, for which the material model parameters
have been previously identified by a uniaxial extension test, material forces are evaluated quantitatively.
The influence of each contribution (internal variables, temperature field and dynamics)
is illustrated and compared to the overall material force response.
Single and multiscale aspects of the modeling of curing polymers
Alexander Bartels ∗ , Sandra Klinge ∗ , Klaus Hackl ∗ , Paul Steinmann ∗∗ (Universität Bochum, Universität
Erlangen-Nürnberg)
Within this presentation, the focus is placed on the simulation of the isochoric behavior of polymers
during the curing process. To this end, a model based on the assumption for the free
energy in the form of a convolution integral is applied. Since this allows the implementation of
the time dependent material parameters, the free energy is interpreted as the total-accumulated
energy. Different from this, the strain energy is related to the current state of deformation and
used to define the temporary stiffness. In order to avoid volume locking effects typical for isochoric
materials, the free energy is furthermore split into a volumetric and a deviatoric part. A
multifield description depending on the displacements, volume change and hydrostatic pressure
is introduced as well. The model is implemented within a single- and multiscale FE program
and used to simulate the behavior of homogeneous and microheterogeneous polymers. The main
property of the multiscale concept used here is that the modeling of a heterogeneous body is
performed by simultaneously solving two boundary value problems: one related to the behavior
of the macroscopic body and the other one dealing with the analysis of the representative volume
element.
Influence of nano-particle interactions on the mechanical behavior of colloidal structures
in polymeric solutions
Roozbeh Dargazany, Ngoc Khiêm Vu , Mikhail Itskov (RWTH Aachen)
Colloidal structures inside solutions are usually considered as rigid bodies or linear springs. Howe-

Section 6: Material modelling in solid mechanics 11
ver, recent experimental results show a strongly nonlinear mechanical response of large clusters.
In this contribution, the nonlinear elastic behavior of the colloidal structures inside polymeric
solutions is studied. So far, the influences of initial length and fractal dimension on the elastic response
of colloidal structures have mostly been considered by scaling theory. Here, we additionally
take into account a deformation induced evolution of the aggregate structure which is mainly influenced
by inter-particle interactions. To this end, central and lateral (non-central) inter-particle
forces are considered separately. Next, the directional stiffness of the colloidal structure is evaluated
by using the concept of a backbone chain. The backbone chain is a unique path between two
ends of the colloidal structure that carries the main portion of load. The mechanical response of
the backbone chain depends on aggregate geometry, deformation history and moreover, on the
nature and the strength of the inter-particle interactions. The aggregate geometry is described by
means of the angular averaging concept. The so-obtained model can further be generalized for all
types of colloidal structures with central and lateral inter-particle forces.
S6.5: Phase Transformations I Wed, 13:30–15:30
Chair: Markus Lazar, Wolfgang Dreyer S1|01–A03
A thermodynamically consistent framework for martensitic phase transformations
interacting with plasticity
Thorsten Bartel, Andreas Menzel (TU Dortmund)
We propose constitutive relations for martensitic phase transformations at large strains which
captures the interactions between phase transformations, plasticity and the local heating of the
material due to the inelastic processes. For the kinematics of finite deformations we make use
of logarithmic Hencky-strains. The total strain is additively decomposed into elastic, plastic and
transformation related parts, where the latter quantities are motivated by crystallographic considerations.
The thermodynamically consistent framework is based on a representative macroscopic
energy density and an inelastic potential in terms of a dual dissipation functional. With these
quantities at hand, thermodynamical driving forces as well as rate-independent evolution equations
are derived in a canonical way. In this regard, the proposed model states an alternative to the
models described in [1,2]. Referring to [3], the local heating of the material is realised by a consistently
derived temperature evolution equation capturing the effect of self heating. The solution
of the obtained local system of equations is challenging and demands a sophisticated algorithmic
treatment. To this end, we present a scheme which avoids cumbersome and time-consuming active
set searches by the use of Fischer-Burmeister NCP functions and furthermore prevents the
occurence of redundant equations by a specific condensation strategy. The numerical examples
emphasize the capabilities of the proposed model which is, among other aspects, exemplified by
a transformation induced anisotropy. Furthermore, special attention is paid to the simulation of
the complex material behaviour of, e.g., TRIP-steels.
[1] T. Bartel, A. Menzel, B. Svendsen, Thermodynamic and relaxation-based modeling of the
interaction between martensitic phase transformations and plasticity, J. Mech. Phys. Sol.
Vol. 59, 1004-1019, 2011,
[2] R. Ostwald, T. Bartel, A. Menzel, A one-dimensional computational model for the interaction
of phase-transformations and plasticity, Int. Journal of Structural Changes in Solids Vol. 3,
63-82, 2011,
[3] J. C. Simo, C. Miehe, Associative coupled thermoplasticity at finite strains: Formulation,
numerical analysis and implementation, Comp. Meth. Appl. Mech. Engrg. Vol. 98, 41-104,

12 Section 6: Material modelling in solid mechanics
1992
Deformation induced martensite transformation in a cold-worked forming process of
austenitic stainless steel
Tim Dally, Kerstin Weinberg (Universität Siegen)
Within the last years the goal of industrial manufacturing processes - such as tube forming - has
shifted towards an optimization of technological as well as mechanical properties of the manufactured
structures. For example, during the forming procedure of sheets made of austenitic stainless
steel X5CrNi18-10, the content of strain-induced martensite needs to be controlled. In order to
achieve optimal structural properties of the manufactured tube with respect to very high-cycle
fatigue (VHCF), a martensite ratio of approximately 25% needs to be obtained.
On the basis of experimental investigations our contribution deals with the numerical simulation
of the forming process with special consideration of the martensite ratio c as a function of
temperature and deformation field:
c = c(T, ε p ).
In particular, we study the interaction of processing temperature, friction, plastic deformation
and stress state during forming. We will further present different approaches of modelling the
martensite evolution as well as the extension of an existing martensite model on polyaxial states
of stress and compare experimental results and numerical simulations for the modified model.
Additionally a facility to calculate the hardening due to martensitic phase transformation will be
presented.
Finally, we will propose a strategy to control the martensite evolution during the tube-forming
process that enables us to achieve the optimal c mentioned above.
Micromechanical modeling of bainitic phase transformation
A. Schneidt, R. Mahnken (Universität Paderborn), T. Antretter (Montanuniversität Leoben)
We develop a micromechanical material model for phase transformation from austenite to bainite
for a polycrystalline low alloys steel. In this material (e.g. 51CrV4) the phase changes from
austenite to perlite-ferrite, bainite or martensite, respectively. This work is concerned with phase
transformation between austenite and n-bainite variants in different orientated grains. Characteristic
of bainite are the combination of time-dependent transformation kinetics and lattice shearing
in the microstructure. These effects are considered on the microscale and by means of homogenisation
scale in the polycrystalline macroscale with stochastically orientated grains. Furthermore, the
numerical implementation of our model with a Newton projection algorithm into a finite-element
program is presented, based on the algorithm in [1].
[1] R. Mahnken and S. Wilmanns, A projected Newton algorithm for simulation of multivariant
textured polycrystalline shape memory alloys. Computational Materials Science 50,
25352548, 2011.
Effects of heat treatment on phase transformation in powder metallurgical multifunctional
coating
Reza Kebriaei, Jan Frischkorn, Stefanie Reese (RWTH Aachen)

Section 6: Material modelling in solid mechanics 13
Heat treatment is an indispensable part of the manufacturing of metallic products, especially in
powder coating process. It provides an efficient way to manipulate the properties of the metal as
e.g. hardness, yield stress and tensile stress by controlling the rate of diffusion and the rate of
cooling within the microstructure.
The process-integrated powder coating by radial axial rolling of rings is a new hybrid production
technique which is introduced in [1]. It takes advantage of the high temperatures and high
forces of the ring rolling process not only to increase the rings diameter, but also to integrate
the application and compaction of powder metallurgical multi-functional coatings to the solid
substrate rings within the same process [2]. The applied temperatures in hot rolling are within
the range of austenitizing temperatures for the investigated steels. Therefore, controlled cooling
can be conducted directly from process heat subsequent to the deformation process.
The talk is concerned with the finite element (FE) simulation of the process-integrated powder
coating by radial axial rolling of rings and the integration of heat treatment of the rolled ring
into the subsequent cooling process. Finally parameter studies are performed to analyse the
temperature profile and phase transformation in the rings cross-section.
[1] J. Frischkorn, S. Reese, Modelling and Simulation of Process-integrated Powder Coating by
Radial Axial Rolling of Rings, Archive of Applied Mechanics 81 (2011)
[2] R. Kebriaei, J. Frischkorn, S. Reese, Influence of Geometric Parameters on Residual Porosity
in Process-integrated Powder Coating by Radial Axial Rolling of Rings, Steel Research
International 163 (2011)
Simulation of phase-transformations based on numerical minimization of intersecting
Gibbs energy potentials
Richard Ostwald, Thorsten Bartel (TU Dortmund), Andreas Menzel (U Dortmund / Lund University)
We present a novel approach for the simulation of solid to solid phase-transformations in polycrystalline
materials. To facilitate the utilization of a non-affine micro-sphere formulation with
volumetric-deviatoric split, we introduce Helmholtz free energy functions depending on volumetric
and deviatoric strain measures for the underlying scalar-valued phase-transformation model.
As an extension of affine micro-sphere models [3], the non-affine micro-sphere formulation with
volumetric-deviatoric split allows to capture different Young’s moduli and Poisson’s ratios on the
macro-scale [1]. As a consequence, the temperature-dependent free energy assigned to each individual
phase takes the form of an elliptic paraboloid in volumetric-deviatoric strain space, where
the energy landscape of the overall material is obtained from the contributions of the individual
constituents.
For the evolution of volume fractions, we use an approach based on statistical physics—taking
into account actual Gibbs energy barriers and transformation probabilities [2]. The computation
of individual energy barriers between the phases considered is enabled by numerical minimization
of parametric intersection curves of elliptic Gibbs energy paraboloids. The framework provided
facilitates to take into account an arbitrary number of solid phases of the underlying material,
though we restrict ourselves to the simulation of three phases, namely an austenitic parent phase
and a martensitic tension and compression phase. It is shown that the model presented nicely
reflects the temperature-dependent effects of pseudo-elasticity and pseudo-plasticity, and thus
captures experimentally observed behaviour at different temperatures.

