## 2 Equivalent Model of Spiral Corrugated FG-CNTRC Cylindrical Shells  
Consider a spiral corrugated nanocomposite **FG-CNTRC** cylindrical shell of longitudinal length \(L\), thickness \(h\), circumferential radius \(R\), reinforced by aligned SWCNTs. In accordance with the manufacture technology, the spiral corrugated cylindrical shell is made by coiling a corrugated loop coil. Corrugation is considered in two cases of round and trapezoidal corrugations as shown in Figure 1a. The CNT is along the corrugated loop coil direction and the corrugated loop coil is defined in the local coordinate system \(n,s,z\). By using the geometrical relation, the number of corrugation waves of a loop coil N is related with corrugation angle by the expression  
\[\theta=\arcsin\biggl{(}\frac{Nc}{\pi R}\biggr{)}, \tag{1}\]  
where \(N\) is the wave number of corrugation of a loop coil, and \(c\) is the length of the half wave of corrugation (Figure 1b). The geometrical parameters of corrugations are assumed to be small in comparison with those of the cylindrical shell. Therefore, a homogeneous technique for corrugated structure may be applied for modeling the global buckling behavior of shells. The spiral corrugated cylindrical shell is defined in a global coordinate system \(x,y,z\). The origin is located at an end of the shell and on the middle surface of corrugations. The axes \(x\) and \(y=R\theta\) are in-plane coordinates in axial and circumferential directions of the shell, respectively, and the axis \(z\) is in the direction of inward normal to the mid-surface, as shown in Figure 1a. The SWCNTs are reinforced into an isotropic polymer matrix through uniform distribution (UD) or functionally graded (FG) distribution in which the volume fraction of CNT continuously varies in the thickness direction of corrugation skin with two different types referred to as FG-O and FG-X (Figure 1c). The mid-surface of  
Figure 1: Geometrical properties and coordinate system of spiral corrugated functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells. corrugation skin is CNT-rich for FG-O type and the internal surface of corrugation skin is enriched by CNTs in case of FG-X type. The CNT is along the spiral direction of corrugation. For the case of ring corrugated cylindrical shells, CNT is in the circumferential direction of shells (\(y\) direction). The effective Young's moduli and shear modulus of the non-corrugated CNTRC shell panels are determined according to the extended rule of mixture as [21]  
\[\frac{\eta_{2}}{E_{11}}=\frac{V_{CNT}}{E_{11}^{CNT}}+\frac{V_{m}}{E^{m}},\ E_{22}=\eta_{1}V_{CNT}E_{22}^{CNT}+V_{m}E^{m},\ \frac{\eta_{3}}{G_{12}}=\frac{V_{CNT}}{G_{12}^{CNT}}+\frac{V_{m}}{G^{m}}, \tag{2}\]  
where \(E_{11}^{CNT}\), \(E_{22}^{CNT}\) and \(G_{12}^{CNT}\) are the [PERSON]'s moduli and shear modulus of the carbon nanotube, respectively, \(E^{m}\) and \(G^{m}\) are the [PERSON]'s modulus and shear modulus of the isotropic matrix, respectively. **Note that:** Equation (2) is applied for non-corrugated CNTRC shell panels. The coefficients \(\eta_{j}\) (\(j=1,2,3\)), are known as the CNT efficiency parameters. \(V_{CNT}\) and \(V_{m}\) are the volume fractions of CNTs and matrix, respectively, and they are related to  
\[V_{CNT}+V_{m}=1. \tag{3}\]  
The variation functions of the CNT volume fraction \(V_{CNT}\) through the thickness of shell for three types of CNT distributions are considered as [26]  
\[V_{CNT}=\left\{ \begin{array}{ll}V_{CNT}^{*},&(UD),\\ 2\big{(}1-\frac{2\big{[}x\big{]}}{h}\big{)}V_{CNT}^{*},&(FG-O),\\ 2\big{(}\frac{2\big{[}x\big{]}}{h}\big{)}V_{CNT}^{*},&(FG-X),\end{array} \right. \tag{4}\]  
where \(V_{CNT}^{*}\) is the total volume fraction of CNT. The [PERSON]'s ratio depends on the thickness and can be determined by [26]  
\[\
u_{12}=V_{CNT}^{*}\
u_{12}^{CNT}+V_{m}\
u^{m}, \tag{5}\]  
where \(\
u_{12}^{CNT}\) and \(\
u^{m}\) are [PERSON]'s ratios of CNT and the isotropic matrix, respectively.