On Finite Element Uniqueness Studies for Coulomb’s Frictional Contact Model

We are interested in the finite element approximation of Coulomb’s frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε| log(h)|−1, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case) suggests a minor dependence of the mesh size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the nonregularized case. It can be shown that one, multiple or an infinity of solutions may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.