Advanced Material Models for the Crystal Plasticity Finite Element

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Transcript of Advanced Material Models for the Crystal Plasticity Finite Element

  • Advanced Material Modelsfor the

    Crystal Plasticity Finite Element Method

    Development of a general CPFEM framework

    Von der Fakultt fr Georessourcen und Materialtechnikder Rheinisch-Westflischen Technischen Hochschule Aachen genehmigte

    Habilitationsschrift

    von Dr. rer. nat. Franz Roters aus Warstein.

    Gutachter: Univ.-Prof. Dr. rer. nat. G. Gottsteinprof. dr. ir. M. G. D. Geersprof. P. Van Houtte

    Tag der Habilitation: 29. Juni 2011

  • Acknowledgments

    First off all I want to thank Prof. Dierk Raabe. About ten years ago he offered me agroup leader position at the Max-Planck-Institut fr Eisenforschung and introduced meto the Crystal Plasticity Finite Element Method (CPFEM). Since then he gave to meall the necessary support to continue my research work in this amazingly versatile fieldof material science.

    Second, I thank all the post-docs and PhD students with whom I had the chance towork during these last ten years. I start the list with Zisu Zhao, he was the one withwhom I took my first steps in CPFEM, then there were Anxin Ma my first PhDstudent, Jui-Chao Kuo, Yanwen Wang, Hyeon S. Jeon-Haurand, Ilja Tikhovskiy, NaderN. Zaafarani, Chung-Souk Han, Claudio Zambaldi*, Wiliam A. Counts*, HongyangLi, Duancheng Ma, Olga Dmitrieva, Luc Hantcherli*, Eralp Demir, Christoph Kords*,Alankar Alankar*, Denny D. Tjahjanto*, and finally Philip Eienlohr*, with whom I runthe joint Max-PlanckFraunhofer group on Computational Mechanics of Polycrystals,CMCn, and who is the main co-contributor to the general CPFEM framework presentedin this thesis. Actually all people marked by * contributed in one or the other way tothe new code. Most of the others contributed to the application examples of CPFEMincluded in this thesis. Besides these people mentioned by name I want to thank all myother former and present colleagues at the Max-Planck-Institut fr Eisenforschung fortheir everlasting support and the good working atmosphere they provide.

    I also especially thank my colleagues Stefanie Sandlbes and Philip Eisenlohr for proof-reading the manuscript.

    Then, there is the research world outside the Max-Planck-Institut fr Eisenforschung. Ialso want to thank a number of collaborators, who used or use my code and therebyor 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 vanPutten, 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 areprobably not aware of it, this is the most important contributions of all!

    iii

  • Contents

    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

    v

  • vi 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 111

  • CONTENTS 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 StressStrain 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

  • viii 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

    11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

    11.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

    11.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

    11.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

    12 Conclusions and Outlook 209