## 2 Theoretical Model
### Functionally Graded Carbon Nanotube Reinforced Composite Structures  
The characteristic shape of CNTs is one of the main factors that causes the agglomeration of these particles when scattered in a polymer matrix [24; 25; 26; 27; 32]. Thus, in some regions of the Carbon Nanotube-reinforced ply, the concentration of CNTs is higher than the average volume fraction in theply itself. According to the micromechanical model presented by [PERSON] et al. [32], these areas can be assumed to be spherically shaped inclusions with different elastic properties from the surrounding matrix, as depicted in Figure 3a. Therefore, CNTs are contained both in the matrix and in the spherical inclusions. The overall volume of the lamina \(W\) is given by the following relation  
\[W=W_{r}+W_{m} \tag{41}\]  
in which \(W_{m}\) is the volume of the matrix. \(W_{r}\) denotes the volume of CNTs embedded in the lamina, which can be further separated as  
\[W_{r}=W_{r}^{in}+W_{r}^{m} \tag{42}\]  
where \(W_{r}^{in}\) and \(W_{r}^{m}\) indicate the volume of CNTs scattered in the inclusions and in the matrix, respectively. The mass fraction of CNTs, \(w_{r}\), and of the matrix, \(w_{m}\), can be expressed by the theory of mixture as  
\[w_{r}=\frac{M_{r}}{M_{r}+M_{m}},\quad w_{m}=\frac{M_{m}}{M_{r}+M_{m}} \tag{43}\]  
Analogously, the volume fraction can be expressed as  
\[V_{r}=\frac{W_{r}}{W},\quad V_{m}=\frac{W_{m}}{W} \tag{44}\]  
In which the relation \(V_{r}+V_{m}=1\) must hold. In this paper, Functionally Graded Carbon Nanotube (FG-CNT)-reinforced composite is considered. This term specifies a structure with the reinforcing  
Figure 2: Doubly-curved panel representation and description. phase made of CNTs, which has a gradual distribution along the thickness direction. Further on, if the FG-CNT-reinforced composite is to be considered, the volume fraction of CNTs \(V_{r}\) can be characterized by a continuous gradual variation from the bottom to the top surface of the lamina assuming  
\[V_{r}(\zeta)=V_{r}^{*}V_{C}(\zeta) \tag{45}\]  
where \(V_{r}^{*}\) is the CNT volume fraction value and \(V_{C}(\zeta)\) is through-the-thickness distribution. Depending on the CNT mass fraction \(w_{r}\) and on the density of both CNT \(\rho_{r}\) and the matrix \(\rho_{m}\), CNT volume fraction value \(V_{r}^{*}\) is given as  
\[V_{r}^{*}=\left(\frac{\rho_{r}}{w_{r}\rho_{m}}-\frac{\rho_{r}}{\rho_{m}}+1 \right)^{-1} \tag{46}\]  
On the other hand, the function \(V_{C}(\zeta)\) introduced in (45) can be defined by various distributions as described in [75; 83]. The superscript \(k\) will be introduced in the following to specify that the through-the-thickness distribution \(V_{C}(\zeta)\) should be defined for each layer, when laminated configurations are analyzed. In other words, the quantity at hand will be indicated as \(V_{C}^{(k)}(\zeta)\). In this paper, the five-parameter exponential law (5P), two-parameter exponential (2P-E) and two-parameter Weibull (2P-W) functions are considered. To describe through the thickness CNTs distribution the following notation has been used  
\[FG-CNT_{\rm bottom}^{\rm top}({\rm distribution})(a^{(k)}/b^{(k)}/...) \tag{47}\]  
where superscript \"top\" indicates the material on the upper surface of the lamina, while the subscript \"bottom\" indicates the material on the lower surface of the lamina. The subscript \"(distribution)\" designates instead the through-the-thickness distribution used to describe the volume fraction \(V_{C}^{(k)}(\zeta)\) as (5P), (2P-E) or (2P-W), if the distribution is, five parameter exponential law, two parameter exponential function or two parameter Weibull function, respectively. Finally, the expression \"\((a^{(k)}/b^{(k)}/\dots)\)\" specifies the parameters that describe these distributions and control the volume fraction profile along the thickness \(h_{k}\) of the \(k\)-th lamina. Depending on the material used on the top or bottom surfaces of the lamina, different expressions have to be used for all of the stated distributions.