Asymptotic behaviour of a pile-up of infinite walls of edge dislocations

TL;DR

This paper analyzes the asymptotic behavior of a system of infinite dislocation walls, characterizing the limiting energy and states as the number of walls grows, revealing diverse regimes and implications for upscaled models.

Contribution

It provides a comprehensive Gamma-convergence analysis of dislocation wall systems, identifying five regimes based on a key parameter and connecting to existing upscaled models.

Findings

Five distinct asymptotic regimes identified

Characterization of stationary states in each regime

Existing models emerge as special cases

Abstract

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x=0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number n of walls tends to infinity, and characterise this behaviour in terms of Gamma-convergence. There are five different cases, depending on the asymptotic behaviour of a single dimensionless parameter. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter.