## 1 Introduction  
Functionally Graded Materials (FGMs) are well-known composite materials whose properties change smoothly along one or multiple directions. FGMs were studied for the first time by [PERSON] et al. [1] in 1984 for aerospace applications, and have been increasingly adopted in the literature as multi-function materials for different engineering applications. These materials are produced by blending two materials such as metals and ceramics together, and are able to eliminate stress discontinuities commonly present in layered composites. Thus far, different studies have focused on the mechanical behavior of FGMs. [PERSON] and [PERSON] [2], for example, proposed an analytical study for the free vibration of doubly-curved Functionally Graded (FG) shallow shells with stiffeners, under a thermal environment. [PERSON] et al. [3] investigated the free vibration of free-form doubly-curved FG shells. In two papers, [PERSON] et al. [4, 5] investigated the static and dynamic behavior of a FGM doubly-curved panel. [PERSON] and [PERSON] [6] also presented an analytical solution for the free vibration and dynamic problem of thick doubly-curved FG smart panels. Porosities and micro-voids are common technical issues that may be introduced during the manufacturing process, with a consecutive reduction of their mechanical properties [7]. This possible relationship between porosity and mechanical properties of materials has increased the interest of the scientific community in obtaining more precise results both experimentally and theoretically. [PERSON] et al. [7] investigated the vibration behavior of FG porous rectangular plates in a thermal environment. [PERSON] and [PERSON] [8; 9] studied the buckling behavior of FG tapered and simple beams made by porous materials, and verified that an increasing porosity volume fraction yields to a decrease in stiffness and buckling load. A general nonlocal approach based on the strain-gradient elasticity model was developed by [PERSON] and [PERSON] [10] for the vibration analysis of porous nano plates resting on elastic substrates. Many applications of single- or doubly-curved shells have increased the scientific attention to the mechanical behavior of these useful structural components. Several studies on doubly-curved shells are available in the literature. A brief review is presented here for the sake of completeness. [PERSON] [11] and [PERSON] et al. [12; 13] investigated the free vibrations of anisotropic doubly-curved shells of revolution defined by free-form meridians. [PERSON] et al. [14] and [PERSON] et al. [15; 16] employed a higher-order formulation, based on an equivalent single layer approach, for the mechanical analysis of laminated composite shell structures characterized by curved surfaces as reference domains. A layer-wise formulation was presented instead in the paper by [PERSON] et al. [17] for the same purpose. A Moving Least Squares Differential Quadrature (MLSDQ) method based on Radial Basis Functions (RBFs) was developed by [PERSON] et al. [18] to perform the free vibration analysis of laminated composite doubly-curved shells. The weak form of the governing equations of doubly-curved shells was solved by [PERSON] et al. [19] to obtain the natural frequencies of shell structures characterized by distorted domains. For this purpose, an approach based on Non Uniform Rational Basis-Splines (NURBS) was developed. [PERSON] et al. [20] employed a local version of the Generalized Quadrature Method (GDQ) for the dynamic analysis of FGM sandwich shells with variable thickness. The topic of shell structures with variable thickness was studied also by [PERSON] [21], who proposed a nonlinear higher-order shear deformation theory. A third-order shear deformation theory was developed by [PERSON] [22; 23] to investigate the nonlinear mechanical behavior of both isotropic and laminated doubly-curved shells.