## Application of Perturbation Method
### Perturbation Solution on \(W_{m}\)  
Now, we introduce the following perturbation parameter  
\[W_{m}=\left(W\right)_{\eta=1}=\left(\frac{w}{t}\right)_{t=0}=\frac{w_{0}}{t} \tag{45}\]where \(w_{0}\) is the center deflection of the plate, that is, the maximum deflection. If \(W_{m}\) is selected as the perturbation parameter, \(P\), \(W\), and \(S\) in the governing equations may be expanded as  
\[\left\{ \begin{array}{l}P/16=P_{1}W_{m}+P_{3}W_{m}^{3}+P_{5}W_{m}^{5}+\cdots\\ W=W_{1}(\eta)W_{m}+W_{3}(\eta)W_{m}^{3}+W_{5}(\eta)W_{m}^{5}+\cdots\\ S=S_{2}(\eta)W_{m}^{2}+S_{4}(\eta)W_{m}^{4}+S_{6}(\eta)W_{m}^{6}+\cdots\end{array} \right. \tag{46}\]  
where \(W_{i}(i=1,3,5,\ldots)\) and \(S_{i}(i=2,4,6,\ldots)\) are unknown functions of \(\eta\), and \(P_{i}(i=1,3,5,\ldots)\) is unknown constants, i.e., \(P=P(W_{m})\), \(W=W(W_{m},\eta)\) and \(S=S(W_{m},\eta)\). By substituting the expressions into the governing Equations (39) and (40), as well as into the boundary conditions (42) and (44), we may obtain a series of decomposed differential equations and the corresponding boundary conditions used for the solution of \(P\), \(W\), and \(S\). In this case, the even exponential terms with respect to \(W_{m}\) in the expansion of \(P\) and \(W\), and the odd terms in \(S\) are not considered because they will be eliminated during the perturbation. By comparing the coefficients of \(W_{m}^{1}\) in Equation (39), we obtain the differential equation for \(W_{1}\) and \(P_{1}\)  
\[\frac{d^{2}}{d\eta^{2}}\left[(1-\eta)\frac{dW_{1}}{d\eta}\right]=-\frac{P_{1}} {K} \tag{47}\]  
which may be solved under the boundary conditions  
\[\left\{ \begin{array}{l}W_{1}=0,\;\lambda_{1}\frac{d^{2}W_{1}}{d\eta^{2}}- \frac{dW_{1}}{d\eta}=0,\;\mbox{at}\;\eta=0\\ W_{1}=1,\;\frac{dW_{1}}{d\eta}\
eq\infty,\;\mbox{at}\;\eta=1\end{array} \right.. \tag{48}\]  
Thus, we obtain  
\[W_{1}=\frac{\eta^{2}+2\lambda_{1}\eta}{2\lambda_{1}+1},\;P_{1}=\frac{4K}{2 \lambda_{1}+1} \tag{49}\]  
By comparing the coefficients of \(W_{m}^{2}\) in Equation (40), we obtain the differential equation used for \(S_{2}\)  
\[\frac{d^{2}}{d\eta^{2}}[(1-\eta)S_{2}]+\frac{V}{2}\left(\frac{dW_{1}}{d\eta} \right)^{2}=0 \tag{50}\]  
which may be solved under the boundary conditions  
\[\left\{ \begin{array}{l}\lambda_{2}\frac{dS_{2}}{d\eta}-S_{2}=0,\;\mbox{at} \;\eta=0\\ S_{2}\
eq\infty,\;\mbox{at}\;\eta=1\end{array} \right.. \tag{51}\]  
Using the determined \(W_{1}\), we obtain  
\[S_{2}=\frac{V}{6(2\lambda_{1}+1)^{2}}[\eta^{3}+(4\lambda_{1}+1)\eta^{2}+(6 \lambda_{1}{}^{2}+4\lambda_{1}+1)\eta+\lambda_{2}(6\lambda_{1}{}^{2}+4 \lambda_{1}+1)] \tag{52}\]  
Similarly, by comparing the coefficients of \(W_{m}^{3}\) in Equation (39), we have the differential equation for \(W_{3}\) and \(P_{3}\)  
\[\frac{d^{2}}{d\eta^{2}}\left[(1-\eta)\frac{dW_{3}}{d\eta}\right]=\frac{S_{2}} {4K}\frac{dW_{1}}{d\eta}-\frac{P_{3}}{K} \tag{53}\]  
which may be solved by the boundary conditions as follows,  
\[\left\{ \begin{array}{l}W_{3}=0,\;\lambda_{1}\frac{d^{2}W_{3}}{d\eta^{2}}- \frac{dW_{3}}{d\eta}=0,\;\mbox{at}\;\eta=0\\ W_{3}=0,\;\frac{dW_{3}}{d\eta}\
eq\infty,\;\mbox{at}\;\eta=1\end{array} \right.. \tag{54}\]Using the determined \(W_{1}\) and \(S_{2}\), we obtain  
\[\left\{ \begin{array}{l}W_{3}=\frac{V}{4320K(2{\lambda_{1}}+1)^{4}} \left\{\begin{array}{l}-2(2{\lambda_{1}}+1)\eta^{6}-6(6{\lambda_{1}}^{2}+5{ \lambda_{1}}+1)\eta^{5}\\ -15(10{\lambda_{1}}^{3}+13{\lambda_{1}}^{2}+6{\lambda_{1}}+1)\eta^{4}\\ -20[12{\lambda_{1}}^{4}+24{\lambda_{1}}^{3}+19{\lambda_{1}}^{2}+7{\lambda_{1}} +1\\ +{\lambda_{2}}(12{\lambda_{1}}^{3}+14{\lambda_{1}}^{2}+6{\lambda_{1}}+1)]\eta^{3 }\\ -[120{\lambda_{1}}^{3}+255{\lambda_{1}}^{2}+178{\lambda_{1}}+43\\ +20{\lambda_{2}}(6{\lambda_{1}}^{2}+4{\lambda_{1}}+1)](\eta^{2}+2{\lambda_{1}} \eta)\end{array}\right.\end{array} \right.