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 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 zero-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.

The gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimisingmovement schemes when the time step converges to 0. The study presented here is a first step towards a generalization of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.