Vibrations of three-dimensionally graded nanobeams

This paper presents an investigation on the free vibration of three-directional functionally graded material (TDFGM) Euler-Bernoulli nano-beam, with small scale effects .To survey the small scale effects on natural frequencies of nano-beam, the nonlocal strain gradient elasticity theory is applied. The material properties except Poisson's ratio are assumed to be graded in all three axial, thickness and width directions, which can vary according to an arbitrary function. Governing equation and boundary conditions are derived, using Hamilton's principle. These equations are solved by employing generalized differential quadrature method (GDQM). The effects of some parameters such as material constant and small-scale parameters are investigated. The results show that in nonlocal strain gradient theory, the natural frequency can either be greater than the natural frequency of classical theory or smaller. These results are also compared to the results reported in the literature, which shows consistency. To the best of the researchers' knowledge, in the literature, there is no study carried out into non-local strain gradient theory for free vibration analysis of TDFGM Euler-Bernoulli nano-beams with arbitrary functions.