## 2 Material Properties in FG-CNTRC Cylinders  
A CNTRC cylindrical shell is considered, with the inner radius \(r_{i}\), and the outer radius \(r_{o}\), as shown in Figure 1, along with the coordinate system \((x,y,z)\). The cylinder is made of a mixture of wavy single walled CNT (SWCNT) along the radial direction and an isotropic polymer matrix phase. The wavy SWCNT reinforcement is either uniformly distributed (UD) or FG in the radial direction. Thus, we apply the rule of mixture as proposed in the literature by [51] and here modified by introducing an efficiency parameter \(\eta^{*}\) in order to account for wavy fibers uniformly dispersed in the polymer matrix phase, namely, the carbon nanotube (CNT) aspect ratio (\(A_{R}\)) and waviness (\(w\)). The effective mechanical properties for the CNTRC cylindrical shell, obtained according to the micro mechanical model [20], take the following form:  
\[ \begin{array}{l}E_{1}\;=\;\eta_{1}V_{CNT}E_{1,\eta*}+V_{m}E^{m}\\ \frac{\eta_{2}}{E_{2}}\;=\;\frac{V_{CNT}}{E_{2,\eta*}}\;+\;\frac{V_{m}}{E^{m}} \\ \frac{\eta_{3}}{\Omega_{12}}\;=\;\frac{V_{CNT}}{G_{12,\eta*}}\;+\;\frac{V_{m}}{ G^{m}}\\ v_{12}\;=\;V_{CN}v_{12}^{CNT}\;+\;V_{m}v^{m}\\ \alpha_{11}\;=\;\frac{V_{CN}E_{11}^{CNT}a_{11}^{CNT}+V_{m}E^{m}a_{m}}{V_{CN}E_ {11}^{CNT}+V_{m}E^{m}}\\ \alpha_{22}\;=\;\big{(}1\;+\;v_{12}^{CNT}\big{)}V_{CN}a_{22}^{CNT}\;+\;(1\;+\;v ^{m})V_{m}a^{m}\;-\;v_{12}\alpha_{11}\end{array}  \tag{1}\]  
where  
\[ \begin{array}{l}E_{i,\eta*}\;=\;\eta^{*}E_{i}^{CNT}\;\text{for}\;i\;=\;1,2,3 \\ \eta^{*}\;=\;1\;-\;\frac{\tanh(KA_{R}/(1\;+\;\langle\cdot\rangle))}{KA_{R}/(1\; +\;\langle\cdot\rangle)}\\ K\;=\;\sqrt{\Big{(}-\frac{2}{1+v_{m}}\Big{)}/\Big{(}\frac{E^{CN}}{E_{m}}\ln(V _{CNT})\Big{)}}\end{array}  \tag{2}\]  
whereas \(E_{1}^{CNT},E_{2}^{CNT}\) refer to the longitudinal and transverse elastic modulus of the CNT, respectively, and \(G_{12}^{CNT},v_{12}^{CNT},E_{\eta^{*}},\langle\cdot\rangle\) refer to its shear modulus, Poisson's ratio, effective reinforcement modulus, average number of contacts per particle, respectively. The main properties for the matrix are referred to as \(E^{m}\), \(G^{m}\) and \(v^{m}\) in terms of elastic modulus, shear modulus and Poisson's ratio, respectively. Moreover, \(V_{CNT}\) and \(V_{m}\) represent the volume fractions of the fiber (CNT) and matrix, respectively, such that \(V_{CNT}\;+\;V_{m}=1\). We finally define \(\eta_{j}\;(j=1,2,3)\) as the CNT efficiency parameters, which are introduced to account for the size-dependent material properties of the shell. These parameters can be computed combining the elastic modulus of CNTRCs as predicted by the molecular dynamics (MD) and the analytical estimations based on Equation (1). For three different volume fractions CNTs, \(V_{CNT}^{*}\), as selected hereafter, the efficiency parameters \(\eta_{j}\) assume the following values:  
\[ \begin{array}{l}\eta_{1}=0.137,\;\eta_{2}=1.022\;\text{for}\;V_{CNT}^{*}=0.12\\ \eta_{1}=0.142,\;\eta_{2}=1.626\;\text{for}\;V_{CNT}^{*}=0.17\\ \eta_{1}=0.141,\;\eta_{2}=1.585\;\text{for}\;V_{CNT}^{*}=0.28\\ \eta_{3}=0.7\,\eta_{2}\end{array} \]  
Figure 1: Geometrical scheme of a stiffened cylindrical shell. The average number \(\langle c\rangle\) of contacts per CNT is also defined as a function of the aspect ratio \(A_{R}\) and waviness \(w\) as follows [51]:  
\[\langle c\rangle=wV_{CNT}\Bigg{(}4+\frac{3A_{R}^{2}}{3A_{R}+2}\Bigg{)} \tag{3}\]  
which, in turn, accounts for the actual curvature of CNTs within the composite material. The accuracy of our approach is verified against the available literature. It is worth noticing that the approach here considered for the analyses can be commonly applied for the only reinforcing phase made of CNTs with a straight and aligned direction inside the matrix. More complex models, however, could be applied in a further research in order to account for the effect of CNTs agglomeration and waviness on the mechanical properties of the reinforced composite (see e.g., [21]).