  We propose an energy-consistent mathematical model for motion of dislocation
curves in elastic materials using the idea of phase field model. This reveals a
hidden gradient flow structure in the dislocation dynamics. The model is
derived as a gradient flow for the sum of a regularized Allen-Cahn type energy
in the slip plane and an elastic energy in the elastic body. The obtained model
becomes a 3D- 2D bulk-surface system and naturally includes the Peach-Koehler
force term and the notion of dislocation core. We also derive a 2D-1D
bulk-surface system for a straight screw dislocation and give some numerical
examples for it.
