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 -