Dynamic Peierls-Nabarro equations for elastically isotropic crystals

TL;DR

This paper derives dynamic Peierls-Nabarro equations for dislocation cores in isotropic elastic media, incorporating non-local kernels and ensuring consistency with established static models at constant velocities.

Contribution

It introduces a dynamic, integro-differential formulation of Peierls-Nabarro equations for screw and edge dislocations in isotropic crystals, extending static models to finite velocities.

Findings

Derived equations include non-local kernels in space and time.

Equations reduce to Weertman's static models at constant velocities.

Differences from previous models include an additional instantaneous term for screw dislocations.

Abstract

The dynamic generalization of the Peierls-Nabarro equation for dislocations cores in an isotropic elastic medium is derived for screw, and edge dislocations of the `glide' and `climb' type, by means of Mura's eigenstrains method. These equations are of the integro-differential type and feature a non-local kernel in space and time. The equation for the screw differs by an instantaneous term from a previous attempt by Eshelby. Those for both types of edges involve in addition an unusual convolution with the second spatial derivative of the displacement jump. As a check, it is shown that these equations correctly reduce, in the stationary limit and for all three types of dislocations, to Weertman's equations that extend the static Peierls-Nabarro model to finite constant velocities.