Normal indentation of elastic half-space with a rigid frictionless axisymmetric punch

The contact of a simply, connected axisymmetric punch with an elastic half-space is examined. The problem is mathematically formulated by using potential theory and complex variable analysis. The final solution of these equations is obtained by assuming a polynomial punch profile. The conditions for complete contact and incomplete contact are also derived. The solutions give the pressure profile at the punch-elastic half-space interface for any polynomial punch profile, even for noninteger power polynomials, as long as the contact region is simply connected. The results show that some classic solutions in linear elasticity are special cases of the derived solution and determine the range of validity for those solutions.