A continuum model for dislocation dynamics incorporating Frank-Read sources and Hall-Petch relation in two dimensions

One of the main targets in the development of dislocation based continuum crystal plasticity theories is to establish continuum constitutive relations which approximately summarize the underlying discrete dislocation dynamics (DDD). However, rigorously transiting from discrete to continuum in describing the evolution of dislocation system is extremely challenging for complex networks of curved dislocations and their interactions at multiple length scales. To address this difficulty, a coarse-grained disregistry function (CGDF) was proposed to represent the continuous distributions of curved dislocations. In this paper, we present a dislocation based continuum model for crystal plasticity incorporating the Frank-Read sources, which serves as a crucial step towards systematically building a three-dimensional dislocation based continuum plasticity theory. The continuum model is derived accurately from the DDD model, and is validated by comparisons of the results with theoretical predictions and DDD simulations conducted under the same conditions. Furthermore by considering dislocation loop pileups within a rectangular grain, we derive analytical formulas which generalize the traditional Hall-Petch relation into two dimensions without any adjustable parameters. It is shown that the yield stress of a rectangular grain depends not only on the grain size, but also on the grain aspect ratio whose exact form is associated with the harmonic mean of the length and width of the rectangle. The derived formulas of the yield stress are shown in excellent agreement with the results by our continuum model and DDD simulations.