Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation

Abstract

This paper analytically investigates the nonlinear free vibration and snap-through instability of nanocomposite shallow arches reinforced with carbon nanotubes (CNTs). The functionally graded (FG) carbon nanotube reinforced composite (CNTRC) arch with shallow curvature is analysed under uniformly distributed transverse loading. The arch is assumed to rest on a three-parameter nonlinear elastic foundation in a uniform temperature field. Thermo-mechanical properties of the arch are graded through the thickness and are considered to be temperature-dependent. The equations of motion are established based on a general high-order shear deformation theory. These nonlinear equations are derived by Hamilton’s principle within the framework of the von Kármán assumption. The static equilibrium equations and dynamic equations of motion are solved analytically for the arch with immovable simply supported edges. The two-step perturbation technique and Galerkin method are implemented to obtain the closed-form solutions. The novel results illustrate the influences of CNT distribution pattern, foundation stiffness and geometrical parameters on the linear/nonlinear frequency and snap-through instability of the arch. It is shown that the minimum value of limit loads/frequencies belongs to the FG-O type of nanocomposite arches and the maximum ones are obtained in FG-X pattern.