variational formulations and numerical analysis of some problems in small strain elastoplasticity

Description

In this thesis we study the mathematical structure and numerical approximation of two boundary-value problems in small strain elastoplasticity. The first problem, which we call the incremental holonomic problem, is based on a consistent incremental holonomic constitutive law, which in turn derives from the notion of extremal paths in stress and strain space as originally proposed by PONTER & MARTIN (1972); the second problem which we study is the classical rate problem. We show that both problems can be formulated as variational inequalities, with internal variables being included explicitly in the formulation. Corresponding minimisation problems follow naturally from standard results in convex analysis. Perturbed minimisation problems are introduced, in which the original functionals J are replaced by perturbed functionals J e: which depend on a parameter e: > 0 • In the rate problem e: is a penalty parameter; here J e: differs from J by a term e: -l j( •) where j( •) is a penalty functional which allows the non-negativity constraint on the plastic multipliers to be removed. In the incremental holonomic problem the non-differentiable plastic work function wP ( •) is regularised, and replaced by a differentiable function WP(~) • e: In both problems the perturbed functionals form the basis for fintte element approximations, the error in the approximate solutions now depending on both mesh size and on the magnitude of e: • Numerical algorithms are proposed, and implemented in two computer programs. On the basis of preliminary numerical experiments we conclude that the penalty-rate formulation is useful in a limited class of elastic-plastic problems, and that the incremental holonomic formulation has exceptional potential, without any apparent limitations.