Gradient plasticity model: Formulation and implementation using vector finite elements.

Abstract

Multi-scale material models for determining onset of ductile fracture in polycrystalline metals, and the deformation behavior of micro-systems with dimensions on the scale of their inherent microstructure, have generated significant interest within the plasticity community. Strain gradient plasticity models, in particular, can capture physical length scale effects associated with fracture and micro-systems deformation behavior. The formulation, variational equations and implementation of such a plasticity model with strict anti-plane shear kinematics is presented. These kinematics allow a simpler analysis of the plasticity model and its finite element implementation. The strain gradient in the plasticity model stems from the curl of the elastic part of the deformation gradient representing the elastic lattice curvature due to geometrically necessary dislocations (GNDs) at grain or sub-grain cell boundaries. This plasticity model leads to a partial differential equation, written in variational form for finite element implementation, resulting in coupled governing equations and an additional field to be solved for. The implementation uses vector finite elements, originally used in solving electromagnetic governing equations, to model the curl of the deformation gradient. Vector elements assign degrees of freedom to the edges rather than to the nodes and help in overcoming several problems associated with standard node based elements such as the occurrence of spurious solutions and difficulties in enforcing boundary conditions. A convergence study is carried out for the purpose of understanding how well these methods represent the curl terms and for comparison with standard finite element methods.