Surface effects in an elastic solid with nanosized surface asperities

The effects of surface elasticity and surface tension on the stress field near nanosized surface asperities having at least one dimension in the range 1–100  nm is investigated. The general two-dimensional problem for an isotropic stressed solid with an arbitrary roughened surface at the nanoscale is considered. The bulk material is idealized as an elastic semi-infinite continuum. In accordance with the Gurtin–Murdoch model, the surface is represented as a coherently bonded elastic membrane. The surface properties are characterized by the residual surface stress (surface tension) and the surface Lame constants, which differ from those of the bulk. The boundary conditions at the curved surface are described by the generalized Young–Laplace equation. Using a specific approach to the boundary perturbation technique, Goursat–Kolosov complex potentials, and Muskhelishvili representations, the boundary value problem is reduced to the solution of a hypersingular integral equation. Based on the first-order approximation, some numerical results in the case of a periodic shape of the surface and the analysis of the influence of surface stress, surface tension, the surface shape, and the size of the asperity on the hoop stress at the surface are presented. It is found that the surface tension alone produces a high level of stress concentration, much more than can be reduced by surface stress arising as a result of deformation. The stress formula obtained by Gao (1991) for sinusoidal surfaces at the macrolevel is extended to nanosized surface asperities.