## 2 Theoretical Model
### Functionally Graded Carbon Nanotube Reinforced Composite Structures
In particular, if the top material is CNTs and the bottom material is polymer matrix (PM), the through-the-thickness distributions \(V_{C}^{(k)}(\zeta)\) can be written as follows  
_Five-parameter exponential law_\(FG-CNT_{PM(5P)(a^{(k)}/b^{(k)}/c^{(k)}/d^{(k)}/p^{(k)})}\)  
\[V_{C}^{(k)}(\zeta)=\left(d^{(k)}-a^{(k)}\bigg{(}\frac{\zeta_{k+1}-\zeta}{h_{k }}\bigg{)}+b^{(k)}\bigg{(}\frac{\zeta_{k+1}-\zeta}{h_{k}}\bigg{)}^{c^{(k)}} \right)^{p^{(k)}} \tag{48}\]  
_Two-parameter exponential function_\(FG-CNT_{PM(2P-E)(a^{(k)}/b^{(k)})}^{CNT}\)  
\[V_{C}^{(k)}(\zeta)=\left(\frac{\exp\left(a^{(k)}\bigg{(}\frac{\zeta-\zeta_{k }-h_{k}/2}{h_{k}}+\frac{1}{2}\bigg{)}\right)-1}{\left(\exp\left(\frac{a^{(k)} }{2}\right)-1\right)\left(\exp\left(a^{(k)}\frac{\zeta-\zeta_{k}-h_{k}/2}{h_{ k}}\right)+1\right)}\right)^{b^{(k)}} \tag{49}\]  
_Weibull function_\(FG-CNT_{\rm PM(2P-W)(a^{(k)}/b^{(k)})}^{CNT}\)  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]0Moreover, if top and bottom materials are reversed, the following distributions apply  
_Five-parameter exponential law_ \(FG-CNT_{CNT(5P)(a^{(k)}/b^{(k)}/c^{(k)}/d^{(k)}/p^{(k)})}^{\text{PM}}\)  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]1  
_Two-parameter exponential function_ \(FG-CNT_{CNT(2P-E)(a^{(k)}/b^{(k)})}^{\text{PM}}\)  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]2  
_Weibull function_ \(FG-CNT_{CNT(2P-W)(a^{(k)}/b^{(k)})}^{\text{PM}}\)  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]3  
Further on, the total volume of the reinforcing phase \(V_{r}\) is separated as follows  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]4  
where \(V_{r}^{in}\) is the volume of CNTs in the inclusions and \(V_{r}^{m}\) is the volume of the nanoparticles scattered in the matrix. Two parameters that characterize the agglomeration of CNTs have to be introduced  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]5  
The parameter \(\mu_{1}\) specifies the effective volume of the inclusions \(W_{in}\) with respect to the overall volume of the layer \(W\). Therefore, if \(\mu_{1}=1\), there is no agglomeration, and the CNTs are uniformly scattered in the polymer matrix. The second parameter \(\mu_{2}\) defines the portion of CNT volume embedded in the inclusions \(W_{r}^{in}\) with respect to the total volume of CNTs \(W_{r}\). Therefore, if \(\mu_{2}=1\), all the nanoparticles are allocated in the spherical inclusions. To obtain the intermediate cases between the two limit cases, the following limitation must be introduced  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]6  
It should be specified that it is possible to increase the CNT spatial heterogeneity for a generic value of \(\mu_{1}<1\) if higher values of \(\mu_{2}\) are taken, meeting the requirement defined in (56). By the use of the relations (44), (54) and (55), the following relations that correlate the agglomeration parameters can be obtained  
\[ \begin{array}{c}\frac{W_{in}^{in}}{W_{in}}=\frac{V_{r}\mu_{2}}{\mu_{1}}\\ \frac{W_{r}^{in}}{W-W_{in}}=\frac{V_{r}(1-\mu_{2})}{1-\mu_{1}}\end{array}  \tag{57}\]  
for \(\mu_{2}>\mu_{1}\). After defining the micromechanics of the particle agglomeration, next step is to define properties of the CNT-reinforced composite layer. In order to do so, the mechanical properties of the polymer matrix and the hybrid inclusions have to be evaluated. For this purpose, in this paper, the Eshelby-Mori-Tanaka approach is considered. Readers can find more information about other methods in the work by [PERSON] et al. (2018). This method assumes that CNTs are randomly oriented in the inclusions, and are made of a transversely isotropic material. The bulk modulus \(K_{in}\) and the shear modulus \(G_{in}\) of the spherical inclusions are given by  
\[ \begin{split} K_{in}(\zeta)&=K_{m}+\frac{V_{r}\mu_{2} (\delta_{r}-3K_{m}\alpha_{r})}{3(\mu_{1}-V_{r}\mu_{2}+V_{r}\mu_{2}\alpha_{r})}\\ G_{in}(\zeta)&=G_{m}+\frac{V_{r}\mu_{2}(\eta_{r}-2G_{m} \beta_{r})}{2(\mu_{1}-V_{r}\mu_{2}+V_{r}\mu_{2}\beta_{r})}\end{split}  \tag{58}\]  
The same moduli for the hybrid matrix are given by  
\[