| Title:
|
Study of a viscoelastic frictional contact problem with adhesion (English) |
| Author:
|
Touzaline, Arezki |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
257-272 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
viscoelastic |
| Keyword:
|
normal compliance |
| Keyword:
|
adhesion |
| Keyword:
|
frictional |
| Keyword:
|
variational inequality |
| Keyword:
|
weak solution |
| MSC:
|
47J20 |
| MSC:
|
49J40 |
| MSC:
|
74M10 |
| MSC:
|
74M15 |
| idZBL:
|
Zbl 1240.74011 |
| idMR:
|
MR2849048 |
| . |
| Date available:
|
2011-05-17T08:39:02Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141499 |
| . |
| Reference:
|
[1] Andersson L.-E.: Existence result for quasistatic contact problem with Coulomb friction.Appl. Math. Optim. 42 (2000), 169–202. MR 1784173, 10.1007/s002450010009 |
| Reference:
|
[2] Bonetti E., Bonfanti G., Rossi R.: Global existence for a contact problem with adhesion.Math. Methods Appl. Sci. 31 (2008), 1029–1064. Zbl 1145.35301, MR 2419088, 10.1002/mma.957 |
| Reference:
|
[3] Bonetti E., Bonfanti G., Rossi R.: Well-posedness and long-time behaviour for a model of contact with adhesion.Indiana Univ. Math. J. 56 (2007), 2787-2820. Zbl 1145.35027, MR 2375702, 10.1512/iumj.2007.56.3079 |
| Reference:
|
[4] Bonetti E., Bonfanti G., Rossi R.: Thermal effects in adhesive contact: Modelling and analysis.Nonlinearity 22 (2009), 2697–2731. Zbl 1185.35122, MR 2550692, 10.1088/0951-7715/22/11/007 |
| Reference:
|
[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). MR 2746376 |
| Reference:
|
[6] Cangémi L.: Frottement et adhérence: modèle, traitement numérique application à l'interface fibre/matrice.Ph.D. Thesis, Univ. Méditerranée, Aix Marseille I, 1997. |
| Reference:
|
[7] Chau O., Fernandez J.R., Shillor M., Sofonea M.: Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion.J. Comput. Appl. Math. 159 (2003), 431–465. Zbl 1075.74061, MR 2005970, 10.1016/S0377-0427(03)00547-8 |
| Reference:
|
[8] Chau O., Shillor M., Sofonea M.: Dynamic frictionless contact with adhesion.J. Appl. Math. Phys. 55 (2004), 32–47. Zbl 1064.74132, MR 2033859, 10.1007/s00033-003-1089-9 |
| Reference:
|
[9] Cocou M., Rocca R.: Existence results for unilateral quasistatic contact problems with friction and adhesion.Math. Model. Num. Anal. 34 (2000), 981–1001. MR 1837764, 10.1051/m2an:2000112 |
| Reference:
|
[10] Cocou M., Schryve M., Raous M.: A dynamic unilateral contact problem with adhesion and friction in viscoelasticity.Z. Angew. Math. Phys. 61 (2010), 721–743. MR 2673333, 10.1007/s00033-009-0027-x |
| Reference:
|
[11] Duvaut G., Lions J.-L.: Les inéquations en mécanique et en physique.Dunod, Paris, 1972. Zbl 0298.73001, MR 0464857 |
| Reference:
|
[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. MR 2128865 |
| Reference:
|
[13] Fernandez J.R., Shillor M., Sofonea M.: Analysis and numerical simulations of a dynamic contact problem with adhesion.Math. Comput. Modelling 37 (2003), 1317–1333. MR 1996040, 10.1016/S0895-7177(03)90043-4 |
| Reference:
|
[14] Frémond M.: Adhérence des solides.J. Méc. Théor. Appl. 6 (1987), 383–407. |
| Reference:
|
[15] Frémond M.: Equilibre des structures qui adhèrent à leur support.C.R. Acad. Sci. Paris Sér. II 295 (1982), 913–916. MR 0695554 |
| Reference:
|
[16] Frémond M.: Non-smooth Thermomechanics.Springer, Berlin, 2002. MR 1885252 |
| Reference:
|
[17] Hild P.: An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction.C.R. Math. Acad. Sci. Paris 337 (2003), 685–688. Zbl 1035.35090, MR 2030112, 10.1016/j.crma.2003.10.010 |
| Reference:
|
[18] Jarušek J., Sofonea M.: On the solvability of dynamic elastic-visco-plastic contact problems.Z. Angew. Math. Mech. 88 (2008), 3–22. MR 2376989, 10.1002/zamm.200710360 |
| Reference:
|
[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. MR 2665220 |
| Reference:
|
[20] Kočvara M., Mielke A., Roubíček T.: A rate-independent approach to the delamination problem.Math. Mech. Solids 11 (2006), 423–447. MR 2245202, 10.1177/1081286505046482 |
| Reference:
|
[21] Nassar S.A., Andrews T., Kruk S., Shillor M.: Modelling and simulations of a bonded rod.Math. Comput. Modelling 42 (2005), 553–572. Zbl 1121.74428, MR 2173474, 10.1016/j.mcm.2004.07.018 |
| Reference:
|
[22] Point N.: Unilateral contact with adherence.Math. Methods Appl. Sci. 10 (1988), 367–381. Zbl 0656.73052, MR 0958479, 10.1002/mma.1670100403 |
| Reference:
|
[23] Raous M., Cangémi L., Cocu M.: A consistent model coupling adhesion, friction, and unilateral contact.Comput. Methods Appl. Mech. Engrg. 177 (1999), 383–399. MR 1710458, 10.1016/S0045-7825(98)00389-2 |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[26] Rossi R., Roubíček T.: Thermodynamics and analysis of rate-independent adhesive contact at small strains.preprint arXiv:1004.3764 (2010). MR 2793554 |
| Reference:
|
[27] Roubíček T., L. Scardia L., C. Zanini C.: Quasistatic delamination problem.Cont. Mech. Thermodynam. 21 (2009), 223–235. MR 2529453, 10.1007/s00161-009-0106-4 |
| Reference:
|
[28] Shillor M., Sofonea M., Telega J.J.: Models and Variational Analysis of Quasistatic Contact.Lecture Notes in Physics, 655, Springer, Berlin, 2004. 10.1007/b99799 |
| Reference:
|
[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. MR 2183435 |
| Reference:
|
[30] Sofonea M., Hoarau-Mantel T.V.: Elastic frictionless contact problems with adhesion.Adv. Math. Sci. Appl. 15 (2005), 49–68. Zbl 1085.74036, MR 2148278 |
| Reference:
|
[31] Sofonea M., Matei A.: Variational inequalities with applications.Advances in Mathematics and Mechanics, 18, Springer, New York, 2009. Zbl 1195.49002, MR 2488869 |
| Reference:
|
[32] Touzaline A.: Frictionless contact problem with adhesion for nonlinear elastic materials.Electron. J. Differential Equations 2007, no. 174, 13 pp. Zbl 1133.35051, MR 2366067 |
| Reference:
|
[33] Touzaline A.: Frictionless contact problem with adhesion and finite penetration for elastic materials.Ann. Pol. Math. 98 (2010), no. 1, 23–38. MR 2607484, 10.4064/ap98-1-2 |
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