14 Section 6: Material modelling in solid mechanics
[1] I. Carol and Z. Bažant, Damage and plasticity in microplane theory, Int. J. Sol. Struct. 34,
3807–3835 (1997)
[2] S. Govindjee and G.J. Hall, A computational model for shape memory alloys, Int. J. Sol.
Struct. 37, 735–760 (2000)
[3] R. Ostwald, T. Bartel and A. Menzel, A computational micro-sphere model applied to the
simulation of phase-transformations, J. Appl. Math. Mech. 90(7-8), 605–622 (2010)
S6.6: Elasticity, Viscoelasticity, -plasticity I Wed, 13:30–15:30
Chair: Merab Svanadze, Daniel Balzani S1|01–A1
Thermoviscoplasticity deduced from enhanced rheological models
Christoph Bröcker, Anton Matzenmiller (Universität Kassel)
In the concept of rheological models, basic elements like springs (ideal elastic), dashpots (ideal
viscous) or friction elements (ideal plastic) are assembed to networks for representing complex
material behaviour [1]. In case of viscoelastic material models, the phenomenological constitutive
equations are usually deduced directly from a rheological network. However, in the case of elastoplasticity,
the rheological models are often used only to visualise the fundamental structure of
the related material model.
In the presentation, a new ideal body is defined for isotropic hardening besides minor modifications
of some well–known basic elements from the literature [2, 3]. Hence, a rheological model of
thermoviscoplasticity may be assembled with linear isotropic and kinematic hardening and nonlinear
strain rate sensitivity. The related constitutive equations including the yield function and the
flow rule are directly deduced from the kinematics and the stress equilibrium of the rheological
network and results in a well–known model. By evaluating the dissipation inequality, the heat
conduction equation is obtained with the dissipative power term, driven by plastic deformations.
Moreover, nonlinear isotropic and kinematic hardening as well as an improved description
of energy storage and dissipation are accomplished by introducing several additional dissipative
strain elements into the viscoplastic arrangement of the rheological model [see also 4], which
corresponds to a generalization of an ideal body already used in [2, 3]. Again, the constitutive
equations may be deduced from the rheological network, its kinematics, and the stress equilibrium.
For that purpose, however, the dissipation inequality has to be utilized.
[1] M. Reiner, Rheologie in elementarer Darstellung, Carl Hanser Verlag, 1969.
[2] A. Krawietz, Materialtheorie, Springer, 1986.
[3] A. Lion, Constitutive modelling in finite thermoviscoplasticity: a physical approach based on
nonlinear rheological models, Int. J. Plast. 16 (2000), 469–494.
[4] A. Matzenmiller, C. Bröcker, Modelling and simulation of coupled thermoplastic and thermoviscous
structuring and forming processes, In: Maier et al. (eds.): Functionally graded
materials in industrial mass production, Verlag Wissenschaftliche Skripten, Auerbach, 2009,
235–250.

Section 6: Material modelling in solid mechanics 15
Steady vibrations problems in the theory of viscoelasticity for Kelvin-Voigt materials
with voids
Maia M. Svanadze (Universität Göttingen)
Viscoelastic materials play an important role in many branches of engineering, technology and
biomechanics. The modern theories of viscoelasticity and thermoviscoelasticity for materials with
microstructure have been a subject of intensive study in recent years. Recently, the theory of
thermoviscoelastic materials with voids is constructed by Iesan (2011).
In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with voids is considered
and the basic internal and external boundary value problems of steady vibrations are investigated.
The formulae of integral representations of regular vectors are obtained. The single-layer,
double-layer and volume potentials are constructed and their basic properties are established.
The uniqueness and existence of regular solutions of the boundary value problems are proved by
means of the potential method.
Modelling of predeformation- and frequency-dependent material behavior of filled
rubber under large predeformations superimposed with harmonic deformations of
small amplitudes
D. Wollscheid, A. Lion (Universität der Bundeswehr München)
Viscoelastic materials show a frequency- and predeformation-dependent behavior under loadings
that consist of large predeformations with superimposed harmonic deformations of small amplitudes.
In order to consider this materialbehavior, some static and dynamic experiments are
developed. Based on Haupt & Lion [1] and Lion, Retka & Rendek [2] we introduce a recently
developed constitutive approach of finite viscoelasticity in the frequency domain that is able to
describe the frequency- and predeformation-dependent materialbehavior with respect to storageand
loss-modulus. The constitutive equations are evaluated in the frequency domain and geometrically
linearized in the neighbourhood of the predeformation. Furthermore a formulation for
incompressible material behavior is introduced and the corresponding dynamic modulus tensors
are derived. Besides constitutive modelling and experiments, parameter identification and some
numerical simulations are presented.
[1] P. Haupt, A. Lion, On finite linear viscoelasticity of incompressible isotropic materials, Acta
Mechanica 159 (2002), 87 – 124.
[2] A. Lion, J.Retka, M.Rendek On the calculation of predeformation-dependent dynamic modulus
tensors in finite nonlinear viscoelasticity, Mechanics Research Communications 36
(2009), 653 – 658.
A multi-scale modelling approach for bituminous asphalt
Thorsten Schüler, Ralf Jänicke, Holger Steeb (Universität Bochum)
Bituminous asphalt is a standard material e.g. in road constructions. However, the term asphalt
involves an extremly broad class of complex multi-scale and multi-phase materials. Typically,
asphalt consists of a mineral filler (e.g. crushed rock), a bituminous binding agent (possibly including
further additive compounds) and pores. The various constituents are to be adapted for
the particular application.
Nowadays, requirements on noise reduction of road constructions become more and more important.
In our ongoing research activities, we focus on the solid-borne acoustic properties of the

16 Section 6: Material modelling in solid mechanics
asphalt cover layer, i.e. the top layer of the entire asphalt-construction. In order to predict the
macro-scale effective material properties of asphalt we apply a numerical homogenisation scheme
based on volume averaging techniques. The main advandtage of this numerical approach is, that
the effective material properties can be determined in knowledge of the micro-scale properties.
Hence, the first step towards an overall model for bituminous asphalt is the quantification of
the micro-scale mechanical properties. Against the background of solid-borne acoustical properties
we restrict ourselves to a geometrically linear description involving only small deformations
on both, macro- and micro-scale. Doing so, the linear-elastic properties of the stiff, granular filler
can be adopted from relevant literature. The viscoelastic behaviour of the bituminous phase is
characterized by rheological experiments (dynamic shear rheometer with plate-plate geometry).
The numerical implementation on the micro-scale is realized using a generalized Zener model (3d).
Making use of the numerical homogenization approach, the effective viscoelastic properties on
the macro-scale are investigated within transient experiments (relaxation/creep test). In particular
we are interested in the particular relaxation mechanisms and the related characteristic frequencies
to be observed on the macro-scale. The influence of micro-scale boundary conditions will be
taken into account. In order to study the interaction between micro-constituents as well as their
geometrical morphology on the one hand and the effective viscoelastic properties on the other,
we introduce artificially produced periodic unit cells based on simplified geometries with varying
volume and surface fractions of the mineral filler.
Jumps of the critical tracking loadings for viscoelastic beams with vanisihing internal
viscosity
S.A.Agafonov (Moscow State Technical University), D.V.Georgievskii (Moscow State University)
Transverse vibrations of a viscoelastic beam under action of a tracking loading are considered.
The constitutive relation represents the following non-linear connection of stress σ(t), strain ε(t)
and strain rate ˙ε(t):
σ = Eε +
where E is the Young modulus, k (n)
i
N
n
n=1 i=1
k (n)
i ε2(n−i) ˙ε 2i−1 , N ≥ 1
> 0 are the coefficients of internal viscosity.
Analytical and numerical investigation of dynamic stability in case N = 3, k (1)
1 = 0, k (2)
1 = 0,
k (1)
2 = 0 shows that if k (3)
α → 0 (k (3)
β
= 0, k(3)
γ = 0; (α, β, γ) may be some permutation of (1, 2, 3))
then three critical values of tracking loading corresponding to each nonzero coefficient of viscosity
k (3)
α less than the value for elastic system by a finite quantity.
Numerical modeling of a non-linear viscous flow in order to determine how parameters
in constitutive relations influence the entropy production
Wolfgang H. Müller, B. Emek Abali (TU Berlin)
Some rheological materials like melting polymers, cosmetic creams, ketchup, toothpaste can be
modeled as non-Newtonian fluids by using a non-linear constitutive relation. Flow of this kind
of amorphous matter can be considered as a thermodynamic process, and a solution of pressure,
velocity and temperature fields describe it fully. Since flow processes are generally irreversible,
entropy is produced leading to dissipation in the system. This energy loss can be measured
indirectly in a cone/plate viscometer which is used to determine viscosity of a Bingham fluid.
While dissipation is a measurable quantity we want to be able to calculate it. Thus the goal of
this work is to explain how to calculate entropy production using balance equations in a spatial

