Size and Foundation Effects on the Vibration of Buckled Functionally Graded Microplates Within the Modified Couple Stress Theory Framework

In the present paper, the vibration behavior of a buckled functionally graded (FG) microplate lying on a nonlinear elastic foundation is studied. The modified couple stress theory is utilized to capture the size effect of the FG microplate, and the Mindlin plate theory with the von Karman's geometric nonlinearity is adopted to describe its deflection behavior. Based on these assumptions and Hamilton's principle, the governing equations and associated boundary conditions are derived for the FG microplate. By linearizing the governing equations around a pre-buckling/post-buckling state, linear perturbation equations are obtained. After substituting the pre-buckling/post-buckling deformation and assumed vibration mode into the linear perturbation equations and applying the Galerkin method, an eigenvalue problem is obtained, from which the free vibration frequency of the FG microplate around its pre-buckling/post-buckling state can be determined analytically. Based on the obtained closed-form solutions, numerical examples are also presented to investigate the effects of the material length scale parameter to thickness ratio, the power law index, and the stiffness of the elastic foundation on the vibration behavior of the buckled FG microplate.