## 2 Theoretical Model
### Functionally Graded Carbon Nanotube Reinforced Composite Structures
\begin{split} K_{out}(\zeta)&=K_{m}+\frac{V_{r}(1- \mu_{2})(\delta_{r}-3K_{m}\alpha_{r})}{3(1-\mu_{1}-V_{r}(1-\mu_{2})+V_{r}(1-\mu_ {2})\alpha_{r})}\\ G_{out}(\zeta)&=G_{m}+\frac{V_{r}(1-\mu_{2})(\eta_{ r}-2G_{m}\beta_{r})}{2(1-\mu_{1}-V_{r}(1-\mu_{2})+V_{r}(1-\mu_{2})\beta_{r})} \end{split}  \tag{59}\]  
where \(K_{m}\) and \(G_{m}\) denote the bulk modulus and the shear modulus of the isotropic matrix. It should be mentioned that the subscripts \"\(in\)\" and \"\(out\)\" are associated with the mechanical properties of the inclusions and of the matrix enriched with scattered CNTs, respectively. From the theory of elasticity, it is known that  
\[K_{m}=\frac{E_{m}}{3(1-2\
u_{m})},\quad G_{m}=\frac{E_{m}}{2(1+\
u_{m})} \tag{60}\]  
where \(E_{m}\) is the elastic modulus and \(\
u_{m}\) the Poisson's ratio. The rest of the unknown quantities from (58) and (59) are defined as  
\[ \begin{split}\alpha_{r}&=\frac{3(K_{m}+G_{m})+k_{ r}+l_{r}}{3(G_{m}+k_{r})}\\ \beta_{r}&=\frac{1}{5}\Big{(}\frac{4G_{m}+2k_{r}+l_{ r}}{3(G_{m}+k_{r})}+\frac{4G_{m}}{G_{m}+p_{r}}+\frac{2(G_{m}(3K_{m}+G_{m})+G_{m}(3 K_{m}+7G_{m}))}{G_{m}(3K_{m}+G_{m})+n_{r}(3K_{m}+7G_{m})}\Big{)}\\ \delta_{r}&=\frac{1}{3}\Big{(}n_{r}+2l_{r}+\frac{(2 k_{r}+l_{r})(3K_{m}+G_{m}-l_{r})}{G_{m}+k_{r}}\Big{)}\\ \eta_{r}&=\frac{1}{5}\Big{(}\frac{2}{3}(n_{r}-l_{r}) +\frac{8G_{m}p_{r}}{C_{m}+p_{r}}+\frac{2(k_{r}-l_{r})(2G_{m}+l_{r})}{3(G_{m}+ k_{r})}+\frac{8m_{r}G_{m}(3K_{m}+4G_{m})}{3K_{m}(n_{r}+G_{m})+G_{m}(7m_{r}+G_{m})} \Big{)}\end{split}  \tag{61}\]  
where \(k_{r}\), \(l_{r}\), \(m_{r}\), \(n_{r}\), \(p_{r}\) are the Hill's elastic moduli of the nanoparticles. The reader can find further descriptions of these quantities in [27]. In brief, the mechanical characterization of a single CNT is given by the so-called Hill's elastic moduli, since it is assumed to be an equivalent continuum cylindrical shell, as shown in Figure 3b [7]. Values for these quantities are given for different Single-Walled Carbon Nanotubes (SWCNTs) for various chiral indices that can be found in literature. In this paper, only armchair-type SWCNTs are considered, as shown in Table 1 with the notation SWCNT(\(\Delta,\Delta\)), where \(\Delta\) stands for the chiral index. After obtaining values from (58) and (59), the Mori-Tanaka method gives the effective bulk and shear modulus of the CNT-reinforced layer as  
\[ \begin{split} K(\zeta)&=K_{out}\Bigg{(}1+\frac{ \mu\Big{(}\frac{K_{in}}{K_{out}}-1\Big{)}}{1+(1-\mu)\Big{(}\frac{K_{in}}{K_{out} }-1\Big{)}\frac{1+\
u_{out}}{5-3\
u_{out}}}\Bigg{)}\\ G(\zeta)&=G_{out}\Bigg{(}1+\frac{\mu\Big{(}\frac{G_{in }}{G_{out}}-1\Big{)}}{1+(1-\mu)\Big{(}\frac{G_{in}}{G_{out}}-1\Big{)}\frac{8-10 \
u_{out}}{15-15\
u_{out}}}\Bigg{)}\end{split}  \tag{62}\]  
where \(\
u_{out}\) is the Poisson's ratio of the hybrid matrix defined as  
\[\
u_{out}(\zeta)=\frac{3K_{out}-2G_{out}}{6K_{out}+2G_{out}} \tag{63}\]The resulting reinforced layer, as specified in the works [24], is isotropic. Therefore, the Young modulus \(E(\zeta)\) and the Poisson's ratio \(\
u(\zeta)\) are evaluated through the following expressions  
\[E(\zeta)=\frac{9 KG}{3K+G},\quad\
u(\zeta)=\frac{3K-2G}{6K+2G} \tag{64}\]  
At the end, density \(\rho(\zeta)\) of the reinforced layer is evaluated by Mixture theory as  
\[\rho(\zeta)=(\rho_{r}-\rho_{m})V_{r}+\rho_{m} \tag{65}\]  
where \(\rho_{r}\) is the density of the CNTs and \(\rho_{m}\) density of the polymer matrix. Further details concerning the present approach are illustrated in depth in the works by [PERSON] et al. [24; 25; 26; 27].