Nonlinear vibrations of functionally graded Mindlin microplates based on the modified couple stress theory

This paper investigates the size-dependent vibrational behavior of functionally graded (FG) rectangular Mindlin microplates including geometrical nonlinearity. The FG Mindlin microplate is considered to be made of a mixture of metal and ceramic according to a power law distribution. To this end, based on the modified couple stress theory (MCST) and Hamilton’s principle, the governing equations of motion and associated boundary conditions are derived. In the solution procedure, the set of nonlinear partial differential equations is discretized through the generalized differential quadrature (GDQ) method. Afterwards, the numerical Galerkin scheme is employed to convert the discretized partial differential equations of motion to the Duffing-type ordinary differential equations. The periodic time differential operators introduced based on the derivatives of a periodic base function are used to discretize differential equations on the time domain. Finally, the pseudo arc-length continuation method is utilized to numerically solve the set of nonlinear algebraic parameterized equations. The effects of the important parameters including material gradient index, length-to-thickness ratio, length scale parameter, and boundary conditions on the vibrational characteristics of rectangular Mindlin microplate are thoroughly discussed.