# Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates
## 3 Closed Form Solutions of Thermal Buckling  
After establishing the thermal buckling governing equation, the material model and temperature fields, the buckling solutions including thermal buckling mode function and critical buckling temperature are obtained below in separation-of-variable form. Assuming that Equation (12) has the following separation-of-variable solution:  
\[w(x,y)=\phi(x)\psi(y) \tag{24}\]  
where \(w\) is the buckling mode function, and  
\[\phi(x)=A\mathrm{e}^{\theta x},\psi(y)=B\mathrm{e}^{\eta y} \tag{25}\]  
where \(\theta\) and \(\eta\) are unknown spatial eigenvalues. Substituting Equations (24) and (25) into Equation (12) yields the relation of \(\theta\), \(\eta\) and \(\dot{G}_{0}\) as \[\Big{(}\hat{G}_{0}+\theta^{2}+\eta^{2}\Big{)}\Big{(}\theta^{2}+\eta^{2}\Big{)}=0 \tag{26}\]  
From Equation (26), we have the following relations  
\[ \begin{split}\hat{\sigma}_{1,2}&=\pm\alpha_{2}=\pm \sqrt{-\eta^{2}}\\ \hat{\sigma}_{3,4}&=\pm\mathrm{i}\alpha_{1}=\pm \mathrm{i}\sqrt{\eta^{2}+\hat{G}_{0}}\\ \eta_{1,2}&=\pm\beta_{2}=\pm\sqrt{-\theta^{2}}\\ \eta_{3,4}&=\pm\mathrm{i}\beta_{1}=\pm\sqrt{\theta^ {2}+\hat{G}_{0}}\end{split}  \tag{27}\]  
Therefore, the functions \(\phi(x)\) and \(\psi(y)\) have the following closed forms:  
\[ \begin{split}\phi(x)&=A_{2}\cos\alpha_{1}x+B_{2} \sin\alpha_{1}x+C_{2}\cosh\alpha_{2}x+D_{2}\mathrm{sinh}\alpha_{2}x\\ \psi(y)&=A_{1}\cos\beta_{1}y+B_{1}\sin\beta_{1}y+C_ {1}\cosh\beta_{2}y+D_{1}\mathrm{sinh}\beta_{2}y\end{split}  \tag{28}\]  
where the five unknown variables \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta_{2}\) and \(\hat{G}_{0}\) are involved; then, four equations are needed for a unique solution. From Equation (27), one can obtain three relations of these five variables:  
\[ \begin{split}\alpha_{1}^{2}+\alpha_{2}^{2}&=\hat{G} _{0}\\ \beta_{1}^{2}+\beta_{2}^{2}&=\hat{G}_{0}\end{split}  \tag{29}\]  
\[ \begin{split}\alpha_{1}^{2}+\beta_{1}{}^{2}=\theta^{2}+\eta^{2}+2 \hat{G}_{0}\end{split}  \tag{30}\]  
Substituting \(\hat{\sigma}=\pm\mathrm{i}\alpha_{1}\), \(\eta=\pm\mathrm{i}\beta_{1}\) into Equation (30) yields  
\[ \begin{split}\alpha_{1}{}^{2}+\beta_{1}{}^{2}=\hat{G}_{0}\end{split}  \tag{31}\]  
Another two transcendental eigenvalue equations can be derived by using the two pairs of boundary conditions of four edges. The boundary conditions used in this study are presented in Table 1 where \(n\) denotes the outer normal direction of the edge. The buckling mode functions have the separation-of-variable form as shown in Equation (24); then, the two transcendental eigenvalue equations can be derived separately by considering the boundary conditions with respect to \(x\) and \(y\), respectively. Substituting the two eigenfunctions in Equation (28) into the boundary conditions given in Table 1 yields the transcendental eigenvalue equations which are shown in Table 2[25; 26; 27]. By using Equation (29), Equation (31) and two transcendental eigenvalue equations corresponding to specific boundary conditions of a rectangular plate, we can solve the five variables \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta_{2}\) and \(\hat{G}_{0}\). For clarity of discussion, the solved \(\hat{G}_{0}\) from eigenvalue equations is denoted by \(\hat{G}_{0}\). \begin{table}
\begin{tabular}{c c c} \hline Simple Support (S) & \(w=0\) & \(M_{n}=0\) \\ Guide (G) & \(\dfrac{\partial w}{\partial n}=0\) & \(Q_{n}=0\) \\ Clamp (C) & \(w=0\) & \(\dfrac{\partial w}{\partial n}=0\) \\ \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions of a rectangular plate. Note that \(\hat{G}_{0\text{cr}}\) is a critical thermal buckling parameter in which the critical temperature is involved, but in general it is hard or impossible to directly extract the explicit form of the critical buckling temperature \(T_{\text{cr}}\) from \(\hat{G}_{0\text{cr}}\). In the present study, a novel method of determining \(T_{\text{cr}}\) is proposed.