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The main subject of this study is the dynamic and static pull-in analyses of rectangular nanoplates made of functionally graded materials based on the nonlocal strain gradient theory compared with the strain gradient theory, Eringen’s differential method, and classical theory considering disparate boundary conditions (BCs). To this end, the governing equation is derived based on Kirchhoff’s plate theory and then, the Galerkin method is implemented to arrive at an ordinary differential equation of second-order in the time domain. The homotopy analysis method is used as an analytical solution methodology to solve the nonlinear ordinary differential equation which is under different nonlinear forces such as intermolecular, electrostatic and hydrostatic one. Likewise, an efficient technique is exerted for omitting secular terms. Accordingly, the dynamic pull-in voltage is obtained for different non-classical continuum theories, material gradient indices, nonlocal parameters and internal length scale parameters. The dynamic and static pull-in instabilities are also compared in the same situation. Additionally, the effects of the nonlocal parameter and internal length scale parameter on the non-dimensional frequency against the initial amplitude, electrostatic force and hydrostatic actuation are explored in disparate boundary conditions. Communicated by Eleonora Tubaldi.

Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory