This book goes back a long way. There is a tradition of research and teaching in inelasticity at Stanford that goes back at least to Wilhelm Flugge and Erastus Lee.I joined the faculty in 1980, and shortly thereafter the Chairman of the Applied Mechanics Division, George Herrmann, asked me to present a course in plasticity.I decided to develop a new two-quarter sequence entitled Theoretical and Computational Plasticity which combined the basic theory I had learned as a graduate student at the University of California at Berkeley from David Bogy, James Kelly,Jacob Lubliner, and Paul Naghdi with new computational techniques from the finite-element literature and my personal research. I taught the course a couple of times and developed a set of notes that 1 passed on to Juan Simo when he joined the faculty in 1985.1 was Chairman at that time and I asked Juan to furthdr develop the course into a full year covering inelasticity from a more comprehensive per-spectiye.

Preface

Chapter I Motivation. One-Dimensional Plasticity and Viscoplasticity.

 1.1 Overview

 1.2 Motivation. One-Dimensional Frictional Models

1.2.1 Local Governing Equations

1.2.2 An Elementary Model for (Isotropic) Hardening Plasticity

1.2.3 Aitemative Form of the Loading/Unloading Conditions

1.2.4 Further Refinements of the Hardening Law

1.2.5 Geometric Properties of the Elastic Domain

 1.3 The Initial Boundary-Value Problem

1.3.1 The Local Form of the IBVP

1.3.2 The Weak Formulation of the IBVP

1.3.3 Dissipation. A priori Stability Estimate

1:3.4 Uniqueness of the Solution to the IBVE Contractivity

1.3.5 Outline of the Numerical Solution of the IBVP

 1.4 Integration Algorithms for Rate-Independent Plasticity

1.4.1 The Incremental Form of Rate-Independent Plasticity

1.4,2 Return-Mapping Algorithms.1sotropic Hardening

1..3 Discrete Variational Formulation. Convex Optimization

1.4.4 Extension to the Combined lsotropic/Kinematic Hardening Model

 1.5 Finite-Element Solution of the Elastoplastic IBVP. An Illustration

1.5.1Spatial Discretization. Finite-Element Approximation

1.5.2 Incremental Solution Procedure

 1.6 Stability Analysis of the Algorithmic IBVP

1.6.1 Algorithmic Approximation to the Dynamic Weak Form

 1.7 One-'Dimensional Viscoplasticity

1.7.1 One-Dimensional Rheological Model

1.7.2 Dissipation. A Priori Stability Estimate

1.7.3 An Integration Algorithm for Viscoplasticity

Chapter 2 Classical Rate-independent Plasticity and Viscoplasticity.

