Effect of a Four-Parameter Kerr-Type Elastic Foundation on the Large-Amplitude Free Vibration of the Nanocomposite Beams: A Dual-FG Modeling

Abstract

This study examines the free vibration of a unique nanocomposite beam, characterized by large-amplitude motions. The composition of the material comprises functionally graded (FG) polymers fortified with graphene platelet (GPL) nanofillers, forming a nanocomposite structure with a dual FG designation. The arrangement of graphene platelet nanofillers within the dual-phase matrix follows one of five linear functionally graded models. To ascertain the behavior of the nanocomposite, the homogenization process employs the Voigt rule of mixtures for the power law-based FG matrix, while the Halpin–Tsai homogenization procedure is employed for the functionally graded graphene platelet (FG-GPL) reinforced nanocomposite. The beam is supported by a novel four-parameter nonlinear Kerr elastic foundation, located at the beam’s lower surface, which governs the linear and nonlinear frequencies. Employing the Karama higher-order shear deformation theory, the beam’s displacements are estimated, and the strains are described by the Von-Kármán geometrically nonlinear relations. Solving the problem necessitates the utilization of the generalized differential quadrature (GDQ) method, coupled with a bilateral iterative displacement control (BIDC) technique. Notably, the equations are rendered independent of the time variable implementing the weighted residual Galerkin procedure. The obtained frequencies of the GPL-reinforced beam with an isotropic homogeneous matrix are compared against those documented in existing literature, ensuring validation. Subsequently, parametric results are presented to assess the impact of the novel nanocomposite features on the linear and nonlinear frequencies of beams.