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

Raz  Kupferman; Cy  Maor

doi:10.3934/jgm.2015.7.361

Article Contents

2015, Volume 7, Issue 3: 361-387. Doi: 10.3934/jgm.2015.7.361

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

Raz Kupferman^1, and
Cy Maor^1,

1.
Institute of Mathematics, The Hebrew University, Jerusalem 91904

Received: September 2014

Revised: May 2015

Published: September 2015

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Abstract

We present a rigorous homogenization theorem for distributed edge-dislocations. We construct a sequence of locally-flat 2D Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenböck 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 Weitzenböck manifolds, which is relevant to this class of homogenization problems.

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Mathematics Subject Classification: Primary: 53Z05; Secondary: 58K99.

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