$\Gamma$-Limit of a Phase-Field Model of Dislocations
Adriana Garroni, Stefan G. Müller · SIAM Journal on Mathematical Analysis · 2005
We study, by means of $\Gamma$-convergence, the asymptotic behavior 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 nonlocal term and a nonlinear potential which penalizes noninteger 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).