A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates

Abstract

In this work, the size-dependent geometrically nonlinear free vibration of functionally graded (FG) microplates is investigated. For this purpose, with the aid of Hamilton’s principle, a nonclassical rectangular microplate model is developed based on Mindlin’s strain gradient theory, Mindlin’s plate theory and the von Karman geometric nonlinearity. For some specific values of the length scale material parameters, the simple form of size-dependent mathematical formulation based on the modified strain gradient theory (MSGT) and modified couple stress theory is obtained. The generalized differential quadrature method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are utilized to determine the geometrically nonlinear free vibration characteristics of FG microplates with different boundary conditions. The parametric effects of thickness-to-material length scale ratio, material gradient index, length-to-thickness ratio, length-to-width ratio and boundary conditions on the nonlinear free vibration characteristics of FG microplates are studied through various numerical examples presented. It is found that a considerable difference exists between the results of various elasticity theories at small values of length scale parameter. A more precise prediction can be provided by using the size-dependent plate model based on the MSGT.