Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy Ec modelling the interactions—at a typical length scale of 1/c— of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy by Γ-convergence in the limit c→∞. While the first term in the expansion, the Γ-limit of Ec, captures the ‘bulk’ profile of the density of dislocation walls in the pile-up domain, the next-order term in the expansion is a ‘boundary-layer’ energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.