# Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates
## 5 Numerical Results and Discussion  
The static and free vibration analyses were performed for FG-CNT-reinforced rhombic plate under different combination of end support, volume fraction, and several geometric parameters. The above-discussed formulation has been incorporated into a computer code. The nine-noded isoparametric elements with seven degrees of freedom per node were chosen for the present model for discretizing the FG-CNT-reinforced rhombic plate. Poly([(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]), basically known as PmPV [40], was chosen as the matrix and the armchair (10, 10) SWCNTs were considered as the reinforcing material. The material properties of the matrix were taken as \(E^{\text{m}}\) = 2.1 GPa, \(\rho^{\text{m}}\) = 1150 kg/m\({}^{3}\) and \(\
u^{\text{m}}\) = 0.34 at room temperature (300 K). The material properties of (10,10) SWCNTs at 300K are tabulated in Table 1. Three types volume fraction were used in present study. In the case of \(V^{\text{*}}\)CNT = 0.11, \(\eta_{1}\) = 0.934 and \(\eta_{2}\) = 0.149, in the case of \(V^{\text{*}}\)CNT = 0.14, \(\eta_{1}\) = 0.150 and \(\eta_{2}\) = 0.941, and for \(V^{\text{*}}\)CNT = 0.17, \(\eta_{1}\) = 0.149 and \(\eta_{2}\) = 1.381. We assume that \(\eta_{2}\) = \(\eta_{3}\) and \(G_{12}\) = \(G_{13}\) = \(G_{23}\). The abovementioned values are used for the following numerical results.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature (K)** & \(\mathbf{E_{11}^{CNT}}\) (**TPa**) & \(\mathbf{E_{22}^{CNT}}\) (**TPa**) & \(\mathbf{G_{12}^{CNT}}\) (**TPa**) & \(\mathbf{f_{11}^{CNT}}\) (\(\mathbf{10^{-6}/K}\)) & \(\mathbf{f_{22}^{CNT}}\) (\(\mathbf{10^{-6}/K}\)) \\ \hline
300 & 5.6466 & 7.0800 & 1.9445 & 3.4584 & 5.1682 \\
500 & 5.5308 & 6.9348 & 1.9643 & 4.5361 & 5.0189 \\
700 & 5.4744 & 6.8641 & 1.9644 & 4.6677 & 4.8943 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Temperature-dependent material properties of (10, 10) SWCNT (L = 9.26 nm, R = 0.68 nm, h = 0.067 nm, \(\rho_{12}^{CNT}\) = 0.175).  
Figure 2: The coordinate system of the rhombic plate.  
The non-dimensional quantities used are:  
* For the bending analysis \[\overline{w}=\frac{w}{h},\,\overline{\sigma}_{\rm x}=\sigma_{\rm x}\left(\frac{a}{ 2},\frac{b}{2},z\right)\frac{h^{2}}{q_{0}a^{2}}.\] (33)
* For the free vibration analysis \[\overline{\omega}=\omega\Big{(}a^{2}/h\Big{)}\sqrt{\rho^{m}/E^{m}}.\] (34) The loading patterns used for the bending analysis are: \[\text{For uniform loading }q=q_{0};\] \[\text{For sin-sin loading }q=q_{0}\sin\bigl{(}\tfrac{\gamma \gamma_{\rm x}}{a}\bigr{)}\sin\bigl{(}\tfrac{\gamma\gamma}{b}\bigr{)};\] (35) \[\text{For cos-cos loading }q=q_{0}\cos\bigl{(}\tfrac{\gamma \gamma_{\rm x}}{a}\bigr{)}\cos\bigl{(}\tfrac{\gamma\gamma}{b}\bigr{)}.\] The boundary conditions taken in the present analysis are as mentioned below:  
1. Simply supported (SSSS): \[v=w=\theta_{\rm y}=\psi_{\rm y}=0\text{ at x}=0,a\] \[u=w=\theta_{\rm x}=\psi_{\rm x}=0\text{ at }y=0,b.\]
2. Clamped (CCCC): \[u=v=w=\theta_{\rm x}=\theta_{\rm y}=\psi_{\rm x}=\psi_{\rm y}=0\text{ at x}=0,a\ \ \text{ and }y=0,b.\]
3. Clamped and simply supported (CCSS): \[u=v=w=\theta_{\rm x}=\theta_{\rm y}=\psi_{\rm x}=\psi_{\rm y}=0 \text{ at x}=0,a\] \[u=w=\theta_{\rm x}=\psi_{\rm x}=0\text{ at }y=0,b.\]