Advanced material models for the crystal plasticity finite element method [Elektronische Ressource] : development of a general CPFEM framework / Franz Roters

F. Roters <[e-mail]>

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2011
• Nombre de lectures: 34
• Langue: English
• Poids de l'ouvrage: 31 Mo

Extrait

Advanced Material Models
for the
Crystal Plasticity Finite Element Method
Development of a general CPFEM framework
Von der Fakultät für Georessourcen und Materialtechnik
der Rheinisch-Westfälischen Technischen Hochschule Aachen genehmigte
Habilitationsschrift
von Dr. rer. nat. Franz Roters aus Warstein.
Gutachter: Univ.-Prof. Dr. rer. nat. G. Gottstein
prof. dr. ir. M. G. D. Geers
prof. P. Van Houtte
Tag der Habilitation: 29. Juni 2011Acknowledgments
First off all I want to thank Prof. Dierk Raabe. About ten years ago he offered me a
group leader position at the Max-Planck-Institut für Eisenforschung and introduced me
to the Crystal Plasticity Finite Element Method (CPFEM). Since then he gave to me
all the necessary support to continue my research work in this amazingly versatile field
of material science.
Second, I thank all the post-docs and PhD students with whom I had the chance to
work during these last ten years. I start the list with Zisu Zhao, he was the one with
whom I took my first steps in CPFEM, then there were Anxin Ma ’my’ first PhD
student, Jui-Chao Kuo, Yanwen Wang, Hyeon S. Jeon-Haurand, Ilja Tikhovskiy, Nader
N. Zaafarani, Chung-Souk Han, Claudio Zambaldi*, Wiliam A. Counts*, Hongyang
Li, Duancheng Ma, Olga Dmitrieva, Luc Hantcherli*, Eralp Demir, Christoph Kords*,
Alankar Alankar*, Denny D. Tjahjanto*, and finally Philip Eienlohr*, with whom I run
the joint Max-Planck–Fraunhofer group on Computational Mechanics of Polycrystals,
nCMC , and who is the main co-contributor to the general CPFEM framework presented
in this thesis. Actually all people marked by ’*’ contributed in one or the other way to
the new code. Most of the others contributed to the application examples of CPFEM
included in this thesis. Besides these people mentioned by name I want to thank all my
other former and present colleagues at the Max-Planck-Institut für Eisenforschung for
their everlasting support and the good working atmosphere they provide.
I also especially thank my colleagues Stefanie Sandlöbes and Philip Eisenlohr for proof-
reading the manuscript.
Then, there is the research world outside the Max-Planck-Institut für Eisenforschung. I
also want to thank a number of collaborators, who used or use ’my’ code and thereby
or in some other way contributed to its improvement. These people are Kurt Helming,
Martin Kraska, Maria Doig, Dmitrij Tikhomirov, Kim Kose, Victotia J. Chen, Koos van
Putten, Thomas Hochrainer, Aruna Prakasch, Raphael Twardowksi, Thomas R. Bieler,
and Koenraad Jansens.
Finally, I thank my family, my wife Martina, my daughter Sophia, and my son Christian.
They provide the private background and support for my work. Even though they are
probably not aware of it, this is the most important contributions of all!
iiiContents
1 Introduction 1
1.1 Crystalline anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 CPFEM as a multi-physics framework . . . . . . . . . . . . . . . . . . . 6
1.3 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 The Crystal Plasticity Finite Element Method 15
2.1 Concise Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Continuum Mechanical Framework . . . . . . . . . . . . . . . . . . . . . 17
2.3 Phenomenological Constitutive Equations . . . . . . . . . . . . . . . . . 18
I Crystal Plasticity at Small Microstructural Scales 21
3 Special Aspects of Small Scale Plasticity Simulations 23
4 Constitutive Models for Small Scale Simulations 27
4.1 A Constitutive Model Based on Unsigned Dislocation Densities . . . . . . 27
4.1.1 The Kinetic Equation of State . . . . . . . . . . . . . . . . . . . . 28
4.1.2 The State Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.3 Geometrically Necessary Dislocations – Strain Gradients . . . . . 30
4.1.4 Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 A Constitutive Model Based on Signed Dislocation Densities . . . . . . . 35
4.2.1 Microstructural State Variables . . . . . . . . . . . . . . . . . . . 35
4.2.2 Microstructure Evolution . . . . . . . . . . . . . . . . . . . . . . . 38
vvi CONTENTS
4.2.3 Dislocation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.4 Finite Volume Discretization . . . . . . . . . . . . . . . . . . . . . 41
