Dynamics of coarse grained dislocation densities from an effective free energy

A continuum description of the time evolution of an ensemble of parallel straight dislocations has recently been derived from the equations of motion of individual dislocations. The predictions of the continuum model were compared to the results of discrete dislocation dynamics (DDD) simulations for several different boundary conditions. It was found that it is able to reproduce all the features of the dislocation ensembles obtained by DDD simulations. The continuum model, however, is systematically established only for single slip. Due to the complicated structure of the equations extending the derivation procedure for multiple slip is not straightforward. In this paper an alternative approach based oil a thermodynamics-like principle is proposed to derive continuum equations for single slip. An effective free energy is introduced even for zero physical temperature, which yields equilibrium conditions giving rise to Debye-like screening; furthermore, it generates dynamical equations along the lines of phase field theory. It is shown that this leads essentially to the same evolution equations as obtained earlier. In addition, it seems that this framework is extendable to Multiple slip as well.