Gradient dependent plasticity and the finite difference method

Abstract

Solutions to complex plasticity problems by numerical methods normally require discretisation of the geometry and approximation of the governing equations. In classical plasticity theories the material post-peak strength depends on a strain softening parameter in terms of an equivalent plastic strain measure. When modelling strain-softening, classical (local) continuum theories suffer from mesh dependency problems because of the lack of an internal length scale. As a result of this deficiency, the critical condition for localisation coincides with the condition for the loss of ellipticity of the governing differential equations. This mathematical ill-conditioning can be prevented by using higher order (non-local) continuum theories. Among the theories, the gradient-dependent plasticity theory, has been successfully implemented into a few finite element algorithms for the stress analysis of strain-softening materials. The addition of a gradient term in the material failure criterion preserves ellipticity condition and consequently removes mesh-dependency after the strain-softening regime has been entered. In this paper, an implementation of the gradient-dependent plasticity theory to general finite difference codes is discussed.