On the nonlocal free vibration analysis of functionally graded porous doubly curved shallow nanoshells with variable nonlocal parameters

In this paper, the free vibration behavior of functionally graded (FG) porous doubly curved shallow nanoshells with variable nonlocal parameters is investigated. The classical Eringen's nonlocal elasticity theory is modified and applied to capture the small size effect of naturally discrete FG nanoshells. The effective material properties of the FG porous doubly curved shallow nanoshells including the nonlocal parameters are graded continuously through the thickness direction via the rule of mixture. A combination of the first-order shear deformation theory and the modified nonlocal elasticity theory is developed to describe the kinematic and constitutive relations of the FG doubly curved shallow nanoshells. The Hamilton's principle is employed to establish the governing equations of motion of FG porous doubly curved shallow nanoshells and then solved analytically using the Navier's solution. The accuracy and correctness of the proposed algorithm are demonstrated by comparing its results with those available from other researchers in the existing literature. Moreover, a comprehensive parametric study is presented and discussed in detail to show the effects of the geometric parameters, material properties, porosity, and the variation of the nonlocal parameter on the free vibration behavior of the FG porous doubly curved shallow nanoshells. Especially, the numerical results showed that the variation of the nonlocal parameters has significant effects on the free vibration behavior of the FG porous doubly curved shallow nanoshells. Some new results are also reported which will serve as a benchmark for future analysis of FG porous doubly curved shallow nanoshells.