**1. Introduction**

Functionally graded carbon nanotube-reinforced composites were first proposed by Shen [1] and have been widely accepted as a new advanced material. In functionally graded carbon nanotube-reinforced composite (FG-CNTRC) structures, the CNTs are assumed to be distributed and functionally graded with certain rules along the desired direction to improve the mechanical properties of the structures. Due to the curvature effect, doubly curved shell structures possess increased structural stiffness as compared to flat ones. Therefore, doubly curved shells are often employed to fabricate structural elements of modern constructions made of advanced materials in various engineering disciplines such as aerospace, civil, marine and mechanical engineering. It is thus significant and very meaningful to explore the mechanical response of doubly curved shells made of laminated FG-CNTRC.

Due to its simplicity and effectiveness, the equivalent single-layer model is used for multi-layer composite materials. Among the equivalent single layer models, the model based on the classical theory (CPT) [2] only provides accurate results for the thin shell because it completely neglects the effect of shear deformation. To overcome the limitations of CPT, the model based on the first-order shear deformation theory (FSDT) [3] takes into account the shear deformation effects and provides relatively accurate results for both thin and moderately thick shells, but it has to use shear correction factor. Therefore, the model based on the higher-order shear deformation theory (HSDT) [4–6] is often desirable. However, it is not convenient to use HSDT because the equations of motions based on HSDT are complicated and difficult to solve. Therefore, the development of simple HSDT is needed. In addition to these, a four-variable deformation theory [7–11] has been developed and applied recently. In this theory model, the transverse shear stresses are satisfied to be parabolic and to be zero on free surfaces. Furthermore, it has only four unknowns, thus the governing equations can be reduced to four.

Based on the above-mentioned theories, various studies have been done to investigate the bending, buckling and vibration responses of FG-CNTRC shells and panels. Using the third-order shear deformation theory, Mehrabadi and Aragh [12] investigated static behavior of FG-CNTRC cylindrical shells. Aragh et al. [13] and Yas et al. [14] studied free vibration of FG-CNTRC cylindrical panels. Alibeigloo [15] analyzed the free vibration behavior of the FG-CNTRC cylindrical panel embedded in piezoelectric layers based on the three-dimensional theory of elasticity and the state-space technique. Lei et al. [16] presented the first-known dynamic stability of FG-CNTRC cylindrical panels under static and periodic axial force. Rasool el al. [17] analyzed the stress wave propagation of FG-CNTRC cylinders subjected to an impact load by using an element-free method. In [18], Shen and Zhang investigated thermal post-buckling of FG-CNTRC cylindrical shells subjected to a uniform temperature rise. Based on a HSDT with a von Kármán-type of kinematic nonlinearity, Shen [19] presented the thermal post-buckling and torsional post-buckling of FG-CNTRC cylindrical shells. Furthermore, Shen and Xiang also performed research on nonlinear vibration [20], and post-buckling [21] behavior of FG-CNTRC cylindrical shells in the thermal environment. A post-buckling analysis of FG-CNTRC cylindrical panels subjected to axial compression was also presented by Liew et al. [22]. In this study, Liew et al. used a meshless approach and arc-length method combined with the modified Newton–Raphson method to trace the post-buckling path. Using the element-free kp-Ritz method, Lei et al. [23] investigated free vibration of FG-CNTRC rotating cylindrical panels. Based on the generalized differential quadrature method (GDQM)and the finite element (FE) method, Tornabene et al. [24] and Thomas et al. [25], respectively, investigated free vibration of FG-CNT-reinforced laminated composite doubly curved shells.

The purpose of this paper is to develop a new four-variable refined shell theory for free vibration analysis of multi-layered functionally graded carbon nanotube-reinforced composite doubly curved panels. The present theory has only four unknowns but it satisfies the stress-free boundary conditions on the top and bottom surface without using shear correction factors. The distribution of the carbon nanotube (CNT) through the thickness of each layer may be functionally graded or uniformly distributed. The resultant eigenvalue system is solved to obtain the frequencies and mode shapes of the anti-symmetric, cross-ply laminated panels by Navier solution. The accuracy of the presented formulation is investigated by comparing the obtained natural frequencies with existing results in the literature. Also, a novelty parameter study of the laminated FG-CNTRC doubly-curved panels of which the geometrical parameters, CNTs distributions, the volume fraction of CNTs, as well as the number of layers are also reported in detail.

#### **2. Theoretical Formulations**
