An analytical study on the buckling and free vibration of rectangular nanoplates using nonlocal third-order shear deformation plate theory

In this paper, analytical closed-form solutions in explicit forms are presented to investigate small scale effects on the buckling and the transverse vibration behavior of Lévy-type rectangular nanoplates based on the Reddy's nonlocal third-order shear deformation plate theory. Two other edges of Lévy-type rectangular nanoplates 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, buckling load ratios, non-dimensional fundamental natural frequencies and non-dimensional buckling loads of rectangular nanoplates with different combinations of boundary conditions are presented for various values of nonlocal parameters, aspect ratios and thickness to length ratios. It is observed that except for SFSF rectangular nanoplates, as the aspect ratio increases, buckling load and natural frequency decreases, while keeping all other parameters fixed. For SFSF rectangular nanoplates, by increasing the aspect ratio, the values of the buckling load and frequency ratio increase.