3D DISCRETE DISLOCATION DYNAMICS APPLIED TO ... - NUMODIS

INSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE
THESE
pour obtenir le grade de
DOCTEUR DE L’INPG
N o attribué par la bibliothèque
Spécialité : « SCIENCE ET GENIE DES MATERIAUX »
préparée au laboratoire Génie Physique et Mécanique des Matériaux (GPM2)
dans le cadre de l’Ecole Doctorale «MATERIAUX ET GENIE DES PROCEDES »
présentée et soutenue publiquement
par
Chansun SHIN
le 25 novembre 2004
Titre :
3D DISCRETE DISLOCATION DYNAMICS APPLIED TO
DISLOCATION-PRECIPITATE INTERACTIONS
Directeur de thèse :
Marc FIVEL
JURY
M. A. PINEAU ,Président, Rapporteur
M. F. LOUCHET ,Examinateur
M. M. FIVEL ,Directeur de thèse
M. K. H. OH ,Co-encadrant
M. H. N. HAN ,Rapporteur
M. C. ROBERTSON ,Invité
M. M. VERDIER ,Invité

3D DISCRETE DISLOCATION DYNAMICS APPLIED TO
DISLOCATION-PRECIPITATE INTERACTIONS
The 3D Discrete Dislocation Dynamics (DDD) method has been applied to investigate the effects of
precipitates on the plasticity of FCC single crystals.
A method to represent the internal interfaces by a series of facets with a pre-defined strength has been
proposed. For a full account of the mutual elastic interactions between dislocations and second-phase
particles, the coupling method with a finite element method is extended. In order to accelerate the
computing time, the serial 3D DDD algorithm has been improved by revisiting the ’box method’ and a new
parallel code has been developed using the standard Message passing Interface (MPI).
The image stresses due to a three-dimensional particle were computed using the FEM/DDD coupling
code. The numerical results have been compared to the corresponding analytical solutions. The ef-
fect of the elastic modulus mismatch on the flow stress and the subsequent hardening behavior has
then been analyzed. The image stresses were found to affect significantly the work hardening and
the local events such as cross slip and climb. Finally, the fatigue of precipitate-hardened materials
was simulated using the new parallel DDD code. The effects of shearable and non-shearable particles
on the fatigue properties were well reproduced by the simulations, and the numerical results showed
good agreements with the available experimental observations in a qualitative way. The mechanism of
the intense slip band formation is proposed from the observation of the simulated dislocation microstructure.
KEY WORDS: DISLOCATION, PRECIPITATE, PLASTICITY, FATIGUE, IMAGE FORCES,
DAMAGE, DYNAMICS, PARALLELIZATION
DYNAMIQUE DES DISLOCATIONS DISCRETES APPLIQUEE AUX
INTERACTIONS ENTRE DISLOCATIONS ET PRECIPITES
La dynamique des dislocations discrètes (DDD) a été appliquée pour examiner les effets des précipités sur
la plasticité des monocristaux de structure CFC.
Les précipités sont modélisés par un assemblage de facettes franchissable pour une contrainte donnée.
Afin de tenir compte des interactions élastiques entre les dislocations et les particules, un couplage avec la
méthode des éléments finis (MEF) a été utilisé. Afin d’accélérer les temps de calculs, la ’méthode des boites’
a été revisitée et une version parallele du code a été développée en utilisant le standard du programmation
’Message Passing Interface (MPI)’.
Dans un premier temps, les contraintes images créées par une particule 3D ont été calculées grâce un cou-
plage entre la MEF et le code de DDD. Les résultats numériques ont été comparés aux solutions analytiques
correspondantes. L’effet de la différence des modules d’Young sur la limite élastique et le comportement
durcissant qui en découle ont ensuite été étudiés numériquement. Nous avons montré que les contraintes
image ont un effet significatif sur le durcissement et les événements locaux tels que le glissement dévié et la
montée. Finalement, la fatigue des matériaux durcis par des précipités cisaillables et non-cisaillables a été
simulée avec le nouveau code parallèle de DDD. Les résultats obtenus grâce à nos simulations sont en accord
avec nos observations experimentales et les données de la littérature. Un mécanisme de formation des ban-
des de glissement intense a été proposé à partir de l’observation des microstructures obtenues par simulation.
MOTS CLES: DISLOCATION, PRECIPITE, PLASTICITE, FATIGUE, FORCES IMAGES, EN-
DOMMAGEMENT, DYNAMIQUE, PARALLELISATION
Laboratoire Génie Physique et Mécanique des Matériaux (GPM2), ESA5010,
ENSPG, 101 Rue de la Physique, BP46, 38402 Saint Martin d’Hères Cedex

Acknowledgements
First of all, I express my big thanks to my advisor Marc Fivel. Five years ago, he kindly replied
to my audacious e-mail, which could be easily neglected considering the content, and gave me an
opportunity to visit him. This short visit led to the three-year Ph.D program between INP Greno-
ble and Seoul National University, and from the moment we shook hands for the first time, to the
moment we shook hands after the thesis defence, he has been my mentor on both work and life.
I am also grateful to Professor Kyu Hwan Oh, whom I have been working with since I began my
Master study eight years ago. He gave me many opportunities to experience in research, and kept
giving me much good advice.
I owe my special thanks to Marc Verdier (LTPCM) and Christian Robertson (CEA Saclay).
They guided me and advised me as an unofficial co-advisor, from the start of the thesis work to
the rehearsal of the thesis presentation with great patience and encouragement. And I cannot help
attributing some of my work to the fantastic tools of Christophe Déprés, who started and finishes
the Ph.D study with me.
I want to thank Professor André Pineau (ENS Mines Paris) for serving both as ’Président’ and
’Rapporteur’ for my thesis defence. From the moment I met him for the first time at the meeting
of the project ’FAMICRO’ 1 , that supported my work on fatigue simulations, I was fascinated with
his enthusiasm for research and with his boundless memory.. he is a walking library! I also thank
Professor François Louchet (LGGE) and Heung Nam Han (SNU) for serving on my thesis
committee and for their useful suggestions and critical assessment on my work.
My work has been supported by EGIDE 2 , and I want to thank the CNOUS at Grenoble and the
French Embassy in Korea for their efficient professional services.
I am much grateful to all the members of GPM2 Laboratory for pleasant daily life, in the blue room:
Julien Chaussidon, Thomas Nogaret, computing room: David Rodney, Valérie Quatela
and on the playground with a soccer ball: Dider Bouvard, Rémy Dendievel, Luc Salvo,
Charles Josserond, Franck Pelloux and Shigesato Genechi.
And finally, I want to express my thanks and love to my wife Suejung, who is both my great
supporter and best friend, for her love and devotion to our family, and to my little daughter Yvine,
who likes to play with my laptop, for laughter and happiness we all share in our growing family.
1 Modélisation de la durée de vie en Fatigue de matériaux métalliques structuraux, à partir de mécanismes physiques
microscopiques
2 Bourse Pasteur du ministère des affaires étrangères

Abstract
The 3D Discrete Dislocation Dynamics (DDD) method has been applied to investigate the effects
of precipitates on the plasticity of FCC single crystals.
A method to represent the internal interfaces by a series of facets with a pre-defined strength
has been proposed. For a full account of the mutual elastic interactions between dislocations and
second-phase particles, the coupling method with a finite element method is extended. In order
to accelerate the computing time, the serial 3D DDD algorithm has been improved by revisiting
the ’box method’ and a new parallel code has been developed using the standard Message passing
Interface (MPI).
The image stresses due to a three-dimensional particle were computed using the FEM/DDD coupling
code. The numerical results have been compared to the corresponding analytical solutions. The
effect of the elastic modulus mismatch on the flow stress and the subsequent hardening behavior has
then been analyzed. The image stresses were found to affect significantly the work hardening and
the local events such as cross slip and climb. Finally, the fatigue of precipitate-hardened materials
was simulated using the new parallel DDD code. The effects of shearable and non-shearable par-
ticles on the fatigue properties were well reproduced by the simulations, and the numerical results
showed good agreements with the available experimental observations in a qualitative way. The
mechanism of the intense slip band formation is proposed from the observation of the simulated
dislocation microstructure.

Résumé
La dynamique des dislocations discrètes (DDD) a été appliquée pour examiner les effets des précip-
ités sur la plasticité des monocristaux de structure CFC.
Les précipités sont modélisés par un assemblage de facettes franchissable pour une contrainte don-
née. Afin de tenir compte des interactions élastiques entre les dislocations et les particules, un
couplage avec la méthode des éléments finis (MEF) a été utilisé. Afin d’accélérer les temps de
calculs, la ’méthode des boites’ a été revisitée et une version parallele du code a été développée en
utilisant le standard du programmation ’Message Passing Interface (MPI)’.
Dans un premier temps, les contraintes images créées par une particule 3D ont été calculées grâce un
couplage entre la MEF et le code de DDD. Les résultats numériques ont été comparés aux solutions
analytiques correspondantes. L’effet de la différence des modules d’Young sur la limite élastique
et le comportement durcissant qui en découle ont ensuite été étudiés numériquement. Nous avons
montré que les contraintes image ont un effet significatif sur le durcissement et les événements lo-
caux tels que le glissement dévié et la montée. Finalement, la fatigue des matériaux durcis par des
précipités cisaillables et non-cisaillables a été simulée avec le nouveau code parallèle de DDD. Les
résultats obtenus grâce à nos simulations sont en accord avec nos observations experimentales et
les données de la littérature. Un mécanisme de formation des bandes de glissement intense a été
proposé à partir de l’observation des microstructures obtenues par simulation.

Contents
Acknowledgements iii
Abstract v
1 Introduction 1
1.1 Computational methods in plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dislocation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 2D simulations of dislocation dynamics . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 3D simulations of dislocation dynamics . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Scope of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Description of the simulation method 9
2.1 Representation of the dislocation lines in FCC metals . . . . . . . . . . . . . . . . . . 10
2.1.1 Preparation of the simulation space . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Discretization of the dislocation lines . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 Existence of a subnetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.4 Comments on other crystal structures and dislocation dynamics models . . . 13
2.2 Computation of stresses and displacements of dislocations . . . . . . . . . . . . . . . 14
2.2.1 Evaluation of the driving force . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Computation of displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Motion of dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Dislocation mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 Dislocation-dislocation interactions . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.4 Cross-slip of screw dislocation segments . . . . . . . . . . . . . . . . . . . . . 28

viii CONTENTS
2.3.5 Plastic strain due to dislocation movement . . . . . . . . . . . . . . . . . . . . 29
2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Internal interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Acceleration of the DDD code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.1 Problem description and review of literatures . . . . . . . . . . . . . . . . . . 35
2.5.2 The Box method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.3 Speedup and Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.4 Boxes and Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . 46
2.6 Computation procedure of the DDD program . . . . . . . . . . . . . . . . . . . . . . 48
3 Parallelization of the Discrete Dislocation Dynamics method 51
3.1 An introduction to Supercomputing . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.2 Classification of hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.3 Parallel programming models . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.4 Classification of parallel languages . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.5 Supercomputers in France and Korea . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 Towards a parallel DDD code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Basic Steps of Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Writing a parallel program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Parallelization of the serial DDD program . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Initialization of parallel environments . . . . . . . . . . . . . . . . . . . . . . 67
3.3.2 Long-distance stresses computations . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.3 Short-distance stresses computation . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.4 Data structures for distributing and the gathering segments . . . . . . . . . . 71
3.3.5 Motion of segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.6 Summary and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Performance improvment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Measure of performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.2 Conditions for good performance . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.3 Performance tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

CONTENTS ix
3.4.4 Load balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4.5 Comparison of simulation results between the serial and parallel DDD code . 88
3.5 Application to Stage I-II transition simulation . . . . . . . . . . . . . . . . . . . . . . 88
3.5.1 Stress-strain curves of FCC single crystals . . . . . . . . . . . . . . . . . . . . 88
3.5.2 Simulation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4 Dislocation-precipitate interactions 95
4.1 Image stresses due to a 3D particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.1 Motivations and review of the literature . . . . . . . . . . . . . . . . . . . . . 95
4.1.2 Interaction of an edge dislocation with a circular cylindrical particle . . . . . 97
4.1.3 Interaction of an edge dislocation with a spherical particle . . . . . . . . . . . 99
4.1.4 Interaction of an edge and a screw dislocation with a cubical particle . . . . . 102
4.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 A simple case of dislocation-particle interaction . . . . . . . . . . . . . . . . . . . . . 105
4.2.1 Motivation and review of literatures . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.2 Calculation procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.3 Flow stress of impenetrable particles with a different shear modulus . . . . . 107
4.2.4 Increment in hardening stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 Fatigue simulations of materials hardened by particles . . . . . . . . . . . . . . . . . 116
4.3.1 Motivation and review of literatures . . . . . . . . . . . . . . . . . . . . . . . 116
4.3.2 Description of the simulation method . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.3 Evolution of the dislocation microstructure during the fatigue tests . . . . . . 123
4.3.4 Mechanical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.3.5 Surface slip markings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3.6 Fatigue properties of materials containing particles with a bimodal size dis-
tribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Conclusions and perspectives 149

Chapter 1
Introduction
1.1 Computational methods in plasticity
A dislocation is a line defect within a crystal, which represents permanent deviations of atoms
from their original crystallographic periodicity. The dislocation glide gives rise to macroscopic
deformation of metals. A dislocation thus is a microscopic carrier of the metallic plasticity.
Modeling the plasticity of metals involves understanding the nature of dislocations, which is defined
at the atomistic scale and also evaluating the deformation behaviors at the macroscopic scale. Many
models have been developed to understand the plasticity of metals. Since the features of plasticity
vary much in size and time, the models also vary largely in length and time scales. Out of a range of
models, most attention is given in this section on Molecular Dynamics (MD), Dislocation Dynamics
(DD) and continuum mechanics.
Atoms are the basic constituent elements of MD simulations. Atoms interact with each other through
an interatomic potential. The temporal trajectory of an ensemble of atoms under an external loading
is simulated by minimizing the total potential energy of the system. The deviations of the position of
the atoms from the lattice sites implicitly represent the dislocations. The atomistic scale topology of
a dislocation line thus can be investigated by MD. MD simulations are employed mostly in studying
physical properties of a single or a few dislocation lines due to the constraints of the simulation size
(< (200nm) 3 ).
In DD methods, dislocation lines are represented explicitly. The collective evolution of a large
number of interacting dislocations is simulated under an external loading. Properties of dislocations
such as mobility, junction strength etc., are input parameters of DD simulations, and dislocation
glide results in plastic strain in the simulation volume. The stress-strain behavior is thus an output

2 Introduction
Time (sec)
10 3
1
10 -3
10 -6
10 -9
10 -12
Molecular
Dynamics
10 -11 10 -10 10 -9
10 -8
10 -7
10 -6
Single crystal
models
Dislocation
Dynamics
10 -5
Space (m)
Homogenization technique
Polycrystal
models
Continuum mechanics
Figure 1.1: Figure illustrating length and time scales of each model. Solid lines represent the limit
ranges imposed by the intrinsic physics of the model. Dashed lines represent the limit imposed by
the available computing power.
of the DD simulations.
Continuum mechanics treat the behavior of a continuum medium by a set of equations and boundary
conditions. There are a wide range of numerical techniques which can solve the equations. Finite
difference and finite element methods are two broad subsets of such techniques. In these methods,
a continuum domain of interest is subdivided into discrete cells or elements, in which the values of
certain physical quantities are determined by solving a system of equations. The output of a typical
application to the metallic plasticity is the deformation behavior of the simulated volume, for which
a governing constitutive equation is assumed.
As introduced briefly above, MD, DD and continuum methods have their own characteristic length
and time scale. Fig. 1.1 shows such ranges of length and time scales of each method. As the
performance of each numerical method is improved, the volume and the physical time which can
be simulated are increasing (top and right domain limit of each method in Fig. 1.1). Recently the
length and time scales of the various methods begin to be overlapped. This gives a great impetus
to exchange information in order to build up a unified model of the metallic plasticity, which would
be able to predict the behavior of a material from the fundamental properties of the material.
10 -4
10 -3
10 -2
10 -1
1

1.2 Dislocation dynamics 3
(a) Weak obstacles (b) Hard obstacles
Figure 1.2: 2D simulations of dislocations moving through a random array of point obstacles: Effects
of obstacles’ strength ([Foreman & Makin 66])
1.2 Dislocation dynamics
1.2.1 2D simulations of dislocation dynamics
Based on the well understood elementary properties of a single dislocation, numerical DD methods
have been developed first in 2D.
Dislocation dynamics in 2D can be further divided in terms of the crystallographic orientation of
the plane used for the simulations: (i) parallel and (ii) perpendicular to dislocation lines. In the
case (i), the plane of the simulations is parallel to the glide plane of dislocation lines, thus nei-
ther cross-slip nor climb of dislocations are allowed. This configurations have been applied initially
to study line tension and the shape of a dislocation under stress ([Brown 64]). The dynamical
movements of dislocations have also been simulated in the case of a glide plane containing random
distribution of point obstacles ([Foreman & Makin 66]). The effects of obstacles’ strength on the
initial flow-stress have been studied, and some of the simulation results are shown in Fig. 1.2. This
type of 2D simulations is still in use to study the effect of particles’ parameters on the flow stresses,
see for example [Mohles & Nembach 01].
In the case (ii), dislocations are perpendicular to the simulation plane, that is, dislocations are
infinite, parallel to each and have the same character. This configuration can simulate the multi-
plication, annihilation, cross-slip and climb of dislocations. It is, however, difficult to include the

4 Introduction
line tension effect explicitly. This kind of configuration has been used to simulate the spontaneous
microstructure formation ([Lépinoux & Kubin 87]). Because of its simplicity, this 2D method
can simulate dislocation motion up to relatively large strains. This method is still largely under
development and applied to several studies, see for example [Cleveringa et al. 97].
1.2.2 3D simulations of dislocation dynamics
The motivation of a 3D DD can be summarized as the needs
• to include the 3D nature of the dislocation behavior, cross-slip, junction formation, ...
• to explain the formation of dislocation structures during the plastic deformation
The first simulation in 3D is proposed in [Canova & Kubin 91]. Since then, the proposed method
has been developed and applied to investigate the collective motion of dislocations under various
conditions by two leading groups 1 in France. This method is based on the representation of dislo-
cation lines by segments in an integer space. Other versions of DD in 3D have emerged since the
end of 1990s, as will be detailed in Sec. 2.1.4. Due to the development of simulation methods and
the increased computing power, these simulation methods have strengthened their positions in the
field of crystal plasticity. The 3D discrete dislocation dynamics (DDD) method has proven to be a
powerful tool to investigate the plasticity of metals and been expected to serve as a link between
atomistic and continuum scale simulations (see Fig. 1.1).
1.3 Scope of Thesis
This thesis aims at applying the 3D DDD method to both rigorous computations of dislocation-
precipitate interactions and studying the effects of precipitates on the fatigue properties of metals.
For the rigorous computations, we extended the code coupled with a finite element method ([Fivel 97])
in order to incorporate 3D precipitates with a differing elastic modulus. The interaction forces due
to a second phase particle are computed and the effects of these forces on the flow stress and the
subsequent hardening are investigated.
1 Génie Physique et Mécanique des Matériaux (GPM2) and Laboratoire d’Etude Métallurgique (LEM)

1.3 Scope of Thesis 5
Recently the 3D DDD method are applied successfully to the study of early fatigue crack initi-
ation of 316L stainless steels ([Déprés 04]). The critical role of cross-slip was pointed out, which
demonstrates the advantages of the 3D DD simulations over the 2D simulations. Inspired by this
study, we applied the 3D DDD method to simulate the fatigue behavior of materials hardened by
precipitates. It was found, however, that the feasible volume fraction of precipitates is quite small
considering the performance of the currently available computing machines with a single processor
and the computing efficiency of the serial 3D DDD code. This is due to the additional computa-
tional loads induced when many precipitates are introduced in the 3D DDD simulations, which are
already computationally demanding because of the long-ranged stress field of a dislocation segment
and the need to handle the dislocation interactions during the segment motions. Because of the
inherent computational load of the 3D DDD simulations, a maximum strain which can be simulated
in a reasonable time still remains in the order of 10 −3 in multislip condition.
The easiest way to suffice the computational demands of the fatigue simulations of precipitation-
hardened materials would be to waite until a faster single processor is available. Considering the
relatively short period of a doctorate, however, it cannot be a good way to choose notwithstanding
the speed of a single processor has improved tremendously 2 .
The other way is to increase the computational capacity by collecting single processors and making
them work together, that is, parallel computing. A parallel computer simply comprises a number of
processors that solve a problem together to reduce the elapsed computation time. In fact, parallel
computing has been widely adopted in many research fields to resolve the increase of the compu-
tational demands, which arises due to many reasons, e.g. encompassing sophisticated boundary
conditions, involving nonlinear material behaviors and many unknowns. Evident successes of the
parallel computing in the field of the computational plasticity can be found in both MD meth-
ods ([Abraham 97]) and continuum mechanics ([Demmel et al. 93], [Fahrat & Roux 94]). The
parallel codes have enabled each model to perform large scale simulations in reasonable time. In
MD simulations, for example, a volume of 0.01nm 3 can be treated over a period of time in the order
of 10 −12 seconds using massively parallel machines ([Abraham 97]).
As can be seen in many references including the few examples cited above, the subject of parallel
2 Semiconductor technology has been known to increase a processor clock rate by double in 18 months up to now.
This is known as the Moore’s law first published in 1965 ([Moore 65]) and which still holds true today. Intel expects
that it will continue at least through the end of this decade. In the end, the performance of a single processor
computing device will reach an upper limit due to the physical limits of semiconductor technology

6 Introduction
computing has been investigated extensively and is now a well established field. From the success in
atomistic and continuum parallel simulations, we came to the conclusion that parallel computation
will be the best and the only choice in order to include a relatively high volume fraction of precip-
itates in the simulation volume, because a dramatic increase in computational power can only be
met through it.
A parallel DDD code thus has been developed and applied to the fatigue simulations containing a
large number of particles. The effects of particles on the fatigue properties are studied focusing on
the irreversibility of slips and the formation of the intense slip bands during the cyclic deformation.
The parallel DDD code developed would be of benefit to not only small scale simulations which
involve a large number of internal defects but also large scale simulations which would make a com-
parison with the macroscopic simulations possible. The parallel DDD code would hence reinforce
the role of the 3D DDD method in the series of the plasticity simulation methods.
This thesis is organized as follows.
Chapter 1 Introduction (this chapter)
Chapter 2 summarizes the theoretical backgrounds and methodologies of the 3D discrete disloca-
tion dynamics method. The computation of the displacement fields of a dislocation loop is
introduced. Several boundary conditions are explained, such as the implementation of internal
interfaces and the periodic boundary conditions. The numerical efficiency of the serial DDD
algorithm is increased by revisiting the so-called box method ([Verdier et al. 98]).
Chapter 3 presents the parallel algorithm developed to parallelize the 3D DDD program used
in this work. The parallel version of the DDD program aims at simulating fatigue tests of
materials containing large number of particles in reasonable time using parallel computers.
The new parallel DDD program is tested and improved in performance by balancing the load
dynamically, and then applied to stage I-II simulations. This chapter also contains general
introduction to parallel computing.
Chapter 4 contains three applications of the method developed and detailed in the preceding
chapters. Image stresses by a cylindrical, spherical and cubical particle are computed. The
effects of image stresses on flow stresses and hardening are investigated. The FEM/DDD

1.3 Scope of Thesis 7
coupling method presented in Sec. 2.4.2 is used for these applications. Finally, the new
parallel program is used for fatigue simulations of precipitate-hardened metals.
Chapter 5 gives concluding remarks and perspectives.

Chapter 2
Description of the simulation method
THE discrete dislocation dynamics (DDD) method initially proposed in [Canova & Kubin 91] has
been improved much in its numerical precision and applicability to problems involving complex
boundary conditions over the past 15 years. The purpose of this chapter is to review the theoretical
backgrounds and methodologies of the DDD method, and also to describe the author’s contributions
: computation of displacement fields, implementation of internal interfaces and the periodic boundary
conditions and acceleration of the code using the revised box method.
The DDD method used in this thesis only deals with perfect dislocations in face-centered cubic (FCC)
metals. Sec. 2.1 introduces the simulation lattice and the discretization of a dislocation line of the DDD
model, and the model is compared with other dislocation dynamics models. Although the focus is given
on the FCC lattice, the methodology is quite general. The extension of the method to the other cubic
crystals is also discussed briefly.
Computation of the effective stress of each dislocation segment is presented in Sec. 2.2. The method
used for computing the displacement field of a dislocation loop is detailed also, and the extension of the
method to more general dislocation structures can be found in Sec. 4.3.5. The stress and displacement
solutions are all based on the theory of linear elasticity in isotropic frame.
Sec. 2.3 introduces the motion of dislocation segments. This section includes a description of the several
local rules needed to handle interactions between dislocations.
New boundary conditions are explained in Sec. 2.4. Representation of internal interfaces is discussed
in both a simple method using facets and a more rigorous way with full elastic interactions. The
implementation of periodic boundary conditions is also detailed.
The performance of the DDD code is improved by revising the box method which was first described in
[Verdier et al. 98]. The computational efficiency of the method is significantly increased by using the

10 Description of the simulation method
linked-list of segments. The methodology and the performance of the box method is described in Sec.
2.5. The overall flowchart of the code is presented at the end of this chapter.
2.1 Representation of the dislocation lines in FCC metals
2.1.1 Preparation of the simulation space
The lattice of the simulation volume is homothetic to that of FCC metals. The lattice spacing
of the simulation lattice is adopted from an experimental measurement of the athermal critical
self annihilation distance between edge dislocations 1 . The experiments of Essmann and Mughrabi
([Essmann & Mughrabi 79]), for example, show that no edge dislocations coexist within the
distance of the order of 1.5 nm in their copper specimens at room temperature. Thus the shortest
distance of two edge dislocation in the simulation is set to this critical distance.
An inter-planar distance of two adjacent {111} planes equals to a/ √ 3, with ’a’ the lattice spacing
(see Fig. 2.2(a)). If ye denotes the critical self annihilation distance, a can be expressed as Eq. 2.1
by equating a/ √ 3 to 2ye.
a = 2 √ 3ye
A typical value of the simulation lattice spacing xl(= a/2) is around 2.598 nm with the value of
ye = 1.5 nm.
The reader should note that xl is the value of the order of 10b, where b is the magnitude of the
Burgers vector. This is certainly larger than the dislocation core radius (∼ 2b). The use of the
lattice spacing larger than the core radius has two effects on the simulation method.
1. Linear elastic solutions of stress and displacement of a dislocation is valid all over the simu-
lation network (Sec. 2.2).
2. It requires to express the core properties of a dislocation in a phenomenological manner (Sec.
2.3.3 & 2.3.4).
2.1.2 Discretization of the dislocation lines
Only perfect dislocations in FCC metals are considered in this work and no dissociation into partials
is allowed. It is probable that the width of spacing of two partial dislocations is smaller than the
1 Screw dislocations annihilate more easily than edge ones by the cross-slip mechanism, thus the critical distance
of edge dislocations defines the lattice spacing of the simulation lattice.
(2.1)

2.1 Representation of the dislocation lines in FCC metals 11
[-12-1]
[111]
[-101]
Edge segment
Screw segment
2 xl
Figure 2.1: Representation of a curved dislocation line with a link of pure edge and screw segments:
The dots represent lattice points on (111) slip plane. Unit lengths of edge ( √ 6xl) and screw segment
( √ 2xl) are shown.
lattice spacing used (∼ 10b), because the stacking fault energy γ is about 140mJm −2 for aluminium,
40mJm −2 for copper and 20mJm −2 for silver, which gives the corresponding width of staking-fault
ribbons of √ 2b, 5 √ 2b and 7 √ 2b for aluminum, copper and silver respectively for the case of Poisson’s
ratio being zero ([Hull & Bacon 83]).
A curved dislocation line is represented as a connected set of discrete dislocation segments of a pure
edge and a pure screw type. This is why the method is called as the edge-screw model. Fig. 2.1
schematically shows the discretization of a dislocation line by a succession of orthogonal edge and
screw segments of the same Burgers vector on the same slip plane 2 .
Maximum length of a segment is set to the discretization length ld and any segment with a length
lseg longer than ld is subdivided further into lseg/ld segments.
The edge (< 112 > type) and screw (< 110 > type) vectors for each of the 12 slip systems used
in the DDD simulations are shown in Table 2.1 3 . Each screw direction is associated to two edge
directions, Edge1 and Edge2, defining the two glide systems, (Screw, Edge1) and (Screw, Edge2),
which share the same Burgers vector. The line directions of the 6 screw vectors (or Burgers vectors)
6 xl
2 Edge segments move along the screw vector direction and vice versa. Edge segments of the line vector [¯12¯1] in
Fig. 2.1, for example, move along ±[¯101], and screw segments of [¯101] move along either ±[¯12¯1] or ±[¯1¯2¯1] direction
(the cross-slip mechanism (see Sec.2.3.4)).
3 The notation of Schmid and Boas [Schmid & Boas 35] is written with the system number.

12 Description of the simulation method
are adopted from the Thompson tetrahedron given in [Hirth & Lothe 92], p319. The signs of the
vectors are defined from the following 2 rules:
1. Edge × Screw = n, where n is the outgoing normal of the Thompson tetrahedron
2. Edge1 × Edge2 = b so that any prismatic loop is unambiguously defined 4
System 1 (B4) 2 (D4) 3 (D1) 4 (C1) 5 (B5) 6 (C5)
Screw [¯101] [011] [1¯10]
Edge [¯12¯1] [¯1¯2¯1] [¯2¯11] [2¯11] [¯1¯12] [¯1¯1¯2]
Plane normal (111) (¯11¯1) (¯11¯1) (¯1¯11) (111) (¯1¯11)
System 7 (D6) 8 (A6) 9 (A2) 10 (B2) 11 (C3) 12 (A3)
Screw [¯1¯10] [0¯11] [101]
Edge [1¯1¯2] [1¯12] [211] [¯211] [1¯2¯1] [12¯1]
Plane normal (¯11¯1) (1¯1¯1) (1¯1¯1) (111) (¯1¯11) (1¯1¯1)
Table 2.1: Vectors of line and glide directions of dislocation segments used in the DDD code.
Each segment is represented numerically by a set of integers that are the three coordinates of the
starting point, the length and the two indexes of the line and the moving vector. The coordinates
are expressed in units of the simulation lattice parameter xl. The length is in unit of the norm of
the line vector. The connection of a line is built through a pointer of segments index.
2.1.3 Existence of a subnetwork
There exist certain sets of slip planes in which mutual dislocation interactions cannot be treated
properly. We shall call each set as subnetwork. This is due to the fact that in the edge-screw model,
a unit line vector of an edge dislocation is , whose length is √ 6xl (Table 2.1).
An edge dislocation [11¯2] on a (111) plane, for example, is shown in Fig. 2.2(b). There are two
(¯1¯11) slip planes which intersect with the [11¯2] edge dislocation in a unit cell of a simulation volume
as illustrated in Fig. 2.2(a). The lattice points along the intersecting lines are shown with filled
and hollow points for each plane in Fig. 2.2(b). One of the planes cuts the unit edge segments in
the middle, which is not permitted 5 .
4 the Right Hand Final to Start(RHFS) rule is adopted, which can be seen in Fig.2.10
5 It is noted that this improper intersection happens between two planes with the same Burgers vector.

2.1 Representation of the dislocation lines in FCC metals 13
[100]
xl
[001]
a
(a) The unit cell
[010]
xl[-110]
xl[11-2]
(b) Subnetwork
Unit edge segment
Figure 2.2: The unit cell of the simulation space and the existence of subnetworks: The lattice is
homothetic to that of FCC crystal, where xl is usually taken as ∼ 10b. There exist subnetworks
due to the definition of dislocation line vectors in Tab. 2.1
This indicates that there exist subnetworks which cannot be used simultaneously. Attention should
thus be given to the initial dislocation configurations of the simulations so that segments in two
slip planes of the same Burgers vector share a common point on one of the planes. In practice each
starting point of the dislocation segments is described in the elementary basis (Screw, Edge1, Edge2)
so that the origin point (0,0,0) is the same for the two involved slip systems. The subnetwork also
imposes certain restrictions while applying periodic boundary conditions (see Sec.2.4.1).
2.1.4 Comments on other crystal structures and dislocation dynamics models
Although it is not treated in this thesis, dislocations in other cubic crystal structures can be rep-
resented in a similar manner. For example, in the body-centered cubic (BCC) crystal structure,
slip occurs in close packed directions. The crystallographic slip planes are {110},{112} and
{123}. By the same analogy as the construction of Table 2.1, the slip systems of {110} or
{112} can be defined by 4 screw and 12 edge line vectors respectively. The {123} 6
slip system involves 4 screw and 24 edge line vectors. Dislocation dynamics models using the BCC
crystal structure can be found in [Devincre & Roberts 96] and [Tang et al. 98].
There exist several dislocation dynamics models. The difference comes mainly from how to dis-
6 The {123} slip system is less closed packed, thus at low temperatures, it would be sufficient to take only
{110} and {112} slip systems into account.

