Nano-Scaled Plate Free Vibration Analysis by Nonlocal Integral Elasticity Theory

In the current study, a finite element method is developed using the principle of total potential energy based on nonlocal integral elasticity theory to investigate the free vibration behavior of nano-scaled plates. The classical plate theory is considered for deriving the formulations of the plate. The eigenvalue problem is extracted by using the variational principle, and corresponding natural frequencies of free vibration are obtained using a numerical method. Different boundary conditions, various geometries, and kernel types can now be appropriately analyzed by using the nonlocal finite element method proposed in the current article. The results of the present study are compared with those available in the literature. Then the effects of nonlocal parameters, geometrical parameters, surface effects and various boundary conditions, including all sides clamped, all sides simply supported and clamped free, on the free vibration behavior of nano-scaled plates are investigated. It is concluded that considering the nonlocal effect results in a reduction of the natural frequencies. Natural frequencies decrease by increasing the length of the square plate and increasing the aspect ratio of the rectangular plate. Besides, it has been seen that variations of natural frequencies with the nonlocal parameter become less noticeable for plates with more considerable lengths.