The object of the chapter In the absence of plastic strain, the problem of brittle fracture by extension of cracks can be presented in a thermodynamic framework, analogous to that of elastoplasticity. This means that the fracture criterion (or the criterion of crack propagation) replaces the plasticity criterion. One important notion is the notion of mechanical field singularity (displacement, stresses).

Introduction and elementary notions

We are interested in the problem of fracture, a phenomenon that occurs, more or less violently, under monotonic loading (whereas fatigue concerns cyclical loading). More specifically, we are interested in the problem of cracking, that is, the progagation of macroscopic cracks (of size of the order of one millimetre), whereas the beginning of cracking belongs to the microscopic and to the metal analyses which will not be examined here. (Microscopic cracks are one cause of damage – see Chapter 10.) The aim of this study is to arrive at a formulation of the crack-propagation laws, based upon fracture criteria and the definition of the conditions that may insure resistance to this fracture. We are certainly aware of the interest that such a subject implies for industry; it suffices to think about aeronautical engines and nuclear installations. Actually, our main interest is brittle fracture, that is, the kind that occurs without considerable plastic strain (i.e. the separation mechanism of crystallographic facets through cleavage), whereas ductile rupture is produced by different mechanisms accompanied by great plastic strains).

The object of the chapter This chapter provides a short introduction to the notion of homogenization (i.e., determining the parameters of a unique fictitious material that ‘best’ represents the real heterogeneous material or composite) and then, at some length, its application to the case where all or some of the constitutive components have an elastoplastic behaviour. The essential notions are those of representative volume element, procedure of localization, and the representation of some microscopic effects by means of internal variables. Composites with unidirectional fibres, polycrystals and cracked media provide examples of application.

Notion of homogenization

Homogenization is the modelling of a heterogeneous medium by means of a unique continuous medium. A heterogeneous medium is a medium of which material properties (e.g., elasticity coefficients) vary pointwise in a continuous or discontinuous manner, in a periodic or nonperiodic way, deterministically or randomly. While, obviously, homogenization is a modelling technique that applies to all fields of macroscopic physics governed by nice partial differential equations, we focus more particularly on the mechanics of deformable bodies with a special emphasis on composite materials (as used in aeronautics) and polycrystals (representing many alloys.) Most of the composite materials developed during the past three decades present a brittle, rather than ductile behaviour. As emphasized in Chapter 7, the elastic behaviour then prevails and there is no need to consider the homogenization of an dastoplastic behaviour.

The object of the chapter The technical difficulties faced in solving plasticity problems, which are free-boundary problems, are such that sooner or later one has to use a numerical implementation. While works fully devoted to numerical methods in solid mechanics give general solution techniques, here we focus on the specificity of the incremental or evolutionary nature of elastoplasticity problems and on Moreau's implicit scheme which is particularly well suited to this.

Introduction

Save for a few exceptions (see Appendix 3) the analytic solution of a problem of elastoplasticity is a formidable task since it involves a free boundary which is none other than the border between elastic and plastic domains, in general an unknown in the problem. In addition, by the very nature of elastoplasticity, the corresponding problems are nonlinear and the nature of certain plasticity criteria does not improve the situation. The relevant question at this point is: what is the quasi-static evolution of an elastoplastic structural member? The very nature of elastoplasticity and the corresponding incremental formulation are well suited to the study of general features of such a mechanical behaviour (see Chapters 4 to 6) and, indeed, via both spatial and temporal discretizations, to a numerical solution for real problems that involve complex geometries, somewhat elaborated plasticity criteria, and complex loading paths (including both loading and unloading). The most appropriate method for the spatial problem obviously is the one of finite elements (for short FEM).

The present book is an outgrowth of my lecture notes for a graduate course on ‘Plasticity and fracture’ delivered for the past five years to students in Theoretical Mechanics and Applied Mathematics at the Pierre-et-Marie Curie University in Paris. It also corresponds to notes prepared for an intensive course in modern plasticity to be included in a European graduate curriculum in Mechanics. It bears the imprint of a theoretician, but it should be of equal interest to practitioners willing to make an effort on the mathematical side. The prerequisites are standard and include classical (undergraduate) courses in applied analysis and Cartesian tensors, a basic course in continuum mechanics (elasticity and fluid mechanics), and some knowledge of the strength of materials (for exercises with a practical touch), of numerical methods, and of elementary thermodynamics. More sophisticated thermodynamics and elements of convex analysis, needed for a good understanding of the contents of the book, are recalled in Appendices.

The book deals specifically with what has become known as the mathematical theory of plasticity and fracture as (unduly) opposed to the physical theory of these fields. The first expression is reserved for qualifying the macroscopic, phenomenological approach which proposes equations abstracted from generally accepted experimental facts, studies the adequacy of the consequences drawn from these equations to those facts, cares for the mathematical soundness of these equations (do they have nice properties?), and then, with some confidence, provides useful tools to designers and engineers.

The object of the chapter In this chapter we are interested in the ruin of perfectly-plastic–elastic structures and we introduce the notions of limit load and of maximum admissible load, the determination of which constitutes the essential object of every engineer's office computations. We shall only attempt an introduction to this type of calculation, which will be illustrated by two examples. Certain minimum principles apply to velocities and to stresses. The static and dynamic methods in the determination of the maximum admissible load are only given in a rough draft.

The notion of limit load

The object of our attention is the notion of limit load and the collapse of perfectly plastic structures under unrestrained plastic strains. What do we mean by that? As certain deformable structures evolve, we observe that the elastoplastic response is produced in three stages. The first phase is elastic, the material being elastic everywhere. This phase lasts until the appearance of the first yielding. But the fact that the criterion of plasticity f(σ) = 0 is reached at one point does not necessarily mean that there is collapse. If the strain rate is still controlled, the plastic strain rate is not unlimited, since it is expressed in terms of έ (Section 4.2). We say then that the plastic strain is still controlled. The second phase of the response corresponds to the appearance and the extension of one or more regions of the structure, generally called plastic zones; in these, the plasticity criterion is satisfied at all the points.