14 Description of the simulation method
(a) Edge-screw model (b) Pure-mixed model (c) Nodal model
Figure 2.3: Discretization of a curved dislocation line in edge-screw, pure-mixed and nodal model
cretize dislocation lines. Zbib et al. ([Zbib et al. 98]), for example, has approximated the dis-
location curves by series of mixed straight segments of an arbitrary length and orientation. The
scheme, which parameterizes a dislocation line by a set of nodes, is often called as ’nodal dislocation
dynamics’. Some of the nodal dislocation dynamics can even treat dislocation splitting into partials
([Shenoy et al. 00], [Weygand et al. 01]). The nodal model has advantages in the numerical pre-
cision. The nodal model is, however, much complex in dealing with topological aspects of segments,
because it involves more degrees of freedom in segment types as compared to the edge-screw model.
Thus the nodal model is used preferably to investigate phenomena involving a small number of
dislocations and a high precision in the dislocation topology ([Schwarz 99], [Ghoniem et al. 00]).
Recently, there has been an attempt to increase numerical accuracy by introducing one more seg-
ment type in the edge-screw model. It is called as the ’pure-mixed’ model. This model incorporates
additional line directions, i.e. ±60 o characters. The model aims at an accurate description of a
curved dislocation line with a minimum number of segments ([Devincre et al. 01], [Madec 01]).
In Fig. 2.3, the discretization description methods for a curved dislocation line used in the edge-
screw, pure-mixed and nodal model are compared side by side.
2.2 Computation of stresses and displacements of dislocations
2.2.1 Evaluation of the driving force
The velocity of each segment is governed by the effective stress τe acting on the segment. The
effective stress is given by τe = fg/b, where b is the magnitude of the Burgers vector and fg is the
magnitude of the glide force per unit length. fg is computed at the center of each segment and

2.2 Computation of stresses and displacements of dislocations 15
includes four contributions:
(i) the force due to the internal stress field produced by all the other dislocation segments in the
simulation volume except by two neighboring segments and the considered segment itself
(ii) the force due to applied stress fields
(iii) the force due to the line tension
(iv) the force due to the Peierls stress
The forces due to atomistic-level interactions, such as dragging forces by solute atoms or jogs, are not
treated explicitly. They can be included implicitly, however, by modifying the motion rule which de-
fines the relation between the glide velocity and the effective shear stress of a segment (see Sec. 2.3).
Internal stresses
To compute the internal stresses at the center of a segment, the expression of the stress field of a
single finite straight segment is required. This problem has been addressed by Li ([Li 64]). Li has
found an interesting fact from the stress solution of an angular dislocation made of two semi-infinite
dislocations joined together at one point. According to Li, the stress field of an angular dislocation
is the sum of the stress fields of each dislocation arm, i.e., a semi-infinite dislocation. Although the
stress field of a semi-infinite dislocation does not obey the equations of equilibrium, the sum of the
stress fields of two semi-infinite dislocations satisfies the equilibrium.
If a semi-infinite dislocation lies in the positive z axis running into the origin, O, the stress field
produced at a point r(x,y,z) has the following components ([Li 64]).
σxx(r) = −bxy−byx
r(r−z) − x2 (bxy−byx)(2r−z))
r 3 (r−z) 2
σyy(r) = bxy+byx
r(r−z) − y2 (bxy−byx)(2r−z))
r 3 (r−z) 2
σzz(r) = z(bxy−byx)
r 3
σyz(r) = y(bxy−byx)
r 3
σzx(r) = x(bxy−byx)
r 3
σxy(r) = bxx−byy
r(r−z)
− 2ν(bxy−byx)
r(r−z)
− νbx
r
+ νby
r
+ (1−ν)bzx
r(r−z)
− (1−ν)bzy
r(r−z)
− xy(bxy−byx)(2r−z)
r 3 (r−z) 2
In Eq. 2.2, the stresses are given in unit of µ/4π(1 − ν) with µ and ν being the shear modulus and
the Poisson ratio respectively. r is the distance to the point r(x,y,z) as shown in Fig. 2.4. The
stress field of a dislocation segment lying on the z axis running from z2 into z1 is obtained from
(2.2)

16 Description of the simulation method
X
Z
z2
z1
O
Dislocation
segment (z2-z1)
r
r(x,y,z)
Y
=
X
Z
z1
O
Semi-infinite
dislocation
Y
-
X
Z
z2
O
Semi-infinite
dislocation
Figure 2.4: A configuration of a semi-infinite dislocation and a calculation of a stress field of a
dislocation segment
that of two semi-infinite dislocations as shown in Fig. 2.4. The stress field is constructed by using
Eq. 2.2 twice, and substituting z in the equation for z − z1 and z − z2 respectively.
σij(r) = σij(r)z−z1 − σij(r)z−z2
(2.3)
The expressions of Li (Eq.2.2) are derived such that a semi-infinite dislocation line lies on the z
axis. A rotation of the stress tensor would be necessary for an arbitrary segment in order to bring
the segment into the reference coordinate.
The compact formulae of de Wit [dewit 67], on the other hand, are given with respect to an
arbitrary Cartesian coordinate system. Thus the expressions of de Wit can be used without any
rotation of the coordinate system. The final form is shown in Eq. 2.4, which has been derived by
Devincre in [Devincre 95].
σij(r) = µ
πY 2
[bYt] s ij − 1
1 − ν [btY]s
(b, Y, t)
ij − δij + titj +
2(1 − ν)
2
Y 2
ρiYj + ρjYi + L
R YiYj
(2.4)
The vectors in Eq. 2.4 are shown on a dislocation line of a line vector t and Burgers vector b
in Fig. 2.5. The vectors and the scalars are defined as R = r − r ′ , L = R · t, ρ=R − Lt and
Y = R + Rt. δij is the Kronecker delta and (b, Y, t) is the mixed product. [abc] s ij
Y
is defined as
1
2 ((a × b)icj + (a × b)jci). The stress field of a dislocation segment between two points A and B
is determined by inserting Eq. 2.4 in Eq. 2.3 and substituting r ′ for r ′ A and r′ B
in Eq. 2.4.

2.2 Computation of stresses and displacements of dislocations 17
X
Z
O
r
r’
R
t
Y
ρ
L
Infinite
dislocation
Figure 2.5: Definitions of the geometry of Eq. 2.4
The formulae of both Li and de Wit are derived within the frame of the isotropic elasticity theory.
A numerical method for stress fields in anisotropic elasticity has been developed recently by Rhee
et al. in [Rhee et al. 01]. The difference between the isotropic and anisotropic solution was found
to have an important effect within only about 15b from the distorted hexagon they used for the
calculations. The difference, however, becomes smaller as the distance from the hexagon increases,
therefore it is sufficient to use the solution of the isotropic elasticity for long-range interactions.
The stress field of a prismatic loop represented by successive straight segments is shown to exhibit a
satisfactory accuracy comparing with the corresponding exact analytical solution ([Khraishi et al. 00a]).
The computation of the segment stress fields shows no anomaly even near the joint of two rectan-
gular segments. The contour of the resolved shear stress on the (¯11¯1) plane is shown in Fig. 2.6(b)
and the corresponding dislocation segments in Fig. 2.6(a).
Computation of internal stresses is the most computationally demanding spot in the DDD algo-
rithm. A method to increase the efficiency of computation will be discussed in Sec. 2.5.
Applied stresses
External stresses are applied in two ways, depending on the boundary conditions involved.
In the first case, the simulation volume represents a small element in a single crystal or a grain of
a polycrystal. In this case, the external stress field is assumed to be homogeneous throughout the
simulation volume. The same stress tensor is applied to each segment in the volume. The magnitude
of this tensor is updated according to a certain rule, constant stress or strain rate ([Fivel 97]).
In the second case, the simulation volume represents a finite volume with free surfaces, thus exter-

18 Description of the simulation method
1 µ m
Dislocation
segments
1 µ m
3 µ m
n=[111]
b=[110]
(a) Dislocation segments configuration (b) Contour of the resolved shear
Figure 2.6: A planar set of dislocation segments and the contour of the resolved shear stress on the
glide plane: The stress is computed at the corner where two orthogonal segments meet (shaded area
stress
of 1 µm × 1 µm). The resolved shear stress shows no anomaly.
nal stresses produce inhomogeneous stress fields in the volume. This inhomogeneity of the applied
stresses can be incorporated using a code coupled with a finite element method ([Fivel et al. 98]).
The more general cases which include internal interfaces, e.g. second phase particles or multilayer
films are treated in Sec. 2.4.2.
Line tension
The mutual effect between two adjacent segments, which is not considered in the internal stress
computation, is accounted for by a local line tension computation. The line tension T (θ) creates
a force τlt = T (θ)/(bR) along the center of a dislocation arc with a radius of curvature R. T (θ) is
given by the energy of a dislocation line E(θ) with θ being the angle that the Burgers vector makes
with the dislocation line direction.
T (θ) = E(θ) + d2 E(θ)
dθ 2
The simplest form of the line tension would be obtained by assuming that edge, screw and mixed
segments have the same energy per unit length, i.e., E = αµb 2 . The line tension of an arc of
dislocation then becomes τlt = αµb
R
from Eq. 2.5.
The energy of a dislocation is dependent on the character however: a screw dislocation has lower
energy than an edge one. This explains why a dislocation line shape is approximately elliptical with
a major axis parallel to the Burgers vector. To include the variation of the energy with a segment
character, the analytical equation of line tension suggested by Foreman [Foreman 67] (Eq. 2.6) is
(2.5)

2.2 Computation of stresses and displacements of dislocations 19
used.
b θ L
τlt τ’ lt
Dislocation line vector
Figure 2.7: Definition of the geometry of the line tension calculation.
τlt =
R
µb
4π(1 − ν)R (1 − 2ν + 3ν cos2
L
θ) ln − ν cos(2θ)
2b
µ and ν stand for the shear modulus and the Poisson ratio respectively. R is the radius of a circle
defined by the three center points of segments. L is the length of a segment and θ is the angle
between the Burgers vector b and the dislocation line vector. The dislocation line vector is taken
as parallel to the vector of two center points of the neighbor segments as illustrated in Fig. 2.7.
τlt is, in fact, the magnitude of the line tension along the direction to the center of the circle. τlt
projected to the glide direction of a segment is finally taken as the line tension acting on a segment.
The Peierls force
The Peierls stress refers to the applied resolved shear stress required to make a dislocation glide in
an otherwise perfect crystal. This effect arises as a direct consequence of the periodic structure of
the crystal lattice and acts as a friction to the dislocation motion. In the DDD, which cannot treat
atomistic effects explicitly because of the lattice parameter xl of the order of 10b (Sec.2.1.1), the
Peierls stress is simply implemented as a frictional force τp and contributes to the effective stress as
a back stress to motion of a segment. In practice, the frictional force τp includes all the chemical
effect, the impurities, and solutes etc. identified on experiments ([Déprés et al. 04]). In the case
of FCC metals, τp is the order of 10 −5 µ, thus is expected to have a minute effect on the simulation
results.
(2.6)

20 Description of the simulation method
Effective stresses
After the internal (σint) and the applied stresses (σapp) are computed, the force on a slip system
is defined by the Peach-Koehler equation and a projection along the glide direction g as shown in
Eq.2.7.
, where l is the unit vector tangent to the dislocation line.
τg b = {[(σint + σapp) · b] × l} .g (2.7)
It should be noted that σint and σapp are computed at the center of a given segment on the
assumption that the stress field variations are small over the segment length. The effective stress
τe is then computed by summing all the contributions as τe = τg + τlt − τp. Then, the velocity of
the dislocation segment is given by Eq. 2.13.
2.2.2 Computation of displacements
The computation of the displacement field of dislocations is very useful not only in analyzing surface
deformation induced by dislocations, but also in imposing displacement boundary conditions in a
coupling method with a finite element method (Sec. 2.4.2).
The displacement solution of any closed curved dislocations can be found from the Burgers formula
in the frame of elastic isotropy. The Burgers equation is given in terms of line and area integrals as
shown in Eq. 2.8 in a vector form.
u(r) = − b
1 b × dl
Ω −
4π 4π C
′
R +
1
8π(1 − ν) ∇
(b × R) dl
C
′
R
b is the Burgers vector and ν is the Poisson ratio. Ω is the solid angle through which the positive
side of a loop is seen and is defined as follows.
RdA
Ω = −
A R3 The parameters for the computation Eq. 2.8 and Eq. 2.9 are shown for the configuration of a closed
loop in Fig. 2.8.
An analytical solution of the displacement fields can be obtained using Eq. 2.8 for the case of
simple dislocation loops 7 . The solutions of complex dislocation loops are generally difficult to be
resolved analytically. The general way of computing a displacement field of an arbitrary dislocation
loop is to decompose the loop into triangular loops as illustrated in Fig. 2.8. The methodology
to construct a displacement field from triangular loops was first presented by Hirth and Lothe (see
7 Khraishi et al. ([Khraishi et al. 00b]) have found a closed-form analytical solution of a circular dislocation loop.
(2.8)
(2.9)

2.2 Computation of stresses and displacements of dislocations 21
b
C
A
Field point
Ω
R
dl’
n Slip plane normal
Dislocation
loop
b
Triangular loop
n
Dislocation
segments
Figure 2.8: The parameters in the Burgers equation (Eq. 2.8) and decomposition of a dislocation
loop by triangular dislocation loops
[Hirth & Lothe 92]). Special care, however, should be taken at evaluating the inverse trigono-
metric functions, as the author experienced. Barnett ([Barnett 85]) has developed a formula more
suitable for numerical computation, which will be detailed below.
The displacement at a field point P(r) generated by a triangular dislocation loop with points
A(rA), B(rB) and C(rC) are expressed as Eq. 2.10. The triangular dislocation loop ABC and a
field point are shown in Fig. 2.9.
u(r) = − b
4π Ω + FAB + FBC + FCA
(2.10)
Ω is the solid angle associated with the triangle ABC, which generates a discontinuity of ∆u = b
in traversing the cut surface ABC. F ij(i,j=A,B or C) is a continuous displacement field term ex-
cept on the dislocation line. The solid angle Ω and the continuous terms Fij are given as follows
([Barnett 85]).
s
s − a s − b s − c
Ω = −sign (Ri.n) 4 arctan tan tan tan tan
2 2
2
2
Fij = −
1 − 2ν
8π(1 − ν) (b × tij) ln Rj + Rj.tij
Ri + Ri.tij
+
1
8π(1 − ν) (b.nij)
Rj
Rj
The vectors and the constants in Eq. 2.11 and Eq. 2.12 are listed below.
− Ri
× nij
Ri
(2.11)
(2.12)

22 Description of the simulation method
n
B(r
)
B
A
(r )
A
RA
RB
P (r)
RC
C (r C )
Triangular loop
Figure 2.9: A geometric configuration of a triangular loop and the parameters for the computation
of displacements using Eq. 2.10
⎧
⎪⎨
⎪⎩
s = a+b+c
2
a = arccos (rB−r).(rC−r)
rB−rrC−r
b = arccos (rA−r).(rC−r)
rA−rrC−r
c = arccos (rA−r).(rB−r)
rA−rrB−r
Ri = ri − r
tij = rj−ri
rj−ri
nij = Ri×Rj
RiRj
The displacements at any field point by a dislocation loop are obtained by the summation of the
displacements of triangular loops which comprise the dislocation loop. As an example, the displace-
ment field of a interstitial prismatic loop (Fig. 2.10(a)) computed by Eq. 2.11 and Eq. 2.12 is
shown in Fig. 2.10(b). It can be seen that the interstitial prismatic loop induces the maximum
displacement of 0.5b on the plane just above the loop.
Displacement computation of more general cases of dislocation loops will be presented in Sec. 4.3.5,
where the presented computation method is applied to the analysis of surface deformation during
fatigue tests (see Sec. 4.3.5).

2.2 Computation of stresses and displacements of dislocations 23
Computation plane
(a) Schematic of the deformation around an interstitial prismatic
loop
b
e 1
e2
Probing line
Surface, b
Probing line, µ m
(b) Computed displacement field around an interstitial prismatic loop
Figure 2.10: Computations of displacements induced by an interstitial prismatic loop using Eq. 2.10

24 Description of the simulation method
2.3 Motion of dislocations
2.3.1 Preliminaries
The stress field of a moving dislocation is, in fact, not equivalent to that of a static dislocation.
Under most dynamic conditions of practical interest, however, dislocations move in such a way
that the dynamic stresses and displacements can be approximated quite accurately by the static
solutions, e.g., the stress equations presented in Sec. 2.2.
Only dislocation glide on a slip plane is considered in the current DDD code. No climb mechanisms 8
are implemented here. Theoretically, diffusion theories could be incorporated in the DDD code to
treat climb event properly, because climb involves interactions between dislocations and point defects
(vacancies or interstitial atoms). Numerically, it would be necessary to include a new line vector
and a glide direction into Tab. 2.1, because climb involves the nucleation and motion of jogs.
Dislocation mobility is dependent on the applied shear stress and temperature. It varies also with the
crystal purity and the dislocation type 9 . There are a number of forms for the relations between glide
velocity and the effective shear stress, including power law forms and expressions with an activation
term in an exponential function to represent the temperature-dependency ([Hirth & Lothe 92],
[Kocks et al. 75]). A simple power law form is adopted in this work for convenience sake, but any
forms of equation can be readily adopted.
2.3.2 Dislocation mobility
The simple power law relation (v ∝ (τ) m ) is used to compute the dislocation velocity. A linear form
of the equation, m=1, is known to predict well the case of glide over the Peierls barrier in FCC
metals.
The velocity of a dislocation segment is given by
vi = τe|b|
B
(2.13)
with the effective stress of segment (τe), the Burgers vector (b) and the phonon drag coefficient
(B) 10 . At room temperature, the coefficient B is found to be of the order of 10 −4 Pa·s for aluminium
8 a process by which an edge dislocation can move out of its slip plane by diffusion
9 In BCC single crystals, for example, a pure screw dislocation is more difficult to move than a mixed one at low
temperature, since a screw dislocation has a complex core structure ([Urabe & Weertman 75]).
10 Damping forces, which oppose dislocation motion, arise from the scattering of lattice vibrations (phonons) or
electrons.

2.3 Motion of dislocations 25
([Mason 68]) and 1.5 · 10 −4 Pa·s for copper ([Fusenig & Nembach 75]). The coefficient B, in
fact, changes with the velocity of a dislocation as B = B0
1−v 2 /c 2 . For simplicity, a constant value of B
is used by putting a limit on the velocity of dislocations as vmax, so that v 2 /c 2 becomes relatively
small.
Using the velocity of a segment given by Eq. 2.13, the next position of the segment is solved by
explicit integration such that x t+∆t
i
= xt i + vi∆t, where xt i is the position of the segment at time t
and ∆t is the time step. As is a feature of the forward explicit algorithm, the use of a larger value
of ∆t causes a numerical instability. In the DDD method, a dislocation segment may oscillate,
because a large time increment causes a segment to move over a too large distance. This brings a
significant change in the local curvature, and in turn, produces an increase of the back stress (the
line tension). The segment oscillates consequently. The use of a constant value of ∆t in the range
from 0.5 × 10 −9 to 1. × 10 −9 has been verified successful in practice, but ∆t has to be adapted for
each simulation. The maximum velocity vmax is imposed so as to prevent the segments to glide over
a too large distance.
2.3.3 Dislocation-dislocation interactions
A segment can interact with other segments during the glide. The task is then to search any possible
intersection with segments within a virtual glide area of the gliding segment, which is defined by
the length of the segment, Li and the free flight distance, vi∆t. The nearest intersection point of
the possible interaction events is found from simple geometry of two finite lines (segments). The
type of interaction is, then, determined by the relation of the Burgers vectors and the slip systems
of the two intersecting segments.
The types of possible dislocation-dislocation interactions considered in the DDD model are catego-
rized as follows:
a. coplanar cases in which two dislocation segments glide on the same plane
b. non-coplanar cases in which two dislocation segments glide on different planes
(a) Coplanar cases
The portion of intersection of two segments with the same Burgers but opposite in direction (oppo-
site sign) is deleted and the links of the rest segments are rebuilt as shown in Fig. 2.11(a). In case
of the same sign, no interaction is realized, since it is elastically repulsive. Only discretization of a
segment is done for the next step as illustrated in Fig. 2.11(b).

26 Description of the simulation method
Opposite sign
Same sign
(a) Annihilation
(b) Repulsion
Annihilation
Discretization
Figure 2.11: Interaction between two segments in the same glide plane: Segments are annihilated
if the sign is opposite, and discretized if the sign is same.
No explicit handling is done in the case of two different Burgers vectors in the same plane, which
corresponds to the a copla
1
(b) Non-coplanar cases
case explained below.
Before introducing interaction handling schemes for non-planar cases, dislocation junctions are
presented, because such interactions result in the formation of junctions. In the frame of the
hardening theory, five different forms of dislocation junctions are usually considered:
(i) a coli
1
(ii) a ortho
1
(iii) a copla
1
for which b1 = b2 on different slip planes
for which b1 ⊥ b2 on different slip planes
for which b1 = b2 on the same slip plane
(iv) a2 for which b1 + b2 is glissile on either of the planes
(v) a3 for which b1 + b2 is sessile on either of the planes
The junctions formed between slip systems are tabulated in Table 2.2 for the 12 slip systems defined
in Table 2.1.
(i) a coli
1
is represented in the DDD by changing neighboring arms between two interacting seg-
ments 11 . Fig. 2.12 shows the intersection of two dislocation segments, which glide on a slip plane
11 Its role in dislocation-hardening can be found in [Madec et al. 03].

2.3 Motion of dislocations 27
A2 A3 A6 B2 B4 B5 C1 C3 C5 D1 D4 D6
A2 a0 a copla
1
a copla
1
a coli
1 a2 a2 a ortho
1 a2 a3 a ortho
1 a3 a2
A3 a0 a copla
1 a2 aortho 1 a3 a2 acoli 1 a2 a3 aortho 1
A6 a0 a2 a3 aortho 1 a3 a2 aortho 1 a2 a2 acoli 1
B2 a0 a copla
1
a copla
1
a ortho
1 a3 a2 a ortho
1 a2 a3
B4 a0 a copla
1 a3 aortho 1 a2 a2 acoli 1
B5 a0 a2 a2 acoli 1 a3 a2 aortho 1
C1 a0 a copla
1
a copla
1
a2
a2
a coli
1 a2 a2
C3 a0 a copla
1 a2 aortho 1
C5 symmetric a0 a2 a3 a ortho
1
D1 a0 a copla
1
a3
a copla
1
D4 a0 a copla
1
D6 a0
Table 2.2: Hardening coefficients
and its deviate plane respectively. Segments change its neighbors upon intersection and make an
angular dislocation with θ = 70.53 ◦ .
(ii) No explicit treatment is done on a ortho
1
(iii) No explicit treatment is done on a copla
1
(known as Hirth lock).
as explained in the coplanar cases above.
(iv) & (v) a2 (known as Glissile junction) and a3 (known as Lomer-Cottrell lock) are implicitly
adopted with the simple energy analogy in Eq. 2.14 Two segments of the Burgers vector b1 and b2
n prim
b
devi
n
Figure 2.12: Changing of neighbor arms between two segments in primary and deviate planes.
n prim
b
devi
n

28 Description of the simulation method
b
(a)
b
(b)
Figure 2.13: Cross-slip of a screw segment
are considered to form a junction if a simple energy criterion
b 2 1 + b 2 2 > (b1 + b2) 2
b
(c)
(2.14)
is satisfied. The energy of a dislocation is assumed to be proportional to |b| 2 and has no dependence
on the line character in this criterion. Once a junction is formed, it is given a certain breaking
strength τjunc. The junction can be broken afterward only if the effective stress of a component
segment of the junction is larger than τjunc, thus the junction acts as a pinning point for the
dislocation motion.
Detailed studies of the dislocation junctions ([Shin et al. 01], [Rodney & Phillips 99]) show that
they are formed due to the elastic fields of two dislocation lines and the strength of junction is
governed by the "unzipping" mechanism. Thus the properties of junctions can be treated by the
elastic stress fields of the involved dislocations. The local breaking strength of a junction will serve
as pinning points in so-called mass simulations.
2.3.4 Cross-slip of screw dislocation segments
The cross slip of a screw segment (Fig. 2.13) is implemented in a stochastic manner accounting for
its thermally activated character. A cross slip probability P over each time step is computed first
using the equation
P = β l
δt
L0 t0
τd − τIII
exp V
κT
(2.15)
,where β is a normalization coefficient, l is the length of the particular screw segment, L0 is 1 µm,
t0 is 1 sec, V is the activation volume, τd is the resolved shear stress in the cross slip system and
τIII is a threshold stress. A random number r is generated and the dislocation cross slip occurs
only if r is lower than P . As an example, the values used for V and τIII in copper are V = 350eV
and τIII = 32 MP a.

2.4 Boundary conditions 29
2.3.5 Plastic strain due to dislocation movement
Dislocation motion results in plastic strain. The plastic strain of a simulation volume is determined
by summing up the slipped area taking place in each slip system. The slip γ (s) of a slip system ’s’
is computed as
γ (s) = |b|A(s)
V
(2.16)
with b being the Burgers vector, V the volume of the simulation box and A (s) the area swept by
all the mobile dislocations of the slip system s over a time step. A (s) is defined as
A (s) =
Livi∆t (2.17)
i
where the summation is done over all the segments of the system s and Livi∆t is the area of glide
of a segment i with the length Li. The components of the plastic strain tensor are given by
ɛij =
12
s=1
1
n
2
(s)
i b(s)
j + n(s)
j b(s)
i γ (s)
(2.18)
with n (s)
i and b (s)
i being the component of the slip plane normal and the Burgers vector of the slip
system s respectively.
2.4 Boundary conditions
2.4.1 Periodic Boundary Conditions
Motivation and review of literatures
Typically simulation volume are 10 3 − 15 3 µm 3 large simulation volume for a so-called mass simu-
lation addressing work hardening or dislocation cell formation. In order to compare the simulations
to experiments, it is desirable to build a simulation volume representative of a small element taken
out from the single crystal or from the grain of a ploycrystal.
Periodic boundary conditions (PBC) forcing segments to cross a boundary between two cells to
emerge in all cells at the equivalent position on the opposite boundary, are extensively used to
avoid undesirable size effects due to the finite dimensions of a simulation volume. PBC can be
easily applied in 2D case, for example, by subtracting a simulation volume size (Lx, Ly) from coor-
dinates of dislocation segments leaving the initial volume. The simulations of Gómez-García et al.
[GG et al. 00] have shown many advantages of PBC compared to free boundary surfaces which are
bound to have artificial dislocation losses and undesirable size effects.

30 Description of the simulation method
0 1 ... ... L-2 L-1
L-1
Figure 2.14: Property of p.b.c
PBC in 3D was considered to be difficult because of the complexities related to the connectivity
of the dislocation lines segments after exiting from one boundary. Since [Bulatov et al. 01] has
demonstrated that PBC can be applied to dislocation dynamics, attentions have been given on the
stress calculation, initial configuration of dislocations and balancing the incoming and outgoing dis-
location fluxes. Madec et al. ([Madec et al. 04]) have reported that portions of dislocation loops
may self-annihilate with replicas having emerged after a certain number of boundary crossings.
This self-annihilation reduces the mean free-path of dislocations and consequently leads to spurious
self-interactions, because a short effective mean free-path affects the density of mobile dislocations
and their storage rate and, hence, both the microstructure arrangements and the strain hardening
properties. The artifact of self-annihilation can be avoided by using an orthorhombic simulation
volume.
A numerical method that Madec et al. ([Madec et al. 04]) have used to apply PBC in 3D is to
translate all the segments about a selected segment by shifting it to the center of a volume, and
apply "MODULO" operations so that the segment coordinates x larger than the simulation size Lx
are replaced by the remainder of x/Lx.
Numerical implementation
A perpendicular line joining two opposite boundaries is actually equivalent to a line on a bracelet
(Fig. 2.14). A quantity ic(i) with an integer argument i in the range of [0 : L − 1] has the
property as given by Eq.(2.19) under PBC. Eq. (2.19) is merely another expression of Fig. 2.14 in
L-2
...
0
...
1

2.4 Boundary conditions 31
a mathematical form.
ic(L + i) = ic(i)
ic(−i) = ic(L − i) for i = 0, . . . (2.19)
The array ipc can be used to redirect a segment coordinates, which has left an initial volume lattice,
to an equivalent position in the initial lattice. Then it is possible to apply PBC with a simple
array reference. The orthorhombic simulation volume is readily realizable by changing the range of
periodicity according to the maximum length of the simulation volume along each axis.
Because of the subnetwork in the simulation volume (see Sec. 2.1.3), the periodicity should be a
multiple of 4xl along each axis.
2.4.2 Internal interfaces
Motivations and review of the literature
The collective behavior of dislocations in a single crystal can be simulated with the stress com-
putation and motion treatments as explained in the previous sections. More rigorous boundary
conditions need to be implemented on the method in order to treat more general cases, such as
a crystal with free surfaces, a crystal containing particles of a second-phase or a polycrystal with
grain boundaries.
A dislocation experiences forces near an interface because the dislocation energy is different in the
two mediums involved. The dislocation is attracted towards a free surface, for example, and repelled
by a rigid surface layer. These image stresses can be treated by using the superposition principle.
The effects of free surfaces were treated by Fivel et al. ([Fivel & Canova 99]). The forces ex-
erted on a free surface by dislocations are computed assuming that the dislocations are embedded
in an infinite medium. These forces are then reversed and changed into the appropriate point
forces to enforce the traction free surface condition. Applications of this method can be found in
[Fivel et al. 98].
The image stresses by a free surface is, in fact, a special case of the more general situation in which
an interface separates two materials of differing elastic constants, e.g., oxide layers and particles.
The image stresses on a dislocation in the presence of a second phase particle can be computed also
by the superposition principle. The formulation follows that of Van der Giessen and Needleman
([Giessen & Needleman 95]). Previous applications of this method to 2D cases can be found in
Cleveringa et al. ([Cleveringa et al. 97]).

