Dynamics of curved dislocation ensembles

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 dipolelike approximation for the orientation-dependent density variables is applied. This leads to a closed set of kinematic evolution equations of total dislocation density, the geometrically necessary dislocation 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.