Computational Inelasticity

by Simo, J. C.; Hughes, Thomas J. R.

• ISBN13: 9780387975207
• ISBN10: 0387975209
• Format: Hardcover
• Copyright: 1998-08-01
• Publisher: Springer Verlag

Summary

This book describes the theoretical foundations of inelasticity, its numerical formulation and implementation. The subject matter described herein constitutes a representative sample of state-of-the- art methodology currently used in inelastic calculations. Among the numerous topics covered are small deformation plasticity and viscoplasticity, convex optimization theory, integration algorithms for the constitutive equation of plasticity and viscoplasticity, the variational setting of boundary value problems and discretization by finite element methods. Also addressed are the generalization of the theory to non-smooth yield surface, mathematical numerical analysis issues of general return mapping algorithms, the generalization to finite-strain inelasticity theory, objective integration algorithms for rate constitutive equations, the theory of hyperelastic-based plasticity models and small and large deformation viscoelasticity. Computational Inelasticity will be of great interest to researchers and graduate students in various branches of engineering, especially civil, aeronautical and mechanical, and applied mathematics.

Table of Contents

Preface	v
Chapter 1 Motivation. One-Dimensional Plasticity and Viscoplasticity	1	(70)
1.1 Overview	1	(1)
1.2 Motivation. One-Dimensional Frictional Models	2	(19)
1.2.1 Local Governing Equations	2	(7)
1.2.2 An Elementary Model for (Isotropic) Hardening Plasticity	9	(4)
1.2.3 Alternative Form of the Loading/Unloading Conditions	13	(4)
1.2.4 Further Refinements of the Hardening Law	17	(2)
1.2.5 Geometric Properties of the Elastic Domain	19	(2)
1.3 The Initial Boundary-Value Problem	21	(11)
1.3.1 The Local Form of the IBVP	22	(2)
1.3.2 The Weak Formulation of the IBVP	24	(2)
1.3.3 Dissipation. A priori Stability Estimate	26	(3)
1.3.4 Uniqueness of the Solution to the IBVP. Contractivity	29	(2)
1.3.5 Outline of the Numerical Solution of the IBVP	31	(1)
1.4 Integration Algorithms for Rate-Independent Plasticity	32	(14)
1.4.1 The Incremental Form of Rate-Independent Plasticity	33	(2)
1.4.2 Return-Mapping Algorithms. Isotropic Hardening	35	(4)
1.4.3 Discrete Variational Formulation. Convex Optimization	39	(4)
1.4.4 Extension to the Combined Isotropic/Kinematic Hardening Model	43	(3)
1.5 Finite-Element Solution of the Elastoplastic IBVP. An Illustration	46	(7)
1.5.1 Spatial Discretization. Finite-Element Approximation	46	(3)
1.5.2 Incremental Solution Procedure	49	(4)
1.6 Stability Analysis of the Algorithmic IBVP	53	(4)
1.6.1 Algorithmic Approximation to the Dynamic Weak Form	54	(3)
1.7 One-Dimensional Viscoplasticity	57	(14)
1.7.1 One-Dimensional Rheological Model	58	(6)
1.7.2 Dissipation. A Priori Stability Estimate	64	(2)
1.7.3 An Integration Algorithm for Viscoplasticity	66	(5)
Chapter 2 Classical Rate-Independent Plasticity and Viscoplasticity	71	(42)
2.1 Review of Some Standard Notation	72	(3)
