INTRODUCTION TO FINITE STRAIN THEORY FOR CONTINUUM ELASTO-PLASTICITY -- Contents -- Preface -- Series Preface -- Introduction -- 1 Mathematical Preliminaries -- 1.1 Basic Symbols and Conventions -- 1.2 Definition of Tensor -- 1.2.1 Objective Tensor -- 1.2.2 Quotient Law -- 1.3 Vector Analysis -- 1.3.1 Scalar Product -- 1.3.2 Vector Product -- 1.3.3 Scalar Triple Product -- 1.3.4 Vector Triple Product -- 1.3.5 Reciprocal Vectors -- 1.3.6 Tensor Product -- 1.4 Tensor Analysis -- 1.4.1 Properties of Second-Order Tensor -- 1.4.2 Tensor Components -- 1.4.3 Transposed Tensor -- 1.4.4 Inverse Tensor -- 1.4.5 Orthogonal Tensor -- 1.4.6 Tensor Decompositions -- 1.4.7 Axial Vector -- 1.4.8 Determinant -- 1.4.9 On Solutions of Simultaneous Equation -- 1.4.10 Scalar Triple Products with Invariants -- 1.4.11 Orthogonal Transformation of Scalar Triple Product -- 1.4.12 Pseudo Scalar, Vector and Tensor -- 1.5 Tensor Representations -- 1.5.1 Tensor Notations -- 1.5.2 Tensor Components and Transformation Rule -- 1.5.3 Notations of Tensor Operations -- 1.5.4 Operational Tensors -- 1.5.5 Isotropic Tensors -- 1.6 Eigenvalues and Eigenvectors -- 1.6.1 Eigenvalues and Eigenvectors of Second-Order Tensors -- 1.6.2 Spectral Representation and Elementary Tensor Functions -- 1.6.3 Calculation of Eigenvalues and Eigenvectors -- 1.6.4 Eigenvalues and Vectors of Orthogonal Tensor -- 1.6.5 Eigenvalues and Vectors of Skew-Symmetric Tensor and Axial Vector -- 1.6.6 Cayley-Hamilton Theorem -- 1.7 Polar Decomposition -- 1.8 Isotropy -- 1.8.1 Isotropic Material -- 1.8.2 Representation Theorem of Isotropic Tensor-Valued Tensor Function -- 1.9 Differential Formulae -- 1.9.1 Partial Derivatives -- 1.9.2 Directional Derivatives -- 1.9.3 Taylor Expansion -- 1.9.4 Time Derivatives in Lagrangian and Eulerian Descriptions -- 1.9.5 Derivatives of Tensor Field.

1.9.6 Gauss's Divergence Theorem -- 1.9.7 Material-Time Derivative of Volume Integration -- 1.10 Variations and Rates of Geometrical Elements -- 1.10.1 Variations of Line, Surface and Volume -- 1.10.2 Rates of Changes of Surface and Volume -- 1.11 Continuity and Smoothness Conditions -- 1.11.1 Continuity Condition -- 1.11.2 Smoothness Condition -- 2 General (Curvilinear) Coordinate System -- 2.1 Primary and Reciprocal Base Vectors -- 2.2 Metric Tensors -- 2.3 Representations of Vectors and Tensors -- 2.4 Physical Components of Vectors and Tensors -- 2.5 Covariant Derivative of Base Vectors with Christoffel Symbol -- 2.6 Covariant Derivatives of Scalars, Vectors and Tensors -- 2.7 Riemann-Christoffel Curvature Tensor -- 2.8 Relations of Convected and Cartesian Coordinate Descriptions -- 3 Description of Physical Quantities in Convected Coordinate System -- 3.1 Necessity for Description in Embedded Coordinate System -- 3.2 Embedded Base Vectors -- 3.3 Deformation Gradient Tensor -- 3.4 Pull-Back and Push-Forward Operations -- 4 Strain and Strain Rate Tensors -- 4.1 Deformation Tensors -- 4.2 Strain Tensors -- 4.2.1 Green and Almansi Strain Tensors -- 4.2.2 General Strain Tensors -- 4.2.3 Hencky Strain Tensor -- 4.3 Compatibility Condition -- 4.4 Strain Rate and Spin Tensors -- 4.4.1 Strain Rate and Spin Tensors Based on Velocity Gradient Tensor -- 4.4.2 Strain Rate Tensor Based on General Strain Tensor -- 4.5 Representations of Strain Rate and Spin Tensors in Lagrangian and Eulerian Triads -- 4.6 Decomposition of Deformation Gradient Tensor into Isochoric and Volumetric Parts -- 5 Convected Derivative -- 5.1 Convected Derivative -- 5.2 Corotational Rate -- 5.3 Objectivity -- 6 Conservation Laws and Stress (Rate) Tensors -- 6.1 Conservation Laws -- 6.1.1 Basic Conservation Law -- 6.1.2 Conservation Law of Mass.

