# Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates
## 2 Carbon Nanotube-Reinforced Laminates  
Figure 1 depicts the geometry of the FG-CNT-reinforced plate used in the present study. The length and width of the plate are taken as \(a\) and \(b\) with the total thickness \(h\). The middle section was taken  
as a reference for the material coordinates (x, y, z) of the rhombic plate. Four types of distribution were considered, as adopted by [PERSON] et al. [5], namely UD, FG-O, FG-X and FG-V of CNTs inside a polymer matrix within CNT-reinforced composite plate in the thickness direction. The ultimate material properties of the FG-CNT-reinforced rhombic plate were determined in accordance with the rule of mixture [36; 37].  
The volume fraction of uniform distribution and functionally graded distributions of the CNTs along the thickness direction of the CNT-reinforced rhombic plates shown in Figure 1 was assumed to be as follows:  
\[\begin{array}{l}V_{\text{CNT}}(\text{z})=\left\{\begin{array}{ll}V_{\text {CNT}}^{*}&(\text{UD})\\ 2\left(1-\frac{2|x|}{h}\right)V_{\text{CNT}}^{*}&(\text{FG-O})\\ 2\left(\frac{2|x|}{h}\right)V_{\text{CNT}}^{*}&(\text{FG-X})\\ \left(1+\frac{2x}{h}\right)V_{\text{CNT}}^{*}&(\text{FG-V})\\ \end{array}\right.\\ \text{where }V_{\text{CNT}}^{*}=\frac{\text{m}_{\text{CNT}}}{w_{\text{CNT}}+ \left(\rho^{\text{CNT}}/\rho^{\text{m}}\right)-\left(\rho^{\text{CNT}}/ \rho^{\text{m}}\right)w_{\text{CNT}}},\end{array} \tag{1}\]  
where \(w_{\text{CNT}}\) is the mass fraction of the CNTs in the CNT-reinforced rhombic plates, whereas \(\rho^{\text{m}}\) and \(\rho^{\text{CNT}}\) are the densities of the polymer matrix and carbon nanotubes, respectively.  
Figure 1: The geometry and configuration of carbon nanotube reinforced plate.  
In line with the rule of mixture, the effective material properties of FG-CNT-reinforced plates were employed by introducing the CNT efficiency parameters; thus, the final properties can be written as follows [38]:  
\[E_{11}=\eta_{1}V_{\text{CNT}}E_{11}^{\text{CNT}}+V_{\text{m}}E^{\text{m}} \tag{2}\]  
\[\frac{\eta_{2}}{E_{22}}=\frac{V_{\text{CNT}}}{E_{22}^{\text{CNT}}}+\frac{V_{ \text{m}}}{E^{\text{m}}} \tag{3}\]  
\[\frac{\eta_{3}}{G_{12}}=\frac{V_{\text{CNT}}}{G_{12}^{\text{CNT}}}+\frac{V_{ \text{m}}}{G^{\text{m}}} \tag{4}\]  
\[\
u_{12}=V_{\text{CNT}}^{*}v_{12}^{\text{CNT}}+V_{\text{m}}v^{\text{m}} \tag{5}\]  
\[\rho_{12}=V_{\text{CNT}}^{*}\rho_{12}^{\text{CNT}}+V_{\text{m}}\rho^{\text{m}}, \tag{6}\]  
where \(E_{11}^{\text{CNT}}\) and \(E_{22}^{\text{CNT}}\) are [PERSON]'s moduli and \(G_{12}^{\text{CNT}}\) is the shear modulus of singly walled CNTs, respectively. \(E^{\text{m}}\) and \(G^{\text{m}}\) are known as [PERSON]'s modulus and shear modulus of the isotropic matrix. \(\
u_{12}^{\text{CNT}}\) and \(\
u^{\text{m}}\) represent the [PERSON]'s ratio of CNTs and matrix respectively. \(V_{\text{m}}\) and \(V_{\text{CNT}}\) are the volume fractions of the matrix and carbon nanotubes, respectively, and the sum of both volume fractions equals to unity. \(\eta_{1}\), \(\eta_{2}\), \(\eta_{3}\) are the scale-dependent material properties and they can be calculated by matching the effective properties of CNT-reinforced composite obtained from the MD simulations with those from the rule of mixture.