32 Description of the simulation method
Facet
,τ facet
Glide plane
Intersection
Segment, τ e
Figure 2.15: Geometries of a facet and construction of a sphere by square facets.
In this section, complete method to treat internal interfaces is described. Firstly, an interface is
represented by a set of facets, which have a certain strength thus, can act as barriers to dislocation
motion. The application of this method can be found in Sec. 4.3. Secondly, a full account of elastic
interaction with dislocations or image stresses is presented using a coupling method with a finite
element method. The method is applied to compute the image stresses in Sec. 4.1.
Internal interfaces represented by facets
A curved 3D boundary is approximated by a series of facets, in the same way as a surface is repre-
sented by a finite element meshing. Each facet is defined by indexes of its nodes or vertices, whose
coordinates are stored separately.
Intersection events between a segment and a facet is detected by determining the nearest intersec-
tion point between a facet and a virtual glide plane of a segment (Fig. 2.15).
Each facet is given a strength τfacet. Only segments whose effective stress are greater than τfacet
are allowed to cross the facet, i.e. a facet acts as a barrier to dislocation motion. The strength
is further specified by τ +
facet
and τ −
facet
depending on the relation between moving direction of a
segment (g) and the normal direction of a facet (n), which makes the application of facets more
flexible. A facet of τ +
facet
being 100 MP a and τ −
facet
being 0, for example, means that the facet
blocks all the segments of g · n > 0 if the effective stress is lower than 100 MP a, but is transparent
to segments moving along −n direction, that is, g · n < 0.
The facet method enables simple geometrical barriers to dislocation motion in dislocation dynamics.
When applied to treat the case of dislocation-precipitate interactions, this method can be consid-

2.4 Boundary conditions 33
ered as a first-order approximation in the sense that elastic interactions between dislocations and
interfaces are not considered. The application of the facet model to a particle-strengthened crystal
is presented in Sec 4.3, where the hypothesis used to derive the strength τfacet are explained in
detail for the unshearable and shearable particle cases.
Full account of elastic interactions
The problem of a finite volume containing dislocations is decomposed into two problems:
1. the problem of dislocations in an infinite elastic isotropic medium
2. the complementary problem of a finite dislocation-free volume, which compensates for the
proper boundary conditions.
The problem decomposition is shown schematically in Fig. 2.16. The stress and strain field of the
current state of the body, σ and ɛ are determined by the governing equations
∇ · σ = 0, ɛ = ∇u
σ = LM : ɛ in VM and σ = LP : ɛ in VP
with moduli LM, LP of matrix and particle respectively. Boundary conditions are
u = Uap on Su, n · σ = Fap on St.
After the decomposition, the stress and strain fields are written as the superposition of two fields:
ɛ = ɛ1 + ɛ2, σ = σ1 + σ2 (2.20)
In the first problem (denoted by 1), it is assumed that the dislocations are in an infinite elastic
isotropic medium. The stress(σ1) /strain(ɛ1) relationship is expressed as σ1=LM : ɛ1 in the whole
volume. The forces FD and displacements UD on the virtual boundaries can be computed by the
expressions presented in Sec. 2.2.
In the second problem, the simulation volume contains no dislocations. The fields in the second
problem (denoted by 2) are the fields needed to correct the actual boundary conditions as well as for
the presence of the inclusions. The governing equations for the complementary problem becomes:
σ2 = LM : ɛ2 in VM
σ2 = LP : ɛ2 + σcorrec in VP (2.21)

34 Description of the simulation method
The complementary problem has a correction term such as σcorrec = (LP −LM) : ɛ1 in VP . With this
correction term, the current stress field in the particle volume can be constructed by the superposi-
tion of two fields (σ = σ1 +σ2 in VP ). By replacing ɛ1 with L −1
M : σ1, this correction term equals to
(LP : L −1
M − I) : σ1, where I represents the fourth-order unit tensor. (LP : L −1
M − I) is expressed by
the Young’s modulii and the Poisson’s ratios of the matrix (E, ν) and the particle (E ∗ , ν ∗ ) as follows.
(LP : L −1
M
⎛
⎞
a1 − 1
⎜
− I) = ⎜
⎝
a2
a1 − 1
a2
a2
a1 − 1
0
0
0
a3 − 1
0
0
0
0
a3 − 1
0
0
0
0
0
a3 − 1
⎟
⎠
with a1 being E∗ (1−ν∗−2νν ∗ )
E(1+ν∗ )(1−2ν∗ ) , a2 being
E∗ (ν∗−ν) E(1+ν∗ )(1−2ν∗ ) and a3 being E∗ (1+ν)
E(1+ν∗ ) .
The complementary problem is solved using CAST∃M( 12 ), a finite element code developed in France
by the Commissariat à l’Energie Atomique. The FEM formulation in the particle volume of the
second problem can be written as follows with a strain-displacement matrix (B) :
VP
B T
· L · BdV · u +
VP
B T · LP : L −1
M − I · σ1dV = 0 (2.22)
In order to compute the body force-like term (right-hand side of Eq. 2.22) within the precipitate
volume, first the stresses σ1 due to the dislocations at the points in VP , e.g., at the stress integration
Gauss points, are computed. Then the correction term is computed within each finite element inside
the particle. The computed stresses σcorrec are then changed into a nodal body force field, f b . This
can be easily done with the ’BSIG’ operator in CAST∃M. These forces are applied to VP and the
FEM gives the solution of a two phases boundary problem where forces Fap −FD and displacements
Uap − UD are imposed at the boundary and body forces f b are applied inside the particle. The
continuity of displacement and normal stresses at the particle/matrix interface are enforced by the
FEM.
12 Finite element code developed by Commissariat à l’Energie Atomique, CEA-DRN/DMT/SEMT

2.5 Acceleration of the DDD code 35
Figure 2.16: Decomposition of the problem into the problem of dislocations in infinite media and
the complementary problem of inhomogeneous finite volume without dislocations. Forces(Fap) and
displacements(Uap) are applied on the boundary. In the complementary problem, boundary condi-
tions are modified with forces(FD), displacements(UD) and a nodal body force field(f b ) generated
by dislocations.
2.5 Acceleration of the DDD code
2.5.1 Problem description and review of literatures
Internal stress computation is the most computationally intensive part in the DDD method. This
is due to the fact that the stress field at a distance r from a dislocation line is proportional to 1/r.
The stress field of a dislocation line is thus long-ranged. Another time consuming part in the DDD
method is handling the dislocation segments interactions. Segment motion involves examination of
possible interactions, between dislocations or between dislocation and internal interfaces.
In a programming perspective, the two parts can be represented as follows in pseudo-code.
Internal stress computation
DO I=1,Nsegm
DO J=1, Nsegm
if(J=I and I’s neighbor)
ENDDO
ENDDO
Compute σ int
I←J
Nsegm: Number of segments
Nfacets: Number of facets
Segment motion
DO I=1,Nsegm
DO J=1, Nsegm
Examine interaction with segment J
ENDDO
DO K=1, Nfacets
Examine interaction with facet K
ENDDO
Move segment I
ENDDO

36 Description of the simulation method
Both parts need the order of N 2 segm computation with Nsegm being the number of dislocation seg-
ments. As for the segment motion, Nsegm × Nfacets additional computations are required. It should
be noted that in the ’Segment motion’, each segment is treated and moved sequentially. In addition,
each individual segment displacement generates a new dislocation configuration. In complex situ-
ation, changing the computing order of the segments may slightly change the resulting dislocation
structure.
In Molecular Dynamics simulations, the stress computation favors the use of a cut-off distance, be-
yond which the stress is called a long-distance stress and neglected, because the interatomic stress
field is short-ranged. This cut-off scheme reduces the cost of the stress computation with a minor er-
ror. In Dislocation Dynamics simulations, however, the cut-off distance scheme may cause a spurious
formation of cells ([Gullouglu et al. 89]). The study of Devincre et al. ([Devincre et al. 01])
has shown, however, that neglecting the long-distance stresses does not affect much the yield stress
and hardening properties of FCC single crystal. It should be noted that the study was dealing with
dislocation patterning in multislip conditions, where cross-slip of dislocations, which is governed by
a short-distance stress, is supposed to play an important role. However, it would be difficult to
generalize Devincre et al.’s observation to other situations. Thus it is generally required to take into
account all the dislocations in the simulation volume to compute the internal stresses.
Reasonable approximations in the computation of the internal stresses have been made to overcome
this severe computational limitation. In [Verdier et al. 98], the simulation volume is decomposed
into boxes and short- and long-distance stresses are classified by the topology of boxes. The com-
putational cost can be reduced by updating the long-distance stresses less frequently. The concept
of superdislocation has been adopted in stress computation by [Zbib et al. 98]. The idea is to re-
place a large number of dislocation segments beyond a certain distance into a limited number of
superdislocations, which have a modified Burgers vector magnitude. This method is based on the
multipolar expansion of the elastic field of a 2D dislocation array and extended in 3D by a simple
’projection-extension’ method.
2.5.2 The Box method
The box method proposed by [Verdier et al. 98] is based on the fact that a dislocation micro-
structure does not change rapidly by comparison with the short time step used in the simulation
(O(10 −9 sec.)). Thus stress fields of the long-distance segments could be updated with a certain
frequency and between updates the previous values could be used with an acceptable error.

2.5 Acceleration of the DDD code 37
(a) Dislocations in a cubical simulation volume (b) Division of the simulation volume into boxes
Figure 2.17: Decomposition of a simulation volume into boxes: (a) A typical dislocation structure
in a cubical simulation volume (b) Simulation volume divided into 10 × 10 × 10 boxes
The simulation volume is first decomposed into boxes. For the sake of simplicity of the computation
scheme, each side of the simulation volume is divided into M boxes. Hence the simulation volume
comprises M 3 homologous boxes. Fig. 2.17(a) shows a typical simulation volume with dislocation
segments. Fig.2.17(b) is an example of the same simulation volume decomposed into 10 3 boxes.
To facilitate the identification of the segments in the box ib, linked-lists of segments are constructed.
As shown in Fig. 2.18, the mid-point of a segment (imid(i)) is used to determine the box index (ib)
to which it belongs.
ib = 1 + imid(1)
N1
M
+ imid(2)
N2
M
M + imid(3)
M N3
M
2
(2.23)
N1, N2 and N3 are the sizes of the orthorhombic simulation volume along x, y and z axis. The array
indexb(ib) saves the number of segments belonging to the box ib. An array isbox(ib, 2) saves the
index of the first segment in the box ib. The linked-list of segments is implemented with the array
isbox(is, 1 : 2): isbox(is, 1) indicates the index of segment prior to is and isbox(is, 2) reserves the
index of segment posterior to is. The identification of the segments in box ib is shown schematically
in Fig. 2.18. A segment can be easily added or subtracted by switching the array isbox.
Note that the number of boxes is limited since the box size should be big enough. The minimum size
of the boxes is chosen so that the first neighboring boxes include the maximum free-flight distance of
a segment. The criterion defining the minimum edge length of a to adopt for a box can be expressed

38 Description of the simulation method
as Eq.(2.24).
ib1 ib2
1
ib3 3 ib4
5
15
14
7
11
16
6
212
9
10
8
13
4
indexb
ib1 7 1
ib2 10 4
ib3 3 5
ib4 12 2
Figure 2.18: Linked list of segments
min a ≥ 1
√ ld, a ≥
2 2
√ vmaxδt
6
ib1 ib2 ib3 ib4
2 4 5 5
6 16
14 11 15
13 9 8
(2.24)
The first term in Eq.(2.24) states that neighbors of a segment are located inside the first neighbor-
hood, hence it is given by the discretization length ld. The second term states that a segment is not
allowed to move across the first neighboring boxes in one step, thus it is a function of the maximum
velocity vmax (see Sec 2.3.2) and time step ∆t. Fig. 2.19 shows the criterion of the minimum edge
length of a box.
The use of the box sizes larger than the minimum size of the box reduces computing cost in ’Segment
motion’ part. This is because only interactions within a maximum free-flight distance of a segment
need to be considered instead of taking all the segments and facets into consideration. This will
reduce the number of segments and facets to be inspected without any approximation.
The internal stress acting on a segment is divided into a long-distance stress (σ LR ), which varies
rather slowly over time steps and a short-distance stress (σ SR ), which shows large fluctuation over
single time steps. σ SR of a segment in the box ib is computed by taking all the segments into
account in the L th neighboring boxes at every simulation step. σ LR is adopted by the stress at the
center point of ib from all the segments outside of the L th neighboring boxes. All segments in the
same box, therefore, have the same σ LR . This approximation is valid if σ LR has a wave length
larger than the box size. The computation of σ LR for all the boxes is updated every f step.
The parameters involved in the box method are listed in Table 2.3. The maximum number of boxes
M is given by the minimum box size (Eq.(2.24)) and the simulation volume size. The other param-
eters should be chosen based on the numerical accuracy and the speedup, and will be the issue of
the following section.

2.5 Acceleration of the DDD code 39
[100]
[001]
Slip direction,
[112]
(1)
(2)
Slip plane,
(11-1)
Screw
segment
Figure 2.19: Minimum box size
Parameter Description
[010]
M number of boxes along each side of a simulation volume
f frequency of σ LR update
L number of layers for σ SR
Table 2.3: Parameters of the box method
The pseudo-code of internal stress computation in Sec. 2.5.1 is then substituted by the following
pseudo-code.
Internal stress computation
DO I=1,Nsegm
Identify the box ’ib’ of the segment ’I’
Compute the stresses σ SR by segments
within short-distance boxes
Add σ LR (ib)
ENDDO
Long-distance stress computation (every f
step)
DO iz=1,M
DO iy=1,M
DO ix=1,M
compute the box index ’ib’
compute the long-distance stresses σ LR (ib)
ENDDO
ENDDO
ENDDO
at the center of ’ib’

40 Description of the simulation method
And the pseudo-code of the segment motion in Sec. 2.5.1 is replaced by
Segment motion
DO I=1,Nsegm
Identify the box ’ib’ of the segment ’I’
Examine interaction with segments and facets
within short-distance boxes
Move the segment ’I’
ENDDO
2.5.3 Speedup and Error
Optimum values of M, L and f in Table 2.3 should be chosen so as to minimize errors and maximize
speedup. There exist two sources of errors in the internal stress computation, i.e. a spatial and a
temporal error. The spatial error occurs because σ LR is computed at the center point of a box and
assigned to all the segments in that box. The temporal error is induced by updating σ LR with a
frequency f so that σ LR of the previous computation is used during f steps.
Speedup
The speedup is defined as the ratio between the execution time of the box method and that of the
original method. It is used to measure the relative algorithm performance. To facilitate an analyt-
ical relation, the execution time is assumed to be proportional to the number of computations.
Nsegm segments are assumed to be homogeneously distributed over the simulation volume de-
composed by M 3 boxes. When not using the box method, the number of computations of in-
ternal stress (n orig
s ) is N 2 segm 13 . Using the box method, the number of computation for σ SR is
(2L + 1) 3 Nsegm
M 3 Nsegm and M 3 − (2L + 1) 3 Nsegm
M 3
M 3
f for σLR . The speedup of the box method is
then given as Eq. 2.25 14
Speedup = norig s
nbox s
=
(2L + 1) 3 N 2 segm
M 3
N 2 segm
+ (M 3 −(2L+1) 3 )Nsegm
f
(2.25)
Solid lines in Fig. 2.20 show Eq. 2.25 as a function of M for Nsegm = 10, 000, 20, 000 and 90, 000.
The number of layers L is set to 1 and σ LR update frequency f is 20. There exist maxima in
13 The number of computations is Nsegm(Nsegm − 3) precisely because two neighbor segments and itself are not
considered in the internal stress computation. For simplicity, it is approximated by N 2 segm
14 It should be noted that the equation is derived with the assumption that periodic boundary conditions are applied
as detailed in Sec. 2.5.4

2.5 Acceleration of the DDD code 41
Speedup
140
120
100
80
60
40
20
0
N segm =90k
N segm =20k
N segm =10k
N segm =20k(measured)
N segm =10k(measured)
5 10 15 20 25
M
Figure 2.20: Evolution of the speedup of domain decomposition method as a function of the number
of boxes (M) and number of segments (N) in the condition that L = 1, f = 20
speedup depending on the number of segments. Solid dots represent the actual data measured with
a 3.0-GHz Intel Pentinum 4 processor and 1 GB of memory. Only the elapsed time for computing
the internal stress is measured. The measured data reflects well the characteristic of Eq. 2.25, even
though the segments are not distributed perfectly homogeneously.
The effect of f on the speedup is shown in Fig. 2.21(a), and that of increasing L is shown in Fig.
2.21(b). The optimum value of M is dependent on the value of f and L.
There is always gain in speedup regarding the segment motion by increasing the number of boxes.
Assuming Nsegm segments and Nfacets facets, the speedup in examining the interactions can be
represented as Eq. 2.26.
Speedup = norig o
nbox o
= N 2 segm + NsegmNfacets
N 2 segm
M 3
+ NsegmNfacets
M 3
= M 3
(2.26)

42 Description of the simulation method
Speedup
80
70
60
50
40
30
20
10
0
f=30
f=20
f=10
5 10 15 20 25
M
(a) Effect of f (L = 1, Nsegm = 20, 000)
Speedup
70
60
50
40
30
20
10
0
L=1
L=2
L=3
5 10 15 20 25
M
(b) Effect of L (f = 20, Nsegm = 20, 000)
Figure 2.21: Effect of the number of layers (L) and frequency (f) on the speedup of stress compu-
tations
Spatial error
A large box size or small M is bound to have a large spatial error at the end of the diagonal due
to a large deviation from σ LR computation position (center of the box)(see Fig. 2.22(a)). A small
box size or large M has also a large spatial error (Fig. 2.22(b)), but in this case the reason is due to
that the front boxes for σ LR computation are too close. Segments in the front boxes would generate
highly inhomogeneous stress fields.
In Fig. 2.23, relative spatial error along a diagonal of the central box is shown for each M. Here,
the simulation volume is cubic shape with the edge length of 16.4 µm containing 22, 210 segments
(ρ 2.5 × 10 12 m −2 ), which is taken from a tensile simulation along [001]. ɛr is defined as Eq. 2.27
with σ exact computed at each point on the diagonal and σ approx computed at the center point of
the box.
6
|σ
i=1
ɛr =
exact
i
6
σexact i
i=1
− σ approx
i |
(2.27)
From the figure, it can be seen that both small and large values of M increase the relative spatial
error. To compare the curves, ɛr is averaged and shown in Fig.2.24 as a function of M . There is a
certain value M that has a minimum spatial error.
The most effective way to minimize the spatial error would use the smallest box size with a certain
number of layers L for σ LR as shown in Fig. 2.22(c) with L = 3. Indeed, the mean spatial stress

2.5 Acceleration of the DDD code 43
is
Long-distance
stress
(a) M=5, L=1
Short-distance
stress
is
Long-distance
stress
(b) M=15, L=1
Short-distance
stress
is
Long-distance
stress
(c) M=15, L=3
Short-distance
stress
Figure 2.22: Effect of the number of boxes (M) and layers (L) on the accuracy of stress computations
Relative spatial Error
0.25
0.2
0.15
0.1
0.05
M=21
M=15
M=7
0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Position on the diagonal(microns)
Figure 2.23: Relative spatial error along a diagonal of the central box. (M=number of boxes along
each axis, L=1)

44 Description of the simulation method
Mean Relative Spatial Error
0.1
0.08
0.06
0.04
0.02
0
6 8 10 12 14 16 18 20 22
Figure 2.24: Mean relative spatial error of stress computation as a function of the number of boxes
(M) and the number of layer (L)
M
L=2
L=3
could be decreased down to 2% using L = 3 as shown in Fig.2.24.
Temporal error
It is difficult to evaluate a temporal error because it is strongly related to how fast a dislocation
structure changes, which is governed by both the type of mechanical test simulated and the time
step. So it is difficult to set f a priori. In order to evaluate the effect of f, a simple tensile test
has been done in a constant strain-rate condition (˙ɛ = 10 3 sec −1 ) with 22, 210 initial segments. The
time step is set to 2.10 −10 second and M = 21, L = 3. During the test, internal stress is recorded
at the center point of a simulation volume. A relative temporal error is defined as in Eq. 2.27.
σ exact denotes the internal stress at the central point computed over each time step and σ approx is
the stress with σ LR updated at the frequency f. In Fig. 2.25, the relative temporal error is shown
for the cases f = 20, 40, and 60. In the case f = 60, the maximum error level observed reaches 5%.
The update frequency of 60, however, induces a negligible effect on the overall stress-strain curve
as shown in Fig. 2.26.
In conclusion, the use of a maximum number of boxes is favorable, although speedup analysis in
stress computation indicates that there exists an optimum number of boxes. One reason is that

2.5 Acceleration of the DDD code 45
Temporal error
0.06
0.05
0.04
0.03
0.02
0.01
0
1200 1250 1300 1350 1400
Step number
f=60
f=40
f=20
Figure 2.25: Effect of the frequency of long-distance stress computation on the relative temporal
error. (f=σ LR update frequency)
Stress[MPa]
180
170
160
150
140
130
120
110
Reference
f=20
f=40
100
f=60
0.0e0 4.0e-5 8.0e-5 1.2e-4 1.6e-4 2.0e-4
Strain
Figure 2.26: Stress-strain curves of simulations with the σ LR update frequency f

46 Description of the simulation method
Shifted
boundaries
Original
boundaries
Figure 2.27: Computation of stresses under periodic boundary conditions and the box method
a large number of M always delivers an advantage on segment motion by the factor of 27/M 3 .
Another one is related to the parallelization scheme, which will be detailed in Sec. 3.3.
2.5.4 Boxes and Periodic boundary conditions
An efficient method to apply PBC is presented in Sec. 2.4.1. When the simulation volume is divided
into boxes and the internal stresses are decomposed into long- and short-distance stresses, attention
should be paid to the segments in the boundary boxes. As shown in Fig. 2.27, some of the boxes
(especially along the boundaries) may need to account for segments inside so-called image boxes
for internal stress computation and segment motion. The segments’ coordinates in the image boxes
are determined by translating the segments coordinates from the appropriate boxes. This operation
can be performed by a simple array reference and addition/subtraction.
Fig. 2.28 shows the example of the activation of a Frank-Read source in the cubic and the or-
thorhombic simulation volume, and the number of segments is recorded in Fig. 2.29. In the case of
the cubic simulation volume, the self annihilation of segments occurs and the number of segments
oscillates as shown in Fig. 2.29, whereas the dislocation density increases in the orthorhombic sim-
ulation volume. It is desirable to use the orthorhombic simulation volume to remove the artificial
self annihilation of dislocations due to the periodic boundary conditions.

2.5 Acceleration of the DDD code 47
[100]
[001]
[010]
(a) The cubic and the orthorhombic simu-
lation volume
(b) Dislocation structure seen at (110) di- (c) Dislocation structure seen at (110) di-
rection in the cubic simulation volume
rection in the orthorhombic simulation vol-
Figure 2.28: Activation of a Frank-Read source in the cubic and the orthorhombic simulation volume
under periodic boundary conditions
ume

48 Description of the simulation method
Number of segments
1800
1600
1400
1200
1000
800
600
400
200
Orthorhombic
Cubic
0
2500 3000 3500 4000 4500
Step number
Figure 2.29: Change of the number of segments with respect to the simulation steps. In the case of
the cubic simulation volume, self annihilation of dislocations occurs.
2.6 Computation procedure of the DDD program
The serial DDD program using the box method can be subdivided into the following tasks.
a. Initialization
b. Discretization of the segments
c. Construction of the linked-lists
d. Updating of the long-distance stresses every f steps
e. Computation of the short-distance stresses
f. Motion of segments
g. Updating the external stresses
h. Save of outputs
A simulation initialization (a) requires to set parameters such as the number of time steps, the
number of boxes, the material property constants and the loading conditions etc. It also reads the
initial segment configurations, geometries of the simulation box and internal interfaces from external

2.6 Computation procedure of the DDD program 49
files.
Operations from ’b’ to ’h’ are executed sequentially over each time step. The segments which are
larger than a maximum length (defined explicitly at initialization) are further discretized in the task
’b’. Linked-lists of segments in each box are constructed (’c’). Long-distance stresses are computed
every f steps at the center of each box (’d’) as described in Sec. 2.5.2. Short-distance stresses are
computed using linked-list of segments (’e’). Once stresses on each segment are known, the effective
stresses are computed using Eq. 2.7, and all the segments are moved to the next positions after
examination of all possible interactions (’f’). The external stresses are updated according to the
loading conditions (’g’), and output data like current stresses, strains and dislocation configurations
are saved in external files (’h’). This completes one time step and the same procedure is performed
at the next time step.
Key points
• The DDD method used in this work discretizes perfect dislocations into discrete dislocation
segments of a pure edge and screw type in a volume homothetic to the FCC structure with
the lattice spacing of ∼ 10b.
• The effective stress acting on a segment is computed, accounting for the internal, applied
stresses, line tension and the Peierls stress in the frame of linear isotropic elasticity.
The displacements of dislocation loops are computed using Barnett’s expressions.
• A linear relation between the effective stress and the velocity of segments is used. Dislocation
interactions are taken into account by local rules. The cross-slip of a screw segment is
implemented in a stochastic manner.
• Periodic boundary conditions are applied in an orthorhombic simulation volume.
• Internal interfaces are represented either by simple facets with certain strengths or by a
coupled method with a finite element method.
• The box method is revisited in order to increase the computing efficiency of the DDD code.
A speedup of 50 with errors lower than 3% is obtained in the typical situation of 20, 000
segments submitted to tensile loading (L=1, f=20, M=15)

Chapter 3
Parallelization of the Discrete
Dislocation Dynamics method
Although the numerical efficiency of the serial DDD method has been improved by using the box
method (see Sec. 2.5), the code is still insufficient to deal with a large density of dislocations or
dislocations interacting with thousands of precipitates. It is often said that there exists the gulf between
the desired problem size and the available computing power, since computational demands usually exceed
the performance of currently available computing hardware.
A parallel version of the DDD program has been developed in an attempt to simulate the interactions
between dislocations and a large number of precipitates within a reasonable time using parallel comput-
ers 1 . The object of this chapter is to present the development of the new parallel DDD program and its
performance.
In Sec. 3.1, parallel computing hardware is listed and various models and programming languages suit-
able for each hardware are reviewed. This section is intended to explain the reason of the parallel model
chosen in this work.
In Sec. 3.2, the hot spots of the serial DDD program are analyzed focusing on the data flow dependen-
cies. Based on the flow dependencies, the existing parallel algorithms are reviewed to help in establishing
a parallelization strategy.
Sec. 3.3 describes the parallelization of the serial DDD code from a programming perspective. The par-
allelization algorithm of each of the hot spots of the serial DDD code is presented using pseudo-codes.
An attempt to increase the performance of the new parallel DDD program is presented in Sec. 3.4. The
1 A parallel computer refers to several computers that are interconnected to increase computing power.

52 Parallelization of the Discrete Dislocation Dynamics method
performance of the program is quantified and issues such as the load balance are investigated.
Although the DDD code used here is the edge-screw model presented in Chapter 2, the parallelization
scheme is quite general and may be applied to any DDD code or finite difference methods which have
similar data dependencies.
3.1 An introduction to Supercomputing
3.1.1 Overview
The different types of parallel computer need to be reviewed before attempting to create a parallel
program, since a programming model and a programming language should be chosen depending on
the selection of an architecture.
A parallel computer is in fact a subset of a supercomputer which is defined as a computer that
performs at or near the currently highest operational rate for computers. Computation using a
supercomputer is often called supercomputing or high performance computing.
In the following sections, the technological trend of supercomputer is reviewed using data from the
top 500 list 2 . The top 500 list compiles information regarding the top 500 fastest supercomputers
in the world 3 .
3.1.2 Classification of hardware
All the current supercomputers use multiple processors and memories. There exist many classifi-
cation methods according to the usage of multiple processors and memories and their interactions.
Supercomputers are usually classified as follows:
(1) by processor type: scalar and vector processor
(2) by memory type: shared and distributed memory
(1) by processor types
Processor architectures can be divided into two principal types: scalar and vector processors. The
main difference between the two types relates to the number of operations performed by a single
2 visit at www.top500.org
3 It is published twice a year, in June at the International Supercomputer Conference and in November at the
ACM/IEEE Supercomputing Conference. The list has been compiled since 1993, when the first top 500 list have
published at the International Supercomputer Conference, Mannheim.

3.1 An introduction to Supercomputing 53
instruction.
Scalar processors perform a single operation for each single instruction. An addition instruction
(a + b), for example, results in the addition of two numbers. This type refers to a general-purpose
processor and is widely used 4 .
In vector processors, a single instruction results in identical operations being performed on differ-
ent data. It means that the addition of two arrays of data (A(i) + B(i)) can be performed in a
single instruction. Vector processors are developed for a high performance numerical computation
of vector data or arrays and are relatively expensive as compared to scalar processors 5 .
Examples of hardware for each processor type are listed in Table 3.1. Vector processors are known to
Processor classification Example
Scalar processor Intel x86, DEC Alpha, PowerPC, IBM Power
Vector processor Cray vector, NEC vector, Fujitsu VPP
Table 3.1: Processor classification
have an excellent effective performance and facilitate the development of parallelization algorithms.
However, they are less frequently used due to their high cost and limited scalability. Fig. 3.1, which
plots the share of each processor types over the past ten years, shows this trends clearly. In June
1993, vector processor architectures accounted for 66.8% of the top 500. That proportion decreased
to 5% in June 2004, whereas the share of scalar processors increases to 95%. The advantage of scalar
processors would be their relatively low price and excellent scalability, though they have a poor ef-
fective performance. Because the majority of supercomputers are using multiple scalar processors,
parallel computing (parallel computer) are often used instead of supercomputing (supercomputer).
4 Scalar processor is divided further into two groups: CISC(Complex Instruction Set Computer) and RISC(Reduced
Instruction Set Computer). The CISC group comprises Motorola 680x0, Intel x86 processors whereas DEC Alpha,
PowerPC and IBM POWER processors are within the RISC group.
5 Vector processors do perform parallel operations in a way that is sometimes described as ’data parallel’, though
they are not a parallel computer in the sense of many machines working together.

54 Parallelization of the Discrete Dislocation Dynamics method
Share %
(2) by memory types
100
90
80
70
60
50
40
30
20
10
0
Scalar processor
Vector processor
Jun 93
Nov 93
Jun 94
Nov 94
Jun 95
Nov 95
Jun 96
Nov 96
Jun 97
Nov 97
Jun 98
Nov 98
Jun 99
Nov 99
Jun 00
Nov 00
Jun 01
Nov 01
Jun 02
Nov 02
Jun 03
Nov 03
Jun 04
Figure 3.1: A change of processor types used in major supercomputers.
Memory architectures can be classified into two principal types: shared and distributed memory.
In shared memory systems, memories and processors are typically all interconnected by a common
bus or switching network. Each processor can access all the memories of the system and a processor
can directly load or store any shared address. In other words, the data movements are transparent to
the user. This provides an easy and powerful model for creating and managing a parallel program.
The shared memory systems can be further grouped into UMA (Uniform Memory Access) 6 and
NUMA (Non Uniform Memory Access) 7 depending on whether the main memory is a single physical
or a logical one. A schematic diagram of processors and memory in UMA system is shown in Fig.
3.2(a) and NUMA system in Fig. 3.2(b). The NUMA model has been developed to overcome a
technical difficulty of a UMA system, which limits the possible number of processors. Because a
NUMA system uses physically distributed memories in several systems as a single logically shared
memory, the access time to a certain memory could be different depending on whether a memory
is a local or a remote one to a specific processor.
In distributed memory systems or MPP (Massively Parallel Processor) systems, several computers,
6 Intel Dual CPU system, Compaq ES40, Sun E10000 and HP N-class belong to the UMA category.
7 Machines such as Compaq GS320, HP Superdome and SGI Origin 3000 belong to this category.

3.1 An introduction to Supercomputing 55
MEMORY
. . .
P P P P P
(a) A UMA system
MEMORY
. . .
P P
Logical Memory interconnect
MEMORY
. . .
P P
. . .
(b) An NUMA system
MEMORY
. . .
P P
Figure 3.2: (a) Schematics of UMA systems (b) Schematics of NUMA systems
M
P
Communication Network
M
P
. . .
M
P
(a) An MPP system
MEMORY
. . .
P P
Communicaton Network
UMA UMA UMA
MEMORY
. . .
P P
. . .
(b) A SMP cluster system
MEMORY
. . .
P P
Figure 3.3: (a) Schematics of MPP systems (b) Schematics of SMP systems
where a single processor has its own memory resource, are interconnected by a bus or network, and
processors access to distributed memories through a network. Fig. 3.3(a) shows a configuration of
processors and memories of such an MPP system. In this model, parallel processing is facilitated by
explicit message passing, since each processor has its own memory resource which cannot be directly
accessed by other processors in the MPP machine. Individual processors could all be of the same
type such as a network or cluster of workstations or PCs, which could work independently or in
unison. A heterogeneous networks of various platforms (vector processors, parallel supercomputers
etc.) could also be assembled in principle. IBM P690 architecture, which is used for some of the
results presented in this work, consists of several UMA machines as shown Fig. 3.3(b) 8 .
Various architectures of each memory classification are summarized in Table 3.2. In the early 1990s,
Memory classification Type
Shared memory UMA, NUMA
Distributed memory MPP, clusters of PCs, clusters of UMAs
Table 3.2: Memory classification
8 The UMA cluster system looks similar to a NUMA system, but as memories are not shared between nodes, the
user should explicitly assign data movement like a distributed memory system.

56 Parallelization of the Discrete Dislocation Dynamics method
Share %
100
90
80
70
60
50
40
30
20
10
0
Shared memory Distributed memory
SMP
SIMD
Single processor
Constellation
Jun 93
Nov 93
Jun 94
Nov 94
Jun 95
Nov 95
Jun 96
Nov 96
Jun 97
Nov 97
Jun 98
Nov 98
Jun 99
Nov 99
Jun 00
MPP
Cluster
Nov 00
Jun 01
Nov 01
Jun 02
Nov 02
Jun 03
Nov 03
Jun 04
Figure 3.4: The transition of the supercomputer structures for the past ten years. ’Cluster’ and
’MPP’ belong to the distributed memory system and ’SMP’ , ’single processor’, ’SIMD’ and ’Con-
stellation’ belongs to the shared memory system
most of the top 500 was shared memory architecture. However, the mainstream has changed to
distributed memory systems since the late 1990s (see Fig. 3.4).
Merits and demerits of each supercomputer type
Supercomputers are classified by a processor and a memory type. Each type has merits and demerits
which come from the different architectures used. They are summarized in Table. 3.3. In the case
Vector
Scalar
Shared Distributed
Ease of use, Good effective performance
High cost, Limited scalability
Ease of use Excellent cost/peak performance
Limited scalability Poor effective performance
Table 3.3: Merits and demerits of each processor and memory type
of a shared memory using vector processors (left-top of Table. 3.3), one can expect an excellent

3.1 An introduction to Supercomputing 57
effective performance and that it is relatively easy to vectorize a code using a compiler. But on the
other hand, the system is relatively expensive and shows limited scalability.
A distributed memory system using scalar processors has a relatively low price and yields a good
scalability. It is often said however, that it needs high skills to parallelize a code and the system
shows generally poor effective performance.
3.1.3 Parallel programming models
The main goal of parallel programming is to minimize the elapsed time of a program by utilizing
several processors. Since there is no single programming model that can be used in any architecture,
it is necessary to adopt different programming models for the different architectures summarized in
Sec. 3.1.2. This section is intended to provide programming models used in shared and distributed
memory systems.
Shared memory based
A program is made threads, each of which contains a work (computation) and a memory data
(object of work). A single-threaded program processes a data sequentially. The main idea of shared
memory based models is to create multiple threads and let each thread compute a portion of data
simultaneously. All the threads share the same address space and it is easy to reference data
that other threads have updated. So multi-thread programs are best fit with the shared memory
architecture in which all the memory spaces are shared.
This model is often called as the ’fork-join’ model as shown in Fig. 3.5(a). The single-thread
program processes S1 through S2, where S1 and S2 are inherently sequential parts. In the multi-
thread program, the first thread forks two more threads and the three threads process P1 through P3
in parallel. They are joined to the first thread once finishing the work. The compiler automatically
parallelizes certain types of ’DO’ loops, or else one can add some directives to tell the compiler how
to divide a work. OpenMP is one of such a compiler, and will be briefly reviewed in Sec. 3.1.4.
Distributed memory based
If the address space is not shared among the different nodes, parallel processors have to transmit
data over an interconnecting network in order to access data that other processors have updated.
Fig. 3.5(b) illustrates how a message-passing program runs. Each processor computes its own part
and the processors communicate with each other during the execution of the parallelizable part,

58 Parallelization of the Discrete Dislocation Dynamics method
t s
Single thread
S1
P1
P2
P3
S2
S1
P1
S2
Multi thread
Fork
P2 P3
Join
(a) Single-thread process and Multi-
thread process
t
s
Serial Parallel
S1
P1
P2
P3
S2
S1
P1
S2
S1
P2
S2
Communications
S1
P3
S2
(b) Message passing between processors
Figure 3.5: Parallel programming models for the shared and distributed memory architectures
P1-P3 (S1 and S2 are inherently sequential parts.). The figure shows data passing between only
two processors adjacent to each other. But in general, each processor communicates with all the
other processors. Due to the communication overhead 9 , time spent for processing each of P1-P3
is generally longer in the message-passing program than in the serial program. So only a modest
fraction is achieved of the capacity of several interconnected processors in practice 10 .
3.1.4 Classification of parallel languages
Different types of parallel computing hardware and the corresponding parallel models have been
reviewed in the preceding sections: shared memory-fork/join model, distributed memory-message
passing model. The choice of a parallel language is largely determined by the hardware type and
the parallel model to be used. Table 3.4 shows possible programming languages of each hardware-
model pair. In this section, two main parallel programming languages are outlined, OpenMP (Open
Message Passing) and MPI (Message Passing Interface).
Hardware-Model Parallel programming languages
Shared memory-fork/join model OpenMP, Pthread
Distributed memory-message passing model MPI, PVM
Table 3.4: Parallel hardware-model pairs and corresponding languages
9 and work load unbalance, and synchronization as shown in Fig. 3.16
10 Theoretically, the computational power should increase linearly with the number of interconnected processors.
t
p

3.1 An introduction to Supercomputing 59
OpenMP
OpenMP is a set of compiler directives and callable runtime libraries that extend the Fortran and
C languages to allow the development of scalable parallel programs on shared memory machines.
OpenMP provides access to the strengths of the shared memory parallel computation without an
excessive programming effort. For example, a single loop can be parallelized by simply inserting
standard directives, ’!$OMP PARALLEL DO’, as follows.
DO I=1, 100 DO I=1, 100
!$OMP PARALLEL DO
C(I)=A(I)+B(I) ⇒ C(I)=A(I)+B(I)
ENDDO ENDDO
!$OMP END PARALLEL DO
The directive, ’!$OMP PARALLEL DO’ creates multi threads (Fork) as schematically shown in
Fig.3.5(a). If four threads are created for example, the second thread would perform the addition
from I = 26 to 50 in the above code. ’!$OMP END PARALLEL DO’ collects results to master
thread (Join). Programming with OpenMP is relatively simple and it shows good efficiency if most
of the program execution time is dominated by a single, simple ’DO’ loop. But the efficiency of
this type of parallelization becomes poor as the data dependency inside the loops becomes com-
plex. OpenMP is also bound to the limit of the shared memory architecture, such as the number
of processors, size of memory, and it lacks portability between different platforms.
MPI
MPI enables message passing programming model in distributed memory architectures. As de-
scribed in the previous sections, distributed memory machine holds all the variables in local memory
space. The work shared across the different processors requires communication and message-passing
is the context in which this communication takes place. MPI is a parallel language which facilitates
message-passing between separated processors. Some of the implementations of MPI are listed in
Table 3.5. MPI will be reviewed in Sec. 3.2.1.