6.1.3 Conservation Law of Linear Momentum -- 6.1.4 Conservation Law of Angular Momentum -- 6.2 Stress Tensors -- 6.2.1 Cauchy Stress Tensor -- 6.2.2 Symmetry of Cauchy Stress Tensor -- 6.2.3 Various Stress Tensors -- 6.3 Equilibrium Equation -- 6.4 Equilibrium Equation of Angular Moment -- 6.5 Conservation Law of Energy -- 6.6 Virtual Work Principle -- 6.7 Work Conjugacy -- 6.8 Stress Rate Tensors -- 6.8.1 Contravariant Convected Derivatives -- 6.8.2 Covariant-Contravariant Convected Derivatives -- 6.8.3 Covariant Convected Derivatives -- 6.8.4 Corotational Convected Derivatives -- 6.9 Some Basic Loading Behavior -- 6.9.1 Uniaxial Loading Followed by Rotation -- 6.9.2 Simple Shear -- 6.9.3 Combined Loading of Tension and Distortion -- 7 Hyperelasticity -- 7.1 Hyperelastic Constitutive Equation and Its Rate Form -- 7.2 Examples of Hyperelastic Constitutive Equations -- 7.2.1 St. Venant-Kirchhoff Elasticity -- 7.2.2 Modified St. Venant-Kirchhoff Elasticity -- 7.2.3 Neo-Hookean Elasticity -- 7.2.4 Modified Neo-Hookean Elasticity (1) -- 7.2.5 Modified Neo-Hookean Elasticity (2) -- 7.2.6 Modified Neo-Hookean Elasticity (3) -- 7.2.7 Modified Neo-Hookean Elasticity (4) -- 8 Finite Elasto-Plastic Constitutive Equation -- 8.1 Basic Structures of Finite Elasto-Plasticity -- 8.2 Multiplicative Decomposition -- 8.3 Stress and Deformation Tensors for Multiplicative Decomposition -- 8.4 Incorporation of Nonlinear Kinematic Hardening -- 8.4.1 Rheological Model for Nonlinear Kinematic Hardening -- 8.4.2 Multiplicative Decomposition of Plastic Deformation Gradient Tensor -- 8.5 Strain Tensors -- 8.6 Strain Rate and Spin Tensors -- 8.6.1 Strain Rate and Spin Tensors in Current Configuration -- 8.6.2 Contravariant-Covariant Pulled-Back Strain Rate and Spin Tensors in Intermediate Configuration.

