Introduction to Finite Strain Theory for Continuum Elasto-Plasticity

Koichi Hashiguchi, Daiichi University, Japan, & Yuki Tamakawa, Tohuku University, JapanKoichi Hashiguchi is Professor, Daiichi University and Emeritus Professor of Kyushu University), Japan. He has taught applied mechanics for undergraduate and postgraduate students for 40 years and has supervised 34 Doctorates of applied mechanics.Current research in the field of plasticity includes the development of constitutive modelling of elastoplastic materials such as metals and soils which have been widely studied as elastoplastic materials for the last forty years. He has published circa 50 refereed journal papers on elastoplasticity since 2000.Yuki Tamakawa is Associate Professor, Dept. Civil and Environmental Eng., Tohoku University. He has taught applied mechanics for undergraduate and postgraduate students for 12 years, and his research interests include elastoplasticity, nonlinear mechanics, material and structural instability, and bifurcation.

• Preface xiSeries Preface xvIntroduction xvii1 Mathematical Preliminaries 11.1 Basic Symbols and Conventions 11.2 Definition of Tensor 21.2.1 Objective Tensor 21.2.2 Quotient Law 41.3 Vector Analysis 51.3.1 Scalar Product 51.3.2 Vector Product 61.3.3 Scalar Triple Product 61.3.4 Vector Triple Product 71.3.5 Reciprocal Vectors 81.3.6 Tensor Product 91.4 Tensor Analysis 91.4.1 Properties of Second-Order Tensor 91.4.2 Tensor Components 101.4.3 Transposed Tensor 111.4.4 Inverse Tensor 121.4.5 Orthogonal Tensor 121.4.6 Tensor Decompositions 151.4.7 Axial Vector 171.4.8 Determinant 201.4.9 On Solutions of Simultaneous Equation 231.4.10 Scalar Triple Products with Invariants 241.4.11 Orthogonal Transformation of Scalar Triple Product 251.4.12 Pseudo Scalar, Vector and Tensor 261.5 Tensor Representations 271.5.1 Tensor Notations 271.5.2 Tensor Components and Transformation Rule 271.5.3 Notations of Tensor Operations 281.5.4 Operational Tensors 291.5.5 Isotropic Tensors 311.6 Eigenvalues and Eigenvectors 361.6.1 Eigenvalues and Eigenvectors of Second-Order Tensors 361.6.2 Spectral Representation and Elementary Tensor Functions 401.6.3 Calculation of Eigenvalues and Eigenvectors 421.6.4 Eigenvalues and Vectors of Orthogonal Tensor 451.6.5 Eigenvalues and Vectors of Skew-Symmetric Tensor and Axial Vector 461.6.6 Cayley–Hamilton Theorem 471.7 Polar Decomposition 471.8 Isotropy 491.8.1 Isotropic Material 491.8.2 Representation Theorem of Isotropic Tensor-Valued Tensor Function 501.9 Differential Formulae 541.9.1 Partial Derivatives 541.9.2 Directional Derivatives 591.9.3 Taylor Expansion 621.9.4 Time Derivatives in Lagrangian and Eulerian Descriptions 631.9.5 Derivatives of Tensor Field 681.9.6 Gauss’s Divergence Theorem 711.9.7 Material-Time Derivative of Volume Integration 731.10 Variations and Rates of Geometrical Elements 741.10.1 Variations of Line, Surface and Volume 751.10.2 Rates of Changes of Surface and Volume 761.11 Continuity and Smoothness Conditions 791.11.1 Continuity Condition 791.11.2 Smoothness Condition 801.12 Unconventional Elasto-Plasticity Models 812 General (Curvilinear) Coordinate System 852.1 Primary and Reciprocal Base Vectors 852.2 Metric Tensors 892.3 Representations of Vectors and Tensors 952.4 Physical Components of Vectors and Tensors 1022.5 Covariant Derivative of Base Vectors with Christoffel Symbol 1032.6 Covariant Derivatives of Scalars, Vectors and Tensors 1072.7 Riemann–Christoffel Curvature Tensor 1122.8 Relations of