Arezki Touzaline. Quasistatic Bilateral Contact Problem With Adhesion and Nonlocal Friction for Viscoelastic Materials[J]. Applied Mathematics and Mechanics, 2010, 31(5): 591-601. doi: 10.3879/j.issn.1000-0887.2010.05.010
 Citation: Arezki Touzaline. Quasistatic Bilateral Contact Problem With Adhesion and Nonlocal Friction for Viscoelastic Materials[J]. Applied Mathematics and Mechanics, 2010, 31(5): 591-601.

# Quasistatic Bilateral Contact Problem With Adhesion and Nonlocal Friction for Viscoelastic Materials

##### doi: 10.3879/j.issn.1000-0887.2010.05.010
• Received Date: 1900-01-01
• Rev Recd Date: 2009-11-23
• Publish Date: 2010-05-15
• A mathematical model which describes a contact problem between a de form able body and a foundation was considered. The contact was bilateral and was modelled with non local friction law in which adhesion was taken in to account. The evolution of the bonding field was described by a firstorder differential equation and the material. s behavior was modelled with an on linear viscoe lastic constitutive law. A variational formulation of the mechanical problem was derived and the existence and uniqueness result of the weak so lution were proved if the coefficien to ffriction was sufficiently small. The proof is based on arguments of time-dependent variationa line qualities, differential equations and Banach fixed-point theorem.
•  [1] Duvaut G, Lions J-L. Les Inéquations en Mécanique et en Physique[M]. Paris: Dunod, 1972. [2] Sofonea M, Han W, Shillor M. Analysis and Approximations of Contact Problems With Adhesion or Damage[M]. Pure and Applied Mathematics.L76.Boca Raton, Florida: Chapman & Hall / CRC Press, 2006. [3] Awbi B, Chau O, Sofonea M. Variational analysis of a frictional contact problem for viscoelastic bodies [J]. Int Math J, 2002, 1(4): 333-348. [4] Chau O, Fernandez J R, Shillor M, et al. Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion[J]. Journal of Computational and Applied Mathematics, 2003, 159(2): 431-465. [5] Chau O, Shillor M, Sofonea M. Dynamic frictionless contact with adhesion[J]. J Appl Math Phys(ZAMP), 2004, 55(1): 32-47. [6] Fernandez J R, Shillor M, Sofonea M. Analysis and numerical simulations of a dynamic contact problem with adhesion[J]. Math Comput Modelling, 2003, 37(12): 1317-1333. [7] Sofonea M, Hoarau-Mantel T V. Elastic frictionless contact problems with adhesion[J]. Adv Math Sci Appl, 2005, 15(1): 49-68. [8] Cangémi L. Frottement et adhérence: modèle, traitement numérique et application  l’interface fibre/matrice[D]. PhD Thesis. Aix Marseille I: Univ Méditerranée, 1997. [9] Frémond M. Adhérence des solides[J]. J Mécanique Théorique et Appliquée, 1987, 6: 383-407. [10] Frémond M. Equilibre des structures qui adhèrent  leur support[J]. C R Acad Sci, Série Ⅱ, 1982, 295: 913-916. [11] Raous M, Cangémi L, Cocu M. A consistent model coupling adhesion, friction, and unilateral contact[J]. Comput Meth Appl Mech Engng, 1999, 177(3/4): 383-399. [12] Rojek J, Telega J J. Contact problems with friction, adhesion and wear in orthopeadic biomechanics—Ⅰ: general developements[J]. J Theor Appl Mech, 2001, 39: 655-677. [13] Shillor M, Sofonea M, Telega J J. Models and Variational Analysis of Quasistatic Contact[M]. Lecture Notes Physics. Vol 655.Berlin: Springer, 2004. [14] Sofonea M, Arhab R, Tarraf R. Analysis of electroelastic frictionless contact problems with adhesion[J]. Journal of Applied Mathematics, 2006: 1-25. ID 64217. [15] Nassar S A, Andrews T, Kruk S, et al. Modelling and simulations of a bonded rod[J]. Math Comput Modelling, 2005, 42(5/6): 553-572. [16] Brezis H. Equations et inéquations non linéaires dans les espaces vectoriels en dualité[J]. Annales Inst Fourier, 1968, 18(1): 115-175. doi: 10.5802/aif.280 [17] Cocou M, Rocca R. Existence results for unilateral quasistatic contact problems with friction and adhesion[J]. Math Model Num Anal, 2000, 34(5): 981-1001. [18] Frémond. Non Smooth Thermomechanics[M]. Berlin: Springer, 2002.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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