Three-dimensional free vibration analysis of functionally graded graphene platelets-reinforced composite toroidal shells

In this work, based on the weak form of three-dimensional (3D) theory of elasticity, we develop a mixed iso-parametric finite element method (IFEM) for the analysis of the free vibration behavior of freely-supported, functionally graded (FG) graphene platelets-reinforced composite (GPLRC) toroidal shells. The functions of five distribution patterns of graphene platelets (GPLs) through the thickness direction are considered, for which the effective Young's modulus is estimated using the Halpin-Tsai model, and the effective Poisson's ratio and the effective mass density are estimated using the rule of mixtures. By incorporating Hamilton's principle and Reissner's strain energy functional, we derive the system equations of the mixed IFEM. Based on the mixed IFEM, the 3D solutions for the natural frequency parameters of the FG-GPLRC toroidal shells converge rapidly, and the convergent solutions are in excellent agreement with the accurate solutions for both homogeneous isotropic toroidal shells and FG transversely isotropic toroidal shells which are reported in the literature. Numerical results show that the impacts of the GPL distribution, the weight fraction of the GPL, and the radius-to-thickness ratio on the lowest natural frequencies of the FG-GPLRC toroidal shells is significant.