Plasticity

by Han, Weimin; Reddy, B. Dayanand

• ISBN13: 9780387987040
• ISBN10: 0387987045
• Format: Hardcover
• Copyright: 1999-04-01
• Publisher: Springer Verlag

Summary

The theory of elastoplastic media is now a mature branch of solid and structural mechanics, having experienced significant development during the latter half of this century. This monograph focuses on theoretical aspects of the small-strain theory of hardening elastoplasticity. It is intended to provide a reasonably comprehensive and unified treatment of the mathematical theory and numerical analysis, exploiting in particular the great advantages to be gained by placing the theory in a convex analytic context. The book is divided into three parts. The first part provides a detailed introduction to plasticity, in which the mechanics of elastoplastic behavior is emphasized. The second part is taken up with mathematical analysis of the elastoplasticity problem. The third part is devoted to error analysis of various semi-discrete and fully discrete approximations for variational formulations of the elastoplasticity. The work is intended for a wide audience: this would include specialists in plasticity who wish to know more about the mathematical theory, as well as those with a background in the mathematical sciences who seek a self-contained account of the mechanics and mathematics of plasticity theory.

Table of Contents

Continuum Mechanics and Elastoplasticity Theory	1	(95)
Preliminaries	3	(12)
Introduction	3	(2)
Some Historical Remarks	5	(3)
Notation	8	(7)
Continuum Mechanics and Linear Elasticity	15	(26)
Kinematics	16	(7)
Balance of Momentum; Stress	23	(5)
Linearly Elastic Materials	28	(2)
Isotropic Elasticity	30	(3)
A Thermodynamic Framework for Elasticity	33	(4)
Initial-Boundary and Boundary Value Problems for Linear Elasticity	37	(1)
Thermodynamics with Internal Variables	38	(3)
Elastoplastic Media	41	(30)
Physical Background and Motivation	41	(7)
Three-Dimensional Elastoplastic Behavior	48	(13)
Examples of Yield Criteria	61	(5)
Hardening Laws	66	(5)
The Plastic Flow Law in a Convex-Analytic Setting	71	(24)
Some Results from Convex Analysis	72	(11)
Basic Plastic Flow Relations of Elastoplasticity	83	(12)
II The Variational Problems of Elastoplasticity	95	(108)
Results from Functional Analysis and Function Spaces	97	(28)
Results from Functional Analysis	98	(9)
Function Spaces	107	(18)
The Spaces Cm(Ω), Cm(Ω), and Lp(Ω)	108	(4)
Sobolev Spaces	112	(8)
Spaces of Vector-Values Functions	120	(5)
Variational Equations and Inequalities	125	(26)
Variational Formulation of Elliptic Boundary Value Problems	125	(12)
Elliptic Variational Inequalities	137	(9)
Parabolic Variational Inequalities	146	(5)
The Primal Variational Problem of Elastoplasticity	151	(26)
The Primal Variational Problem	151	(7)
Qualitative Analysis of an Abstract Problem	158	(9)
Analysis of the Primal Problem	167	(5)
Stability Analysis	172	(5)
The Dual Variational Problem of Elastoplasticity	177	(26)
The Dual Variational Problem	178	(4)
Analysis of the Stress Problem	182	(13)
Analysis of the Dual Problem	195	(5)
Rate Form of Stress-Strain Relation	200	(3)
III Numerical Analysis of the Variational Problems	203	(152)
Introduction to Finite Element Analysis	205	(18)
Basics of the Finite Element Method	207	(3)
Affine Families of Finite Elements	210	(4)
Local Interpolation Error Estimates	214	(6)
Global Interpolation Error Estimates	220	(3)
Approximation of Variational Problems	223	(14)
Approximation of Elliptic Variational Equations	224	(3)
Approximation of EVI of the First Kind	227	(2)
Approximation of EVI of the Second Kind	229	(6)
Approximation of Parabolic Variational Inequalities	235	(2)
Approximations of the Abstract Problem	237	(34)
Spatially Discrete Approximations	238	(2)
Time-Discrete Approximations	240	(6)
Fully Discrete Approximations	246	(7)
Convergence Under Minimal Regularity	253	(18)
Numerical Analysis of the Primal Problem	271	(48)
Error Analysis of Discrete Approximations of the Primal Problem	272	(9)
Solution Algoithms	281	(12)
Convergence Analysis of the Solution Algorithms	293	(9)
Regularization Technique and A Posteriori Error Analysis	302	(8)
Fully Discrete Schemes with Numerical Integration	310	(9)
Numerical Analysis of the Dual Problem	319	(36)
Time-Discrete Approximations of the Stress Problem	321	(6)
Time-Discrete Approximations of the Dual Problem	327	(4)
Fully Discrete Approximations of the Dual Problem	331	(13)
Predictor-Corrector Iterations	344	(9)
Computation of the Closest Point Projections	353	(2)
Bibliography	355	(10)
Index	365