Γ-limit of a phase-field model of dislocations
Adriana Garroni and Stefan Müller
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Submission date: 10. Nov. 2003
published in: SIAM journal on mathematical analysis, 36 (2005) 6, p. 1943-1964
DOI number (of the published article): 10.1137/S003614100343768X
MSC-Numbers: 82B26, 31C15, 49J45
Keywords and phrases: dislocations, phase transitions, capacity, gamma convergence
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We study, by means of -convergence, the asymptotic behaviour of a variational problem modeling a dislocation ensemble moving on a slip plane through a discrete array of obstacles. The variational problem is a two dimensional phase transition type energy given by a non local term and a non linear potential which penalizes non integer values. In this paper we consider a regime corresponding to a diluted distribution of obstacles. In this case the leading term of the energy can be described by means of a cell problem formula defining an appropriate notion of capacity (that we call dislocation capacity).