# A quasistatic bilateral contact problem with adhesion and friction for viscoelastic materials

• Volume: 51, Issue: 1, page 85-97
• ISSN: 0010-2628

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## Abstract

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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.

## How to cite

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Touzaline, Arezki. "A quasistatic bilateral contact problem with adhesion and friction for viscoelastic materials." Commentationes Mathematicae Universitatis Carolinae 51.1 (2010): 85-97. <http://eudml.org/doc/37746>.

@article{Touzaline2010,
abstract = {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.},
author = {Touzaline, Arezki},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {viscoelastic materials; adhesion; Tresca's friction; fixed point; weak solution; contact problem; viscoelastic material; Tresca's friction; adhesion; existence; unicity; Banach fixed point theorem},
language = {eng},
number = {1},
pages = {85-97},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A quasistatic bilateral contact problem with adhesion and friction for viscoelastic materials},
url = {http://eudml.org/doc/37746},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Touzaline, Arezki
TI - A quasistatic bilateral contact problem with adhesion and friction for viscoelastic materials
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 1
SP - 85
EP - 97
AB - 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.
LA - eng
KW - viscoelastic materials; adhesion; Tresca's friction; fixed point; weak solution; contact problem; viscoelastic material; Tresca's friction; adhesion; existence; unicity; Banach fixed point theorem
UR - http://eudml.org/doc/37746
ER -

## References

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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. Zbl1002.74074MR1846749
2. Brezis H., 10.5802/aif.280, Annales Inst. Fourier 18 (1968), 115–175. Zbl0169.18602MR0270222DOI10.5802/aif.280
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., 10.1016/S0377-0427(03)00547-8, J. Computational and Applied Mathematics 159 (2003), 431–465. Zbl1075.74061MR2005970DOI10.1016/S0377-0427(03)00547-8
5. Chau O., Shillor M., Sofonea M., 10.1007/s00033-003-1089-9, Z. Angew. Math. Phys. 55 (2004), 32–47. Zbl1064.74132MR2033859DOI10.1007/s00033-003-1089-9
6. Cocu M., Rocca R., 10.1051/m2an:2000112, Math. Model. Numer. Anal. 34 (2000), 981–1001. Zbl0984.74054MR1837764DOI10.1051/m2an:2000112
7. Duvaut G., Lions J.-L., Les inéquations en mécanique et en physique, Dunod, Paris, 1972. Zbl0298.73001MR0464857
8. 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
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. MR0695554
11. Frémond M., Non-smooth Thermomechanics, Springer, Berlin, 2002. MR1885252
12. 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
13. 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
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., 10.1007/b99799, Lecture Notes in Physics, 655, Springer, Berlin, 2004. DOI10.1007/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. MR2183435
17. Sofonea M., Hoarau-Mantel T.V., Elastic frictionless contact problems with adhesion, Adv. Math. Sci. Appl. 15 (2005), no. 1, 49–68. Zbl1085.74036MR2148278
18. Sofonea M., Arhab R., Tarraf R., Analysis of electroelastic frictionless contact problems with adhesion, J. Appl. Math. 2006, ID 64217, pp.1–25. Zbl1143.74042MR2251808

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