Homogenization of the Peierls-Nabarro model for dislocation dynamics

TL;DR

This paper studies the homogenization of a complex integro-differential model for dislocation dynamics, resulting in an effective non-local Hamilton-Jacobi equation that describes plastic behavior in materials.

Contribution

It introduces a homogenization result for a dislocation model involving anisotropic Lévy operators and periodic potentials, deriving a simplified effective equation.

Findings

Derivation of a non-local Hamilton-Jacobi limit equation

Extension of homogenization techniques to integro-differential dislocation models

Characterization of effective plastic laws for dislocation densities

Abstract

This paper is concerned with a result of homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic L\'{e}vy operator of order 1 and a potential depending periodically on $u/\ep$. The limit equation is a non-local Hamilton-Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane.