## 1 Introduction  
Functionally graded materials (FCMs) can reduce or even eliminate the interface between components by continuously controlling the content distribution of each component to meet the different requirements of different parts of structural components for material performance. When piezoelectric materials are subjected to mechanical loading, they generate an electric charge, which is called the positive piezoelectric effect. In contrast, the piezoelectric material will produce mechanical deformation under the electric field effect, which is called the inverse piezoelectric effect. Functionally graded piezoelectric materials (FGPMs) have been developed in recent years to meet the requirements of special conditions and extend the service life. As a new material design concept, FGPMs have very broad application prospects [1; 2; 3; 4; 5; 6; 7; 8; 9].  
The functional gradient plate structures composed by a variety of materials combine the excellent performance of a variety of component materials, so the functional gradient plates are important in the fields of aerospace, transportation engineering, construction and other fields. The interference of the plate structures is taken from external conditions in the process of work, and many mechanical behaviors such as bending and vibration, so many scholars have made a lot of research on the mechanical behaviors of the plates and have achieved rich research results [10; 11; 12; 13; 14; 15; 16]. [PERSON] [17] investigated the modal performance of a sandwich composite plate and provided a new design idea for dynamic behaviors of sandwich plates. [PERSON] [18] et al. introduced FGPMs plates for the modal analysis of ceramic and proposed hierarchic models. [PERSON] [19] et al. analyzed the free vibration of a circular plate composed of FGMs based on classical plate theory. [PERSON] [20] et al. studied vibration behaviors of rotating composite nano-annular plates based on different plate theories and explored the effects of the thickness-to-radius ratio and angular velocity on the natural frequencies. [PERSON] [21] et al. proposed a new generalized 5-variable shear deformation theory and calculated the static response of FGMs plates. [PERSON] [22] et al. studied the bending analysis of a FGPMs plate by using a simple quasi-3D sinusoidal shear deformation theory under simply supported edge conditions.  
In the fields of engineering, during the design of the plate structures, it is often placed on an elastic foundation, such as the high-speed shock absorbing layers of the train, which can be considered an elastic foundation. However, an elastic foundation may have an important impact on the mechanical properties of the plates, so many scientific research workers have studied the mechanical behaviors of the plates on an elastic foundation and achieved important research results [23; 24; 25; 26; 27].  
There are a lot of research results of mechanical behaviors of the FGPMs plates so far but few have studied the mechanical-electrical-thermal coupling of FGPMs. In this paper, the results of a transversely isotropic FGPMs plate under mechanical-electrical-thermal coupling on an elastic foundation are studied, and some references are provided for the design of FGPMs.