3.2.6. Effect of Number of Layers

The influence of number of layers (*n* is a couple of layers (0/90)) on ωˆ, and PCF are depicted in Figure 7a,b, respectively. Here, the geometrical dimensions of the panels are taken as *a*/*b* = 1, *a*/*h* = 20, *Rx* = *Ry* = *R*, and *V*<sup>∗</sup> *CNT* = 0.17, FG-X, (0/90)n. As the Figures show, with a fixed value of total thickness, the non-dimensional frequencies and the percentage change of frequency of laminated FG-CNTRC panels are strongly affected by the number of layers, changing from one layer to two layers. However, these two dimensionless parameters vary very slightly for the number of layers greater than three. This is compatible with the investigations of Reddy [31], for conventional fiber reinforced composites.

**Figure 7.** Effect of number of layers (n) on free vibration of FG-CNTRC shell panels (*a*/*b* = 1, *Rx* = *Ry* = *R*; *V*∗ *CNT* = 0.17, (0/90)n: (**a**) For the frequency parameter ωˆ = ω*a* - ρ*<sup>m</sup> Em* ; (**b**) for the percentage change of frequency *PCF*.

#### 3.2.7. Effect of Different Wave Numbers

Table 4 listed non-dimensional frequencies for two-layered (0/90) FG-CNTRC doubly curved shell panels (*a*/*b* = 1; *R*/*a* = 5, *a*/*h* = 50, FG-X,*V*∗ *CNT* = 0.17) for different wave numbers. It can be seen that at the small value of wave numbers (*n*, *m*) the SPH panels have highest non-dimensional frequencies while the HPR panels have lowest ones. However, it also can be seen that the non-dimensional frequencies of all three types of doubly curved panels will approximately have more wave numbers.


**Table 4.** Non-dimensional frequencies ω for two-layered (0/90) FG-CNTRC doubly curved shell panels for different wave numbers (*a*/*b* = 1; *R*/*a* = 5, *a*/*h* = 50, , FG-X, *V*∗ *CNT* = 0.17).

Figures 8–10 depict the first six mode shapes of the simply supported laminated FG-CNTRC CYL, SPH and HPR shell panels, respectively. Geometric characteristics of the panels are *a*/*b* = 1, *R*/*a* = 5 and *a*/*h* = 50. Type of CNT distribution is FG-X and volume faction of CNT is *V*∗ *CNT* = 0.17. It can be noticed from these Figures, that in CYL panels, mode (*m* = 2, *n* = 1) is higher than mode (*m* = 1, *n* = 2), while in SPH and HPR panels, mode (*m* = 1, *n* = 2) and mode (*m* = 2, *n* =1) are the same order. This is because the CYL panel only has the curvature in *x* direction while SPH and HPR panels have the curvature

in both *x* and *y* directions. These mode shapes can help to understand vibration characteristics of laminated FG-CNTRC doubly curved shell panels.

**Figure 8.** The first six mode shapes of simply supported laminated FG-CNTRC CYL panels.

**Figure 9.** The first six mode shapes of simply supported laminated FG-CNTRC SPH panels.

**Figure 10.** The first six mode shapes of simply supported laminated FG-CNTRC HPR panels.

#### **4. Conclusions**

In this paper, an analytical solution based on a new four-variable refined shell theory for free vibration analysis of the laminated FG-CNTRC doubly curved shell panels was developed. The accuracy and efficiency of the present model are validated through a review of comparison studies. The influences of several parameters such as FG-CNTRC parameters, curvature, curvature ratio, thickness ratio, aspect ratio and the number of layers on free vibration responses of the panels are explored. The results revealed that the shell panels become stiffer with increasing curvature, conversely, the stiffness of the panels is reduced as the aspect ratio increases. FG-X CNTRC panels have the highest frequency, while FG-O CNTRC panels have the smallest frequency regarding all inlet studied parameters.

The present theory is accurate and efficient in solving free vibration behaviours of doubly curved laminated FG-CNT reinforced composite panels and may be useful in the study of similar composite structures.

