Nonlinear vibration of a shear deformable functionally graded plate

In this paper, the nonlinear partial differential equations of nonlinear vibration for a functionally graded plate in a general state of non-uniform initial stress is presented. The material properties of a functionally graded plate were graded continuously in the direction of thickness. The variation of properties followed a simple power-law distribution in terms of the volume fractions of the constituents. The equations that include the effects of transverse shear deformation and rotary inertia are derived. With the derived governing equations, the nonlinear vibration of an initially stressed functionally graded plate was studied. The governing nonlinear partial differential equations are transformed into ordinary nonlinear differential equations using the Galerkin method and the nonlinear and linear frequencies obtained using the Runge-Kutta method. The linear frequency was calculated by neglecting the nonlinear terms of the ordinary nonlinear differential equations and the von Karman assumption. The nonlinear vibration of a simply supported ceramic/metal functionally graded plate was solved. The initial stress was a combination of a pure bending stress and an extensional stress in the plane of the plate. It was found that both the initial stresses and the volume fractions of constituents greatly changed the behavior of nonlinear vibration.