## 2 The Governing Equation and General Solution
### The Analytical Solution of the FGM-Coated Circular Plate  
The six unknown constants including \(A_{11}\),\(A_{12}\) of the homogeneous circular portion, and \(A_{21},\ A_{22},\ A_{23},\ A_{24}\) of the FGM-coated annular portion can be obtained by considering the boundary conditions and continuity equations:  
\[ \begin{array}{l}w(r=R)=0,\ M_{r}(r=R)=0,\ w(r=R_{1}^{-})=w(r=R_{1}^{+}),\ \left.\frac{dw}{dr}\right|_{r=R_{1}^{-}}=\left.\frac{dw}{dr}\right|_{r=R_{1}^{+}},\\ M_{r}(r=R_{1}^{-})=M_{r}(r=R_{1}^{+}),\ V_{r}(r=R_{1}^{-})=V_{r}(r=R_{1}^{+}). \end{array}  \tag{24}\]  
Equation (24) provides six simultaneous equations to solve six unknowns as follows:  
1. For \(n=0\)  
When \(n=0\), the FGM-coated circular plate degenerates to the homogeneous circular plate with [PERSON]'s modulus \(E_{0}\). The six unknowns \(A_{11}\), \(A_{12}\), \(A_{21}\), \(A_{22}\), \(A_{23}\), \(A_{24}\) for the case \(n=0\) are evaluated from Equation (24). Substituting the obtained coefficients into Equation (17) or Equation (19), one can find the bending deflection of a homogeneous circular plate under thermal load \(T(r,z)=T_{0}(1+\overline{r}^{m})\overline{z}^{2P+1}\) as: \[w(r)=\frac{DR^{2}}{2(1+\
u)}\Bigg{\{}\frac{m+3-v}{m+2}\Big{(}1-\overline{r}^{ 2}\Big{)}+\frac{2(1+v)}{(m+2)^{2}}\Big{(}1-\overline{r}^{m+2}\Big{)}\Bigg{\}}.\] (25)  
Notably, Equation (25) coincides with that presented by [PERSON] (2000). Subsequently, the stresses and bending moments of the homogeneous circular plate under thermal loads are  
\[ \begin{array}{l}\sigma_{r}=\frac{z_{6}D}{(m+2)(1-v^{2})}[(m+3-v)+(m+1+v) \overline{r}^{m}]-\frac{E_{0}A_{1}}{1-v}(1+\overline{r}^{m})\overline{z}^{2P+ 1},\\ \sigma_{\theta}=\frac{z_{6}D}{(m+2)(1-v^{2})}\{(m+3-v)+(mv+v+1)\overline{r}^{m }\}-\frac{E_{0}A_{1}}{1-v}(1+\overline{r}^{m})\overline{z}^{2P+1},\\ M_{r}=\frac{E_{0}B}{12(m+2)(1+v)}(1-\overline{r}^{m}),\\ M_{\theta}=\frac{E_{0}B^{2}D}{12(m+2)(1+v)}[1-(m+1)\overline{r}^{m}].\end{array}  \tag{26}\]
2. For \(n=2\)  
The coefficients \(A_{11}\), \(A_{12}\), \(A_{21}\), \(A_{22}\), \(A_{23}\), \(A_{24}\) for \(n=2\) determined from the simultaneous equations of boundary and continuity conditions are expressed in the form as follows:  
\[\{A_{11},\ A_{12},\ A_{21},\ A_{22},\ A_{23},\ A_{24}\}^{T}=[K]^{-1}\{K_{1}\}\]  
herein  
\[[K]= \begin{bmatrix}0&0&-1&-\ln R&-R^{\lambda_{3}}&-R^{\lambda_{4}}\\ 0&0&0&(1-\
u)&-\lambda_{3\
u}R^{\lambda_{3}}&-\lambda_{4\
u}R^{\lambda_{4}}\\ 1&R_{1}^{2}&-1&-\ln R_{1}&-R_{1}^{\lambda_{3}}&-R_{1}^{\lambda_{4}}\\ 0&2R_{1}^{2}&0&-1&-\lambda_{3}R_{1}^{\lambda_{3}}&-\lambda_{4}R_{1}^{\lambda_{ 4}}\\ 0&2(1+v)R_{1}^{2}&0&(1-\
u)&-\lambda_{3\
u}R_{1}^{\lambda_{3}}&-\lambda_{4\
u}R _{1}^{\lambda_{4}}\\ 0&0&0&2(1-\
u)&0&0\\ \end{bmatrix} , \tag{27}\]  
\[\{K_{1}\}= \begin{bmatrix}A_{25}R^{2}+A_{26}R^{m+2}\\ 2 DR^{2}+2(1+v)A_{25}R^{2}+m_{\
u 1}A_{26}R^{m+2}\\ A_{25}R_{1}^{2}+(A_{26}-A_{16})R_{1}^{m+2}\\ 2A_{25}R_{1}^{2}+(m+2)(A_{26}-A_{16})R_{1}^{m+2}\\ 2(1+v)A_{25}R_{1}^{2}+m_{\
u 1}(A_{26}-A_{16})R_{1}^{m+2}\\ Y_{1}\end{bmatrix}  \tag{28}\]  
where \(Y_{1}=2D(1+\overline{R}_{1}^{m})R_{1}^{2}+4(1+v)A_{25}R_{1}^{2}-m{(m+2)}^{2}A_{ 16}R_{1}^{m+2}+m_{\
u 3}A_{26}R_{1}^{m+2}\) with \(m_{\
u 3}=(m+2)[m^{2}+4m+2(1+v)]\). (3) For \(n\
eq 0\) or \(n\
eq 2\)  
Moving forward, let us investigate the influence of the material index \(n\) on mechanical behavior. Figure 3 presents the stresses for \(P=0\), \(m=2\) and \(n=0\), \(0.5\), \(1\), \(2\), \(3\). It is evident that as the value of \(n\) increases, indicating a decrease in the overall Young's modulus, the stresses decrease accordingly. These results showcased in Figure 3 highlight the effectiveness of utilizing the FGM-coated layer in reducing the maximum stress in the circular plate. Furthermore, Figure 4 depicts the mechanical behavior with fixed values of \(P\) and \(n\) while varying the parameter \(m\). The plots in Figure 4 demonstrate that the stresses in the homogeneous portion of the FGM-coated circular plate follow a function of \(r^{m}\), by Equation (18).