Size Effect on the Nonlinear Thickness-Shear Vibration of an Elliptical Piezoelectric Plate

Many studies have found that when the piezoelectric devices are with the sizes of micrometer/nanoscale level, another significant electromechanical coupling characteristic, namely the flexoelectric effect, also plays important role in the performance of devices. In order to investigate the effect of flexoelectricity on the thickness-shear vibration, a mathematical model of a finite piezoelectric plate with slowly varying thickness and elliptical crosssection is presented with the consideration of flexoelectric effect in this paper. The proposed model uses the first order McLaughlin series to approximate the elliptical geometric function and the its first derivative is ignored since we assume that the plate thickness varies slowly along the length. The displacement function we adopted is assumed to be antisymmetric through the thickness due to the thicknessshear vibration mode. In addition, regarding to the thin and long flexoelectric plate, only the shear strain gradient through the thickness is considered in the mathematical model. The nonlinear governing equations, which incorporate geometric nonlinearity, are converted into differential equations as function of time by the Galerkin method. The method of multiple scales is employed to obtain the solution of nonlinear governing equation with the first order approximation. The obtained frequencies of thickness-shear mode are expressed in terms of vibration amplitude, eccentricity and flexoelectric coefficients. The results show that the nonlinear thickness-shear vibration is size dependent and flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro thin plates. The eccentricity and geometric nonlinearity also have great influence on the nondimensional fundamental frequencies. Thus, for accurate design of nano/micro scale high precision and high frequency flexoelectric/piezoelectric devices with elliptical cross-section, flexoelectricity must be taken into consideration.