A Discontinuous Galerkin Method for an Incompatibility-Based Strain Gradient Plasticity Theory

Abstract

We consider a recent strain gradient plasticity theory based on incompatibility of plastic strain due to the nature of lattice distortion around a dislocation (J. Mech. Phys. Solids, 52, 2545–2568, 2004). The key features of this theory are an explicit treatment of the Burgers vector, a microforce balance that leads to a classical yield condition, and the inclusion of dissipation from plastic spin. The flow rule involves gradients of the plastic strain, and is therefore a partial differential equation. We apply recently-developed ideas on discontinuous Galerkin finite element methods to treat this higher-order nature of the yield condition, while retaining considerable flexibility in the mathematical space from which the plastic strain is drawn. In particular, despite the higher-order continuity apparent in the yield condition, it is possible to use plastic strain interpolations that are discontinuous across element edges. Two distinct approaches are outlined: the Interior Penalty Method and the Lifting Operator Method. The numerical implementation of the Interior Penalty Method is discussed, and a numerical example is presented.