Abstract or Additional Information
We consider a phase field model for dislocations introduced by Koslowski, Cuitino, and Ortiz in 2002. The model describes a single slip plane and consists of a Peierls potential penalizing non-integer slip and a long range interaction modeling elasticity. Forest dislocations are introduced as a restriction to the allowable phase field functions: they have to vanish at the union of a number of small disks in the plane. Garroni and Müller proved large scale limits of these models in terms of Gamma-convergence, obtaining a line-tension energy for the dislocations and a bulk term penalizing slip. This bulk term is a capacity stemming from the forest dislocations. In the present work, we show that the contribution of the forest dislocations to the viscous gradient flow evolution is small. In particular it is much slower than the timescale for other effects like elastic attraction/repulsion of dislocations, which, by recent results due to del Mar Gonzales and Monneau/Patrizi and Valdinoci is already slower than the time scale from line tension energy. Overall, this leads to an effective behavior like a gradient flow in a wiggly potential. On the other hand, when adding a driving force in the direction of increasing slip, one needs to spend the energy to overcome the obstacles. The forest dislocations thus act like a dissipation for increasing the amount of slip, but their effect on the propagation is absent when the amount of slip is decreasing.