On three-dimensional nonlocal elasticity: Free vibration of rectangular nanoplate

In Ritz and Galerkin methods based on classical elasticity theory, mostly polynomials are employed as approximating functions. However, these methods in their common form are not suitable to be used in three-dimensional nonlocal analysis. This is because the Galerkin approach has a problem with free boundary conditions in nonlocal elasticity theory, and in three-dimensional analysis of any plate top and bottom surfaces are considered as free boundaries. In this research two simple cases of three-dimensional free vibration of simply-supported nanoplate and wave propagation in three-dimensional infinite nonlocal solid are investigated. To achieve that, suitable potential functions for Helmholtz displacement vector representation of these cases are presented. To deal with other boundary conditions, novel trigonometric series are developed to be used as approximating functions in a Galerkin based approach. Considering the numerical results, the effects of length to thickness ratio, aspect ratio, nonlocal parameter, and different boundary conditions on the non-dimensional natural frequencies of nanoplates are studied. It can be shown that nonlocal elasticity theory and Aifantis' strain gradient elasticity theory are equivalent, therefore, the results of this investigation can be extended to that theory. The results show that the difference between both behavior and magnitude of frequencies obtained from two- and three-dimensional nanoplate analysis is noticeable.