A phase field model for electromechanical fracture in flexoelectric solids

Flexoelectricity describes the electromechanical coupling between the stain gradient and the electric polarization, which can play an important role in the fracture behavior of piezoelectrics since large strain gradients exist near the crack tip. In this paper, in order to investigate this effect, a phase field (PF) model is developed for electromechanical fracture in brittle flexoelectric solids, where the crack surfaces are assumed to be either electrically impermeable or electrically permeable. The damage driving force of the PF model is derived from the non-negative energy dissipation rate for both kinds of crack surface conditions, where the mechanical part is regarded as the driving force for the crack evolution. A decomposition method of the mechanical driving force is proposed to characterize the asymmetric fracture behavior under tension and compression. The governing equations are solved numerically with finite element method. The finite element formulation is implemented in Abaqus through the user defined subroutine UEL. The PF model and its numerical implementation are validated by studying benchmark problems. The simulation results indicate that the proposed PF model can well characterize the flexoelectric effect and predict complex crack propagation paths in flexoelectric solids. In addition, the numerical results show that whether a negative applied electric field promotes the crack propagation or retard the crack propagation is closely related to the electric field strength and the flexoelectric coefficients, which can reconcile the contradictory phenomena observed in previous experimental studies. The influences of the applied electric field and the length scale parameters on the electromechanical fracture behavior of flexoelectric solids as well as the size effect are discussed in detail in the paper.