Three-dimensional free vibration analysis of rectangular nanoplates based on nonlocal theory of elasticity

Conventional Ritz and Galerkin methods based on local theory of elasticity employ polynomials as their approximating functions, however these methods are not convenient to use in three-dimensional nonlocal analysis. In the present study, to conquer this difficulty, a type of weighted residual approach with a set of trigonometric approximating functions were developed. By using appropriate trigonometric approximating functions, it is possible to consider the effect of various edges boundary condition on frequency behavior of nanoplate. Validation of present formulation is carried out by comparing numerical result with the published results. It is concluded that the effect of nonlocal parameter on natural frequencies is significant especially in higher modes due to the lower wavelength of the mode. The research shows that in nonlocal elasticity there are distinct discrepancies between behaviors of two and three-dimensional results. In addition, the difference between the two- and three-dimensional results in local elasticity is not as noticeable as in nonlocal elasticity. Furthermore, the effects of length to thickness ratio, aspect ratio, nonlocal parameter and different boundary conditions on fundamental natural frequency of nanoplates were studied. This benchmark solution can be used to assess the accuracy of conventional two-dimensional theories.