Skip to main content
main-content

Über dieses Buch

This book focuses on the theoretical aspects of small strain theory of elastoplasticity with hardening assumptions. It provides a comprehensive and unified treatment of the mathematical theory and numerical analysis. It is divided into three parts, with the first part providing a detailed introduction to plasticity, the second part covering the mathematical analysis of the elasticity problem, and the third part devoted to error analysis of various semi-discrete and fully discrete approximations for variational formulations of the elastoplasticity. This revised and expanded edition includes material on single-crystal and strain-gradient plasticity. In addition, the entire book has been revised to make it more accessible to readers who are actively involved in computations but less so in numerical analysis. Reviews of earlier edition: “The authors have written an excellent book which can be recommended for 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.” (ZAMM, 2002) “In summary, the book represents an impressive comprehensive overview of the mathematical approach to the theory and numerics of plasticity. Scientists as well as lecturers and graduate students will find the book very useful as a reference for research or for preparing courses in this field.” (Technische Mechanik) "The book is professionally written and will be a useful reference to researchers and students interested in mathematical and numerical problems of plasticity. It represents a major contribution in the area of continuum mechanics and numerical analysis." (Math Reviews)

Inhaltsverzeichnis

Frontmatter

Continuum Mechanics and Elastoplasticity Theory

Frontmatter

1. Preliminaries

Abstract
The theory of elastoplastic media is now a mature branch of solid and structural mechanics, having experienced significant development during the latter half of the twentieth century. In particular, the classical theory, which deals with small-strain elastoplasticity problems, has a firm mathematical basis, and from this basis further developments, both mathematical and computational, have evolved. Small-strain elastoplasticity is well understood, and the understanding of its governing equations can be said to be almost complete. Likewise, theoretical, computational, and algorithmic work on approximations in the spatial and time domains are at a stage at which approximations of desired accuracy can be achieved with confidence.
Weimin Han, B. Daya Reddy

2. Continuum Mechanics and Linearized Elasticity

Abstract
We will be concerned with bodies that at the macroscopic level may be regarded as being composed of material that is continuously distributed. By this it is meant, first, that such a body occupies a region of three-dimensional space that may be identified with R3. The region occupied by the body will of course vary with time as the body deforms.
Weimin Han, B. Daya Reddy

3. Elastoplastic Media

Abstract
In this chapter we begin to look at the features that characterize elastoplastic materials and at how these physical features are translated into a mathematical theory. The theory has grown steadily during the last century, with the impetus for development coming alternately from physical understanding of such materials and from insight into how the physical attributes might be modeled mathematically. We will eventually arrive, towards the end of this chapter, at a theory that is now regarded as classical and that incorporates all the main features of elastoplasticity. This theory may be further generalized, and placed in a unifying framework, if the ideas and techniques of convex analysis are employed. While the entire theory could easily have been developed ab initio in such a framework, we have chosen instead to focus first on giving a clear outline of the main features of the mathematical theory, without introducing any sophisticated ideas from convex analysis. In this way, we hope that the connection between physical behavior and its mathematical idealization may be more readily seen. Once such a theory is in place, the business of abstraction and generalization, using the tools of convex analysis, may begin.
Weimin Han, B. Daya Reddy

4. The Plastic Flow Law in a Convex-Analytic Setting

Abstract
The previous chapter has been devoted to the presentation of the basic theory of elastoplasticity in a fairly classical manner. While this theory is adequate in its own right, it is highly advantageous from the point of view of carrying out a mathematical and numerical analysis of the ensuing initial–boundary value problem to recast the constitutive theory in a convex-analytic framework. That will be the aim of this chapter.
Weimin Han, B. Daya Reddy

The Variational Problems of Elastoplasticity

Frontmatter

5. Basics of Functional Analysis and Function Spaces

Abstract
The focus of Part II of this monograph will be, firstly, on the construction of variational formulations of the initial–boundary value problem of elastoplasticity, and, secondly, on the well-posedness of these variational problems. There are a number of tools from functional analysis that are called upon in the course of such analyses, and naturally the variational problems themselves are posed on particular function spaces. For these reasons we begin Part II by reviewing, in this chapter, those results from functional analysis that are pertinent to subsequent developments. We also collect in one place a number of results pertaining to function spaces, especially Sobolev spaces.
Weimin Han, B. Daya Reddy

