Exact solutions for free vibration of lévy-type rectangular nanoplates via nonlocal third-order plate theory

In this paper, exact closed-form solutions in explicit forms are presented to investigate small scale effects on the transverse vibration behavior of Lévy-type rectangular nanoplates based on the Reddy’s nonlocal third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton’s principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on frequency ratios and non-dimensional fundamental natural frequencies of rectangular nanoplates with different combinations of boundary conditions are tabulated for various values of nonlocal parameters, aspect ratios and thickness to length ratios. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. Also, the present study may be useful for static and dynamic analysis of thicker nano scale plate-like structures, multi-layer graphene and graphite as composite or sandwich structures.