Undamped frequency of elastically supported porous PU foam hemispherical shells reinforced with GNPs and MWCNTs using generalized Halpin–Tsai scheme

Polyurethane (PU) foam combined with Graphene Nanoplatelets (GNPs) and Multi-Walled Carbon Nanotubes (MWCNTs) offers a promising solution for providing improved mechanical properties such as higher stiffness, improved energy absorption, and enhanced damping. In this study, to homogenize the mechanical and physical properties of hybrid nanocomposite, the generalized Halpin-Tsai (GHT) scheme along with Biot's theory is extended. In addition, three different porosity distribution functions are employed to consider the effect of porosity in the homogenized hybrid nanocomposite. This research aims to illuminate the circumferential vibrational behavior of Annular Hemispherical Shells (AHSs) made from these nanocomposites, particularly under different Boundary Conditions (BCs). The dynamic behavior of AHSs using Donnell's shell theory and First-order Shear Deformation Theory (FSDT) is examined. Furthermore, the elastic foundation is modeled using the two-parameter Winkler-Pasternak theory, which considers both the normal and shear interactions between the shell and the supporting medium. The governing equations are derived through Hamilton's principle and discretized using the Generalized Differential Quadrature Method (GDQM) to achieve high accuracy in capturing the dynamic features of these composite structures. A key novelty of this research lies in the dynamic analysis of AHSs made from GNP/MWCNT-reinforced porous PU foam, subjected to arbitrary boundary conditions and modeled using an advanced homogenization framework and the GDQM. The study also looks into how Skempton's coefficient, porosity, nanocomposite combination ratio, elastic medium stiffness, BCs and Circumferential Wave Number (CWN) affect the vibrational response.