Mathematical analysis of a two degree-of-freedom frictional contact problem with discontinuous solutions

In this paper, we consider a generalized formulation of a quasistatic frictional contact problem for a material point, subject to a system of linear springs and constrained to remain in a half plane. The unilateral Signorini law and the Coulomb law of friction govern the contact with the boundary of the forbidden region. The formulation we consider here is generalized, because it is given in terms of a variable different from the original time variable. The new variable (arc-length type) allows us to spread out the behavior of the system along time discontinuities that are expected in this kind of problem (in terms of the new variable, the solution is absolutely continuous). The mathematical formulation is given by means of a differential inclusion, which admits branches of solutions. Among them, assuming that the applied force does not oscillate with infinite frequence, it is possible to select one that will make it possible to reconstruct a solution to the mechanical problem.