A unified geometric treatment of material defects

TL;DR

This paper introduces a comprehensive geometric framework for understanding material defects by unifying smooth and singular descriptions using currents, with potential for broader defect structures and practical examples.

Contribution

It develops a unified geometric theory of material defects based on currents, integrating smooth and singular defect descriptions and exploring weaker dislocation structures.

Findings

Unified geometric framework for defects using currents

Inclusion of both smooth and singular defect descriptions

Discussion of weaker dislocation structures

Abstract

A unified theory of material defects, incorporating both the smooth and the singular descriptions, is presented based upon the theory of currents of Georges de Rham. The fundamental geometric entity of discourse is assumed to be represented by a single differential form or current, whose boundary is identified with the defect itself. The possibility of defining a less restrictive dislocation structure is explored in terms of a plausible weak formulation of the theorem of Frobenius. Several examples are presented and discussed.