# Thickness-Stretch Vibration of an Infinite Piezoelectric Plate with Flexoelectricity
## 1 Introduction  
Piezoelectric materials are widely used in various functional devices due to their electromechanical coupling capabilities. They play an important role in the fields of communications, electronics, aerospace and aviation structures, and energy systems. With the rapid development of modern technology, piezoelectric devices are developing towards miniaturization and functionalization, and their requirements for material properties and structural dimensions are getting higher and higher. Nanoscale composite materials and structures will have important application potentials in future multifunctional and smart devices, due to their excellent material properties and small structure size. At the nanoscale, these materials exhibit new mechanical behaviors that are different from traditional macroscopic materials. At the nanoscale, the strain gradient of piezoelectric materials is very significant, which cannot be ignored in the design of nanoscale materials [1]. This is due to another electromechanical coupling effect, flexoelectricity, which is an electromechanical coupling effect caused by a strain gradient or non-uniform deformation and widely exists in various dielectric materials [2; 3; 4]. In fact, such non-uniformity will lead to a significant change in the macroscopic mechanical behavior of micro/nano scale piezoelectric structures with flexoelectricity. Recent studies have found that flexoelectricity has significant impacts on the performance of micro/nano piezoelectric devices, such as Rayleigh wave propagation in semi-infinite flexoelectric dielectrics [5], Lamb wave propagation with flexoelectricity and strain gradients [6], and thickness-twist waves in nanoplates with flexoelectricity [7]. There are also many studies regarding the vibration and bending of piezoelectric nanoplatesand nanobeams which consider flexoelectricity. [PERSON] et al. [8] investigated the converse flexoelectric effect in comb electrode piezoelectric microbeams. [PERSON] [9] studied the size-dependent bending and vibration behaviors of piezoelectric circular nanoplates. [PERSON] [10] investigated the flexoelectric effect on bending stresses in piezoelectric laminate under an inhomogeneous electric field. Due to the miniaturization of high-frequency piezoelectric devices, high-frequency vibrations are also affected by flexoelectricity when the devices are thin enough [11]. As one of the main functioning modes for piezoelectric resonators and transducers, thickness-stretch mode can be adopted for film bulk acoustic resonators (FBAR) as a working mode. Two-dimensional plate equations governing the extensional and thickness-stretch coupling motions were developed by [PERSON] and [PERSON] [12]. A single equation for thickness-stretch waves can be achieved by reducing the two-dimensional plate equations of coupled extension and thickness-stretch vibration in this model. [PERSON] [13] developed an approximate method for thickness-stretch waves in an elastic plate. He also developed an energy harvesting model based on the thickness-stretch mode [14]. [PERSON] [15] studied the propagation of long thickness-stretch waves in piezoelectric plates by simplifying the two-dimensional coupled equations with five vibration modes. [PERSON] et al. [16] studied the nonlinear coupling between the thickness-shear and thickness-stretch modes in a rotated Y-cut quartz resonator. [PERSON] [17] investigated the thickness-stretch vibration of a crystal plate carrying a micro-rod array fixed on the top surface, with rods modeled by the one-dimensional structural theory for extensional vibration. In addition to these works, there are also many studies regarding nonlinear thickness-stretch vibrations [18], the thickness-stretch vibration of a magnetoelectric plate with electrodes [19], a plate under biasing field [20], the electroelastic effect [21], spurious mode analysis [22], and so on. In this work, we present an investigation of the flexoelectric effect on the thickness-stretch vibration of an infinite piezoelectric plate.