Section 6: Material modelling in solid mechanics 17
frame.
Starting from balance of mass, linear momentum, internal energy and employing method
of weighted residuals, we get a non-linear coupled set of partial differential equations in space
and time. A space discretization in finite elements method and a time discretization in finite
difference method leads to an approximation after a successful linearization. We achieve to solve
the problem without any stabilization or usage of specific type of elements and show here the
entropy production and its variation subject to material parameters for the sake of a better
intuitive understanding of dissipation. This may lead to an inverse problem where the calculated
dissipation is measured and material parameters are determined out of it, which is left to future
research.
S6.7: Phase Transformations II Wed, 16:00–18:00
Chair: Wolfgang Dreyer S1|01–A03
Nonsingular Dislocation Loops in Gradient Elasticity
Markus Lazar (TU Darmstadt)
This work studies the fundamental problem of nonsingular dislocations in the framework of the
theory of gradient elasticity. A general theory of nonsingular dislocations is developed for linearly
elastic, infinitely extended, homogeneous, isotropic media. Using gradient elasticity, we give the
nonsingular fields produced by arbitrary dislocation loops. We present the ‘modified’ Mura, Peach-
Koehler and Burgers formulae in the framework of gradient elasticity theory. These formulae are
given in terms of an elementary function, which regularizes the classical expressions, obtained
from the Green tensor of generalized Navier equations. Using the mathematical method of Green’s
functions and the Fourier transform, we found exact, analytical and nonsingular solutions. The
obtained dislocation fields are nonsingular due to the regularization of the classical singular fields.
On a paradox within the phase field modeling of storage systems and its resolution
Clemens Guhlke, Wolfgang Dreyer (WIAS Berlin)
We study two import storage problems: The storage of lithium in an electrode of a lithium-ion
battery and the storage of hydrogen in hydrides.
When foreign atoms are reversibly stored in a crystal, there may be a regime where two coexisting
phases with low and high concentration of the stored atoms occur. Furthermore hysteretic
behavior can be observed, i.e. the processesof loading and unloading follow different paths.
We apply a viscous Cahn-Hilliard model with mechanical coupling to calculate the voltagecharge
diagram of the battery, respectively the pressure-charge diagram of a hydrogen system.
The diagrams exhibit phase transition and hysteresis. However, we show that the model can only
describe the observed phenomena for fast but nor for slow loading.
We relate the reason for failure to the microstructure of modern storage systems. In fact these
consist of an ensemble of nano-sized interconnected storage particle. Each particle is described by
a non-monotone chemical potential function but on the time scale of slow loading of the ensemble,
the coexisting phases are unstable within an individual particle and cannot be observed on the
time scale of the loading.
In the slow loading regime, the occurence of two coexisting phases phases is a many-particle
effect of the ensemble. The nucleation and evolution of the phases is embodied by a nonlocal
conservation law of Fokker-Planck type.
Numerical Simulation of Coarsening in Metallic Alloys
Uli Sack, Carsten Gräser, Ralf Kornhuber (FU Berlin)

18 Section 6: Material modelling in solid mechanics
Phase separation phenomena in alloys, such as spinodal decomposition and Ostwald ripening
can be described by phasefield models of Cahn-Hilliard-type. Realistic models based on thermodynamically
correct logarithmic free energies contain highly nonlinear and singular terms as
well as drastically varying length scales (cf. [1]). We present a globally convergent nonsmooth
Schur-Newton Multigrid method for vector-valued Cahn-Hilliard-type equations ([2, 3]) and a
numerical study of coarsening in a binary eutectic AgCu alloy taking into account elastic effects.
Vector-valued Cahn-Hilliard computations open the perspective to multicomponent simulations.
[1] W. Dreyer, W.H. Müller, F. Duderstadt and T. Böhme, “Higher Gradient Theory of Mixtures”,
WIAS-Preprint No 1286 (2008)
[2] C. Gräser and R. Kornhuber, “Nonsmooth Newton Methods for Set-valued Saddle Point
Problems”, SIAM J. Numer. Anal. (2009) 47 (2)
[3] C. Gräser, R. Kornhuber, “Schur-Newton Multigrid Methods for Vector-Valued Cahn–Hilliard
Equations”, in prep
A Phase Field Model for Martensitc Transformations
Regina Schmitt, Ralf Müller, Charlotte Kuhn (TU Kaiserslautern)
Considering the microscopic level of steel, there are different structures with different mechanical
properties. Under mechanical deformation the metastable austenitic face-centered cubic phase
transforms into the tetragonal martensitic phase at which transformation induced eigenstrain
arises. On the other hand, the mircostructure affects the macroscopic mechanical behavior of
the specimen. In order to take the complex interactions into account, a phase field model for
martensitic transformation is developed. Within the phase field approach, an order parameter
is introduced to indicate different material phases. Its time derivative is assumed to follow the
time-dependent Ginzburg-Landau equation. The coupled field equations are solved using finite
elements together with an implicit time integration scheme. With the aid of this model, the effects
of the elastic strain minimization on the formation of microstructure can be studied so that the
evolution of the martensitic phase is predictable. The applicability of the model is illustrated
through different numerical examples.
Investigation of the strain localization behavior with application of the phase transition
approach
M. Ievdokymov, H. Altenbach, V.A. Eremeyev (Universität Magdeburg)
Metal foams found recently many applications in civil, airspace and mechanical engineering. In
particular, foams have a good energy absorption property. This property relates to the phenomenon
of localization of strains in foams under loading. In porous materials the localization of
strains leads to change of mass density of material and appearance of areas with low and high
densities separated by sharp interface. This behavior is similar to the phase transitions in solids
with sharp interfaces.
In this paper we use the methods of modelling of phase transitions in solids to the description of
strain localization in foams. We assume that the foam consist of two phases with different densities
and mechanical properties. The results of modelling of one- and two-dimensional problems are
discussed. The calculations are performed by Abaqus and script language Python.
S6.8: Elasticity, Viscoelasticity, -plasticity II Wed, 16:00–18:00
Chair: Ismail Caylak S1|01–A1

Section 6: Material modelling in solid mechanics 19
A New Continuum Approach to the Coupling of Shear Yielding and Crazing with
Fracture in Glassy Polymers
Lisa Schänzel, Christian Miehe (Universität Stuttgart)
Over the past decades, considerable effort was made to develop constitutive models that account
for finite viscoplasticity and failure in glassy polymers. Recently, we developed a new model
of ductile thermoviscoplasticity of glassy polymers in the logarithmic strain space [1]. However,
depending on thermal and loading rate conditions to which the material is subjected, the response
might change from ductile to brittle. This brittle response is characterized by inelastically deformed
zones, so-called crazes, having the thickness of micrometers and spanning at some fractions of
a millimeter [2]. The crazing is associated with considerable dilatational plasticity, containing a
dense array of fibrils interspersed with elongated voids. The shear yielding and crazing are not
completely independent excluding each other. In this lecture, we outline an extension of the ductile
plasticity model [1] towards the description of (i) volumetric directional plasticity effect due to
crazing and (ii) the modeling of the local failure due to fracture. To this end, the first extension
accounts for a (i) dilatational plastic deformation mechanism in the direction of the maximum
principle tensile stress. The ultimate amount of this volumetric plastic craze strain is bounded
by a limiting value, where failure occurs. Then, in a second step, the (ii) modeling of subsequent
failure mechanisms is realized by the introduction of a fracture phase field, characterizing via an
auxiliary variable the crack topology. Here, we adopt structures of a recently developed continuum
phase field model of fracture in brittle solids [3], and modify it for a fracture driving term related
to the volumetric plastic deformation of the crazes. We demonstrate the performance of proposed
formulation by means of representative boundary value problems.
[1] Miehe, C.; Mendez, J.; Göktepe, S.; Schänzel, L. [2011]: Coupled thermoviscoplasticity of
glassy polymers in the logarithmic strain space based on the free volume theory. International
Journal of Solids and Structures, 48: 1799–1817.
[2] Kramer, E. J.[1983]:Microscopic and Molecular Fundamentals of Crazing. Advances in Polymer
Science, 52/53: 1–56.
[3] Miehe, C; Hofacker, M.; Welschinger, F. [2010]: A phase field model for rate-independent
crack propagation: Robust algorithmic implementation based on operator splits. Computer
Methods in Applied Mechanics and Engineering, 199: 2765-2778.
Anisotropic finite strain hyperelasticity based on the multiplicative decomposition of
the deformation gradient
Raad Al-Kinani, Kaveh Talebam, Stefan Hartmann (TU Clausthal)
Frequently, the case of finite strain anisotropy, particularly, the case of transversal isotropy, is
applied to biological applications or to model fiber-reinforced composite materials. In this article
the multiplicative decomposition of the deformation gradient into one part constrained in the direction
of the axis of anisotropy and one part describing the remaining deformation is proposed.
Accordingly, a form of additively decomposed strain-energy function is proposed. This leads to a
clear assignment of deformation and stress states in the direction of anisotropy and the remaining
part. The decomposition of the case of transversal isotropy is explained. The behavior of the
model is investigated at different simple analytical examples, such as uniaxial tension along and
perpendicular to the axis of anisotropy, and simple shear. In addition, the model is also studied for
the case of a thick-walled tube under internal pressure, where a second order ordinary differential

20 Section 6: Material modelling in solid mechanics
equation (two-point boundary-value problem) is obtained.
Experimental characterization of the viscoelastic behaviour of discontinuous glass
fibre reinforced thermoplastics
B. Brylka, T. Böhlke (KIT)
In automotive applications short and long glass fibre reinforced thermoplastics are commonly used
for non-structural parts. Due the versatile possibilities of manufacturing, forming, joining and
recycling, thermoplastic matrix based composites are increasingly used also for semi-structural
parts. Thermoplastics like e.g. polypropylene show a high temperature and strain-rate dependency,
especially in the temperature and strain-rate ranges which are relevant for automotive
applications. Additionally, the non-linear influence of the viscoelastic behaviour of the matrix
material on the effective material behaviour of the composite is of high interest.
The dynamic mechanical analysis (DMA) technique is an effective method to investigate the
elastic and viscoelastic stiffness response of materials under cyclic loading. After a short introduction
into the DMA technique, experimental results for a polypropylene and polypropylene based
composite material will be presented. The composite under consideration is an discontinuous glass
fibre reinforced polypropylene. In the manufacturing process, which is commonly compression or
injection moulding, the flow of the mould induces an heterogeneous and anisotropic distribution
of fibre orientations. Therefore, the effective properties as well as the temperature and strain-rate
dependency has been investigated taking into account an anisotropic material behaviour. The
comparison of the elastic and viscoelastic material response of the matrix and the composite will
be discussed in detail. Additionally, a parameter identification for common viscoelastic material
models will be presented.
[1] Middendorf, P.: Viskoelastisches Verhalten von Polymersystemen, Fortschritt-Berichte VDI,
Reihe 5, VDI Verlag (2002).
[2] Deng, S., Hou, M., Ye, L.: Temperature-dependent elastic moduli of epoxies measured by
DMA and their correlations to mechanical testing data, Polym. Test., 26, 803-813 (2007).
[3] Schledjewski, R., Karger-Kocsis, J.: Dynamic mechanical analysis of glass mat-reinforced
polypropylene (GMT-PP), J. Thermoplast. Compos., 7, 270-277 (1994).
A solid-shell finite element for fibre reinforced composites
J.-W. Simon, B. Stier, S. Reese (RWTH Aachen)
Fibre reinforced composites are typically characterized by high Young’s modulus at low density,
which makes them very attractive for lightweight constructions. The fibre composites considered
here consist of several layers, each of which is composed of a woven fabric embedded in a matrix
material. This structure makes the constitutive behavior of fibre composites anisotropic. Moreover,
it is generally highly nonlinear, and the materials’ response in tension and compression can
differ significantly. In order to describe this rather complex behavior, we use a modification of
a micromechanically motivated model proposed by Reese [1]. Therein, an anisotropic model has
been presented for the hyperelastic material behavior of membranes reinforced with roven-woven
fibres, which is particularly suitable for the present fibre reinforced composites.
The use of a fully three-dimensional material model strongly suggests using solid elements.
On the other hand, fibre composites are mostly applied in thin shell-like structures, where shell

Section 6: Material modelling in solid mechanics 21
elements should usually be preferred. Therefore, we use a solid-shell element presented by Schwarze
and Reese [2] which combines the advantages of both solid elements and shell elements at the
same time. This element allows for displaying realistically the three-dimensional geometry while
still providing the suitable shape for thin structures.
In addition, the present solid-shell formulation utilizes a reduced integration scheme within the
shell plane using one integration point, whereas a full integration is used in thickness direction.
Thus, an arbitrary number of integration points can be chosen over the shell thickness. Thereby,
different fibre orientations of the layers can be taken into account easily, since the material
parameters can be defined for each integration point separately.
Moreover, the proposed solid-shell formulation removes all potential locking phenomena. In
particular, volumetric locking in case of (nearly) incompressible materials as well as Poisson
thickness locking in bending problems of shell-like structures are eliminated by use of the enhanced
assumed strain (EAS) concept. In addition, to cure the transverse shear locking which is present
in standard eight-node hexahedral elements, the assumed natural strain (ANS) method is applied.
[1] S. Reese. A micromechanically motivated material model for the thermo-viscoelastic material
behaviour of rubber-like polymers, Int J Plast, 19, 909–940, 2003.
[2] M. Schwarze, S. Reese. A reduced integration solid-shell finite element based on the EAS and
the ANS concept - large deformation problems, Int J Numer Methods Engng, 85, 289–329,
2011.
On consistent tangent operator derivation and comparative study of rubber-like material
models at finite strains
Mokarram Hossain, Paul Steinmann (Universität Erlangen-Nürnberg)
The overall micro-structure of rubber-like materials can be idealized by chain-like macromolecules
which are connected to each other at certain points via entanglements or cross-links. Such
special structure leads to a completely random three-dimensional network [2,3]. To model the
mechanical behaviour of such randomly-oriented micro-structure, several phenomenological and
micro-mechanically motivated network models for nearly incompressible hyperelastic polymeric
materials have been proposed in the literature. To implement these models for polymeric material
(undoubtedly with widespread engineering applications) in finite element method, one would
require two important ingredients, e.g. the stress tensor and the consistent fourth-order tangent
operator where the latter is the result of linearization of the former.
In this contribution, an extensive overview on several hyperelastic rubber-like material models
has been presented. Special focus is given particularly to derive the accurate stress tensors and
tangent operators which yield quadratic convergence when the governing nonlinear equations for
a boundary value problem are solved by the Newton-like iterative schemes. A simple but efficient
algorithm will be demonstrated to testify the correctness of the tangent operator locally of a
particular model without going into details of the finite element implementation [1].
[1] Steinmann P, Hossain M, Possart G (2011), Hyperelastic models for rubber-like materials:
Consistent tangent operators and suitability for Treloar’s data. Archive of Applied Mechanics,
In review (2011)
[2] Boyce MC, Arruda EM (2000), Constitutive models of rubber elasticity: a review. Rubber
Chemistry and Technology, 73: 504-523, 2000

22 Section 6: Material modelling in solid mechanics
[3] Marckmann G, Verron E (2006), Comparison of hyperelastic models for rubber-like materials.
Rubber Chemistry and Technology, 79 (2006) 835-858
Accelerating constitutive modeling by automatic tangent generation
Steffen Rothe, Stefan Hartmann (TU Clausthal)
Nowadays models become more and more complex. Within the framework of finite elements a
material, i.e. consistent, tangent is required for the overall Newton-like method. Obtaining these
derivatives is time consuming and error-prone which contradicts to the goal of changing the model
during the developing process. On the one hand the calculation by hand can be very expensive
and on the other hand also the implementation has to be done with high diligence. Therefore, a
fast and safe way of consistent tangent generation will be presented with the help of automatic
differentiation (AD) techniques.
Analytical tangents have the advantage that they are exact. However during the constitutive
modeling process a change of the model is natural. Thus, the effort is very high to calculate
the derivative after every modification. Numerical tangents computed by finite differences are
easy to compute, but can lead to a significant slowdown or even to a failure of the simulation
due to numerical errors. Tangents generated by OpenAD have no round-off errors and are easily
computed by the help of automatic differentiation.
These three methods (analytical, numerical and automatic differentiation) for tangent generation
will be analyzed concerning the simulation time and applicability for a number of different
constitutive models.
S6.9: Microheterogeneous Materials Thu, 13:30–15:30
Chair: Johannes Schnepp, Eleni Agiasofitou S1|01–A03
The boundary value problems of the full coupled theory of poroelasticity for materials
with double porosity
Merab Svanadze (Ilia State University, Tbilisi)
Porous materials play an important role in many branches of engineering, e.g., the petroleum industry,
chemical engineering, geomechanics, and, in recent years, biomechanics. The construction
and the intensive investigation of the theories of continua with microstructures arise by the wide
use of porous materials into engineering and technology.
The general 3D theory of poroelasticity for materials with single porosity was formulated by
Biot (1941). The double porosity model was first proposed by Barenblatt and coauthors (1960).
The quasi-static theory of poroelasticity for materials with double porosity in the framework of
mixture theory was presented by Aifantis and his co-workers (1982).
In this paper the full coupled theory of poroelasticity for materials with double porosity is
presented. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation
with double porosity. The boundary value problems (BVPs) of the steady vibrations are investigated.
The fundamental solution of system of equations of steady vibrations is constructed. The
basic properties of plane waves and the radiation conditions for regular vector are established. The
uniqueness theorems of the internal and external BVPs of steady vibrations are proved. The basic
properties of elastopotentials are established. The representation of general solution of equations
of steady vibrations is obtained. The existence of regular solution of the BVPs by means of the
boundary integral method and the theory of singular integral equations are proved.