2.1.1 The Local Form of the IBVP. Elasticity	73	(2)
2.2 Classical Rate-Independent Plasticity	75	(14)
2.2.1 Strain-Space and Stress-Space Formulations	75	(1)
2.2.2 Stress-Space Governing Equations	76	(6)
2.2.3 Strain-Space Formulation	82	(3)
2.2.4 An Elementary Example: 1-D Plasticity	85	(4)
2.3 Plane Strain and 3-D, Classical J(2) Flow Theory	89	(2)
2.3.1 Perfect Plasticity	89	(1)
2.3.2 J(2) Flow Theory with Isotropic/Kinematic Hardening	90	(1)
2.4 Plane-Stress J(2) Flow Theory	91	(4)
2.4.1 Projection onto the Plane-Stress Subspace	92	(1)
2.4.2 Constrained Plane-Stress Equations	92	(3)
2.5 General Quadratic Model of Classical Plasticity	95	(3)
2.5.1 The Yield Criterion	96	(1)
2.5.2 Evolution Equations. Elastoplastic Moduli	96	(2)
2.6 The Principle of Maximum Plastic Dissipation	98	(7)
2.6.1 Classical Formulation. Perfect Plasticity	98	(3)
2.6.2 General Associative Hardening Plasticity in Stress Space	101	(2)
2.6.3 Interpretation of Associative Plasticity as a Variational Inequality	103	(2)
2.7 Classical (Rate-Dependent) Viscoplasticity	105	(8)
2.7.1 Formulation of the Basic Governing Equations	105	(1)
2.7.2 Interpretation as a Viscoplastic Regularization	105	(1)
2.7.3 Penalty Formulation of the Principle of Maximum Plastic Dissipation	106	(4)
2.7.4 The Generalized Duvaut-Lions Model	110	(3)
Chapter 3 Integration Algorithms for Plasticity and Viscoplasticity	113	(41)
3.1 Basic Algorithmic Setup. Strain-Driven Problem	114	(1)
3.1.1 Associative plasticity	115	(1)
3.2 The Notion of Closest Point Projection	115	(5)
3.2.1 Plastic Loading. Discrete Kuhn-Tucker Conditions	116	(2)
3.2.2 Geometric Interpretation	118	(2)
3.3 Example 3.1. J(2) Plasticity. Nonlinear Isotropic/Kinematic Hardening	120	(5)
3.3.1 Radial Return Mapping	120	(2)
3.3.2 Exact Linearization of the Algorithm	122	(3)
3.4 Example 3.2. Plane-Stress J(2) Plasticity. Kinematic/Isotropic Hardening	125	(14)
3.4.1 Return-Mapping Algorithm	126	(1)
3.4.2 Consistent Elastoplastic Tangent Moduli	127	(1)
3.4.3 Implementation	128	(3)
3.4.4 Accuracy Assessment. Isoerror Maps	131	(2)
3.4.5 Closed-Form Exact Solution of the Consistency Equation	133	(6)
3.5 Interpretation. Operator Splits and Product Formulas	139	(4)
3.5.1 Example 3.3. Lie's Formula	139	(1)
3.5.2 Elastic-Plastic Operator Split	140	(1)
3.5.3 Elastic Predictor. Trial Elastic State	140	(1)
3.5.4 Plastic Corrector. Return Mapping	141	(2)
3.6 General Return-Mapping Algorithms	143	(6)
3.6.1 General Closest Point Projection	143	(2)
3.6.2 Consistent Elastoplastic Moduli. Perfect Plasticity	145	(3)
3.6.3 Cutting-Plane Algorithm	148	(1)
3.7 Extension of General Algorithms to Viscoplasticity	149	(5)
3.7.1 Motivation. J(2)-Viscoplasticity	150	(1)
3.7.2 Closest Point Projection	151	(1)
3.7.3 A Note on Notational Conventions	151	(3)
Chapter 4 Discrete Variational Formulation and Finite-Element Implementation	154	(44)
4.1 Review of Some Basic Notation	155	(6)
4.1.1 Gateaux Variation	156	(2)
4.1.2 The Functional Derivative	158	(1)
4.1.3 Euler-Lagrange Equations	159	(2)
4.2 General Variational Framework for Elastoplasticity	161	(7)
