Elastic layer under axisymmetric surface loads and influence of surface stresses

Abstract

This paper presents the analysis of an infinite, rigid-based, elastic layer under the action of axisymmetric surface loads, taking the surface energy effects into account. The corresponding boundary value problems for the bulk and the surface are formulated based on a classical theory of linear elasticity and a complete Gurtin–Murdoch constitutive relation. An analytical technique using Love's representation and the Hankel integral transform is adopted to derive an explicit integral-form solution for both the displacement and stress fields. A selected numerical quadrature is subsequently applied to efficiently evaluate all involved integrals. After conducting an extensive parametric study, the surface stresses show strong influences on the responses in the region relatively close to the surface and when a length scale of the problem is comparable to the intrinsic length of the surface. Such influence is more evident when the contribution of the residual surface tension is taken into account. Results for general axisymmetric surface loads are then used to derive fundamental solutions for a unit normal concentrated load, a unit normal ring load, and a unit tangential ring load. Such basic results constitute the essential basis for the development of boundary integral equations governing other related problems, such as contact and nano-indentation problems.