## 1 Introduction
[PERSON] and [PERSON] [14] proposed a theoretical framework for analyzing the response of piezoelectric actuators and sensor composite laminates pre- and post-buckling. Based on the Euler-Bernoulli beam theory, [PERSON] et al. [15] investigated the free transverse vibration and buckling characteristics of a double beam system under compressive axial loads. The free buckling and vibration of a set of parallel Timoshenko beams under the axial load was studied by [PERSON] et al. [16]. In this study, a nonlinear and axial load are considered while analyzing the vibration and bending properties of the sandwich piezoelectric nanobeam. Piezoelectric materials have received increasing attention because of their excellent electromechanical coupling properties, and are widely used in the field of energy harvesting. When there is mechanical energy in the environment where piezoelectric materials are located, such as wind energy, noise, mechanical vibration, these mechanical energies can cause the deformation of piezoelectric materials. Because of the piezoelectric effect, the voltage potential is generated on the surface of piezoelectric nanomaterials, so the transformation of mechanical energy to electrical energy is realized. It is worth noting that at micro- and nanoscale, the electromechanical properties of piezoelectric nanomaterials and nanostructures are significantly different from those of macroscopic piezoelectric materials. It provides guidance for the development of a piezoelectric micro energy harvester. Piezoelectric materials show a significant flexoelectric effect when the size is at nanoscale. The flexoelectricity is the electromechanical coupling effect present in a dielectric, which is similar to piezoelectricity. It not only exhibits the polarization response and strain gradient, but also exhibits the electromechanical coupling effect [17]. The flexoelectricity is the basic property of dielectric materials, which not only exists in piezoelectric materials, but also almost exists in all dielectrics. It is noteworthy that the electricity caused by the flexoelectric effect is much larger compared with the piezoelectric effect of nanomaterials. Therefore, the flexoelectric effect is usually considered when studying nanostructures [18; 19; 20; 21; 22]. [PERSON] et al. [18] studied the vibration of piezoelectric sandwich nanobeams based on the theory of nonlocal strain gradients and discussed the flexoelectric effect in detail. [PERSON] et al. [23] modeled the beam as a laminated beam and investigated the effect of flexoelectric layer stiffness on the dynamic response of the structure. [PERSON] et al. [24] investigated the combined piezoelectric and flexoelectric effects in the resonant dynamics of nanocantilever beams. Since the effect of pliability generated in piezoelectric nanostructures cannot be ignored, this study takes into account the flexoelectric effect while analyzing the vibration of functionally graded sandwich piezoelectric nanobeams. The sandwich structure has excellent properties such as high stiffness and light weight, so it is widely used in engineering fields, such as biomedicine, architecture, automotive and aerospace. The sandwich structure is a composite structure formed by connecting two thin surface layers and one core layer. The porous material is a kind of lightweight material that widely exists in nature, such as metal foams. The porous material has the advantages of high strength and low density, and is often used as energy absorption, noise reduction, flame retardant, heat insulation [25; 26; 27; 28], and is therefore widely used in the automotive industry, aerospace, and other fields. The composite structure with porous materials as the core layer of the sandwich structure is called the sandwich porous structure. Generally, the sandwich porous structure cannot only ensure enough stiffness, but also reduce the quality of the whole structure.