## Appendix A  
The first variation of kinetic energy can be achieved as:  
\[\delta K=\int\limits_{0}^{a}\int\limits_{0}^{b}\left[ \begin{array}{c}\left(-I _{0}\frac{\partial^{2}u_{0}}{\partial t^{2}}-I_{1}\frac{\partial^{2}\phi_{y}} {\partial t^{2}}+I_{3}c_{1}\Big{(}\frac{\partial^{2}\phi_{y}}{\partial t^{2}} +\frac{\partial^{3}w}{\partial t^{2}\partial y}\Big{)}\right)\delta\phi_{x}\\ +\Big{(}-I_{1}\frac{\partial^{2}v_{0}}{\partial t^{2}}-I_{2}\frac{\partial^{2} \phi_{y}}{\partial t^{2}}+I_{4}c_{1}\Big{(}\frac{\partial^{2}\phi_{y}}{ \partial t^{2}}+\frac{\partial^{3}w}{\partial t^{2}\partial y}\Big{)}\Big{)} \delta\phi_{y}\\ +\Big{(}c_{1}I_{3}\frac{\partial^{2}v_{0}}{\partial t^{2}}+c_{1}I_{4}\frac{ \partial^{2}\phi_{y}}{\partial t^{2}}-I_{6}c_{1}{}^{2}\Big{(}\frac{\partial^{ 2}\phi_{y}}{\partial t^{2}}+\frac{\partial^{3}w}{\partial t^{2}\partial y} \Big{)}\Big{)}\delta\phi_{y}\\ +\Big{(}c_{1}I_{3}\frac{\partial^{2}v_{0}}{\partial t^{2}}+c_{1}I_{4}\frac{ \partial^{2}\phi_{y}}{\partial t^{2}}-I_{6}c_{1}{}^{2}\Big{(}\frac{\partial^{ 2}\phi_{y}}{\partial t^{2}}+\frac{\partial^{3}w}{\partial t^{2}\partial y} \Big{)}\Big{)}\delta\phi_{y}\\ +\Big{(}-c_{1}I_{3}\frac{\partial^{2}v_{0}}{\partial y\partial t^{2}}-c_{1}I_{4 }\frac{\partial^{3}\phi_{y}}{\partial y\partial t^{2}}+I_{6}c_{1}{}^{2}\Big{(} \frac{\partial^{3}\phi_{y}}{\partial y\partial t^{2}}+\frac{\partial^{4}w}{ \partial t^{2}\partial y^{2}}\Big{)}\Big{)}\delta\omega_{0}\\ +\Big{(}-I_{0}\frac{\partial^{2}w_{0}}{\partial t^{2}}\Big{)}\delta w_{0}\end{array}  dA\right. \tag{16}\]  
in which  
\[\{I_{i}\}=\int\limits_{-h/2}^{h/2}\rho\Big{\{}z^{i}\Big{\}}(1+\frac{z}{R_{1}}) (1+\frac{z}{R_{2}})dz,\quad i=1:6 \tag{17}\]  
The parameters that are used in Equation (10) can be formulated as follows:  
\[\begin{array}{l}\Big{\{}N_  
where:  
\[ \begin{split}&\{A_{11},B_{11},C_{11},D_{11},E_{11},F_{11},G_{11}\}= \int\limits_{-h/2}^{h/2}Q_{11}\times\left\{1,z,z^{2},z^{3},z^{4},z^{5},z^{6} \right\}\times(1+z/R_{1})\times(1+z/R_{2})dz,\\ &\{A_{12},B_{12},C_{12},D_{12},E_{12},F_{12},G_{12}\}=\int\limits_ {-h/2}^{h/2}Q_{12}\times\left\{1,z,z^{2},z^{3},z^{4},z^{5},z^{6}\right\}\times(1 +z/R_{1})\times(1+z/R_{2})dz,\\ &\{A_{22},B_{22},C_{22},D_{22},E_{22},F_{22},G_{22}\}=\int\limits_ {-h/2}^{h/2}Q_{22}\times\left\{1,z,z^{2},z^{3},z^{4},z^{5},z^{6}\right\}\times(1 +z/R_{1})\times(1+z/R_{2})dz,\\ &\{A_{44},B_{44},C_{44},D_{44},E_{44}\}=\int\limits_{-h/2}^{h/2}Q_ {44}\times\left\{1,z,z^{2},z^{3},z^{4}\right\}\times(1+z/R_{1})\times(1+z/R_{ 2})dz,\\ &\{A_{55},B_{55},C_{55},D_{55},E_{55}\}=\int\limits_{-h/2}^{h/2}Q_ {55}\times\left\{1,z,z^{2},z^{3},z^{4}\right\}\times(1+z/R_{1})\times(1+z/R_{ 2})dz,\\ &\{A_{66},B_{66},C_{66},D_{66},E_{66}\}=\int\limits_{-h/2}^{h/2}Q_ {66}\times\left\{1,z,z^{2},z^{3},z^{4}\right\}\times(1+z/R_{1})\times(1+z/R_{ 2})dz.\end{split}  \tag{100}\]  
The governing equations can be derived as below:  
\[\begin{split}&\delta u_{0}:\left(1-\
abla^{2}l^{2}\right)\left( \frac{\partial N_{\text{xx}}}{\partial x}+\frac{\partial N_{\text{xy}}}{ \partial y}\right)=\left(1-\
abla^{2}\mu^{2}\right)\left(l_{0}\frac{\partial^{ 2}u_{0}}{\partial l^{2}}+I_{1}\frac{\partial^{2}\phi_{x}}{\partial l^{2}}-3c_{ 1}\left(\frac{\partial^{2}\phi_{x}}{\partial l^{2}}+\frac{\partial^{2}w_{0}}{ \partial l^{2}}\right)\right),\\ &\delta v_{0}:\left(1-\
abla^{2}l^{2}\right)\left(\frac{\partial N _{\text{yy}}}{\partial y}+\frac{\partial N_{\text{xy}}}{\partial x}\right)= \left(1-\
abla^{2}\mu^{2}\right)\left(l_{0}\frac{\partial^{2}w_{0}}{\partial l ^{2}}+I_{1}\frac{\partial^{2}\phi_{y}}{\partial l^{2}}-3c_{1}\left(\frac{ \partial^{2}\phi_{y}}{\partial l^{2}}+\frac{\partial^{2}w_{0}}{\partial l^{2} \partial y}\right)\right),\\ &\delta w_{0}:\left(1-\
abla^{2}l^{2}\right)\left(