## 6 Numerical Examples
### Parametric Studies and Discussions  
In the following examples, the DIR is traced as a dimensionless excitation frequency \(\Omega=\omega b^{2}\sqrt{\rho h/D_{11}}\) in which \(D_{11}=E_{1}h^{3}/12(1-\
u_{12}\
u_{21})\) against the dynamic load factor \(\beta\). It was supposed, unless otherwise explained, that \(a/b=1\) (zigzag sheet I) and \(\mu=0.5\) nm\({}^{2}\). Figure 2 investigates the influence of the temperature change on the dynamic instability of orthotropic graphene sheet without elastomeric foundation. As clearly visible in Figure 2, an increase in temperature change reduces the excitation frequencies and shifts the origins of the DIR to the left side and also decreases gradually its width. The temperature rise leads to an increased compressive prestress in the graphene sheet, thus weakening the nanoplate stiffness. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mu\) (nm\({}^{2}\)) & \(\mathbf{K_{W}}\) & \(\mathbf{K_{P}}\) & **[PERSON] [40]** &
\begin{tabular}{c} **[PERSON] and** \\ **[PERSON] [41]** \\ \end{tabular} & **Present** & **Error (\%)** \\ \hline
0 & 0 & 0 & 1.93861 & 1.93861 & 1.93851 & 0.0051 \\  & 100 & 0 & 2.18396 & 2.18396 & 2.18386 & 0.0045 \\  & 100 & 20 & 2.96017 & - & 2.96010 & 0.0023 \\
1 & 0 & 0 & 1.17816 & 1.17816 & 1.17809 & 0.0059 \\  & 100 & 0 & 1.54903 & 1.54903 & 1.54898 & 0.0032 \\  & 100 & 20 & 2.52831 & - & 2.52827 & 0.0015 \\
2 & 0 & 0 & 0.92261 & 0.92261 & 0.92257 & 0.0043 \\  & 100 & 0 & 1.36479 & 1.36479 & 1.36475 & 0.0029 \\  & 100 & 20 & 2.41979 & - & 2.41976 & 0.0012 \\
3 & 0 & 0 & 0.78347 & 0.78347 & 0.78343 & 0.0051 \\  & 100 & 0 & 1.27485 & 1.27485 & 1.27482 & 0.0023 \\  & 100 & 20 & 2.37020 & - & 2.37018 & 0.0008 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensionless natural frequencies of the square graphene sheet for different nonlocal and foundation parameters (\(a/h=10\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(a\) (nm) & **MD Results [42]** & **DQM [43]** & **FSM [19]** & **Present** \\ \hline
4.99 & 1.0837 & 1.0749 & - & 1.0711 \\
8.08 & 0.6536 & 0.6523 & 0.6592 & 0.6514 \\
10.77 & 0.4331 & 0.4356 & 0.4387 & 0.4353 \\
14.65 & 0.2609 & 0.2645 & 0.2656 & 0.2643 \\
18.51 & 0.1714 & 0.1751 & 0.1757 &As mentioned before, in this study we analyze the effect of the temperature on the mechanical properties of both the nanostructure and elastic foundation. Thus, Figure 3a,b plot the DIR of the SLGS resting on the elastomeric foundation under the twofold assumption of temperature-independent and temperature-dependent material properties. A comparative evaluation of the results is illustrated in Figure 3a,b, for two different values of temperature, i.e., \(T=500\) K and \(T=700\) K, respectively. As visible in Figure 3a,b, when temperature-dependent properties are used, the origins of the instability regions occur at smaller excitation frequencies. The difference between the two cases becomes even more pronounced as the temperature is enhanced. This means that it is significant to account for the effect of temperature on the material properties in order to obtain more reliable results. The surrounding elastic foundation effect on the DIR of SLGS is displayed in Figure 4a,b, for \(T=300\) K and \(T=700\) K, respectively. As visible in Figure 4a,b, the presence of an elastic foundation increases the excitation frequency and hence, the origin of unstable region moves to the right side. The existence of an elastomeric foundation magnifies the nanoplate stiffness which yields an increased stability of the nanoplate. In addition, by considering an elastic foundation, the instability region of the graphene sheet tends to become wider, especially at increased levels of temperature. Moreover, it can be observed that the effect of the elastomeric Pasternak medium (K\({}_{\rm W}\
eq 0\) and K\({}_{\rm P}\
eq 0\)) is higher  
Figure 3: Comparison between the DIR of SLGS resting on the elastomeric foundation for constant and temperature-dependent properties.