Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model

TL;DR

This paper derives the 1-D Groma-Balogh equations from the Peierls-Nabarro model by analyzing dislocation dynamics and their macroscopic density evolution, including dislocation collisions.

Contribution

It provides a formal derivation of the Groma-Balogh equations from a microscopic dislocation model, accounting for dislocation orientations and collisions.

Findings

Rescaled solutions converge to a nonlinear integro-differential equation.

Derivation completes previous work by including dislocation orientation.

Handles dislocation collisions in the derivation.

Abstract

We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations \cite{groma}, a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of \cite{patsan}. The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them.