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

Abstract

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 [Formula: see text] modelling the interactions — at a typical length-scale of [Formula: see text] — of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy [Formula: see text] in powers of [Formula: see text] by [Formula: see text]-convergence, in the limit [Formula: see text]. While the zeroth-order term in the expansion, the [Formula: see text]-limit of [Formula: see text], 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.