On the contact problem with slip displacement dependent friction in elastostatics

The problem of the static contact with slip dependent friction between an elastic body and a rigid foundation is considered. The existence of a solution is proved. The link between the equilibrium and the minimum of the energy is deduced and sufficient conditions for stability are given. The qualitative behaviour of the solution is decided by the competition between two parameters which involve the geometry, the normal stress, the elastic properties and the slip weakening of the friction force. We distinguish two cases; In the first one we prove that the solution is unique and depends continuously upon the load. In the second case we point out the loss of uniqueness and the discontinuous dependence upon the load of the solution. The existence of an homogeneous sliding is analysed and its local uniqueness is studied. We give sufficient conditions. for the bifurcation of nonhomogeneous sliding from homogeneous sliding. An application to anti-plane problem is also given.