A phase-field fracture model for flexoelectric structures

The distribution and propagation of cracks play a critical role in the design, fabrication, and optimization of high-sensitivity flexoelectric materials. Despite significant progress, the elucidation of the flexoelectric fracture mechanism remains an unresolved and challenging issue. In this study, a novel phase-field model addressing electromechanical fracture in both piezoelectric and flexoelectric materials is developed. Gradient elasticity and flexoelectricity are incorporated into the phase-field theory, with governing equations derived in the framework of thermodynamics. In light of the stress projection and the energetic failure criterion, the unsymmetric electromechanical fracture behavior can be reproduced. Different energy degradation strategies are introduced to account for the energy contributions from strain and strain gradients, enabling the accurate simulation of fractures influenced by flexoelectricity. Compared with conventional fracture models, this model reproduces the flexoelectric crack-tip effect during crack propagation for the first time. We performed numerical simulations of the crack evolution process in various flexoelectric materials and structures. The findings indicate that the gradient and flexoelectric effects can hinder the failure of solids for tensile fracture. Meanwhile, flexoelectricity can promote crack deflection toward the direction of the local electric field around the crack tip. Moreover, crack growth under mechanical loading is consistently accompanied by a continuous increase in the efficiency of energy conversion from mechanical to electrical form. However, excessive crack propagation weakens the capacity for dielectric energy storage in the material. This model not only provides a new perspective for the optimal design of efficient flexoelectric structures, but also offers a new framework for investigating the fracture behavior of electro-mechanical coupling materials with size effects.