An energy-consistent model of dislocation dynamics in an elastic body
  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. 
Vladimir Chalupecky and Masato Kimura
math.NA