Free vibration of nanoplates based on a nonlocal two-variable refined plate theory

The application of a two-variable refined plate theory is extended to the free vibration of nanoplates. The theory takes into account the effects of small scale and the quadratic variation of the transverse shear strains through the thickness of the nanoplate and hence, it does not require the use of shear correction factors. The equations of motion and the related boundary conditions are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined plate theory via Hamilton’s principle. The obtained equations are only dynamically coupled. Exact solution for the simply supported nanoplates is obtained to verify the theory by comparing its results with other available solutions in the open literature. Then, using the differential quadrature method (DQM) as an accurate and computationally efficient numerical method, the effect of small scale parameter on the frequency parameters of the nanoplates with some different types of boundary conditions are studied.