4.2.1 Variational Characterization of Plastic Response	162	(1)
4.2.2 Discrete Lagrangian for elastoplasticity	163	(2)
4.2.3 Variational Form of the Governing Equations	165	(2)
4.2.4 Extension to Viscoplasticity	167	(1)
4.3 Finite-Element Formulation. Assumed-Strain Method	168	(10)
4.3.1 Matrix and Vector Notation	168	(2)
4.3.2 Summary of Governing Equations	170	(1)
4.3.3 Discontinuous Strain and Stress Interpolations	170	(1)
4.3.4 Reduced Residual. Generalized Displacement Model	171	(1)
4.3.5 Closest Point Projection Algorithm	172	(2)
4.3.6 Linearization. Consistent Tangent Operator	174	(2)
4.3.7 Matrix Expressions	176	(1)
4.3.8 Variational Consistency of Assumed-Strain Methods	176	(2)
4.4 Application. B-Bar Method for Incompressibility	178	(5)
4.4.1 Assumed-Strain and Stress Fields	178	(1)
4.4.2 Weak Forms	179	(1)
4.4.3 Discontinuous Volume/Mean-Stress Interpolations	180	(1)
4.4.4 Implementation 1. B-Bar-Approach	181	(1)
4.4.5 Implementation 2. Mixed Approach	182	(1)
4.4.6 Examples and Remarks on Convergence	183	(1)
4.5 Numerical Simulations	183	(15)
4.5.1 Plane-Strain J(2) Flow Theory	184	(4)
4.5.2 Plane-Stress J(2) Flow Theory	188	(10)
Chapter 5 Nonsmooth Multisurface Plasticity and Viscoplasticity	198	(21)
5.1 Rate-Independent Multisurface Plasticity. Continuum Formulation	199	(7)
5.1.1 Summary of Governing Equations	199	(2)
5.1.2 Loading/Unloading Conditions	201	(3)
5.1.3 Consistency Condition. Elastoplastic Tangent Moduli	204	(1)
5.1.4 Geometric Interpretation	204	(2)
5.2 Discrete Formulation. Rate-Independent Elastoplasticity	206	(9)
5.2.1 Closest Point Projection Algorithm for Multisurface Plasticity	206	(2)
5.2.2 Loading/Unloading. Discrete Kuhn-Tucker Conditions	208	(1)
5.2.3 Solution Algorithm and Implementation	209	(3)
5.2.4 Linearization: Algorithmic Tangent Moduli	212	(3)
5.3 Extension to Viscoplasticity	215	(4)
5.3.1 Motivation. Perzyna-Type Models	216	(1)
5.3.2 Extension of the Duvaut-Lions Model	217	(1)
5.3.3 Discrete Formulation	217	(2)
Chapter 6 Numerical Analysis of General Return Mapping Algorithms	219	(21)
6.1 Motivation: Nonlinear Heat Conduction	220	(8)
6.1.1 The Continuum Problem	221	(4)
6.1.2 The Algorithmic Problem	225	(1)
6.1.3 Nonlinear Stability Analysis	226	(2)
6.2 Infinitesimal Elastoplasticity	228	(11)
6.2.1 The Continuum Problem for Plasticity and Viscoplasticity	228	(7)
6.2.2 The Algorithmic Problem	235	(2)
6.2.3 Nonlinear Stability Analysis	237	(2)
6.3 Concluding Remarks	239	(1)
Chapter 7 Nonlinear Continuum Mechanics and Phenomenological Plasticity Models	240	(36)
7.1 Review of some Basic Results in Continuum Mechanics	240	(22)
7.1.1 Configurations. Basic Kinematics	241	(4)
7.1.2 Motions. Lagrangian and Eulerian Descriptions	245	(3)
7.1.3 Rate of Deformation Tensors	248	(2)
7.1.4 Stress Tensors. Equations of Motion	250	(2)
7.1.5 Objectivity. Elastic Constitutive Equations	252	(7)
7.1.6 The Notion of Isotropy. Isotropic Elastic Response	259	(3)
7.2 Variational Formulation. Weak Form of Momentum Balance	262	(7)
7.2.1 Configuration Space and Admissible Variations	262	(7)
