The continuous theory of dislocations for a material containing dislocations to one Burgers vector only

TL;DR

This paper reviews the mathematical foundations of the continuous dislocation theory, introducing a space of dislocation measures and an evolution equation based on fundamental conditions.

Contribution

It provides a rigorous mathematical framework for the continuous theory of dislocations, including the definition of dislocation measure spaces and evolution equations.

Findings

Defined a space of dislocation measures including Hausdorff measures

Derived an evolution equation for dislocation measures from fundamental conditions

Enhanced the mathematical understanding of dislocation dynamics

Abstract

We review the continuous theory of dislocations from a mathematical point of view using mathematical tools, which were only partly available when the theory was developed several decades ago. We define a space of dislocation measures, which includes Hausdorff measures representing the dislocation measures of single dislocation curves. The evolution equation for dislocation measures is defined on this space. It is derived from four basic conditions, which must be satisfied by the model.