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  • Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page I le-tex

    Franz Roters, Philip Eisenlohr,Thomas R. Bieler, and Dierk Raabe

    Crystal Plasticity Finite ElementMethods

  • Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page II le-tex

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  • Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page III le-tex

    Franz Roters, Philip Eisenlohr, Thomas R. Bieler,and Dierk Raabe

    Crystal Plasticity Finite Element Methods

    in Materials Science and Engineering

    WILEY-VCH Verlag GmbH & Co. KGaA

  • Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page IV le-tex

    The Authors

    Dr. Franz RotersMPI fr Eisenforschung GmbHAbt. MikrostrukturphysikMax-Planck-Str. 140237 DsseldorfGermany

    Dr.-Ing. Philip EisenlohrMPI fr Eisenforschung GmbHAbt. MikrostrukturphysikMax-Planck-Str. 140237 DsseldorfGermany

    Prof. Dr. Thomas R. BielerMichigan State UniversityCollege of EngineeringChemical Engineering and Materials ScienceEast Lansing, MI 48824USA

    Prof. Dr. Dierk RaabeMPI fr Eisenforschung GmbHAbt. MikrostrukturphysikMax-Planck-Str. 140237 DsseldorfGermany

    All books published by Wiley-VCH are carefullyproduced. Nevertheless, authors, editors, andpublisher do not warrant the informationcontained in these books, including this book, tobe free of errors. Readers are advised to keep inmind that statements, data, illustrations,procedural details or other items mayinadvertently be inaccurate.

    Library of Congress Card No.: applied for

    British Library Cataloguing-in-Publication Data:A catalogue record for this book is availablefrom the British Library.

    2010 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim

    All rights reserved (including those of translationinto other languages). No part of this book maybe reproduced in any form by photoprinting,microfilm, or any other means nor transmittedor translated into a machine language withoutwritten permission from the publishers. Regis-tered names, trademarks, etc. used in this book,even when not specifically marked as such, arenot to be considered unprotected by law.

    Typesetting le-tex publishing services GmbH,LeipzigPrinting and Binding Fabulous Printers PteLtd, SingaporeCover Design Formgeber, Eppelheim

    Printed in SingaporePrinted on acid-free paper

    ISBN 978-3-527-32447-7

  • Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods Chap. roters9419f01 2010/7/23 page V le-tex

    V

    Notation

    As a general scheme of notation, vectors are written as boldface lowercase letters(e. g., a, b), second-order tensors as boldface capital letters (e. g., A, B), and fourth-order tensors as blackboard-bold capital letters (e. g., A, B). For vectors and tensors,Cartesian components are denoted as, ai , A i j , and A i j k l respectively. The action ofa second-order tensor upon a vector is denoted as A b (in components A i j b j , withimplicit summation over repeated indices) and the action of a fourth-order tensorupon a second-order tensor is designated as AB (A i j k l Bk l). The composition oftwo second-order tensors is denoted as AB (A i j B j l ). The tensor (or dyadic) productbetween two vectors is denoted as ab (ai b j ). All inner products are indicated by asingle dot between the tensorial quantities of the same order, for example, ab (ai b i)for vectors and AB (A i j Bi j ) for second-order tensors. The cross-product of a vectora with a second-order tensor A, denoted by a A, is a second-order tensor definedin components as (a A)i j D i k l ak A l j , where is the LeviCivita permutationmatrix. The transpose, AT, of a tensor A is denoted by a superscript T, and theinverse, A1, by a superscript 1. Additional notation will be introduced whererequired.

