3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

Abstract

• Development of a model based on the 3-D elasticity and modified couple stress theories for FG micro/nano plates. • Exponential law is used for variation of material properties through the plate thickness. • Exact closed-form solutions are presented for the bending of functionally graded micro/nano plates. • The effects of the length scale parameter and gradient index on the bending of FG micro/nano plates are investigated. This paper addresses a 3D elasticity analytical solution for static deformation of a simply-supported rectangular micro/nanoplate made of both homogeneous and functionally graded (FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler–Pasternak elastic foundation, and its modulus of elasticity is assumed to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the boundary value problem (BVP) of plate system, including its governing partial differential equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.