Nonlinear free vibration of skew nanoplates with surface and small scale effects

• As a first endeavor, investigating the nonlinear free vibration of skew nanoplates.
• Considering the influences of free surface energy and size effect simultaneously.
• The formulations are derived based on the nonlocal and surface elasticity theories.
• Using the differential quadrature method as an efficient and accurate method.
• Studying the effects of different parameters on the nonlinear frequency parameters.

The nonlinear free flexural vibration of skew nanoplates is studied by considering the influences of free surface energy and size effect (small scale) simultaneously. The formulations are derived based on classical plate theory (CPT) in conjunction with nonlocal and surface elasticity theories using Hamilton's principle. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The free surfaces are modeled as two-dimensional membranes adhering to the underlying bulk material without slipping. The solution algorithm is based on the transformation of the governing differential equation from the physical domain to a rectangular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The effect of small scale parameter and surface effect together with the geometrical parameters and boundary conditions on the nonlinear frequency parameters of the skew nanoplates are studied.