Section 6: Material modelling in solid mechanics 23
Application of an anisotropic growth and remodelling formulation to computational
topology optimisation
Tobias Waffenschmidt (TU Dortmund), Andreas Menzel (TU Dortmund / Lund University)
A classical topology optimisation problem consists of a problem-specific objective function which
has to be minimised in consideration of particular constraints with respect to design and state
variables. In this contribution we present a conceptually different approach for the optimisation or
rather improvement of the topology of a structure which is not based on a classical optimisation
technique. Instead, we establish a constitutive micro-sphere-framework in combination with an
energy-driven anisotropic microstructural growth formulation, which was originally proposed for
the simulation of adaptation and remodelling phenomena in hard biological tissues such as bones.
The key aspect of this contribution is to investigate this anisotropic growth formulation with
a special emphasis on its topology-optimising characteristics or rather topology-improving properties.
To this end, several illustrative three-dimensional benchmark-type boundary value problems
are discussed and compared qualitatively with the results obtained by classic topologyoptimisation
strategies. The simulation results capture the densification effects and clearly identify
the main load bearing regions. It turns out, that even though making use of this conceptually
different growth formulation as compared to the procedures used in the more classic topologyoptimisation
context, we identify qualitatively very similar topologies. Moreover, in contrast to
common topology optimisation strategies, which mostly aim to optimise merely the structure,
i.e. size, shape or topology, our formulation also contains the optimisation or improvement of the
material itself, whichapart from the structural improvementresults in the generation of problemspecific
local material anisotropy and textured evolution.
Amplification damping properties of multiphase composites with spherical and fibers
inclusions
Sergey Lurie (Institute of Applied Mechanics, RAS), Natalia Tuchkova (Dorodnicyn Computing
Centre, RAS), Juri Soliaev (Institute of Applied Mechanics, RAS)
We consider composite materials reinforced with spherical and fibrous inclusions coated with a
layer of lossy viscoelastic material. For the coating layers, typical viscoelastic properties of a polymer
at and well above the glass transition region are assumed. It is shown that the remarkable loss
amplification mechanism is also operative in such particulate-morphology materials. The aim of
our study is to determine the effective loss modulus composites, which formally defines the rate
of energy dissipation per unit volume. We use a combination of modeling and numerical tools
to study composite materials consisting of a matrix filled with spherical and fibrous inclusions
coated with a layer of lossy viscoelastic material.
The method of the four phases is used to describe the damping properties of composites
filled with multiphase spherical inclusions and monolayer with multiphase fibrous inclusions. The
constituent phases are supposed to be isotropic. Using the Eshelby method the effective moduli
are determined self-consistently, by requiring that the average strain in the composite inclusion
is the same as the macroscopic strain imposed at infinity. The analytical solution of this problem
were received. The effective dynamic modulus and loss modulus were found using a viscoelastic
analogy.
It is shown[1] that the analytical give very similar, practically indistinguishable predictions
from the numerical solutions which were received using finite element method. Hence, when
studying such systems one can rely on the predictions of the four-phase sphere model, which
are much easier to achieve than the finite element ones. a polymer at and well above the glass
transition region are assumed. It is shown that by optimizing the thickness of the layers, one can
achieve multiphase materials with effective loss characteristics significantly exceeding those of the

24 Section 6: Material modelling in solid mechanics
individual materials constituents. It was found that for the considered composites have place the
additional peak damping properties, which is realized for very small thicknesses of the viscoelastic
phase. This peak is still about a factor 20 or so compared to the loss modulus of the pure matrix.
We demonstrated that by introducing thin layers of a viscoelastic material, one can significantly
increase the loss characteristics of discrete morphology multiphase materials. The layered fibers
composites are also considered. It is shown that for such materials may be implemented at the
same time as high elastic properties, and abnormally high damping properties.
[1] A.A.Gusev , S.A. Lurie, Loss amplification effect in multiphase materials with viscoelastic
interfaces. Macromolecules (2009) 42,14, 5372 – 5377.
Modeling of composite structural elements made from aramid fiber using the method
of features objects
Michał Majzner, Andrzej Baier (Silesian University of Technology)
The use of modern materials such as composite materials, enabling the production of new or modifying
existing design solutions, improve their technical characteristics, while use in the process
of design and engineering and manufacturing, will allow the modification of endurance, physical
and chemical properties to match the features and functionality that are comply with. In research
studies, it is proposed to systematize and formalize elementary objects in the context of modeling
and fabrication of objects created on the basis of structural fiber composites. Application of features
objects methods was shows on an example of modeling the structural element, in the form
of an existing manufactured from steel, which was modified and converted with aramid fiber. It
was necessary to carry out research in the form of numerical analysis, examining the strength of
the modified object.
On the fibres shape effect for non-linear and unidirectional stationary heat conduction
in two-phase hollow cylinder with radially graded material properties
Piotr Ostrowski (Technical University of Lodz)
The main aim of this paper is to consider unidirectional and stationary heat conduction in the
infnite two-phase hollow cylinder with temperature dependent material properties. The deterministic
microstructure of this composite is periodic (for a fixed radius) along the angular axis
and has slowly varying effective properties in the radial direction. Therefore, we deal here with
a special case of functionally graded materials, FGM, c.f. Suresh, Mortensen (1998). One of the
components is called fibre, which is arranged in considered hollow cylinder with circular pattern.
The physical phenomenon of the heat transfer is described by well known Fourier’s equation
c ˙ θ − ∇(K∇θ) = 0, (1)
which contains temperature dependent (in this case), highly oscillating and discontinuous coefficients
of K = K(θ) - heat conduction tensor, and c = c(θ) - specific heat. To this macroscopic
model the tolerance averaging approximation will be used, cf. Wozniak, Wierzbicki (2000). The
general approach to the description of longitudinally graded stratified media can be found in
[Wozniak, Michalak, Jedrysiak 2008]. The fibres width function g = g(r) of radius r will be examined
and its effects on the temperature field. The averaged differential equation has smooth
and slowly varying coefficients, hence in some special cases, for boundary value problem, analytical
solution can be obtained. In other cases, numerical methods have to be used. This model
takes into account the effect of microstructure size on the overall heat transfer behaviour.

Section 6: Material modelling in solid mechanics 25
[1] S. SURESH, A. MORTENSEN, Fundamentals of functionally graded materials, Cambridge,
The University Press, 1998.
[2] Cz. WOZNIAK, B. MICHALAK, J. JEDRYSIAK (eds), Thermomechanics of micro-heterogeneous
solids and structures. Tolerance averaging approach, Wydawnictwo Politechniki Lodzkiej,
Lodz, 2008.
S6.10: Elasticity, Viscoelasticity, -plasticity III Thu, 13:30–15:30
Chair: Daniel Balzani S1|01–A1
Network evolution model: thermodynamics consistency, parameter identification and
finite element implementation
Vu Ngoc Khiêm, Roozbeh Dargazany, Mikhail Itskov (RWTH Aachen)
In this contribution, the previously proposed network evolution model [1] for carbon black filled
elastomers is further studied. First, we show that the model does not contradict the second law of
thermodynamics and is thus thermodynamically consistent. On the basis of new experimental data
the influence of filler concentration on the material parameters is further examined. Accordingly,
this influence concerns only three material parameters and is approximated by phenomenological
relations. These relations enable one to simulate rubbers based on the same compound with
various filler concentrations. Finally, the model is implemented to the FE-Software ABAQUS and
illustrated by a number of numerical examples. The examples demonstrate good agreement with
experimental results with respect to the Mullins-Effect, permanent set and induced anisotropy.
[1] R. Dargazany and M. Itskov, A network evolution model for the anisotropic Mullins effect
in carbon black filled rubbers, International Journal of Solids and Structures 46 (2009),
2967–2977.
Shakedown analysis of periodic composites with kinematic hardening material model
Min Chen, A. Hachemi, D. Weichert (RWTH Aachen)
Lower-bound limit and shakedown analysis of periodic composites with the consideration of kinematic
hardening are investigated on the representative volume element. With the combination of
homogenization theory, the homogenized macroscopic admissible loading domains are evaluated.
Furthermore, the strengths of periodic composites of elastic-perfectly plastic, unlimited and linear
limited kinematic hardening material models are calculated and compared in this paper.
Theory of mixture based material modeling - An inelastic material model for a 12%chromium
steel
Andreas Kutschke, Konstantin Naumenko, Holm Altenbach (Universität Magdeburg)
Components composed of advanced heat resistant steels face a complex loading of mechanical and
thermal stresses under cyclic and long-term conditions. Additionally, a failure of these components
usually has serious outcome, because of the extreme working conditions. From this follows the
need of a reliable material model for the construction process.
Classical technical guidelines are historically grown and the experience of engineers contributed
to their development, but it turns out that these classical guidelines face their limit of use. Recently
more sophisticated material models are developed to reach the real limit of the materials and to
predict their behavior with more accuracy.