60 Parallelization of the Discrete Dislocation Dynamics method
Acronym Developers
MPI/Pro MPI Software Technology
IBM MPI IBM product implementation for the SP and RS/6000 workstation clusters
MPICH Argonne National Lab and Mississippi State University
UNIFY Mississippi State University
CHIMP Edinburgh Parallel Computing Center
LAM Ohio Supercomputer Center
Table 3.5: Various version of MPI
3.1.5 Supercomputers in France and Korea
Before finishing Sec. 3.1, the states of supercomputers in Korea and France in June 2004 are listed.
Table 3.6 and 3.7 shows the rank in top 500, machine specs, Rmax 11 and Rpeak 12 of the top 5
supercomputers in Korea and France, respectively. At the date of this thesis (summer 2004), Korea
possesses 9 supercomputers in the top 500 and France does 16 machines.
Rank Site/Year Computer(manufacturer)/processors Rmax/Rpeak
48 KIST/2003 xSeries Cluster Xeon(IBM)/1024 3067/4915.2
113 KISTI/2004 xSeries Cluster Xeon(IBM)/512 1762/2867
115 KISTI/2003 pSeries 690 (IBM)/544 1760/3699.2
233 SNU/2002 Pegasus P4 Xeon cluster(Self-made)/400 1011/1843
310 KT/2004 Integrity Superdome,HPlex(HP)/176 844/1056
Table 3.6: Top 5 supercomputers in June 2004, Korea
3.2 Towards a parallel DDD code
3.2.1 Basic Steps of Parallelization
In case of parallelizing an existing serial program, the basic steps could be summarized as follows
([Aoyama & Nakano 99]).
11 Maximal LINPACK performance achieved
12 Theoretical peak performance

3.2 Towards a parallel DDD code 61
Rank Site/Year Computer(manufacturer)/processors Rmax/Rpeak
28 CEA/2001 AlphaServer SC45(HP)/2560 3980/5120
120 TotalFinaElf/2003 xSeries Cluster Xeon(IBM)/1024 1755/4915.2
124 SG SGBI/2003 xSeries Cluster Xeon(IBM)/968 1685.49/4646.4
132 CNRS/IDRIS/2004 eServer pSeries 690(IBM)/384 1630/2611.2
149 CNRS/IDRIS/2004 eServer pSeries 655(IBM)/384 1477/2611.2
1. Tune the serial program
Table 3.7: Top 5 supercomputers in June 2004, France
The performance of a parallel program is bound to that of a serial program from which the
parallel program is written. The first step thus is to tune the hot spots of the serial program
and make the serial program as efficient as possible.
2. Consider the outline of the parallelization
It needs to get the profile of the tuned serial program and know which part or parts consume
most of the CPU time. It might be sufficient to parallelize most time consuming parts only.
At the same time, it is necessary to select the hardware on which the program is parallelized.
3. Determine the strategy for the parallelization
Depending on the hardware chosen and the data dependencies of the program, a parallel
algorithm should be made. For this, the existing strategies can be adopted if a pattern of
parallelization is similar, or else it needs to create a new algorithm. Then it should be decided
which scalar variables and arrays must be transmitted.
4. Parallelize the program
The strategy chosen is then realized using an appropriate parallel language.
The procedure of parallelization which has been selected in this work is summarized below according
to the basic steps mentioned above.
Step 1: Tune the serial program
The numerical efficiency of the serial DDD code has been increased using the box method and
the linked-list of segments (see Sec. 2.5). The internal stresses are approximated by the long- and
short-distance stresses, and there is no approximation in handling the dislocations interactions. The

62 Parallelization of the Discrete Dislocation Dynamics method
speedup is dependent on the parameters (in Table 2.3) chosen. The speedup of 50 is attained in the
case of L = 1, M = 15 and f = 20 with Nsegm = 20, 000 (Fig. 2.20).
Step 2: Consider the outline of the parallelization
In this work, a parallel DDD code has been written on distributed memory machines using the
MPI. The choice of the distributed memory systems and the MPI has several advantages such as
popularity, portability and extendability even though they are not the most efficient and the easiest
combination. As already shown in Fig. 3.4, most of the top 500 are a distributed memory system
nowadays, and distributed memory systems becomes more popular and widely used as individual
laboratories purchase parallel computers made of several PCs and workstations connected through
a network.
Computation of the internal stresses and handling of the dislocations interactions are still the most
time consuming parts.
Step 3: Determine the strategy for the parallelization
Before developing a parallel algorithm suitable for the DDD method, a few characteristics of the
method are summarized. First, the number of dislocation segments is not constant. Dislocation
segments can be created or annihilated with time. Next, the DDD method has highly complex flow
dependence in that a movement of a segment modifies not only its own position and connection, but
also the surrounding dislocation configurations. This is because dislocation lines are represented as
connected sets of segments and segments’ connections are often changed by cutting the dislocation
lines.
The stress computation has no flow dependence. If the computation load is distributed over P
processors, ideally the elapsed time for stress calculation will decrease by a factor of 1/P . To fully
make use of the box method as described in the previous chapter, it is pertinent to distribute the
stress computation in the boxes to several processors.
On the other hand, segment positions updating has a highly complex flow dependency as mentioned
before. This dependency can be shown as follows. a(i, j, k) represents the quantity of the segments
(e.g. the number of segments) in a box (i, j, k) indexed along x, y and z direction. In order to
update a(i, j, k), all the information from the first neighboring boxes are needed because the segment
interactions needs to take into account all the segments in the first neighboring boxes. In addition,
any quantity inside the first neighbors are susceptible to modification by the motion of segments

3.2 Towards a parallel DDD code 63
a(i+1,j-1) a(i+1,j)
a(i,j-1)
a(i-1,j-1)
a(i,j)
a(i-1,j)
a(i+1,j+1)
a(i,j+1)
a(i-1,j+1)
Figure 3.6: Dependence on neighbors: The center element a(i,j) is being computed. All of the
surrounding elements are used in the computation and also are modified after computing the center
element.
in the (i, j, k) box. This dependence is represented in Fig.3.6 in a simple 2D configuration. Thus a
special attention should be paid in handling segment interactions so that no boxes are overlapped
between processors when updating the segment positions. A specific sequence is required to avoid
updating adjacent boxes in two different processors concurrently.
Among the existing parallel strategies, that of molecular dynamics and of a finite difference method
are of particular interest because inter-dislocation stress computation is similar to inter-atomic stress
computation and the box method divides a simulation volume with 3D arrays of boxes, which is
similar to a matrix in a finite difference method.
In molecular dynamics programs, computation of forces on atoms usually accounts for most of the
CPU time. For each atom i, the total force exerted by the other atoms is computed using a double-
nested loop in which both loops are running from i = 1 to i = Natom, with Natom being the total
number of atoms. These loops are often parallelized, for example, by distributing the atoms among
the different processors. Each processor then computes forces of the resident atoms only. This
method is referred to data decomposition.
Matrix often represent physical data at grid points in a finite difference method. A parallel program
breaks up these matrix and distributes the parts across the processors. This method is called
domain decomposition. Domain decomposition simply refers to the subdivision or partitioning of

64 Parallelization of the Discrete Dislocation Dynamics method
a problem over a number of processors in a parallel program. Various method such as red-black
ordering and multi-color schemes have been proposed to deal with the data dependency between
adjacent grids. Further information can be found in [Dongarra et al. 98]. The main point in the
domain decomposition is how to specify the order of communication among processors to provide
the necessary data. The number of inter-processor communication and the order are determined by
the data dependencies of a problem.
Step 4: Parallelize the program
MPI is reviewed in more detail because it is the parallel library used in this work and it is used to
explain the newly developed parallel code as presented in Sec. 3.3.
The Message Passing Interface Forum (MPIF) has been organized to develop a standard library
for writing message-passing programs in 1992. The MPIF comprised more than 40 organizations
and endeavored to make the standard practical, efficient and flexible. Practically, it means that the
standard should allow convenient C and Fortran bindings and define an interface not too different
from the practice at that time, e.g., the Parallel Virtual Machine (PVM). The standard aimed at
efficient communication on a reliable communication interface so that the users need not struggle
with communication failures. Flexibility is guaranteed by defining an interface that can be imple-
mented on many vendor’s platforms with no significant changes and allowing usage in heterogeneous
environments. The first draft of the standard was published in 1994 and revised in 1997 (MPI-2).
The standard MPI provides descriptions of the parallel tasks as subroutines in Fortran and functions
in C. Only the Fortran version of MPI is presented here.
There exist around 192 subroutines in the MPI. All of them facilitate the parallel tasks of a MPI
program, which could be summarized as i) specifying a group of processors, ii) extracting a rank or
processor ID and iii) defining message passings between or among processors. It needs not to know
all of the subroutines since only about a dozen of the subroutines are frequently used to parallelize
a program.
The MPI subroutines could be categorized into three main groups as follows:
• Environment Management Subroutines
This group controls the overall environment of a MPI program. It includes initialization and
finalization of a parallel environment. It also includes creation of a communicator or a group
of processors.
A general MPI program would look like as follows.

3.2 Towards a parallel DDD code 65
PROGRAM parallel
INCLUDE ’mpif.h’
CALL MPI_INIT(ierr)
CALL MPI_COMM_SIZE(MPI_COMM_WORLD, nprocs, ierr)
CALL MPI_COMM_RANK(MPI_COMM_WORLD, myrank, ierr)
Computations here . . .
CALL MPI_FINALIZE(ierr)
END
Line 2 includes ’mpif.h’, which defines MPI-related parameters such as MPI_INTEGER and
MPI_COMM_WORLD. All Fortran procedures that use MPI subroutines have to include
this file. Line 3 calls ’MPI_INIT’ for initializing an MPI environment. ’MPI_INIT’ must
be called only once before calling any other MPI subroutines. In Fortran, ’ierr’ is the return
code of every MPI subroutine and ’0’ if successful or a non zero if failed. The subroutine
’MPI_COMM_SIZE’ in line 4 returns the number of processors(nprocs) belonging to the
communicator(MPI_COMM_WORLD). ’nprocs’ is given as the environmental variable of
the parallel work. ’MPI_COMM_WORLD’ is an identifier associated with a group of pro-
cessors and represents the group consisting of all the processors participating in the parallel
job. Each processor in a communicator has its unique rank, which is in the range [0,nprocs-
1]. The subroutine, ’MPI_COMM_RANK’ in line 5 returns the rank of the process within
the communicator. In line 6, each processor does some work on its data, and line 7 calls
’MPI_FINALIZE’. ’MPI_FINALIZE’ terminates MPI processing and no other MPI call can
be made afterwards.
• Point-to-point Communication Subroutines
This group specifies data exchange between two processors in the communicator. There exist
blocking and non-blocking communication subroutines. Details are not discussed here and the
interested reader is advised to refer [Aoyama & Nakano 99].
As an example of using non-blocking send/receive subroutines, which are used in the paral-
lelization of the DDD code by the author, let us consider two processors that need to exchange
data with each other.
IF (myrank==0) THEN

66 Parallelization of the Discrete Dislocation Dynamics method
CALL MPI_ISEND(a, 1, MPI_REAL8, 1, itag, MPI_COMM_WORLD, ireq1, ierr)
CALL MPI_IRECV(b, 1, MPI_REAL8, 0, itag, MPI_COMM_WORLD, ireq2, ierr)
ELSEIF (myrank==1) THEN
CALL MPI_ISEND(a, 1, MPI_REAL8, 0, itag, MPI_COMM_WORLD, ireq1, ierr)
CALL MPI_IRECV(b, 1, MPI_REAL8, 1, itag, MPI_COMM_WORLD, ireq2, ierr)
ENDIF
CALL MPI_WAIT(ireq1, istatus, ierr)
CALL MPI_WAIT(ireq2, istatus, ierr)
In this example, the processor of rank 0 sends a variable ’a’, which is one element and a real
number to the processor of rank 1 in the communicator ’MPI_COMM_WORLD’. ’itag’ is
the message tag. ’ireq1’ and ’ierr’ are subroutine return values. The syntax of the subroutine
’MPI_IRECV’ can be understood in the similar way, and the processor of rank 1 saves the
received data from rank 0 to variable ’b’.
• Collective Communication Subroutines
This group allows the user to exchange data among a group of processors. It happens fre-
quently that data in one processor need to be shared with all the processors in the commu-
nicator, or, inversely, data in each processor need to be collected to one processor. It would
not be efficient to use point-to-point communication in this case considering communication
latency of a network. The syntax of subroutines in this group comprise a sending and/or
receiving data array, its size and type and a rank of a processor which send or receive data
from a communicator. The subroutine, MPI_BCAST, for example, has the following syntax,
CALL MPI_BCAST(buffer, count, datatype, root, MPI_COMM_WORLD, ierr)
, where ’buffer’ is broadcasted from a processor of rank root to all processors in the commu-
nicator, MPI_COMM_WORLD.
In addition to the groups categorized above, there exist subroutines relating to managing processor
groups, defining data types and controlling input and output files. The MPI standard also provide
supports for profiling interface and file management, etc.

3.3 Parallelization of the serial DDD program 67
3.2.2 Writing a parallel program
To save efforts to parallelize the serial DDD program, the serial subroutines are kept intact or made
minor modifications if possible. In the following, parts that need to be modified for parallelization
are indicated in bold characters on the general computation procedure of the serial code from Sec.
2.6).
a. Initialization of parallel environments
b. Discretization of the segments
c. Construction of the linked-lists
d. Updating of the long-distance stresses every f steps
e. Computation of the short-distance stresses
f. Motion of segments
g. Updating the external stresses
h. Save of outputs
The modified ’a’ is needed to build a parallel environment involved in partitioning the computation
into a selected number of concurrent processors. The internal stress computation steps (’d’-’e’)
need minor modifications. The computation step ’f’ needs complex interactions between processors
because of the flow dependencies. In the following section, the parallelization scheme is detailed in
programming perspectives.
3.3 Parallelization of the serial DDD program
3.3.1 Initialization of parallel environments
The boxes which decompose a simulation volume (Fig. 3.7(a)) are partitioned into parallel-piped
subsystems (Fig. 3.7(b)). The processors in a parallel computer are then logically arranged accord-
ing to the topology of the physical subsystems, and assigned to each subsystem.
A processor of rank ’p’ is assigned to a parallel-piped subsystem assuming that the subsystems are
arranged in a 3D array of dimensions P1, P2 and P3. The total number P of processors required
is then given by P1 × P2 × P3. Vector IDs of each subsystem on the cartesian system is stored in
the array nid(:), that is, nid(1) is, for example, in the range of [0 : P 1 − 1]. A processor p is then
assigned to each subsystem as defined as Eq. 3.1. Remember that each processor is given a unique

68 Parallelization of the Discrete Dislocation Dynamics method
(a) Cubic simulation volume using the box method
X
Z
Y
Proc 1
Proc 0
Proc 3
(b) Parallel-piped subsystems
Figure 3.7: Domain decomposition of the simulation volume (a) into parallel-piped subsystems (b).
The use of four processors is assumed and each parallel-piped is allocated to each processor.
processor identification number (rank) p in the range of [0 : P − 1].
p = nid(1) + nid(2)P1 + nid(3)P2P3
For each processor, the six face-shared neighbor processors are identified by a sequential array,
nni(:), and given automatically by the following equation.
X
Z
Y
Proc 2
(3.1)
nni(k) = ick(1) + ick(2)P1 + ick(3)P2P3, k = 1, 6 (3.2)
nni(:) will be used to identify neighbor processors for message-passing.
In Eq. 3.2, ick(i) is a vector ID of neighbor processor k, and is written in Eq. 3.3 using an
array iv(:, :) defined in Table. 3.8 and the ’MODULO’ operation. A torus connection between the
processors is considered in Eq. 3.3 to enforce periodic boundary conditions.
ick(i) = MOD(nid(i) + iv(i, k) + Pi, Pi) i = 1, 3 & k = 1, 6 (3.3)
If there exist M boxes along each axis, the boxes are distributed as follows. Suppose when M is
divided into Pi (number of processors), the quotient is q and the remainder is r, that is, M = qPi+r.
Processors whose vector ID, nid(i) is smaller than r are assigned q+1 boxes and the other processors
are assigned q boxes. Total number of boxes along i axis are kept as (q + 1)r + q(Pi − r) equals to

3.3 Parallelization of the serial DDD program 69
y
12
x
Neighbor ID,k 1 2 3 4 5 6
iv(3,k) -1 0 0 1 0 0 0 -1 0 0 1 0 0 0 -1 0 0 1
Table 3.8: The relative location of each neighbor processor
13
14
15
8 9 10 11
4 5 6 7
0 1 2 3
(a)
y
x
3 4
Figure 3.8: Top view of 20 × 20 × 20 subboxes being assigned to 4 × 4 × 1 processors. Numbers
represent processor identification.
M. This distribution method is useful when the number of boxes M is not divisible by the number
of processors, Pi.
Fig.3.8(a) shows the decomposition of 20 × 20 boxes into 4 × 4 subsystems. For simplicity, the
configuration is considered in 2D, which is equivalent to 3D with P3 being 1, for example. All the
subsystems have the equal number of the boxes. In Fig. 3.8(a), the number of boxes is not identical
in each subsystem because M = 20 is not divisible by P1, P2 = 3.
IDs of the boxes which bound the subsystem of processor p are stored in the array ibs(:): ibs(1), ibs(3)
and ibs(5) save the first box number along x, y and z direction respectively, and ibs(2), ibs(4) and
ibs(6) represent the last box number along each direction. Processor 6 in Fig. 3.8(a), for exam-
ple, is bounded by ibs(1) = 11, ibs(2) = 15, ibs(3) = 6, ibs(4) = 10 and neighbor processors are
nni(1) = 5, nni(2) = 7, nni(3) = 2, nni(4) = 10.
6
0
7
1
(b)
8
5
2

70 Parallelization of the Discrete Dislocation Dynamics method
3.3.2 Long-distance stresses computations
The serial version computes long-distance stresses as follows. The boxes that are situated at long-
distance relative to one give box are recognized by a topological relation. The stresses due to the
segments in the long-distance labeled boxes are computed at the center point of the given box. Thus
one processor scans all the boxes and computes the long-distance stresses in the serial program.
In a parallel program, the work is divided into several processors, since each processor is responsible
for a fraction of the boxes. Boxes in each processor are distinguished by the array ibs(6), and each
processor computes the long-distance stresses of boxes only in its subsystem.
The serial and the parallel version are compared in the following.
Serial version
DO iz=1, M
DO iy=1, M
DO ix=1, M
the box ’ib’
compute the box index ’ib’
compute the long-distance stresses of
ENDDO
ENDDO
ENDDO
⇒
Parallel version
DO iz=ibs(5), ibs(6)
DO iy=ibs(3), ibs(4)
DO ix=ibs(1), ibs(2)
compute the box index ’ib’
compute the long-distance stresses of the
box ’ib’
ENDDO
ENDDO
ENDDO
The parallel version uses most of the serial codings and only the range of the loops are slightly
modified. And it should be noted that each subsystem shares all the segments information at the
time of computing the long-distance stresses.
3.3.3 Short-distance stresses computation
The following pseudo-code explains how the short-distance stresses are computed both in the serial
and in the parallel DDD code.

3.3 Parallelization of the serial DDD program 71
Serial version
DO is=1, Nsegm
segment ’is’
Identify the box ’ib’ containing the
Compute the short-distance stresses
due to the segments
within the short-distance boxes
ENDDO
⇒
Parallel version
DO is=1, iscnt(p)
Identify the box ’ib’ containing the seg-
ment ’is’
Compute the short-distance stresses due
to the segments
within the short-distance boxes
ENDDO
As for the computation of the long-distance stresses, only small modifications are made to the
serial coding: in the case of the serial program, all the segments (Nsegm) are processed by a single
processor. In the parallel program, on the contrary, segments are distributed among several proces-
sors, and a processor p computes stresses of iscnt(p) segments only. The construction of iscnt(p)
will be discussed in Sec. 3.3.4.
Since the stress on a segment can be computed without regard to the stress on the other segments,
all processors can work independently. The elapsed time for stress computation decreases by a
factor of 1/P (the number of processors), if the number of segments of each processor is the same.
Otherwise, the overall elapsed time for the stress computation is determined by the busiest proces-
sor, because the other processors have to wait until the latest processor finishes the computation to
move the segments. For higher efficiency, the segments have to be distributed uniformly over the
different processors. This can be realized by shifting the subsystem boundaries, which changes the
ibs array and consequently iscnt. This load balancing issue will be addressed in Sec. 3.4.4.
3.3.4 Data structures for distributing and the gathering segments
The processors do not work entirely independently in a parallel program. At some point of a pro-
gram, it needs to collect all the information to one processor or to distribute the data to all the
processors. An obvious example of gathering information is when data are written in external files.
One processor normally takes charge of writing files, and the data to be written are sent to that
processor from the other processors.
In a parallel DDD program, segments’ information including coordinates, neighbors, linked-list and
the effective stress etc. need to be communicated. The segments are identified by a vector of integer
numbers. To send segments’ data to the other processor, the list of segments to be sent should
be shared between the sender and the receiver processors. The arrays iswork(:,:) and iscnt(;) are

72 Parallelization of the Discrete Dislocation Dynamics method
Proc 2 Proc 3
6
3
14
8
11
7
13
5
10
4
12
1 2
Proc 0 Proc 1
9
Figure 3.9: List of segments
p 0 1 2 3
iscnt(p) 2 2 5 5
iswork(i,p) 1 9 8 10
i 7 2 14 4
3 13
11 12
6 5
constructed to facilitate this process and contain the list of segment identification number and the
number of segments in each processor respectively. In Fig. 3.9, for example, four processors treat
fourteen segments in a 2D configuration. The values of the arrays iswork and iscnt are written in
the figure as an example.
The segments in processor p can be recognized by scanning the processor box content and using the
linked-lists, indexb(ib) and isbox(ib, 2) as described in Sec. 2.5.2. The arrays iswork and iscnt can
be constructed as follows.
DO iz = ibs(5), ibs(6)
DO iy = ibs(3), ibs(4)
DO ix = ibs(1), ibs(2)
compute box number ’ib’ from ix,iy and iz
call Bliste(ib, isliste)
DO is=1, indexb(ib)
iscnt(p)=iscnt(p)+1
iswork(iscnt(p),p)=isliste(is)
ENDDO
ENDDO
ENDDO
ENDDO
The subroutine Bliste generates the list isliste of segments in the box ib, and indexb(ib) contains

3.3 Parallelization of the serial DDD program 73
the number of segments inside this box. For a given processor p, the number of segments and the
list are saved in iscnt(p) and iswork(:,p) respectively. The arrays iscnt(p) and iswork(:,p) can then
be shared among all the processors by using the MPI_BCAST subroutine,
DO irank=0, nprocs-1
call MPI_BCAST(iscnt(irank), 1, MPI_INTEGER,
ENDDO
DO irank=0, nprocs-1
irank, MPI_COMM_WORLD, ierr)
call MPI_BCAST(iswork(1,irank), iscnt(irank), MPI_INTEGER,
ENDDO
with nprocs being the total number of processors.
irank, MPI_COMM_WORLD, ierr)
Now all the processors in the MPI_COMM_WORLD communicator share the list of segments in
each processor, segments’ information gathering or distributing can be realized using the lists.
3.3.5 Motion of segments
The segment motion induces interactions with the other dislocation segments. The dislocation in-
teractions involve complex dependencies as shown in Fig. 3.6. The key idea of handling dislocation
interactions is to avoid any overlap of neighbor boxes of concurrently updated boxes. The handling
of the dislocation interactions is managed by first dividing the boxes inside a processor p into three
groups according to the topology of the neighboring boxes: inner boxes (IB), boundary boxes (BB)
and corner boxes (CB) (Fig. 3.10). It should be noted that at least three boxes are required along
each axis in each subsystem to categorize the boxes into these three groups.
The inner boxes have all their neighboring boxes in the same processor, thus the motion of the
segments in the inner boxes modifies the segments located in the same processor only. Because
all the information needed to handle the dislocation interactions are stored in the local memory,
and there is no overlap of the neighboring boxes between the adjacent processors, the positions
of the segments in the inner boxes of the different processors can be updated simultaneously and
independently and involve no message passing.
The boundary boxes and the corner boxes, on the other hand, have a lack of neighboring boxes
in the same processor. Thus it needs message passing between processors to obtain the segments’

74 Parallelization of the Discrete Dislocation Dynamics method
y
12
x
13
14
15
8 9 10 11
4 5 6 7
0 1 2 3
(a) Parallel-piped subsystems
CB
BB
BB
CB
IB BB
CB BB CB
(b) Categorization of boxes
Figure 3.10: Three category of boxes in a processor p: Inner boxes (IB) have all the neighbor boxes
in the same processor, thus need no communications. Boundary boxes (BB) have lack of neighbor
boxes and need communications with a neighbor processor. Corner boxes (CB) have lack of neighbor
boxes in three different processors.
information from their neighboring boxes and to send back the information modified by the dislo-
cation interactions.
In the case of the boundary boxes, all the missing neighboring boxes are situated in the neighboring
processors, therefore a message passing only with the adjacent processor is sufficient to provide the
missing information. The corner boxes, however, have neighboring boxes scattered in more than
four different processors including itself (in a 2D configuration), and thus are bound to involve
complex message passings.
Updating the positions of the segments is performed in the following three steps.
• In the first step, all the segments in the inner boxes of each processor are updated indepen-
dently and simultaneously.
• In the second step, the segments in the boundary boxes are updated involving message passing
with the their respective neighboring processors. The order of computation is from right to
left in the x,y and z direction order (see Fig. 3.11).
• In the final step, all the information of segments are collected into one processor (Master
processor) and segment positions updating in the corner boxes are made in that processor
only. This procedure avoids at least complex message passing between the different processors.

3.3 Parallelization of the serial DDD program 75
y
y
12 13 14 15
8 9 10 11
4 5 6 7
0 1 2 3
x
x
(a) Inner boxes
(d) Boundary boxes y+
y
y
x
(b) Boundary boxes x+
x
(e) Boundary boxes y-
Figure 3.11: Overall procedure of motion of segments
y
y
x
(c) Boundary boxes x-
x
(f) Corner boxes
The overall procedure is drawn in Fig. 3.11. Fig. 3.11 shows that the simulation volume is
subdivided into nine processors. In the first step, all the inner boxes of each processor are updated
and the updated boxes are represented as shaded one. The boundary boxes are then updated in the
order of x, y and z direction, and at the right (plus) and the left (minus) position of each direction
sequentially. After updating all the boundary boxes, the corner boxes are treated by one processor
exclusively. In what follows, the details and the corresponding message passing of each step are
discussed.
Inner boxes
The segments in the inner boxes can be identified easily by performing the loops over [ibs(1) + 1,
ibs(2) − 1], [ibs(3) + 1, ibs(4) − 1] and [ibs(5) + 1, ibs(6) − 1]. The same segment motion algorithm
as in the serial program can be used to update the list of segments in the inner boxes.

76 Parallelization of the Discrete Dislocation Dynamics method
A special care should be taken on the label numbers assigned to the different segments, in order to
avoid giving duplicate numbers to the newly created segments, in the different processors. Duplicate
labels can generate confusion when all the segment information are gathered into one processor. A
new label list can be generated consistently sending to each processor the number of the new
segments created inside all the other processors. The procedure for renumbering the new segments
is shown in Fig. 3.12 in the case of four processors. The key point is that the label numbers of the
newly created segments are updated in real time and the new segments are renumbered in ascending
orders of processor ranks.
For label renumbering, the array isnewcnt(p) is increased by one whenever a new segment is created
inside a given processor p. After all the processors have finished treating the segment motion and
related interactions, this array is synchronized and used to renumber the newly created segments
as follows.
DO irank=0, nprocs-1
call MPI_BCAST(isnewcnt(irank), 1, MPI_INTEGER,
ENDDO
iadd=0
DO irank=0, p-1
iadd=iadd+isnewcnt(irank)
ENDDO
DO is=nsegm, nsol+1, -1
isnew=is+iadd
irank, MPI_COMM_WORLD, ierr)
Shifting segment information from ’is’ to ’isnew’
ENDDO
nsegm is the local number of segments of each processor, and nsol is the global number of segments
before to treat the segment motion. After renumbering, the global number of segments can be
computed by summing the array isnewcnt:.
Boundary boxes
Fig.3.13 shows the sequence of message passings to update the segment positions inside the bound-
ary boxes, on the +x direction. Before sending the information, arrays concerning the number of

3.3 Parallelization of the serial DDD program 77
Proc 2
6
3
14
8
11
Proc 3
13
5
10
4
12
Proc 2 6
3
14
8
11
15
16
16
Proc 3
13 17
15
5
10
4
12
7
1 2
9
7
17
16
1
15 15
2
9
17
16
Proc 0 Proc 1 Proc 0
Proc 1
isnewcnt(0)=3
isnewcnt(1)=3
isnewcnt(2)=2
isnewcnt(3)=3
Synchronize
Proc 2 6
3
14
8
11
21
22
24 Proc 3
13 25
23
5
10
4
12
7
9
17 20
16
1
15 18
2
19
Proc 0 Proc 1
Figure 3.12: Label assignment to the newly created segments
segments ibcnt and the list of segments ibwork to be sent and be received are synchronized between
a sender and a receiver processor. The arrays ibcnt and ibwork are constructed and synchronized
in a similar way as the arrays iscnt and iswork are processed in Sec. 3.3.4.
Information of segments, e.g. coordinates, neighbor segments and effective stresses, etc., are packed
in one dimensional buffer arrays of the integer, real and logical types.
The buffer arrays are then sent to the next processor inext and received from the previous processor
iprev using the subroutine MPI_ISEND and MPI_IRECV. An example code is shown below.
call MPI_ISEND(bufsi(1), ibcnt(p)*11, MPI_INTEGER,
inext, itag, MPI_COMM_WORLD, ireqs, ierr
call MPI_IRECV(bufri(1), ibcnt(iprev)*11, MPI_INTEGER,
call MPI_WAIT(ireqs, istatus, ierr)
call MPI_WAIT(ireqr, istatus, ierr)
iprev, itag, MPI_COMM_WORLD, ireqr, ierr
Buffer arrays which are received, e.g. bufri in the above example, then are unpacked in the in-

78 Parallelization of the Discrete Dislocation Dynamics method
Send buffer arrays of
this column to nni(1)
Receive buffer arrays of
this column from nni(2)
To be
updated
(a) Before updating the boundary boxes
Receive modified
column from nni(1)
Updated
Send modified
column to nni(2)
(b) After updating the boundary boxes
Figure 3.13: A sequence of message passings to update the positions in the boundary boxes located
at +x position (dark grey). Two message passing steps are involved: (a) Send segment information
in the leftmost column to the neighboring processor in the −x direction and receive information
from the neighboring processor in the +x direction (b) Send segment information, which is modified
due to updating the boundary boxes, back to the processor in the +x direction and receive from
the processor in the −x direction.
verse sense of the packing using synchronized ibcnt, ibwork.
When all the necessary information from the neighboring boxes are collected, segment positions in
the boundary boxes are updated. The segment motion in the boundary boxes also modifies the
segment configuration in the received boxes. In order to properly synchronize this modification,
information of the received boxes are then repacked and resent to the original processor. This
completes the boundary boxes updating in the +x direction, and likewise all the updating in the
−x, ±y, ±z directions are completed.
Corner boxes
Once all the boundary boxes are updated, information of segments of each processor is sent to one
processor (Master) and the segments in the corner boxes are treated by the Master processor only.
After finishing the motion of segments in the corner boxes, only one processor (Master) contains
the final configuration of the segments of the current time step. Before running the next time step,
the information concerning the dislocation segments are sent to all the processors from the Master
processor.

