Revisiting Volterra defects: Geometrical relation between edge dislocations and wedge disclinations

TL;DR

This paper develops a comprehensive geometric model for Volterra defects, establishing a rigorous mathematical link between edge dislocations and wedge disclinations using advanced differential geometry and complex potentials.

Contribution

It introduces a novel geometric framework that generalizes classical Volterra defect theory and provides analytical expressions for stress fields, enhancing understanding of defect relations.

Findings

Established a geometric relation between edge dislocations and wedge disclinations.

Derived analytical stress fields consistent with existing results.

Utilized Riemannian holonomy and complex potentials to analyze topological defect properties.

Abstract

This study presents a comprehensive mathematical model for Volterra defects and explores their relations using differential geometry on Riemann--Cartan manifolds. Following the standard Volterra process, we derived the Cartan moving frame, a geometric representation of plastic fields, and the associated Riemannian metric using exterior algebra. Although the analysis naturally defines the geometry of three types of dislocations and the wedge disclination, it fails to classify twist disclinations owing to the persistent torsion component, suggesting the need for modifications to the Volterra process. By leveraging the interchangeability of the Weitzenb\"ock and Levi-Civita connections and applying an analytical solution for plasticity derived from the Biot--Savart law, we provide a rigorous mathematical proof of the long-standing phenomenological relationship between edge dislocations and wedge disclinations. Additionally, we showcase the effectiveness of novel mathematical tools, including Riemannian holonomy for analysing the Frank vector and complex potentials that encapsulate the topological properties of wedge disclinations as jump discontinuities. Furthermore, we derive analytical expressions for the linearized stress fields of wedge disclinations and confirm their consistency with existing results. These findings demonstrate that the present geometrical framework extends and generalizes the classical theory of Volterra defects.