Consistent predictors and the solution of the piecewise holonomic incremental problem in elasto-plasticity

Abstract

Algorithms for the solution of the classical incremental problems in static elasto-plasticity are an important feature in large-scale finite element programs for nonlinear stress analysis. The algorithms are based on the Newton-Raphson method, where a linearized iteration is used. In each iteration a predictor step is employed to compute a new, improved estimate of the kinematic variables, and a corrector step recomputes the stresses and the internal forces associated with the improved solution.Backward difference methods have been shown empirically to provide an effective and a stable means of implementing the corrector step. Very recently it has also been shown that so-called consistent predictors should be used together with the backward difference correctors in order to ensure the quadratic rate of convergence normally expected with the Newton-Raphson method. Very considerable success has been achieved with the use of consistent predictors, both in terms of stability and rate of convergence.In this paper we present a rational formulation of the incremental problem. It is shown that the adoption of a backward difference corrector is equivalent to the reformulation of the problem as a piecewise holonomic problem. This permits the problem to be written as an unconstrained convex nonlinear programming problem. It is further shown that the consistent predictor can easily be identified within this formulation, and as a consequence it can be interpreted as a rational step in the algorithm for the solution of the nonlinear programming problem.