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

Weimin Han; Mircea Sofonea

Applicationes Mathematicae (1999)

  • Volume: 26, Issue: 4, page 415-435
  • ISSN: 1233-7234

Abstract

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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.

How to cite

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Han, Weimin, and Sofonea, Mircea. "Analysis and numerical approximation of an elastic frictional contact problem with normal compliance." Applicationes Mathematicae 26.4 (1999): 415-435. <http://eudml.org/doc/219249>.

@article{Han1999,
abstract = {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.},
author = {Han, Weimin, Sofonea, Mircea},
journal = {Applicationes Mathematicae},
keywords = {normal compliance; finite element approximation; Coulomb's law; error estimates; frictional contact; variational inequality; fixed point},
language = {eng},
number = {4},
pages = {415-435},
title = {Analysis and numerical approximation of an elastic frictional contact problem with normal compliance},
url = {http://eudml.org/doc/219249},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Han, Weimin
AU - Sofonea, Mircea
TI - Analysis and numerical approximation of an elastic frictional contact problem with normal compliance
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 4
SP - 415
EP - 435
AB - 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.
LA - eng
KW - normal compliance; finite element approximation; Coulomb's law; error estimates; frictional contact; variational inequality; fixed point
UR - http://eudml.org/doc/219249
ER -

References

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  1. [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. Zbl0903.73065
  2. [2] C P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. Zbl0383.65058
  3. [3] M. Cocu, Existence of solutions of Signorini problems with friction, Int. J. Engrg. Sci. 22 (1984), 567-581. Zbl0554.73096
  4. [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. 
  5. [5] D G. Duvaut, Loi de frottement non locale, J. Méc. Théor. Appl., Special issue 1982, 73-78. 
  6. [6] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. Zbl0331.35002
  7. [7] H W. Han, On the numerical approximation of a frictional contact problem with normal compliance, Numer. Funct. Anal. Optim. 17 (1996), 307-321. Zbl0856.73066
  8. [8] I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford Univ. Press, Oxford, 1993. Zbl0787.73005
  9. [9] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988. Zbl0685.73002
  10. [10] A. Klarbring, A. Mikelič and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engrg. Sci. 26 (1988), 811-832. Zbl0662.73079
  11. [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. Zbl0672.73079
  12. [12] J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elastoplastic Bodies : An Introduction, Elsevier, Amsterdam, 1981. 
  13. [13] P P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser, Basel, 1985. Zbl0579.73014
  14. [14] M. Rochdi, M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with normal compliance and friction, J. Elasticity 51 (1998), 105-126. Zbl0921.73231
  15. [15] M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with friction, Int. J. Engrg. Sci., to appear. Zbl1210.74132
  16. [16] S N. Strömberg, Continuum Thermodynamics of Contact , Friction and Wear, Ph.D. Thesis, Linköping University, 1995. 
  17. [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. Zbl0926.74012
  18. [18] Z E. Zeidler, Nonlinear Functional Analysis and its Applications. IV : Applications to Mathematical Physics, Springer, New York, 1988. Zbl0648.47036

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