Free and Forced Vibration Analyses of Functionally Graded Graphene-Nanoplatelet-Reinforced Beams Based on the Finite Element Method

The finite element method (FEM) is used to investigate the free and forced vibration characteristics of functionally graded graphene-nanoplatelet-reinforced composite (FG-GPLRC) beams. The weight fraction of graphene nanoplatelets (GPLs) is assumed to vary continuously along the beam thickness according to a linear, parabolic, or uniform pattern. For the FG-GPLRC beam, the modified Halpin–Tsai micromechanics model is used to calculate the effective Young’s modulus, and the rule of mixture is used to determine the effective Poisson’s ratio and mass density. Based on the principle of virtual work under the assumptions of the Euler–Bernoulli beam theory, finite element formulations are derived to analyze the free and forced vibration characteristics of FG-GPLRC beams. A two-node beam element with six degrees of freedom is adopted to discretize the beam, and the corresponding stiffness matrix and mass matrix containing information on the variation of material properties can be derived. On this basis, the natural frequencies and the response amplitudes under external forces are calculated by the FEM. The performance of the proposed FEM is assessed, with some numerical results obtained by layering method and available in published literature. The comparison results show that the proposed FEM is capable of analyzing an FG-GPLRC beam. A detailed parametric investigation is carried out to study the effects of GPL weight fraction, distribution pattern, and dimensions on the free and forced vibration responses of the beam. Numerical results show that the above-mentioned effects play an important role with respect to the vibration behaviors of the beam.