Analysis of dislocation pile-ups using a dislocation-based continuum theory

The increasing demand for materials with well-defined microstructure, accompanied by the advancing miniaturization of devices, is the reason for the growing interest in physically motivated, dislocation-based continuum theories of plasticity. In recent years, various advanced continuum theories have been introduced, which are able to described the motion of straight and curved dislocation lines. The focus of this paper is the question of how to include fundamental properties of discrete dislocations during their motion and interaction in a continuum dislocation dynamics (CDD) theory. In our CDD model, we obtain elastic interaction stresses for the bundles of dislocations by a mean-field stress, which represents long-range stress components, and a short range corrective stress component, which represents the gradients of the local dislocation density. The attracting and repelling behavior of bundles of straight dislocations of the same and opposite sign are analyzed. Furthermore, considering different dislocation pile-up systems, we show that the CDD formulation can solve various fundamental problems of micro-plasticity. To obtain a mesh size independent formulation (which is a prerequisite for further application of the theory to more complex situations), we propose a discretization dependent scaling of the short range interaction stress. CDD results are compared to analytical solutions and benchmark data obtained from discrete dislocation simulations.