Derivation of Orowan's law from the Peierls-Nabarro model

TL;DR

This paper derives Orowan's law from the Peierls-Nabarro model, showing that dislocation velocity is proportional to effective stress, leading to a relation between plastic strain rate, dislocation density, and stress.

Contribution

It provides a rigorous derivation of Orowan's law from the Peierls-Nabarro model in a one-dimensional setting.

Findings

Dislocations move at velocity proportional to effective stress

Orowan's law is validated in the model

Dislocation density influences plastic strain rate

Abstract

In this paper we consider the time dependent Peierls-Nabarro model in dimension one. This model is a semi-linear integro-differential equation associated to the half Laplacian. This model describes the evolution of phase transitions associated to dislocations. At large scale with well separated dislocations, we show that the dislocations move at a velocity proportional to the effective stress. This implies Orowan's law which claims that the plastic strain velocity is proportional to the product of the density of dislocations by the effective stress.