3.3 Parallelization of the serial DDD program 79
Initialization of
parallel environments
Discretization of
the segments
Linked-lists of
the segments
Computation of the
long-distance stresses
Computation of the
short-distance stresses
Motion of
the segments
Update external stresses
and save outputs
Inner boxes
Boundary boxes
Corner boxes
(1)
(2)
Send/Receive
Gather (3)
Broadcast
Figure 3.14: The overall flow chart of the new parallel DDD code
3.3.6 Summary and comments
The overall flow chart of the new parallel DDD code is shown in Fig. 3.14. The ’Motion of segments’
step is composed of three parts which correspond to inner, boundary and corner boxes. The message
passing addresses are also indicated.
It should be noted that all the processors begin each time step with the same segment information
(marked as ’(1)’ in Fig. 3.14), although it is not indicated explicitly in the previous sections.
After the segment discretization, each processor computes and thus alters its local segments’ data
independently up to ’Inner boxes’ step (’(2)’). While updating information in the boundary boxes,
two adjacent processors mutually send and receive data and then send the local data to one processor
(as indicated ’Gather’ in Fig. 3.14). The Master processor then updates all the information in the
corner boxes and broadcast data to the other processors (as indicated ’Broadcast’ in Fig. 3.14).
Hence all the processors share the same segment information.
Thus, there is no gain in a memory aspect of the program by using several processors in the present
parallel version. The parallel code can further be improved by decomposing the data space, i.e. by
making each processor to use only the necessary and sufficient amount of memory. This would save
memory space for parallel computation, and would also decrease the communication overhead and
eventually increase the performance of the code.

80 Parallelization of the Discrete Dislocation Dynamics method
3.4 Performance improvment
3.4.1 Measure of performance
It is needed to measure the performance of the new parallel program in terms of gain of the elapsed
time. The following measure is often used.
Speedup(P ) = t0
, where t0 is the elapsed time of the serial program and tp is that of the parallel program using P
processors. The speedup indicates what the practical advantage is by using the parallel program
instead of the serial program. t0 can be replaced by the elapsed time of the parallel program
run with one processor. Then the speedup parameter shows the advantage of the use of several
processors because both the numerator and the denominator contain overhead for initializing a
parallel environment. This is often called the algorithmic speedup ratio. The speedup results
presented in the following section are measured using the algorithmic speedup ratio, because it
is difficult to compare directly the serial and the parallel DDD program compiled using different
compilers and run in different platforms.
The efficiency of a parallel program is a measure of the effectiveness of the hardware usage. The
efficiency is represented as the ratio of the speedup on P processors to P , that is, Speedup(P )/P .
An efficiency close to 1 indicates an excellent scalability.
3.4.2 Conditions for good performance
In the ideal case, the speedup is a linear function of the number of processors P , i.e. Speedup(P ) =
P . This case is hardly achievable in practice because of the moderate fraction of the parallelizable
part of an algorithm, the communication overhead and the load unbalance.
Suppose that a fraction fp of a serial program can be parallelized and that the remaining 1 − fp
cannot be parallelized. The Speedup(P ) can be written as the following equation, supposing a
perfect load balancing and no communication overhead.
Speedup(P ) =
tp
1
(1 − fp) + fp/P
Eq. 3.5 is plotted in Fig. 3.15 for fp = 1, 0.99, 0.9 and 0.5. The ideal case (Speedup(P ) = P ) is
only possible if fp equals to 1, and a maximum speedup of only 10 is expected where fp = 0.9.
A parallel program involves communication overhead to send and receive data, which does not exist
(3.4)
(3.5)

3.4 Performance improvment 81
Speedup
50
45
40
35
30
25
20
15
10
5
0
f p =1.00
f p =0.99
f p =0.90
f p =0.50
5 10 15 20 25 30 35 40 45 50
Number of CPUs
Figure 3.15: Ideal speedup of a program with the number of processors when only a fraction fp of
the program is parallelized
in a serial program. In general situations, the performance of a parallel program is worse since the
load is not perfectly balanced among the different processors. The performance of general parallel
programs is shown in Fig. 3.16. It is assumed that only 80% of a serial program is parallelized, and
the effects of the communication overheads and the load unbalance are shown as well.
From the figure, it is obvious that good performance can be achieved by good load balancing among
processors, minimizing communication overhead and increasing the parallelisable fraction f of a
serial program. Note that the communication overhead can be decreased both by minimizing the
amount of communication (good algorithm) and by using a fast network (good hardware).
3.4.3 Performance tests
A simple speedup model of our algorithm is made and compared to the actual timing results. It
is assumed that the simulation volume is decomposed in M × M × M boxes and that the total
number of processors used is P , dividing boxes into a 2D array of P 1/2 × P 1/2 × 1 or into a 3D array
of P 1/3 × P 1/3 × P 1/3 . The elapsed time by using a single processor is approximately the sum of
the time needed for the stress computation (t s stress) and that used to update the positions (t s update ).
Assuming that t s update is a fraction of ts stress (t s update = αts stress), the total elapsed time t s then is

82 Parallelization of the Discrete Dislocation Dynamics method
Serial
Parallel
cpu 1
cpu 2
cpu 3
cpu 4
20 80
20 20
Load unbalance
Unparallelizable part
Parallelizable part
Communications
Figure 3.16: Load unbalance and communication overhead of general parallel programs
written as Eq.(3.6).
t s = t s stress + t s update = (1 + α)ts stress
The number of inner boxes (BI), boundary boxes (BB) of each processor and the total corner boxes
(BC) can be expressed using M and P as Eq.(3.7) in the case of a 2D array and as Eq.(3.8) in the
case of a 3D array of processors. It is assumed that every processors have the same number of boxes
in its subsystem.
BI =
M
BI = M − 2
P 1/2
M
− 2
P 1/3
3
2
(3.6)
M
; BB = 4M − 2 ; BC = 4MP (3.7)
P 1/2
2
M
; BB = 6 − 2 ; BC = P 12
P 1/3 M
− 16
P 1/3
If dislocation segments are homogeneously distributed over all the processors, the elapsed time for
the stress computation of each processor (t p
stress ) is merely a division of ts stress by P . Considering
that the elapsed time for updating segments’ positions of a box is t s update /M 3 , the elapsed time of a
processor (t p ) for both the stress computation and the segment motion can be expressed as Eq.(3.9).
t p = t p
stress + tp
update
(3.8)
= ts stress
P + ts update
M 3 (BI + BB + BC) + tc (3.9)
The elapsed time for updating BC is included on each processor, because every processors wait until
the updates of BC by the Master processor are finished. tc represents the time needed for message

3.4 Performance improvment 83
Speedup
70
60
50
40
30
20
10
0
Ideal case
3D, t c =0
2D, t c =0
3D, t c =0.02t s stress
2D, t c =0.02t s stress
10 20 30 40 50 60
Number of CPUs
Figure 3.17: Speedup model of the algorithm (Eq.(3.9)) with M = 21, t s update = 0.02ts stress for 2D
array of processors (2D) and 3D array of processors (3D)
passings.
A speedup (t s /t p ) is plotted in Fig.3.17 using Eq.(3.9) with M = 21, α = 0.02 and tc/t s stress =
0., 0.02. The curve is drawn up to P = 49 in the case of 2D array of processors. Note that a
maximum of 49 processors can be used with M = 21 in the 2D array of processors, since there
should be at least three boxes along any coordinate axis.
The speedup of the algorithm is strongly dependent on the network speed. If the network is fast
enough (tc 0), the algorithm speedup can be as high as 23 using 25 processors with the 2D array
of processors.
It seems that the 3D array of processors have an advantage over the 2D array if the same number
of processors are used. In reality it is controversial, because a 3D array of processors involves more
messages passing than a 2D array. A 3D array needs message passings along all of three coordinates,
whereas a 2D array needs message passings along only two coordinates. The size of each message,
however, is smaller in the case of a 3D array of processors.
Dislocation structures with 13185, 37182, 57605 and 77198 segments are extracted from a simple
tensile test of a single crystal with M = 20. Then execution time for 100 steps with zero applied
stress is measured and the elapsed time per step is averaged by dividing the execution time by

84 Parallelization of the Discrete Dislocation Dynamics method
Time per step(seconds)
160
140
120
100
80
60
40
20
0
0 5 10 15 20 25 30 35 40
Number of CPUs
N=77198
N=57605
N=37182
N=13185
Figure 3.18: Elapsed time per step, in seconds, as a function of the number of processors for 13185,
37182, 57605 and 77198 segments.
100. Fig.3.18 shows the average elapsed time required to complete one time step using up to
36 processors in a 2D array of processors on the IBM p690 architecture with 1.7GHz POWER4
processors 13 . Fig.3.19 shows the speedup of each number of processors and compares the actual
data with the speedup model. Measured data agrees well with the model except in the 13185
segments case. The speedup decrease in the 13185 segments case for large values of P is due to the
the proportion of the computation time over the communication time decreases with the number of
processors.
3.4.4 Load balancing
As pointed out in Sec. 3.4.2, good load balancing is crucial to achieve high performance of a parallel
computation. In many cases, DDD simulations involve highly heterogeneous dislocation structures.
An example is the formation of intense slip bands in fatigue simulations as shown in Fig. 3.21 (see
Sec. 4.3). Fig. 3.21 shows the worst case in load balancing, due to the inherent highly heterogeneous
13 The author would like to acknowledge the support from KISTI (Korea Institute of Science and Technology
Information) under "the 5th Strategic Supercomputing Applications Support Program’ with Dr. Sangmin LEE as
the technical supporter. The use of the computing system of the Supercomputing Center is also greatly appreciated.

3.4 Performance improvment 85
Speedup
40
35
30
25
20
15
10
5
0
Ideal case
N=37182
N=57605
N=77198
N=13185
t c =0.015t s stress
0 5 10 15 20 25 30 35 40
Number of CPUs
Figure 3.19: Speedup by using P processors in 2D array for 13185, 37182, 57605 and 77198 segments
(on IBM p690 architecture)
Efficiency
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Ideal case
N=37182
N=57605
N=77198
N=13185
0 5 10 15 20 25 30 35 40
Number of CPUs
Figure 3.20: Efficiency by using P processors in 2D array for 13185, 37182, 57605 and 77198
segments (on IBM p690 architecture)

86 Parallelization of the Discrete Dislocation Dynamics method
(a) Intense slip bands of fatigue tested volume con-
taining bimodal-sized particles
[010]
6 7 8
3 4 5
0 1 2
[100]
(b) Decomposition of the simulation volume by 3 × 3 × 1 proces-
sors
Figure 3.21: An example taken from fatigue tests of cylindrical simulation volume containing parti-
cles of bimodal size distribution (see Sec.4.3). Load is highly unbalanced among processors due to
the highly heterogeneous dislocation microstructure
dislocation microstructure in fatigue and the geometry of the simulation volume.
If the simulation volume is decomposed into equal sized subsystems as shown in Fig. 3.21(b), there
is a high discrepancy in the number of segments between the different processors, and consequently
in the computation time. A load-balance method is thus highly desirable to equilibrate the proces-
sor loadings.
One obvious way to better balance the loads is to shift the boundaries of each subsystem, or the
array ibs, so that each processor has approximately the same number of segments, since the com-
putation time is usually proportional to the number of segments. In surface grain simulations,
however, the number of segments may not be a good yardstick because some segments are treated
as virtual ones and need no internal stress computation, which will be detailed in Sec. 4.3. Hence
the actual elapsed time for stress computations is taken as an indication for load balancing.
The elapsed time is measured by using, for example, the MPI timer function MPI_WTIME(). Load
balancing is processed every fbalan steps. To minimize the overhead of load balancing, the processor
which having the minimum elapsed time is determined one step before the load balancing. This
processor then takes charge of shifting the boundaries. This makes the overhead of load balancing
to be hidden in the process of overall stress computation.

3.4 Performance improvment 87
Y
Z
Initial
boundaries
Current
boundaries
Figure 3.22: Shifting of subsystem boundaries to balance load among processors
The elapsed time for the stress computation of each processor is gathered to the processor in charge.
The processor adds the elapsed time of processors in the same column along x, y, and z axis in the
processor array, and the average elapsed time of the columns is calculated. By comparing the
elapsed time of a column to the average time, the boundary is shifted by one box so that the size
of a column increases or decreases. The boundary can move until the number of boxes has reached
the minimum number of boxes of a subsystem on any axis (3 boxes).
The boundary adjustment procedure during a fatigue test is shown in Fig. 3.22. Dislocations in
the top-right part of the cubic simulation volume are ’virtual’, so less computation time is needed
to treat them. The boundaries of each subsystem thus move toward bottom-left of the simulation
volume until they share a comparable number of real dislocation segments.
The load balancing by parallel-piped subsystems has the following limitations: (i) The different
subsystems on the same column should have the same width, thus the computing load is balanced
among the columns, not among the processors. (ii) There should be at least three boxes along
each axis of each subsystem, thus load concentration smaller than three boxes can not be balanced
further.
During simulations, the number of segments can change dramatically. It is not unusual that an
initial Frank-Read source produces millions of segments. When the number of segments is small,
X

88 Parallelization of the Discrete Dislocation Dynamics method
the efficiency of the new parallel DDD program is quickly decreasing with the number of processors
(Fig. 3.20), and the speedup could be even reversed by using more processors (Fig. 3.19). One way
to guarantee a maximum efficiency and to prevent inverse speedup of multiple processors would be
to change the number of processors dynamically based on the current number of segments. This
can be done, for example, by creating a new communicator of n [1 : N] processors in the initial
communicator of N processors.
3.4.5 Comparison of simulation results between the serial and parallel DDD
code
The simulation results of a parallel program should not be significantly different from that of a
serial program when addressing the same problem. There could be a slight difference, however, due
to the parallelization because the order of computations might be changed.
In DDD simulations, the segments are moved sequentially, and two different orders of segments
can results in different dislocation configurations even though the applied stresses are the same.
Nevertheless the overall stress-strain relation and the dislocation density in the simulation should
be consistent when using the parallel or the serial DDD program.
Fig. 3.23(a) shows the stress-strain curves of a simple tensile test along [001] direction of a single
crystal. It can be verified that the curve of the parallel program is consistent with that obtained
using the serial program. The number of dislocations is slightly different by using the two programs
but the error is negligible as compared to the overall evolution of dislocations (Fig. 3.23(b)).
3.5 Application to Stage I-II transition simulation
In this section, resorting to the performance of the new parallel DDD code, it is attempted to
simulate the transition from Stage I to Stage II in the stress-strain curves of FCC single metals
subjected to a uniaxial tension.
3.5.1 Stress-strain curves of FCC single crystals
In the general case, when a FCC single crystal is subjected to tensile tests, the stress-strain curves
represent three distinctive stages, I, II and III. Fig. 3.24 shows stress-strain curves from experiments
in copper crystals covering a wide range of orientations. Stage I or ’easy glide’ is a region of low

3.5 Application to Stage I-II transition simulation 89
Stress[MPa]
180
160
140
120
100
80
60
40
20
Parallel
0
Serial
0.0e0 1.0e-4 2.0e-4 3.0e-4 4.0e-4
Strain
(a) Stress-strain curves
Number of segments
40000
35000
30000
25000
20000
15000
10000
5000
Parallel
0
Serial
0 500 1000 1500 2000 2500 3000
Step number
(b) Number of segments
Figure 3.23: Comparison of (a) stress-strain curves and (b) number of segments of the serial and
the parallel DDD program in a tensile test simulation
linear hardening (θ = ∂τ
∂γ
G ∼ 300 ) and is observed at the beginning of deformation. Stage II or ’linear
hardening’ is a second linear region with a much greater rate of work hardening (θ ∼ G
30 ), then is
followed by Stage III or ’parabolic hardening’, which represents a period of decreasing rate for the
hardening. Fig. 3.24 also shows that the shape of the curves is strongly dependent on the crystal
orientation, e.g. the orientations close to the [001] − [¯111] side show a short or no Stage I, whereas
the orientations far from the boundaries of the standard triangle show a long Stage I.
The slip system on which the resolved shear stress is the highest is called the primary system, and
the deformation commences on this system involving a low work hardening rate (Stage I). The
accumulated slip rotates the orientation of the crystal, and subsequently the resolved shear stress
on different slip systems is modified as the slip direction rotates towards the tension axis. When
the tensile axis arrives the [001] − [¯111] side, another slip system (the conjugate system), which has
been inactive initially, begins to activate, and the interactions of two slip systems initiates Stage II.
3.5.2 Simulation conditions
The initial dislocation configuration is made of 5.65 µm-long Frank-Read sources homogeneously
spread over the 12 slip systems (see Fig. 3.25(a) 14 ). The orthorhombic simulation box has been
used with the ratio of the axis’ length close to 40 : 30 : 31, and the periodic boundary conditions
are applied along all the axes. The simulation volume is around 577 µm 3 and the initial dislocation
14 Different colors represent dislocations on different slip systems

90 Parallelization of the Discrete Dislocation Dynamics method
Figure 3.24: Resolved shear sress/shear strain curves of copper crystals as a function of orientation.
([Diehl 56])
density is 8.82 × 10 11 m −2 (90 sources).
The materials parameters of copper are used : G = 42000 MPa (Shear modulus), ν = 0.324
(Poisson’s ratio), b = 2.56 Å(Burgers vector magnitude), B = 10 −5 Pa s (Viscous drag coefficient),
V/b 3 = 350 (Activation volume) and τIII = 32 MPa (Threshold stress).
The initial tensile axis (T) has been chosen to [ ¯14 15 25], so that the initial configuration is in
single glide close to the double glide axis. The resulting Schmid factor is shown in Fig. 3.25(b).
The primary system is the system B4: (111)[¯101] and the conjugate system is C1: (¯1¯11)[011] (the
notations of Schmid and Boas are recalled in Table 2.1). The simulation runs in the constant strain-
rate condition of ˙ɛ = 100sec −1 . The rotation tensor of the crystal is computed every time step using
Eq. 3.10, and the new tensile stress axis (T ′ ) is updated as Eq. 3.11
dW =
12
s=1
1
2 (ms ⊗ n s − n s ⊗ m s ) (3.10)
T ′ = (I + dW) T (3.11)
This way of updating the orientation of the tensile axis reproduces the experimental tensile tests
where the crossheads of the tensile machine are unconstrained, i.e. the rotations of the crosshead
are allowed.

3.5 Application to Stage I-II transition simulation 91
(010) Y
(100) X
(001) Z
(a) Initial dislocation configuration
3.5.3 Simulation results
Schmid factor
0.3
0.2
0.1
0
B4 D4 D1 C1 B5 C5 D6 A6 A2 B2 C3 A3
(b) The initial distributions of the Schmid fac-
tors
Figure 3.25: Initial conditions of the simulation
In Fig. 3.26(a), the tensile stress-strain curve is plotted. The curve shows no significant hardening
up to the strain level of 1.3%. The dislocation configuration at a cumulated tensile strain of about
1.% are shown in Fig. 3.27, and the dislocation structure shows that the dislocations in the primary
system are mainly activated, thus the simulation is still in Stage I at this moment.
The stress-strain curve shows an insignificant hardening or even a negative hardening after, say,
0.5% cumulated strain. This phenomenon seems to be related to the enhanced cross slip of the
primary dislocations due to the spurious dipoles generated by the periodic boundary conditions.
Indeed, the evolution of the dislocation densities plotted in Fig. 3.26(b) shows that the density of
the cross slip dislocations of the primary system is significant even though the Schmid factor and
the observed shear strain on the deviate (or cross-slip) system are negligible.
However, the rotation of the tensile axis is well accounted for in the simulation. Fig. 3.28(a) shows
the rotation of the stress axis plotted within the standard stereographic triangle. The stress axis
rotates toward the [001] − [¯111] boundary. Subsequently, the Schmid factors are modified and the
ratio between the primary and the conjugate system increases toward 1 as plotted in Fig. 3.28(b).
Despite the spurious softening, the shear stress-strain curves of the primary and the conjugate
system (see Fig. 3.29) demonstrate that the hardening is decreased in the primary system due to
the rotation of the axis, whereas the hardening is more pronounced in the conjugate primary system.
This typical simulation of the behavior of a bulk crystal of copper is a first tentative of a massive

92 Parallelization of the Discrete Dislocation Dynamics method
σ 11
30
25
20
15
10
5
0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
ε11 (a) The tensile stress-strain curve
ρ[10 11 m-2]
140
120
100
80
60
40
20
B4
D4
D1
C1
B5
C5
D6
A6
A2
B2
C3
A3
0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
ε11 (b) The evolution of the dislocation densities
Figure 3.26: The tensile stress-strain curve and the evolution of the dislocation densities
Figure 3.27: The dislocation configuration at a cumulated tensile strain of about 1.%
(010) Y
(100) X
(001) Z

3.5 Application to Stage I-II transition simulation 93
τ(B4) [MPa]
12
10
8
6
4
2
[-14 15 25]
[-111]
[001] [011]
(a) The rotation of the tensile axis
τ(C1)/τ(B4)
0.985
0.98
0.975
0.97
0.965
0.96
0.955
0.95
0.945
0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05 7e+05
Number of steps
(b) The ratio between τ(C1) and τ(B4)
Figure 3.28: The rotation of the tensile axis and the modified Schmid factors
0
0e+00 5e-03 1e-02 1e-02 2e-02 3e-02 3e-02 4e-02
γ(B4)
(a) The shear stress-strain curve of the primary system
τ(D4) [MPa]
12
10
8
6
4
2
0
0e+00 1e-04 2e-04 3e-04 4e-04 5e-04 6e-04
γ(D4)
(b) The shear stress-strain curve of the conjugate system
Figure 3.29: The shear stress-strain curves of the primary and the conjugate system
simulation using the new parallel DDD code. The amount of cumulated shear strain on the primary
system reaches 3% and the number of segments at the end of the simulation was close to 43, 000.
More investigations are needed to understand the reason of the softening observed during Stage I.
The transition to the stage II should nevertheless be observed in a few more steps, when the Schmid
factor on the conjugate system (C1) will be high enough to multiply its dislocation density, and
hinder the dislocation motion on the primary system (B4).

94 Parallelization of the Discrete Dislocation Dynamics method
Key points
• Parallel models and languages are strongly dependent on the different types of parallel
computers. For shared memory machines, fork-join model is usually applied using
OpenMP or Pthread. The message passing model is adequate for distributed memory
architectures using the MPI as the programming language.
• Distributed memory system and MPI have been chosen here for the development of
a parallel DDD code. To handle the dislocation interactions during the segments
motion is the most complex part for parallelization. Attention is focused on the fact
that the 3D array of boxes physically dividing the cubic simulation volume is similar
to a matrix in the computer memory space.
• The cubic simulation volume is decomposed into parallel-piped subsystems which are
mapped to processors. The internal stress computation involves small modifications
in the serial DDD code. The segment motion algorithms have been developed for the
different box types categorized into inner, boundary and corner boxes. Attention is
paid to avoid any overlap of the neighboring boxes between the processors.
• The performance of the new parallel DDD program is measured and compared to a
simple speedup model. The boundaries of each parallel-piped system are dynamically
moved to balance the computing load among the processors. Simulation results of
the parallel DDD version corresponds well to those obtained with the serial version.
A speedup of around 17 is found when using 25 cpus to handle more than 30, 000
segments
• The new parallel code has been applied to simulate the Stage I-II transition of single
FCC metals subjected to a uniaxial tension. The simulation runs up to the 3.2%
cumulated shear strain at this moment and still remains in Stage I. However, the
rotation of the axis is well accounted for, as measured on the evolution of the Schmid
factors. The softening observed on the mechanical response is attributed to avalanche
of cross-slip events which may be induced by the periodic boundary conditions.

Chapter 4
Dislocation-precipitate interactions
4.1 Image stresses due to a 3D particle
4.1.1 Motivations and review of the literature
The image forces need to be considered when one wants to study the behavior of dislocations near
a free surface. Another level of complexity arises when there are internal interfaces in metals, such
as voids, second phase particles and microcracks. The magnitude of the interaction forces is at least
required to understand the dislocation behaviors around the internal interfaces. There have been
many studies to obtain the interaction both in analytical and numerical ways.
Free surfaces
The effect of a free surface can be easily treated in 2D by introducing the mirror images so that the
traction on the free surface would be forced to zero. The problem is more complex in 3D because it is
almost impossible to find analytically the image dislocation for a finite dislocation segment that is not
parallel to the free surface. This problem can be solved using either the solution of the Boussinesq
problem ([Fivel et al. 96]) or the superposition principles using FEM ([Fivel et al. 98]). The
main idea of this method is to apply point forces on the free surface so that these forces nullify the
surface stress field generated by a dislocation in an infinite medium on the free surface. Using this
method, the dislocation depletion near a free surface can be computed and the direct comparison
can be made between the dislocation structure calculated in dislocation dynamics simulation and
the experimental observations, e.g. TEM (Transmission Electron Microscopy).
The dislocation-free zone near a crack tip ([Kobashi & Ohr 80]) and the plastic zone yielding from

96 Dislocation-precipitate interactions
a crack ([Vitek 75]) are other fields where the image force acting on a dislocation is an important
factor to consider. There are several analytical solutions on the interaction of a dislocation line
and a hole or rigid inclusion. To make the problem to be a simple 2D case, it is generally assumed
that a infinitely long dislocation line is interacting with an infinitely long cylindrical inclusion
(hole or rigid) ([Santare & Keer 86], [Zhou & Lung 88], [Chen et al. 99]). The authors used
a complex potential approach in plane strain restriction for an isotropic medium. They solved the
elastic solution satisfying the stresses and displacements continuity at the interface. The application
of these 2D solutions is rather limited to the case of fiber-strengthened composite or microcracks
with large aspect ratio.
Particles
Even for a simple geometrical shape like a spherical particle, the calculation of the interaction force
between a dislocation line and a particle satisfying the stress and displacement continuity across
the interface is not an easy task. The elastic problem to satisfy the rigorous boundary conditions is
too complex to solve in an analytical manner. Instead of the exact solution, approximate solutions
have been obtained for 3D shapes of particle. One method consists of using the interaction energy
between a dislocation line and a particle. It is assumed that the interaction energy is equal to
the change of the energy density of a dislocation line by the presence of a particle volume. The
interaction force or the image force is obtained by differentiating the interaction energy. This ap-
proach tells that the ratio of the image force acting on an edge dislocation and a screw dislocation
is equal to the ratio of the energy density of a dislocation of the respective type. Using this method,
an analytical equation is obtained for the force acting on a screw dislocation line near a cubical
particle ([Melander & Persson 78]) and the force on an edge and a screw dislocation near a
spherical particle are calculated numerically ([Nembach 83]). The long-range interaction between
a screw dislocation and a spherical inclusion has been treated assuming that a straight disloca-
tion line is located far from a spherical particle so that the particle disturbs the uniform stress
field([Weeks et al. 69],[Comninou & Dundurs 72]).
The interaction force due to a second phase particle with an elastic modulus mismatch is considered
as negligible compared with the lattice mismatch and the stacking fault energy mismatch effect in
the case of a penetrable particle ([Nembach 97]). The interaction forces by an elastic modulus
mismatch, however, increase with the number of dislocations around the particles 1 . The image
1 This type of interaction is referred to the paraelastic interaction ([Nembach 97]).

