The discrete dislocation dynamics of multiple dislocation loops

TL;DR

This paper models the evolution of multiple dislocation loops in crystalline structures using a nonlocal reaction-diffusion equation, showing that as a small parameter tends to zero, the dislocations evolve independently according to mean curvature flow.

Contribution

It connects the Peierls-Nabarro dislocation model to fractional Allen-Cahn equations and demonstrates the independent evolution of multiple dislocation interfaces in the limit.

Findings

Dislocation dynamics are governed by a fractional Allen-Cahn equation after rescaling.

Multiple dislocation interfaces evolve independently as mean curvature flows.

The model captures the asymptotic behavior of dislocation loops as the phase parameter vanishes.

Abstract

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls-Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in $\mathbb{R}^n$, $n \geq 2$. After suitably rescaling the equation with a small phase parameter $\varepsilon>0$, the rescaled solution solves a fractional Allen-Cahn equation. We show that, as $\varepsilon \to 0$, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.