Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory

Abstract

In this paper, the nonlinear bending and free vibration responses of a simply supported functionally graded (FG) microplate lying on an elastic foundation are studied within the framework of the modified couple stress theory and the Kirchhoff/Mindlin plate theory together with the von Karman’s geometric nonlinearity. The equations of motion and boundary conditions for the FG microplate are derived from the Hamilton’s principle. Due to introducing the physical neutral surface, there is no stretching-bending coupling in the constitutive equations, and the equations of motion become simpler. By using the Galerkin method, the equations of motion are reduced to nonlinear algebraic equations and ordinary differential equations (ODEs) for the bending and vibration problems respectively. By solving the algebraic equations, closed-form solutions for the nonlinear bending deflection of the microplate are derived. Closed-form solutions for the nonlinear vibration frequency are also obtained by applying He’s variational method to the ODEs. Based on the obtained closed-form solutions, numerical examples are further presented to investigate the effects of the material length scale parameter to thickness ratio, the length to thickness ratio, the power law index and the elastic foundation on the nonlinear bending and free vibration responses of the microplate.