It was noted that a continuum description of the time-evolution of an ensemble of parallel straight dislocations had recently been derived from the equations of motion of individual dislocations. The predictions of the continuum model were compared with the results of discrete dislocation dynamics simulations for several boundary conditions. It was found that it was able to reproduce all of the features of the dislocation ensembles which were obtained by discrete dislocation dynamic simulation. The continuum model was systematically established only for single slip. Due to the complicated structure of the equations, extension of the derivation procedure to multiple slip was not easy. An alternative approach, based upon a thermodynamic-like principle, was proposed here in order to derive continuum equations for single slip. An effective free energy was introduced even for zero temperature; yielding equilibrium conditions which gave rise to Debye-like screening. It also generated dynamic equations as in phase field theory. It was shown that this led essentially to the same evolution equations as those obtained previously. In addition, this method was perhaps applicable to multiple slip.

Dynamics of Coarse Grained Dislocation Densities from an Effective Free Energy. I.Groma, G.Györgyi, B.Kocsis: Philosophical Magazine, 2007, 87[8-9], 1185-99