Vibration analysis of nonlocal strain gradient embedded single-layer graphene sheets under nonuniform in-plane loads

This paper develops a nonlocal strain gradient plate model for vibration analysis of graphene sheets under nonuniform in-plane mechanical loads. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton's principle. Galerkin's method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as in-plane loading, load factor, nonlocal parameter, length scale parameter, elastic foundation, and boundary conditions on vibration characteristics of graphene sheets are examined.