Discrete contact problems with friction: a stress-based approach

Many three-dimensional contact problems with friction between linearly elastic bodies and rigid supports can be successfully modelled by using Signorini’s law of unilateral contact and generalized Coulomb’s law of anisotropic friction. The discrete stress-based static formulation of such a class of nonlinear problems leads to quasi-variational inequalities, whose unknowns, after condensation on the initial contact area, are the normal and tangential contact forces. In this paper a block-relaxation solution algorithm of these inequalities is studied, at the typical step of which two subproblems are solved one after the other. The former is a problem of friction with given normal forces, the latter is a problem of unilateral contact with prescribed tangential forces. For friction coeffcient values smaller than an explicitly-computable limit value, every step of the iteration is shown to be a contraction. The contraction principle is used to establish the well-posedness of the discrete formulation, to prove the convergence of the algorithm, and to obtain an estimate of the convergence rate. An example shows the sub-optimality of the obtained limit value of the friction coefficient.