 2.1 Review of Some Standard Notation

2.1.1 The Local Form of the IBVP. Elasticity

 2.2 Classical Rate-Independent Plasticity

2.2.1 Strain-Space and Stress-Space Formulations

2.2.2 Stress-Space Governing Equations

2.2.3 Strain-Space Formulation

2.2.4 An Elementary Example: I-D Plasticity

 2.3 Plane Strain and 3-D, Classical /2 Flow Theory

2.3.1 Perfect Plasticity

2.3.2 -/2 Flow Theory with lsotropic/Kinematic Hardening

 2.4 Plane-Stress -/2 Flow Theory

  2.4.1 Projection onto the Plane-Stress Subspace

  2.4.2 Constrained Plane-Stress Equations

 2.5 General Quadratic Model of Classical Plasticity

2.5.1 The Yield Criterion

2.5.2 Evolution Equations. Elastoplastic Moduli

 2.6 The Principle of Maximum Plastic Dissipation

2.6.1 Classical Formulation. Perfect Plasticity

2.6.2 General Associative Hardening Plasticity in Stress Space

2.6.3 Interpretation of Associative Plasticity as a Variational Inequality

 2.7 Classical (Rate-Dependent) Viscoplasticity

2.7.1 Formulation of the Basic Governing Equations

2.7.2 Interpretation as a Viscoplastic Regularization

2.7.3 Penalty Formulation of the Principle of Maximum Plastic Dissipation

2.7.4 The Generalized Duvaut-Lions Model

Chapter 3 Integration Algorithms for Plasticity and Viscoplasticity

 3.1 Basic Algorithmic Setup. Strain-Driven Problem

3.1.1 Associative plasticity

 3.2 The Notion of Closest Point Projection

3.2.1 Plastic Loading. Discrete Kuhn--Tucker Conditions

3.2.2 Geometric Interpretation

 3.3 Example 3.1. J2 Plasticity. Nonlinear Isotropic/KinematicHardening

3.3.1 Radial Return Mapping

3.3.2 Exact Linearization ofthe Algorithm

 3.4 Example 3.2. Plane-Stress ./2 Plasticity. Kinematic/Isotropic Hardening

3.4.1 Return-Mapping Algorithm

3.4.2 Consistent Elastoplastic Tangent Moduli

3.4.3 Implementation

3.4.4 Accuracy Assessment.1soerror Maps

3.4.5 Closed-Form Exact Solution of the Consistency Equation.

 3.5 Interpretation. Operator Splits and Product Formulas

3.5.1 Example 3.3. Lie's Formula

3.5.2 Elastic-Plastic Operator Split

3.5.3 Elastic Predictor. Trial Elastic State

3.5.4 Plastic Corrector. Return Mapping

 3.6 General Return-Mapping Algorithms

3.6.1 General Closest Point Projection

3.6.2 Consistent Eiastoplastic Modul1. Perfect Plasticity

3.6.3 Cutting-Plane Algorithm

 3.7 Extension of General Algorithms to Viscoplasticity

3.7.1 Motivation. J2-Viscoplasticity

3.7.2 Closest Point Projection

3.7.3 A Note on Notational Conventions

Chapter 4 Discrete Variational Iormulation and Finite-Element Implementation

 4.1 Review of Some Basic Notation

4.1.1 Gateaux Variation

4.1.2 The Functional Derivative

4.1.3 Euler-Lagrange Equations

 4.2 General Variational Framework for Elastoplasticity

4.2.1 Variational Characterization of Plastic Response

4.2.2 Discrete Lagrangian for elastoplasticity

4.2.3 Variational Form of the Governing Equations

4.2.4 Extension to Viscoplasticity

 4.3 Finite-Element Formulation. Assumed-Strain Method

4.3.1 Matrix and Vector Notation

4.3.2 Summary of Governing Equations

4.3.3 Discontinuous Strain and Stress Interpolations

4.3.4 Reduced Residual. Generalized Displacement Model

4.3.5 Closest Point Projection Algorithm

4.3.6 Linearization. Consistent Tangent Operator

4.3.7 Matrix Expressions

4.3.8 Variational Consistency of Assumed-Strain Methods

 4.4 Application. B-Bar Method for Incompressibility

4.4.1 Assumed-Strain ,and Stress Fields

4.4.2 Weak Forms

4.4.3 Discontinuous Volume/Mean-Stress Interpolations

4.4.4 Implementation 1. B-Bar-Approach

4.4.5 Implementation 2. Mixed Approach

4.4.6 Examples and Remarks on Convergence

 4.5 Numerical Simulations

4.5.1 Plane-Strain J2 Flow Theory

4.5.2 Plane-Stress /2 Flow Theory

Chapter S Nonsmooth Multisurface Plasticity and Viscoplasticity

 5.1 Rate-Independent Multisurface Plasticity. Continuum Formulation

5.1.1 Summary of Governing Equations

5.1 .2 Loading/Unloading Conditions

5.1.3 Consistency Condition. Elastoplastic Tangent Moduli

5.1.4 Geometric Interpretation

 5.2 Discrete Formulation. Rate-Independent Elastoplasticity

