Axi-symmetric contact of a punch on an elastic half-space in the presence of tangential tractions

A solution of the axi-symmetric contact problem when an elastic half-space is loaded by both normal and tangential tractions over a circular region is presented. It is assumed that the indenter is rigid and has a profile which can be specified as a polynomial with even powers. Both the spherical and flat-ended cylindrical punches are considered. ae for the contact tractions, total load and the penetration depth of the tip of the indenter are found. When the tangential traction vanishes, the well-known solutions for the frictionless contact are recovered.