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We consider the gradient flow evolution of a phase-field model for crystaldislocations in a single slip system in the presence of forest dislocations.The model consists of a Peierls-Nabarro type energy penalizing non-integer slipand elastic stress. Forest dislocations are introduced as a perforation of thedomain by small disks where slip is prohibited. The $\Gamma$-limit of thisenergy was deduced by Garroni and M\"uller (2005 and 2006). Our main resultshows that the gradient flows of these $\Gamma$-convergent energy functionalsdo not approach the gradient flow of the limiting energy. Indeed, the gradientflow dynamics remains a physically reasonable model in the case of non-monotoneloading. Our proofs rely on the construction of explicit sub- andsuper-solutions to a fractional Allen-Cahn equation on a flat torus or in theplane, with Dirichlet data on a union of small discs. The presence of theseobstacles leads to an additional friction in the viscous evolution whichappears as a stored energy in the $\Gamma$-limit, but it does not act as adriving force. Extensions to related models with soft pinning and non-viscousevolutions are also discussed. In terms of physics, our results explain how inthis phase field model the presence of forest dislocations still allows forplastic as opposed to only elastic deformation.

The Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Plastic Slip