## 1 Introduction
A key point of the static and dynamic response of FG shell structures is related to the presence of porosities, which can form during a fabrication process, with possible effects on the global structural response. Indeed, an increasing attention to this aspect has been devoted in the scientific community for a correct interpretation of the mechanical performances of FG materials and structures. For example, in a recent work, [PERSON] and [PERSON] [34] assessed the possible effect of porosity on the vibration behavior and static response of FG magneto-electro-elastic plates, with a clear reduction of the natural frequencies for an increased porosity within the material. In another study, [PERSON] et al. [35] analyzed the effect of porosity on the structural behavior of skew FG magneto-electro-elastic plates. [PERSON] et al. [36] performed the buckling analysis of higher order graded smart piezoelectric plates with porosities resting on an elastic foundation. It was found that the buckling behavior of piezoelectric plates is significantly influenced by the porosity distribution. In the further works by [PERSON] et al. [37; 38], the authors studied the vibration response of longitudinally traveling FG plates with porosities [37] while considering the thermo-mechanical coupled response in [38]. A similar free vibration problem was studied in [39; 40] for FG cylindrical shells, by means of the sinusoidal shear deformation theory and the Rayleigh-Ritz method, accounting for the possible presence of defects and porosities. In the context of nanomaterials and nanostructures, some modified couple stress theories have been recently proposed as efficient theoretical tools to study their coupled thermomechanical vibration behavior, also in presence of different levels of porosity, see [41; 42; 43; 44; 45; 46; 47], among others. Up to date, however, there is a general lack of works in literature focusing on the dynamic buckling response of bi-directional (BD)-FG cylindrical shell embedded in a Winkler-Pasternak foundation, including the simultaneous effect of porosity. To this end, we propose the third-order shear deformation theory (TSDT) to model the cylindrical shells with porosities, subjected to an axial compressive excitation. The [PERSON]'s principle will be employed to determine the governing partial equations of motion, whereby the generalized differential quadrature (GDQ) method is adopted to solve the problem together with the associated boundary conditions into a system of Mathieu-Hill equations. Afterward, the Bolotin method is employed to determine the boundaries of the dynamic instability region (DIR) of BD-FG cylindrical shells. A systematic study focuses on the sensitivity of the dynamic stability behavior to different dimensionless ratios, i.e., the thickness-to-radius ratio, or the length-to-radius ratio, as well as to different boundary conditions, transverse and longitudinal power indexes, porosity volume fractions and foundation constants. The paper is organized as follows. In Section 2 we determine the governing equations of the problem for porous BD-FG cylindrical shells, which are solved numerically according to the GDQ and Bolotin methods, as detailed in Section 3. Section 4 aims at validating the proposed approach and shows the main results from a broad numerical investigation, whereas the final remarks are discussed in Section 5.