## 2 Governing Equations  
Considering a rectangular FGPMs plate on an elastic foundation with a length, width and height of \(l_{a},l_{b},h\), the foundation coefficient is \(K\), as shown in Figure 1. In this paper, PZT-4 and PZT-5 piezoelectric ceramic materials are applied to constitute a functional gradient composite plate. Since most scholars in past studies have used these two materials when studying the mechanical properties of composite piezoelectric structures, a comparison can be made during the analysis of the data, and the two materials can be used to study the changes of the material gradient index. The resulting conclusions have guided significance for the mechanical properties of the plates formed by other materials. Assume that the materials of a plate are composed of the piezoelectric material PZT \(-5\) on the lower surface, changing continuously to the material PZT \(-4\) component on the upper surface, and the material properties gradient along the thickness direction, which can be expressed as [28]:  
\[P(z)=P_{5}+(P_{4}-P_{5})\bigg{(}\frac{1}{2}+\frac{z}{h}\bigg{)}^{k} \tag{1}\]  
where \(P(z)\) represents physical parameters of the functionally graded material, \(P_{4}\) represents physical parameters of the material PZT \(-4\), \(P_{5}\) represents physical parameters of the material PZT \(-5\), and \(k\) represents functional graded index. Figure 1: Functionally graded piezoelectric plate on elastic foundation. Assuming that the physical middle position \(z=z_{0}\) of the FGPMs plate is not coincident with the geometric middle position, it can be obtained according to the literature [29]. \[z_{0}=\frac{\int_{-h/2}^{h/2}zc_{11}(z)\mathrm{d}z}{\int_{-h/2}^{h/2}c_{11}(z) \mathrm{d}z}=\frac{(c_{11(\mathrm{PZT-4})}-c_{11(\mathrm{PZT-5})})kh}{2(2+k)(c _{11(\mathrm{PZT-4})}+kc_{11(\mathrm{PZT-5})})} \tag{2}\]  
where \(c_{11(\mathrm{PZT-4})},c_{11(\mathrm{PZT-5})}\) represent the elastic modulus of the material PZT \(-4\) and the material PZT \(-5\), respectively. According to [PERSON]'s plate theory, the displacement of any point in the functionally graded piezoelectric plate can be expressed as  
\[\left\{ \begin{array}{l}u(x,y,z,t)=-(z-z_{0})\frac{\partial w}{\partial x}\\ v(x,y,z,t)=-(z-z_{0})\frac{\partial w}{\partial y}\\ w(x,y,z,t)=w(x,y,t)\end{array} \right. \tag{3}\]  
The relationship between the strains and displacement \(w\) of the piezoelectric plate can be obtained from geometric relations  
\[\left\{ \begin{array}{l}\varepsilon_{xx}=-(z-z_{0})\frac{\partial^{2}w}{ \partial x^{2}}\\ \varepsilon_{yy}=-(z-z_{0})\frac{\partial^{2}w}{\partial x^{2}}\\ \varepsilon_{xy}=-(z-z_{0})\frac{\partial^{2}w}{\partial x\partial y}\end{array} \right. \tag{4}\]  
For the Kirchhoff functionally graded piezoelectric plate model, the constitutive relations under mechanical-electro-thermal coupling can be expressed as  
\[\left\{ \begin{array}{l}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}C_{11}&C_{12}&0\\ C_{12}&C_{11}&0\\ 0&0&C_{66}\end{array} \right]\left\{ \begin{array}{l}\varepsilon_{xx}\\ \varepsilon_{yy}\\ 2\varepsilon_{xy}\end{array} \right\}-\left[ \begin{array}{ccc}0&0&e_{31}\\ 0&0&e_{31}\\ 0&0&0\end{array} \right]\left\{ \begin{array}{l}E_{x}\\ E_{y}\\ E_{z}\end{array} \right\}-\left\{ \begin{array}{l}\beta_{1}\\ \beta_{1}\\ 0\end{array} \right\}\Delta T \tag{5}\]  
\[\left\{ \begin{array}{l}D_{x}\\ D_{y}\\ D_{z}\end{array} \right\}=\left[ \begin{array}{ccc}0&0&0\\ 0&0&0\\ e_{31}&e_{31}&0\end{array} \right]\left\{ \begin{array}{l}\varepsilon_{xx}\\ \varepsilon_{yy}\\ 2\varepsilon_{xy}\end{array} \right\}+\left[ \begin{array}{ccc}s_{11}&0&0\\ 0&s_{11}&0\\ 0&0&s_{33}\end{array} \right]\left\{ \begin{array}{l}E_{x}\\ E_{y}\\ E_{z}\end{array} \right\}+\left\{ \begin{array}{l}0\\ 0\\ 0\\ a_{3}\end{array}