Free Vibration Analysis of Polymer/Graphene Nanoplatelet/Fiber Truncated Conical Shells Embedded in an Elastic Foundation

In this paper, a semi-analytical solution is presented for the free vibration analysis of a three-phase polymer-based truncated conical shell reinforced with Graphene NanoPlatelets (GNPs) and glass fibers, embedded in an elastic foundation. The conical shell is modeled based on the First-order Shear Deformation Theory (FSDT), and the elastic foundation is modeled using the Pasternak model. The effective mechanical properties of the three-phase polymer/GNP/fiber composite are estimated utilizing the rule of mixture, Halpin-Tsai model, and the micromechanical relations. The set of the governing equations and associated boundary conditions are derived using Hamilton’s principle, and are solved analytically in the circumferential direction using trigonometric functions and numerically in the meridional direction via the Differential Quadrature Method (DQM). The natural frequencies and corresponding mode shapes are derived for various boundary conditions, including different combinations of clamped, simply supported, and free edges at both ends of the shell. Convergence of the presented numerical solution is examined, the accuracy of the presented results is confirmed, and the effects of various parameters on the natural frequencies of the shell are investigated including the circumferential wave number, semi-vertex angle of the cone, weight fraction of the fibers, weight fraction of the GNPs, and the boundary conditions.