General teleparallel geometric theory of defects

TL;DR

This work addresses fundamental shortcomings of conventional metric-affine defect theories by proposing a generalized teleparallel geometric theory of defects (GTPG) in which dislocations, disclinations, and point defects are encoded as torsion and non-metricity traces, supplemented by a free scalar torsion degree of freedom $S$. The authors develop the formalism in an Eulerian exterior-algebra framework, derive kinetic relations for defect densities from the Bianchi identities, and introduce a quadratic, second-order Lagrangian that couples torsion, non-metricity, and their irreducible components via constants $\kappa_i$, ensuring a metric formulation free from Ostrogradsky instabilities. A clear mapping from geometric quantities to defect densities (Burgers, Frank vectors, and point defects) is established, with explicit expressions for $T^a$ and $Q_{ab}$ and their associated defect dynamics. The paper outlines future directions, including deriving explicit field equations from the Lagrangian, solving for the deformed metric to predict defect configurations, and exploring symmetric teleparallel variants as potential simplifications or gauge-theoretic extensions.

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.

Figures (1)

• Figure 1: Continuum mechanics. $\mathcal{B}$ is the body manifold, $\mathcal{S}$ is the spatial manifold.