# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 2 CNT Beams Material Properties  
Let us consider a coupled curved CNT beam, with two curved beams connected with a layer of linear springs. The geometry of beams is described by a radius \(R\), length \(L\), central angle \(\theta\), and transverse width and height, \(b\) and \(h\), respectively, of the beam section. Different combinations of boundary conditions can be considered per beam, namely, clamped-clamped (C-C), clamped-simply supported (C-S), clamped-free (C-F) or simply-simply supported (S-S). Figure 1 shows a simply supported (S-S/S-S) curved system with \(\theta=45^{\circ}\) made of a composite material reinforced with CNTs.  
More specifically, the selected example considers a PMMA matrix and SWCNTs (10,10) as the reinforcement phase. Five types of reinforcement patterns are used here, including a uniform distribution of CNTs (UD-CNTs), as well as FG distributions of CNTs of type V, \(\Lambda\), O, and X, as depicted in Figure 2.  
From an analytical standpoint, for example, a \(\Lambda-\)type variation of a function in the z direction can be defined as [29]  
\[F(z)=\left(\frac{1}{2}-\frac{z}{h}\right)^{n}\quad-\frac{h}{2}\leq z\leq\frac{ h}{2} \tag{1}\]  
In the particular case of CNTs volume fraction varying with a \(\Lambda\) distribution, it is  
\[V_{CNT}(z)=\left(n+1\right)\left(\frac{1}{2}-\frac{z}{h}\right)^{n}V_{CNT}^{*} \tag{2}\]  
which reverts to a linear distribution throughout the thickness for \(n=1\), namely  
\[V_{CNT}(z)=\left(1-2\frac{z}{h}\right)V_{CNT}^{*} \tag{3}\]  
Based on the assumption of linear distribution, the mathematical definitions of all the selected CNT patterns along the \(z\) direction are summarized in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Pattern Type** & **Distribution Factor** \\ \hline UD & \(V^{*}_{CNT}\) \\ FG-V & \((1+2\frac{z}{h})\)\(V^{*}_{CNT}\) \\ FG-\(\Lambda\) & \((1-2\frac{z}{h})\)\(V^{*}_{CNT}\) \\ FG-O & \(2\left(1-2\frac{z}{h}\right)\)\(V^{*}_{CNT}\) \\ FG-X & \((4\frac{\|z}{h})\)\(V^{*}_{CNT}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mathematical definition of different CNTs patterns throughout the thickness.  
Figure 1: A scheme of the coupled system, (**a**) general scheme, (**b**) simply supported (S-S/S-S) configuration of the coupled curved beam.  
Figure 2: Different types of CNTs patterns in the beam section.  
Based on the role of mixture, we determine the equivalent mechanical properties of the reinforced beam as follows [29]:  
\[V_{CNT}+V_{Matrix}=1 \tag{4}\]  
\[E_{11}=\eta_{1}\,V_{CNT}\,E_{11}^{CNT}+V_{Matrix}\,E^{Matrix} \tag{5}\]  
\[E_{22}=\frac{\eta_{2}}{\frac{V_{CNT}}{E_{22}^{CNT}}+\frac{V_{Matrix}}{E_{Matrix}}} \tag{6}\]  
\[G_{12}=\frac{\eta_{3}}{\frac{V_{CNT}}{G_{12}^{CNT}}+\frac{V_{Matrix}}{G_{12}^ {Matrix}}} \tag{7}\]  
\[v_{CNT}=v_{CNT}\,V_{CNT}+v_{Matrix}\,V_{Matrix} \tag{8}\]  
\[\rho_{CNT}=\rho_{CNT}\,V_{CNT}+\rho_{Matrix}\,V_{Matrix} \tag{9}\]  
where \(V_{CNT}\) and \(V_{Matrix}\) refer to the CNTs and matrix volume fraction, respectively; \(E_{11}^{CNT}\), \(E_{22}^{CNT}\) and \(G_{12}^{CNT}\) indicate the normal and shear elasticity moduli of SWCNTs; \(E^{Matrix}\) and \(G^{Matrix}\) stand for the elasticity moduli of the matrix, respectively; \(\eta_{i}\,(i=1,\,2,\,3)\) is a factor that considers the size-dependent material properties.