4.2.5 Test Cases Using Simplified Geometries . . . . . . . . . . . . . . . 51
5 Small Scale Application Examples 57
5.1 Simulation of Single and Bicrystal Shear . . . . . . . . . . . . . . . . . . 58
5.1.1 Experimental and Simulation Setup . . . . . . . . . . . . . . . . . 58
5.1.2 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.3 Size Dependence of the Non-local Model . . . . . . . . . . . . . . 60
5.1.4 Von Mises Strain Distributions and Crystal Orientation . . . . . . 61
5.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Bicrystal Shear Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Single Crystal Micro-Compression . . . . . . . . . . . . . . . . . . . . . . 72
5.3.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . 72
5.3.2 Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.3 Theoretical Study on Pillar Stability . . . . . . . . . . . . . . . . 75
5.3.4 Prediction of Active Slip Systems in Micro-Pillar Compression . . 90
5.4 Rotation Patterns Below Nanoindents . . . . . . . . . . . . . . . . . . . . 93
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.2 Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . 94
5.4.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . 97
5.4.4 Comparison Between Experiment and Simulation . . . . . . . . . 97
5.4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
II Crystal Plasticity at Large Microstructural Scales 109
6 Special Aspects of Large Scale Plasticity Simulations 111CONTENTS vii
7 Advanced Models for Large Scale Simulations 113
7.1 Macro Texture Discretization . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1.1 The Texture Component Method . . . . . . . . . . . . . . . . . . 115
7.1.2 The Hybrid IA Scheme . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Homogenization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2.1 The Isostrain and Isostress Scheme . . . . . . . . . . . . . . . . . 133
7.2.2 Weighted-Taylor Homogenization Scheme . . . . . . . . . . . . . . 134
7.2.3 The Relaxed Grain Cluster Scheme . . . . . . . . . . . . . . . . . 136
7.3 Twinning as Additional Deformation Mechanism . . . . . . . . . . . . . . 148
7.3.1 Microstructural state variables . . . . . . . . . . . . . . . . . . . . 149
7.3.2 Microstructure Evolution . . . . . . . . . . . . . . . . . . . . . . . 151
7.3.3 Dislocation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.4 Shear Rate due to Twinning . . . . . . . . . . . . . . . . . . . . . 156
7.3.5 A Modified CPFE Framework Including Deformation Twinning . 156
8 Large Scale Application Examples 159
8.1 Simulation of Deep Drawing . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.1.1 Earing Behavior of AA3104 Hot and Cold Band . . . . . . . . . . 160
8.1.2 Effect of Texture Gradients on Earing Behavior of X6Cr17 . . . . 162
8.1.3 Optimization of Earing Behavior . . . . . . . . . . . . . . . . . . 164
8.1.4 Cup Drawing of Dual Phase Steel . . . . . . . . . . . . . . . . . . 167
8.2 Lankford (R-) Value Simulation . . . . . . . . . . . . . . . . . . . . . . . 172
8.3 Stress–Strain Curves of an Fe-23%Mn TWIP Steel . . . . . . . . . . . . . 173
8.4 The Virtual Laboratory – RVE Simulations . . . . . . . . . . . . . . . . 177
8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.4.2 The Virtual Specimen (RVE) . . . . . . . . . . . . . . . . . . . . 177
8.4.3 Stamping and trimming simulation . . . . . . . . . . . . . . . . . 181
8.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184viii CONTENTS
III A Multiscale Crystal Plasticity Implementation 185
9 Structure of the General Multiscale CPFEM Framework 187
10 The Integration Scheme 191
10.1 Explicit Versus Implicit Integration Methods . . . . . . . . . . . . . . . . 192
10.2 The Integration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.2.1 Stress Level Iterations . . . . . . . . . . . . . . . . . . . . . . . . 194
10.2.2 Material State Iterations . . . . . . . . . . . . . . . . . . . . . . . 195
10.2.3 Solution Scheme for Non-Local Models . . . . . . . . . . . . . . . 196
10.2.4 Homogenization Iterations . . . . . . . . . . . . . . . . . . . . . . 196
10.3 Parallelization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
11 Material Representation 201
11.1 <homogenization> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.2 <microstructure> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11.3 <crystallite> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.4 <phase> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.5 <texture&