Internal variable formulations of elastic-plastic dynamic problems

Abstract

The paper reviews work that has been carried out in recent years in the Centre for Research in Computational and Applied Mechanics (CERECAM) at the University of Cape Town on the compact internal variable formulation of the problem of an elastic-plastic body subject to incremental loading. The fundamental problem is expressed as a convex nonlinear programming problem. The simplest two step algorithm for the solution of this programming problem is shown to be the standard Newton-Raphson algorithm, and the formulation permits a discussion of the convergence of the iterative solution procedure. It is shown that this static formulation is readily extended to include inertia and linear damping terms, leading to an identical basic formulation in which the stiffness matrix and the residual are redefined for the dynamic problem. The static problem, and hence any software written for its solution, can thus be simply modified to include dynamic effects. The essential analogy between the static and dynamic problems was first presented as a master's thesis (R.D. Isted, 1988); subsequently there have been a number of additional contributions to the basic formulation for static problems, in which the internal variable framework was extended and clarified. Further contributions have also been made in respect of the dynamic problem, and these developments are summarised. Some of this later work has been directed at improvements in the implicit solution algorithms. These aspects will not be discussed in detail, however, the main purpose of this review is to demonstrate the formal relationship between the static and dynamic problems.