## 2 The Governing Equation and General Solution
### The FGM Annular Portion of the FGM-Coated Circular Plate: \(R_{1}\leq r\leq R\)  
The general solutions of the FGM annular portion can be directly established from Equations (12) and (14), and they are  
\[w(r)=\begin{cases}A_{21}+A_{22}r^{2}+A_{23}\ln r+A_{24}r^{2}\ln r+A_{26}r^{m+2},&\text{for $n=0$}\\ A_{21}+A_{22}r^{\lambda_{2}}\ln r+A_{23}r^{\lambda_{3}}+A_{24}r^{\lambda_{4}}+A_ {25}r^{2}+A_{26}r^{m+2},&\text{for $n=2$}\\ A_{21}+A_{22}r^{\lambda_{2}}+A_{23}r^{\lambda_{3}}+A_{24}r^{\lambda_{4}}+A_{25}r^{ 2}+A_{26}r^{m+2},&\text{for $n\
eq 0$ or $n\
eq 2$}\end{cases} \tag{19}\]  
where \(A_{21},\ A_{22},\ A_{23},\ A_{24}\) are unknown constants; \(A_{25}\) and \(A_{26}\) are the coefficients of a particular solution, related to Equation (15) as  
\[A_{25}=\frac{-D}{2(1+\
u)},\ A_{26}=\frac{-(m+n)^{2}D}{R^{m}\xi}. \tag{20}\]  
Consequently, the stresses and bending moments of the FGM annular portion evaluated according to Equations (1)-(3) are  
\[\sigma_{j}=\sigma_{j1}-\sigma_{\text{T1}},\ M_{j}=\frac{h^{3}}{12z}\sigma_{j1 }-M_{\text{T1}},j=r,\theta \tag{21}\]  
where  
\[\begin{array}{l}\sigma_{r1}=-\frac{zE_{0}}{1-z^{2}}\big{\{}2(1+\
u)A_{22}+( \
u-1)A_{23}r^{-2}+[2(1+v)\ln r+\
u+3]A_{24}+m_{\
u 1}A_{26}r^{m}\big{\}},\\ \sigma_{\theta 1}=-\frac{zE_{0}}{1-z^{2}}\big{\{}2(1+v)A_{22}+(1-v)A_{22}r^{-2}+[2(1+v) \ln r+3v+1]A_{24}+m_{\
u 2}A_{26}r^{m}\big{\}},\\ \sigma_{\text{T1}}=\frac{E_{0}r^{2}}{1-v}(1+\overline{r}^{m})\overline{z}^{2 +D+1},\ M_{\text{T1}}=\frac{E_{0}h^{3}D}{12(1-v^{2})}(1+\overline{r}^{m})\end{array}\]  
for \(n=0\),  
\[\sigma_{j}=\sigma_{j2}-\sigma_{\text{T2}},\ M_{j}=\frac{h^{3}}{12z}\sigma_{j2 }-M_{\text{T2}},j=r,\theta \tag{22}\]  
where  
\[\begin{array}{l}\sigma_{r2}=-\frac{zE_{0}\overline{r}^{2}}{1-v^{2}}\big{\{} -(1-v)A_{22}r^{-2}+\lambda_{3v}A_{23}r^{\lambda_{3}-2}+\lambda_{4v}A_{24}r^{ \lambda_{4}-2}+2(1+v)A_{25}+m_{\
u 1}A_{26}r^{m}\big{\}},\\ \sigma_{\theta 2}=-\frac{zE_{0}\overline{r}^{2}}{1-v^{2}}\big{\{}(1-v)A_{22}r^{-2}+ \lambda_{3v}A_{23}r^{\lambda_{3}-2}+\lambda_{4v}A_{24}r^{\lambda_{4}-2}+2(1+v )A_{25}+m_{\
u 2}A_{26}r^{m}\big{\}},\\ \sigma_{\tau 2}=\frac{E_{0}\overline{r}^{2}A_{T0}}{1-v}(1+\overline{r}^{m}) \overline{z}^{2D+1},\ M_{\text{T2}}=\frac{E_{0}\overline{r}^{2}h^{3}D}{12(1-v ^{2})}(1+\overline{r}^{m})\end{array}\]  
with \(\lambda_{3}=-\sqrt{2-2v},\lambda_{4}=\sqrt{2-2v}\) for \(n=2\), and  
\[\sigma_{j}=\sigma_{j3}-\sigma_{\text{T3}},\ M_{j}=\frac{h^{3}}{12z}\sigma_{j3 }-M_{\text{T3}},j=r,\theta \tag{23}\]  
where  
\[\begin{array}{l}\sigma_{r3}=-\frac{zE_{0}\overline{r}^{m}}{1-v^{2}}\big{\{} \lambda_{2v}A_{22}r^{\lambda_{2}-2}+\lambda_{3v}A_{23}r^{\lambda_{3}-2}+ \lambda_{4v}A_{24}r^{\lambda_{4}-2}+2(1+v)A_{25}+m_{\
u 1}A_{26}r^{m}\big{\}},\\ \sigma_{\theta 3}=-\frac{zE_{0}\overline{r}^{2}}{1-v^{2}}\big{\{}\lambda_{2v}^{2}A_{22 }r^{\lambda_{2}-2}+\lambda_{3v}^{*}A_{23}r^{\lambda_{3}-2}+\lambda_{4v}^{*}A_ {24}r^{\lambda_{4}-2}+2(1+v)A_{25}+m_{\
u 2}A_{26}r^{m}\big{\}},\\ \sigma_{T3}=\frac{E_{0}\overline{r}^{2}A_{T0}}{1-v}(1+\overline{r}^{m}) \overline{z}^{2D+1},\ M_{\text{T3}}=\frac{E_{0}\overline{r}^{2}h^{3}D}{12(1-v ^{2})}(1+\overline{r}^{m})\end{array}\]  
with \(\lambda_{2v}=\lambda_{2}(\lambda_{2}-1+v),\ \lambda_{3v}=\lambda_{3}(\lambda_{3}-1+v),\ \lambda_{4v}=( \lambda_{4}-1+v),\) and \(\lambda_{2v}^{*}=\lambda_{2}(\
u\lambda_{2}-\
u+1),\ \lambda_{3v}^{*}=\lambda_{3}(\
u\lambda_{3}-v+1),\ \lambda_{4v}^{*}=\lambda_{4}(\
u\lambda_{4}-v+1)\) for \(n\
eq 0\) or \(n\
eq 2\).  
Importantly, a notable observation is that in the special case of \(P=0\), which signifies a linear temperature variation along the thickness direction, the relationships \(M_{Tj}=h^{3}\sigma_{Tj}/(12z),j=1,\ 2,\ 3\) in Equations (21)-(23) are valid. Consequently, the thermal stresses and moments exhibit the relationships \(M_{j}=h^{3}\sigma_{j}/(12z),\ j=r,\ \theta\) for this specific case of \(P=0\).