Strain Gradient Finite Element Formulation of Flexoelectricity in Ferroelectric Material Based on Phase-Field Method

Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics, regardless of point group symmetry. However, the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect. In this study, we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect. A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions. The microstructure evolution of domains is described by an independent order parameter, namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory. The model is developed based on a thermodynamic framework, in which a set of microforces is introduced to construct the constitutive relation and evolution equation. For the flexoelectric part of electric enthalpy, the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space. The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution. The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids. A comparison is made between the polarization distribution in dielectrics and ferroelectrics. The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube, and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect.