Phase field model of dislocation networks

In phase field model of dislocations, the short-range, non-linear core-core interactions are characterized through the crystalline energy and the gradient energy. In this article we extend and generalize the approximations of these energies employed in previous phase field models to account for dislocation reactions leading to network formation. In order to characterize dislocation activities involving all slip planes, we suggest the crystalline energy to be a function of a general plastic strain tensor produced by arbitrary linear combinations of simple shears associated with each slip system. For the four {111} slip planes in an fcc crystal, a particular form of such a crystalline energy is formulated by simple linear superposition of the interplanar potential of each individual slip plane. A more detail and general form of the gradient energy is derived from the consideration of the total Burgers vector dependence of dislocation line energy. Examples of applications are presented for interactions between two dislocation loops expanding on either a single slip plane or two intersecting slip planes, as well as for more complex reactions taking place in dislocation networks. It is shown that the generalized expressions are able to handle self-consistently reactions among dislocations of all slip systems in accord with Frank's rule. These extensions are necessary steps toward advanced applications of the phase field method to dislocation substructure formation and coarsening. (C) 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.