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

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.