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
MSC-Numbers: 82B26, 31C15, 49J45
Keywords and phrases: phase transition, capacity, gamma-convergence, line tension
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We study the interaction of a singularly perturbed multiwell energy (with an anisotropic nonlocal regularizing term of formula5 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 formula7-limit of the energy in all scaling regimes for the number formula9 of obstacles. The most interesting regime is formula11, where formula13 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 formula5 regularization.

18.10.2019, 02:12