## 4 Results and Discussions
### Bimodular Effects of FGMs on Deformation
Thus, \(T_{1},T_{2},\lambda_{1},\lambda_{2},K,V\) may be computed via those previous expressions [41], such that,  
\[T_{1}=\frac{-1+\mu^{+}\pm\sqrt{(1-\mu^{+})(1-\mu^{-})}}{\mu^{+}-\mu^{-}}=0.46 41,\,T_{2}=1-T_{1}=0.5359 \tag{82}\]  
\[K=K^{+}+K^{-}=\frac{1}{1-(\mu^{+})^{2}}\left[\left(\frac{T_{1}^{2}}{\alpha_{1 }}-\frac{2T_{1}}{\alpha_{1}^{2}}+\frac{2}{\alpha_{1}^{3}}\right)e^{\alpha_{1} T_{1}}-\frac{2}{\alpha_{1}^{3}}\right]+\frac{1}{1-(\mu^{-})^{2}}\left[-\left( \frac{T_{2}^{2}}{\alpha_{2}}+\frac{2T_{2}}{\alpha_{2}^{2}}+\frac{2}{\alpha_{ 2}^{3}}\right)e^{-\alpha_{2}T_{2}}+\frac{2}{\alpha_{2}^{3}}\right] \tag{83}\]  
and  
\[\lambda_{1}=\frac{2K}{K^{+}(1+\mu^{+})+K^{-}(1+\mu^{-})},\lambda_{2}=\frac{2} {1-\mu^{+}}=\frac{10}{3},\,\,V=\frac{e^{\alpha_{1}}-1}{\alpha_{1}e^{\alpha_{1 }T_{2}}}. \tag{84}\]  
With the help of the known \(\lambda_{1},\lambda_{2},K,V\), the final relations of load vs central deflection under four different edge constraints are obtained in four cases of elastic moduli, as shown in Table 2.  
Figure 3: Variation of \(E^{+}(z)\) and \(E^{-}(z)\) with the thickness direction in four cases. The variation curves of \(P_{m}-W_{m}\) in four cases of elastic moduli are plotted in Figure 4, in which four different boundary conditions including rigidly clamped, provably clamped, simply supported and simply supported, are considered, as shown in Figure 4a-d. It is easy to find that, among four different edge constraints, the central deflection of the rigidly clamped plate is the least one, next the movably clamped plate, and then simply hinged plate; the central deflection of simply supported plate is the maximum one. This phenomenon agrees well with the results obtained from classical problems that the stronger edge constraint tends to be, the smaller the deformation magnitude becomes. Figure 4: Load vs central deflection in four cases of elastic moduli. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(a_{1}=1\), & \(a_{1}=-1\), & \(a_{1}=1\), \(a_{2}=-0.5\) & \(a_{1}=-1\), \(a_{2}=0.5\) \\  & \(a_{2}=0.5\) & \(a_{2}=-0.5\) & \(E^{+}(z)>E_{0}\), & \(E^{+}(z)<E_{0}\), \\  & \(E^{+}(z)>E_{0}>E^{-}(z)\) & \(E^{+}(z)<E_{0}<E^{-}(z)\) & \(E^{-}(z)>E_{0}\) & \(E^{-}(z)<E_{0}\) \\ \cline{2-5}  & \(K=0.1002\) & \(K=0.0935\) & \(K=0.1218\) & \(K=0.0719\) \\  & \(V=1.0054\) & \(V=1.0803\) & \(V=1.0054\) & \(V=1.0803\) \\  & \(\lambda_{1}=1.5237\) & \(\lambda_{1}=1.5872\) & \(\lambda_{1}=1.5473\) & \(\lambda_{1}=1.5647\) \\  & \(\lambda_{2}=3.3333\) & \(\lambda_{2}=3.3333\) & \(\lambda_{2}=3.3333\) & \(\lambda_{2}=3.3333\) \\ \hline rigidly clamped & \(P_{m}=0.4008W_{m}\) & \(P_{m}=0.3740W_{m}\) & \(P_{m}=0.4872W_{m}\) & \(P_{m}=0.2876W_{m}\) \\ edge & \(+0.2231W_{m}^{3}\) & \(+0.2397W_{m}^{3}\) & \(+0.2231W_{m}^{3}\) & \(+0.2397W_{m}^{3}\) \\ \hline movably clamped & \(P_{m}=0.4008W_{m}\) & \(P_{m}=0.3740W_{m}\) & \(P_{m}=0.4872W_{m}\) & \(P_{m}=0.2876W_{m}\) \\ clamped edge & \(+0.0680W_{m}^{3}\) & \(+0.0730W_{m}^{3}\) & \(+0.0680W_{m}^{3}\) & \(+0.0730W_{m}^{3}\) \\ \hline simply hinged & \(P_{m}=0.0990W_{m}\) & \(P_{m}=0.0896W_{m}\) & \(P_{m}=0.1190W_{m}\) & \(P_{m}=0.0696W_{m}\) \\ edge & \(+0.1953W_{m}^{3}\) & \(+0.2106W_{m}^{3}\) & \(+0.1956W_{m}^{3}\) & \(+0.2104W_{m}^{3}\) \\ \hline simply supported edge & \(P_{m}=0.0990W_{m}\) & \(P_{m}=0.0896W_{m}\) & \(P_{m}=0.1190W_{m}\) & \(P_{m}=0.0696W_{m}\) \\  & \(+0.0260W_{m}^{3}\) & \(+0.0277W_{m}^{3}\) & \(+0.0259W_{m}^{3}\) & \(+0.0278W_{m}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(P_{m}-W_{m}\) relations in four cases of elastic moduli. Figure 4a also shows in the four cases of elastic moduli, the capacity resisting deformation of bimodular FGM thin plates are, in turn, from the stronger to the weaker, case (c) \(E^{+/-}(z)>E_{0}\), case (a) \(E^{+}(z)>E_{0}>E^{-}(z)\), case (b) \(E^{+}(z)<E_{0}<E^{-}(z)\) and case (d) \(E^{+/-}(z)<E_{0}\), in which cases (a) and (b) are close to each other. The same phenomenon may be found in Figure 4b-d.