5.2.1 Closest Point Projection Algorithm for Multisurface Plasticity

5.2.2 Loading/Unloading. Discrete Kuhn-Tucker Conditions

5.2.3 Solution Algorithm and Implementation

5.2.4 Linearization: Algorithmic Tangent Moduli

 5.3 Extension to Viscoplasticity

5.3.1 Motivation. Perzyna-Type Models

5.3.2 Extension of the Duvaut-Lions Model

5.3.3 Discrete Formulation

Chapter 6 Numerical Analysis of General Return Mapping Algorithms

 6.1 Motivation: Nonlinear Heat Conduction

6.1.1 The Continuum Problem

6.1.2 The Algorithmic Problem

6.1.3 Nonlinear Stability Analysis

 6.2 Infinitesimal Elastoplasticity

6.2.1 The Continuum Problem for Plasticity and Viscoplasticity..

6.2.2 The Algorithmic Problem

6.2.3 Nonlinear Stability Analysi

 6.3 Concluding Remarks

Chapter 7 Nonlinear Continuum Mechanics and Phenomenological Plasticity Models

 7.1 Review of Some Basic Results in Continuum Mechanics

7.1.1 Configurations. Basic Kinematics .

7.1.2 Motions. Lagrangian and Eulerian Descriptions

7.1.3 Rate of Deformation Tensors

7.1.4 Stress Tensors. Equations of Motion

7.1.5 Objectivity. Elastic Constitutive Equations

7.1.6 The Notion of lsotropy, lsotropic Elastic Response

 7.2 Variational Formulation. Weak Form of Momentum Balance

7.2.1 Configuration Space and Admissible Variations

7.2.2 The Weak Form of Momentum Balance

7.2.3 The Rate Form of the Weak Form of Momentum Balance.__

 7.3 Ad Hoc Extensions of Phenomenological Plasticity Based on Hypoelastic Relationships

7.3.1 Formulation in the Spatial Description

7.3.2 Formulation in the Rotated Description

Chapter 8 Objective Integration Algorithms for Rate Formulations of Elastoplasticity

 8.1 Objective Time-Stepping Algorithms

8.1.1 The Geometric Setup

8.1.2 Approximation for the Rate of Deformation Tensor

8.1.3 Approximation for the Lie Derivative

8.1.4 Application: Numerical Integration of Rate Constitutive Equations

 8.2 Application to J2 Flow Theory at Finite Strains

8.2.1 A ./2 Flow Theory

 8.3 Objective Algorithms Based on the Notion of a Rotated Configuration

8.3.1 Objective Integration of Eiastoplastic Models

8.3.2 Time-Stepping Algorithms for the Orthogonal Group

Chapter 9 Phenomenological Plasticity Models Based on the Notion of an Intermediate Stress-Free Configuration

 9.1 Kinematic Preliminaries. The (Local) Intermediate Configuration

9.1.1 Micromechanical Motivation. Single-Crystal Plasticity

9.1 .2 Kinematic Relationships Associated with the Intermediate Configuration

9.1.3 Deviatoric-Volumetric Multiplicative Split

 9.2 Flow Theory at Finite Strains. A Model Problem

9.2.1 Formulation of the Governing Equations

 9.3 Integration Algorithm for J2 Flow Theory

9.3.1 Integration of the Flow Rule and Hardening Law

9.3.2 The Return-Mapping Algorithm

9.3.3 Exact Linearization of the Algorithm

 9.4 Assessment of the Theory. Numerical Simulations

Chapter 10 Viscoelasticity

 10.1 Motivation. One-Dimensional Rheologicai Models

10.1.1 Formulation of the Constitutive Model

10.1.2 Convolution Representation

10.1.3 Generalized Relaxation Models

 10.2 Three-Dimensional Models: Fohnulation :Restricted to Linearized Kinematics

10.2.1 Fg.rDulation of the Model

10.2.2 Thermodynamic Aspects, Dissipation

 10.3 Integration Algorithms

10.3.1 Algorithmic Internal Variables and Finite-Element Database

10.3.2 One-Step, Unconditionally Stable and Second-Order,Accurate Recurrence Formula

10.3.3 Linearization. Consistent Tangent Moduli

 10.4 Finite Elasticity with Uncoupled Volume Response

10.4.1 Volumetric/Deviatoric Multiplicative Split

10.4.2 Stored-Energy Function and Stress Response

10.4.3 Elastic Tangent Moduli

 10.5 A Class of Nonlinear, Viscoelastic, Constitutive Models

10.5.1 Formulation of the Nonlinear Viscoelastic Constitutive Model

 10.6 Implementation of Integration Algorithms for Nonlinear Viscoelasticity

10.6.1 One-Step, Second-Order Accurate Recurrence Formula.

10.6.2 Configuration Update Procedure

10.6.3 Consistent (Algorithmic) Tangent Moduli

References

Index