  We consider the gradient flow evolution of a phase-field model for crystal
dislocations in a single slip system in the presence of forest dislocations.
The model consists of a Peierls-Nabarro type energy penalizing non-integer slip
and elastic stress. Forest dislocations are introduced as a perforation of the
domain by small disks where slip is prohibited. The $\Gamma$-limit of this
energy was deduced by Garroni and M\"uller (2005 and 2006). Our main result
shows that the gradient flows of these $\Gamma$-convergent energy functionals
do not approach the gradient flow of the limiting energy. Indeed, the gradient
flow dynamics remains a physically reasonable model in the case of non-monotone
loading. Our proofs rely on the construction of explicit sub- and
super-solutions to a fractional Allen-Cahn equation on a flat torus or in the
plane, with Dirichlet data on a union of small discs. The presence of these
obstacles leads to an additional friction in the viscous evolution which
appears as a stored energy in the $\Gamma$-limit, but it does not act as a
driving force. Extensions to related models with soft pinning and non-viscous
evolutions are also discussed. In terms of physics, our results explain how in
this phase field model the presence of forest dislocations still allows for
plastic as opposed to only elastic deformation.
