Elastic behavior of a half-space with a Steigmann–Ogden boundary under nanoscale frictionless patch loads

Abstract

In this paper, the linearized version of Steigmann–Ogden theory of surface elasticity is employed to solve the elastic fields in a half-space subjected to nanosized normal loads. The half-space boundary is modeled as a mathematical membrane of vanishing thickness that can resist both strain and bending deformation. The method of Boussinesq displacement potentials and Hankel integral transforms are coupled to develop general solutions to the problem under frictionless patch loads confined inside a circular area on the plane boundary. As examples, the elastic fields due to five representative traction loads that are typically encountered in classical contact mechanics are derived and expressed in series representations of improper integrals involving rational functions, powers, exponentials, and Bessel functions. Extensive numerical experiments are implemented to show and compare the significances of both Gurtin–Murdoch and Steigmann–Ogden theories of surface mechanics. For surface material parameters with physically interpretable signs and orders of magnitude, the results suggest the equal importance of both strain and curvature dependences of the half-space boundary. Benefitting from the incorporation of surface elasticity, both surface models report much more smooth displacement and stress variations near the loading perimeter. In particular, the Steigmann–Ogden model of surface elasticity predicts lower maximum displacements and stresses (in absolute values) as compared with those by both the Gurtin–Murdoch and the classical model.