Integrating finite element modeling with high-order shear deformation theory for nonlinear free vibration of CNT/SiO2 nanocomposite plates

Abstract

This study establishes a unified framework to investigate the mechanical properties and geometrically nonlinear free vibration of polymer nanocomposite rectangular plates containing carbon nanotubes (CNTs) and silica nanoparticles (SiO2). A micromechanics-based finite element approach is employed to predict the effective mechanical properties of the hybrid nanocomposite, accounting for factors such as volume fraction, nanofiller geometries, interphase characteristics, and clustering effects. The geometrically nonlinear governing equations are derived using Reddy’s third-order shear deformation plate theory, von Kármán-type nonlinear strain–displacement relations, and Hamilton’s principle. In order to solve the governing equations, a multistep numerical methodology is applied, incorporating the generalized differential quadrature scheme, Galerkin approach, time-periodic differential scheme, pseudo-arc-length continuation algorithm, and modified Newton–Raphson method. A comprehensive assessment of nonlinear frequency response curves is performed with consideration of microstructure-level parameters and boundary condition variations. It is concluded that increasing nanofiller content, leveraging elongated CNTs and fine SiO2, and optimizing nanofiller dispersion patterns significantly enhance both linear and nonlinear frequencies, with fully clamped boundary conditions exhibiting the highest frequencies. When the length (width)-to-thickness ratio of a 1 vol.% CNT/5 vol.% SiO2/polymer nanocomposite plate is 12, the non-dimensional linear frequencies incorporating aligned, randomly oriented, and agglomerated nanofillers are 0.6765, 0.5814, and 0.5237, respectively, under simply supported edges, and 1.187, 1.0196, and 0.9179, respectively, under fully clamped edges. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2025.