## 2 The Governing Equation and General Solution
### Governing Equations  
While the through-the-radius FGM circular plates exhibit heterogeneity, their stretching and bending behaviors are decoupled due to mid-surface symmetry. Considering small deformations about the plate's thickness, the strains in the \((r,\theta,z)\) directions of the FGM circular plate expressed as bending displacements \(w\) are \(\{\varepsilon_{r},\,\varepsilon_{\theta},\,\gamma_{r\theta}\}=z\{\kappa_{r}, \kappa_{\theta},\kappa_{r\theta}\}\), where \(\kappa_{r},\,\kappa_{\theta},\,\kappa_{r\theta}\) are curvatures of the FGM plate. For the non-homogeneous elastic FGM plate with [PERSON]'s modulus \(E(r)\) and [PERSON]'s ratio \(\upsilon\), the stress-strain relation under thermal loading \(T(r,z)\) based on the assumptions of small deformation can be described as follows [2]:  
\[\left\{ \begin{aligned} &\sigma_{r}\\ &\sigma_{\theta}\\ &\tau_{r\theta}\end{aligned} \right\}=\frac{E(r)}{1-\upsilon^{2 }} \begin{bmatrix}1&\
u&0\\ \upsilon&1&0\\ 0&0&(1-\upsilon)/2\end{bmatrix} \left\{ \begin{aligned} & \varepsilon_{r}\\ &\varepsilon_{\theta}\\ & \tau_{r\theta}\end{aligned} \right\}-\frac{E(r)\kappa T(r,z)}{1-\upsilon} \left\{ \begin{aligned} & 1\\ & 0\end{aligned} \right\} \tag{1}\]  
where \(\alpha\) is the thermal expansion coefficient. The definitions of the bending moments \((M_{r},M_{\theta},M_{r\theta})=\int_{-h/2}^{h/2}(z\sigma_{r},z\sigma_{ \theta},z\tau_{r\theta})dz\) yield  
\[\left\{ \begin{aligned} & M_{r}\\ & M_{\theta}\\ & M_{r\theta}\end{aligned} \right\}=\frac{E(r)h^{3}}{12(1-\upsilon^ {2})} \begin{bmatrix}1&\
u&0\\ \
u&1&0\\ 0&0&(1-\upsilon)/2\end{bmatrix} \left\{ \begin{aligned} & \kappa_{r}\\ &\kappa_{\theta}\\ & \kappa_{r\theta}\end{aligned} \right\}-\left\{ \begin{aligned} & M^{\Delta T}\\ & M^{\Delta T}\\ & 0\end{aligned} \right\}, \tag{2}\]where the entry \(M^{\Delta T}\) is of the following form:  
\[M^{\Delta T}=\frac{\alpha E(r)}{1-v}\int_{-h/2}^{h/2}zT(r,z)dz. \tag{3}\]  
The equilibrium equations in the \(r\)-, \(\theta\)-, and z-directions of the elastic solid subjected to the transverse load \(q_{z}\) are [34]  
\[\frac{\partial N_{r}}{\partial r}+\frac{\partial N_{r\theta}}{\partial\theta} +\frac{N_{r}-N_{\theta}}{r}=0,\ \frac{\partial N_{r\theta}}{\partial r}+\frac{ \partial N_{\theta}}{r\partial\theta}+\frac{2}{r}N_{r\theta}=0,\ \frac{\partial V_{r}}{\partial r}+\frac{V_{r}}{r}+\frac{1}{r}\frac{ \partial V_{\theta}}{\partial\theta}=-q_{z} \tag{4}\]  
where \(V_{r}\) and \(V_{\theta}\) are the transverse shear forces acting on the radial and tangential surfaces, respectively, and they are  
\[V_{r}=\frac{\partial M_{r}}{\partial r}+\frac{M_{r}-M_{\theta}}{r}+\frac{1}{ r}\frac{\partial M_{r\theta}}{\partial\theta},\ \ \ V_{\theta}=\frac{\partial M_{r\theta}}{\partial r}+2\frac{M_{r\theta}}{r}+ \frac{1}{r}\frac{\partial M_{\theta}}{\partial\theta}. \tag{5}\]  
Substituting Equation (5) into Equation (4) gives the equilibrium equation in the \(z\)-direction in terms of the moments:  
\[\frac{\partial^{2}M_{r}}{\partial r^{2}}+\frac{2}{r}\frac{\partial^{2}M_{r \theta}}{\partial r\partial\theta}+\frac{1}{r^{2}}\frac{\partial^{2}M_{ \theta}}{\partial\theta^{2}}+\frac{2}{r}\frac{\partial M_{r}}{\partial r}+ \frac{2}{r^{2}}\frac{\partial M_{r\theta}}{\partial\theta}-\frac{1}{r}\frac{ \partial M_{\theta}}{\partial r}=-q_{z}. \tag{6}\]  
Further, assume that the FGM circular plate exhibits radial variations in its Young's modulus following the power function:  
\[E(r)=E_{0}(\frac{r}{R})^{n}. \tag{7}\]  
In the absence of a transverse load \(q_{z}\), the substitution of Equations (2)-(6), along with the assistance of Equation (3), results in the bending equilibrium equation for FGM circular plates with a through-the-radius variation of Young's modulus \(E(r)=E_{0}(r/R)^{n}\), expressed in terms of displacement:  
\[