Scardia L.: Asymptotic behaviour of a pile-up of infinite walls of edge dislocations

Abstract. We consider a system of parallel straight edge dislocations and we analyse its as-ymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional hori-zontal positions xi> 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number n of walls tends to infinity, and charac-terise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter βn, corresponding to βn 1/n, 1/n βn 1, and βn 1, and the two critical regimes βn ∼ 1/n and βn ∼ 1. As a con-sequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter βn. 1