## 1 Introduction
[PERSON] [24] and [PERSON] and [PERSON] [25] included the stretching effect and shear strains in their higher-order formulation for the large-amplitude vibrations of laminated doubly curved shells. [PERSON] et al. [3] analyzed the free vibration of free-form doubly-curved shells made by FGMs, while applying some higher-order theories. [PERSON] et al. [26] investigated the agglomeration effect on the natural frequency of FG carbon nano-tube-reinforced laminated composite doubly-curved shells. A further study on nonlinear vibrations of doubly-curved shallow shells can be found in [PERSON] [27]. [PERSON] et al. [28] investigated the free vibrations of FG spherical shell panel using an innovative exact closed-form solution. A higher-order facet shell element was developed by [PERSON] et al. [29] for the free vibration analysis of composite and sandwich laminates. Few studies conducted thus far, however, have focused on the mechanical behavior of doubly-curved shells made by FGM in the presence of porosities. This problem is tackled here numerically, while focusing on the free vibration of FG porous doubly-curved shell with simply supported boundary conditions. This is the main novel aspect introduced in the paper, since previous works were focused only on beam structures and one-dimensional models. An improved third-order shear deformation theory was employed by [PERSON] et al. [30] to formulate the governing equations for the free vibrations of functionally graded beams including porosities. The same analyses were performed by [PERSON] and [PERSON] [31], including the effect of nonlinearities on the structural response. In their paper, the influence of porosities on the structural response was proven through a set of experimental analyses. [PERSON] and [PERSON] [32] studied the flexural vibration of functionally graded beams based on Timoshenko theory by means of the Chebyshev collocation method. The Timoshenko beam theory was employed by [PERSON] et al. [33] to investigate the vibrational behavior of two-dimensional functionally graded nano and microbeams includingporosities. To this aim, the Eringen's nonlocal elasticity and the modified couple stress theories were employed. [PERSON] and [PERSON] [34] investigated the large-amplitude nonlinear vibration characteristics of functionally graded Timoshenko beams made of porous constituents. Finally, [PERSON] investigated the vibration and stability analyses of porous FG sandwich beams resting on elastic foundations [35]. For this purpose, various hierarchical refined exponential, polynomial, and trigonometric higher-order beam theories were developed in a generalized manner. An empirical approach has been used in all the aforementioned papers concerning porous and imperfect media to evaluate the overall mechanical properties of the composites at issue [30; 31; 32; 33; 34; 35]. For this purpose, the well-known rule of mixture has been modified to deal with porosities. Since this approach has been validated by other authors, as illustrated in [30; 31; 32; 33; 34; 35], the same empirical scheme is employed in the present paper to deal with FGM structures. The methodology in hand is also extended to the case of shell structures, since the previous studies are mainly focused on beams and plates with porosities. In other words, the authors do not aim to propose a new micromechanical scheme for porous and imperfect media, but their purpose is to apply an existing homogenization technique, the rule of mixture, to easily include and investigate the presence of porosities in a functionally graded material. The First-order Shear Deformation Theory (FSDT), combined with the Hamilton's principle, is applied to obtain the governing equations of the problem [36; 37; 38; 39], which is defined by means of five dynamic parameters. The accuracy of the proposed formulation is thus verified through a comparative evaluation between our results and some predictions available in the literature.