Rigid frictionless indentation on half-space with surface stresses

Abstract

The formulation and solutions of a boundary value problem of an axisymmetric, rigid, frictionless indentation acting on an elastic half-space is presented. The novel feature of the formulation is associated with the treatment of surface energy effects by employing a complete Gurtin-Murdoch continuum model for surface elasticity. With use of standard Love’s representation and Hankel integral transform, such boundary value problem can be reduced into a set of dual integral equations associated with the mixed boundary conditions on the surface of the half-space. Once these dual integral equations are transformed into a Fredholm integral equation of the second kind, selected numerical procedures based upon solution discretization and a collocation technique are proposed to construct its solution numerically. After solving a system of linear algebraic equation arising from the discretization, numerical results for elastic fields are obtained and compared for indentations of different profiles and contact radii at various depths to show the size-dependency and influence of surface free energy on bulk stresses and displacements especially in the vicinity of the free surface. The significant contribution of the residual surface tension suggests that the surface energy effects have to be accounted for characterization of soft elastic solids and nanoscale material properties.