## 2 Foppl-von Karman Equations of Bimodular FGM Thin Plates
### Consistency Equation  
Since the _xoy_ plane is placed on the neutral surface, the displacement components in the plate along \(x\), \(y\) and \(z\) direction, \(u\), \(v\) and \(w\) will take the following pattern, according to the classical [PERSON] plate theory,  
\[\left\{ \begin{array}{l}u=u_{0}-z\frac{\partial w}{\partial x}\\ v=v_{0}-z\frac{\partial w}{\partial y}\\ w=w(x,y)\end{array} \right., \tag{9}\]  
where \(u_{0}\) and \(v_{0}\) are the displacement of the neutral surface. The strain components in the plate come from two different aspects, one is the tensile strain on the neutral surface and another is from the bending strain in the small defection theory, that is  
\[\{\varepsilon\}=\{\varepsilon_{0}\}+z\{\varepsilon_{1}\} \tag{10}\]  
where  
\[\{\varepsilon\}=\{\varepsilon_{x},\varepsilon_{y},\gamma_{xy}\}^{T},\{ \varepsilon_{0}\}=\{\varepsilon_{x0},\varepsilon_{y0},\gamma_{xy0}\}^{T},\{ \varepsilon_{1}\}=\{\varepsilon_{x1},\varepsilon_{y1},\gamma_{xy1}\}^{T} \tag{11}\]  
and  
\[\{\varepsilon_{0}\}=\left\{\frac{\partial u_{0}}{\partial x}+\frac{1}{2} \bigg{(}\frac{\partial w}{\partial x}\bigg{)}^{2},\frac{\partial v_{0}}{ \partial y}+\frac{1}{2}\bigg{(}\frac{\partial w}{\partial y}\bigg{)}^{2}, \frac{\partial v_{0}}{\partial x}+\frac{\partial u_{0}}{\partial y}+\frac{ \partial w}{\partial y}\frac{\partial w}{\partial x}\right\}^{T} \tag{12}\]  
\[\{\varepsilon_{1}\}=\left\{-\frac{\partial^{2}w}{\partial x^{2}},-\frac{ \partial^{2}w}{\partial y^{2}},-2\frac{\partial^{2}w}{\partial x\partial y} \right\}^{T}. \tag{13}\]  
From Equation (12), the consistency equation expressed in terms of strain components gives  
\[\frac{\partial^{2}\varepsilon_{x0}}{\partial y^{2}}+\frac{\partial^{2} \varepsilon_{y0}}{\partial x^{2}}-\frac{\partial^{2}\gamma_{xy0}}{\partial x \partial y}=\left(\frac{\partial^{2}w}{\partial x\partial y}\right)^{2}-\frac{ \partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}}. \tag{14}\]  
Next, we need to derive the consistency equation expressed in terms of of \(w\), \(N_{x}\), \(N_{y}\) and \(N_{xy}\). Note that corresponding to the strain components, the stress components \(\sigma_{x}\), \(\sigma_{y}\), and \(\tau_{xy}\) are also from two different aspects, one is the tensile stress or membrane stress and another is from the bending stress in the small deflection theory. Thus, for the tensile zone \(0\leq z\leq t_{1}\), we have  
\[\left\{ \begin{array}{l}\sigma_{x}^{+}=\frac{E^{+}(z)}{1-(\mu^{+})^{2}}\left[ \varepsilon_{x0}+\mu^{+}\varepsilon_{y0}-z\left(\frac{\partial^{2}w}{\partial x ^{2}}+\mu^{+}\frac{\partial^{2}w}{\partial y^{2}}\right)\right]\\ \sigma_{y}^{+}=\frac{E^{-}(z)}{1-(\mu^{+})^{2}}\left[\varepsilon_{y0}+\mu^{+} \varepsilon_{x0}-z\left(\frac{\partial^{2}w}{\partial y^{2}}+\mu^{+}\frac{ \partial^{2}w}{\partial x^{2}}\right)\right]\\ \tau_{xy}^{+}=\frac{E^{+}(z)}{2(1+\mu^{+})}\left(\gamma_{xy0}-2z\frac{ \partial^{2}w}{\partial x\partial y}\right)\end{array} \right.\] (15a) and for the compressive zone \[-t_{2}\leq z\leq 0\], \[\left\{ \begin{array}{l}\sigma_{x}^{-}=\frac{E^{-}(z)}{1-(\mu^{-})^{2}}\left[ \varepsilon_{x0}+\mu^{-}\varepsilon_{y0}-z\left(\frac{\partial^{2}w}{\partial x ^{2}}+\mu^{-}\frac{\partial^{2}w}{\partial y^{2}}\right)\right]\\ \sigma_{y}^{-}=\frac{E^{-}(z)}{1-(\mu^{-})^{2}}\left[\varepsilon_{y0}+\mu^{-} \varepsilon_{x0}-z\left(\frac{\partial^{2}w}{\partial y^{2}}+\mu^{-}\frac{ \partial^{2}w}{\partial x^{2}}\right)\right]\\ \tau_{xy}^{-}=\frac{E^{-}(z)}{2(1+\mu^{-})}\left(\gamma_{xy0}-2z\frac{ \partial^{2}w}{\partial x\partial y}\right)\end{array} \right.. \tag{15b}\]  
Theoretically speaking, the axial forces \(N_{x}\) and \(N_{y}\) should be the sum of integrals in tensile and compressive zones, such that  
\[\left\{ \begin{array}{l}N_{x}=\int_{0}^{t_{1}}\sigma_{x}^{+}dz+\int_{-t_{2}} ^{0}\sigma_{x}^{-}dz\\ N_{y}=\int_{0}^{t_{1}}\sigma_{y}^{+}dz+\int_{-t_{2}}^{0}\sigma_{y}^{-}dz\end{array}