**Author Contributions:** Formal analysis, Software, V.V.T.; Writing-original draft, Investigation, T.H.Q.; Supervision-editing, T.M.T.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

#### **Appendix A**

Detailed steps to construct the new shape function:

The transverse strains associated with the displacement field in Equation (1) are:

$$\begin{array}{lcl}\gamma\_{xz} &=& \frac{1}{1+z/R\_x} \left[ \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} - \frac{u\_0}{R\_x} \right] =& \frac{1}{1+z/R\_x} \left[ \frac{\partial w\_b}{\partial x} + \frac{\partial w\_s}{\partial x} + \frac{u\_0}{R\_x} - \frac{\partial w\_b}{\partial x} - f'(z) \frac{\partial w\_s}{\partial x} - \frac{u\_0}{R\_x} \right] \\ &=& \frac{1}{1+z/R\_x} \left[ (1-f'(z)) \frac{\partial w\_b}{\partial x} \right] \\ \gamma\_{yz} &=& \frac{1}{1+z/R\_y} \left[ \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} - \frac{v\_0}{R\_y} \right] =& \frac{1}{1+z/R\_y} \left[ \frac{\partial w}{\partial y} + \frac{\partial w\_s}{\partial y} + \frac{v\_0}{R\_y} - \frac{\partial w\_b}{\partial y} - f'(z) \frac{\partial w\_s}{\partial y} - \frac{v\_0}{R\_y} \right] \\ &=& \frac{1}{1+z/R\_y} \left[ (1-f'(z)) \frac{\partial w\_s}{\partial y} \right] \end{array} \tag{A1}$$

For shells under bending, the transverse shear stresses σ*xz*, σ*yz* must be vanished at the top and bottom surfaces. These conditions lead to the requirement that the corresponding transverse strains on these surfaces have to be zero. From γ*xz <sup>x</sup>*, *<sup>y</sup>*,±*<sup>h</sup>* 2 = γ*yz <sup>x</sup>*, *<sup>y</sup>*,±*<sup>h</sup>* 2 = 0, we obtain:

$$\gamma\_{xz} = \frac{1}{1 + z/R\_x} \left[ (1 - f'(z)) \frac{\partial w\_s}{\partial x} \right] = 0 \text{ at } z = \pm \frac{h}{2} \tag{A2}$$

$$\gamma\_{yz} = \frac{1}{1 + z/R\_y} \left[ (1 - f'(z)) \frac{\partial w\_s}{\partial y} \right] = 0 \text{ at } z = \pm \frac{h}{2} \tag{A3}$$

From Equations (A2) and (A3), we have:

$$f'(z) = 1 \text{ at } z = \pm \frac{h}{2} \tag{A4}$$

Function *f*(*z*) satisfies the condition (5) can be selected as a polynomial, trigonometric, and exponential, ... function. In our study, we chose *f*(*z*) as a cubic polynomial: *f*(*z*) = *az* + *<sup>b</sup> <sup>h</sup>*<sup>2</sup> *<sup>z</sup>*3, thus:

$$f'(z) = a + \frac{3b}{h^2}z^2 = 1\tag{A5}$$

Some authors have chosen the value of the pair a, b to satisfy Equation (A5). In this study, we chose: *a* = −1/8, *b* = 3/2. Thus:

$$f(z) = -\frac{1}{8}z + \frac{3}{2}\frac{z^3}{h^2},\ f'(z) = \left(-\frac{1}{8} + \frac{3.3}{2}\frac{z^2}{h^2}\right)|\_{z=\pm\frac{h}{2}} = 1\tag{A6}$$

#### **Appendix B**

Matrix elements of Equation (25):

$$s\_{11} = -\left(\overline{A}\_{11} + 2\frac{\overline{B}\_{11}}{R\_{\text{x}}} + \frac{\overline{D}\_{11}}{R\_{\text{x}}^2}\right)\alpha\_m^{-2} - \left(\hat{A}\_{\theta\theta} + 2\frac{\mathcal{B}\_{\theta\theta}}{R\_{\text{x}}} + \frac{\mathcal{D}\_{\theta\theta}}{R\_{\text{x}}^2}\right)\theta\_m^{-2} \tag{A7}$$

