## 4 Results and Discussions
### Bimodular Effects of FGMs on Deformation  
Due to the fact that the functionally graded materials with bimodular effect in this study are defined as in tensile zone, \(E^{+}(z)=E_{0}e^{\alpha_{1}z/t}\) for \(0\leq z\leq t_{1}\), and in compressive zone, \begin{table}
\begin{tabular}{c c} \hline \hline
**Perturbation Solution on \(W_{m}\)** & **Perturbation Solution on \(P_{m}\)** \\ \hline \(\left\{\begin{array}{ll}P_{m}=P_{1}W_{m}+P_{3}W_{m}^{3}&(\text{a})\\ W=W_{1}W_{m}+W_{3}W_{m}^{3}&(\text{b})\\ S=S_{2}W_{m}^{2}+S_{4}W_{m}^{4}&(\text{c})\end{array}\right.\) & \(\left\{\begin{array}{ll}W_{m}=\overline{W}_{m1}P_{m}+\overline{W}_{m3}P_{m}^{ 3}&(\text{d})\\ W=\overline{W}_{1}P_{m}+\overline{W}_{3}P_{m}^{4}&(\text{e})\\ S=\overline{S}_{2}P_{m}^{2}+\overline{S}_{4}P_{m}^{4}&(\text{f})\end{array}\right.\) \\ \hline \end{tabular} (a) and (d) are equivalent, which has been demonstrated from (74) to (78).  
\begin{tabular}{c c} \hline Substitute (a) into (e): & Substitute (d) into (b): \\ \hline \(W=P_{1}\overline{W}_{1}W_{m}+(P_{3}\overline{W}_{1}+P_{1}^{3}\overline{W}_{3} )W_{m}^{3}\) & \(W=W_{1}\overline{W}_{m1}P_{m}+(W_{1}\overline{W}_{m3}+W_{3}\overline{W}_{m1}^{ 3})P_{m}^{3}\) \\ \multicolumn{2}{c}{Satisfy:} & Satisfy: \\ \(W_{1}=P_{1}\overline{W}_{1}\), \(W_{3}=P_{3}\overline{W}_{1}+P_{1}^{3}\overline{W}_{3}\) & \(\overline{W}_{1}=W_{1}\overline{W}_{m1}\), \(\overline{W}_{3}=W_{1}\overline{W}_{m3}+W_{3}\overline{W}_{m1}^{3}\) \\ \hline \multicolumn{2}{c}{Result: (b) and (e) are equivalent.} \\ \hline Substitute (a) into (f): & Substitute (d) into (c): \\ \hline \(S=P_{1}^{2}\overline{S}_{2}W_{m}^{2}+(2P_{1}P_{3}\overline{S}_{2}+P_{1}^{4} \overline{S}_{4})W_{m}^{4}\) & \(S=S_{2}\overline{W}_{m1}^{2}P_{m}^{2}+(2\overline{S}_{2}\overline{W}_{m1} \overline{W}_{m3}+S_{4}\overline{W}_{m1}^{4})P_{m}^{4}\) \\ \multicolumn{2}{c}{Satisfy:} & Satisfy: \\ \(S_{2}=P_{1}^{2}\overline{S}_{2}\), \(S_{4}=2P_{1}P_{3}\overline{S}_{2}+P_{1}^{4}\overline{S}_{4}\) & \(\overline{S}_{2}=S_{2}\overline{W}_{m1}^{2}\), \(\overline{S}_{4}=2S_{2}\overline{W}_{m1}\overline{W}_{m3}+S_{4}\overline{W}_{m1} ^{4}\) \\ \hline \multicolumn{2}{c}{Result: (c) and (f) are equivalent.} \\ \hline \hline \end{tabular} (c) Conclusion: Two perturbation solutions are equivalent.  
\end{table} 
Table 1: Equivalence proof of perturbation solutions on \(W_{m}\) and \(P_{m}\). for \(-t_{2}\leq z\leq 0\), if the positive or negative signs of grade parameters \(\alpha_{1}\) and \(\alpha_{2}\) are properly selected, it is easily found that the relation among the tensile modulus \(E^{+}(z)\), the compressive modulus \(E^{-}(z)\) and the modulus at the neutral layer \(E_{0}\) is definite. Basically, there are four cases, and they are (a) \(\alpha_{1}>0\), \(\alpha_{2}>0\), (b) \(\alpha_{1}<0\), \(\alpha_{2}<0\), (c) \(\alpha_{1}>0\), \(\alpha_{2}<0\) and (d) \(\alpha_{1}<0\), \(\alpha_{2}>0\), respectively, as shown in Figure 3.  
Here, we focus our discussions on the relations of load vs central deflection which is very important in the analysis and design of the bimodular FGM thin plates. For this purpose, the unknown neutral surface needs to be determined first. According to our previous study [41], the tensile thickness of the plate may be computed using a simplified formula expressed in terms of \(\mu^{+}\) and \(\mu^{-}\), thus, we begin with the given values of \(\mu^{+}\) and \(\mu^{-}\), as well as the given values of \(\alpha_{1}\) and \(\alpha_{2}\), for example, \(\mu^{+}=0.4\) and \(\mu^{-}=0.2\), and \(\alpha_{1}=\pm 1\) and \(\alpha_{2}=\pm 0.5\) are also given for four different cases of elastic moduli.