Study of a viscoelastic frictional contact problem with adhesion

Arezki Touzaline

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 2, page 257-272
  • ISSN: 0010-2628

Abstract

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We consider a quasistatic frictional contact problem between a viscoelastic body with long memory and a deformable foundation. The contact is modelled with normal compliance in such a way that the penetration is limited and restricted to unilateral constraint. The adhesion between contact surfaces is taken into account and the evolution of the bonding field is described by a first order differential equation. We derive a variational formulation and prove the existence and uniqueness result of the weak solution under a certain condition on the coefficient of friction. The proof is based on time-dependent variational inequalities, differential equations and Banach fixed point theorem.

How to cite

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Touzaline, Arezki. "Study of a viscoelastic frictional contact problem with adhesion." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 257-272. <http://eudml.org/doc/247225>.

@article{Touzaline2011,
abstract = {We consider a quasistatic frictional contact problem between a viscoelastic body with long memory and a deformable foundation. The contact is modelled with normal compliance in such a way that the penetration is limited and restricted to unilateral constraint. The adhesion between contact surfaces is taken into account and the evolution of the bonding field is described by a first order differential equation. We derive a variational formulation and prove the existence and uniqueness result of the weak solution under a certain condition on the coefficient of friction. The proof is based on time-dependent variational inequalities, differential equations and Banach fixed point theorem.},
author = {Touzaline, Arezki},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {viscoelastic; normal compliance; adhesion; frictional; variational inequality; weak solution; viscoelasticity; normal compliance; adhesion; friction; variational inequality; weak solution},
language = {eng},
number = {2},
pages = {257-272},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Study of a viscoelastic frictional contact problem with adhesion},
url = {http://eudml.org/doc/247225},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Touzaline, Arezki
TI - Study of a viscoelastic frictional contact problem with adhesion
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 257
EP - 272
AB - We consider a quasistatic frictional contact problem between a viscoelastic body with long memory and a deformable foundation. The contact is modelled with normal compliance in such a way that the penetration is limited and restricted to unilateral constraint. The adhesion between contact surfaces is taken into account and the evolution of the bonding field is described by a first order differential equation. We derive a variational formulation and prove the existence and uniqueness result of the weak solution under a certain condition on the coefficient of friction. The proof is based on time-dependent variational inequalities, differential equations and Banach fixed point theorem.
LA - eng
KW - viscoelastic; normal compliance; adhesion; frictional; variational inequality; weak solution; viscoelasticity; normal compliance; adhesion; friction; variational inequality; weak solution
UR - http://eudml.org/doc/247225
ER -

