*2.2. Material Properties of Functionally Graded Carbon Nanotube-Reinforced Composite*

In the present study, the lamina is assumed to be perfectly bonded at layer interfaces. As shown in Figure 2, five types of functionally graded distributions of CNTs in each layer are taken into consideration, named as UD, FG-A, FG-V, FG-X and FG-O.

**Figure 2.** Configurations of the FG-CNTRC panels: (**a**) UD; (**b**) FG-A; (**c**) FG-V; (**d**) FG-X; (**e**) FG-O.

For these cases, the CNT volume fractions are given as [26]:

$$\begin{aligned} \text{LID:} & \quad V\_{\text{CNT}}(z) = V\_{\text{CNT}}^\*; \\\\ \text{rG} - V: & \quad V\_{\text{CNT}}(z) = 2V\_{\text{CNT}}^\* \frac{z - z\_k}{z\_{k+1} - z\_k}; \\\\ \text{rG} - A: & \quad V\_{\text{CNT}}(z) = 2V\_{\text{CNT}}^\* \frac{z\_{k+1} - z}{z\_{k+1} - z\_k}; \\\\ \text{rG} - O: & \quad V\_{\text{CNT}}(z) = 2V\_{\text{CNT}}^\* \left(1 - \frac{|2z - z\_k - z\_{k+1}|}{z\_{k+1} - z\_k}\right); \\\\ \text{rG} - X: & \quad V\_{\text{CNT}}(z) = 2V\_{\text{CNT}}^\* \left(\frac{|2z - z\_k - z\_{k+1}|}{z\_{k+1} - z\_k}\right) \end{aligned} \tag{1}$$

where *zk* and *zk*<sup>+</sup><sup>1</sup> are the coordinates of the *k-*th layer to the reference plane (*z* = 0). *V*<sup>∗</sup> *CNT* is the given volume fraction of CNTs and can be calculated as:

$$W\_{\rm CNT}^\* = \frac{w\_{\rm CNT}}{w\_{\rmCNT} + (\rho^{\rm CNT}/\rho^m) - (\rho^{\rm CNT}/\rho^m)w\_{\rmCNT}} \tag{2}$$

in which, *wCNT* is the mass fraction of the carbon nanotube, ρ*<sup>m</sup>* and ρ*CNT* are mass densities of the matrix and the CNT, respectively. The effective material properties of FG-CNTRC of each layer can be expressed by the extended rule of the mixture as follows [27]:

$$\begin{aligned} E\_{11}(z) &= \eta\_1 V\_{\text{CNT}}(z) E\_{11}^{\text{CNT}} + V\_m(z) E^m \\\\ \frac{\eta\_2}{E\_{22}(z)} &= \frac{V\_{\text{CNT}}(z)}{E\_{22}^{\text{CNT}}} + \frac{V\_m(z)}{E^m}; \\\\ \frac{\eta\_3}{G\_{12}(z)} &= \frac{V\_{\text{CNT}}(z)}{G\_{12}^{\text{CNT}}} + \frac{V\_m(z)}{G^m}; \\\\ \rho(z) &= V\_{\text{CNT}}(z) \rho^{\text{CNT}} + V\_m(z) \rho^m; \\\\ \nu\_{12} &= V\_{\text{CNT}}^\* \nu\_{12}^{\text{CNT}} + V\_m(z) \nu^m \end{aligned} \tag{3}$$

where *ECNT* <sup>11</sup> , *ECNT* <sup>22</sup> , *Em* and *<sup>G</sup>CNT* <sup>12</sup> , *<sup>G</sup><sup>m</sup>* are the Young's moduli and shear modulus of CNT and matrix; η1, η<sup>2</sup> and η<sup>3</sup> are CNT/matrix efficiency parameters; *VCNT*(*z*) and *Vm*(*z*) are volume fractions of CNT and matrix, and are related by *VCNT*(*z*) + *Vm*(*z*) = 1; *vCNT* <sup>12</sup> and *<sup>v</sup><sup>m</sup>* are Poisson's ratio of CNT and matrix.
