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).