## 1 Introduction  
Circular cylindrical shells have an essential role in various fields of engineering applications such as aircraft, pressure vessels, gas turbines, and many other industrial purposes because of their excellent performance. Due to the advancement of the knowledge and technology, in recent years a new category of materials with interesting properties, named as functionally graded materials (FGMs) has been successfully applied. Conventional types of these materials are made of two or more different constituent phases, namely, the ceramic and metal phases, which are distributed gradually according some fixed functions. Since FGMs have some extraordinary properties, namely, a high temperature and a corrosion resistance, as well as an improved residual stress distribution, they are widely studied in many field of the applied sciences and they are adopted as structural components in military, medical, or aerospace industries, as well as in power plants or vessels. Thus, due to their special privileges in comparison with traditional materials, most industries make effort to exert such materials in lieu of ordinary ones [1; 2; 3; 4]. A large number of studies in literature has focused on the thermo-mechanical and buckling behavior of FGMs for shell and plate structures. In this context, only some research works associated with FG cylindrical shells will be reviewed here, in line with the perspective developed in the present work. [PERSON] et al. [5] investigated the nonlinear forced vibration response of FG cylindrical thin shells, and used the perturbation method and the numerical Poincare maps to solve the governing equations of the problem. [PERSON] et al. [6] studied the vibration of FG cylindrical shells with ring supports. It was found that symmetric and asymmetric boundary conditions affect significantly the vibrations of the structure, with a general increase or decrease, respectively. In a recent work, [PERSON] et al. [7] have studied the agglomeration effect of FG hybrid single-walled carbon nanotubes on the vibration of hybrid laminated cylindrical shell structures. [PERSON] et al. [8] performed the nonlinear static and dynamic buckling of imperfect eccentrically stiffened FG thin circular cylindrical shells subjected to an axial compression. [PERSON] et al. [9] presented a novel formulation based on a modified couple stress theory to study simply supported FG circular cylindrical shells in the framework of thin shell structures, whereby the vibration behavior based on a classical continuum was found to be quite unaffected by the length scale parameters. [PERSON] et al. [10] studied the nonlinear vibrations of a simply supported fluid-filled FG cylindrical shell subjected to a lateral time-dependent load and an axial static preloading condition. [PERSON] and [PERSON] [11] applied the displacement method to study the nonlinear vibration of FG circular cylindrical shells subjected to an axial and transverse mechanical loading. [PERSON] et al. [12] introduced an analytical solution for the deformation and stress response of axisymmetric clamped-clamped thick FG cylindrical shells with variable thickness, while applying the first-order shear deformation theory and the perturbation theory, based on the [PERSON]'s nonlinear large deflection theory. To date, many analytical and numerical approaches have been proposed in literature to handle simple and coupled vibration problems of cylindrical shell structures, including thermo-elastic, piezoelectric, and thermo-piezoelectric multi-field problems (see refs. [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27], among others). As far as FGMs are concerned, many recent studies about the free vibration and buckling response of conventional and bi-directional FG cylindrical shells have been recently performed in literature [28; 29; 30; 31; 32; 33].