Introduction to the Geometric Theory of Defects

TL;DR

This paper introduces a geometric framework using Riemann-Cartan geometry to model defects like dislocations and disclinations in materials, linking curvature and torsion to physical defect densities and connecting elasticity theory with gauge fields.

Contribution

It presents a unified geometric approach to describe defects in solids, integrating nonlinear elasticity and gauge theories, and clarifies the geometric interpretation of defect densities.

Findings

Curvature and torsion represent surface densities of defects.

The geometric model reduces to classical elasticity without defects.

The approach connects defect theory with gauge field equations.

Abstract

We describe defects - dislocations and disclinations - in the framework of Riemann-Cartan geometry. Curvature and torsion tensors are interpreted as surface densities of Frank and Burgers vectors, respectively. Equations of nonlinear elasticity theory are used to fix the coordinate system. The Lorentz gauge yields equations for the principal chiral SO(3)-field. In the absence of defects the geometric model reduces to the elasticity theory for the displacement vector field and to the principal chiral SO(3)-field model for the spin structure.