Asymptotic analysis of a pile-up of edge dislocation

The idealised problem of a pile-up of dislocation walls (that is, of planes each containing an infinite number of parallel and identical dislocations) was presented by Roy et al. (Mater. Sci. Eng. A 486:653-661, 2008) as a proto-type for understanding the importance of discrete dislocation interactions in dislocation-based plasticity models. They noted that analytic solutions for the dislocation wall density are available for a pile-up of screw dislocation walls, but that numerical methods seem to be necessary for investigating edge dislocation walls. In this paper, we use the techniques of discrete-to-continuum asymptotic analysis to obtain a detailed description of a pile-up of edge dislocation walls. To leading order, we find that the dislocation wall density is governed by a simple differential equation and that boundary layers are present at both ends of the pile-up