Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

We present a family of phase field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack face boundary conditions in sharp models) into the phase field framework. The proposed models are verified through comparisons with the corresponding sharp crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.