The emergence of torsion in the continuum limit of distributed edge-dislocations

TL;DR

This paper rigorously demonstrates how a dense distribution of edge dislocations in a material leads to a continuum limit characterized by a flat manifold with non-zero torsion, revealing the emergence of torsion in the homogenization process.

Contribution

It introduces a new convergence notion for Weitzenb"ock manifolds and proves a homogenization theorem connecting discrete dislocations to a continuum with torsion.

Findings

Dense dislocation distributions converge to a flat manifold with torsion.

Introduces a new convergence concept for Weitzenb"ock manifolds.

Establishes a rigorous link between microscopic dislocations and macroscopic torsion.

Abstract

We present a rigorous homogenization theorem for distributed dislocations. We construct a sequence of locally-flat Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenb\"ock manifold, i.e. a flat manifold endowed with a metrically-consistent connection with zero curvature and non-zero torsion. In the process, we introduce a new notion of convergence of Weitzenb\"ock manifolds, which is relevant to this class of homogenization problems.