## 2 Foppl-von Karman Equations of Bimodular FGM Thin Plates
### Axisymmetric Case  
Let the polar radius and the polar rotation in a polar coordinates system be \(r\) and \(\theta\), we have the polar coordinates form of Equations (22) and (23) as follows  
\[ \begin{array}{l}D^{*}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}}\right)^{2}w\\ =q+\frac{\partial^{2}w}{\partial r^{2}}\left(\frac{1}{r}\frac{\partial}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}\varphi}{\partial\theta^{2}} \right)-2\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial w}{ \partial\theta}\right)\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{ \partial\varphi}{\partial\theta}\right)+\left(\frac{1}{r}\frac{\partial w}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}\right) \frac{\partial^{2}\varphi}{\partial r^{2}}\end{array}  \tag{27}\]  
and  
\[\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r }+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}\right)^{2}\varphi=E_{ 0}t\frac{e^{\alpha_{1}}-1}{\alpha_{1}e^{\alpha_{1}t_{2}/t}}\Bigg{\{}\left[ \frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial w}{\partial\theta} \right)\right]^{2}-\frac{\partial^{2}w}{\partial r^{2}}\left(\frac{1}{r}\frac {\partial w}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2} }\right)\Bigg{\}} \tag{28}\]  
Specially, in an axisymmetric problem, \(w=w(r)\) and \(\varphi=\varphi(r)\). Let the axial force per unit width along the directions of \(r\) and \(\theta\) be \(N_{r}\) and \(N_{\theta}\), respectively, the relation of the stress function and the axial force is simplified as  
\[N_{r}=\frac{1}{r}\frac{d\varphi}{dr},N_{\theta}=\frac{d^{2}\varphi}{dr^{2}}= \frac{d}{dr}(rN_{r}) \tag{29}\]At the same time, we have the axisymmetric conditions at the center,  
\[\frac{dw}{dr}=0(\
eq\infty)\text{ and }N_{r}\
eq\infty,\text{ at }r=0 \tag{30}\]  
which may be used for the simplification of the equations. Thus, the axisymmetric form of Foppl-von Karman equations of bimodular FGM thin plates becomes  
\[D^{*}\bigg{(}\frac{d^{3}w}{dr^{3}}+\frac{1}{r}\frac{d^{2}w}{dr^{2}}-\frac{1}{r ^{2}}\frac{dw}{dr}\bigg{)}-N_{r}\frac{dw}{dr}=\frac{q}{2}r \tag{31}\]  
and  
\[r^{2}\frac{d^{2}N_{r}}{dr^{2}}+3r\frac{dN_{r}}{dr}+\frac{E_{0}t}{2}\frac{e^{ \alpha_{1}}-1}{\alpha_{1}e^{\alpha_{1}t_{2}/t}}\bigg{(}\frac{dw}{dr}\bigg{)}^{ 2}=0 \tag{32}\]  
which is used for the solution of two unknown functions, \(w(r)\) and \(N_{r}\). The boundary conditions of a thin circular plate with radius \(a\) are considered to be rigidly clamped, movably clamped, simply hinged, and simply supported, respectively. For rigidly clamped,  
\[w=0,\text{ \ }\frac{dw}{dr}=0,\text{ \ }u_{r}=0\text{ at }r=a\] (33a) for movably clamped, \[w=0,\text{ \ }\frac{dw}{dr}=0,\text{ \ }N_{r}=0\text{ at }r=a\] (33b) for simply hinged, \[w=0,\text{ \ }M_{r}=0,\text{ \ }u_{r}=0\text{ at }r=a\] (33c) and for simply supported, \[w=0,\text{ \ }M_{r}=0,\text{ \ }N_{r}=0\text{ at }r=a,\] (33d) where \[u_{r}\] is the radial displacement and \[N_{r}=\frac{1}{r}\frac{d\varphi}{dr},N_{\theta}=\frac{d^{2}\varphi}{dr^{2}}= \frac{d}{dr}(rN_{r}) \tag{29}\]0 is the radial bending moment, and they need to be further changed as the boundary conditions expressed in terms of \[N_{r}=\frac{1}{r}\frac{d\varphi}{dr},N_{\theta}=\frac{d^{2}\varphi}{dr^{2}}= \frac{d}{dr}(rN_{r}) \tag{29}\]1 and \[N_{r}=\frac{1}{r}\frac{d\varphi}{dr},N_{\theta}=\frac{d^{2}\varphi}{dr^{2}}= \frac{d}{dr}(rN_{r}) \tag{29}\]2. For this purpose, we first use the geometric equation in the case of axisymmetric problem \[N_{r}=\frac{1}{r}\frac{d\varphi}{dr},N_{\theta}=\frac{d^{2}\varphi}{dr^{2}}= \frac{d}{dr}(rN_{r}) \tag{29}\]3  
where \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\) are the radial strain and circumferential strain, respectively. At the same time, the axisymmetric form of Equation (17) yields  
\[\left\{ \begin{array}{l}N_{r}=A_{11}\varepsilon_{r}+A_{12}\varepsilon_{ \theta}\\ N_{\theta}=A_{12}\varepsilon_{r}+A_{11}\varepsilon_{\theta}\end{array}