Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
viscoelastic materials; adhesion; Tresca's friction; fixed point; weak solution
viscoelastic materials; adhesion; Tresca's friction; fixed point; weak solution
Summary:
We consider a mathematical model which describes a contact problem between a deformable body and a foundation. The contact is bilateral and is modelled with Tresca's friction law in which adhesion is taken into account. The evolution of the bonding field is described by a first order differential equation and the material's behavior is modelled with a nonlinear viscoelastic constitutive law. We derive a variational formulation of the mechanical problem and prove the existence and uniqueness result of the weak solution. The proof is based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.
References:
[1] Awbi B., Chau O., Sofonea A.: Variational analysis of a frictional contact problem for viscoelastic bodies. Int. Math. J. 1 (2002), no. 4, 333–348. MR 1846749 | Zbl 1002.74074
[2] Brezis H.: Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Annales Inst. Fourier 18 (1968), 115–175. DOI 10.5802/aif.280 | MR 0270222 | Zbl 0169.18602
[3] Cangémi L.: Frottement et adhérence: modèle, traitement numérique et application à l'interface fibre/matrice. Ph.D. Thesis, Univ. Méditerranée, Aix Marseille I, 1997.
[4] Chau O., Fernandez J.R., Shillor M., Sofonea M.: Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. J. Computational and Applied Mathematics 159 (2003), 431–465. DOI 10.1016/S0377-0427(03)00547-8 | MR 2005970 | Zbl 1075.74061
[5] Chau O., Shillor M., Sofonea M.: Dynamic frictionless contact with adhesion. Z. Angew. Math. Phys. 55 (2004), 32–47. DOI 10.1007/s00033-003-1089-9 | MR 2033859 | Zbl 1064.74132
[6] Cocu M., Rocca R.: Existence results for unilateral quasistatic contact problems with friction and adhesion. Math. Model. Numer. Anal. 34 (2000), 981–1001. DOI 10.1051/m2an:2000112 | MR 1837764 | Zbl 0984.74054
[7] Duvaut G., Lions J.-L.: Les inéquations en mécanique et en physique. Dunod, Paris, 1972. MR 0464857 | Zbl 0298.73001
[8] 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
[9] Frémond M.: Adhérence des solides. J. Méc. Théor. Appl. 6 (1987), 383–407.
[10] Frémond M.: Equilibre des structures qui adhèrent à leur support. C.R. Acad. Sci. Paris Sér. II 295, (1982), 913–916. MR 0695554
[11] Frémond M.: Non-smooth Thermomechanics. Springer, Berlin, 2002. MR 1885252
[12] 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
[13] 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
[14] 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.
[15] 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
[16] Sofonea M., Han W., 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
[17] Sofonea M., Hoarau-Mantel T.V.: Elastic frictionless contact problems with adhesion. Adv. Math. Sci. Appl. 15 (2005), no. 1, 49–68. MR 2148278 | Zbl 1085.74036
[18] Sofonea M., Arhab R., Tarraf R.: Analysis of electroelastic frictionless contact problems with adhesion. J. Appl. Math. 2006, ID 64217, pp.1–25. MR 2251808 | Zbl 1143.74042
Search
Partner of
