## 1 Introduction  
In the mid-1980s, the notion of functionally graded material (FGM) was proposed by Japanese scientists to develop high-performance heat-resisting composites for the Space Shuttle in high-temperature environments [1]. Thereafter, FGMs have been applied in various engineering fields, such as nuclear reactors and chemical plants, where high-performance heat-resisting materials are required. The heat-resisting composites used in the early days were lamination-type, which suffers from fatal thermal stress concentration at the lamina interfaces. This fatal stress concentration was caused by the sharp material discontinuity across the lamina interfaces and could frequently cause the microcracking and debonding. Owing to this problem, the conventional lamination-type composites encounter limitations in their use in high-temperature environments. The main idea of FGM was to enforce the material continuity across the lamina interfaces, which was achieved by introducing a graded layer between two homogeneous layers. The particles within the graded layer are mixed according to the desired volume fraction distribution with the gradient in a specific spatial direction, while satisfying the overall material continuity. Functionally graded volume fraction distribution within the graded layer is designed to maximize the target performance using the numerical optimization technique [2,3]. Due to the unlimited potential for being a next-generation advanced material, FGMs have been spotlighted and have attracted extensive research. A tremendous number of papers on material characterization and modeling, analysis and design, and fabrication and testing have been published [4; 5; 6]. The most research on FGMs was performed based upon the implicit assumption that FGMs are perfect without porosity. However, it had been reported that micropores and porosity may be generated during the fabrication of FGMs owing to the difference in solidification temperatures of constituent materials [7; 8; 9]. Thereafter, a number of attempts have been made to model porous materials and investigate their mechanical responses. A simplified mixture rule was proposed to predict the material properties of porous materials and validated through the experiment of [PERSON] et al. [10], where the predicted effective material properties were in good agreement with the experiment. [PERSON] et al. [11; 12] investigated static bending, buckling, free, and forced vibration responses of shear deformable FG porous beams using Timoshenko beam theory. [PERSON] [13] examined the effect of porosity on the static bending and free vibration of simply supported FG porous plates using the first-order shear deformation plate theory (SDPT). [PERSON] and [PERSON] [14] investigated the free vibration of FG porous cylindrical micro shell subjected to thermal load using the first-order shear deformation shell and modified couple stress theories. [PERSON] et al. [15] numerically investigated the free and forced vibration responses of FG porous beam-type structures by the finite element method (FEM) incorporated with Timoshenko and Euler-Bernoulli beam theories. [PERSON] et al. [16] analyzed the nonlinear bending and vibration characteristics of FG porous tubes within the framework of nonlocal strain gradient theory. [PERSON] et al. [17] proposed an analytical method for primary resonance analysis of FG porous cylindrical shells including the damping effect. [PERSON] et al. [18] examined the effect of porosity on the nonlinear vibration of FG porous nano beams using von Karman nonlinearity and the Euler-Bernoulli beam theory. [PERSON] et al. [19] examined the bending, free vibration and buckling responses of FG porous micro plates using the first-order SDPT. [PERSON] et al. [20] investigated the free vibration of FG porous rectangular plates with uniform elastic boundary conditions using an improved Fourier series method. [PERSON] et al.