Confinement of dislocations inside a crystal with a prescribed external strain

TL;DR

This paper investigates how specific boundary conditions and external strains can confine multiple screw dislocations within a crystal, using asymptotic analysis and an iterative scheme to understand their energetic favorability and confinement behavior.

Contribution

It introduces a boundary condition framework that causes dislocation confinement and analyzes the asymptotic behavior as core size vanishes, laying groundwork for large-scale dislocation limits.

Findings

External strain with circulation n times lattice spacing confines n dislocations.

Imposing boundary integral conditions results in energetic favorability for dislocation confinement.

Asymptotic analysis as core size approaches zero supports the confinement mechanism.

Abstract

A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied in the framework of linear elasticity. Imposing a suitable boundary condition for the strain, namely requesting the non-vanishing of its boundary integral, results in a confinement effect. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The result is obtained by formulating the problem via the core radius approach and by studying the asymptotics as the core size vanishes. An iterative scheme is devised to prove the main result. This work sets the basis for studying the upscaling problem, i.e., the limit as $n\to\infty$, which is treated in [17].