An energy-consistent model of dislocation dynamics in an elastic body

TL;DR

This paper introduces an energy-consistent phase field model for dislocation dynamics in elastic materials, revealing a gradient flow structure and incorporating dislocation core and Peach-Koehler force, with numerical examples for straight screw dislocations.

Contribution

It develops a novel 3D-2D bulk-surface gradient flow model for dislocation motion that includes core effects and force terms, advancing the mathematical understanding of dislocation dynamics.

Findings

Model reveals a hidden gradient flow structure.

Includes dislocation core and Peach-Koehler force.

Provides numerical examples for straight screw dislocations.

Abstract

We propose an energy-consistent mathematical model for motion of dislocation curves in elastic materials using the idea of phase field model. This reveals a hidden gradient flow structure in the dislocation dynamics. The model is derived as a gradient flow for the sum of a regularized Allen-Cahn type energy in the slip plane and an elastic energy in the elastic body. The obtained model becomes a 3D- 2D bulk-surface system and naturally includes the Peach-Koehler force term and the notion of dislocation core. We also derive a 2D-1D bulk-surface system for a straight screw dislocation and give some numerical examples for it.