## 2 Foppl-von Karman Equations of Bimodular FGM Thin Plates
### Consistency Equation
\right.. \tag{16}\]  
However, we should note that from Equation (15), the integral of the items of \(z\) has been determined as zero, which is exactly the conditions used for the determination of the unknown neutral surface, as reported in our previous study [41]. Thus, we have  
\[\left\{ \begin{array}{l}N_{x}=A_{11}\varepsilon_{x0}+A_{12}\varepsilon_{y0} \\ N_{y}=A_{12}\varepsilon_{x0}+A_{11}\varepsilon_{y0}\\ N_{xy}=A_{33}\gamma_{xy0}\end{array} \right.. \tag{17}\]  
where the coefficients \(A_{11}\), \(A_{12}\) and \(A_{33}\) are the integrals relative to the properties of the materials, and may be computed as  
\[\left\{ \begin{array}{l}A_{11}=\int_{-t_{2}}^{t_{1}}\frac{E^{+}(z)}{1-(\mu^{+ })^{2}}dz=\int_{-t_{2}}^{t_{1}}\frac{E_{0}e^{\mu_{1}z/t}}{1-(\mu^{+})^{2}}dz= \frac{E_{0}t}{1-(\mu^{+})^{2}}\frac{e^{\mu_{1}}-1}{\alpha_{1}e^{\mu_{1}z/t}} \\ A_{12}=\int_{-t_{2}}^{t_{1}}\frac{\mu^{+}E^{+}(z)}{1-(\mu^{+})^{2}}dz=\mu^{+}A _{11}\\ A_{33}=\int_{-t_{2}}^{t_{1}}\frac{E^{+}(z)}{2(1+\mu^{+})}dz=\frac{1-\mu^{+}}{2} A_{11}\end{array} \right.. \tag{18}\]  
Due to the fact that what we considered here is the membrane stress or the tensile stress, all integrals along the whole height should be carried out only to the corresponding tensile quantities. This mathematical treatment agrees with the physical phenomenon and has been adopted in the study of previous bimodular problems [21; 22; 23]. Solving the strain components from Equation (17) and substituting into Equation (14), we have  
\[\frac{A_{11}}{A_{11}^{2}-A_{12}^{2}}\left(\frac{\partial^{2}N_{x}}{\partial y^ {2}}+\frac{\partial^{2}N_{y}}{\partial x^{2}}\right)-\frac{A_{12}}{A_{11}^{2 }-A_{12}^{2}}\left(\frac{\partial^{2}N_{x}}{\partial x^{2}}+\frac{\partial^{ 2}N_{y}}{\partial y^{2}}\right)-\frac{1}{A_{33}}\frac{\partial^{2}N_{xy}}{ \partial x\partial y}=\left(\frac{\partial^{2}w}{\partial x\partial y}\right)^{ 2}-\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}} \tag{19}\]  
Noting the relation among the coefficients \(A_{11}\), \(A_{12}\) and \(A_{33}\), we have  
\[\frac{\partial^{2}N_{x}}{\partial y^{2}}+\frac{\partial^{2}N_{y}}{\partial x^{2}} -\mu^{+}\left(\frac{\partial^{2}N_{x}}{\partial x^{2}}+\frac{\partial^{2}N_{y} }{\partial y^{2}}\right)-2(1+\mu^{+})\frac{\partial^{2}N_{xy}}{\partial x \partial y}=A_{11}[1-(\mu^{+})^{2}]\left[\left(\frac{\partial^{2}w}{\partial x \partial y}\right)^{2}-\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{ \partial y^{2}}\right] \tag{20}\]Now, Equations (4), (7), and (20) constitute the basic equations of the problem expressed in terms of \(w\), \(N_{x}\), \(N_{y}\) and \(N_{xy}\). We further introduce the following stress function  
\[N_{x}=\frac{\partial^{2}\varphi}{\partial y^{2}},N_{y}=\frac{\partial^{2} \varphi}{\partial x^{2}},N_{xy}=-\frac{\partial^{2}\varphi}{\partial x \partial y}. \tag{21}\]  
Obviously, Equation (4) is naturally satisfied and Equations (7) and (20) become  
\[D^{*}\
abla^{4}w=q+\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2} \varphi}{\partial y^{2}}-2\frac{\partial^{2}w}{\partial x\partial y}\frac{ \partial^{2}\varphi}{\partial x\partial y}+\frac{\partial^{2}w}{\partial y^{2 }}\frac{\partial^{2}\varphi}{\partial x^{2}} \tag{22}\]  
and  
\[\
abla^{4}\varphi=E_{0}t\frac{e^{\alpha_{1}}-1}{\alpha_{1}e^{\alpha_{1}t_{2}/t }}\left[\left(\frac{\partial^{2}w}{\partial x\partial y}\right)^{2}-\frac{ \partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}}\right], \tag{23}\]  
where \(A_{11}\) is replaced as the definite expression from Equation (18). Thus, we obtain Foppl-von Karman equations of bimodular FGM thin plates expressed in terms of \(w\) and \(\varphi\). Here we note that  
\[\lim_{\alpha_{1}\to 0}\frac{e^{\alpha_{1}}-1}{\alpha_{1}e^{\alpha_{1}t_{2}/t }}=\lim_{\alpha_{1}\to 0}\frac{e^{\alpha_{1}}}{\alpha_{1}t_{2}/t}+\alpha_{1}(t_{2}/t )e^{\alpha_{1}t_{2}/t}=1 \tag{24}\]  
which indicates that the regression is satisfied.