  A phase field model of a crystalline material at the mesoscale is introduced
to develop the necessary theoretical framework to study plastic flow due to
dislocation motion. We first obtain the elastic stress from the phase field
free energy and show that it obeys the stress strain relation of linear
elasticity. Dislocations in a two dimensional hexagonal lattice are shown to be
composite topological defects in the amplitude expansion of the phase field,
with topological charges given by the Burgers vector. This allows us to
introduce a formal relation between dislocation velocity and the evolution of
the coarse grained envelopes of the phase field. Standard dissipative dynamics
of the phase field crystal model is shown to determine the velocity of the
dislocations. When the amplitude equation is valid, we derive the Peach-Koehler
force on a dislocation, and compute the associated defect mobility. A numerical
integration of the phase field crystal equations in two dimensions is used to
compute the motion of a dislocation dipole, and good agreement is found with
the theoretical predictions.
