# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 6 Parametric Investigation
\begin{split} E_{CNT}^{11}&=6.565376-1.76156\,\left( \frac{T}{T_{0}}\right)+1.13347\left(\frac{T}{T_{0}}\right)^{2}-0.32260\left( \frac{T}{T_{0}}\right)^{3}+0.03193\left(\frac{T}{T_{0}}\right)^{4}\text{TPa}\\ E_{CNT}^{22}&=8.22710-2.19725\,\left(\frac{T}{T_{0}} \right)+1.41176\left(\frac{T}{T_{0}}\right)^{2}-0.40125\,\left(\frac{T}{T_{0}} \right)^{3}+0.03964\,\left(\frac{T}{T_{0}}\right)^{4}\text{TPa}\\ G_{CNT}^{12}&=1.10442+1.88427\,\left(\frac{T}{T_{0}} \right)-1.47623\left(\frac{T}{T_{0}}\right)^{2}+0.49029\,\left(\frac{T}{T_{0}} \right)^{3}-0.05829\,\left(\frac{T}{T_{0}}\right)^{4}\text{TPa}\\ \rho_{CNT}&=1400\,\,\frac{kg}{cm^{2}}\text{ and }v_{CNT}=0.175\end{split}  \tag{43}\]  
where \(T\) and \(T_{0}\) represent the environment and room temperatures, respectively. The thermal-dependent Young's modulus for PMMA can be obtained at any temperature by employing the following equation:  
\[E_{Matrix}(T)=\left\{3.52-0.0034\,(300+\Delta T)\right\}\text{GPa} \tag{44}\]  
The density and Poisson's ratio in PMMA are also 1150 kg/m\({}^{3}\) and 0.34, respectively, in thermal environment. Table 8 summarizes the variation of the mechanical properties for a SWCNT (10,10) and PMMA, with an overall monotonic decrease of all properties for an increased temperature. \begin{table}
\begin{tabular}{c c c c c} \hline \hline T & \(\text{E}_{Matrix}\) & \(\text{E}_{11}\) & \(\text{E}_{22}\) & \(\text{G}_{12}\) \\ \hline \({}^{\circ}\)K & GPa & TPa & TPa & TPa \\ \hline
300 & 2.50 & 5.6466 & 7.080 & 1.9445 \\
400 & 2.16 & 5.5679 & 6.9816 & 1.9704 \\
600 & 1.48 & 5.5062 & 6.9039 & 1.9578 \\
800 & 0.80 & 5.4253 & 6.8026 & 1.9813 \\
1000 & 0.12 & 5.2814 & 6.6219 & 1.9454 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Properties of PMMA and SWCNT (10,10) at different temperatures. Figure 6: Effect of \(\theta\) on the first dimensionless frequency of system in different boundary conditions. It is worth observing that the mechanical properties of a pure PMMA matrix are more affected by temperature than the corresponding composite SWCNT (10,10). It is observed that the mechanical properties of PMMA and SWCNT decreases as the temperature increases. The following size-dependent parameters are here used in the rule of mixture to obtain the equivalent properties of beams in a thermal environment:  
\[ \begin{array}{l}V_{CNT}^{*}=0.12\rightarrow\eta_{1}=0.137,\;\eta_{2}=1.022,\; \textit{and}\;\eta_{3}=0.715\\ V_{CNT}^{*}=0.17\rightarrow\eta_{1}=0.142,\;\eta_{2}=1.626,\;\textit{and}\; \eta_{3}=1.138\\ V_{CNT}^{*}=0.28\rightarrow\eta_{1}=0.141,\;\eta_{2}=1.585,\;\textit{and}\; \eta_{3}=1.109\end{array}  \tag{45}\]  
In this section, we ignore the possible effect of \(\alpha\Delta\)T in the theoretical formulation, where we compute the dimensionless frequencies as  
\[\overline{\omega}=\omega\;R\;\sqrt{\frac{I_{BM}}{\Delta_{BM}}} \tag{46}\]  
where \(I_{BM}\) and \(A_{BM}\) are computed at a room temperature (T = 300 \({}^{\circ}\)K). Table 9 presents the first ten dimensionless frequencies of the system made of a pure PMMA matrix. Results show that an increased environmental temperature does not affect the frequency response of S-S/S-S and C-C/C-C structures. Table 10 summarizes the first six frequencies of the system at different temperatures. Five values of temperature (300\({}^{\circ}\), 400\({}^{\circ}\), 600\({}^{\circ}\), 800\({}^{\circ}\), and 1000 \({}^{\circ}\)K) are assumed to evaluate the thermal sensitivity of the vibration response. The fraction volume of CNTs in both beams kept equal to 0.12 under the assumption of uniform distribution for the reinforcement phase. The geometry properties of the beam are still assumed as R\({}_{1}\) = R\({}_{2}\) = 10, R/h = 10, and \(L_{1}=L_{2}=2\;R\;\sin(\theta/2)\). The stiffness factor and the curvature angle are assumed equal to 100 and 45\({}^{\circ}\), respectively.