Convected and Cartesian Coordinate Descriptions 1153 Description of Physical Quantities in Convected Coordinate System 1173.1 Necessity for Description in Embedded Coordinate System 1173.2 Embedded Base Vectors 1183.3 Deformation Gradient Tensor 1213.4 Pull-Back and Push-Forward Operations 1234 Strain and Strain Rate Tensors 1314.1 Deformation Tensors 1314.2 Strain Tensors 1364.2.1 Green and Almansi Strain Tensors 1364.2.2 General Strain Tensors 1414.2.3 Hencky Strain Tensor 1444.3 Compatibility Condition 1454.4 Strain Rate and Spin Tensors 1464.4.1 Strain Rate and Spin Tensors Based on Velocity Gradient Tensor 1474.4.2 Strain Rate Tensor Based on General Strain Tensor 1524.5 Representations of Strain Rate and Spin Tensors in Lagrangian and Eulerian Triads 1534.6 Decomposition of Deformation Gradient Tensor into Isochoric and Volumetric Parts 1585 Convected Derivative 1615.1 Convected Derivative 1615.2 Corotational Rate 1655.3 Objectivity 1666 Conservation Laws and Stress (Rate) Tensors 1796.1 Conservation Laws 1796.1.1 Basic Conservation Law 1796.1.2 Conservation Law of Mass 1806.1.3 Conservation Law of Linear Momentum 1816.1.4 Conservation Law of Angular Momentum 1826.2 Stress Tensors 1836.2.1 Cauchy Stress Tensor 1836.2.2 Symmetry of Cauchy Stress Tensor 1876.2.3 Various Stress Tensors 1886.3 Equilibrium Equation 1946.4 Equilibrium Equation of Angular Moment 1976.5 Conservation Law of Energy 1976.6 Virtual Work Principle 1996.7 Work Conjugacy 2006.8 Stress Rate Tensors 2036.8.1 Contravariant Convected Derivatives 2036.8.2 Covariant–Contravariant Convected Derivatives 2046.8.3 Covariant Convected Derivatives 2046.8.4 Corotational Convected Derivatives 2046.9 Some Basic Loading Behavior 2076.9.1 Uniaxial Loading Followed by Rotation 2076.9.2 Simple Shear 2156.9.3 Combined Loading of Tension and Distortion 2207 Hyperelasticity 2257.1 Hyperelastic Constitutive Equation and Its Rate Form 2257.2 Examples of Hyperelastic Constitutive Equations 2307.2.1 St. Venant–Kirchhoff Elasticity 2307.2.2 Modified St. Venant–Kirchhoff Elasticity 2317.2.3 Neo-Hookean Elasticity 2327.2.4 Modified Neo-Hookean Elasticity (1) 2337.2.5 Modified Neo-Hookean Elasticity (2) 2347.2.6 Modified Neo-Hookean Elasticity (3) 2347.2.7 Modified Neo-Hookean Elasticity (4) 2348 Finite Elasto-Plastic Constitutive Equation 2378.1 Basic Structures of Finite Elasto-Plasticity 2388.2 Multiplicative Decomposition 2388.3 Stress and Deformation Tensors for Multiplicative Decomposition 2438.4 Incorporation of Nonlinear Kinematic Hardening 2448.4.1 Rheological Model for Nonlinear Kinematic Hardening 2458.4.2 Multiplicative Decomposition of Plastic Deformation Gradient Tensor 2468.5 Strain Tensors 2498.6 Strain Rate and Spin Tensors 2528.6.1 Strain Rate and Spin Tensors in Current Configuration 2528.6.2 Contravariant–Covariant Pulled-Back Strain Rate and Spin Tensors in Intermediate Configuration 2548.6.3 Covariant Pulled-Back Strain Rate and Spin Tensors in Intermediate Configuration 2568.6.4 Strain Rate Tensors for Kinematic Hardening 2598.7 Stress and Kinematic Hardening Variable Tensors 2618.8 Influences of Superposed Rotations: Objectivity 2668.9 Hyperelastic Equations for Elastic Deformation and Kinematic Hardening 2688.9.1 Hyperelastic Constitutive Equation 2688.9.2 Hyperelastic Type Constitutive Equation for Kinematic Hardening 2698.10 Plastic Constitutive Equations 2708.10.1 Normal-Yield and Subloading Surfaces 2718.10.2 Consistency Condition 2728.10.3 Plastic and Kinematic Hardening Flow Rules 