Dislocations via incompatibilities in phase-field models of microstructure evolution

We develop a phase-field model that describes the elastic distortion of a ferroelastic material with cubic anisotropy due to an arbitrary dislocation network and a uniform external load. The dislocation network is characterized using the Nye tensor and enters the formulation via a set of incompatibility constraints for the internal strain field. The long-range elastic response of the material is obtained by minimization of the free energy that accounts for higher-order terms of the order parameters and symmetry-adapted strain gradients. The influence of dislocations on the microstructure is studied using a static equilibrium analysis of a material without dislocations and with a random array of parallel edge dislocations. A minimal continuum dislocation dynamics is then used to investigate the simultaneous evolution of the network of geometrically necessary dislocations and the internal strain field. The model developed here is directly applicable to single-phase cubic crystals with an arbitrary degree of anisotropy as well as to ferroelastic materials undergoing temperature-driven cubic-to-tetragonal phase transitions.