26 Section 6: Material modelling in solid mechanics
Starting with the theoretical concept of Prandtl, who introduced the idea of a hardening
substance in a material, and the investigations of Mughrabi, who established a composite model
for chromium steels, the advanced steels are treated as a composition of at least two continua, in
the sense of the theory of mixture.
Therefore the general balance equations for a mixture will be presented. The example of a
12% chromium steel will be used to illustrate this approach and to derive a material model for
a mixture of two constituents. It will be shown that the model predicts tension-, compression-,
cyclic-creep, uniaxial monotonic tension und low-cycle-fatigue loading in a satisfying manner in
comparison with experimental results.
Zur Abtragssimulation beim Strömungsschleifen (AFM)
Joachim Schmitt, Stefan Diebels (Universität des Saarlandes)
Im Zuge konkurrenzfähiger Produktionsverfahren werden seit einigen Jahren schwer zugängliche
Kanten im Inneren von komplex aufgebauten Fertigungsteilen (wie beispielsweise die Gehäuse von
Einspritzanlagen) mittels des Strömungsschleifens (Abrasive flow machining - AFM) verrundet
bzw. oberflächenvergütet. Dazu wird eine mit Schleifpartikeln versetzte Silikonpaste mehrfach
durch das Bauteil gepresst. Das viskose Verhalten der Paste ist dabei für die effektive Wirkung
dieses Verfahrens sehr wichtig. Beim Überströmen von Kanten steigt die Schergeschwindigkeit und
damit auch die Viskosität je nach Materialmodell überproportional an. Die so geänderten Druckund
Geschwindigkeitsverhältnisse an der Bauteiloberfläche verändern auch den Materialabtrag
durch die Schleifpaste.
Im Rahmen dieser Studie werden zunächst die viskosen Eigenschaften der Paste anhand von
Experimenten untersucht und die Modellparameter bestimmt. Im zweiten Teil wird die Paste
hinsichtlich ihrer abrasiven Eigenschaften vorgestellt und eine Abschätzung des Materialabriebs
vorgenommen. Ein daraus gebildetes einfaches Abtragsmodell wird mittels einer FEM-Simulation
an elementaren Bauteilgeometrien umgesetzt und qualitativ überprüft.
Material parameter identification using model reduction to uniaxial tensile tests
Stephan Krämer, Steffen Rothe, Stefan Hartmann (TU Clausthal)
Uniaxial tensile tests are commonly used for material parameter identification. It is also common
to use one-dimensional formulations of a constitutive model to identify the corresponding
material constants, although these material parameters are not necessarily identical to the material
parameters in the three-dimensional material model. We present an easy way to reduce a
three-dimensional material routine to the case of uniaxial tension and to use the reduced form
to derive material parameters using common trust-region algorithms. With this approach one
is able to use the full three-dimensional model for parameter identification. Instead of using a
full finite element software, one is able to use the material driver routine, which leads to a more
straight forward calculation of material parameters. In this respect, it is shown that the classical
structure of stress algorithms and consistent tangent operators are necessary within the threeto
one-dimensional problem reduction. This procedure is also extendable to further homogeneous
stress- and strain-states.
Systematic representation of the yield criteria for isotropic materials
Vladimir A. Kolupaev (DKI Darmstadt), Holm Altenbach (Universität Magdeburg)
The theory of plasticity operates with different flow criteria of incompressible material behavior.
These criteria have hexagonal symmetry in the π-plane and do not distinguish between tension
and compression (non-SD-effect). Many tasks in the engineering practice are treated on the basis

Section 6: Material modelling in solid mechanics 27
of these criteria and the flow rule.
Selection of an appropriate criterion for a specific material is challenging. The models of Tresca
and Schmidt-Ishlinsky, representing two regular hexagons in the π-plane, define accordingly the
lower and the upper limit of convexity. The criteria of Sokolovskij and Ishlinsky-Ivlev describe
the regular dodecagons in the π-plane and have only geometrical meaning. In difference to these
four criteria, the model of von Mises has no singular corners and delivers unique results by the
strain rates calculation.
The evaluation of the measurements shows that the material behavior differs from the idealized
models. The models proposed by Drucker, Dodd-Naruse, Edelman-Drucker and Hershey aim to
better adapt the flow surface. These models are the functions of one parameter. They have no
claims on the generality and are used only for the approximation of the measurements.
Three models with one parameter are known as generalized models: Unified Yield Criterion
(UYC) of Yu, Bi-Cubic Model (BCM), and Multiplicative Ansatz (MA) [1]. These models include
the models of Tresca and Schmidt-Ishlinsky. They allow approximating of existing measurements
better than other models.
This work compares the flow criteria. For this aim their geometries in the π-plane will be
considered in polar coordinates R and ϕ. The radii of the surface at the angles of ϕ = 15 and
30 ◦ are related to the radius at ϕ = 0 ◦ : h = R(15 ◦ )/R(0 ◦ ), k = R(30 ◦ )/R(0 ◦ ). On the basis of
these two relations, well-known criteria will be systematized and shown in the h − k–diagram.
New criteria will be introduced. The convexity limits for them will be stated.
From the h−k–diagram, it is clear that UYC and MA set left and right boundary of convexity.
Thus, the extreme solutions for parts can be found. The two models UYC and MA are the
functions of the parameter k. The linear combination of UYC and MA provides a universal model
with two parameters k and ξ ∈ [0, 1] describing all convex surfaces of incompressible material
behavior of hexagonal symmetry in the π-plane.
The proposed consideration of the flow criteria simplifies the selection of the model and is
suitable for didactic purposes. All known and new flow criteria can be described by universal
model, and thus can be omitted.
[1] Kolupaev, V. A., Altenbach, H.: Considerations on the unified strength theory due to Mao-
Hong Yu, Forschung im Ingenieurwesen 74(3), (2010).
S6.11: Special Methods in Material Modeling Thu, 16:00–18:00
Chair: Bernhard Eidel, Markus Scholle S1|01–A03
Modeling of Carbon Nanotubes by Molecular Mechanics
Oliver Eberhardt, Thomas Wallmersperger (TU Dresden)
Carbon Nanotubes (CNTs) are structures in the nanoscale consisting of carbon atoms which are
arranged in a hexagonal lattice. Hence, they can be imagined as a plane sheet of graphene rolled
into a seamless tube. By doing so we obtain a so called Single Wall Carbon Nanotube (SWCNT).
Besides Single Wall Carbon Nanotubes also Double Wall Carbon Nanotubes (DWCNT) and, in
general, Multi Wall Carbon Nanotubes (MWCNT) exist. Their high stiffness and active deformation
of approx. 1 % at applied low electric voltages (approx. 1 V) make the Carbon Nanotubes a
very promising material for applications e.g. in new classes of composites and actuators/sensors.
In this research the approach to model Carbon Nanotubes is made by using molecular mechanics.
The aim of these efforts is to determine the mechanical properties, for example the Young’s