*J. Compos. Sci.* **2019**, *3*, 104

$$s\_{12} = -\left(A\_{12} + A\_{66} + (B\_{12} + B\_{66})\left(\frac{1}{R\_x} + \frac{1}{R\_y}\right) + \frac{1}{R\_x R\_y}(D\_{12} + D\_{66})\right)\theta\_{ll}\alpha\_m\tag{A8}$$

$$s\_{14} = \left(\overline{B}\_{11}^{\circ} + \frac{\overline{D}\_{11}^{\circ}}{R\_{\circ}}\right) \alpha\_{tt}{}^{3} + \left(\left(B\_{12}^{\circ} + B\_{66}^{\circ} + B\_{66}^{\circ} + \frac{D\_{66}^{\circ}}{R\_{\circ}} + \frac{\hat{D}\_{66}^{\circ}}{R\_{\circ}} + \frac{D\_{12}^{\circ}}{R\_{\circ}}\right) \beta\_{t}{}^{2} + \frac{\overline{A}\_{11}}{R\_{\circ}} + \frac{A\_{12}}{R\_{\circ}} + \frac{\overline{B}\_{11}}{R\_{\circ}^{2}} + \frac{B\_{12}}{R\_{\circ}R\_{\circ}}\right) \alpha\_{tt} \tag{A9}$$

$$\hat{s}\_{22} = -\left(\overline{A}\_{66} + 2\frac{\overline{B}\_{66}}{R\_y} + \frac{\overline{D}\_{66}}{R\_y^2}\right)\hat{a}\_m^{\;2} - \left(\hat{A}\_{22} + 2\frac{\hat{B}\_{22}}{R\_y} + \frac{\hat{D}\_{22}}{R\_y^2}\right)\hat{\rho}\_m^{\;2} \tag{A10}$$

$$\begin{cases} s\_{23} = \left(\frac{A\_{12}}{R\_x} + \frac{\hat{A}\_{22}}{R\_y} + \frac{1}{R\_y} \left(\frac{B\_{12}}{R\_x} + \frac{B\_{22}}{R\_y}\right)\right) \beta\_n + \left(\hat{B}\_{22} + \frac{D\_{22}}{R\_y}\right) \beta\_n{}^3 + \\\ \left(B\_{12} + B\_{66} + \overline{B}\_{66} + \frac{1}{R\_y} \left(D\_{12} + D\_{66} + \overline{D}\_{66}\right)\right) \beta\_n \alpha\_m \end{cases} \tag{A11}$$

$$\begin{cases} s\_{24} = \left(\frac{A\_{12}}{\mathcal{R}\_x} + \frac{\hat{A}\_{22}}{\mathcal{R}\_y} + \frac{1}{\mathcal{R}\_y} \left(\frac{B\_{12}}{\mathcal{R}\_x} + \frac{B\_{22}}{\mathcal{R}\_y}\right)\right) \beta\_n + \left(\hat{B}\_{22}^s + \frac{D\_{22}^s}{\mathcal{R}\_y}\right) \beta\_n^{\ast 3} + \\\ \left(B\_{12}^s + B\_{66}^s + \overline{B}\_{66}^s + \frac{1}{\mathcal{R}\_y} \left(D\_{12}^s + D\_{66}^s + \overline{D}\_{66}^s\right)\right) \beta\_n \alpha\_m^{\ast 2} \end{cases} \tag{A12}$$

 $s\_{33} = -\frac{\overline{A}\_{11}}{R\_x^2} - 2\frac{A\_{12}}{R\_x R\_y} - \frac{\hat{A}\_{22}}{R\_y^2} - 2\left(\frac{B\_{12}}{R\_x} + \frac{\hat{B}\_{22}}{R\_y}\right)\theta\_n^{-2} - 2\left(\frac{\overline{B}\_{11}}{R\_x} + \frac{B\_{12}}{R\_y}\right)\alpha\_m^{-2} - \frac{\overline{D}\_{11}}{D\_{11}}\tag{A13}$  $\overline{D}\_{11}\alpha\_m^{-4} - \left(2D\_{12} + 2D\_{66} + \overline{D}\_{66} + D\_{66}\right)\alpha\_m^{-2}\beta\_n^{-2} - \hat{D}\_{22}\beta\_n^{-4}$ 

$$\begin{split} s\_{34} &= -\frac{\overline{A}\_{11}}{R\_x^2} - 2\frac{A\_{12}}{R\_x R\_y} - \frac{\dot{A}\_{22}}{R\_y^2} - \left( \frac{\overline{B}\_{11}}{R\_x} + \frac{\overline{B}\_{11}^\*}{R\_x} + \frac{B\_{12}}{R\_y} + \frac{B\_{12}^\*}{R\_y^2} \right) a\_m^2 - \left( \frac{B\_{12}}{R\_x} + \frac{B\_{12}^\*}{R\_x} + \frac{B\_{22}}{R\_y} + \frac{B\_{22}^\*}{R\_y} \right) \theta\_n^2 - \\ &\left( 2D\_{12}^e + 2D\_{66}^e + \overline{D}\_{66}^e + \dot{D}\_{66}^e \right) a\_m^2 \beta\_n^2 - \overline{D}\_{11}^\* a\_m \,^4 - D\_{22}^e \beta\_n \,^4 \end{split} \tag{A14}$$

