Geometric Aspects of Singular Dislocations

TL;DR

This paper explores the geometric and topological structure of singular dislocations within the framework of differential geometry and currents, providing a rigorous mathematical foundation for dislocation theory.

Contribution

It introduces a geometric formulation of singular dislocations using de Rham currents on manifolds, deriving Frank's rules from boundary operator properties.

Findings

Dislocations are modeled as boundaries of currents on manifolds.

Frank's rules are derived from the nilpotency of the boundary operator.

Provides a unified geometric framework for dislocation theory.

Abstract

The theory of singular dislocations is placed within the framework of the theory of continuous dislocations using de Rham currents. For a general $n$-dimensional manifold, an $(n-1)$-current describes a local layering structure and its boundary in the sense of currents represents the structure of the dislocations. Frank's rules for dislocations follow naturally from the nilpotency of the boundary operator.