28 Section 6: Material modelling in solid mechanics
modulus. The Molecular Mechanics method originates in the investigation of the properties of
big molecules which due to their size cannot be handled by quantum mechanical methods. In
Molecular Mechanics the behavior of the covalent bonds in a Single Wall Carbon Nanotube is
represented by a set of potentials. Here every single potential expression describes a corresponding
bond deformation. As a result of the description of the bonds by potentials, the force acting
between the atoms can be calculated as the derivatives of the potentials with respect to their
corresponding displacement variable. Hence, the Carbon Nanotube is modeled by point masses
representing the carbon atoms and a set of springs or beam elements representing the bonds. Regarding
Multi Wall Carbon Nanotubes this model can be extended by adding the van-der-Waals
interactions, described by another potential.
During the process of modeling and also during the evaluation of the results we have to rise
several challenges. Since the Molecular Mechanics method is a so called semi-empiric method we
have to choose between different sets of potentials. Some of these potentials - leading to linear
or nonlinear behavior of the interaction forces between the atoms - are investigated regarding
their advantages and disadvantages. In order to illustrate this we compare the (theoretical) results
available in literature with our numerical results, which we obtain by using the model to
conduct a virtual tensile test. The dependance of the Young’s modulus on type and diameter of
the Carbon Nanotubes is discussed.
Dislocation Dynamics in Quasicrystals
Eleni Agiasofitou, Markus Lazar (TU Darmstadt), Helmut Kirchner (Leibniz Institut für neue
Materialien, Saarbrücken)
In this work, we present a theoretical framework of dislocation dynamics in quasicrystals [1] according
to the continuum theory of dislocations. Quasicrystals have been discovered by Shechtman
et al [2] in 1982 opening a new interdisciplinary research field. A comprehensive presentation of
the current state of the art of this research field, focused on the mathematical theory of elasticity,
can be found in the recently published book of Fan [3].
We start presenting the fundamental theory of moving dislocations in quasicrystals giving
the dislocation density tensors and introducing the dislocation current tensors for the phonon
and phason fields, including the Bianchi identities. In the literature, there exist different versions
of generalized linear elasticity theory of quasicrystals. The difference of these versions lies in the
dynamics of phonon and phason fields. In the present work, we deal with the elastodynamic as well
as the elasto-hydrodynamic model of quasicrystals. According to the first model, the equations of
motion are of wave-type for both phonons and phasons while according to the second model the
equations of motion for the phonons are equations of wave-type and for the phasons are equations
of diffusion-type. Therefore, we give the equations of motion for the incompatible elastodynamics
as well as for the incompatible elasto-hydrodynamics of quasicrystals.
We continue with the derivation of the balance law of pseudomomentum thereby obtaining
the generalized forms of the Eshelby stress tensor, the pseudomomentum vector, the dynamical
Peach-Koehler force density and the Cherepanov force density for quasicrystals. Moreover, we
deduce the balance law of energy that gives rise to the generalized forms of the field intensity
vector and the elastic power density of quasicrystals. The above balance laws are produced for
both models. The differences between the two models and their consequences are revealed. The
influences of the phason fields as well as of the dynamical terms are also discussed.
[1] E. Agiasofitou, M. Lazar and H. Kirchner, Generalized dynamics of moving dislocations in
quasicrystals, J. Phys.: Condens. Matter 22 (2010), 495401 (8pp).
[2] D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Metallic phase with long-range orienta-

Section 6: Material modelling in solid mechanics 29
tional order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951 – 1953.
[3] T. Fan, Mathematical Theory of Elasticity of Quasicrystals and its Applications, Science
Press Beijing and Springer-Verlag Berlin, 2011.
On inverse form finding based on an ALE formulation
Sandrine Germain, Paul Steinmann (Universität Erlangen-Nürnberg)
A challenge in the design of functional parts in forming processes is the determination of the
initial, undeformed shape such that under a given load a part will obtain the desired deformed
shape. Two numerical methods might be used to solve this problem, which is inverse to the
standard kinematic analysis in which the undeformed shape is known and the deformed shape
unknown.
The first method deals with the formulation of an inverse mechanical problem, where the
spatial (deformed) configuration and the mechanical loads are given. Hence the objective is to
find the inverse deformation map that determines the (undeformed) material configuration.
The second method deals with shape optimization that predicts the initial shape in the sense
of an inverse problem via successive iterations of the direct problem. In [1] a nodes-based shape
optimization approach for elastoplastic materials based on logarithmic strains is presented. An
update of the reference configuration is considered in order to avoid mesh distortions, which often
occur in nodes-based optimization problems. The principal drawback is the high computational
costs. An alternative is an Arbitrary-Lagrangian-Eulerian (ALE) formulation [2,3], which is neither
purely Lagrangian (the nodes are not attached to the material) nor purely Eulerian (the nodes
are not fixed in space). The nodes are free to move in space independently of the material.
In this contribution we review the ALE formulation [2,3] for anisotropic hyperelastic materials
and its application in shape optimization. Several examples illustrate the ALE approach in
hyperelasticity. Results and computational costs are compared with the ones obtained with the
approach in [1].
This work is supported by the German Research Foundation (DFG) within the Collaborative
Research Centre SFB Transregio 73.
[1] S. Germain and P. Steinmann. Towards form finding methods for a sheet-bulk-metal (DC04),
15th ESAFORM, Key Engineering Materials submitted (2012).
[2] E. Kuhl et al., An ALE formulation based on spatial and material settings of continuum
mechanics. Part 1: Generic hyperelastic formulation, Comput. Methods Appl. Mech. Engrg.
193 (2004), 4207 – 4222.
[3] H. Askes et al., An ALE formulation based on spatial and material settings of continuum
mechanics. Part 2: Classification and applications, Comput. Methods Appl. Mech. Engrg.
193 (2004), 4223 – 4245.
On the direct connection of rheological elements in nonlinear continuum mechanics
Ralf Landgraf, Jörn Ihlemann (TU Chemnitz)
The direct connection of rheological elements is a widely used concept to develop material models
for one-dimensional deformation processes at small strains. This concept is based on the decomposition
of the total strain into several subparts and the formulation of single material laws for

30 Section 6: Material modelling in solid mechanics
idealized phenomena (e.g. nonlinear elastic behaviour and viscous or plastic yielding). Several
of those elements are then assembled to one complex material model. This can be achieved by
enforcing the equilibrium of stresses on the connecting points between rheological elements.
Within the scope of nonlinear continuum mechanics there also exists the concept of rheological
elements. The kinematics is described by the deformation gradient which gets multiplicatively
decomposed into several sub deformation gradients. For the definition of the material behaviour,
a common approach is to formulate a free energy function as a sum of sub free energy functions
attributed to the defined sub deformation gradients. By the evaluation of the Clausius-Duheminequality
and under consideration of constitutive assumptions a system of equations describing
a thermodynamically consistent material behaviour can be derived. Depending on particular definitions,
those materials can include a mixture of elastic, viscous and plastic material behaviour.
An alternative approach has been presented by Ihlemann (2006). It is based on the additive
decomposition of the stress power density and leads to a system of equations for the derivation
of the stress equilibrium on defined intermediate configurations as well as the derivation of the
total stresses. Special attention has to be given to the definition of accurate stress measures on
the intermediate configurations. By this approach, a formal procedure for the direct connection of
single elements at large deformation processes can be obtained. The procedure itself is independent
of the concrete material behaviour.
The concept of direct connection of rheological elements in nonlinear continuum mechanics
and some examples for its application will be presented. Furthermore, a numerical strategy for
the direct implementation of this concept will be demonstrated.
[1] J. Ihlemann (2006), Beobachterkonzepte und Darstellungsformen der nichtlinearen Kontinuumsmechanik,
Habilitation, Universität Hannover
A stochastic model for the direct and the inverse problem of adhesive materials
N. Nörenberg, R. Mahnken (Universität Paderborn)
This work deals with the generation of artificial data [1] based on experimental data for adhesive
materials and the application of this data to the inverse and the direct problem. In reality there are
only a very limited number of experimental data available. Therefore, the prediction of material
behaviour is difficult and a statistical analysis with a stochastic proved thesis is nearly impossible.
In order to increase the number of tests a method of stochastic simulation based on time series
analysis [2] is applied. With artificial data an arbitrary number of data is available and the
process of the parameter identification can be statistically analysed. Additionally, two examples
are shown, which adapt the analysed material parameter to the direct problem. The stochastic
finite element method [3] is used to take into account the distribution and deviation of the fracture
strain.
[1] S. Schwan, Identifikation der Parameter inelastischer Werkstoffmodelle: Statistische Analyse
und Versuchsplanung. Diss. Shaker, 2000.
[2] P. J. Brockwell and R. A. Davis, Time series: theory and methods. Springer, 2009
[3] I. Babuška, R. Tempone and G. E. Zouraris, Galerkin finite element approximations of stochastic
elliptic partial differential equations. SIAM Journal on Numerical Analysis 42(2)
(2005), 800 – 825

