A multi-phase field model of planar dislocation networks

In this paper we extend the phase-field model of crystallographic slip of Ortiz (1999 J. Appl. Mech. ASME 66 289–98) and Koslowski et al (2001 J. Mech. Phys. Solids 50 2957–635) to slip processes that require the activation of multiple slip systems, and we apply the resulting model to the investigation of finite twist boundary arrays. The distribution of slip over a slip plane is described by means of multiple integer-valued phase fields. We show how all the terms in the total energy of the crystal, including the long-range elastic energy and the Peierls interplanar energy, can be written explicitly in terms of the multi-phase field. The model is used to ascertain stable dislocation structures arising in an array of finite twist boundaries. These structures are found to consist of regular square or hexagonal dislocation networks separated by complex dislocation pile-ups over the intervening transition layers.