General teleparallel geometric theory of defects

TL;DR

This paper introduces a novel teleparallel geometric framework for modeling defects in materials, addressing shortcomings of previous theories by avoiding hierarchy issues and instabilities, and providing a more consistent and complete description.

Contribution

The authors develop a generalized teleparallel geometric theory of defects that overcomes conceptual and technical limitations of earlier models, including hierarchy inconsistency and instabilities.

Findings

The new theory is free from hierarchy and instability issues.

Dislocations are modeled with torsion trace, disclinations with non-metricity trace.

The framework is formulated in a coordinate-independent, Eulerian approach.

Abstract

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.