Section 6: Material modelling in solid mechanics 31
Analysis of nanoindentation experiments by means of rheological models
Holger Worrack, Wolfgang H. Müller (TU Berlin)
The nanoindentation technique, which is similar to the instrumented indentation hardness according
to MARTENS, is established in the field of material characterization at small dimensions.
It is daily practice to analyze nanoindentation data with an almost classical formula based on
the publications by Oliver and Pharr and Fischer-Cripps. In this formula the gradient at the
beginning of the unloading curve is used to determine Youngs modulus of the tested material,
which is one of the material parameters of interest.
The procedure works well for elastic time-independent plastic material behavior, for example
copper and the calibration material fused silica, even at higher test temperatures. However, low
melting solder materials are susceptible to creep behavior, especially at the higher indentation
temperatures where the homologous temperature is > 0.5. As a result of the creep effects the
beginning of the unloading curve often shows a bulge. This discrepancy from the ideal unloading
curve complicates the correct determination of the unloading stiffness and finally yields to
incorrect results for Youngs modulus.
For this reason, additional analysis procedures are required to determine the material parameters
more precisely. In this paper the authors want to give an introduction to an enhanced
analysis of nanoindentation data based on rheological models, which are often used to describe
the time-dependence of material response. Two examples of such models are the MAXWELL- and
the KELVIN-body. In these models springs and dashpots connected in series and/or in parallel are
used to describe the time-dependent material response. The viscous parameters, determined from
the recorded data during the nanoindentation experiment, can be used to identify the mechanical
material parameters. The authors present miscellaneous viscoeleastic and viscoplastic rheological
models in connection with the corresponding equations which are used to extract the material
properties from the recorded data. Results of the analysis are presented and discussed in context
with the results from the classical Oliver and Pharr procedure and with the material parameters
published in the literature. Finally, the models are assessed by their applicability of modeling the
time-dependent material response of low melting solder materials.
S6.12: Special Material Behavior Thu, 16:00–18:00
Chair: Lurie Sergey, Jaan-Willem Simon S1|01–A1
Carbon Fibre Prepregs: Simulation of a Thermo-Mechanical-Chemical Coupled Problem
F. Hankeln, R. Mahnken (Universität Paderborn)
In automotive industry research is done to replace high strength steel by combinations of steel
and carbon-fibre prepregs (pre-impregnated fibres). It is planned to form both steel and uncured
prepregs in one step followed by the curing process under pressure in the forming die [1]. The
ability to simulate the mechanical behaviour during forming and curing would allow more economical
processes. The simulation of prepregs must regard highly anisotropic, viscoelastic and
thermal- chemical properties. For this the model is split into an anisotropic elastic part, which
represents the fibre fraction and an isotropic, viscoelastic part, representing the matrix. This part
also contains curing, causing a dependency on time and temperature. During deep-drawing large
deformations are occurring, so a large strain model regarding anisotropy[2], viscoelasticity [3]
and curing [4] has been developed. Also experiments were made to validate this model. Current
progress is the identification of material parameters.

32 Section 6: Material modelling in solid mechanics
[1] Homberg, W., Dau, J., Damerow, U. Combined Forming of Steel Blanks with Local CFRP
Reinforcement, in: G. Hirt, A. E. Tekkaya (Eds.), steel research int., Special Edition: Proceedings
of the 10th International Conference on Technology of Plasticity, Wiley-VCH,
Weinheim, 2011, pp. 441-446
[2] Menzel, A., Modelling and Computation of Geometrically Nonlinear Anisotropic Inelasticity,
Dissertation, University of Kaiserlautern, 2002
[3] Tschoegl, N.W. The Phenomenologial Theory of Linear Viscoelastic Behaviour, Springer
Verlag, 1989
[4] Lion, A. and Höfer, P., On the phenomenological representation of curing phenomena in
continuum mechanics, Arch. Mech. 59, 2007
On the axially compressed multiple walled carbon nanotubes
Ligia Munteanu (Institute of Solid Mechanics of Romanian Academy)
A new approach is proposed in this paper, which combines elements of Toupin-Mindlin strain
gradient theory and the Molecular Mechanics to include the inlayer van der Waals atomistic
interactions for axially compressed multiple walled carbon nanotubes. The neighboring walls of
a multiwalled nanotube are coupled through van der Waals interactions, and the shell buckling
would initiate in the outermost shell, when nanotubes are short. The load-unloaded-displacement
curve, the critical buckling and the appropriate values for elastic moduli are obtained. The theoretical
results obtained show a good agreement with the experimental data reported by Waters,
Gudury, Jouzi and Xu (2005). The size dependence of the hardness with respect to the depth
and the radius of the indenter is also investigated. Results show a peculiar size influence on the
hardness, which is explained via the shear resistance between the neighboring walls during the
buckling of the multiwalled nanotubes. The present method can be further extended to investigate
the stress-strain relations and the fracture behaviors of S/MWCNT
From vortices to dislocations: How fluid mechanics can inspire solid mechanics
Markus Scholle (Hochschule Heilbronn)
Although it is known for nearly a century that production and movement of dislocations are the
elementary processes responsible for the plastic deformation of a solid body, all attempts to derive
a continuum theory of plasticity from the discrete micro–theory of dislocations did not succeed
in universally valid field equations like e.g. Navier–Stokes equations in fluid dynamics.
In this paper an approach is presented based on an analogy between dislocations and vortex
lines in fluid flow. Since the dynamics of the latter ones is well understood in terms of the Navier–
Stokes equations, it is near at hand to make use of this analogy in order to establish a kind of
’Navier–Stokes equations for the distorsion tensor’.
Starting from Clebsch’s variational formulation for inviscid flow, a relation beween the symmetries
of the Lagrangian and the balance of vorticity resulting from it is shown giving rise for
the construction of another Lagrangian from which the respective dislocation balance results.
The present state of the theory is discussed.
On the connection of dissipation and deviatoric stress in the continuum theory of
defects
Johannes Schnepp (RWTH Aachen)

Section 6: Material modelling in solid mechanics 33
Continuous distributions of defects (e. g. dislocations) can be described as differential geometric
properties of the material manifold. Material forces and the Eshelby tensor are also quantities
in this three-dimensional material space. The relation to defects has been treated often in the
literature and various theories for the dynamics of defects have been proposed.
Moving defects lead to differential geometric quantities changing with time. This has motivated
some authors to augment the three space-like coordinates in the material space by a time-like
coordinate. In this way one gets a four-dimensional material space-time manifold, analogous to
the four-dimensional physical space-time in the theory of relativity. A connection of a time-like
material coordinate to temperature can be found in the literature on relativistic elasticity. These
ideas will be brought together in this contribution and some implications will be developed.
A time-like vector field in the material space-time can be associated with temperature and
heat flux. Together with the three lattice vectors a tetrad field is defined and this field determines
the differential geometric properties of the material manifold. If a variational formulation
for a hyper-elastic solid is adopted, the derivatives of the Lagrangian density with respect to the
components of the tetrad can be arranged in a four-dimensional second-order tensor. This tensor
unifies the information about the three-dimensional Eshelby tensor (material momentum current),
the entropy current, and the entropy density. The time-like component of the divergence of this
tensor is the entropy production. Besides the entropy production due to heat transfer the theory
yields entropy production due to defect movement which is proportional to deviatoric stress.
Stress Reduction Factor of Ceramic Plate to Thermal Shock
Weiguo Li, Tianbao Cheng (College of Resources and Environmental Science, Chongqing), Daining
Fang (Peking University)
In this work, through introducing the analytical solution to transient conduction problem for the
thin rectangular plate with convection into the thermal stress field model of the elastic plate, the
stress reduction factor which is useful in determining the thermal stresses in the ceramic plate
subjected to thermal shock was presented in the dimensionless form. The properties and appropriate
conditions of the stress reduction factor were analyzed. Then the definitions, origins and
limitations of the first and second thermal stress fracture resistance parameters which characterize
the thermal shock resistance of brittle ceramics were discussed. Because of the demands in
the calculation of thermal stress, a new stress reduction factor was defined and compared with
the previous one. For the given Biot number, the previous stress reduction factor first increases
and then reaches a maximum value, and afterwards decreases as the Fourier number increases.
However, the new stress reduction factor decreases as the Fourier number increases. The new
factor is more convenient for calculating the thermal stress in the plate subjected to thermal
shock. The presented results are useful for the calculating of the thermal stress and choosing of
the appropriate thermal stress fracture resistance parameter when the ceramic plate is subjected
to thermal shock.
Modeling Deformation Twinning in FeMn Alloys
Shyamal Roy, Rainer Glüge, Albrecht Bertram (Universität Magdeburg)
Multiple twinning [1] in single crystals is modeled by studying the response at each material
point in order to understand micro-structural evolution and mechanical properties of FeMn
(TWIP steel). Here the minimum elastic strain energy approach is used. It is well known that the
elastic strain energy of an individual material is convex. Since twinning leads to isomorphic structures,
the elastic strain energy of a twin material is convex too and hence, forms a non-convex
energy landscape. A smooth strain energy landscape is constructed to avoid the discontinuity
at the transition point from parent to twin material. A controlled viscous relaxation scheme is

34 Section 6: Material modelling in solid mechanics
incorporated since the boundary value problem of the non-convex energy function is ill posed [2].
Dealing with multiple energy wells is numerically very challenging. The solution is not deterministic
as any twin configuration can be activated from the parent configuration, which is unknown
initially. Therefore, to have a complete energy landscape in hand at the beginning is impossible.
The material response is instantaneously calculated depending on which twin configuration is
activated. However, the minimum strain energy approach faces its limitation because of elastic
strain energy invariance, which occurs due to the crystallographic equivalence of possible twin
configurations. Having identified the preferred twin configuration, the secondary and, hence, the
multiple twinning are modeled.
[1] Christian J. W., Mahajan S., Deformation Twinning, Prog Mat Sci, 39, 1-157 (1995)
[2] Carstensen C., Ten Remarks on Non-convex Minimization for Phase Transition Simulations,
Comp Meth App Mech Eng, 194, 169-193 (2005)