7.2.2 The Weak Form of Momentum Balance	264	(2)
7.2.3 The Rate Form of the Weak Form of Momentum Balance	266	(3)
7.3 Ad Hoc Extensions of Phenomenological Plasticity Based on Hypoelastic Relationships	269	(7)
7.3.1 Formulation in the Spatial Description	269	(2)
7.3.2 Formulation in the Rotated Description	271	(5)
Chapter 8 Objective Integration Algorithms for Rate Formulations of Elastoplasticity	276	(24)
8.1 Objective Time-Stepping Algorithms	278	(9)
8.1.1 The Geometric Setup	279	(2)
8.1.2 Approximation for the Rate of Deformation Tensor	281	(2)
8.1.3 Approximation for the Lie Derivative	283	(2)
8.1.4 Application: Numerical Integration of Rate Constitutive Equations	285	(2)
8.2 Application to J(2) Flow Theory at Finite Strains	287	(3)
8.2.1 A J(2) Flow Theory	288	(2)
8.3 Objective Algorithms Based on the Notion of a Rotated Configuration	290	(10)
8.3.1 Objective Integration of Elastoplastic Models	291	(4)
8.3.2 Time-Stepping Algorithms for the Orthogonal Group	295	(5)
Chapter 9 Phenomenological Plasticity Models Based on the Notion of an Intermediate Stress-Free Configuration	300	(36)
9.1 Kinematic Preliminaries. The (Local) Intermediate Configuration	301	(5)
9.1.1 Micromechanical Motivation. Single-Crystal Plasticity	301	(1)
9.1.2 Kinematic Relationships Associated with the Intermediate Configuration	302	(3)
9.1.3 Deviatoric-Volumetric Multiplicative Split	305	(1)
9.2 J(2) Flow Theory at Finite Strains. A Model Problem	306	(5)
9.2.1 Formulation of the Governing Equations	307	(4)
9.3 Integration Algorithm for J(2) Flow Theory	311	(11)
9.3.1 Integration of the Flow Rule and Hardening Law	311	(3)
9.3.2 The Return-Mapping Algorithm	314	(6)
9.3.3 Exact Linearization of the Algorithm	320	(2)
9.4 Assessment of the Theory. Numerical Simulations	322	(14)
Chapter 10 Viscoelasticity	336	(39)
10.1 Motivation. One-Dimensional Rheological Models	337	(10)
10.1.1 Formulation of the Constitutive Model	338	(1)
10.1.2 Convolution Representation	339	(4)
10.1.3 Generalized Relaxation Models	343	(4)
10.2 Three-Dimensional Models: Formulation Restricted to Linearized Kinematics	347	(4)
10.2.1 Formulation of the Model	347	(2)
10.2.2 Thermodynamic Aspects. Dissipation	349	(2)
10.3 Integration Algorithms	351	(7)
10.3.1 Algorithmic Internal Variables and Finite-Element Database	351	(2)
10.3.2 One-Step, Unconditionally Stable and Second-Order Accurate Recurrence Formula	353	(2)
10.3.3 Linearization. Consistent Tangent Moduli	355	(3)
10.4 Finite Elasticity with Uncoupled Volume Response	358	(6)
10.4.1 Volumetric/Deviatoric Multiplicative Split	358	(1)
10.4.2 Stored-Energy Function and Stress Response	359	(2)
10.4.3 Elastic Tangent Moduli	361	(3)
10.5 A Class of Nonlinear, Viscoelastic, Constitutive Models	364	(3)
10.5.1 Formulation of the Nonlinear Viscoelastic Constitutive Model	364	(3)
10.6 Implementation of Integration Algorithms for Nonlinear Viscoelasticity	367	(8)
10.6.1 One-Step, Second-Order Accurate Recurrence Formula	367	(2)
10.6.2 Configuration Update Procedure	369	(1)
10.6.3 Consistent (Algorithmic) Tangent Moduli	370	(5)
References	375	(14)
Index	389