TY - JOUR

T1 - Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities

AU - Kupferman, Raz

AU - Maor, C. Y.

N1 - Publisher Copyright:
© Copyright 2016 Royal Society of Edinburgh.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, 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 modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, 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.

AB - We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, 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 modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, 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.

KW - Gromov-Hausdorff convergence

KW - Weitzenböck manifolds

KW - dislocations

KW - homogenization

KW - torsion

UR - http://www.scopus.com/inward/record.url?scp=84976526062&partnerID=8YFLogxK

U2 - 10.1017/S0308210515000773

DO - 10.1017/S0308210515000773

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AN - SCOPUS:84976526062

SN - 0308-2105

VL - 146

SP - 741

EP - 768

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

IS - 4

ER -