6. Variational Equations and Inequalities

Abstract
In this chapter we review some standard results for boundary value and initial–boundary value problems, paying particular attention to weak or variational formulations. The first section will be concerned with elliptic variational equations, and this will be followed by a review of some material on elliptic variational inequalities. Because the variational form taken by elastoplastic problems resembles that of parabolic variational inequalities, we also include some material on this class of problems.
Weimin Han, B. Daya Reddy

7. The Primal Variational Problem of Elastoplasticity

Abstract
The initial–boundary value problem of elastoplasticity may be formulated in two alternative ways, depending on which of the two forms of the plastic flow law (see Section 4.2) is adopted.We describe as the primal problem the version that takes as its point of departure the flow law that uses the dissipation function, for example as in (4.38), while the dual problem is formulated using the form of the flow law that makes use of the yield function, as in (4.35) for example.
Weimin Han, B. Daya Reddy

8. The Dual Variational Problem of Classical Elastoplasticity

Abstract
This chapter has a purpose parallel to that of Chapter 7, in that the dual variational problem of elastoplasticity will be studied in detail. This problem takes as its point of departure the flow law in the form (4.35), that is, the statement of the flow law that makes use of the yield surface and the normality law.
Weimin Han, B. Daya Reddy

Numerical Analysis of the Variational Problems

Frontmatter

9. Introduction to Finite Element Analysis

Abstract
In the previous two chapters we have formulated and analyzed the primal and dual variational formulations of the elastoplasticity problem. Later on, we will study various numerical methods to solve the variational problems. In all the numerical methods to be considered, we will use finite differences to approximate the time derivative and use the finite element method to discretize the spatial variables. The finite elemfent method is widely used for solving boundary value problems of partial differential equations arising in physics and engineering, especially solid mechanics. The method is derived from discretizing the weak formulation of a boundary value problem. The analysis of the finite element method is closely related to that of the weak formulation of the boundary value problem.
Weimin Han, B. Daya Reddy

10. Approximation of Variational Problems

Abstract
In this chapter we consider the approximation by the finite element method of variational equations and inequalities. In Chapter 6 we have reviewed some standard results for the well-posedness of variational equations and inequalities.
Weimin Han, B. Daya Reddy

11. Approximations of the Abstract Problem

Abstract
As a prelude to the error analysis of various numerical schemes for solving the primal variational problem, we will first give a convergence analysis and derive error estimates for numerical solutions of the abstract problem, introduced in Section 6.4, which includes the primal variational problem as a special case. In the next chapter, we will apply the results presented here to perform an error analysis for various numerical approximation schemes for solving the primal problem. For convenience, let us recall the abstract problem.
Weimin Han, B. Daya Reddy

12. Numerical Analysis of the Primal Problem

Abstract
In this chapter we consider numerical approximations for the primal variational problem of elastoplasticity. We start with the derivation of error estimates for various numerical schemes approximating the solution of the primal variational problem by applying the results for the abstract variational problem proved in the last chapter. We also discuss the convergence property for various schemes under the basic solution regularity condition.
Weimin Han, B. Daya Reddy

13. Numerical Analysis of the Dual Problem

Abstract
In this last chapter we present some results on the numerical analysis for the dual formulation of the elastoplasticity problem. For various numerical approximation schemes, we will derive error estimates under sufficient regularity assumptions on the solution and prove convergence assuming basic regularity of the solution. In Section 13.1 we study a family of generalized midpoint schemes for the stress problem. For the dual problem, we analyze several time-discrete schemes in Section 13.2 and fully discrete schemes in Section 13.3.
Weimin Han, B. Daya Reddy

Backmatter

Weitere Informationen

Premium Partner

    Bildnachweise