Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories

In this paper the free vibration behaviors of the nanoplate made of functionally graded materials with small scale effects are investigated. To study the small scale effects on natural frequencies, the Eringen's nonlocal theory is applied. To gain more accurate results in studying the nanoplate, higher order shear deformation plate theory (HSDT) is required when stocky and short nanoplates are considered. Employing the principle of minimum potential energy the governing equations are obtained. Generalize differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the nonlinear natural frequencies of FG nanoplates. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Comparison between the results of GDQ method and Aghababaei and Reddy paper's for vibration of a simply supported rectangular plate made of isotropic material reveals the accuracy of GDQ method. At the end some numerical results are presented to study the effects of material length scale parameter, plate thickness, aspect ratio, Poisson's ratio boundary condition and side to thickness ratio on size dependent frequency.