  To develop a dislocation-based statistical continuum theory of crystal
plasticity is a major challenge of materials science.During the last two
decades such a theory has been developed for the time evolution of a system of
parallel edge dislocations. The evolution equations were derived by a
systematic coarse-graining of the equations of motion of the individual
dislocations and later retrieved from a functional of the dislocation densities
and the stress potential by applying the standard formalism of phase field
theories. It is, however, a long standing issue if a similar procedure can be
established for curved dislocation systems. An important prerequisite for such
a theory has recently been established through a density-based kinematic theory
of moving curves. In this paper, an approach is presented for a systematic
derivation of the dynamics of systems of curved dislocations in a single slip
situation. In order to reduce the complexity of the problem a dipole like
approximation for the orientation dependent density variables is applied. This
leads to a closed set of kinematic evolution equations of total dislocation
density, the GND densities, and the so-called curvature density. The analogy of
the resulting equations with the edge dislocation model allows one to
generalize the phase field formalism and to obtain a closed set of dynamic
evolution equations.
