New analytical and numerical results for two-dimensional contact profiles

Recently, a generalized Coulomb law for elastic bodies in contact has been developed by the author, which assumes that the tangential traction is the difference of the slip stress of the contact and the stick area, whereby each stick area corresponds to a smaller contact area. It holds for multiple contact regions also. Several applications for elastic half planes, half spaces, thin and thick layers and impact problems have been published. For plane contact of equal bodies with friction, it provides exact solutions, and the interior stress field can be expressed with analytical results in closed form. In this article, a singular superposition of flat punch solutions is outlined, in which the punches are aligned with an edge of the contact area. It is shown that this superposition satisfies Coulomb's inequalities directly, and new results for the Muskhelishvili potentials of several profiles are presented. It is illustrated how problems of singularity and multi-valuedness of complex functions can be solved in closed form, and the Chebyshev approximation used by earlier authors can be avoided. For comparison, some previous solutions for symmetric profiles are appended. Some results for the interior stress field, the pressure, the frictional traction and the surface displacements are compared with FEM solutions of an equivalent problem. The small differences between both methods show characteristic features of the FEM model and the theoretical assumptions, and are shortly explained. Further, this example can be used as benchmark test for FEM and BEM programs.