×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Han, W., A Posteriori Error Analysis via Duality Theory, with Applications in Modeling and Numerical Approximations, (2005), New York, NY: Springer, New York, NY
[2] Han, W.; Sofonea, M., Analysis and numerical approximation of an elastic frictional contact problem with normal compliance, Appl. Math., 26, 4, 415-435, (1999) · Zbl 1050.74639
[3] Haslinger, J.; Makinen, R., Shape optimization of elasto-plastic bodies under plane strains: sensitivity analysis and numerical implementation. Springer-Verlag, Struct. Opt., 4, 3-4, 133-141, (1992)
[4] Hlavaček, I.; Nečas, J., Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Studies in Applied Mechanics 3., (1981), North Holland: Elsevier · Zbl 0448.73009
[5] Benkhira, E.-H.; Essoufi, E.-H.; Fakhar, R., Analysis and numerical approximation of an electro-elastic frictional contact problem, Math. Model. Nat. Phenom., 5, 84-90, (2010) · Zbl 1262.74017
[6] Essoufi, E.-H.; Fakhar, R.; Koko, J., A decomposition method for a unilateral contact problem with tresca friction arising in electro-elastostatics, Num. Funct. Anal. Opt., 36, 12, 1533-1558, (2015) · Zbl 1333.74081
[7] Han, W.; Jensen, S.; Shimansky, I., The kačanov method for some nonlinear problems, Appl. Num. Math., 24, 1, 57-79, (1997) · Zbl 0878.65099
[8] Brezis, H., Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Université de Grenoble, Annales de L’Institut Fourier, 18, 115-175, (1968) · Zbl 0169.18602
[9] Glowinski, R.; Le Tallec, P., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, (1989), Philadelphia, PA: Studies in Applied Mathematics, SIAM, Philadelphia, PA · Zbl 0698.73001
[10] Fortin, M.; Glowinski, R., Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems, Studies in Mathematics and its Applications 15, (1983), North Holland: Elsevier
[11] Koko, J., Uzawa block relaxation method for the unilateral contact problem, J. Comput. Appl. Math., 235, 8, 2343-2356, (2011) · Zbl 1260.74031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.