## 1 Introduction
[9] to study the free vibration and post buckling behavior of multilayer FG porous nanocomposite beams reinforced by GPLs. The authors applied a direct iterative algorithm and Ritz method to investigate the influence of the weight fraction and dispersion pattern within the material, as well as the geometry and dimension of the structure. Furthermore, [PERSON] et al. [10] applied the finite element approach to evaluate the stability of FG graphene platelet-reinforced cylindrical shells, including a cutout. A large parametric investigation was performed numerically to investigate the influence of the weight fraction, geometry and distribution of GPLs, total number of layers, shell dimensions, and location of the cutout on the buckling response. In addition, [PERSON] et al. [11] presented a three dimensional thermo-elastic analysis of axially symmetric FG polymer nanocomposite circular and annular plates reinforced with GPLs subjected to coupled thermal-mechanical loading. The problem was solved based on the generalized method proposed by [PERSON] and [PERSON], according to which a low content of GPLs was required to significantly improve the global stiffness of the plate. [PERSON] et al. [12] applied [PERSON]'s beam theory for the analysis of the free vibration and buckling behavior of FG porous nanocomposite beams, with various distributions of internal pores and graphene platelets. Based on a large numerical investigation, an effective increase of the stiffness was reached for a uniform distribution of the porosity and GPL within the matrix. A different approach based on the first order shear deformation theory and von-Karman nonlinear strain theory was applied by [PERSON] et al. [13] for the nonlinear analysis of the buckling and post buckling response of FG multilayer composite plates reinforced with GPLs. Thus, an eigenvalue problem was solved by the authors using a two-step perturbation technique. A further interesting work on the topic was proposed by [PERSON] et al. [14], who studied the nonlinear bending behavior of multi-layer polymer nanocomposite beams reinforced with graphene platelets,based on the [PERSON]'s beam theory, as well as the von Karman nonlinear relations and the Ritz method. Based on the numerical results, the nonlinear deformations of the nanostructure were unaffected by symmetrical distributions of graphene platelets. Similarly, [PERSON] and [PERSON] [15] applied the von Karman nonlinear strain-displacement relations together with a sinusoidal shear deformation plate approach to investigate the nonlinear bending response of FG multilayer graphene platelet-reinforced polymer composite materials. Many other works in the literature have focused on the static, dynamic, and buckling behavior of FG-reinforced nanobeams and nanostructures [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. More specifically, [PERSON] et al. [16] employed the first-order shear deformation theory for the bending and compressive buckling analysis of FG multilayer GPLs. The effective Young's modulus of the nanocomposites and [PERSON]'s ratio were determined using the Halpin-Tsai micromechanics model and the rule of mixtures, respectively. [PERSON] and [PERSON] [17] studied the free vibration problem of a FG curved nanobeam by applying [PERSON]'s principle. The [PERSON] method was proposed by the authors to solve the governing equations of motion for a simply-supported curved nanobeam, while evaluating the sensitivity of the response to the nonlocal parameter, opening angle, aspect ratio, mode number, and gradient index. A nonlocal strain gradient elasticity theory and higher order hyperbolic refined beam theory were suggested by [PERSON] and [PERSON] [18] as alternative tools to solve the buckling problems of FG curved nanobeams.