## 2 Foppl-von Karman Equations of Bimodular FGM Thin Plates
### Equation of Equilibrium  
A bimodular FGM rectangular thin plate with thickness \(t\) is subjected to a normal uniformly distributed load \(q\), as shown in Figure 1. According to deformation law of thin plates, this causes a concave deformation of the plate, thus generating a so-called tensile zone and compressive zone. We therefore place the _xoy_ plane at the unknown neutral layer, first, and the \(z\) axis is still set along the thickness direction of the plate, as shown in Figure 1, where the tensile and compressive section height of the plate are \(t_{1}\) and \(t_{2}\), the modulus of elasticity of the materials in tensile and compressive zone be \(E^{+}(z)\) and \(E^{-}(z)\), and the Poisson's ratio are two different constants, \(\mu^{+}\) and \(\mu^{-}\), respectively. Figure 1: Scheme of a functionally graded material (FGMs) thin plate with bimodular effect. In order to obtain an explicit solution of the problem, we need to define the functionally graded expressions in the case of bimodulus. For this purpose, an exponential function is used to express the function grade of the material, therefore, \(E^{+}(z)\) and \(E^{-}(z)\) may be written as  
\[E^{+}(z)=E_{0}e^{\alpha_{1}z/t},E^{-}(z)=E_{0}e^{\alpha_{2}z/t}, \tag{1}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are two graded indexes, \(E^{+}(z)=E^{-}(z)=E_{0}\) when \(z=0\) (the neutral layer), as shown in Figure 2. We also note that \(E^{+}(z)=E^{-}(z)=E_{0}\) when \(\alpha_{1}\to 0,\alpha_{2}\to 0\), this indicates that the adoption of \(\alpha_{1}\) and \(\alpha_{2}\) represents the bimodular effect and the functionally graded feature of the materials. In addition, for the convenience of the next description, we define and compute the following integrals  
\[\left\{ \begin{array}{l}A_{1}^{+}=\int_{0}^{t_{1}}ze^{\alpha_{1}z/t}dz=\left( \frac{tt_{1}}{\alpha_{1}}-\frac{t^{2}}{\alpha_{1}^{2}}\right)e^{\alpha_{1}t_{ 1}/t}+\frac{t^{2}}{\alpha_{1}^{2}}\\ A_{1}^{-}=\int_{-t_{2}}^{0}ze^{\alpha_{2}z/t}dz=\left(\frac{tt_{2}}{\alpha_{2} }+\frac{t^{2}}{\alpha_{2}^{2}}\right)e^{-\alpha_{2}t_{2}/t}-\frac{t^{2}}{ \alpha_{2}^{2}}\end{array} \right.\] (2a) and \[\left\{ \begin{array}{l}A_{2}^{+}=\int_{0}^{t_{1}}z^{2}e^{\alpha_{1}z/t}dz= \left(\frac{tt_{1}^{2}}{\alpha_{1}}-2\frac{t^{2}t_{1}}{\alpha_{1}^{2}}+2\frac {t^{3}}{\alpha_{1}^{2}}\right)e^{\alpha_{1}t_{1}/t}-2\frac{t^{3}}{\alpha_{1}^ {3}}\\ A_{2}^{-}=\int_{-t_{2}}^{0}z^{2}e^{\alpha_{2}z/t}dz=-\left(\frac{tt_{2}^{2}}{ \alpha_{2}}+2\frac{t^{2}t_{2}}{\alpha_{2}^{2}}+2\frac{t^{3}}{\alpha_{2}^{2}} \right)e^{-\alpha_{2}t_{2}/t}+2\frac{t^{3}}{\alpha_{2}^{3}}\end{array} \right. \tag{2b}\]  
where these integrals satisfy the following limit conditions  
\[\lim_{\alpha_{1}\to 0}A_{1}^{+}=\frac{t_{1}^{2}}{2},\lim_{\alpha_{2}\to 0}A_{1}^{-}=- \frac{t_{2}^{2}}{2}\] (3a) and \[\lim_{\alpha_{1}\to 0}A_{2}^{+}=\frac{t_{1}^{3}}{3},\lim_{\alpha_{2}\to 0}A_{2}^{-}= \frac{t_{2}^{3}}{3}. \tag{3b}\]  
Let the deflection of the plate be \(w(x,y)\), the axial force and the shear force per unit width along the directions of \(x\) and \(y\) be \(N_{x}\), \(N_{y}\) and \(N_{xy}\), respectively, and the bending moment and torsion moment per unit length be \(M_{x}\), \(M_{y}\) and \(M_{xy}\), respectively. According to the classical [PERSON] plate theory, differential equations of equilibrium can be expressed as  
\[\left\{ \begin{array}{l}\frac{\partial N_{x}}{\partial x}+\frac{\partial N_{ xy}}{\partial y}=0\\ \frac{\partial N_{y}}{\partial y}+\frac{\partial N_{xy}}{\partial x}=0\end{array} \right. \tag{4}\]  
and  
\[\frac{\partial^{2}M_{x}}{\partial x^{2}}+2\frac{\partial^{2}M_{xy}}{\partial x \partial y}+\frac{\partial^{2}M_{y}}{\partial y^{2}}+N_{x}\frac{\partial^{2}w} {\partial x^{2}}+2N_{xy}\frac{\partial^{2}w}{\partial x\partial y}+N_{y}\frac {\partial^{2}w}{\partial y^{2}}+q=0, \tag{5}\]  
Figure 2: Materials model for a bimodular FGM plate. where \(M_{x}\), \(M_{y}\) and \(M_{xy}\) has been determined as, according to our previous study [41],  
\[\left\{