Indentation of an Elastic Half-space by a Rigid Flat Punch as a Model Problem for Analysing Contact Problems with Coulomb Friction

One important problem which still remains to be solved today is the uniqueness of the solution of contact problems in linearized elastostatics with small Coulomb friction. This difficult question is addressed here in the case of the indentation of a two-dimensional elastic half-space by a rigid flat punch of finite width, which has been previously studied by Spence in Proc. Camb. Philos. Soc. 73, 249–268 (1973). It is proved that all the solutions have the same simple structure, involving active contact everywhere below the punch and a sticking interval surrounded by two inward slipping intervals. All these solutions show the same local asymptotics for surface traction and displacement at a border between a sticking and a slipping zone. These asymptotics describe (soft) singularities, which are universal (they hold with any geometry) and are explicitly given. It is also proved that in cases where the friction coefficient is small enough, the sticking intervals present in two distinct solutions, if two distinct solutions exist, cannot overlap.