Local density approximation for the energy functional of three-dimensional dislocation systems

TL;DR

This paper develops a local density approximation for the elastic energy functional of 3D dislocation systems, enabling simplified modeling of dislocation interactions on larger scales.

Contribution

It introduces a systematic method to derive energy functionals for 3D dislocation systems based on dislocation density variables, extending previous 2D models.

Findings

Derived energy functionals for 3D curved dislocation lines.

Illustrated applications including Debye screening and stress term derivations.

Provided a framework for multiscale dislocation modeling.

Abstract

The elastic energy functional of a system of discrete dislocation lines is well known from dislocation theory. In this paper we demonstrate how the discrete functional can be used to systematically derive approximations which express the elastic energy in terms of dislocation density-like variables which average over the discrete dislocation configurations and represent the dislocation system on scales above the spacing of the individual dislocation lines. We study the simple case of two-dimensional systems of straight dislocation lines before we proceed to derive energy functionals for systems of three-dimensionally curved dislocation lines pertaining to a single, as well as to multiple slip systems. We then illustrate several applications of the theory including Debye screening of dislocations in two and three dimensions, and the derivation of back stress and friction stress terms entering the stress balance from the free energy functionals.