Phase field study of the thermo-electro-mechanical fracture behavior of flexoelectric solids

Abstract

Piezoelectrics working in a temperature varying environment are prone to thermo-electro-mechanical fracture due to their brittle nature. In the meantime, flexoelectricity can greatly influence the fracture behavior of piezoelectrics since large strain gradients exist near the crack tip. In this paper, a novel phase field model is developed to investigate the thermo-electro-mechanical fracture behavior of piezoelectrics with consideration of the flexoelectric effect. In the proposed phase field model, the crack surfaces are assumed to be thermally adiabatic and electrically impermeable. The damage driving force is derived from the Clausius-Duhem inequality where the mechanical part is regarded as the driving force for the crack evolution. The mechanical driving force is decomposed to properly characterize the mixed mode crack propagation behavior. The governing equations of the four-field-problem are derived from the variational principle and solved numerically with the mixed finite element method. The fracture behavior of the flexoelectric solid under combined thermal, electrical and mechanical loads is then investigated by phase field modelling. The numerical results indicate that the proposed phase field model can well characterize the multi-physical effects and predict complex crack propagation trajectories in flexoelectric solids. Besides the mechanical and electrical loads, the thermal effect, the flexoelectric effect and the temperature dependence of the flexoelectric coefficients can all have significant influences on the thermo-electro-mechanical fracture behavior of flexoelectric solids. The present phase field model can thus act as a promising numerical tool to predict the reliability of flexoelectric components working in a temperature varying environment.