A Finite Element approach with patch projection for strain gradient plasticity formulations

Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust. For this goal a strain gradient Finite Element approach is suggested where elements with linear shape functions are applied in combination with a patch projection technique well known from error indication and adaptive mesh procedures. This approach is applied to a strain gradient crystal plasticity formulation where strain gradients are incorporated in terms of geometrically necessary dislocation densities. Simulation results of size dependent problems, including laminates in simple shear and a three dimensional contact problem, are presented and discussed to assess the performance of the suggested approach.