WEAK SOLUTIONS VIA LAGRANGE MULTIPLIERS FOR CONTACT MODELS WITH NORMAL COMPLIANCE

Abstract

We consider a 3D elastostatic frictional contact problem with normal compliance, which consists of a systems of partial di erential equations associated with a displacement boundary condition, a traction boundary condition and a frictional contact boundary condition. The frictional contact is modeled by means of a normal compliance condition and a version of Coulomb's law of dry friction. After we state the problem and the hypotheses, we deliver a variational formulation as a mixed variational problem with solution-dependent Lagrange multipliers set. Next, we prove the existence and the boundedness of the weak solutions. 1.

Keywords

• normal compliance contact condition,Coulomb's law of dry friction,Lagrange multiplier

References

1. [1] R. A. Adams. Sobolev spaces, Academic Press, 1975.
2. [2] L.-E. Andersson, A quasistatic frictional problem with normal compliance, Nonlinear Anal- ysis TMA 16 (1991), 347-370.
3. [3] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28 SIAM, Philadelphia, PA, 1999.
4. [4] J. Haslinger, I. Hlavacek and J. Necas, Numerical Methods for Unilateral Problems in Solid Mechanics, in "Handbook of Numerical Analysis", J.-L. L. P. Ciarlet, ed., IV, North-Holland, Amsterdam, 1996, 313{485.
5. [5] P. Hild, Y. Renard, A stabilized Lagrange multiplier method for the nite element approximation of contact problems in elastostatics. Numer. Math. 115 101{129, 2010.
6. [6] S. Hueber, A. Matei, B. Wohlmuth, A contact problem for electro-elastic materials, Journal of Applied Mathematics and Mechanics (ZAMM), Z. Angew. Math. Mech., DOI: 10.1002/zamm.201200235, 93 (10-11) (2013), 789-800. Special Issue: Mathematical Modeling: Contact Mechanics, Phase Transitions, Multiscale Problems.
7. [7] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequal- ities and Finite Element Methods, SIAM, Philadelphia, 1988.
8. [8] A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci. 26 (1988), 811{832.

9. [9] A. Klarbring, A. Mikelic and M. Shillor, A global existence result for the quasistatic frictional contact problem with normal compliance, in G. del Piero and F. Maceri, eds., Unilateral Problems in Structural Analysis Vol. 4, Birkhauser, Boston, 1991, 85-111.
10. [10] J.-L. Lions and E. Magenes, Problemes aux limites non homogenes, Dunod, Paris, 1968.
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 Analysis TMA, 11 (1987), 407-428.
12. [12] A. Matei, On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers, Proceedings of The Royal Society of Edinburgh, Section: A Mathematics, 143(05) (2013), 1047-1059.
13. [13] A. Matei, Weak solvability via Lagrange multipliers for contact problems involving multicontact zones, Mathematics and Mechanics of Solids, DOI: 10.1177/1081286514541577.
14. [14] A. Matei, An existence result for a mixed variational problem arising from Contact Mechanics, Nonlinear Analysis Series B: Real World Application, 20 (2014), 74-81.
15. [15] A. Matei, An evolutionary mixed variational problem arising from frictional contact mechanics, Mathematics and Mechanics of Solids, DOI: 10.1177/1081286512462168, 19(3) (2014), 225 - 241.
16. [16] A. Matei, Weak Solutions via Lagrange Multipliers for a Slip-dependent Frictional Contact Model, IAENG International Journal of Applied Mathematics, 44 (3), 2014, 151-156 (special issue WCE 2014-ICAEM).
17. [17] A. Matei, A mixed variational formulation for a slip-dependent frictional contact model, Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2014, 2-4 July, 2014, London, U.K., pp 750-754 (ISBN: 978-988-19253-5-0, ISSN: 2078-0958).
18. [18] M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity 51 (1998), 105{126.
19. [19] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society, Lecture Note Series 398, 280 pages, Cambridge University Press, 2012.
20. [20] B. Wohlmuth, A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier, SIAM Journal on Numerical Analysis, 38(2000), 989-1012.
21. [21] B.Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, in \Lecture Notes in Computational Science and Engineering", 17, Springer, 2001.