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 nonlocal energy Eγ modelling the interactions — at a typical length-scale of 1/γ — of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy Eγ in powers of 1/γ by Γ-convergence, in the limit γ →∞. While the zeroth-order term in the expansion, the Γ-limit of Eγ, captures the “bulk” profile of the density of dislocation walls in the pile-up domain, the first-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.