Reformulation of continuum defects in terms of the general teleparallel geometry in the language of exterior algebra

TL;DR

This work addresses the geometric modeling of continuum defects by reformulating defect theory in the general teleparallel geometry, where curvature vanishes but non-metricity is allowed. By representing disclination density $\theta^{ab}$ with the symmetric part of the defect 1-form (non-metricity $Q_{ab}$) and dislocation density $\alpha^{ab}$ with its antisymmetric part (torsion), the authors derive explicit continuity equations within an exterior-algebra framework, linking $\theta^{ab}$ to $Q^{bcd}$ and $\alpha^{ab}$ to $T^a$. Key contributions include explicit, metric-formulable relations between defect densities and geometric quantities, the demonstration that the theory reduces to known RCW results in appropriate limits, and the establishment of a coherent, Cartesian-friendly linearization pathway. The significance lies in offering a unified, metric-amenable description of defects with potential for variational formulations, dynamical extensions, and applications to photonic crystals and other ordered media.

Abstract

We discuss the geometric formulation of continuum defects consisting of dislocations and disclinations. After reviewing the metric affine geometry and the present geometric formulation of dislocation and disclination written in terms of torsion and full curvature (together with vanishing non-metricity), we give a new formulation of them in a novel way in terms of torsion and non-metricity (together with vanishing full curvature), the so-called general teleparallel geometry. All calculations are performed by using the exterior algebra. We obtain continuity equations explicitly for dislocation density and disclination density.