Free vibration analysis of functionally graded doubly curved nanoshells using nonlocal first-order shear deformation theory with variable nonlocal parameters

This paper studies the free vibration of the functionally graded doubly curved nanoshells using nonlocal first-order shear deformation theory with variable nonlocal parameters. The novelty of the current work is that the nonlocal parameters vary smoothly through the thickness of the nanoshells which is never investigated in the past. Four types of the doubly curved nanoshells named flat plates, spherical shells, hyperbolic parabolic shells, and cylindrical shells are considered. The first-order shear deformation theory, the nonlocal elasticity theory, and Hamilton's principle are used to establish the governing equations of motion of the functionally graded doubly curved nanoshells. The frequencies of the simply supported functionally graded doubly curved nanoshells are carried out via Navier's solution technique. The numerical results obtained by the proposed formulations are compared with other published results in several special cases to demonstrate the accuracy and efficiency of the developed model. Furthermore, the effects of some parameters such as the nonlocal parameters, the power-law index, the thickness-to-sides ratio, the radius ratio on the free vibration response of the functionally graded doubly curved nanoshells are investigated in detail.