4.1 Image stresses due to a 3D particle 97
(-1-12) Z
(111) Y
(1-10) X
(a) Cylindrical particle
y=0.86Rp
y=0.5Rp
y=0
(111) Y
(-1-12) Z (1-10) X
(b) Spherical particle
Figure 4.1: Computation geometries of (a) a cylindrical particle and (b) a spherical particle
stresses by particles thus could be an appreciable factor in the computation of the energy state of
dislocation structures around a particle and in the phenomena involving several dislocations like
work hardening rate. Moreover in dispersion-strengthened alloys at high temperature, the interac-
tion force on a single dislocation line in climb direction is essential to investigate high temperature
properties, for example creep threshold stresses ([Marquis & Dunand 02]).
Scope of this section
Image forces on a long, straight dislocation line near a particle are computed using the decompo-
sition method as detailed in Sec. 2.4.2. Three cases are considered: a cylindrical (Fig. 4.1(a)), a
spherical (Fig. 4.1(b)) and a cubical particle. The cylindrical particle case can be compared with
analytical solutions of 2D circular particles. Image forces along both a glide and a climb direction
are considered.
4.1.2 Interaction of an edge dislocation with a circular cylindrical particle
Image forces on an edge dislocation around a rigid particle have been solved analytically in 2D by
Santare et al. ([Santare & Keer 86]) and around a void by Vitek ([Vitek 75]) and Chen et al.
([Chen et al. 99]). The analytical solutions are obtained using complex potentials.
In the case of a rigid particle, the image force of an edge dislocation projected along the glide

98 Dislocation-precipitate interactions
y/Rp
1.5
1
0.5
1
0
-1 -0.5 0 0.5 1 1.5
-0.5
-1
5
9
2
3
7
4
1
x/Rp
(a) Analytical solution [Santare & Keer 86]
y/Rp
1.5
1
0.5
0
1
0
-1 -0.5 0 0.5 1 1.5 2
x/Rp
-0.5
-1
(b) Numerical solution
Figure 4.2: Case of a rigid particle (The image force is normalized by µmb 2 /(4π(1 − νRp)) (a)
Analytical solution [Santare & Keer 86] (b) Numerical solution (FEM/DDD)
direction can be simplified as follows ([Santare & Keer 86])
F
µmb 2
4π(1−ν)Rp
= x 4x 4 + k 2 x 2 + 2k 2 x 2 y 2 − 3x 2 − 2kx 2 y 2
(x 2 + y 2 ) 3 (x 2 + y 2 − 1) k
+ x 2k 2 y 4 + 2ky 2 − 4y 4 − k 2 y 2 + 5y 2 − 2ky 4
(x 2 + y 2 ) 3 (x 2 + y 2 − 1) k
with µm, ν being the shear modulus and the Poisson’s ratio of the matrix, Rp being the radius of
the particle and k = (3 − 4ν) for a plane strain condition.
There are two solutions of image forces around a circular void. The solutions are written in Eq. 4.2
([Chen et al. 99]) and Eq. 4.3 ([Vitek 75]).
F
µmb 2
4π(1−ν)Rp
= −2x x 6 + x 4 y 2 + 4x 2 y 2 − x 2 y 4 + 4y 4 − 2y 2 − y 6
(x 2 + y 2 ) 3 (x 2 + y 2 − 1)
F
µmb 2
4π(1−ν)Rp
= −2x 2x 4 − x 2 + 2x 2 y 2 + y 2
(x 2 + y 2 ) 3 (x 2 + y 2 − 1)
Image forces are computed numerically around a cylindrical particle with a height axis [¯1¯12], which
is parallel to an edge dislocation. The shear modulus of the cylinder is set to be 10 3 µm for the
rigid case and 10 −3 µm for the void. The contours of the image forces acting on an edge dislocation
and projected along the glide direction are shown in Fig. 4.2 for the case of a rigid inclusion. The
forces are normalized by µmb 2 /(4π(1 − νRp)). The image force profiles are computed on three lines
45
9
3
2
1
(4.1)
(4.2)
(4.3)

4.1 Image stresses due to a 3D particle 99
F/(µ m b 2 /(4π(1-ν)R p ))
15
10
5
0
-5
-10
y=0.5R p
y=0.86R p
y=0.86R p
y=0.5R p
y=0.0
y=0.0
[SANTARE & KEER 86]
[CHEN et al. 99]
[VITEK 75]
DDD/FEM
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
x/R p
Figure 4.3: Normalized image forces on an edge dislocation situated at x/Rp from the center of a
circular hole or a rigid cylinder along three lines with different stand-off distances (Rp: radius of
cylindrical inclusion) Solid line: Analytical solution (Rigid case: [Santare & Keer 86], Hole case:
[Vitek 75], [Chen et al. 99]) Points: Calculated by FEM/DDD
with a direction of [1¯10] and a stand-off distance of 0., 0.5Rp and 0.87Rp from the plane passing
through the center of the cylinder, and are plotted in Fig. 4.3 and compared with the analytical
solutions. Numerical solutions fit well to the analytical solution(Eq. 4.1) in the rigid particle case.
The computed image forces fall between two analytical solutions (Eq. 4.2 and Eq. 4.3) in the case
of the void.
From the 2D cylindrical circular case, it is thus validated that the image forces can be computed
correctly using the FEM-DDD coupled method.
4.1.3 Interaction of an edge dislocation with a spherical particle
There is no existing analytical solution of image forces in the case of a spherical particle. However, it
is common to encounter a precipitate or an inclusion of a spherical shape. Thus, numerical solutions
of the spherical particle case would be useful in practice.
Before computing image forces, both the issues of convergence and accuracy of the numerical solu-
tions have been addressed. Using 20-node 3D elements, it was verified that the numerical solutions

100 Dislocation-precipitate interactions
F/(µ m b 2 /(4π(1-ν)R p ))
12
10
8
6
4
2
0
y=0.86R p
y=0.5R p
[SANTARE & KEER 86]
y=0.0
DDD/FEM, y=0.0
DDD/FEM, y=0.5R p
DDD/FEM, y=0.86R p
0.6 0.8 1 1.2
x/Rp 1.4 1.6 1.8
Figure 4.4: Normalized image forces on an edge dislocation situated at x/Rp from the center of
a spherical rigid particle. (Rp: radius of a spherical particle) Solid line: Analytical solution for a
cylindrical rigid inclusion [Santare & Keer 86] Points: Calculated by FEM/DDD
are converging by increasing the number of elements and the accuracy of the solutions were checked
using the isobands method ([Bathe 96]). A mesh using 6656 20-node elements is found to be able
to represent the high stress gradients correctly, although the 3D spherical particle problem involves
rough and discontinuous point load distributions on the elements of the particle volume.
Image force profiles are obtained on three lines with a direction of [1¯10] and a stand-off distance
of 0., 0.5Rp and 0.87Rp from the plane passing through the center of the sphere. In Fig. 4.4, the
image force profile is compared with the corresponding 2D analytical solution (Eq. 4.1) for the case
of a rigid particle and in Fig. 4.5 for the void particle with Eq. 4.2.
The magnitude of the image force in the spherical particle case is lower than in the corresponding
cylindrical case and the difference increases with the stand-off distance of the glide plane of the
dislocation. It should be noted that the difference is much significant in the case of a spherical void.
It can be deduced that the interaction volume is smaller in the case of a spherical than that of a
cylindrical particle.
The computed image force profiles are fitted in the form of α/(x/Rp) β . The profiles are divided in
two regions, with a high gradient (up to x = 1.4Rp) and with a moderate gradient. The parameter

4.1 Image stresses due to a 3D particle 101
F/(µ m b 2 /(4π(1-ν)R p ))
0
-1
-2
-3
-4
-5
-6
-7
-8
y=0.86R p
y=0.5R p
y=0.0
[CHEN et al]
DDD/FEM, y=0.0
DDD/FEM, y=0.5R p
DDD/FEM, y=0.86R p
0.6 0.8 1 1.2
x/Rp 1.4 1.6 1.8
Figure 4.5: Normalized image forces on an edge dislocation situated at x/Rp from the center of a
spherical void. (Rp: radius of a spherical particle) Solid line: Analytical solution for a cylindrical
hole [Chen et al. 99] Points: Calculated by FEM/DDD

102 Dislocation-precipitate interactions
β is in the range of 6.42 to 8.89 in the high gradient region and 4.34 to 4.84 in the moderate gradient
region. This result is consistent to the argument of Comninou et al. ([Comninou & Dundurs 72]),
which shows that the interaction force is proportional to (x/Rp) −4 assuming that a straight dislo-
cation line is located far from a spherical particle. Although the authors derived the equation in
the case of a screw dislocation, the scheme can also be applied to an edge dislocation.
4.1.4 Interaction of an edge and a screw dislocation with a cubical particle
A cubical particle on a FCC matrix habit plane of {111} is now considered. It facilitates the problem
because a dislocation line lies parallel to an edge and a face of the cube. The side length 2a of the
cube is set to be 1.612Rp so that the volume of the cube is equal to that of the spherical particle
considered previously. Image forces are computed on three glide planes with the stand-off distance
y=0., y=0.5a, y=0.87a. The shear modulus of the particle (µp) is set to be twice that of the matrix
and the Poisson’s ratio ν is 0.312 for both the particle and the matrix.
This configuration was also proposed by Melander ([Melander & Persson 78]) using the energy
density of a screw dislocation line. Image forces are obtained by differentiating the interaction
energy. The image force of a screw dislocation is given by
F = (µp − µm)b 2
8π 2 a
⎡
⎣ tan−1
(Y −1)
|X−1| − tan−1 |X − 1|
(Y +1)
|X−1|
tan
−
−1
, where X, Y are coordinates normalized by the half side length a.
(Y −1)
|X+1| − tan−1 |X + 1|
(Y +1)
|X+1|
⎤
⎦ (4.4)
Computed image force profiles on an edge dislocation line are shown in Fig. 4.6. The decrease of
the image force with the stand-off distance is not as fast as in the spherical particle case, since the
dislocation line is parallel to one face of the cubical particle and the glide plane is normal to the
face. It is found that the image force is 20% higher in the case of a cubical particle at the stand-off
distance of 0 than in the case of a spherical particle. Image forces on a screw dislocation shows that
the ratio between an edge and a screw dislocation is around 0.68, which is close to (1-ν). However,
Eq. 4.4 fits the edge dislocation case well, even though Eq. 4.4 is solved in the case of a screw
dislocation. It is not clear whether this discrepancy is due to the mesh size or the approximations
used to derive Eq. 4.4.
The image force magnitude along the climb direction of an edge dislocation is given by
τcl =
((b · σ) × t)
|b|
· (b × t)
|b × t|
(4.5)

4.1 Image stresses due to a 3D particle 103
F/((µ p -µ m )b 2 /(4π(1-ν)a))
3.5
3
2.5
2
1.5
1
0.5
0
y=0.87R p
Edge,y=.00Rp
Edge,y=.50Rp
Edge,y=.87Rp
Screw,y=.00Rp
[MELANDER & PERSSON 78]
y=0.0
y=0.5R p
1.1 1.2 1.3 1.4
x/a
1.5 1.6 1.7 1.8
Figure 4.6: Image forces on a dislocation interacting with a cubical particle. solid line: Analytical
solution for a cubical particle [Melander & Persson 78]
with b being the Burgers vector and t being the dislocation line vector. The climb forces draw
attention because it affects the local climb of a dislocation around a particle. Climb forces on an
edge dislocation at the position of x = 1.1a on three different stand-off distances are plotted in Fig.
4.7. The climb forces are negligible up to y = 0.5a. Even at y = 0.877a, the magnitude of the climb
force is only around half that of the image force along the glide direction. The situation is quite
different in the spherical particle case. The magnitude of the climb force is 2 − 3 times higher than
the force along the glide direction y = 0.866Rp. It can be said that the configuration chosen for the
cubical particle is more resistant to dislocation climb.
4.1.5 Discussion
The interaction force of a dislocation line with a circular cylindrical, spherical and cubical particles
with differing elastic modulus was computed using the superposition principle. The complementary
problem was solved using the FEM-DDD coupling code. There have been significant research
interests in case of a long cylindrical inhomogeneity. In the case of a long edge dislocation line
close to a long cylindrical particle, the image force calculated numerically was compared with the
analytical solutions. It showed that good accuracy could be obtained by the superposition method.

104 Dislocation-precipitate interactions
F cl /((µ m b 2 /(4π(1-ν)R p ))
14
12
10
8
6
4
2
0
-2
Sphere(Rigid)
Sphere(Void)
Cube
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
y/R p
Figure 4.7: Climb forces on an edge dislocation at the position of x=1.1Rp(a) along y-axis
The same scheme was applied to the case of a spherical particle interacting with a dislocation line,
which is a more common shape for inhomogeneities observed in metals. The image force acting on
a dislocation at x = 1.1Rp along the glide direction is found to be smaller than the cylindrical case
by a factor of 0.89 for the rigid and 0.58 for the hole case. The interacting volume involved in the
spherical case is thus much smaller than that in the cylindrical case. As for climb forces, it is found
that considerable forces are acting along the climb direction. The climb force is acting on an edge
dislocation in the direction of reducing the extra plane around rigid particle, and in the opposite
direction around a void. As the stand-off distance of the glide plane is increased, the climb force
increases. A dislocation line parallel to one face and one edge of a cubical particle showed that the
image force is higher than the spherical particle by 20% and the climb force is negligible. It can be
said that a cubical particle is more resistant to dislocation climb.

4.2 A simple case of dislocation-particle interaction 105
4.2 A simple case of dislocation-particle interaction
4.2.1 Motivation and review of literatures
The hardening of materials by distributing small particles of another phase is a well-known phe-
nomenon and has been used to develop high strength structural materials. The impediment of
dislocation glide by second phase particles is the basic mechanism of increase in the flow stress.
In the case of impenetrable particles, the back stress exerted by trapped closed loops causes the
subsequent increment of stress, i.e. the work hardening. In addition to the glissile closed loops left
around the particles, prismatic loops have also been observed experimentally to form near a particle
by the cross slip of dislocations ([Humphreys & Martin 67]). The particles in the two phase
materials can be classified by their size, shape, volume fraction, spatial distribution (regular or ran-
dom) and the characteristic of the particle/matrix interface (coherent or incoherent). In addition
to these morphological particle parameters, the stress fields around the particles, which arise by a
difference in the lattice constant, the stress incompatibility due to a difference in the shear modulus,
or the image stress caused by the change in the strain energy of the dislocation near a second phase
particle are important factors needed to describe the mechanical properties of two-phase alloys.
A number of experimental and analytical researches have been focused on finding the relevant param-
eters and their effects in determining the flow stress and work hardening properties of these alloys.
For example, well-designed experiments, in which the volume fraction, size and spacing of particles
can be varied, have been made to measure the effect of such parameters on mechanical behavior of
single crystals containing hard particles ([Ebeling & Ashby 66], [Humphreys & Martin 67]).
Theoretical approaches to account for the effect of relevant parameters have been developed from
the Orowan mechanism, which considers the elementary dislocation-particle interaction and relates
the flow stress with the obstacle spacing, to the methods for dealing with the anisotropy of the
material and the complicated statistics such as random distribution of particles. An overview of
the analytical approaches can be found in the review article of Reppich ([Reppich 93]). To fully
account for the realistic interaction of dislocations and many particles, computer simulations have
also been developed. Foreman and Makin ([Foreman & Makin 66]) had investigated the effect
of a random arrangement of strong and weak point obstacles on the flow stress. In recent simula-
tions more complex and accurate states are treated, e.g. the distribution of finite particles including
shape effect ([Zhu & Starke 99]), the particles with mismatch stress field by a difference in lattice
constant ([Mohles & Nembach 01]). More insights of the effect of each parameter on the flow

106 Dislocation-precipitate interactions
stress can be gained through these simulations. Note that in a simulations mentioned above, it was
generally assumed that the dislocation moves on the single glide plane so that no 3D events such as
cross slip were allowed.
Scope of this section
In this work, the effect of the image stress on the flow stress and work hardening has been studied
using the 3D dislocation dynamics simulation coupled to a finite element code. The simulation
method is detailed in section 4.2.2. The glide of a dislocation line through a channel between two
incoherent, impenetrable spherical particles is considered. The dynamic interaction of a dislocation
with particles and the resulting image stresses are solved in the 3D space. Several dislocations are
forced to move between two spherical particles one by one and the resolved shear stress needed to
bypass the particles having trapped loops is monitored while the cross slip of dislocation is accounted
for. By changing the shear modulus of the particle with other parameters such as particle radius
and the inter-particle spacing fixed, the effect of image stress induced by second phase particles is
evaluated and the relation between the flow stress and the difference in shear modulus is established.
4.2.2 Calculation procedures
The situation considered here consists of two impenetrable rigid spherical particles having radius
Rp, inter-particle distance L and shear modulus Gp. A typical 3D simulation box is shown in
figure 4.8(a). In the dislocation code, the spherical particles are modeled as a set of facets made of
polyhedral surface elements. Each facet of the spherical particle has a certain strength to act as an
obstacle to dislocation motion, i.e. a dislocation is authorized to cross a facet if the local effective
resolved shear stress is above the particle strength. In this study, the strength of the particle has
been chosen high enough so that it represents impenetrable hard obstacles. That way, it is possible
to use a simplified version of the code for which no image stresses are computed. The interaction
between dislocations and particles is then only related to an obstacle effect. This situation is later
referred as the ∆G = 0 case. In order to include the effect of the image forces, one has to solve
the complementary problem described in figure 2.16. To do so, a 3D box containing two spherical
particles has been meshed in the code CAST∃M( 2 ). To represent correctly the high gradient of
stress near the particles, the mesh has been refined near the periphery of the particles. A (111)
section of the 3D mesh taken at the centers of particles is shown in figure 4.8(b). The displacement
2 Finite element code developed by Commissariat à l’Energie Atomique, CEA-DRN/DMT/SEMT

4.2 A simple case of dislocation-particle interaction 107
(a) (b)
Figure 4.8: (a)Simulation Box used in 3D discrete dislocation simulation. Two particles of radius Rp
and inter-particle distance L are shown with dislocation line on (111)[1-10] slip system. (b)Typical
mesh of the simulation box constructed by 4672 20-nodes 3D elements in the finite element code
CAST∃M. Mesh is sectioned on the (111) plane containing the particle centres.
of the bottom surface is set to be zero in the direction normal to the surface and two nodes located
on this surface are fixed in adapted directions in order to remove the trivial rigid body solution.
A dislocation line, which is initially a pure screw segment, is pinned at the two end points. The
position of the pinned points is set to be 0.9 of L/2 from the border of particle so that the portion
of dislocation line which lies between the particles bypasses the particles by bowing out. In section
4.2.3, it will be shown that this fixed point can be used to obtain reliable results for the flow stress.
Single slip loading conditions on the (111)[1-10] system are assumed. The resolved shear stress τ on
the slip plane is increased step by step. After each load increment, the new positions of all segments
are computed as a function of time until the shear strain γ caused by the dislocation motion has
fallen below a pre-selected value. If the dislocation line has reached an equilibrium position, ∆γ is
nearly equal to zero, τ is then increased and ∆γ is monitored again. After the dislocation line has
completely bypassed the particles leaving two Orowan loops, a new dislocation line is introduced
and the subsequent increment of τ, i.e. the work hardening is computed.
4.2.3 Flow stress of impenetrable particles with a different shear modulus
In order to validate the computation method used in this work, the flow stresses induced by im-
penetrable particles with no image stresses have been calculated and compared with the results of

108 Dislocation-precipitate interactions
Bacon et al ([Bacon et al. 73]). As explained in section 4.2.2, the finite element procedure is no
more used here. The particle are modeled by the facets obstacles. The radius of particles are set to
be 0.131(2 9 b), 0.262(2 10 b), 0.524(2 11 b) µm with a fixed inter-particle distance of L = 2.59 µm. The
situation considered here is similar to the models used by Bacon et al ([Bacon et al. 73]) except
for the periodic boundary condition. Figure 4.9 shows the dislocation configurations at the flow
stress for three radius of particles respectively. The dislocation line near the particles is in paral-
lel position due to the self-interaction which pulls the branches on opposite sides of the particle.
The line between the particles is quite symmetric. The results obtained for the flow stresses have
been compared according to the ’effective line tension’ argument proposed by the authors. They
argued that the effective line tension, which properly accounts for the interactions, can be taken as
A(ln(1/(2Rp) + 1/L) −1 + B) where A is 1/(2π) and 1/(2π(1 − ν)) for edges and screws respectively.
Figure 4.10 shows our results obtained for the flow stresses normalized by Gmb/L plotted against
ln(1/(2Rp) + 1/L) −1 . The linear relation is perfectly reproduced and the slope of the fitting line
is about 0.254, which is closed to the expected value of 1/(2π(1 − ν)). Considering these observa-
tions, it can be said that the fixed dislocation source at the point 0.9 of L/2 from the periphery
of the spherical particle correctly reproduces the periodic boundary condition used by Bacon et al
([Bacon et al. 73]).
To investigate the effect of a difference in shear modulus on the flow stress, we have made simulations
of an alloy made of a copper matrix containing two spherical particles of radius Rp = 0.262 µm. A
shear modulus ratio (∆G/Gm) was set to be 1, 3 and 5, where ∆G = Gp − Gm. The inter-particle
distance is fixed to L = 2.59 µm respectively. Figure 4.11 shows the increment in the flow stress
as a function of ∆G/Gm. As the shear modulus of the particles increases, the flow stress increases
due to the fact that the repulsive image stresses on the dislocation line needs higher resolved shear
stress to bypass the dislocation line through particles. The fitting curve shows that the flow stress
changes as (∆G/Gm) 0.6 . The change in the flow stress is small even for the particles with a shear
modulus of 6Gm, for which the shear stress only increases by about 6 percent comparing to the no
image stress case. Actually, the minute effect of ∆G is expected from the short range of the image
stresses. Indeed, calculations show that the image stress exerted on a dislocation line decreases as
|x − x0| α , where x, x0 represents the position of a dislocation line and the centre of a spherical
particle respectively. α is found to be around 6 ∼ 7. So even in the case where ∆G/Gm = 5, the
image stress decreases below the flow stress of hard obstacle (∆G = 0) at a distance of 1.4 × Rp
from the centre of a particle. This effects means that the repulsive interaction between a dislocation

4.2 A simple case of dislocation-particle interaction 109
Figure 4.9: Dislocation configuration at the flow stress. The radius of particles are 0.131, 0.262,
0.524 µm from top to bottom.

110 Dislocation-precipitate interactions
Normalized flow stress
2.45
2.4
2.35
2.3
2.25
2.2
2.15
2.1
6.8 7 7.2 7.4 7.6 7.8 8
ln[1/(2Rp )+1/L] -1
Figure 4.10: τys/(Gmb/L) vs.ln(1/(2Rp) + 1/L) −1 . The line represents the fitting line and the slope
is 0.254.
and a particle will reduce the inter-particle spacing by about 8 percent. The small reduction in the
effective inter-particle spacing will result in the small increase in the flow stress. The effect of Rp
on the flow stress has been calculated with the constant ∆G/Gm value of three. The results are
shown in figure 4.12. The flow stress depends almost linearly on Rp.
The effects of a difference in shear modulus on the flow stress can be summarized as τys ∝
Rp × (∆G/Gm) α , where α is lower than 1. Note that this result is different from the case of
shearable particles. Indeed, the effects of shear modulus of coherent, penetrable particles can be
found in the literature ([Nembach 83]). He has calculated the image force exerted by one spherical
particle of modulus Gp on a straight, infinite dislocation line using the change in the strain energy
density of the dislocation. The shear stress is found to be proportional to ∆G 1.5 and R 0.22
p . Thus,
we observed that ∆G has a weaker effect in the case of hard particles than in the case of shearable
particles.

4.2 A simple case of dislocation-particle interaction 111
Increase in flow stress
0.06
0.05
0.04
0.03
0.02
0.01
0
0 1 2 3 4 5
(Gp-Gm )/Gm Figure 4.11: Increase in flow stress by a difference in shear modulus. ∆τ/τ0 is plotted against
∆G/Gm, where τ0 represents the flow stress of impenetrable obstacle with no image stress. Rp =
0.262 µm. The fitting curve shows ∆τ ∝ (∆G/Gm) 0.6 .
Normalized flow stress
2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2
400 600 800 1000 1200 1400 1600 1800 2000 2200
Rp /b
Figure 4.12: Normalized flow stress (τys/(Gmb/L)) vs. Normalized particle radius (Rp/b).
∆G/Gm = 3.

112 Dislocation-precipitate interactions
4.2.4 Increment in hardening stress
Although a difference in shear modulus has a little effect on the flow stress, its effect is much stronger
on the hardening stress. In this section, the effects of ∆G on the hardening stress are discussed.
The stress required to force a dislocation to glide between particles which have remaining Orowan
loops, are plotted in figure 4.13 both for the case of Gp = 4Gm (filled symbols) and for no image
stress (∆G = 0 : open symbols). The change in the shear modulus of the particles results in an
increased work hardening rate and the effect of ∆G increases with the particle radius. The image
stress fields are the sum of the interactions of the particle and each dislocation present around the
particles. Thus the dislocation lines have to overcome the additional image stress field coming from
the interaction of the residual loops and the particles. This additional stress is directly related to the
number of Orowan loops stored around the particles. As a result, compared to the case of no image
stress, a higher shear stress is required to bypass the particles and this effect is more pronounced as
more dislocation lines are passing, which leads to a higher material hardening. Considering that the
range and the magnitude of the image stress increases as the radius Rp, it can be understood that the
hardening rate is increased as Rp. Fisher et al ([Fisher et al. 53]) have investigated the hardening
of metal crystals induced by precipitate particles. They computed the back stress resulting from
Orowan loops and calculated the effective critical stresses of the Frank-Read sources. The argument
is that the hardening stress (τh) is related to the number of loops (N) by τh = γN, where γ is a
function of the particle radius Rp and the inter-particle distance L. They obtained
γ = 0.65cbGm
1 −
ν
2(1 − ν)
R 2 p
(L + Rp) 3
, where c is the parameter describing the closest distance between a source and a particle. We
obtained the slopes of each graph in figure 4.13 by linear fitting and the dependence of these slopes
on b(Rp) 2 /(L + Rp) 3 as shown in figure 4.14. It is found that the argument of back stress proposed
by Fisher et al ([Fisher et al. 53]) still holds in the case of moving dislocation line through two
particles. The parameter c is around 3.45 for the case of no image stress and 4.43 when Gp = 4Gm,
which means, based on their argument, that the effective distance of a stress source is shorter or
the back stress is higher if the image stress is included.
It is observed experimentally that all the dislocations left around the particles by the gliding dis-
locations are not rigorously confined to a single glide plane, but are rather of the prismatic form
([Humphreys & Martin 67]). If cross-slip is easy, a dislocation may overcome an obstacle in its
glide plane by slipping on another slip plane, with the formation of long jogs. The simulations pre-
(4.6)

4.2 A simple case of dislocation-particle interaction 113
Shear stress increment (MPa)
12
10
8
6
4
2
0
0 1 2 3 4 5
Number of Orowan loops
Figure 4.13: Work hardening of alloy containing two particles of radius 0.131 µm(1), 0.262 µm(2),
0.524 µm(3). τh − τys is plotted against number of Orowan loops around each particle, where τh,
τ0 represents the hardening stress and the flow stress respectively.
sented above to investigate the change of hardening stress have been done under the condition that
the cross slip of dislocation is prohibited by artificially changing the cross slip parameters. When
the normal conditions for cross slip are used, cross slip events have been observed. As an example
for the particles of radius 0.262 µm, cross slip occurs if the number of the Orowan loops reaches
four in case of no image stress and two in case of Gp = 4Gm. Considering that the back stress on
the primary slip plane becomes higher as the accumulation of the Orowan loops proceeds, it is easy
to cross slip to the secondary slip plane and cross slip again (double cross slip) into the primary
plane to bypass the particle. Figure 4.15 shows the bypassing of a dislocation line by double cross
slip. If the shear modulus of the particle is higher than that of the matrix, a high local stress is
generated near the particle and the local event of cross slip is more probable due to the image force.
This demonstrate the importance of including the image stress to investigate local events such as
cross slip.

114 Dislocation-precipitate interactions
Slope
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0 0.02 0.04 0.06 0.08 0.1 0.12
b(Rp ) 2 /(L+2Rp ) 3
Figure 4.14: The slope of fitting line in figure 4.13 vs. bR 2 p/(L + Rp) 3 . ∗ : ∆G = 0, × : ∆G = 3Gm.
Figure 4.15: Bypassing of particles by double cross slip of dislocation line. Dislocation initially
glides on the slip system of (111)[1-10] and changes the system on (11-1)[1-10] and then comes back
to initial slip system.

4.2 A simple case of dislocation-particle interaction 115
4.2.5 Discussion
In this work, we studied the effect of a difference in shear modulus on the flow stress and the
subsequent hardening stress using the 3D discrete dislocation dynamics code. The effect of ∆G on
the flow stress can be summarized by τys ∝ Rp(∆G/Gm) α , where α is lower than 1 and Rp is the
radius of particle. Because the range of the image stress is short, the maximum increment in the flow
stress is only 6 percent in the case of ∆G/Gm = 5 comparing to no image stress case. Nevertheless
the image stress increases as Orowan loops accumulate, resulting in a change of the work hardening
rate. This effect is due to the fact that the image stress fields are the sum of interactions of a
particle and each dislocation present around the particles. As slip accumulates, the dislocation
line feels an additional image stress field coming from the interaction of the residual loops and the
particles. The first order approximation on the work hardening of Fisher et al ([Fisher et al. 53])
is found to be valid even in the simple configuration of two particles and one dislocation line. The
effect of the image stress is that the effective distance of a stress source becomes shorter or the back
stress becomes larger. If dislocation cross slip is allowed in the code, it has been observed that a
dislocation can avoid an obstacle in its glide plane by cross slip into another slip plane. The back
stresses on the glide plane due to the Orowan loops trigger the cross slip event. If the image stress
is included, the cross slip probability increases. The image stress around the particle is large enough
to affect local events such as cross slip.
However, the computation time and effort are too demanding to include the effect of image stress
on simulations of alloy containing a large number of particles. The number of elements used here
in the simple situation of two particles is already about 5000 20-nodes elements. An approximate
way to include the effect of image stress is to introduce an effective radius which can represent
the difference in shear modulus. That way, the facets obstacles alone can be used to reproduce
the precipitate hardening, so that the finite element coupling is no more needed. For example, in
the case of ∆G = 3Gm, the average radius of the first trapped loop is 0.272 µm comparing to
0.264 µm for the case of ∆G = 0. Hence, the difference in shear modulus changes the effective
particle radius by only a few percent. Nevertheless the stress field generated by a second phase
particle is increasing as slip accumulates and the image stress field turns out to be crucial to predict
work hardening magnitude. This problem can then be treated using an empirical solution of the
image stress generated by the interaction between several dislocations and a particle.

116 Dislocation-precipitate interactions
4.3 Fatigue simulations of materials hardened by particles
4.3.1 Motivation and review of literatures
Fatigue in single-phased metals
Strain is usually localized in single-phased metals submitted to cyclic deformation. The imposed
plastic strain amplitude is accommodated by high local strains in strain localization zones called
persistent slip bands (PSBs).
The strain localization results in persistent slip markings (PSMs) at the specimen surface ([Man et al. 02]).
The irreversible character of slip inside PSBs are known to generate permanent surface steps. After
numerous experimental observations, it is generally agreed that fatigue crack initiates at the PSMs
([Mughrabi 85], [Suresh 98]) along the individual PSBs. Fatigue life is thus largely dominated
by the irreversibility of the slip in the PSBs and the associated surface step.
Existing crack initiation models can be categorized in (i) crack initiation due to a surface step larger
than a critical size, (ii) crack initiation due to local decohesion of crystal planes. Understanding of
the intrinsic PSB microstructure is therefore crucial to establish such models.
Transmission electron microscopy (TEM) are used to examine the dislocation microstructure in-
volved in PSBs, and to understand the specific role of dislocations in cyclic deformation. Surface step
displacements can be measured by atomic force microscopy (see for example, [Risbet et al. 03]).
Fatigue in precipitation-hardened materials
Multi-phase materials which contain precipitates often show good static strength compared to single-
phase materials. Under cyclic loading conditions, however, precipitation-hardened materials do not
always insure better fatigue properties.
Typical cyclic properties of materials containing shearable and non-shearable particles are shown in
Fig. 4.16 ([Gerold & Steiner 82]). The cyclic hardening behavior is shown as a function of the
cumulative plastic shear strain for various particle sizes. Fig. 4.16(a) demonstrates that specimens
containing shearable particles often suffer severe softening and even early fatigue failure. Large
cyclic softening is observed after an initial strong hardening up to a maximum shear stress: the
softening rate increases with the particle sizes, and peak-aged alloys (74 Å) show the largest cyclic
softening. In the case of non-shearable particles (Fig. 4.16(b)), the rate of hardening and the stress
drop decreases as the particle radius increases and the shear stresses are saturated.
A few of the characteristic fatigue properties observed in experiments are outlined below for each

4.3 Fatigue simulations of materials hardened by particles 117
(a) Underaged specimens (Shearable particles) (b) Overaged specimens (Non-shearable particles)
Figure 4.16: Cyclic hardening and softening of aged Cu-2at%Co single crystals
case ([Mughrabi 83]).
• Shearable particles
After an initial hardening stage, the cyclic strain becomes localized into persistent slip bands.
A drastic cyclic softening related to the destruction of the precipitation hardening in the PSBs
leads to the early initiation of shear-type fatigue cracks at the PSBs surface intersection.
• Non-shearable particles
The cyclic softening is strongly reduced and the cyclic deformation behavior is much more
stable. Non-shearable particles produce more homogeneous straining.
Numerical simulations of fatigue tests
The dynamical features of the dislocations inside the PSBs during the cyclic deformation are not
easily accessible by experimental observations, e.g. TEM. It is due to the fact that the stresses
are removed during TEM observations, which makes the observed dislocation microstructure to be
different from the microstructure under stresses. Besides, the free surface effects are not negligible.
Another difficulty arises when relating the details of the formation of the surface steps directly
with the dislocation microstructure inside the grains, since the two experiments are often performed
independently, and sample preparation methods are usually destructive. Thus, a complete and
comprehensive scheme for fatigue crack initiation is still missing.
The development of the DDD method and the increase of the computer capabilities enable simula-

118 Dislocation-precipitate interactions
tions to provide crucial information concerning the formation of slip bands. Numerical simulations
make it now possible to observe the details of the PSB formation, and to investigate the relation
between surface steps and the corresponding dislocation microstructure. This knowledge would help
to understand better crack initiation mechanisms and to build a more elaborate fatigue life model.
Scope of this section
The performance of the new parallel DDD program (Chapter 3) makes it feasible to simulate dislo-
cations interacting with thousands of precipitates. Fatigue tests of precipitation-hardened material
are simulated in 3D. The fatigue simulations are similar to the work of [Déprés 04] developed in the
case of 316L stainless steel. The effects of shearable and non-shearable particles on the formation of
PSBs are studied. Dislocation mechanisms for PSBs formation are detailed and some of numerical
results are compared with experimental observations.
4.3.2 Description of the simulation method
Simulation volume geometries
Cylindrical grain geometry has been adopted for the shape of the simulated volume. The volume
is assumed to be a surface grain of a fatigue tested specimen, i.e. the volume consists of one free
surface and grain boundaries. The cylindrical volume is represented by 20 facets as shown in Fig.
4.18(a). The free surface is represented by assigning zero strength to the top facets (see Sec. 2.4.2),
thus dislocations can escape through that surface. All the other facets act as strong obstacles to
the dislocation motion as if for highly disordered grain boundaries.
A virtual volume is prepared on top of the free surface to keep track of the dislocations exiting
the simulated crystal volume (see Fig. 3.21(b)). This virtual volume and virtual dislocations are
introduced to compute deformations of the free surface as will be detailed in Sec. 4.3.5. The virtual
dislocations are set to have no effects on dislocations inside the simulation volume and no return
into the crystal simulation volume is authorized.
The normal vector of the top facet is taken as [110]. The diameter of the cylinder is 10 µm and the
height is 5 µm. The image forces due to the free surface are not taken into account in this work.
Materials parameters
The material’s parameters used in this simulations are those of nickel as listed in Table 4.1.

4.3 Fatigue simulations of materials hardened by particles 119
Poisson’s ratio Shear modulus Burgers vector
magnitude
Activation vol-
ume
Viscous drag co-
efficient
Threshold stress
ν G(GPa) b(Å) (V/b 3 ) B(10 −5 P as) τIII(MPa)
0.276 94.7 2.5 2117 1.06 51.2
3D particle arrangement
Table 4.1: Mechanical and microscopic parameters of nickel
A cylindrical volume containing random distributed particles is constructed as follows. For sim-
plicity, the particles are assumed to have the same radius rp and the associated volume fraction is
vf .
Step 1 Preparing closed packed spheres (Fig. 4.17(a))
Close packed spheres of an arbitrary radius r are constructed in a larger sphere (radius R).
The center of each sphere of radius r is assumed to be the nucleation site of each particle,
and each particle gets their material from the volume of sphere r during the Ostwald ripening
process.
Step 2 Adjusting the volume fraction
The radii of all spheres are reduced by a common factor while their locations remain un-
changed. The factor is given so that the volume fraction of the shrunken spheres is equal to
vf .
Step 3 Adjusting the radius of particles (Fig. 4.17(b))
The radius of the shrunken spheres in Step 2 is scaled to rp, and the coordinates of the centers
are scaled as well.
Step 4 Cutting the cylindrical volume (Fig. 4.17(c))
A cylindrical simulation volume is placed at the center of the outer sphere, and spherical
particles situated inside the cylinder are selected.
In this work, ’Step 1’ is achieved by successive trials of putting spheres of radius ’r’ in the sphere
of radius ’R’. Subsequent trials are accepted only if the new sphere is not intersecting with spheres
already in the volume. Although this method does not generate closely packed arrangement of
particles, the resulting particle arrangement shows a purely random arrangement. Bi-modal size

120 Dislocation-precipitate interactions
(a) Randomly placed spheres
Volume of interest, ri
Particle, rp
Simulation volume
(b) Adjusting the radius of particles (c) Selecting spheres inside the
cylindrical volume
Figure 4.17: Construction of a randomly distributed configuration of particles in the cylindrical
simulation volume (Examples of bimodal size distribution case)
distributions (see Sec. 4.3.6) are constructed using the same procedure except that two radii of
spheres are placed in ’Step 1’ as shown in Fig. 4.17(a).
Radius and volume fraction of particles
Two cases of particle radius, rp = 160 nm and 400 nm, are considered. The volume fraction vf is
fixed to 14% for all the cases. The number of particles generated in the cylindrical volume by the
procedure above is 2510 for rp = 160 nm and 161 for rp = 400 nm case .
Each individual particle is constructed using two pyramids attached at the bottom 3 in an effort
to reduce the number of nodes and facets constituting the particles (reducing the computation
load). The cylindrical simulation volumes are shown in Fig. 4.18, which contain (a) 161 particles
of rp = 400 nm and (b) 2510 particles of rp = 160 nm respectively.
The strength of a particle
Particles are assumed to act as geometrical barriers with a pre-defined strength to the dislocation
motion. The image forces due to the elastic modulus difference are not computed and no stress
fields around particles are considered for the simplification of computing.
The strength of a particle decreases as the particle is sheared by dislocations. This is due to both
the decrease of the effective particle size on the glide plane and the loss of coherency for ordered
precipitates ([Stoltz & Pineau 78]). The evolution of particle’s strength is illustrated in Fig.
4.19(a) for the case where a particle is sheared by successive passages of dislocations in the same
3 One particle involves thus six facets and five nodes.

