## 1 Introduction  
Currently, nanocomposites in which nanofillers such as graphene platelets (GPLs) or carbon nanotubes (CNTs) are reinforced have been attracting much attention [1, 2]. These nanofillers exhibit excellent physical, chemical, and electrical properties so that their introduction can dramatically improve the performances of conventional composites. It was reported that the structural strength of polymeric composites is greatly increased when only a tiny amount of nanofillers are reinforced [3]. In terms of application, the nanocomposites are manufactured in the form of beams, plates, and shells, and their basic mechanical behaviors such as static deflection, free vibration, and buckling have been intensively and continuously investigated [4, 5, 6]. This is because the quantitative investigation of their mechanical responses is needed for their practical design in the specific configuration under consideration, even though the superiority of material grapheme has been qualitatively well-known. The mechanical behaviors of these nanocomposites are strongly affected by the distribution patterns of nanofillers, particularly through the thickness of the composite structures. So, several purposeful thickness-wise distribution patterns have been proposed according to the notion of functionally graded material (FGM), which was introduced in the late 1990s to overcome bi-material-type heat-proof composites [7, 8, 9, 10]. The representative functional distributions are FG-U, FG-O, FG-X, and FG-\(\wedge\), and the nanocomposite structures with these functional distributions of nanofillers are called FG-GPLRC and FG-CNTRC structures. Besides the functional distribution of nanofillers, the mechanical behaviors of functional nanocomposite structures are affected by the porosity distribution. This is because the difference in the solidification temperatures of a matrix and a filler during fabrication produces porosities [11; 12], and both the stiffness and mass of composite structure are remarkably influenced by the amount and distribution pattern of porosity. In this context, recently, porosity has been widely taken into consideration in the parametric investigation of the mechanical behaviors of functional nanocomposites with respect to the porosity coefficient and distribution [13; 14; 15]. Moreover, the combination of nanofiller and porosity distribution patterns leads to more diverse mechanical behaviors of functional nanocomposites because both distribution patterns are usually not the same as one another. Due to their excellent material properties and the low density, functionally graded porous nanocomposites have been attracting much attention for high-strength, lightweight structures [16]. Carbon nanotubes are cylindrical in form so their production costs are relatively higher than that of graphene platelets. In addition, their material properties are not isotropic but orthotropic, with a higher elastic modulus in the direction of CNT alignment. Thus, the mechanical responses of FG-CNTRC structures are sensitive to the CNT alignment direction such that those in the two off-alignment directions are worse than those of FG-GPLRC structures [17; 18; 19]. On the other hand, FG-GPLRC structures are usually modeled as isotropic materials according to the micromechanical homogenization approach [20]. For this reason, together with their relatively low production costs, FG-GPLRC structures nowadays are attracting more attention [18]. Meanwhile, the studies on FG-GPLRC have been based on those of FG-CNTRC because GPLs received attention somewhat later than CNTs. As well, studies on the fundamental mechanical responses of FG-GPLRC structures have mostly included geometry nonlinearity in the displacement field according to the increase in interest on large deflection bending and large amplitude vibration. Regarding studies on the nonlinear free vibration of GPL-reinforced structures, [PERSON] et al.