## 3 The Plate Theories (RPT and IRPT)
\right\}^{(k)} \tag{15}\]  
where \(C_{ij}^{(k)}\) is the three-dimensional elastic constant as follows:  
\[C_{11}^{(k)}=C_{22}^{(k)}=C_{33}^{(k)}=\frac{\Big{(}1-\upsilon_{c}^{(k)} \Big{)}E_{c}^{(k)}}{\Big{(}1-2\upsilon_{c}^{(k)}\Big{)}\Big{(}1+\upsilon_{c}^ {(k)}\Big{)}} \tag{16}\]  
\[C_{12}^{(k)}=C_{13}^{(k)}=C_{23}^{(k)}=\frac{\upsilon_{c}^{(k)}E_{c}^{(k)}}{ \Big{(}1-2\upsilon_{c}^{(k)}\Big{)}\Big{(}1+\upsilon_{c}^{(k)}\Big{)}} \tag{17}\]  
\[C_{44}^{(k)}=C_{55}^{(k)}=C_{66}^{(k)}=\frac{E_{c}^{(k)}}{2\Big{(}1+\upsilon_ {c}^{(k)}\Big{)}} \tag{18}\]  
If the thickness stretching effect is neglected, the stress-strain relationship is shown below. \[\left\{ \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \tau_{xy}\\ \tau_{xz}\\ \tau_{yz}\end{array} \right\}^{(k)}=\frac{E_{c}^{(k)}}{1-\Big{[}\upsilon_{c} ^{(k)}\Big{]}^{2}}\left[ \begin{array}{cccc}1&\upsilon_{c}^{(k)}&0&0&0\\ v_{c}^{(k)}&1&0&0&0\\ 0&0&\frac{1-\upsilon_{c}^{(k)}}{2}&0&0\\ 0&0&0&0&\frac{1-\upsilon_{c}^{(k)}}{2}\end{array} \right]\left\{ \begin{array} []{c}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\\ \gamma_{xz}\\ \gamma_{yz}\end{array} \right\}^{(k)} \tag{19}\]  
The virtual strain energy of the system is as follows:  
\[\delta U=\int_{\Omega}\sum_{k=1}^{N_{l}}\int_{z_{k}}^{z_{k+1}}\Big{(}\sigma_{ xx}^{(k)}\delta\varepsilon_{xx}^{(k)}+\sigma_{yy}^{(k)}\delta\varepsilon_{yy}^{(k)}+ \sigma_{zz}^{(k)}\delta\varepsilon_{zz}^{(k)}+\tau_{xy}^{(k)}\delta\tau_{xy}^{ (k)}+\tau_{zz}^{(k)}\delta\gamma_{xz}^{(k)}+\tau_{yz}^{(k)}\delta\gamma_{yz}^{ (k)}\Big{)}\text{d}z\text{d}x\text{d}y \tag{20}\]  
The variation in work carried out by the external forces \(q\) is as follows:  
\[\delta V=-\int_{\Omega}\sum_{k=1}^{N_{l}}\int_{z_{k}}^{z_{k+1}}q(\delta w_{b} +\delta w_{s}+g(z)\delta w_{z})\text{d}z\text{d}x\text{d}y \tag{21}\]  
where \(q\) represents the external load in the transverse direction. The virtual kinetic energy of the system is as follows:  
\[\delta K=\int_{\Omega}\sum_{k=1}^{N_{l}}\int_{z_{k}}^{z_{k+1}}\rho_{c}^{(k)}( \dot{u}\delta\dot{u}+\dot{v}\delta\dot{v}+\dot{w}\delta\dot{w})\text{d}z\text{ d}x\text{d}y \tag{22}\]  
In this paper, the dot-superscript convention denotes the differential with respect to the time variable. If the effect of thickness stretching is ignored, then:\[\delta U=\int_{\Omega}\sum_{k=1}^{N_{L}}\int_{z_{k}}^{z_{k+1}}\Big{(}\sigma_{xx}^{( k)}\delta\epsilon_{xx}^{(k)}+\sigma_{yy}^{(k)}\delta\epsilon_{gy}^{(k)}+\tau_{xy}^{(k)} \delta\gamma_{xy}^{(k)}+\tau_{zz}^{(k)}\delta\gamma_{xz}^{(k)}+\tau_{yz}^{(k)} \delta\gamma_{yz}^{(k)}\Big{)}\mathrm{d}z\mathrm{d}x\mathrm{d}y \tag{23}\]  
\[\delta V=-\int_{\Omega}\sum_{k=1}^{N_{L}}\int_{z_{k}}^{z_{k+1}}q(\delta w_{b} +\delta w_{s})\mathrm{d}z\mathrm{d}x\mathrm{d}y \tag{24}\]  
\[\delta K=\int_{\Omega}\sum_{k=1}^{N_{L}}\int_{z_{k}}^{z_{k+1}}\rho_{c}^{(k)}( i\omega\dot{u}+\dot{v}\dot{\delta v}+\dot{w}\dot{w})\mathrm{d}z\mathrm{d}x \mathrm{d}y \tag{25}\]  
[PERSON]'s principle is as follows:  
\[\int_{0}^{\mathrm{T}}\delta(U+V-K)\mathrm{d}t=0\]  
Substituting the expressions from Equations (20)-(22) into Equation (26), we obtain the governing equations under IRPT as follows:  
\[ \begin{array}{l}A_{11}\frac{\partial^{2}u}{\partial x^{2}}+A_{66}\frac{ \partial^{2}u}{\partial y^{2}}+(A_{12}+A_{66})\frac{\partial^{2}v}{\partial x \partial y}-B_{11}\frac{\partial^{3}w_{b}}{\partial x^{3}}-(B_{12}+2B_{66}) \frac{\partial^{3}w_{b}}{\partial x\partial y^{2}}-\\ B_{11}^{3}\frac{\partial^{3}w_{b}}{\partial x^{3}}-\big{(}B_{12}^{5}+2B_{66}^{ 5}\big{)}\frac{\partial^{3}w_{b}}{\partial x\partial y^{2}}+X_{13}\frac{ \partial w_{b}}{\partial x}=I_{0}\ddot{u}-I_{1}\frac{\partial\ddot{w}_{b}}{ \partial x}-I_{1}\frac{\partial\ddot{w}_{b}}{\partial x}\end{array}  \tag{27}\]  
\[