

Preprint 76/2004
A variational model for dislocations in the line tension limit
Adriana Garroni and Stefan Müller
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Submission date: 31. Oct. 2004
Pages: 35
published in: Archive for rational mechanics and analysis, 181 (2006) 3, p. 535-578
DOI number (of the published article): 10.1007/s00205-006-0432-7
Bibtex
MSC-Numbers: 82B26, 31C15, 49J45
Keywords and phrases: phase transition, capacity, gamma-convergence, line tension
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Abstract:
We study the interaction of a singularly perturbed multiwell energy
(with an anisotropic nonlocal regularizing term of type) and
a pinning condition. This functional arises in
a phase field model for dislocations which was recently
proposed by Koslowski, Cuitiño and Ortiz but is also
of broader mathematical interest. In the context of the dislocation model
we identify the
-limit of the energy in all scaling regimes
for the number
of obstacles. The most interesting regime
is
, where
is a nondimensional length scale related to the size
of the crystal lattice. In this case
the limiting model is of line tension type. One important feature
of our model is that the set of energy wells is periodic and hence
not compact. A key ingredient in the proof is thus a compactness
estimate (up to a single translation) for finite energy sequences,
which generalizes earlier results of Alberti, Bouchitté and Seppecher
for the two-well problem with an
regularization.