$$\begin{cases} s\_{13} = \left( \frac{\overline{A}\_{11}}{\overline{R}\_{\overline{x}}} + \frac{A\_{12}}{\overline{R}\_{\overline{y}}} + \frac{1}{\overline{R}\_{\overline{x}}} \left( \frac{\overline{B}\_{11}}{\overline{R}\_{\overline{x}}} + \frac{B\_{12}}{\overline{R}\_{\overline{y}}} \right) \Big| \mathcal{S}\_{12} + \mathcal{B}\_{66} + \mathcal{B}\_{66} + \frac{1}{\overline{R}\_{\overline{x}}} \left( D\_{12} + D\_{66} + D\_{66} \right) \Big| \mathcal{S}\_{n} \right) \mathbf{a}\_{m} + \\ \nabla \cdot \left( \overline{B}\_{11} + \frac{\overline{D}\_{11}}{\overline{R}\_{\overline{x}}} \right) \mathbf{a}\_{m}^{-3} \end{cases} \tag{A15}$$

$$\begin{cases} s\_{44} = -\frac{\overline{A}\_{11}}{R\_x^2} - 2\frac{A\_{12}}{R\_x R\_y} - \frac{\hat{A}\_{22}}{R\_y^2} - \left(\hat{A}\_{44}^s + 2\frac{B\_{12}^s}{R\_x} + 2\frac{\hat{B}\_{22}^s}{R\_y}\right)\boldsymbol{\beta}\_n^2 - \left(\overline{A}\_{55}^s + 2\frac{\overline{B}\_{11}^s}{R\_x} + 2\frac{B\_{12}^s}{R\_y}\right)\boldsymbol{\alpha}\_m^2 - \\\ \left(2\overline{E}\_{12}^s + 2\overline{E}\_{66}^s + \overline{E}\_{66}^s + \hat{E}\_{66}^s\right)\boldsymbol{\alpha}\_m^2\boldsymbol{\beta}\_n^2 - \overline{E}\_{11}^5\boldsymbol{\alpha}\_m^4 - \hat{E}\_{22}^s\boldsymbol{\beta}\_n^4 \end{cases} \tag{A16}$$

$$\begin{aligned} m\_{11} &= -\left(\overline{l}\_{0} + 2\frac{\overline{l}\_{1}}{\overline{\mathcal{K}}\_{x}} + \frac{\overline{l}\_{2}}{\overline{\mathcal{K}}\_{x}^{2}}\right); m\_{12} = 0; m\_{13} = \left(\overline{l}\_{1} + \frac{\overline{l}\_{2}}{\overline{\mathcal{K}}\_{x}}\right) \alpha\_{m}; m\_{14} = \left(\overline{l}\_{1} + \frac{\overline{l}\_{2}}{\overline{\mathcal{K}}\_{x}}\right) \alpha\_{m};\\ m\_{11} &= -\left(\overline{l}\_{0} + 2\frac{\overline{l}\_{1}}{\overline{\mathcal{K}}\_{y}} + \frac{\overline{l}\_{2}}{\overline{\mathcal{K}}\_{y}^{2}}\right); m\_{23} = \left(\overline{l}\_{1} + \frac{\overline{l}\_{2}}{\overline{\mathcal{K}}\_{y}}\right) \beta\_{n}; m\_{24} = \left(\overline{l}\_{1} + \frac{\overline{l}\_{2}}{\overline{\mathcal{K}}\_{y}}\right) \beta\_{n};\\ m\_{33} = -\overline{l}\_{0} - \overline{l}\_{2} \left(\alpha\_{m}^{-2} \beta\_{n}^{-2}\right); m\_{34} = -\overline{l}\_{0} - \overline{l}\_{2} \left(\alpha\_{m}^{-2} \beta\_{n}^{-2}\right); m\_{44} = -\overline{l}\_{0} - \overline{\mathcal{K}}\_{1} \left(\alpha\_{m}^{-2} \beta\_{n}^{-2}\right);\end{aligned} \tag{A17}$$
