## Appendix A  
The non-null strain components are defined as  
Figure 15: Dimensionless deflection vs. GPL weight fraction, for different Kerr foundation coefficients (\(b/a=1\), \(a/h=10\), \(GPL-X\), \(p_{0}=10^{5}\) and \(\text{K}_{\text{I}}=10^{5}\)): (**a**) SSSS boundary conditions, (**b**) CCCC boundary conditions. \[\left\{ \begin{array}{l}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{yz}\\ \gamma_{xz}\\ \gamma_{xy}\end{array} \right\}=\left\{ \begin{array}{l}\varepsilon_{xx}^{0}\\ \varepsilon_{yy}^{0}\\ \gamma_{yz}^{0}\\ \gamma_{xz}^{0}\\ \gamma_{xy}^{0}\end{array} \right\}+z\left\{ \begin{array}{l}k_{xx}^{0}\\ k_{yy}^{0}\\ k_{xz}^{0}\\ k_{xz}^{0}\\ k_{xy}^{0}\end{array} \right\}+z^{2}\left\{ \begin{array}{l}\varepsilon_{xx}^{ *}\\ \varepsilon_{yy}^{*}\\ \gamma_{yz}^{*}\\ \gamma_{xz}^{*}\\ \gamma_{xy}^{*}\end{array} \right\}+z^{3}\left\{ \begin{array}{l}k_{xx}^{*} \\ k_{yy}^{*}\\ k_{yz}^{*}\\ k_{xz}^{*}\\ k_{xy}^{*}\end{array} \right\} \tag{10}\]  
where  
\[\left\{ \begin{array}{l}\varepsilon_{xx}^{0}\\ \varepsilon_{yy}^{0}\\ \gamma_{yz}^{0}\\ \gamma_{xy}^{0}\\ \gamma_{xy}^{0}\\ \gamma_{xy}^{0}\end{array} \right\}=\left\{ \begin{array}{l}\frac{\partial u_ {0}}{\partial x}\\ \frac{\partial v_{0}}{\partial y}\\ \gamma_{xz}^{0}\\ \frac{\partial u_{1}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\left\{ \begin{array}{l}k_{xx}^{0}\\ k_{yy}^{0}\\ k_{xz}^{0}\\ k_{xy}^{0}\\ \frac{\partial u_{1}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\right\}\left\{ \begin{array}{l}k_{xx}^{0}\\ k_{yy}^{0}\\ k_{xz}^{0}\\ k_{xy}^{0}\end{array} \right\}=\left\{ \begin{array}{l}\frac{\partial u_{1}}{ \partial x}\\ \frac{\partial v_{1}}{\partial y}\\ \gamma_{yz}^{*}\\ \gamma_{xz}^{*}\\ \frac{\partial u_{1}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\right\}=\left\{ \begin{array}{l}\frac{\partial u_{1}}{\partial x} \\ \frac{\partial v_{1}}{\partial y}\\ 3u_{3}\\ \frac{\partial u_{2}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\cdot\left\{ \begin{array}{l}\varepsilon_{xx}^{*}\\ \varepsilon_{yy}^{*}\\ 3v_{yz}\\ 3u_{3}\\ \frac{\partial u_{3}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\left\{ \begin{array}{l}\varepsilon_{xx}^{*}\\ \varepsilon_{yy}^{*}\\ 3v_{yz}\\ 3u_{3}\\ \frac{\partial u_{1}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\left\{ \begin{array}{l}\varepsilon_{yy}^{*}\\ \gamma_{yz}^{*}\\ \gamma_{xz}^{*}\\ \gamma_{xy}^{*}\end{array} \right\}=\left\{ \begin{array}{l}\frac{\partial u _{2}}{\partial x}\\ \frac{\partial v_{2}}{\partial y}\\ 3v_{3}\\ 3u_{3}\\ \frac{\partial u_{3}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\}\left\{ \begin{array}{l}k_{xx}^{*}\\ k_{yy}^{*}\\ k_{xz}^{*}\\ k_{xy}^{*}\end{array} \right\}=\left\{ \begin{array}{l}\frac{\partial u_{3}}{ \partial x}\\ \frac{\partial v_{3}}{\partial y}\\ 0\\ 0\\ \frac{\partial u_{3}}{\partial y}+\frac{\partial v_{0}}{\partial x}\end{array}  \right\} \tag{11}\]  
The elastic constants used in Equation (7) are introduced as  
\[Q_{11}=\frac{E_{C}}{1-\
u_{C}^{2}},\ Q_{12}=\frac{\
u_{C}E_{C}}{1-\
u_{C}^{2}},Q _{13}=Q_{23}=Q_{12},\ Q_{33}=  
\[H=\frac{Ln\Big{[}\rho\Big{(}\rho+\sqrt{\rho^{2}-1}\Big{)}\Big{]}}{\sqrt{\big{(} \rho^{2}-1\big{)}^{3}}}-\frac{1}{\rho^{2}-1},\] (A22)  
\[K_{x}=\frac{K_{g}}{\frac{2R_{k}K_{g}}{L}+1},\] (A23)  
\[K_{z}=\frac{K_{g}}{\frac{2R_{k}K_{g}}{L}+1},\] (A24)  
\[\rho=\frac{l_{GPL}}{l_{GPL}}.\] (A25)  
The constants in Equation (23) are determined as  
\[C_{11}=\frac{\left| \begin{array}{cc}T_{1}&T_{2}\\ 1&e^{-\sqrt{A_{1}}h}\\ \hline\hline\end{array} \right|}{\left| \begin{array}{cc}1&1\\ e^{\sqrt{A_{1}}h}&e^{-\sqrt{A_{1}}h}\\ \end{array} \right|}\,\ \ C_{22}=\frac{\left| \begin{array}{cc}T_{2}&T_{1}\\ e^{\sqrt{A_{1}}h}&1\\ \hline\end{array} \right|}{\left| \begin{array}{cc}1&1\\ e^{\sqrt{A_{1}}h}&e^{-\sqrt{A_{1}}h}\\ \end{array}