Analytical Solutions for an Isotropic Elastic Half-Plane with Complete Surface Effects Subjected to a Concentrated/Uniform Surface Load

Abstract

Within the context of Gurtin–Murdoch surface elasticity theory, closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load. Both the effects of residual surface stress and surface elasticity are included. Airy stress function method and Fourier integral transform technique are used. The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all. Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane. Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane, while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded. In the remaining situations, combined effects of surface elasticity and residual surface stress should be considered. The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects. The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.