A semi-analytical sinusoidal shear deformation theory for nonlinear dynamic response and vibration of CNT-FGM doubly curved shallow shells

This paper presents a semi-analytical approach for nonlinear vibration and dynamic response of ceramic-metal functionally graded doubly curved shallow shells reinforced by carbon nanotubes under three different types of boundary conditions. This approach utilizes a new sinusoidal shear deformation theory combined with von Karman's geometric nonlinearity. The proposed theory contains the sinusoidal distribution of transverse shear strains and satisfies the conditions of free transverse shear stress on both the top and bottom surfaces of the shell with only four unknown variables. The shells are made of a ceramic-metal matrix reinforced by carbon nanotubes with two types of distributions, both uniform distributions and functionally graded distributions. Equations of motion are derived from Hamilton's principle and then solved by the Galerkin method and Airy's stress function in which the closed-form solutions of the shells with fully simply supported edges, fully clamped edges, and two opposite simply supported and two opposite clamped edges are obtained. The study investigates the effects of boundary conditions, distribution types, volume fraction of carbon nanotubes, and geometrical parameters on the dynamic response, free and forced vibration of doubly curved shallow shells through numerical results.