4.3 Fatigue simulations of materials hardened by particles 121
(a) Particles of rp = 400 nm and vf = 14% (b) Particles of rp = 160 nm and vf =
Figure 4.18: Cylindrical simulation volume containing randomly distributed particles of (a) rp =
400 nm and vf = 14% (b) rp = 160 nm and vf = 14%
Strength of particle
(a) Geometrical effect
Number of
dislocation passage
14%
Strength of particle
(b) Chemical effect
Figure 4.19: Evolution of particle’s strength due to shear-off by dislocation passages
Loss of strength
Number of
dislocation passage
glide plane. Fig. 4.19(b) illustrates that a particle may loose its strength completely before being
totally sheared off due to the surface energy increase and the loss of coherency induced by the
dislocations random chop-up.
In this work, the particle’s strength (or facet’s strength) is decreased linearly with each event of
dislocation passage through a given particle’s facets. As a first order approximation, the strength
of the facet τfacet (see Sec.2.4.2) is decreased linearly from the initial strength by τfacet/(2rp/b)
whenever a dislocation penetrates the facet as shown in Fig. 4.20 4 . The facet strength is set to
zero after a certain number of passages of dislocations to represent the chemical effect shown in Fig.
4.19(b).
4 The magnitude of the strength drop is from the assumption that a particle looses its strength after 2rp/b of
dislocations passages, which corresponds to complete shear-off of the particle as sketched in Fig. 4.19(a)

122 Dislocation-precipitate interactions
Strength of particle
Initial strength,
τ
∗ facet
τ facet
τfacet Slope=
2ri/b
Number of
dislocation passage
Figure 4.20: Evolution of facet’s strength with dislocation passages
Particles of radius 160 nm are assumed to be easily shearable, and the initial strength of facets is
defined as 292 MP a and the final strength τ ∗ facet
being 162 MP a. Particle of radius 400 nm are
assumed to be non-shearable or difficult to be sheared by using the initial strength being 7310 MP a.
The initial configuration
The initial dislocation microstructure of all the simulations is composed of four Frank-Read sources,
in the form of pinned dislocation segments. All the Frank-Read sources are of the edge-type with
the Burgers vector a
2 [¯1¯10] on the slip plane (¯11¯1) (system 7 in Table 2.1). It should be noted that
there is no dislocation nucleation around the particles and also that no dislocations are punched in
from the free surface.
The loading conditions
Fatigue simulations are performed under a plastic strain control with a fully symmetrical push-
pull loading ratio (ɛmax p /ɛmin p = −1), and an applied plastic strain amplitude △ɛp = 1 × 10−3 . In
DDD simulations, only stresses can be applied. Imposed plastic strain conditions are achieved by
monitoring the total slip accumulated in all the active slip systems. The applied stresses then are
increased or decreased by comparing the resulting plastic strain to the pre-selected strain level.
In the fatigue simulations, the plastic strain rate is monitored at each time step. The applied stress
is stepwise increased by 1 MP a if the plastic strain rate is lower than the pre-selected minimum
plastic strain rate, 10 −7 ((1) in Fig. 4.21). The load is kept constant in the case that the plastic

4.3 Fatigue simulations of materials hardened by particles 123
dλ
λ k
εp VM
εp VM
εp max
εp min
1 2 3 4
Figure 4.21: Quasi-static loading condition: stepwise increment and decrement of the applied
stresses
strain rate is between the minimum and the maximum strain rate ((2) in Fig. 4.21), until the
dislocation microstructure is in equilibrium with the external loading. This condition is achieved
by keeping the load constant while performing discrete time steps until the resulting plastic strain
rate becomes lower than the pre-selected minimum strain rate ((3) in Fig. 4.21). The applied stress
is decreased by 1 MP a if the plastic strain rate is higher than the pre-selected maximum plastic
strain rate, 10 −4 ((4) in Fig. 4.21).
4.3.3 Evolution of the dislocation microstructure during the fatigue tests
Pas k
Pas k
Pas k
General features of the formation of the dislocation microstructure
The initial Frank-Read sources begin to expand and generate dislocation loops as the applied stresses
are increased during the first quarter cycle. Parts of the loops are leaving the simulation volume
through the free surface, which prints steps on the free surface as will be detailed in Sec. 4.3.5.
The other parts are piled up along the strong grain boundaries. The screw part of dislocation lines
tends to cross slip owing to the back stresses from the stored dislocations. The cross slip mechanism
spreads slip lines over the whole simulation volume. Particles affect both the dislocation mobility
and the cross slip probabilities which result in quite different microstructures as compared with the
single-phased material case.

124 Dislocation-precipitate interactions
2
3 4
1
Figure 4.22: Evolution of the dislocation microstructure by the cyclic loading (Case of rp = 400 nm).
As the sign of the applied stresses is reversed, the motion of the dislocations is reversed likewise. The
initial dislocation microstructure, however, can not be completely restored due to the irreversible
character of slip. Slip irreversibility is caused by the cross slip, the line reconnection (colinear
junction) and the elimination of dislocation lines by the free surface. It increases with the number
of fatigue cycles.
These general features are illustrated in Fig. 4.22 using the figures taken from the rp = 400 nm
case. The evolution of the dislocation structures is shown with the stress-strain curve of the first
fatigue cycle 5 . The initial expansion of the Frank-Read sources (1) is followed by cross-slip which
spreads slip through the entire simulation volume (2), and by subsequent changing of the sign of
the applied plastic strain, specific dislocation microstructure forms (3)-(4), and even after one cycle
the microstructure (5) is quite different from the initial one (1).
5 Compressive plastic strain is applied first.
5

4.3 Fatigue simulations of materials hardened by particles 125
Dislocation density evolution
The evolution of the total dislocation density, ρtot is monitored for the case of the volume contain-
ing particles with rp = 160 nm , particles with rp = 400 nm and no particles during the cyclic
deformation. ρtot is plotted as a function of the accumulated cyclic Von Mises strain in Fig. 4.23.
The dislocation densities quickly increase and fluctuate according to the cyclic deformation in all
the cases owing to the periodically vanishing cyclic load. Here the saturation of the dislocation
densities is not observed because of the relatively small number of fatigue cycles have been per-
formed (close to 3 cycles 6 for rp = 160 nm and 5 cycles for rp = 400 nm). It is expected, however,
that the dislocation densities would gradually saturate as the fatigue cycles proceed as observed by
[Déprés et al. 04].
The simulation results show that
rp=160 nm
1. ρtot rp=400 nm
> ρtot No particle
> ρtot 2. The rates of the dislocation accumulation after each fatigue cycle are of the same order as the
total densities.
After three fatigue cycles (around ɛ V M 0.006), ρ rp=160 nm
tot
and six times than ρ
No particle
tot
.
is three times larger than ρ rp=400 nm
tot
The simulation volume containing particles with rp = 160 nm has a high resistance related to the
limited glide area per slip plane. This effect is due to the large number of particles that are supposed
to be shearable. To accommodate the applied plastic strain with limited dislocation glide, it is nec-
essary to have a higher density of dislocations. After reversing the stresses, the dislocations still
have difficulties to find an easy glide path and to annihilate each other, thus most of the dislocations
are left inside the volume. For these reasons, it can be deduced that shearable particles give rise to
a high slip irreversibility.
In the case of particles with rp = 400 nm, dislocations have a higher chance to find an easy glide
path because there are fewer particles in the volume. Moreover the particles are more effective
in spreading dislocations through the simulation volume by cross slip because they are not easily
shearable and involved with Orowan loops. The applied plastic strain thus can be accommodated
with a lower dislocation density, and the rate of the dislocation accumulation reduces compared
to the shearable particle case, since it is easier to move reversely by annihilating Orowan loops
6 The simulation has to be stopped just before ɛp reaches zero near 3 fatigue cycles because of the large number of
segments involved

126 Dislocation-precipitate interactions
ρ[m -2 ]
1.8e+13
1.6e+13
1.4e+13
1.2e+13
1e+13
8e+12
6e+12
4e+12
2e+12
0
0.0e0 2.0e-3 4.0e-3 6.0e-3 8.0e-3 1.0e-2 1.2e-2
ε VM *
r p =160nm
r p =400nm
No particle
Figure 4.23: Evolution of the total dislocation density of the volume containing rp = 160 nm,
rp = 400 nm and no particles
left around the particles from the forwarding glide. Thus the irreversibility of slip is significantly
reduced in the case of non-shearable particles.
Strain localization kinematics
In the preceding section, it is shown that the shearable particles favor high ρtot. The next question
to address is whether the simulation can reproduce the localization of the plastic deformation or
strain. Fig. 4.24 shows the dislocation microstructure formed after 3 cycles in the rp = 160 nm
case and after 5 cycles in the rp = 400 nm case, along [110] direction. The figures illustrate clearly
that the dislocation structures are highly heterogenous and intense slip bands are formed on the
primary slip plane due to the cyclic loading. This result is consistent with experimental observations
([Calabrese & Laird 74]). Plastic strain localization is believed to cause fatigue damage, since
the local plastic strain has to be high enough to accommodate all the applied plastic strain. This
process can eventually lead to fatigue crack nucleation.
To demonstrate the statistics of the PSBs formation quantitatively, the spatial distribution of the
dislocation densities is computed as follows at each time step k. The cylindrical simulation volume

4.3 Fatigue simulations of materials hardened by particles 127
(a) Particles with rp = 160 nm (b) Particles with rp = 400 nm
Figure 4.24: Localization of slip by forming intense slip bands
is sliced into finite layers along the slip planes normal [¯11¯1]. Dislocation densities are then computed
in each layer . The heterogeneity of the dislocation density can be shown by plotting the calculated
dislocation densities of each layer along the reference axis [¯11¯1].
Fig. 4.25 shows the evolution of such spatial dislocation density distributions. Three axis of the
coordinate system correspond to the dislocation density, the position of each layer and the cycle
number.
From the figure, the general features of the formation of dislocation microstructure can be confirmed,
i.e. the first increase of the applied plastic strain spreads dislocations over the simulation volume
in all the cases. In the next cycles, the heterogeneous dislocation structure forms and certain zones
accumulate a high dislocation density.
The detailed observation of Fig. 4.25 reveals that the particles affect the slip localization in several
ways :
1. The width wdisl of the dislocation distributions over the simulation volume is the largest in
the case rp = 400 nm (non-shearable particles) and the smallest in the case rp = 160 nm
rp=400 nm
(shearable particles), i.e. wd No particle
> wd > w
rp=160 nm
d
rp=160 nm
2. The maximum local dislocation densities (ρmax) are in the following order, ρmax ρ rp=400 nm
max
> ρ
No particle
max
.
rp=400 nm
3. The intense slip band width is smaller in the case of shearable particles, i.e. db d rp=160 nm
b .
.
>
>

128 Dislocation-precipitate interactions
(a) Particles with rp = 160 nm (b) Particles with rp = 400 nm
(c) No particle
Figure 4.25: Evolution of slip localization
4. There is no clear dislocation localization up to five cycles in the case containing no particles. In
the other cases, dislocation localization has occurred, and at leat one high dislocation density
peak is present through the whole simulated fatigue cycles.
Item 1 demonstrates that non-shearable particles promote the cross-slip due to the back stresses of
Orowan loops around the particles, and the dislocations easily sweep a large area of the simulation
volume as a result. Item 2 and 3 are consistent with the experimental observations, according
to which persistent slip bands (PSBs) are much thinner if particles are shearable, and the local
plastic strain becomes higher as the PSBs gets narrower ([Lee & Laird 83]). Fig. 4.26 shows
some of the experimental data of the PSB thicknesses and the related local plastic shear strain
([Mughrabi 83]). Item 4 can be related to the early initiation of fatigue crack in the case of shear-

4.3 Fatigue simulations of materials hardened by particles 129
Figure 4.26: Relation between the local plastic shear strain amplitude and the thickness of PSBs
able particles (see Fig. 4.16(a)). It is also interesting to note that the number of intense slip bands
is only one in the case of shearable particles but second intense slip band begins to form in the
case of non-shearable particles. This is related to the experimental observation that the number of
cycles till crack initiation is inverse to the average slip band distance ([Graf & Hornbogen 78]),
although higher number of fatigue cycles are necessary to confirm it.
The speed of the slip localization can be quantified by the standard deviation of the spatial dislo-
cation distribution curves at each time step, because the standard deviation becomes larger as the
dislocation structure gets more heterogeneous. The standard deviation at time t is computed as
follows.
σρ(t) =
1
Dg
Dg
0
ρ(t, x (s) ) − ¯ρ(t) 2 dx (s) (4.7)
¯ρ(t) is the average dislocation density, and Dg is the size of the simulation volume.
Fig. 4.27 shows the evolution of σρ(t) for each case, and it shows σρ(t) rp=160 nm > σρ(t) rp=400 nm >
σρ(t) No particle . It can be seen that both the intensity and the speed of the slip localization is the
highest in the case with shearable particles.
Details of the intense slip band
The intense slip bands of the rp = 160 nm case (shearable particles) are shown in Fig. 4.28. The
dislocation microstructure is taken at a cycle number 3 with ɛp ∼ 0. Fig. 4.28(a) shows a 3D image
of the dislocation structure viewed from the orientations normal to [1¯11] (slip plane normal), [110]
(Burgers vector) and [¯112].

130 Dislocation-precipitate interactions
Standard deviation
3e+13
2.5e+13
2e+13
1.5e+13
1e+13
5e+12
0
0.0e0 2.0e-3 4.0e-3 6.0e-3 8.0e-3 1.0e-2 1.2e-2
ε VM *
r p =160nm
r p =400nm
No particle
Figure 4.27: Evolution of the standard deviation σρ(t) (Eq. 4.7) in different simulations
In the plane perpendicular to the primary slip plane (normal to [110]), intense slip bands can be
seen clearly in the form of thin and compact dislocation walls. In the plane parallel to the primary
slip plane (normal to [1¯11]), the dislocation structure is quite heterogeneous, and shows ladder-like
structures along the Burgers vector direction. Dislocation debris and small loops are visible between
the particles. Fig. 4.28(b) shows the isolated intense slip band, and considerable amounts of residual
dislocations are clearly visible. It should be noted that the residual dislocation tangles are different
in size compared to those in the single-phased material case, in which large tangles are observed
together with ladder-like slip bands ([Obrtlik et al. 94], [Déprés et al. 04]).
Small tangles between particles in the case of shearable particles are also observed in experimental
data. Fig. 4.29 shows a dislocation microstructure of fatigue tested Inconel 718 observed in TEM.
Although the material characteristics are quite different 7 , this micrograph clearly shows small high
density tangles of primary residual dislocations.
Fig. 4.30 shows the intense slip bands of non-shearable particle case (rp = 400 nm) at cycle number
5 with ɛp = 0. The wall thickness of the PSB is much larger than the previous case, and the
dislocation densities are variable in the band seen along [110] direction. In the plane normal to
[1¯11], several Orowan loops and dense dislocation tangles are formed around the particles. In the
7 Particle radius=20 − 40 nm, Grain size=20 − 40 µm, Volume fraction> 15%

4.3 Fatigue simulations of materials hardened by particles 131
(a) 3D image of intense slip bands (b) Isolation of intense slip band
Figure 4.28: Details of intense slip band of the shearable particle case(rp = 160 nm). Three layers
of thickness 300 nm are assembled for 3D image in (a)
Figure 4.29: TEM micrograph of fatigue tested Inconel 718 up to 10,000 cycles

132 Dislocation-precipitate interactions
(a) 3D image of intense slip bands (b) Isolation of intense slip band
Figure 4.30: Details of intense slip band of the non-shearable particle case(rp = 400 nm). Three
layers of thickness 300 nm are assembled for 3D image in (a)
space between the particles, however, long dislocation lines are clearly visible and the dislocation
distribution is rather homogeneous as illustrated in Fig. 4.30(b). It is also observed that some of
the complex dislocation structures are formed separated by the same distance as that of between
two close particles.
Intense slip band formation mechanism
In the case of shearable particles (rp = 160 nm), the dislocation density increases rapidly since
dislocations possess a high degree of irreversibility (see Fig. 4.23). The characteristic of this con-
figuration is that there is a limited number of easy glide paths for dislocations. Thus, slip bands
would form along one of the easy glide paths whose thickness is usually limited (order of rp). Upon
load reversal, double cross-slipped dislocations can glide in the opposite direction along path close
to the initial glide path because (i) cross-slipped dislocations also have a limited glide distance and
(ii) the particles in the initial path loose part of their initial strength. This would form closely
spaced edge dipoles, so called vein structures. As the cycling proceeds, the subsequent cross-slipped

4.3 Fatigue simulations of materials hardened by particles 133
log(frequency)
10 0
10 −1
10 −2
10
160 180 200 220 240 260 280 300
−3
τ [MPa]
facet
Figure 4.31: Repartition by the particle strength
screw dislocations due to the cyclic loading react with the edge dipoles and produce prismatic
loops aligned in the Burgers direction or helicoidal structures as observed in single-phased materi-
als ([Li & Laird 94], [Déprés et al. 04]) but with a much smaller size. The prismatic loops can
move along their glide cylinder, and form ladder-like structures. The repeated motion of interfacial
dislocations with the cycles will eventually make particles at the PSBs edges to loose their strength,
and persistent bands will be formed at this place. This process is observed numerically in the sim-
ulations.
Fig. 4.31 shows the statistical distribution of the facet’s residual strength after 3 cycles. The facet
strength are distributed as follows: most of the facets are not sheared and keep their initial strength
(right peak) and a small portion of the facets are completely sheared (left peak). The spatial distri-
bution of the facet’s strength is shown in Fig. 4.32(a) by superimposing colors corresponding to the
magnitude of strength for each facet. A clear channel of sheared particles is visible and its position
corresponds exactly to that of the intense slip band. The dislocation structure is overlapped in
Fig. 4.32(b). It should be noted that the particles near the intense slip band also loose strength,
which possibly demonstrates that there exist interfacial dislocations at the periphery of the slip
band, which move rather freely according to the cyclic load changes. Clear channel is also observed
experimentally in which no dislocations and no particles are visible as shown in Fig. 4.33.
In the case of non-shearable particles, the accommodation of the applied plastic strain is much easier
because dislocations can move over a relatively long distance on the glide plane. During the first
few cycles, the particles are bypassed by the Orowan mechanism. Glissile loops are accumulated
around the particles. When the critical stress is reached, the screw portions of the loops change

134 Dislocation-precipitate interactions
(110)
(a)
τfacet[MPa]
300
280
260
240
220
200
180
160
Figure 4.32: Spatial distribution of particle strength
Figure 4.33: Clear channel containing no particles and no dislocations (Inconel 718)
(b)
τfacet[MPa]
300
280
260
240
220
200
180
160

4.3 Fatigue simulations of materials hardened by particles 135
σ VM [MPa]
300
200
100
0
-100
-200
-300
-6.0e-4 -4.0e-4 -2.0e-4 0.0e0 2.0e-4 4.0e-4 6.0e-4
ε VM
Figure 4.34: Typical stress-strain curve coming from the simulations (No particles case)
their glide plane by cross slip, which contribute both to propagate slips in the simulation volume
by generating dislocations in the secondary plane, and also to the formation of 3D loops around the
particles. Interactions between these loops and the dislocations in the secondary plane eventually
form dense tangles around the particles. As the cyclic deformation proceeds, the tangles around
the particles act as pinning points of dislocations moving between the particles. It favors thus the
formation of dislocation dipoles which are linked by two near particles. The cutting of these dipoles
by freely gliding dislocations, then generate stable dislocation structures. The subsequent formation
of dipoles between the particles and the dislocation interactions make dense dislocation structures
to form between the particles. This mechanism explains the dense dislocation tangles observed
around the particles and complex dislocation structures between pairs of closely spaced particles.
4.3.4 Mechanical behavior
Cyclic stress-strain relation
A typical cyclic stress-strain curve is shown in Fig. 4.34 for the ’No particles’ case. The first
quarter-cycle corresponds to the activation of the initial Frank-Read sources, which is the hardest
part of the cycles.
The cyclic response curves are shown in Fig. 4.35 for the case of rp = 160 nm (shearable), rp =

136 Dislocation-precipitate interactions
Stress[MPa]
500
450
400
350
300
250
r p =160nm, v f =14%
r p =480nm, v f =14%
r p =480nm, v f =8%
No particles
200
0.0e0 2.0e-3 4.0e-3 6.0e-3 8.0e-3 1.0e-2 1.2e-2 1.4e-2
Cumulative plastic strain
Figure 4.35: Cyclic response for rp = 160 nm (vf = 14%) and rp = 400 nm (vf = 8% and 14%)
compared with the single-phased material case
400 nm (non-shearable) and the single-phased material case. The curves obtained with two different
volume fractions (vf = 8, vf = 14%) are plotted for non-shearable particles.
The initial shear stress amplitude is the highest in the case of shearable particles as expected and is
the lowest in the single-phased material case. The simulation volumes containing the non-shearable
particles show intermediate initial stress values, and the stress amplitude increases with the particle
volume fraction.
The short hardening stage is followed by a cyclic softening response in all the cases. The degree of
softening is maximum for the shearable particle case, and increases as the volume fraction in the
non-shearable particle case.
4.3.5 Surface slip markings
Surface displacement computation method
Dislocations that leave the simulation volume print steps on the free surface. The computation of
the surface steps can give valuable information concerning the fatigue life because it is believed that
the fatigue cracks are initiated from these surface steps. In this section, the method to compute
surface steps is presented.

4.3 Fatigue simulations of materials hardened by particles 137
b
P r i m a i r e
Op
D2
Od
D1
D e v i e
(a) Associated problem
b
d
O
b
P r i m a i r e
+...
Op
+
+
Od
D e v i e
∆
(b) Systematic method
Figure 4.36: Computation method of displacements associated with the general case of non-planar
dislocation loops
Displacements of closed loops can be computed by decomposing the loop into as many triangular
dislocation loops as needed using the equations in Sec. 2.2.2. In the case that a part of a dislocation
loop has changed its slip plane by cross slip, an additional operation is necessary [Déprés et al. 03].
Fig. 4.36(a) shows a non-planar dislocation loop. Points Op and Od represent the common point to
construct triangular dislocations in the primary and the cross-slip plane respectively. If triangular
loops are constructed for each dislocation segment in both planes, the non-planar dislocation loop
would miss two triangular loops (OpD2D1 and OdD1D2) as shown in Fig. 4.36(a). The displacement
solution computed from the arbitrary dislocation segments (e.g. OpD1) would be wrong, since they
are never canceled. To remove this artifact, once a segment is found to have a neighbor in a different
plane, a supplementary triangular loop is constructed by three points: a common point (e.g. Op),
the extreme point of the segment (e.g. D1) and the projection point of the common point along the
glide axis (∆). Fig. 4.36(b) shows the procedure, which cancels the effect of the arbitrary segments
generated in constructing triangular dislocation loops.
In fatigue simulations, dislocations can leave the simulation volume. Thus some of the dislocation
loops are cut and left open by the free surface, and the Barnett’s equations (Sec. 2.2.2) can no
longer be used without a special treatment. The displacements of open loops can be solved by
adding virtual dislocations outside the simulation volume ([Weygand et al. 02]). The simulation
volumes are constructed by two distinct parts, one part is the crystal and the other is a virtual
medium, containing the virtual segments as briefly introduced in Sec. 4.3.2. Dislocations are
allowed to leave the crystal volume to the virtual medium in order to keep the dislocation loops
closed. The dislocations however are not allowed to return back into the crystal volume from the
b
+
d
+
O
+...

138 Dislocation-precipitate interactions
Figure 4.37: Examples of surface steps generated by the activation of a single Frank-Read source
in the simulation volume
virtual medium so that the motion of the dislocations in the crystal is not arbitrarily modified by
the virtual dislocations. The activation of the Frank-Read source and the subsequent deformation
of the free surface are shown in Fig. 4.37, which adopts the virtual medium method.
Surface steps and associated dislocation structures
Surface steps are computed and shown in Fig. 4.38(a) for the case of shearable particles after 3
fatigue cycles and in Fig. 4.38(b) for the case of non-shearable particles after 5 fatigue cycles. The
surface steps represent exactly the same characteristics of the associated dislocation structure. The
surface markings are intensively confined in a narrow region in the case of shearable particles as the
slip bands involved are narrow and contain a high density of dislocations (see Fig. 4.24(a)). The
surface markings are dispersed over the free surface in case of non-shearable particles as the wall
thickness of slip bands are relatively wider and there are more than one band inside the crystal as
shown in Fig. 4.24(b).
The differences in the surface markings for each cases can be seen clearly from one dimensional
profiles along a probing line. This displaying method is similar as for experimental results obtained
using atomic force microscopy (AFM). Fig. 4.39 shows such surface profiles along the direction
normal to the primary plane, i.e. [1¯11]. As indicated in Sec. 4.3.3, the simulation of the shearable
particle case has finished just before ɛp reaches zero near 3 fatigue cycles, thus the surface profile
associated with a small plastic strain. The surface marking, however, is significantly wider in the
case of non-shearable particles than in the case of shearable particles as indicated in Fig. 4.39.
Detailed surface morphologies are computed on the surface at the exact location of the intense slip

4.3 Fatigue simulations of materials hardened by particles 139
(a) Case of shearable particles (b) Case of non-shearable particles
Figure 4.38: Surface steps of (a) the simulation volume containing shearable particles after 3 cycles
and (b) the simulation volume containing non-shearable particles after 5 cycles
Suface step (b)
10
5
0
−5
Shearable particle
Non−shearable
particle
−10
0 1 2 3 4 5 6 7 8 9 10 11 12
Probe distant (µm)
Figure 4.39: One dimensional profiles of the surface steps along [1¯11] direction for the case of
shearable particles (dashed curve) and non-shearable particles (solid curve)

140 Dislocation-precipitate interactions
(a) Tongue-like surface slip markings in the case of
shearable particles
(b) Ribbon-like surface slip markings in the case
of non-shearable particles
Figure 4.40: Evolution of detailed surface morphologies computed on the surface of 500 nm width
of (a) the shearable particle case after 1
2
and 4 1
2 cycles
and 2 1
2
cycles (b) the non-shearable particle case after 1
2
bands formed in the volume at ɛp = 0. Fig. 4.40 shows the evolution of the detail of the surface
morphologies from (a) 1
2
4 1
2
cycle to 2 1
2
cycles in the case of sherable particles and (b) 1
2
cycle to
cycles in the case of non-shearable particles. A close examination of these images shows that
the tongue-like slip markings are associated with the intense slip bands in the simulation volumes
containing shearable particles (see Fig. 4.28(b)) and the ribbon-like slip markings are associated
with the intense slip bands of the non-shearable particle case (see Fig.b 4.30(b)). In the shearable
particle case, prismatic loops aligned in the Burgers direction are responsible for the tongue-like slip
markings. The ribbon-like slip markings are related to dislocation structures gliding between the
particles. The length of the ribbon-like marking is closely related to inter-particle distance.
4.3.6 Fatigue properties of materials containing particles with a bimodal size
distribution
Alloys which contain a bimodal size particle distribution are particularly interesting because of
the optimized combination of fatigue properties, i.e. good strength (merit of underaged alloy as
shown in Fig. 4.16(a) ) and cyclic stability (merit of overaged alloy as shown in Fig. 4.16(a)).
Waspaloy ([Clavel & Pineau 82]) is one of the examples which have particles with a bimodal size
distribution.
Three bimodal cases are considered with the same volume fraction of particles vf = 14% but with
different ratio between the number of large (rp = 400 nm) and small (rp = 160 nm) particles. The
number of particles of each size is listed below for the three considered cases, and the simulation
volume is shown in Fig. 4.41 for the ’Bimodal2’ case.

