Analysis and numerical approximation of an elastic frictional contact problem with normal compliance

Han, Weimin ; Sofonea, Mircea

Applicationes Mathematicae, Tome 26 (1999), p. 415-435 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider the problem of frictional contact between an elastic body and an obstacle. The elastic constitutive law is assumed to be nonlinear. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We present two alternative yet equivalent weak formulations of the problem, and establish existence and uniqueness results for both formulations using arguments of elliptic variational inequalities and fixed point theory. Moreover, we show the continuous dependence of the solution on the contact conditions. We also study the finite element approximations of the problem and derive error estimates. Finally, we introduce an iterative method to solve the resulting finite element system.

Détail

Publié le : 1999-01-01

Zbl 1050.74639

EUDML-ID : urn:eudml:doc:219249

Bibliographie

[000] [1] K. T. Andrews, A. Klarbring, M. Shillor and S. Wright, A dynamic thermoviscoelastic body in frictional contact with a rigid obstacle, Eur. J. Appl. Math., to appear. | Zbl 0903.73065

[001] [2] C P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. | Zbl 0383.65058

[002] [3] M. Cocu, Existence of solutions of Signorini problems with friction, Int. J. Engrg. Sci. 22 (1984), 567-581. | Zbl 0554.73096

[003] [4] M. Cocu, Unilateral contact problems with friction for an elastoviscoplastic material with internal state variable, in: Proc. Contact Mechanics Int. Symp., A. Curnier (ed.), PPUR, 1992, 207-216.

[004] [5] D G. Duvaut, Loi de frottement non locale, J. Méc. Théor. Appl., Special issue 1982, 73-78.

[005] [6] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. | Zbl 0331.35002

[006] [7] H W. Han, On the numerical approximation of a frictional contact problem with normal compliance, Numer. Funct. Anal. Optim. 17 (1996), 307-321. | Zbl 0856.73066

[007] [8] I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford Univ. Press, Oxford, 1993. | Zbl 0787.73005

[008] [9] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988. | Zbl 0685.73002

[009] [10] A. Klarbring, A. Mikelič and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engrg. Sci. 26 (1988), 811-832. | Zbl 0662.73079

[010] [11] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal. 11 (1987), 407-428. | Zbl 0672.73079

[011] [12] J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elastoplastic Bodies$:$ An Introduction, Elsevier, Amsterdam, 1981.

[012] [13] P P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser, Basel, 1985. | Zbl 0579.73014

[013] [14] M. Rochdi, M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with normal compliance and friction, J. Elasticity 51 (1998), 105-126. | Zbl 0921.73231

[014] [15] M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with friction, Int. J. Engrg. Sci., to appear. | Zbl 1210.74132

[015] [16] S N. Strömberg, Continuum Thermodynamics of Contact$,$ Friction and Wear, Ph.D. Thesis, Linköping University, 1995.

[016] [17] N. Strömberg, L. Johansson and A. Klarbring, Derivation and analysis of a generalized standard model for contact friction and wear, Int. J. Solids Structures 33 (1996), 1817-1836. | Zbl 0926.74012

[017] [18] Z E. Zeidler, Nonlinear Functional Analysis and its Applications. IV$:$ Applications to Mathematical Physics, Springer, New York, 1988. | Zbl 0648.47036