A two-variable first-order shear deformation theory coupled with surface and nonlocal effects for free vibration of nanoplates

Abstract

A two-variable first-order shear deformation theory in combination with surface free energy and small scale (size-effect) is employed to present a simple and computationally efficient formulation for the free vibration of nanoplates with arbitrary boundary conditions. The free surfaces are modeled as two-dimensional membranes adhering to the underlying bulk material without slipping. To take into account the small scale effect, the nonlocal constitutive relations of Eringen are used. The equations of motion and the related boundary conditions are derived by employing Hamilton’s principle. An analytical solution for the free vibration of simply supported nanoplates is obtained and comparison studies with the results of other two-dimensional theories available in the open literature are performed to validate the proposed formulation. Then, by using the differential quadrature method, an approximate solution for nanoplates surrounded by an elastic media and with arbitrary boundary conditions is developed. Consequently, the effects of surface free energy, the small scale parameter and elastic constant of surrounding media together with geometrical parameters on the natural frequencies of nanoplates are investigated.