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

TL;DR

This paper uses boundary-layer analysis and $ ext{Gamma}$-convergence to understand how dislocation walls behave near obstacles, providing a detailed description of the density profile close to the lock.

Contribution

It introduces a first-order approximation of the energy for dislocation pile-ups, capturing boundary-layer effects near obstacles, advancing the theoretical understanding of dislocation behavior.

Findings

Zero-order term describes bulk dislocation density profile.

First-order term captures boundary-layer profile near the lock.

Provides a rigorous mathematical framework for dislocation-obstacle interactions.

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