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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.

MSC:

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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