Incompatible 4-node element for gradient-dependent plasticity

Abstract

In gradient-dependent plasticity theory, the yield strength depends on the Laplacian of an equivalent plastic strain measure (hardening parameter), and the consistency condition results in a differential equation with respect to the plastic multiplier. The plastic multiplier is then discretized on the mesh, in addition to the usual discretization of the displacements, and the consistency condition is solved simultaneously with the equilibrium equations. The notorious disadvantage is that the plastic multiplier requires a Hermitian interpolation which has four degrees of freedom at each node. However, in this article, an incompatible (trigonometric) interpolation is proposed for the plastic multiplier. This incompatible interpolation uses only the function values of each node, but both the function and first-order derivatives are continuous across element boundaries. It greatly reduces the degrees of freedom for a problem, and is shown through numerical examples on localization to give good results.