Free vibration of symmetric and sigmoid functionally graded nanobeams

Abstract

The objective of this paper was the investigation of vibration characteristics of both nonlinear symmetric power and sigmoid functionally graded nonlocal nanobeams. The volume fractions of metal and ceramic are assumed to be distributed through a beam thickness by sigmoid law distribution and symmetric power function. Structures with symmetric distribution with mid-plane such as ceramic–metal–ceramic and metal–ceramic–metal are proposed. Nonlocal differential Eringen’s elasticity is exploited to incorporate size dependency of nanobeam. The kinematic relations of Euler–Bernoulli beam are proposed, with the assumption of a small strain. A nonlocal equation of motion of nanobeam is derived by using principle of virtual work and then discretized by finite element method to obtain numerical solution. Numerical results show the effects of the function distribution, gradient index and nonlocal parameter on natural frequencies of macro- and nanobeam. This model is helpful in the mechanical design of nanoelectromechanical systems manufactured from FGM.