Elastoplasticity Theory

Title Page

Preface

Contents

Tensor Analysis

Conventions and Symbols

Vector

Tensor

Operations of Second-Order Tensor

Eigenvalues and Eigenvectors

Calculations of Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors of Skew-Symmetric Tensor

Cayley-Hamilton’s Theorem

Positive Definite Tensor

Polar Decomposition

Isotropic Tensor-Valued Tensor Function

Representation of Tensor in Principal Space

Two-Dimensional State

Partial Differential Calculi

Time Derivatives

Differentiation and Integration in Field

Motion and Strain (Rate)

Motion and Deformation

Strain Tensor

Strain Rate and Spin Tensors

Various Simple Deformations

Surface Element, Volume Element and Their Rates

Conservation Laws and Stress Tensors

Conservation Law of Mass

Conservation Law of Momentum

Conservation Law of Angular Momentum

Stress Tensor

Equilibrium Equation

Equilibrium Equation of Moment

Virtual Work Principle

Objectivity and Corotational Rate Tensor

Objectivity

Influence of Rigid-Body Rotation on Various Mechanical Quantities

Rate of State Variable and Corotational Rate Tensor

Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative

Various Objective Stress Rate Tensors

Work Conjugacy

Elastic Constitutive Equations

Hyperelasticity

Cauchy Elasticity

Hypoelasticity

Basic Formulations for Elastoplastic Constitutive Equations

Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate

Conventional Elastoplastic Constitutive Equations

Loading Criterion

Associated Flow Rule

Anisotropy

Incorporation of Tangential-Inelastic Strain Rate

Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

Unconventional Elastoplasticity Model: Subloading Surface Model

Mechanical Requirements

Subloading Surface Model

Salient Features of Subloading Surface Model

On Bounding Surface and Bounding Surface Model

Incorporation of Anisotropy

Incorporation of Tangential Inelastic Strain Rate

Cyclic Plasticity Model: Extended Subloading Surface Model

Classification of Cyclic Plasticity Models

Translation of Subyield Surface(s): Extension of Kinematic Hardening

Extended Subloading Surface Model

Modification of Reloading Curve

Incorporation of Tangential-Inelastic Strain Rate

Viscoplastic Constitutive Equations

History of Viscoplastic Constitutive Equations

Mechanical Response of Ordinary Overstress Model

Modification of Overstress Model: Extension to General Rate of Deformation

Incorporation of Subloading Surface Concept: Subloading Overstress Model

Constitutive Equations of Metals

Isotropic and Kinematic Hardening

Cyclic Stagnation of Isotropic Hardening

On Calculation of the Normal-Yield Ratio

Comparisons of Test Results

Orthotropic Anisotropy

Representation of Isotropic Mises Yield Condition

Plane Stress State

{\it Plane Strain State}

Constitutive Equations of Soils

Isotropic Consolidation Characteristics

Yield Conditions

Isotropic Hardening Function

Rotational Hardening

Extended Subloading Surface Model

Partial Derivatives of Subloading Surface Function

Calculation of Normal-Yield Ratio

Simulations of Test Results

Simple Subloading Surface Model

Super-Yield Surface for Structured Soils in Natural Deposits

Numerical Analysis of Footing Settlement Problem

Corotational Rate Tensor

Hypoelasticity

Kinematic Hardening Material

Plastic Spin

Localization of Deformation

Element Test

Gradient Theory

Shear-Band Embedded Model: Smeared Crack Model

Necessary Condition for Shear Band Inception

Numerical Calculation

Numerical Ability of Subloading Surface Model

Return-Mapping Formulation for Subloading Surface Model

Constitutive Equation for Friction

History of Constitutive Equation for Friction

Decomposition of Sliding Velocity

Normal Sliding-Yield and Sliding-Subloading Surfaces

Evolution Rules of Sliding-Hardening Function and Normal Sliding-Yield Ratio

Relations of Contact Traction Rate and Sliding Velocity

Loading Criterion

Sliding-Yield Surfaces

Basic Mechanical Behavior of Subloading-Friction Model

Extension to Orthotropic Anisotropy

Appendixes

Appendix 1: Projection of Area

Appendix 2: Proof of $\partial(F_{jA} / J) / \partial{x}_{j}$=0

Appendix 3: Euler’s Theorem for Homogeneous Function

Appendix 4: Normal Vector of Surface

Appendix 5: Convexity of Two-Dimensional Curve

Appendix 6: Derivation of Eq. (11.19)

Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State

References

Index