2758.10.4 Plastic Strain Rate 2778.11 Relation between Stress Rate and Strain Rate 2788.11.1 Description in Intermediate Configuration 2788.11.2 Description in Reference Configuration 2788.11.3 Description in Current Configuration 2798.12 Material Functions of Metals 2808.12.1 Strain Energy Function of Elastic Deformation 2808.12.2 Strain Energy Function for Kinematic Hardening 2818.12.3 Yield Function 2828.12.4 Plastic Strain Rate and Kinematic Hardening Strain Rate 2838.13 On the Finite Elasto-Plastic Model in the Current Configuration by the Spectral Representation 2848.14 On the Clausius–Duhem Inequality and the Principle of Maximum Dissipation 2859 Computational Methods for Finite Strain Elasto-Plasticity 2879.1 A Brief Review of Numerical Methods for Finite Strain Elasto-Plasticity 2889.2 Brief Summary of Model Formulation 2899.2.1 Constitutive Equations for Elastic Deformation and Isotropic and Kinematic Hardening 2899.2.2 Normal-Yield and Subloading Functions 2919.2.3 Plastic Evolution Rules 2919.2.4 Evolution Rule of Normal-Yield Ratio for Subloading Surface 2939.3 Transformation to Description in Reference Configuration 2939.3.1 Constitutive Equations for Elastic Deformation and Isotropic and Kinematic Hardening 2939.3.2 Normal-Yield and Subloading Functions 2949.3.3 Plastic Evolution Rules 2959.3.4 Evolution Rule of Normal-Yield Ratio for Subloading Surface 2969.4 Time-Integration of Plastic Evolution Rules 2969.5 Update of Deformation Gradient Tensor 3009.6 Elastic Predictor Step and Loading Criterion 3019.7 Plastic Corrector Step by Return-Mapping 3049.8 Derivation of Jacobian Matrix for Return-Mapping 3089.8.1 Components of Jacobian Matrix 3089.8.2 Derivatives of Tensor Exponentials 3109.8.3 Derivatives of Stresses 3129.9 Consistent (Algorithmic) Tangent Modulus Tensor 3129.9.1 Analytical Derivation of Consistent Tangent Modulus Tensor 3139.9.2 Numerical Computation of Consistent Tangent Modulus Tensor 3159.10 Numerical Examples 3169.10.1 Example 1: Strain-Controlled Cyclic Simple Shear Analysis 3189.10.2 Example 2: Elastic–Plastic Transition 3189.10.3 Example 3: Large Monotonic Simple Shear Analysis with Kinematic Hardening Model 3209.10.4 Example 4: Accuracy and Convergence Assessment of Stress-Update Algorithm 3229.10.5 Example 5: Finite Element Simulation of Large Deflection of Cantilever 3269.10.6 Example 6: Finite Element Simulation of Combined Tensile, Compressive, and Shear Deformation for Cubic Specimen 33010 Computer Programs 33710.1 User Instructions and Input File Description 33710.2 Output File Description 340x Contents10.3 Computer Programs 34110.3.1 Structure of Fortran Program returnmap 34110.3.2 Main Routine of Program returnmap 34310.3.3 Subroutine to Define Common Variables: comvar 34310.3.4 Subroutine for Return-Mapping: retmap 34510.3.5 Subroutine for Isotropic Hardening Rule: plhiso 37710.3.6 Subroutine for Numerical Computation of Consistent Tangent Modulus Tensor: tgnum0 377A Projection of Area 385B Geometrical Interpretation of Strain Rate and Spin Tensors 387C Proof for Derivative of Second Invariant of Logarithmic-Deviatoric Deformation Tensor 391D Numerical Computation of Tensor Exponential Function and Its Derivative 393D.1 Numerical Computation of Tensor Exponential Function 393D.2 Fortran Subroutine for Tensor Exponential Function: matexp 394D.3 Numerical Computation of Derivative of Tensor Exponential Function 396D.4 Fortran Subroutine for Derivative of Tensor Exponential Function: matdex 400References 401Index 409