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 $E_{\gamma }$ modelling the interactions$-$at a typical length-scale of $1\mathrm{/}\gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_{\gamma }$ in powers of $1\mathrm{/}\gamma$ by $\mathrm{\Gamma }$-convergence, in the limit $\gamma \to \mathrm{\infty }$. While the zero-order term in the expansion, the $\mathrm{\Gamma }$-limit of $E_{\gamma }$, 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.