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arXiv · 1810.12359

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

Elihu Olami·Raz Kupferman
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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.

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Elihu Olami, Raz Kupferman. 2018-10-29. Homogenization of edge-dislocations as a weak limit of de-Rham currents. https://arxiv.org/abs/1810.12359

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