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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
1.4.3 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
1.5.1 Spatial 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<
2.7 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
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
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
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References
Index