## 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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