Influence of Surface Stresses on the Deflection of Circular Nanoplate with Two-Parameter Elastic Substrate

Abstract

This paper presents the influence of surface energy effects on the deflection of circular nanoplate with two-parameter elastic substrate. The governing equation for axisymmetric bending of the nanoplate, based on the Gurtin-Murdoch surface elasticity theory, resting on a Winkler-Pasternak elastic foundation is derived from a variational approach based on the concept of minimum total potential energy. The analytical general solution to the governing equation is then obtained in terms of the modified Bessel functions. Finally, closed-form solutions for deflections, bending moment and transverse shear in the nanoplate subjected to normally distributed loading are presented explicitly for the boundary conditions of simple, clamped, and free edges.  A set of numerical solutions are selected to demonstrate the influence of surface material parameters and the substrate moduli on the deflection and bending moment profiles of a silicon nanoplate on Winkler-Pasternak foundation. It is found that the nanoplate clearly shows size-dependent behaviors, and becomes stiffer with the existence of surface stresses.