We introduce a phenomenological theory of dislocation motion appropriate for two-dimensional lattices. A coarse grained description is proposed that involves as primitive variables local lattice rotation and Burgers vector densities along distinguished slip systems of the lattice. We then use symmetry considerations to propose phenomenological equations for both defect energies and their dissipative motion. As a consequence, the model includes explicit dependencies on the local state of lattice orientation, and allows for differential defect mobilities along distinguished directions. Defect densities and lattice rotation need to be determined self-consistently and we show specific results for both square and hexagonal lattices. Within linear response, dissipative equations of motion for the defect densities are derived which contain defect mobilities that depend nonlocally on defect distribution.
Bibliographical noteFunding Information:
This research has been supported by the National Science Foundation under contract DMS-DMREF 1435372 and the Minnesota Supercomputing Institute. We thank Noah Mitchell and Zhi-Feng Huang for many useful and stimulating discussions.