A Novel Analytical Homotopy Solution for Nonlinear Vibration of Porous GPL Reinforced Plate on Nonlinear Elastic Foundation

This study presents a comprehensive analysis of the nonlinear free vibration behavior of porous plates reinforced with graphene platelets resting on a Winkler-Pasternak nonlinear elastic foundation, considering the in-plane and out of plane displacement component. The governing equations of motion of the thin plate are derived based on Von Kármán strain relations and Hamilton's principle. The Galerkin method is employed to transform the partial differential equations into ordinary ones. The system of nonlinear differential equations has been solved using the Homotopy Analysis Method (HAM) for the first time. The time response of the plate vibrations has been found analytically and compared with that predicted by the numerical method for the porous GPL-reinforced plate. It is shown that the results have very good agreement with the numerical solutions that obtained from the fourth-order Runge-Kutta method. The resulting nonlinear frequencies and time histories provide critical insights into the dynamic response of the reinforced plates. The findings reveal that neglecting in-plane displacements leads to significant errors, especially as the initial amplitude increases. Additionally, a symmetrical distribution pattern of graphene platelets substantially enhances the plate's stiffness and nonlinear frequency. The analysis also indicates that an increase in geometric parameters, such as the plate aspect ratio, yields a decrease in nonlinear frequency. These results underscore the importance of considering geometric parameters, boundary conditions, foundation parameters and the distribution patterns of graphene platelets for accurately assessing the nonlinear behavior of graphene-reinforced porous plates.