Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory

A continuum model based on the nonlocal theory of elasticity is developed for buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates under uniform in-plane compression. The nanoplate is considered to be rested on two-parameter Winkler-Pasternak elastic foundation. The principle of virtual work is used to derive the governing vibration and stability equations. The weighted residual statements of the equations of motion are performed and the well-known Galerkin method is employed to obtain the nonlocal “Quadratic Functional” for embedded micro/nano-plates. The Ritz functions are taken to form an expression for transverse displacement which satisfies the kinematic boundary conditions. In this way, the entire nanoplate is considered as a single super-continuum element. Employing the Ritz functions eliminates the need for mesh generation and thus large number of degrees of freedom arising in discretization methods such as finite element (FE). The results show obvious dependency of critical buckling loads on the non-locality of the micro/nano elliptical plate, especially, at very small dimensions.