A mesoscale continuum approach of dislocation dynamics and the approximation by a Runge-Kutta discontinuous Galerkin method

We consider a mesoscale continuum model for the evolution of dislocation density in small-strain crystal plasticity. The model is based on the continuum dislocation dynamics theory and extended by a formulation for impenetrable grain boundaries. We introduce a fully coupled numerical method combining a conforming finite element approximation of elasto-plasticity with an implicit Runge-Kutta discontinuous Galerkin discretization of the dislocation microstructure which allows for 3d computations including multiple slip systems and dislocation interaction. In addition, a numerical representation of grain boundaries impenetrable to dislocation flux is considered within this framework. The formulation is applied to a tricrystal focusing on the analysis of dislocation stress interaction between different grains. The results are compared to discrete dislocation dynamics data from the literature.