Upscaling of screw dislocations with increasing tangential strain

TL;DR

This paper investigates how screw dislocations in materials behave under increasing tangential external strain, deriving a continuum limit that describes their distribution and boundary concentration as strain grows.

Contribution

It characterizes the gamma-limit of the renormalized energy for dislocations under external strain, revealing their boundary concentration and distribution in the continuum limit.

Findings

Dislocations concentrate at the boundary as external strain increases.

The gamma-limit describes the distribution of dislocations in the continuum.

Provides a discrete-to-continuum transition analysis for dislocation systems.

Abstract

The upscaling of a system of screw dislocations in a material subject to an external strain is studied. The $\Gamma$-limit of a suitable rescaling of the renormalized energy is characterized in the space of probability measures. This corresponds to a discrete-to-continuum limit of the dislocations, which, as a byproduct, provides information on their distribution when the circulation of the tangential component of the external strain becomes larger and larger. In particular, dislocations are shown to concentrate at the boundary of the material and to distribute as the limiting external strain.