# $\Gamma$-limit of a phase-field model of dislocations

created on 02 Dec 2003
modified by garroni on 24 Jan 2006

[BibTeX]

Published Paper

Inserted: 2 dec 2003
Last Updated: 24 jan 2006

Journal: SIAM Math. Anal.
Volume: 36
Number: 6
Pages: 1943-1964
Year: 2005

Abstract:

We study, by means of $\Gamma$-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).