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Keywords:
viscoelastic; normal compliance; adhesion; frictional; variational inequality; weak solution
viscoelastic; normal compliance; adhesion; frictional; variational inequality; weak solution
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.
References:
[1] Andersson L.-E.: Existence result for quasistatic contact problem with Coulomb friction. Appl. Math. Optim. 42 (2000), 169–202. DOI 10.1007/s002450010009 | MR 1784173
[2] Bonetti E., Bonfanti G., Rossi R.: Global existence for a contact problem with adhesion. Math. Methods Appl. Sci. 31 (2008), 1029–1064. DOI 10.1002/mma.957 | MR 2419088 | Zbl 1145.35301
[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. DOI 10.1512/iumj.2007.56.3079 | MR 2375702 | Zbl 1145.35027
[4] Bonetti E., Bonfanti G., Rossi R.: Thermal effects in adhesive contact: Modelling and analysis. Nonlinearity 22 (2009), 2697–2731. DOI 10.1088/0951-7715/22/11/007 | MR 2550692 | Zbl 1185.35122
[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
[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.: Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. J. Comput. Appl. Math. 159 (2003), 431–465. DOI 10.1016/S0377-0427(03)00547-8 | MR 2005970 | Zbl 1075.74061
[8] Chau O., Shillor M., Sofonea M.: Dynamic frictionless contact with adhesion. J. Appl. Math. Phys. 55 (2004), 32–47. DOI 10.1007/s00033-003-1089-9 | MR 2033859 | Zbl 1064.74132
[9] Cocou M., Rocca R.: Existence results for unilateral quasistatic contact problems with friction and adhesion. Math. Model. Num. Anal. 34 (2000), 981–1001. DOI 10.1051/m2an:2000112 | MR 1837764
[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. DOI 10.1007/s00033-009-0027-x | MR 2673333
[11] Duvaut G., Lions J.-L.: Les inéquations en mécanique et en physique. Dunod, Paris, 1972. MR 0464857 | Zbl 0298.73001
[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
[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. DOI 10.1016/S0895-7177(03)90043-4 | MR 1996040
[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. MR 0695554
[16] Frémond M.: Non-smooth Thermomechanics. Springer, Berlin, 2002. MR 1885252
[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. DOI 10.1016/j.crma.2003.10.010 | MR 2030112 | Zbl 1035.35090
[18] Jarušek J., Sofonea M.: On the solvability of dynamic elastic-visco-plastic contact problems. Z. Angew. Math. Mech. 88 (2008), 3–22. DOI 10.1002/zamm.200710360 | MR 2376989
[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
[20] Kočvara M., Mielke A., Roubíček T.: A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006), 423–447. DOI 10.1177/1081286505046482 | MR 2245202
[21] Nassar S.A., Andrews T., Kruk S., Shillor M.: Modelling and simulations of a bonded rod. Math. Comput. Modelling 42 (2005), 553–572. DOI 10.1016/j.mcm.2004.07.018 | MR 2173474 | Zbl 1121.74428
[22] Point N.: Unilateral contact with adherence. Math. Methods Appl. Sci. 10 (1988), 367–381. DOI 10.1002/mma.1670100403 | MR 0958479 | Zbl 0656.73052
[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. DOI 10.1016/S0045-7825(98)00389-2 | MR 1710458
[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). MR 2793554
[27] Roubíček T., L. Scardia L., C. Zanini C.: Quasistatic delamination problem. Cont. Mech. Thermodynam. 21 (2009), 223–235. DOI 10.1007/s00161-009-0106-4 | MR 2529453
[28] Shillor M., Sofonea M., Telega J.J.: Models and Variational Analysis of Quasistatic Contact. Lecture Notes in Physics, 655, Springer, Berlin, 2004. DOI 10.1007/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. MR 2183435
[30] Sofonea M., Hoarau-Mantel T.V.: Elastic frictionless contact problems with adhesion. Adv. Math. Sci. Appl. 15 (2005), 49–68. MR 2148278 | Zbl 1085.74036
[31] Sofonea M., Matei A.: Variational inequalities with applications. Advances in Mathematics and Mechanics, 18, Springer, New York, 2009. MR 2488869 | Zbl 1195.49002
[32] Touzaline A.: Frictionless contact problem with adhesion for nonlinear elastic materials. Electron. J. Differential Equations 2007, no. 174, 13 pp. MR 2366067 | Zbl 1133.35051
[33] Touzaline A.: Frictionless contact problem with adhesion and finite penetration for elastic materials. Ann. Pol. Math. 98 (2010), no. 1, 23–38. DOI 10.4064/ap98-1-2 | MR 2607484
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