The boundary value problem of dislocation dynamics

The present work is concerned with the mathematical formulation of the boundary value problem of dislocation dynamics. The boundary conditions for the elastic and dislocation fields are derived for two cases which are currently treated by dislocation dynamics simulation models. In the first case, the crystal space under consideration is a representative subvolume of a much larger crystal undergoing statistically homogeneous plastic distortion. The second is the case of a crystal which is bounded by a free surface. For the former case, the condition of statistical homogeneity of the elastic stress and the plastic distortion fields is used to derive local traction and dislocation flux boundary conditions on the surface of the deforming crystal space. When the crystal is bounded, the formation of slip traces is connected with the dislocation flux and dislocation emission at the free surface. Three theorems are derived, two on conservation of the total Burgers vector and the dislocation flux in a deforming crystal, and the third on image interactions. Some important mathematical ideas related to the geometry of deforming crystals are also discussed.