## 4 Circular Plate Undercoated with Radially FGM
### The Analytical Solution of the FGM-Undercoated Circular Plate  
To obtain the analytical solution for the FGM-undercoated circular plate, it is necessary to ensure that the boundary and continuity conditions at the interfaces between the FGM and the homogeneous layers are met. The boundary conditions are  
\[w(r=R)=0,\ M_{r}(r=R)=0 \tag{34}\]  
and the continuity conditions are  
\[\begin{array}{l}w(r=R_{1}^{-})=w(r=R_{1}^{+}),\ w(r=R_{2}^{-})=w(r=R_{2}^{+ }),\\ \frac{dw}{dr}\Big{|}_{r=R_{1}^{-}}=\frac{dw}{dr}\Big{|}_{r=R_{1}^{+}},\ \frac{dw}{dr}\Big{|}_{r=R_{2}^{-}}=\frac{dw}{dr}\Big{|}_{r=R_{2}^{+}},\\ M_{r}(r=R_{1}^{-})=M_{r}(r=R_{1}^{+}),\ M_{r}(r=R_{2}^{-})=M_{r}(r=R_{2}^{+}), \\ V_{r}(r=R_{1}^{-})=V_{r}(r=R_{1}^{+}),\ V_{r}(r=R_{2}^{-})=V_{r}(r=R_{2}^{+}). \end{array} \tag{35}\]Setting up the simultaneous equations of the boundary and continuity conditions gives the unknown constants \(A_{ij}\) in the form of  
\[\{A_{11},\ A_{12},\ A_{21},\ A_{22},\ A_{23},\ A_{24},\ A_{31},\ A_{32},\ A_{33},\ A_{34}\}^{T}=[K]^{-1}\{K_{1}\}. \tag{36}\]  
The superscript \(T\) in Equation (36) denotes the transform and the quantity \([K]^{-1}\) represents the inverse of the matrix \([K]\). The unknown constants \(A_{ij}\) will be determined according to the different values of the quantity \(n\) as follows:  
(1) For \(n=0\)  
By applying the boundary and continuity conditions for \(n=0\) in Equation (34), we can easily solve for the unknown coefficients \(A_{ij}\). As a result, the bending deflection for the entire circular plate can be determined as follows:  
\[w(r)=\frac{DR^{2}}{2(1+\
u)}\Bigg{\{}\frac{m+3-v}{m+2}\Big{(}1-\vec{r}^{2} \Big{)}+\frac{2(1+v)}{\left(m+2\right)^{2}}\Big{(}1-\vec{r}^{m+2}\Big{)} \Bigg{\}}. \tag{37}\]  
when \(n=0\), the FGM-undercoated circular plate becomes a homogeneous circular plate. Therefore, Equation (37) is equivalent to Equation (25) and aligns with the findings of [PERSON] [34], as anticipated.  
(2) For \(n=2\)  
The matrices \([K]\) and \(\{K_{1}\}\) in Equation (36) for \(n=2\) are established as:  
\[[K]=\begin{bmatrix}0&0&0&0&0&1&R^{2}&\ln R&R^{2}\ln R\\ 0&0&0&0&0&0&2(1+\
u)R^{2}&-(1-\
u)&  
\[[K]=\begin{bmatrix}0&0&0&0&0&0&1&R^{2}&\ln R&R^{2}\ln R\\ 0&0&0&0&0&0&0&2(1+v)R^{2}&-(1-v)&X_{1}R^{2}\\ 1&R_{1}^{2}&-1&-R_{1}^{\lambda_{2}}&-R_{1}^{\lambda_{3}}&-R_{1}^{\lambda_{4}}&0&0 &0\\ 0&0&-1&-R_{2}^{\lambda_{2}}&-R_{2}^{\lambda_{3}}&-R_{2}^{\lambda_{2}}&1&R_{2}^ {2}&\ln R_{2}&R_{2}^{2}\ln R_{2}\\ 0&2R_{1}^{2}&0&-\lambda_{2}R_{1}^{\lambda_{2}}&-\lambda_{3}R_{1}^{\lambda_{3}} &-\lambda_{4}R_{1}^{\lambda_{4}}&0&0&0\\ 0&0&0&-\lambda_{2}R_{2}^{\lambda_{2}}&-\lambda_{3}R_{1}^{\lambda_{3}}&- \lambda_{4}R_{2}^{\lambda_{4}}&0&2R_{2}^{2}&1&X_{3}R_{2}^{2}\\ 0&2(1+v)R_{1}^{2}&0&-\lambda_{2v}R_{1}^{\lambda_{2}}&-\lambda_{3v}R_{1}^{ \lambda_{3}}&-\lambda_{4v}R_{1}^{\lambda_{4}}&0&0&0\\ 0&0&0&-\lambda_{2v}R_{2}^{\lambda_{2}}&-\lambda_{3v}R_{1}^{\lambda_{3}}&- \lambda_{4v}R_{1}^{\lambda_{4}}&0&2(1+v)R_{2}^{2}&-(1-v)&X_{2}R_{2}^{2}\\ 0&0&0&-\lambda_{2v}R_{1}^{\lambda_{2}}&-\lambda_{3v}R_{1}^{\lambda_{3}}&- \lambda_{4v}R_{1}^{\lambda_{4}}&0&0&0\\ 0&0&0&-\lambda_{2v}R_{2}^{\lambda_{2}}&-\lambda_{3v}R_{2}^{\lambda_{3}}&- \lambda_{4v}R_{2}^{\lambda_{4}}&0&0&0\\ 0&0&0&-\lambda_{2v}R_{2}^{\lambda_{2}}&-\lambda_{3v}R_{2}^{\lambda_{3}}&- \lambda_{4v}R_{2}^{\lambda_{4}}&0&0&4R_{2}^{2}\end{bmatrix} \tag{40}\]  
where \(Y_{4}=nD(1+\overline{R}_{2}^{m})R_{2}^{2}+2n(1+v)K_{25}R_{2}^{2}-m(m+2)^{2}K_ {36}R_{2}^{m+2}+m_{v4}K_{26}R_{2}^{m+2}\). Therefore, the unknown constants \(A_{ij}\) for \(n\
eq 0\) or \(n\
eq 2\) can be determined from Equations (36), (40) and (41). Consequently, the deflection, stresses, and moments of the FGM-undercoated circular plate can be evaluated according to the results in Sections 4.1-4.3, after the coefficients \(A_{ij}\) are obtained.