From the Peierls-Nabarro model to the equation of motion of the dislocation continuum

TL;DR

This paper derives a macroscopic equation for dislocation dynamics in crystals from a microscopic integro-differential model, demonstrating convergence and recovering classical laws like Orowan's law.

Contribution

It connects the Peierls-Nabarro model to a continuum dislocation equation, providing a rigorous derivation and analysis of the limit behavior for many dislocations.

Findings

Rescaled solutions converge to the dislocation continuum equation.

The model recovers Orowan's law relating dislocation velocity to stress.

Provides a bridge between atomistic models and macroscopic plasticity.

Abstract

We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian %This model describes the evolution of phase transitions associated to dislocations. 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 well known equation called by Head \cite{H} "the equation of motion of the dislocation continuum". The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan's law which states that dislocations move at a velocity proportional to the effective stress.