An efficient and minimalist scheme for continuum dislocation dynamics

Continuum dislocation dynamics methods have received considerable interests for simulating dislocation microstructures at the meso-scale, serving as potential tools for bridging the gap between micro-and macro-scale models for the plastic behaviours of crystalline materials. Recently, an exact evolution equation for the "all-dislocation"density that represents dislocation quantities over both space and dislocation-character domains has been developed by one of the present authors. The "all-dislocation"representation is superior to representations based on the Nye tensor or geometrically necessary dislocations (GND), since the statistically stored dislocation (SSD) contents will be preserved. In this paper, a numerical scheme is presented to solve the dynamics of the "all-dislocation"density efficiently, with long-range elastic interaction between dislocations accounted for via Mura's formula after singularity removal. The proposed simulation scheme is demonstrated by simulation examples in the multi-scale hierarchy, from intensive microstructures of individual dislocations including Frank-Read source and Orowan looping, to extensive microstructures of coarse-grained dislocation densities in single-and multi-slip in the face-centred cubic crystal structure.