4.3 Fatigue simulations of materials hardened by particles 141
(110)
Unshearable
Shearable
partiicles (r=0.4µ m) partiicles (r=0.16µ m)
Figure 4.41: The simulation volume which contains both rp = 160 nm and rp = 400 nm particles
• Bimodal1 : rp = 160 nm, 2080 particles + rp = 400 nm, 31 particles
• Bimodal2 : rp = 160 nm, 1456 particles + rp = 400 nm, 57 particles
• Bimodal3 : rp = 160 nm, 772 particles + rp = 400 nm, 103 particles
The same volume geometries and material properties are adopted for the three arrangements. The
simulation box is taken as a cylindrical volume, and the particles of each size have the same initial
and final strengths as before (see Sec. 4.3.2). The same loading condition as in the mono-modal
cases is applied, i.e. △ɛp = 1 × 10 −3 and R = −1.
The evolution of the total dislocation densities are compared with the case of the mono-modal size
particles (rp = 160 nm) in Fig. 4.42(a). The total densities and the rates of the dislocation ac-
cumulation decrease as the percentage of the large particles is increased. Fig. 4.42(b) shows that
the slip localization retards with the percentage of large particles. It should be noted that the total
dislocation densities and the slip localization kinetics of all the bimodal cases considered here lie
between the two previously investigated mono-modal cases (see Fig. 4.23 and Fig. 4.27).
A 3D reoresentation of the dislocation structure is shown in Fig. 4.43(a) for the ’Bimodal2’ case
after four fatigue cycles, and details of the associated intense slip bands are shown in Fig. 4.43(b).
As compared with the mono-modal distribution case (rp = 160 nm, see Fig. 4.28(a)), the slip bands
are more diffuse (the band thickness is larger and the local dislocation density is accordingly lower).
The dislocation structure in the (1¯11) plane shown in Fig. 4.43(a) and the intense slip bands plotted

142 Dislocation-precipitate interactions
ρ[m -2 ]
2.5e+13
2e+13
1.5e+13
1e+13
5e+12
r p =160nm
Bimodal 1
Bimodal 2
Bimodal 3
0
0e0 2e-3 4e-3 6e-3 8e-3 1e-2 1e-2 1e-2 2e-2
*
εVM (a) Evolution of the total dislocation density
Standard deviation
3e+13
2.5e+13
2e+13
1.5e+13
1e+13
5e+12
r p =160nm
Bimodal 1
Bimodal 2
Bimodal 3
0
0e0 2e-3 4e-3 6e-3 8e-3 1e-2 1e-2 1e-2 2e-2
*
εVM (b) Evolution of the standard deviation σρ(t) (Eq. 4.7)
Figure 4.42: Effects of the percentage of large particles on the statistics of fatigue tests
in Fig. 4.43(b) demonstrates that the structural characteristics from the two previous mono-modal
cases coexist in the bimodal case: dense dislocation tangles as well as Orowan loops are observed
around the large particles and ladder-like dislocation structures are formed along the Burgers vec-
tor direction. Both tangles of residual dislocations and long dislocation lines with a relatively high
mobility are visible.
These results are consistent with the experimental observations which show a much more homoge-
neous and stable slip mode than in the shearable particle cases ([Martin 80], [Edwards & Martin 82]).
The effective dispersal of slip by the non-shearable particles can explain the formation of more ho-
mogeneous slip mode.
TEM micrographs of intense slip bands formed in fatigue tested Waspaloy are shown in Fig. 4.44.
The positions of a few of the large particles are indicated to facilitate the visualization in Fig.
4.44(b). Although there exists large discrepancy between the simulation and the experiments con-
cerning the particle sizes and the magnitude of the applied plastic strain 8 , the micrographs simulated
dislocation microstructure, i.e. dislocation tangles are formed around the large particles, residual
dislocations are present between the particles (Fig. 4.44(b)) and slip bands are more diffuse com-
pared to the shearable particle case (Fig. 4.44(a)).
Clear channels of totally sheared particles are no more formed due to the effective dispersal of dis-
location by the large particles. Fig. 4.45(a) shows the distribution of the residual strength of the
small particles after seven fatigue cycles (’Bimodal3’ case). The final particle strength distribution
8 Particle radius=15 nm and 80 nm, Grain size∼ 50 µm, Volume fraction> 40%, △ɛp = 10 −2

4.3 Fatigue simulations of materials hardened by particles 143
(a) 3D image of intense slip bands (b) Isolation of intense slip band
Figure 4.43: Details of intense slip band of the ’Bimodal2’ case after four fatigue cycles
(a) Slice normal to the primary plane (b) Slice parallel to the primary plane
Figure 4.44: TEM micrographs of intense slip bands of fatigue tested Waspaloy

144 Dislocation-precipitate interactions
log(frequency)
10 0
10 −1
10 −2
10 −3
10
160 180 200 220 240 260 280 300
−4
τ [MPa]
facet
(a) Repartition by the particle strength
(110)
(b) Spatial distribution of sheared particles
Figure 4.45: Statistical and spatial distribution of the strength of small particles
τfacet[MPa]
is significantly broader as compared to the mono-modal case (see Fig. 4.31) between the initial and
the final strength. The spatial distribution of the sheared particles in Fig. 4.45(b) also shows that
the shearing-off of the small particles does not occur in a confined channel but rather in a more
distributed area as compared to in Fig. 4.32(a).
The addition of large particles decreases the degree of cyclic softening seen in the mono-modal case
(rp = 160 nm), and the stress amplitudes of the bimodal cases lie between the two mono-modal
cases presented before (rp = 160 nm and 400 nm) although the stress differences between the dif-
ferent bimodal cases is rather small as shown in Fig. 4.46.
The deformed surface corresponding to the ’Bimodal2’ case is shown in Fig. 4.47(a) after four
fatigue cycles. As the related slip bands are dispersed in the volume, the steps spread over the
free surface to a larger extent than those observed in the shearable particle case (Fig. 4.38(a)).
In addition, the surface step morphologies in Fig. 4.47(b) are shifted from the tongue-like to the
ribbon-like type, although with a lower extent than in the large particle mono-modal case (Fig.
4.40).
4.3.7 Summary
The simulated fatigue properties of materials hardened by shearable and non-shearable particles
are qualitatively in good agreement with experimental observations. Simple geometries are used for
the simulation volume and the particles. The evolution of the particle’s strength by shearing-off
is assumed also in a simplified manner. The differences of microstructural and mechanical fatigue
features can be summarized as follows for the shearable and non-shearable particle cases.
300
280
260
240
220
200
180
160

4.3 Fatigue simulations of materials hardened by particles 145
Stress[MPa]
500
450
400
350
300
250
Shearable
Bimodal 1
Bimodal 2
Bimodal 3
Non-shearable
200
0.0e0 4.0e-3 8.0e-3 1.2e-2 1.6e-2
Cumulative plastic strain
Figure 4.46: Cyclic response curves of the bimodal cases compared with the mono-modal cases
(a) Deformed surface of ’Bimodal2’
case after four fatigue cycles
(b) Evolution of surface steps above intense slip bands
Figure 4.47: Surface morphology of the ’Bimodal2’ case

146 Dislocation-precipitate interactions
1. Material hardened by shearable particles
• High magnitude and accumulation rate of ρtot
• Formation of thin slip bands with high local dislocation density
• Ladder-like structures along the primary Burgers vector direction, and small-sized tangles
between the particles
• Clear channels of particles of low residual strength
• Tongue-like surface markings
• High initial stress amplitude followed by severe softening
2. Material hardened by non-shearable particles
• Low accumulation rate of ρtot
• Larger slip band thickness and reduced inter-band spacing (more than one slip band
formed in the volume)
• Dense tangles around particles and complex dislocation structure between pairs of closely-
spaced particles
• Ribbon-like surface markings
• Intermediate initial stress and rather stable cyclic response
In the shearable particle case, a detailed investigation of the slip band formation shows that they
are made of closely spaced edge dipolar loops. This is due to the limited glide distance of the
double cross-slipped screws having a limited easy glide path. In the non-shearable particle case,
dense dislocation tangles around the particles are attributed to the formation of Orowan loops and
the subsequent interactions with the gliding dislocations, which act as pinning points for relatively
mobile dislocations and contribute to form complex dislocation structures in the vicinity of the large
particles.
The addition of non-shearable particles (the bimodal cases) promotes dispersion of the slip bands,
which results in retarded slip localization and in more diffused slip bands. The stress amplitude
and the characteristics of slip markings are comprised between two mono-modal cases.
It is observed that the large particles (rp = 400 nm) in the bimodal cases are also sheared off
significantly. In Fig. 4.48, the residual facet strength are shown after 7 fatigue cycles for the
bimodal case (Fig. 4.48(a)) and after 6 cycles for the mono-modal case (Fig. 4.48(b)). It is

4.3 Fatigue simulations of materials hardened by particles 147
(110)
(a) Bimodal case
τfacet[MPa]
7400
6850
6300
5750
5200
4650
4100
3550
3000
(110)
(b) Mono-modal case
τfacet[MPa]
Figure 4.48: Comparison of the strength of particles with the radius rp = 400 nm in the (a) bimodal
case after 7 fatigue cycles and (b)mono-modal case after 6 fatigue cycles
apparent that the large particles are more sheared off in the bimodal case, e.g. the minimum value
of the strength is 5128 MP a in Fig. 4.48(a), and 7227 MP a in Fig. 4.48(b). This implies that
significant softening and damaging effects could eventually take place in the large particles present
in a bimodal-sized particle distribution. The small difference in the number of cycles does not seem
to influence much this observation.
Because of the relatively small number of the simulated fatigue cycles, quantitative analyses on the
slip irreversibility are not presented in this work. The computational limitations of the simulations
lie in the maximum possible number of segments (related to memory capacity) and the poor load
balance characteristic (see Fig. 3.21(b)). As already pointed out in Sec. 3.3.6 and 3.4.4, the
computational performance can be further increased by
1. Decomposing the data space
Each processor use only the necessary and sufficient data for computation, and this will allow
to use larger memory for the simulations.
2. Revising the load balance scheme
Fatigue simulations are poor in load balance due to the highly heterogeneous dislocation
microstructure involved, moreover the geometry of the simulation volume (cylinder) as shown
in Fig. 3.21(b) is not easy to decompose in a set of cubic boxes as needed by the parallelization
scheme. Thus a more efficient load balancing scheme is highly desirable.
7400
6850
6300
5750
5200
4650
4100
3550
3000

148 Dislocation-precipitate interactions
Good qualitative agreements of the microstructural and mechanical features of fatigue between the
simulations and the experiments are, however, very promising for the development of fatigue-life
models. It is generally agreed that the irreversible fraction of the cumulative cyclic strain describes
well the fatigue life-controlling mechanisms. Like experimental efforts to well describe the state of
damage ([Coupeau & Grilhe 99], [Cretegny & Saxena 01]), various parameters are measured
during the simulations. Parameters such as the surface topology, slip band width and separation
distance are direct outputs of the simulations. The evolution of the elastic energy inside of the
slip bands is also accessible by post-processing of the simulated dislocation structures. Although
each simulation needs considerable computing time 9 , the flexibility of simulations makes possible an
extensive study on the effects of various parameters like geometries (grain size etc.), particle char-
acteristics (particle size, volume fraction and strength etc.) and boundary conditions (the applied
plastic strain etc.). The compilation of these information will serve to build fatigue crack nucleation
criteria based on the intrinsic microstructural features involved.
Key points
• Image stresses by a 3D particle are computed using the FEM/DDD coupling code
explained in Sec. 2.4.2. Cylindrical, spherical and cubical particles are considered,
and interaction forces along both glide and climb directions are shown.
• The effect of the elastic modulus difference is investigated focusing on the flow stress
and the subsequent hardening behavior. A simple configuration involving two particles
is used for the computation.
• Fatigue simulations are performed using the new parallel DDD code. The effects
of particles (shearable or non-shearable) on the fatigue properties, like the intense
slip band microstructure, the cyclic mechanical response, and the surface markings
are investigated. The simulated results are compared to the available experimental
observations in a qualitative way. Bimodal particle distributions are also simulated.
The simulations can be used effectively to build fatigue-life models based on the
intrinsic microstructural features involved.
9 It takes 4 − 7 days for the fatigue simulations presented in this work using 9 processors in IBM P690 architecture
supported from KISTI (Korea Institute of Science and Technology Information)

Chapter 5
Conclusions and perspectives
At the beginning of this thesis, we have presented the details of the 3D discrete dislocation dynam-
ics method. Efforts were given to elucidate the theoretical backgrounds and the assumptions lying
under the method in order to ameliorate and expand the applicability of the method and also to
be a good guidance to new comers in this field, especially to whom are not Francophone, since it
is the first thesis on the French group written in English. The method to discretize the simulation
space and the dislocation lines can be readily applied to other crystal structures, and the anisotropic
stress fields and the various forms of dislocation mobility can be adopted according to the need of
research objects.
Besides the compilation of the existing components, new important elements are added to the
3D DDD method, i.e. the computation of the displacement fields of dislocations, the implemen-
tation of the internal interfaces and the periodic boundary conditions. These new features open a
wide range of research areas in which the DDD code can be used.
The computation of the displacement fields has been applied successfully to both the study of
the surface markings during the cyclic loadings and the enforcement of displacement boundary con-
ditions in the code coupled with CAST∃M, although the latter is not presented in this work.
The internal interfaces represented by facets are effectively adopted for the particles in precipitation-
hardened metals. This method can initiate a number of studies which involve the internal interfaces
: the plasticity in a polycrystal and multilayer films ([Verdier 04]) which comprise grain bound-
aries and interfaces between films respectively.
The periodic boundary conditions are applied to the simulation of the Stage I-II transition. It is

150 Conclusions and perspectives
now being applied to study the effect of the polarization of forest dislocations on the critical stress
of a gliding dislocation line, in which the periodicity is forced along the line and the glide direction
of the moving dislocation.
Although it is not extensively studied in this work, the junction formation and its representation
in the dislocation dynamics methods are largely investigated nowadays. Out of many important
issues, the colinear junctions ([Madec et al. 03]) and the glissile junctions 1 are especially of interest.
The usage of the linked-lists of segments and the decomposition of the orthorhombic simulation
volume into homothetic boxes produce a significant increase in the computation efficiency and give
great advantages on computing time with minor errors in the stress computation. This allows mas-
sive simulations of bulk materials under homogeneous loading condition.
Although the computational efficiency of the 3D DDD method has improved significantly by ap-
plying the box method, the code was still infeasible to incorporate many particles in the simulation
volume. A parallel version of the method has thus been developed.
The distributed memory system and the standard MPI have been chosen to develop a parallel
DDD code since the distributed memory architectures are the major stream of the parallel comput-
ers and it will be for the time being.
The scheme of the new parallel code is designed based on the box method: the boxes dividing the
simulation volume are decomposed into parallel-piped subsystems. The advantages of the parallel
scheme developed in this work are several: most of the serial codes can be used without any mod-
ification and a relatively short period of development time was needed (less than 4 months). The
gained speedup is quite satisfactory anyhow. Especially the efficiency of the internal stress compu-
tation is 100%, thus the anisotropic stress solutions can be incorporated with the same computing
expenses as those of the isotropic solutions by using several processors. The requirement that at
least three boxes should exist along each axis of an individual subsystem, however, puts a certain
limit on the number of processors that can be used simultaneously. Better strategies for the load
balancing and the decomposition of data space would be highly desirable to improve the efficiency
and the applicability of the new parallel code.
Parallel to our efforts, there have been several groups which have converted their own dislocation dy-
1 D. Weygand in the conference ’Dislocations 2004’ held at "La Colle-sur-Loup, France", September 13-17, 2004

namics codes into a parallel version, especially in Lawrence Livermore National Laboratory (LLNL)
and University of California in Los Angeles (UCLA). Both of them use the nodal model.
The image stresses due to a 3D particle were computed using the FEM/DDD coupling code. The
interaction of a dislocation line with a circular cylindrical, spherical and cubical particle with differ-
ing elastic modulus was investigated. The computation method was validated by comparing with
the corresponding analytical solutions. It was shown that the image stresses need to be taken into
account especially in the study of the local events around the particles, e.g. the computation of the
energy state around a particle and the calculation of the creep threshold stresses at high tempera-
tures.
In these modeling, it is necessary to mesh the whole simulation volume because the geometrical
symmetries are broken by the heterogeneous force boundary conditions due to a dislocation. Conse-
quently, the cost of the FEM computation is relatively high both in term of cpu time and required
memory. The force profiles fitted from the computed data can used as approximation solutions of
interactions due to the elastic modulus mismatch. For the dynamics, however, the use of a parallel
finite element method would be of benefit and will be served as a good tool in studying the plasticity
of multilayer films, for example.
The effect of the elastic modulus mismatch is investigated focusing on the flow stress and the
subsequent hardening behavior using the simple geometry involving two particles. The characteris-
tics of the image stresses (short-ranged and paraelastic) generate minor effects on the yield stresses
but significant effects on the work hardening rate. The image stresses are also found to affect sig-
nificantly the local events such as cross slip and climb.
The fatigue simulations are performed using the internal interfaces represented by facets and the
new parallel DDD program. The characters of shearable and non-shearable particles and the par-
ticle’s strength evolution by shearing-off were represented in a simplified manner by adjusting the
strength of the facets.
Major features of the fatigue properties of materials hardened by shearable and non-shearable par-
ticles are well reproduced by the simulations, e.g. microstructure of the intense slip bands, the
cyclic mechanical response and the surface markings. The simulated results were compared with
the available experimental observations, and showed good agreements in a qualitative way. The
151

152 Conclusions and perspectives
mechanism of the intense slip band formation is proposed from the observation of the simulated
dislocation microstructure.
The flexibility of the simulations can permit an extensive study on the effects of various parameters
like the geometries (grain size etc.), the characteristic of particles (particle size, volume fraction and
strength etc.) and the applied plastic strain. The compilation of these information will serve to
build fatigue crack nucleation criteria based on the intrinsic microstructural features involved.
To build a reliable fatigue life model, it is however imperative to increase the number of fatigue
cycles of the simulations. For this purpose, the efficiency and performance of the new parallel code
need to be improved by adopting better strategies for the load balancing and decomposing the data
space so as to increase the maximum number of dislocation segments and particles.
In conclusion, the methods that we have developed and verified have been applied to the dislocation-
precipitate interactions and opens many paths to new interesting research areas.

Bibliography
[Abraham 97] Abraham F. F., Portrait of a crack: Rapid fracture mechanics using parallel molec-
ular dynamics, IEEE Computational Science & Engineering, Vol. 4, No.‌ 2, 1997, pp. 66–77.
[Aoyama & Nakano 99] Aoyama Y. & Nakano J., Practical MPI Programming(RS/6000 SP),
IBM Redbooks, Vervante, 1999.
[Bacon et al. 73] Bacon D. J., Kocks U. F. & Scattergood R. O., The effect of dislocation self-
interaction on the orowan stress, Phil. Mag., Vol. 28, 1973, p. 1241.
[Barnett 85] Barnett D. M., The displacment field of a triangular dislocation loop, Phil. Mag.
A, Vol. 51, No.‌ 3, 1985, pp. 383–387.
[Bathe 96] Bathe K. J., Finite Element Procedures, Prentice-Hall International, INC., 1996.
[Brown 64] Brown L. M., The self-stress of dislocations and the shape of extended nodes, Phil.
Mag., 1964, pp. 441–466.
[Bulatov et al. 01] Bulatov V. V., Rhee M. & Cai W., Periodic boundary conditions for disloca-
tion dynamics simulations in three dimensions, Mat. Res. Soc. Symp. Proc., ed. by Kubin L. P.,
Selinger R. L., Bassani J. L. & Cho K., 2001.
[Calabrese & Laird 74] Calabrese C. & Laird C., Cyclic stress-strain response of two-phase
alloys, parts i and ii, Materials Science and Engineering, Vol. 13, 1974, pp. 141–174.
[Canova & Kubin 91] Canova G. R. & Kubin L. P., Dislocation microstructures and plastic flow:
a three dimensional simulaiton, Continuum models and discrete systems, ed. by Maugin G. A.,
1991.
[Chen et al. 99] Chen B. T., Zhang T. Y. & Lee J. K., Interaction of an edge dislocation with an
elliptical hole in a rectilinearly anisotropic body, Mech. of Mat., Vol. 31, 1999, p. 71.

154 BIBLIOGRAPHY
[Clavel & Pineau 82] Clavel M. & Pineau A., Fatigue behaviour of two nickel-base alloys i:
Experimental results on low cycle fatigue, fatigue crack propogation and substructures, Materials
Science and Engineering, Vol. 55, 1982, pp. 157–171.
[Cleveringa et al. 97] Cleveringa H. H. M., Giessen E. Vander. & Needleman A., Comparison of
discrete dislocation and continuum plasticity predictions for a composite material, Acta Materi-
alia, Vol. 45, No.‌ 8, 1997, pp. 3163–3179.
[Comninou & Dundurs 72] Comninou M. & Dundurs J., Long-range interaction between a screw
dislocation and a spherical inclusion, J. Appl. Phys., Vol. 43, 1972, p. 2461.
[Coupeau & Grilhe 99] Coupeau C. & Grilhe J., Quantitative analysis of surface effects of plastic
deformation, Materials Science and Engineering A, Vol. 271, 1999, pp. 242–250.
[Cretegny & Saxena 01] Cretegny L. & Saxena A., Afm characterization of the evolution of
surface deformation during fatigue in polycrystalline copper, Acta Materialia, Vol. 49, No. ‌ 18,
2001, pp. 3647–3887.
[Demmel et al. 93] Demmel J., Heath M. & van der Vorst H., Parallel numerical linear algebra,
Acta Numerica 1993, 1993.
[Déprés 04] Déprés C., Modèlisation physique des stades précurseurs de l’endommagement en
fatigue, Thèse de PhD, Institut National Polytechnique De Grenoble, 2004.
[Déprés et al. 03] Déprés C., Fivel M., Robertson C. F., Fissolo A. & Verdier M., Etude des
stades précurseurs de l’endommagement en fatigue: expériences et simulations à l’échelle des
dislocations, Journal de Physique IV, Vol. 106, 2003, pp. 81–90.
[Déprés et al. 04] Déprés C., Robertson C. F. & Fivel M., Low-strain fatigue in 316l steel surface
grains: a three dimensional discrete dislocation dynamics modelling of the early cycles. part-1:
Dislocation microstructures and mechanical behaviour, Phil. Mag., Vol. 84, No. ‌ 22, 2004, pp.
2257–2275.
[Devincre 95] Devincre B., Three dimensional stress field expressions for straight dislocation
segments, Solid State Communications, Vol. 93, No.‌ 11, 1995, pp. 875–878.
[Devincre et al. 01] Devincre B., Kubin L. P., Lemarchand C. & Madec R., Mesoscopic sim-
ulations of plastic deformation, Materials Science and Engineering, Vol. A309-310, 2001, pp.
211–219.

BIBLIOGRAPHY 155
[Devincre & Roberts 96] Devincre B. & Roberts S., Three-dimensional simulation of
dislocation-crack interactions in b.c.c. metals at the mesoscopic scale, Acta Materialia, Vol. 44,
No.‌ 7, 1996, pp. 2981–2900.
[dewit 67] deWit R., Some relations for straight dislocations, Phys. Stat. Sol., Vol. 20, 1967, pp.
567–573.
[Diehl 56] Diehl J., Z. Metallk., Vol. 47, 1956, p. 331.
[Dongarra et al. 98] Dongarra J. J., Duff I. S., Sorenson D. C. & Vorst H. A., Numerical Linear
Algebra for High Performance Computers, SIAM, Philadelphia, 1998.
[Ebeling & Ashby 66] Ebeling R. & Ashby M. F., Dispersion hardening of copper single crystals,
Phil. Mag., Vol. 13, 1966, p. 805.
[Edwards & Martin 82] Edwards L. & Martin J. W., Proc. of 6th Int. Conf. on the strength of
metals and alloys, ed. by Gifkins R. C., 1982.
[Essmann & Mughrabi 79] Essmann U. & Mughrabi H., Annihilation of dislocations during
tensile and cyclic deformation and limits of dislocation densities, Phil. Mag. A, Vol. 40, No. ‌ 6,
1979, pp. 731–756.
[Fahrat & Roux 94] Fahrat C. & Roux F. X., Implicit parallel processing in structural mechanics,
Computational Mechanics Advances, Vol. 2, No.‌ 1, 1994.
[Fisher et al. 53] Fisher J. C., Hart E. W. & Rry R. H., The hardening of metal crystals by
precipitate particles, Acta Materialia, Vol. 1, 1953, p. 336.
[Fivel 97] Fivel M., Études numériques à différentes échelles de la déformation plastique des
monocristaux de structure CFC, Thèse de PhD, Institut National Polytechnique De Grenoble,
1997.
[Fivel & Canova 99] Fivel M. & Canova G. R., Developing rigorous boundary conditions to
simulations of discrete dislocation dynamics, Modelling Simul. Mater. Sci. Eng., Vol. 7, 1999, pp.
753–768.
[Fivel et al. 96] Fivel M., Gosling T. J. & Canova G. R., Implementing image stresses in a 3d
dislocation simulation, Modelling Simul. Mater. Sci. Eng., Vol. 4, No.‌ 6, 1996, pp. 581–596.

156 BIBLIOGRAPHY
[Fivel et al. 98] Fivel M., Robertson C. F., Canova G. R. & Boulanger L., 3d modeling of indent-
induced plastic zone at a mesoscale, Acta Materialia, Vol. 7, 1998, pp. 6183–6194.
[Foreman 67] Foreman A. J. E., The bowing of a dislocation segment, Phil. Mag., Vol. 15, 1967,
pp. 1011–1021.
[Foreman & Makin 66] Foreman A. J. E. & Makin M. J., Dislocation movement through random
arrays of obstacles, Phil. Mag., Vol. 14, 1966, p. 911.
[Fusenig & Nembach 75] Fusenig K. D. & Nembach E., Dynamic dislocation effects in precipi-
tation hardened materials, Acta metall. mater., Vol. 41, 1975, pp. 3181–3189.
[Gerold & Steiner 82] Gerold V. & Steiner D., Fatigue softening in precipitation-hardened
copper-cobalt, Scripta Metallurgica, Vol. 16, 1982, pp. 405–408.
[GG et al. 00] GómezGarcía D., Devincre B. & Kubin L. P., Forest hardening and boundary con-
ditions in 2d simulations of dislocations dynamics, Mat. Res. Soc. Symp. Proc., ed. by Robertson
I. M., Lassila D. H., Devincre B. & Phillips R., 2000.
[Ghoniem et al. 00] Ghoniem N. M., Singh B. N., Sun L. Z. & de la Rubia T. D., Interaction
and accumulation of glissile defect clusters near dislocations, J. Nucl. Mater., Vol. 276, 2000, pp.
166–177.
[Giessen & Needleman 95] Giessen E. Vander. & Needleman A., Discrete dislocation plasticity:
a simple planar model, Modelling Simul. Mater. Sci. Eng., Vol. 3, 1995, pp. 689–735.
[Graf & Hornbogen 78] Graf M. & Hornbogen E., The effect of inhomogeneity of cyclic strain
on initiation of cracks, Scripta Metallurgica, Vol. 12, 1978, pp. 147–150.
[Gullouglu et al. 89] Gullouglu A. N., Srolovitz D. J., Lesar R. & Lomdahl P. S., Dislocation
distributions in two dimensions, Scripta Metallurgica, Vol. 23, 1989, p. 1347.
[Hirth & Lothe 92] Hirth J. P. & Lothe J., Theory of Dislocations, Krieger Publishing Company,
Malabar, Florida, 1992.
[Hull & Bacon 83] Hull D. & Bacon D. J., Introduction to Dislocations, Pergamon Press, p96,
1983.
[Humphreys & Martin 67] Humphreys F. J. & Martin J. W., Phil. Mag., Vol. 16, 1967, p. 927.

BIBLIOGRAPHY 157
[Khraishi et al. 00a] Khraishi T. A., Zbib H. M., Hirth J. P. & de la Rubia T. D., The stress field
of a general circular volterra dislocation loop: Analytical and numerical approches, Phil. Mag.,
Vol. 80, 2000, pp. 95–105.
[Khraishi et al. 00b] Khraishi T. A., Zbib H. M., Hirth J. P. & Khaleel M., The displacement,
and strain-stress fields of a general circular volterra dislocation loop, Int. J. Eng. Sci., Vol. 80,
2000, pp. 251–266.
[Kobashi & Ohr 80] Kobashi S. & Ohr S. M., Phil. Mag. A, Vol. 42, 1980, p. 763.
[Kocks et al. 75] Kocks U. F., Argon A. S. & Ashby M. F., Thermodynamics and kinetics of slip,
Progress in Materials Science, ed. by Kubin L. P., Selinger R. L., Bassani J. L. & Cho K., 1975.
[Lee & Laird 83] Lee J. K. & Laird C., Strain localization during fatigue of precipitation-hardened
aluminium alloys, Phil. Mag., Vol. 47A, 1983, pp. 579–597.
[Lépinoux & Kubin 87] Lépinoux J. & Kubin L. P., The dynamic organization of dislocation
structures: a simulation, Scripta Metallurgica, Vol. 21, 1987, pp. 833–837.
[Li 64] Li J. C. M., Stress field of a dislocation segment, Phil. Mag., Vol. 10, 1964, pp. 1097–1098.
[Li & Laird 94] Li Y. & Laird C., Cyclic response and dislocation structures of aisi 316l stainless
steel. part 1: Single crystals fatigued at intermediate strain amplitude., Materials Science and
Engineering A, Vol. 186, No.‌ 1–2, 1994, pp. 65–86.
[Madec 01] Madec R., Des intersections entre dislocations a la plasticité du monocristal CFC;
Étude par dynamique des dislocations, Thèse de PhD, Universite Paris XI Orsay, 2001.
[Madec et al. 03] Madec R., Devincre B., Kubin L. P., Hoc T. & Rodney D., The role of collinear
interaction in dislocation-induced hardening, Science, Vol. 301, No.‌ 26, 2003, pp. 1879–1882.
[Madec et al. 04] Madec R., Devincre B. & Kubin L. P., On the use of periodic boundary condi-
tions in dislocation dynamcis simulation, Mesoscopic Dynamics in Fracture Process and Stresngth
of Materials, ed. by Shibutani Y. & Kitagawa H., 2004.
[Man et al. 02] Man J., Obrtlik K., Blochwitz C. & Polák J., Atomic force microscopy of surface
relief in individual grains of fatigued 316l austenitic stainless steel, Acta Materialia, Vol. 50, 2002,
pp. 3767–3780.

158 BIBLIOGRAPHY
[Marquis & Dunand 02] Marquis E. A. & Dunand D. C., Model for creep threshold stress in
precipitation-strengthened alloys with coherent particles, Scripta Materialia, Vol. 47, 2002, p.
503.
[Martin 80] Martin J. W., Micromechanisms in particle-hardened alloys, Cambrideg Solide State
Science Series, ed. by Cahn R. W., Thompson M. W. & Ward I. M., 1980.
[Mason 68] Mason W. P., Dislocation dynamics, MacGraw-Hill, 1968.
[Melander & Persson 78] Melander A. & Persson P. A., The strength of a precipitation hard-
ened alznmg alloy, Acta Materialia, Vol. 26, 1978, p. 267.
[Mohles & Nembach 01] Mohles V. & Nembach E., The peak- and overaged states of particle
strengthened materials: computer simulations, Acta Materialia, Vol. 49, 2001, p. 2405.
[Moore 65] Moore G. E., Cramming more components onto integrated circuits, Electronics,
Vol. 38, No.‌ 8, 1965.
[Mughrabi 83] Mughrabi H., Deformation of multi-phase and particle containing materials, Pro-
ceedings of the 4th Risø International Symposium on Metallurgy and Materials Science, ed. by
Bilde-Sørensen J. B., Hansen N., Horsewell A., Leffers T. & Lilholt H., 1983.
[Mughrabi 85] Mughrabi H., Dislocation Properties in Real Materials, Book No. 323, The Institute
of Metals, London, 1985.
[Nembach 83] Nembach E., Phys. Stat. Sol., Vol. 78, 1983, p. 571.
[Nembach 97] Nembach E., Particle strengthening of metals and alloys, John Wiley and Sons,
1997.
[Obrtlik et al. 94] Obrtlik K., Kruml T. & Polák J., Dislocation structures in 316l stainless steel
cycled with plastic strain amplitudes over a wide interval., Materials Science and Engineering A,
Vol. 187, No.‌ 1, 1994, pp. 1–10.
[Reppich 93] Reppich B., Particle strengthinig, Mater. Sci. Technol., Vol. 6, 1993, pp. 311–357.
[Rhee et al. 01] Rhee M., Stolken J. S., Bulatov V. V., de la Rubia T. D., Zbib H. M. & Hirth
J. P., Dislocation stress fields for dynamic codes using anisotropic elasticity: methodology and
analysis, Materials Science and Engineering, Vol. A309-310, 2001, pp. 288–293.

BIBLIOGRAPHY 159
[Risbet et al. 03] Risbet M., Feaugas X., Guillemer-Neel C. & Clavel M., Use of atomic force
microscopy to quantify slip irreversibility in a nickel-base superalloy, Scripta Materialia, Vol. 49,
2003, pp. 533–538.
[Rodney & Phillips 99] Rodney D. & Phillips R., Structure and strength of dislocation junctions:
an atomic-level analysis, Phy. Rev. Lett., Vol. 82, 1999, pp. 1704–1707.
[Santare & Keer 86] Santare M. H. & Keer L. M., Interaction between an edge dislocation and
a rigid elliptical inclusion, J. Appl. Mech., Vol. 53, 1986, p. 382.
[Schmid & Boas 35] Schmid E. & Boas W., Kristallplastizitat, Springer Verlag(Berlin), 1935.
[Schwarz 99] Schwarz K. W., Simulation of dislocations on the mesoscpoic scale. i. methods and
examples, J. Appl. Phys., Vol. 85, No.‌ 1, 1999, pp. 108–119.
[Shenoy et al. 00] Shenoy V. B., Kukta R. V. & Phillips R., Mesoscopic analysis of structure
and strength of dislocatoin junctions in fcc metals, Phy. Rev. Lett., Vol. 84, No. ‌ 7, 2000, pp.
1491–1494.
[Shin et al. 01] Shin C. S., Fivel M., Rodney D., Phillips R., Shenoy V. B. & Dupuy L., Forma-
tion and strength of junctions in fcc metals : a study by dislocation simulation and atomistic
simulations, Journal de Physique IV, Vol. 11, No.‌ Pr5, 2001, pp. 19–26.
[Stoltz & Pineau 78] Stoltz R. E. & Pineau A., Dislocation-precipitate interaction and cyclic
stress-strain behavior of a γ’-strengthened superalloy, Materials Science and Engineering, Vol. 34,
1978, pp. 275–284.
[Suresh 98] Suresh S., Fatigue of Materials, 2nd edi., Cambridge University Press, 1998.
[Tang et al. 98] Tang M., Kubin L. P. & Canova G. R., Dislocation moility and the mechanical
response of bcc single crystals: a mesoscopic approach, Acta Materialia, Vol. 46, 1998, p. 9.
[Urabe & Weertman 75] Urabe N. & Weertman J., Dislocation mobility in potassium and iron
single crystals, Materials Science and Engineering, Vol. 18, 1975, p. 41.
[Verdier 04] Verdier M., Plasticity in fine scale semi-coherent metallic films and multilayers,
Scripta Materialia, Vol. 50, No.‌ 6, 2004, pp. 769–773.

160 BIBLIOGRAPHY
[Verdier et al. 98] Verdier M., Fivel M. & Groma I., Mesoscopic scale simulation of dislocation
dynamic in fcc metals: Principle and applications, Modelling Simul. Mater. Sci. Eng., Vol. 6, No.
‌ 6, 1998, pp. 755–770.
[Vitek 75] Vitek V., Yielding from a crack with finite root-radius loaded in uniform tension, J.
Mech. Phys. Solids, Vol. 24, 1975, p. 67.
[Weeks et al. 69] Weeks R. W., Pati S. R., Ashby M. F. & Barrand P., The elastic interaction
between a straight dislocation and a bubble or a particle, Acta Metallurgica, Vol. 17, 1969, p.
1403.
[Weygand et al. 01] Weygand D., Friedman L. H., Giessen E. Vander. & Needleman A., Discrete
dislocation modeling in tree-dimensional confined volumes, Materials Science and Engineering,
Vol. A309-310, 2001, p. 420.
[Weygand et al. 02] Weygand D., Friedman L. H. & Giessen E. Vander., Aspect of boundary-
value problem solutions with three-dimensional dislocation dynamics, Modelling Simul. Mater.
Sci. Eng., Vol. 10, 2002, pp. 437–468.
[Zbib et al. 98] Zbib H. M., Rhee M. & Hirth J. P., On plastic deformation and the dynamics of
3d dislocations, Int. J. Mch. Sci., Vol. Nos, No.‌ 2-3, 1998, pp. 113–127.
[Zhou & Lung 88] Zhou S. J. & Lung C. W., An image force expression for the dislocation near
a crack, J. Phys. F: Met. Phys., Vol. 18, 1988, p. 851.
[Zhu & Starke 99] Zhu A. W. & Starke E. A., Computer experiment on superposition of strength-
ening effects of different particles, Acta Materialia, Vol. 47, 1999, p. 3263.