On finite element uniqueness studies for Coulomb’s frictional contact model. (English) Zbl 1041.74070

Summary: We are interested in 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 \(C\varepsilon^2| \log(h) |^{-1}\), where \(h\) and \(\varepsilon\) 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 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.


74S05 Finite element methods applied to problems in solid mechanics
74M15 Contact in solid mechanics
74M10 Friction in solid mechanics
74G35 Multiplicity of solutions of equilibrium problems in solid mechanics
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