Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities

TL;DR

This paper demonstrates that models of edge-dislocations in solids, represented as singularities and as manifolds with torsion, are connected through a homogenization process, bridging classical and modern geometric approaches.

Contribution

It rigorously shows that the smooth torsion model can be derived as a homogenization limit of the singular dislocation model in solids.

Findings

Homogenization limit of dislocation models established

Connection between singular and smooth geometric models proven

Torsion fields emerge as limits of dislocation densities

Abstract

We reconcile between two classical models of edge-dislocations in solids. The first model, dating from the early 1900s models isolated edge-dislocations as line singularities in locally-Euclidean manifolds. The second model, dating from the 1950s, models continuously-distributed edge-dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modeled as a Weitzenb\"ock manifold. We prove, using a weak notion of convergence [KM15], that the second model can be obtained rigorously as a homogenization limit of the first model, as the density of singular edge-dislocation tends to infinity.