    Crystal Plasticity Finite Element Methods.Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk RaabeCopyright 2010 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-32447-7

  • Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page VI le-tex

  • Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page VII le-tex

    VII

    Contents

    Notation V

    Preface XI

    1 Introduction to Crystalline Anisotropyand the Crystal Plasticity FiniteElement Method 1

    Part One Fundamentals 11

    2 Metallurgical Fundamentals of Plastic Deformation 132.1 Introduction 132.2 Lattice Dislocations 142.3 Deformation Martensite and Mechanical Twinning 18

    3 Continuum Mechanics 213.1 Kinematics 213.1.1 Material Points and Configurations 213.1.2 Deformation Gradient 223.1.3 Polar Decomposition 243.1.4 Strain Measures 253.1.5 Velocity Gradient 263.1.6 Elastoplastic Decomposition 273.2 Mechanical Equilibrium 303.3 Thermodynamics 31

    4 The Finite Element Method 354.1 The Principle of Virtual Work 354.2 Solution Procedure Discretization 364.3 Nonlinear FEM 38

    5 The Crystal Plasticity Finite Element Method as a MultiphysicsFramework 41

    Part Two The Crystal Plasticity Finite Element Method 47

    6 Constitutive Models 496.1 Dislocation Slip 49

    Crystal Plasticity Finite Element Methods.Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk RaabeCopyright 2010 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-32447-7

  • Franz Roters, Philip Eisenlohr, Thomas R. Bieler, and Dierk Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page VIII le-tex

    VIII Contents

    6.1.1 Introduction 496.1.2 Phenomenological Constitutive Models 496.1.2.1 Extension to Body-Centered Cubic Materials 516.1.3 Microstructure-Based Constitutive Models 516.1.3.1 Dislocation-Based Constitutive Laws

    in Crystal Plasticity Finite Element Models 526.1.3.2 Introduction of Geometrically Necessary Dislocations 536.1.3.3 Interface Models 566.2 Displacive Transformations 646.2.1 Introduction 646.2.2 Martensite Formation and Transformation-Induced Plasticity

    in CPFE Models 646.2.2.1 Decompositions of Deformation Gradient and Entropy Density 656.2.2.2 Constitutive Relations of StressElastic Strain

    and TemperatureReversible Entropy 676.2.2.3 Driving Forces and Kinetic Relations for Transformation and Plasticity 676.2.3 Mechanical Twinning in CPFE Models 696.2.3.1 A Modified CPFE Framework Including Deformation Twinning 716.2.3.2 Phenomenological Approach to Mechanical Twinning 736.2.4 Guidelines for Implementing Displacive Transformations

    in CPFE Constitutive Models 756.3 Damage 756.3.1 Introduction 756.3.2 Continuum Approaches to Modeling Damage 766.3.3 Microstructurally Induced Damage 776.3.4 Heterogeneous Plastic Deformation 786.3.5 Interfaces 816.3.6 Cohesive Zone Boundary Modeling 826.3.7 Grain Boundary Slip Transfer 856.3.8 Experimental Studies of Fracture-Initiation Criteria 886.3.9 Strain Energy as a Criterion for Damage 896.3.10 Assessment of Current Knowledge about Damage Nucleation 90

    7 Homogenization 937.1 Introduction 937.2 Statistical Representation of Crystallographic Texture 957.3 Computational Homogenization 977.4 Mean-Field Homogenization 997.5 Grain-Cluster Methods 100

    8 Numerical Aspects of Crystal Plasticity Finite Element MethodImplementations 109

    8.1 General Remarks 1098.2 Explicit Versus Implicit Integration Methods 1118.3 Element Types 111

  • Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods 2010/7/23 page IX le-tex

    Contents IX

    Part Three Application 113

    9 Microscopic and Mesoscopic Examples 1159.1 Introduction to the Field

    of Crystal Plasticity Finite Element Experimental Validation 1159.2 Stability and Grain Fragmentation in Aluminum

    under Plane Strain Deformation 1169.3 Texture and Dislocation Density Evolution

    in a Bent Single-Crystalline Copper Nanowire 1179.4 Texture and Microstructure Underneath a Nanoindent

    in a Copper Single Crystal 1199.5 Application of a Nonlocal Dislocation Model

    Including Geometrically Necessary Dislocationsto Simple Shear Tests of Aluminum Single Crystals 120

    9.5.1 Comparisons of von Mises Strain Distributions 1209.5.2 Size Dependence of the Nonlocal Model 1209.5.3 Conclusions 1239.6 Application of a Grain Boundary Constitutive Model

    to Simple Shear Tests of Aluminum Bicrystalswith Different Misorientation 124

    9.7 Evolution of Dis