Boussinesq problem with the surface effect and its application to contact mechanics at the nanoscale

In the literature, it has been demonstrated that residual surface stress and surface elasticity are two equally important parts of surface stress theory and that, generally, neither of these aspects can be neglected. In this paper, we develop a non-classical formulation of the Boussinesq problem with the surface effect, in which both the residual surface stress and the surface elasticity are considered. To take into account the surface effect, a Lagrangian description of the governing equations of the surface is adopted. The theoretical and numerical results in this paper show that the contributions of the residual surface stress and the surface elasticity to the stresses and displacements at the surface are not always equal. The residual surface stress mostly influences the normal stress, whereas the surface elasticity is a dominant factor in the in-plane shear stress. As an application of this formulation, the three-dimensional Hertzian contact problem at the nanoscale is studied. It is concluded that the surface effect strengthens the elastic contact stiffness. The smaller the contact region, the larger the contact stiffness. Finally, in terms of the dimensionless surface parameters, the influences of the residual surface stress and the surface elasticity on the stresses and displacements are further studied, and a simple scaling law for the stresses and displacements at the surface is constructed for the first time.