## 2 Governing Equations
\begin{array}{c}\frac{\partial^{2}M_{xx}}{\partial x^{2}}+\frac{\partial^{2 }M_{yy}}{\partial y^{2}}+2\frac{\partial^{2}M_{xy}}{\partial x\partial y}-E_ {31}^{2}\frac{\partial^{4}w}{\partial x^{4}}-2E_{31}^{2}\frac{\partial^{4}w}{ \partial x^{2}\partial y^{2}}-E_{31}^{2}\frac{\partial^{4}w}{\partial x^{2} \partial y^{2}}-E_{31}^{2}\frac{\partial^{4}w}{\partial y^{4}}+N_{x}\frac{ \partial^{2}w}{\partial x^{2}}+N_{y}\frac{\partial^{2}w}{\partial y^{2}}-kw-q \\ =-I_{0}\frac{\partial^{2}w}{\partial t^{2}}+I_{2}\frac{\partial^{4}w}{ \partial x^{2}\partial t^{2}}+I_{2}\frac{\partial^{4}w}{\partial y^{2} \partial t^{2}}\end{array}  \tag{17}\]  
and the simply supported boundary condition of the plate is deduced as follows:  
\[\left\{ \begin{array}{c}x=0,l_{u}\quad\frac{\partial^{2}w}{\partial x^{2}}= 0\\ y=0,l_{b}\quad\frac{\partial^{2}w}{\partial y^{2}}=0\end{array} \right. \tag{18}\]The bending moments \(M_{xx}\) and \(M_{yy}\), and the expression of torque \(M_{xy}\) in the governing Equation (17) are defined as  
\[\left\{ \begin{array}{c}M_{xx}\\ M_{yy}\\ M_{xy}\end{array} \right\}=\int_{-h/2}^{h/2}\left(z-z_{0}\right)\left\{ \begin{array} []{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{array} \right\}\text{d}z \tag{19}\]  
The expressions of the bending moments and the torque can be derived from matrix Equation (5):  
\[\left\{ \begin{array}{c}M_{xx}\\ M_{yy}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}(-D_{11}+E_{31}^{1})&(-D_{ 12}+E_{31}^{1})&0\\ (-D_{12}+E_{31}^{1})&(-D_{11}+E_{31}^{1})&0\\ 0&0&-2D_{66}\end{array} \right]\left\{ \begin{array}{c}\frac{\partial^{2}w}{ \partial x^{2}}\\ \frac{\partial^{2}w}{\partial x^{2}}\\ \frac{\partial^{2}w}{\partial x\partial y}\end{array} \right\}+\left\{ \begin{array} []{c}M_{e}-M_{T}\\ M_{e}-M_{T}\\ 0\end{array} \right\} \tag{20}\]  
Substituting Equation (20) into Equation (17) to obtain an expression with displacement \(w\) as a variable:  
\[ \begin{array}{l}(-D_{11}+E_{31}^{1}-E_{31}^{2})\frac{\partial^{4}w}{ \partial x^{4}}+(-D_{11}+E_{31}^{1}-E_{31}^{2})\frac{\partial^{4}w}{\partial y ^{4}}+(-2D_{12}+2E_{31}^{1}-2E_{31}^{2}-4D_{66})\frac{\partial^{4}w}{\partial x ^{2}\partial y^{2}}+N_{x}\frac{\partial^{2}w}{\partial x^{2}}+N_{y}\frac{ \partial^{2}w}{\partial y^{2}}\\ -kw-q=-I_{0}\frac{\partial^{2}w}{\partial t^{2}}+I_{2}\frac{\partial^{4}w}{ \partial x^{2}\partial t^{2}}+I_{2}\frac{\partial^{4}w}{\partial y^{2}\partial t ^{2}}\end{array}  \tag{21}\]  
The correlation coefficients in Equations (20) and (21) are expressed as  
\[\left\{ \begin{array}{l}D_{11}=\int_{-h/2}^{h/2}C_{11}(z-z_{0})^{2}\text{d}z \quad D_{12}=\int_{-h/2}^{h/2}C_{12}(z-z_{0})^{2}\text{d}z\\ D_{66}=\int_{-h/2}^{h/2}C_{66}(z-z_{0})^{2}\text{d}z\quad E_{31}^{1}=\int_{-h/ 2}^{h/2}\frac{\varepsilon_{31}}{\varepsilon_{33}}z(z-z_{0})\text{d}z\\ E_{31}^{2}=\int_{-h/2}^{h/2}\frac{\varepsilon_{31}}{\varepsilon_{33}}z(z-z_{ 0})\text{d}z\quad M_{e}=\int_{-h/2}^{h/2}e_{31}(z-z_{0})\frac{\varepsilon_{0} }{h}\text{d}z\\ M_{T}=\int_{-h/2}^{h/2}\beta_{1}\Delta T(z-z_{0})\text{d}z\quad I_{0}=\int_{A }\rho(z)\cdot(z-z_{0})\text{d}A\\ I_{2}=\int_{A}\rho(z)\cdot(z-z_{0})^{2}\text{d}z\end{array} \right. \tag{22}\]  
From Equation (21), we can obtain the free vibration Equation (23) and the static bending Equation (24) of the rectangular plate, respectively:  
\[ \begin{array}{l}(-D_{11}+E_{31}^{1}-E_{31}^{2})\frac{\partial^{4}w}{ \partial x^{4}}+(-D_{11}+E_{31}^{1}-E_{31}^{2})\frac{\partial^{4}w}{ \partial y^{4}}+(-2D_{12}+2E_{31}^{1}-2E_{31}^{2}-4D_{66})\frac{\partial^{4}w} {\partial x^{2}\partial y^{2}}+N_{x}\frac{\partial^{2}w}{\partial x^{2}}+N_{y }\frac{\partial^{2}w}{\partial y^{2}}\\ -kw=-I_{0}\frac{\partial^{2}w}{\partial t^{2}}+I_{2}\frac{\partial^{4}w}{ \partial x^{2}\partial t^{2}}+I_{2}\frac{\partial^{4}w}{\partial y^{2}\partial t ^{2}}\end{array}