| Title:
|
A quasistatic bilateral contact problem with adhesion and friction for viscoelastic materials (English) |
| Author:
|
Touzaline, Arezki |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
85-97 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
viscoelastic materials |
| Keyword:
|
adhesion |
| Keyword:
|
Tresca's friction |
| Keyword:
|
fixed point |
| Keyword:
|
weak solution |
| MSC:
|
47J20 |
| MSC:
|
49J40 |
| MSC:
|
74M10 |
| MSC:
|
74M15 |
| idZBL:
|
Zbl 1224.74089 |
| idMR:
|
MR2666082 |
| . |
| Date available:
|
2010-05-21T12:35:36Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140088 |
| . |
| Reference:
|
[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. Zbl 1002.74074, MR 1846749 |
| Reference:
|
[2] Brezis H.: Equations et inéquations non linéaires dans les espaces vectoriels en dualité.Annales Inst. Fourier 18 (1968), 115–175. Zbl 0169.18602, MR 0270222, 10.5802/aif.280 |
| Reference:
|
[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. |
| Reference:
|
[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. Zbl 1075.74061, MR 2005970, 10.1016/S0377-0427(03)00547-8 |
| Reference:
|
[5] Chau O., Shillor M., Sofonea M.: Dynamic frictionless contact with adhesion.Z. Angew. Math. Phys. 55 (2004), 32–47. Zbl 1064.74132, MR 2033859, 10.1007/s00033-003-1089-9 |
| Reference:
|
[6] Cocu M., Rocca R.: Existence results for unilateral quasistatic contact problems with friction and adhesion.Math. Model. Numer. Anal. 34 (2000), 981–1001. Zbl 0984.74054, MR 1837764, 10.1051/m2an:2000112 |
| Reference:
|
[7] Duvaut G., Lions J.-L.: Les inéquations en mécanique et en physique.Dunod, Paris, 1972. Zbl 0298.73001, MR 0464857 |
| Reference:
|
[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. MR 1996040, 10.1016/S0895-7177(03)90043-4 |
| Reference:
|
[9] Frémond M.: Adhérence des solides.J. Méc. Théor. Appl. 6 (1987), 383–407. |
| Reference:
|
[10] Frémond M.: Equilibre des structures qui adhèrent à leur support.C.R. Acad. Sci. Paris Sér. II 295, (1982), 913–916. MR 0695554 |
| Reference:
|
[11] Frémond M.: Non-smooth Thermomechanics.Springer, Berlin, 2002. MR 1885252 |
| Reference:
|
[12] 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:
|
[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. MR 1710458, 10.1016/S0045-7825(98)00389-2 |
| Reference:
|
[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. |
| Reference:
|
[15] 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:
|
[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 |
| Reference:
|
[17] Sofonea M., Hoarau-Mantel T.V.: Elastic frictionless contact problems with adhesion.Adv. Math. Sci. Appl. 15 (2005), no. 1, 49–68. Zbl 1085.74036, MR 2148278 |
| Reference:
|
[18] Sofonea M., Arhab R., Tarraf R.: Analysis of electroelastic frictionless contact problems with adhesion.J. Appl. Math. 2006, ID 64217, pp.1–25. Zbl 1143.74042, MR 2251808 |
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