Surface effect on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates with various edge supports

• Study of the surface effect on nonlinear forced vibration of rectangular nanoplates.
• Using the Gurtin–Murdoch theory and Hamilton principle to obtain the equations.
• Solving the nonlinear force vibration problem using a numerical approach.
• Comparing results given by classical and nonclassical models in various thicknesses.

Surface stress and surface inertia effects may play a significant role in the mechanical characteristics of nanostructures with a high surface to volume ratio. The objective of this study is to present a comprehensive study on the surface stress and surface inertia effects on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates. To this end, the Gurtin–Murdoch theory, first-order shear deformation theory (FSDT) and Hamilton׳s principle are employed to develop a non-classical continuum plate model capable of taking the surface stress and surface inertia effects and also the rotary and in-plane inertias into account. To solve numerically the geometrically nonlinear forced vibration of nanoplates with different boundary conditions, the generalized differential quadrature (GDQ) method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are employed. The effects of parameters such as thickness, surface residual stress, surface elasticity, surface mass density, length-to-thickness ratio, width-to-thickness ratio and boundary conditions on the nonlinear forced vibration of rectangular nanoplates are fully investigated. The results demonstrate that surface effects on the nonlinear frequency response of aluminum (Al) nanoplate are more prominent in comparison with the silicon (Si) nanoplate.