New analytical solutions for a flat rounded punch compared with FEM

Recently, a generalized Coulomb law for elastic bodies in contact was developed by the author and several applications for elastic half-planes, half-spaces, thin and thick layers and impact problems have been published. This theory assumes that the tangential traction is the difference of the slip stress of the contact and the stick area, and each stick area is a smaller contact area. It holds for multiple contact regions also. 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 symmetric superposition of flat punch solutions is outlined and some useful formulas are listed. The necessary assumptions and simplifications are discussed. It is shown that this superposition satisfies Coulomb's inequalities directly and can also be used for torsion or shift of axisymmetric profiles. A new formula for the Muskhelishvili potential of a two-dimensional flat rounded punch is presented, which avoids the Chebyshev expansion used by other authors. Similar equations can be derived for different profiles. 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 some 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.