Calculus of Variations and Geometric Measure Theory

A. Garroni - P. van Meurs - M. A. Peletier - L. Scardia

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

created by vanmeurs on 08 Sep 2018
modified by scardia on 12 Jun 2020


Published Paper

Inserted: 8 sep 2018
Last Updated: 12 jun 2020

Journal: M3AS
Year: 2015

ArXiv: 1502.05805 PDF


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.