8.6.3 Covariant Pulled-Back Strain Rate and Spin Tensors in Intermediate Configuration -- 8.6.4 Strain Rate Tensors for Kinematic Hardening -- 8.7 Stress and Kinematic Hardening Variable Tensors -- 8.8 Influences of Superposed Rotations: Objectivity -- 8.9 Hyperelastic Equations for Elastic Deformation and Kinematic Hardening -- 8.9.1 Hyperelastic Constitutive Equation -- 8.9.2 Hyperelastic Type Constitutive Equation for Kinematic Hardening -- 8.10 Plastic Constitutive Equations -- 8.10.1 Normal-Yield and Subloading Surfaces -- 8.10.2 Consistency Condition -- 8.10.3 Plastic and Kinematic Hardening Flow Rules -- 8.10.4 Plastic Strain Rate -- 8.11 Relation between Stress Rate and Strain Rate -- 8.11.1 Description in Intermediate Configuration -- 8.11.2 Description in Reference Configuration -- 8.11.3 Description in Current Configuration -- 8.12 Material Functions of Metals -- 8.12.1 Strain Energy Function of Elastic Deformation -- 8.12.2 Strain Energy Function for Kinematic Hardening -- 8.12.3 Yield Function -- 8.12.4 Plastic Strain Rate and Kinematic Hardening Strain Rate -- 8.13 On the Finite Elasto-Plastic Model in the Current Configuration by the Spectral Representation -- 8.14 On the Clausius-Duhem Inequality and the Principle of Maximum Dissipation -- 9 Computational Methods for Finite Strain Elasto-Plasticity -- 9.1 A Brief Review of Numerical Methods for Finite Strain Elasto-Plasticity -- 9.2 Brief Summary of Model Formulation -- 9.2.1 Constitutive Equations for Elastic Deformation and Isotropic and Kinematic Hardening -- 9.2.2 Normal-Yield and Subloading Functions -- 9.2.3 Plastic Evolution Rules -- 9.2.4 Evolution Rule of Normal-Yield Ratio for Subloading Surface -- 9.3 Transformation to Description in Reference Configuration -- 9.3.1 Constitutive Equations for Elastic Deformation and Isotropic and Kinematic Hardening.

9.3.2 Normal-Yield and Subloading Functions -- 9.3.3 Plastic Evolution Rules -- 9.3.4 Evolution Rule of Normal-Yield Ratio for Subloading Surface -- 9.4 Time-Integration of Plastic Evolution Rules -- 9.5 Update of Deformation Gradient Tensor -- 9.6 Elastic Predictor Step and Loading Criterion -- 9.7 Plastic Corrector Step by Return-Mapping -- 9.8 Derivation of Jacobian Matrix for Return-Mapping -- 9.8.1 Components of Jacobian Matrix -- 9.8.2 Derivatives of Tensor Exponentials -- 9.8.3 Derivatives of Stresses -- 9.9 Consistent (Algorithmic) Tangent Modulus Tensor -- 9.9.1 Analytical Derivation of Consistent Tangent Modulus Tensor -- 9.9.2 Numerical Computation of Consistent Tangent Modulus Tensor -- 9.10 Numerical Examples -- 9.10.1 Example 1: Strain-Controlled Cyclic Simple Shear Analysis -- 9.10.2 Example 2: Elastic-Plastic Transition -- 9.10.3 Example 3: Large Monotonic Simple Shear Analysis with Kinematic Hardening Model -- 9.10.4 Example 4: Accuracy and Convergence Assessment of Stress-Update Algorithm -- 9.10.5 Example 5: Finite Element Simulation of Large Deflection of Cantilever -- 9.10.6 Example 6: Finite Element Simulation of Combined Tensile, Compressive, and Shear Deformation for Cubic Specimen -- 10 Computer Programs -- 10.1 User Instructions and Input File Description -- 10.2 Output File Description -- 10.3 Computer Programs -- 10.3.1 Structure of Fortran Program returnmap -- 10.3.2 Main Routine of Program returnmap -- 10.3.3 Subroutine to Define Common Variables: comvar -- 10.3.4 Subroutine for Return-Mapping: retmap -- 10.3.5 Subroutine for Isotropic Hardening Rule: plhiso -- 10.3.6 Subroutine for Numerical Computation of Consistent Tangent Modulus Tensor: tgnum0 -- A Projection of Area -- B Geometrical Interpretation of Strain Rate and Spin Tensors.

C Proof for Derivative of Second Invariant of Logarithmic-Deviatoric Deformation Tensor.