# Free Vibrations and Flutter Analysis of Composite Plates Reinforced with Carbon Nanotubes
## 2 Materials and Methods
### Constitutive Equations  
Classical plate theory with nonlinear von Karman strains is used to characterize many materials subjected to supersonic flow, including CNT/nanocomposite plates [42], metal matrix nanocomposites [43], laminated plates [44; 45; 46], and shells [47]. The aero-thermoelastic instabilities of supersonic CNT/nanocomposite flat panels were studied by [PERSON] et al. [7] using the first-order shear deformation theory with nonlinear von Karman relations and the first-order piston theory. [PERSON] [24] used the third-order shear deformation theory in the structural modeling of a CNT/nanocomposite sandwich panel to investigate flutter and divergence phenomena. Similarly, [PERSON] et al. [48] used third-order shear deformation theory in the piezoelectric flutter control of CNT/nanocomposite plates in supersonic airflow. The higher-order shear deformation theory was applied by [PERSON] et al. [49] to analyze the vibration of CNT/nanocomposite cylindrical shells in supersonic flow. A benchmark for the numerical analysis of composite plates in supersonic flow using a variety of plate theories, including classical, first-order, and higher-order theories has been presented by [PERSON] [50]. The issue of plate flutter is studied here by considering flat, thin nanocomposite plates made of aligned and uniformly distributed carbon nanotubes in a polymer matrix and subjected to a supersonic airflow parallel to the \(x\)-axis that generates aerodynamic pressure. Characteristic dimensions in (\(x,y,z\)) coordinates are plate lengths \(a\), \(b\), and plate thickness \(h\), respectively, as shown in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Material Constant**} & \multicolumn{3}{c}{**CNT’s Volume Fraction**} \\ \cline{2-4}  & **1\%** & **3\%** & **5\%** \\ \hline \(E_{11}\) (GPa) & 12.917 & 32.86 & 52.804 \\ \(E_{22}\) (GPa) & 3.3 & 3.469 & 3.596 \\ \(G_{12}\) (GPa) & 1.113 & 1.158 & 1.205 \\ \(G_{23}\) (GPa) & 1.109 & 1.145 & 1.184 \\ \(v_{21}\) & 0.352 & 0.357 & 0.361 \\ \(v_{23}\) & 0.489 & 0.514 & 0.519 \\ Density (kg/m\({}^{3}\)) & 1202 & 1206 & 1210 \\ \hline \end{tabular}
\end{table} 
Table 2: The effective material properties of CNT/nanocomposites. Figure 2: The mechanical model of the equivalent nanocomposite plate subjected to a supersonic airflow in the \(x\)-direction. The \(z\)-coordinate is determined from the mid-plane of the plate. According to [PERSON]'s plate theory (classical plate theory (CPT)), the displacement field of an arbitrary point of a plate is expressed as follows [51]  
\[ \begin{cases}u\\ v\\ w\end{cases} = \begin{cases}u_{0}-z\frac{\partial u_{0}}{\partial x}\\ v_{0}-z\frac{\partial u_{0}}{\partial y}\\ w_{0}\end{cases} , \tag{3}\]  
where \(u\), \(v\), and \(w\) are displacement components in the Cartesian coordinate system (\(x\), \(y\), \(z\)); \(u_{0}\) and \(v_{0}\) are in-plane displacements in the mid-plane of the plate, and \(w_{0}\) is the out-of-plane displacement. Once the displacements (\(u_{0}\), \(v_{0}\), \(w_{0}\)) in the mid-plane are known, the displacements of any arbitrary point (\(x\), \(y\), \(z\)) can be computed. Assuming small strains and moderate rotations, non-zero nonlinear von Karman strains can be written in terms of the displacement field. Kinematic relations considering the nonlinear von Karman relations are written as follows [51]:  
\[\{\varepsilon\}=\{\varepsilon_{0}\}+z\{\kappa\}= \begin{cases}\frac{\partial u_{ 0}}{\partial x}+\frac{1}{2}\left(\frac{\partial w_{0}}{\partial x}\right)^{2} \\ \frac{\partial v_{0}}{\partial y}+\frac{1}{2}\left(\frac{\partial w_{0}}{ \partial y}\right)^{2}\\ \left(\frac{\partial u_{0}}{\partial y}+\frac{\partial v_{0}}{\partial x} \right)+\left(\frac{\partial u_{0}}{\partial x}\frac{\partial w_{0}}{ \partial y}\right)\\ \end{cases} +z \begin{cases}-\frac{\partial^{2}w_{0}}{\partial x^{2}}\\ -\frac{\partial^{2}w_{0}}{\partial y^{2}}\\ -2\frac{\partial^{2}w_{0}}{\partial xGy}\end{cases}