Homogenization of edge-dislocations as a weak limit of de-Rham currents

TL;DR

This paper demonstrates how continuous distributions of crystalline defects can be derived as limits of singular edge-dislocations using de-Rham currents, revealing the emergence of torsion in the homogenization process.

Contribution

It introduces a rigorous mathematical framework connecting singular dislocation structures with smooth defect models via currents, showing the emergence of torsion in the homogenization limit.

Findings

Continuous defect models are limits of singular dislocation arrays.

Singular torsion currents converge to smooth torsion tensors.

Homogenization leads to the emergence of torsion in material structures.

Abstract

In the material science literature we find two continuum models for crystalline defects: (i) A body with (finite) isolated defects is typically modeled as a Riemannian manifold with singularities, and (ii) a body with continuously distributed defects, which is modeled as a smooth (non-singular) Riemannian manifold with an additional structure of an affine connection. In this work we show how continuously distributed defects may be obtained as a limit of singular ones . The defect structure is represented by layering 1-forms and their singular counterparts - de-Rham (n-1) currents. We then show that every smooth layering $1$-form may be obtained as a limit, in the sense of currents, of singular layering forms, corresponding to arrays of edge dislocations. As a corollary, we investigated manifolds with full material structure, i.e., a complete co-frame for the co-tangent bundle. We define the notion of singular torsion current for manifolds with a parallel structure and prove its convergence to the regular smooth torsion tensor at homogenization limit. Thus establishing the so-called emergence of torsion at the homogenization limit.