A Convergence Analysis of a P-Version Finite Element Method for One-Dimensional Elastoplasticity Problem with Constitutive Laws Based on the Gauge Function Method

Abstract

In this paper, we will give a p-version finite element method for the one-dimensional nonlinear elastoplasticity problem. A family of admissible constitutive laws based on the so-called gauge function method is introduced first, and then a p-version semi-discretization scheme is presented. We show that the existence and uniqueness of the solution for the semi-discrete problem can be guaranteed by using some special properties of the constitutive law. Finally, we show that as the polynomial degree of the elements ${\bf p} \to \infty $ oo the solution of the semi-discrete problem will converge to the solution of the continuous problem. The p-version convergence analysis given here is only for one-dimensional problems with a linear hardening constitutive law hypothesis. However, it also gives us a very useful idea for the study of the hp-version finite element method for the one-dimensional problem with more general constitutive laws or even two-dimensional problems.