## Analysis and numerical approximation of a contact problem involving nonlinear Hencky-type materials with nonlocal Coulomb’s friction law.(English)Zbl 1419.35059

Summary: A static frictional contact problem between an elasto-plastic body and a rigid foundation is considered. The material’s behavior is described by the nonlinear elastic constitutive Hencky’s law. The contact is modeled with the Signorini condition and a version of Coulomb’s law in which the coefficient of friction depends on the slip. The existence of a weak solution is proved by using Schauder’s fixed-point theorem combined with arguments of abstract variational inequalities. Afterward, a successive iteration technique, based on the Kačanov method, to solve the problem numerically is proposed, and its convergence is established. Then, to improve the conditioning of the iterative problem, an appropriate Augmented Lagrangian formulation is used and that will lead us to Uzawa block relaxation method in every iteration. Finally, numerical experiments of two-dimensional test problems are carried out to illustrate the performance of the proposed algorithm.

### MSC:

 35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 49J40 Variational inequalities 74S05 Finite element methods applied to problems in solid mechanics
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### References:

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