References

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  1. Andersson L.-E., 10.1007/s002450010009, Appl. Math. Optim. 42 (2000), 169–202. MR1784173DOI10.1007/s002450010009
  2. Bonetti E., Bonfanti G., Rossi R., 10.1002/mma.957, Math. Methods Appl. Sci. 31 (2008), 1029–1064. Zbl1145.35301MR2419088DOI10.1002/mma.957
  3. Bonetti E., Bonfanti G., Rossi R., 10.1512/iumj.2007.56.3079, Indiana Univ. Math. J. 56 (2007), 2787-2820. Zbl1145.35027MR2375702DOI10.1512/iumj.2007.56.3079
  4. Bonetti E., Bonfanti G., Rossi R., 10.1088/0951-7715/22/11/007, Nonlinearity 22 (2009), 2697–2731. Zbl1185.35122MR2550692DOI10.1088/0951-7715/22/11/007
  5. Bonetti E., Bonfanti G., Rossi R., Long-time behaviour of a thermomechanical model for adhesive contact, (preprint arXiv:0909.2493), Discrete Contin. Dyn. Syst. Ser. S, in print (2010). MR2746376
  6. Cangémi L., Frottement et adhérence: modèle, traitement numérique application à l'interface fibre/matrice, Ph.D. Thesis, Univ. Méditerranée, Aix Marseille I, 1997. 
  7. Chau O., Fernandez J.R., Shillor M., Sofonea M., 10.1016/S0377-0427(03)00547-8, J. Comput. Appl. Math. 159 (2003), 431–465. Zbl1075.74061MR2005970DOI10.1016/S0377-0427(03)00547-8
  8. Chau O., Shillor M., Sofonea M., 10.1007/s00033-003-1089-9, J. Appl. Math. Phys. 55 (2004), 32–47. Zbl1064.74132MR2033859DOI10.1007/s00033-003-1089-9
  9. Cocou M., Rocca R., 10.1051/m2an:2000112, Math. Model. Num. Anal. 34 (2000), 981–1001. MR1837764DOI10.1051/m2an:2000112
  10. Cocou M., Schryve M., Raous M., 10.1007/s00033-009-0027-x, Z. Angew. Math. Phys. 61 (2010), 721–743. MR2673333DOI10.1007/s00033-009-0027-x
  11. Duvaut G., Lions J.-L., Les inéquations en mécanique et en physique, Dunod, Paris, 1972. Zbl0298.73001MR0464857
  12. Eck C., Jarušek J., Krbec M., Unilateral Contact Problems. Variational Methods and Existence Theorems, Pure Appl. Math., 270, Chapman & Hall / CRC, Boca Raton, 2005. MR2128865
  13. Fernandez J.R., Shillor M., Sofonea M., 10.1016/S0895-7177(03)90043-4, Math. Comput. Modelling 37 (2003), 1317–1333. MR1996040DOI10.1016/S0895-7177(03)90043-4
  14. Frémond M., Adhérence des solides, J. Méc. Théor. Appl. 6 (1987), 383–407. 
  15. Frémond M., Equilibre des structures qui adhèrent à leur support, C.R. Acad. Sci. Paris Sér. II 295 (1982), 913–916. MR0695554
  16. Frémond M., Non-smooth Thermomechanics, Springer, Berlin, 2002. MR1885252
  17. Hild P., 10.1016/j.crma.2003.10.010, C.R. Math. Acad. Sci. Paris 337 (2003), 685–688. Zbl1035.35090MR2030112DOI10.1016/j.crma.2003.10.010
  18. Jarušek J., Sofonea M., 10.1002/zamm.200710360, Z. Angew. Math. Mech. 88 (2008), 3–22. MR2376989DOI10.1002/zamm.200710360
  19. Jarušek J., Sofonea M., On the solvability of dynamic elastic-visco-plastic contact problems with adhesion, Ann. Acad. Rom. Sci. Ser. Math. Appl. 1 (2009), 191–214. MR2665220
  20. Kočvara M., Mielke A., Roubíček T., 10.1177/1081286505046482, Math. Mech. Solids 11 (2006), 423–447. MR2245202DOI10.1177/1081286505046482
  21. Nassar S.A., Andrews T., Kruk S., Shillor M., 10.1016/j.mcm.2004.07.018, Math. Comput. Modelling 42 (2005), 553–572. Zbl1121.74428MR2173474DOI10.1016/j.mcm.2004.07.018
  22. Point N., 10.1002/mma.1670100403, Math. Methods Appl. Sci. 10 (1988), 367–381. Zbl0656.73052MR0958479DOI10.1002/mma.1670100403
  23. Raous M., Cangémi L., Cocu M., 10.1016/S0045-7825(98)00389-2, Comput. Methods Appl. Mech. Engrg. 177 (1999), 383–399. MR1710458DOI10.1016/S0045-7825(98)00389-2
  24. Rocca R., Analyse et numérique de problèmes quasi-statiques de contact avec frottement local de Coulomb en élasticité, Thesis, Univ. Aix. Marseille 1, 2005. 
  25. Rojek J., Telega J.J., Contact problems with friction, adhesion and wear in orthopeadic biomechanics I: General developements, J. Theor. Appl. Mech. 39 (2001), 655–677. 
  26. Rossi R., Roubíček T., Thermodynamics and analysis of rate-independent adhesive contact at small strains, preprint arXiv:1004.3764 (2010). MR2793554
  27. Roubíček T., L. Scardia L., C. Zanini C., 10.1007/s00161-009-0106-4, Cont. Mech. Thermodynam. 21 (2009), 223–235. MR2529453DOI10.1007/s00161-009-0106-4
  28. Shillor M., Sofonea M., Telega J.J., 10.1007/b99799, Lecture Notes in Physics, 655, Springer, Berlin, 2004. DOI10.1007/b99799
  29. Sofonea M., Han H., Shillor M., Analysis and Approximations of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics, 276, Chapman & Hall / CRC Press, Boca Raton, Florida, 2006. MR2183435
  30. Sofonea M., Hoarau-Mantel T.V., Elastic frictionless contact problems with adhesion, Adv. Math. Sci. Appl. 15 (2005), 49–68. Zbl1085.74036MR2148278
  31. Sofonea M., Matei A., Variational inequalities with applications, Advances in Mathematics and Mechanics, 18, Springer, New York, 2009. Zbl1195.49002MR2488869
  32. Touzaline A., Frictionless contact problem with adhesion for nonlinear elastic materials, Electron. J. Differential Equations 2007, no. 174, 13 pp. Zbl1133.35051MR2366067
  33. Touzaline A., 10.4064/ap98-1-2, Ann. Pol. Math. 98 (2010), no. 1, 23–38. MR2607484DOI10.4064/ap98-1-2

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