An existence result for Discrete Dislocation Dynamics in three dimensions

TL;DR

This paper develops a rigorous mathematical framework for three-dimensional Discrete Dislocation Dynamics, establishing well-posedness, deriving forces, and proving existence of evolution until dislocation density becomes infinite.

Contribution

It introduces a regularised energy and a dissipative evolution law for curved dislocations, providing the first rigorous existence results for 3D dislocation dynamics.

Findings

Well-posedness of 3D Discrete Dislocation Dynamics established

Derivation of Peach-Koehler force via inner variation

Existence and regularity results up to infinite dislocation density

Abstract

We present a mathematical framework within which Discrete Dislocation Dynamics in three dimensions is well-posed. By considering smooth distributions of slip, we derive a regularised energy for curved dislocations, and rigorously derive the Peach-Koehler force on the dislocation network via an inner variation. We propose a dissipative evolution law which is cast as a generalised gradient flow, and using a discrete-in-time approximation scheme, existence and regularity results are obtained for the evolution, up until the first time at which an infinite density of dislocation lines forms.