# American Institute of Mathematical Sciences

December  2011, 31(4): 1039-1051. doi: 10.3934/dcds.2011.31.1039

## On some frictional contact problems with velocity condition for elastic and visco-elastic materials

 1 University of La Réunion, PIMENT EA4518, 97715 Saint-Denis Messag cedex 9 La Réunion, France, France, France

Received  October 2009 Revised  May 2010 Published  September 2011

We study the evolution of a class of quasistatic problems, which describe frictional contact between a body and a foundation. The constitutive law of the materials is elastic, or visco-elastic: with short or long memory, and the contact is modelled by a general subdifferential condition on the velocity. We derive weak formulations for the models and establish existence and uniqueness results. The proofs are based on evolution variational inequalities, in the framework of monotone operators and $fi$xed point methods. We show the approach of the viscoelastic solutions to the corresponding elastic solutions, when the viscosity tends to zero. Finally we also study the approach to short memory visco-elasticity when the long memory relaxation coefficients vanish.
Citation: Khalid Addi, Oanh Chau, Daniel Goeleven. On some frictional contact problems with velocity condition for elastic and visco-elastic materials. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1039-1051. doi: 10.3934/dcds.2011.31.1039
##### References:
 [1] L.-E. Andersson, A global existence result for a quasistatic contact problem with friction, Advances in Mathematical Sciences ans Applications, 5 (1995), 249-286. [2] B. Awbi, El. H. Essoufi and M. Sofonea, A viscoelastic contact problem with normal damped response and friction, Annales Polonici Mathematici, 75 (2000), 233-246. [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leiden, 1976. [4] N. Bourbaki, "Éléments de Mathématiques," Première Partie, Livre IV, Fonctions d'une variable réelle, Hermann, 1961. [5] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280. [6] H. Brézis, Problèmes unilatéraux, J. Math. Pures et Appli, 51 (1972), 1-168. [7] F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, in "Nonlinear Functional Analysis" (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), 1-308, Amer. Math. Soc., Providence, RI, 1976. [8] O. Chau, R. Oujja and M. Rochdi, A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity, Discrete and Continuous Dynamical Systems, Series S, Volume 1 (2008), 61-70. [9] O. Chau, D. Motreanu and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials, Applications of Mathematics, 47 (2002), 341-360. doi: 10.1023/A:1021753722771. [10] O. Chau and M. Rochdi, On a dynamic bilateral contact problem with friction for viscoelastic materials, Int. J. of Appli. Math. and Mech, 2 (2006), 41-52. [11] P. G. Ciarlet, "Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988. [12] M. Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact, Int. J. Engn. Sci, 34 (1996), 783-798. doi: 10.1016/0020-7225(95)00121-2. [13] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976. [14] C. Eck and J. Jarušek, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Mathematical models and Methods in Applied Sciences, 9 (1999), 11-34. doi: 10.1142/S0218202599000038. [15] H. Fraysse and J. M. Arnaudiès, "Cours de Mathématiques," Dunod, 1999. [16] D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics," Nonconvex Optimization and its Applications, 69, Kluwer Academic Publishers, Boston, MA, 2003. [17] I. Hlaváček, J. Haslinger, J. Necăs and J. Lovíšek, "Solution of Variational Inequalities in Mechanics," Applied Mathematical Sciences, 66, Springer-Verlag, New York, 1988. [18] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999. [19] J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Mathematical Journal, 46 (1996), 475-487. [20] N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, SIAM, Philadelphia, PA, 1988. [21] J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302. [22] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlin. Anal., 11 (1987), 407-428. doi: 10.1016/0362-546X(87)90055-1. [23] J. Nečas and I. Hlavaček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction," Elsevier, Amsterdam, 1981. [24] P. D. Panagiotopoulos, "Inequality Problems in Meechanical and Applications. Convex and Nonconvex Energy Functions," Birkhäuser Boston, Inc., Boston, MA, 1985. [25] P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering," Springer-Verlag, Berlin, 1993. [26] M. Raous, M. Jean and J. J. Moreau, eds., "Contact Mechanics," Plenum Press, New York, 1995. [27] M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity, 51 (1998), 105-126. doi: 10.1023/A:1007413119583. [28] E. Zeidler, "Nonlinear Functional Analysis and its Applications," Springer-Verlag, 1997.

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##### References:
 [1] L.-E. Andersson, A global existence result for a quasistatic contact problem with friction, Advances in Mathematical Sciences ans Applications, 5 (1995), 249-286. [2] B. Awbi, El. H. Essoufi and M. Sofonea, A viscoelastic contact problem with normal damped response and friction, Annales Polonici Mathematici, 75 (2000), 233-246. [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leiden, 1976. [4] N. Bourbaki, "Éléments de Mathématiques," Première Partie, Livre IV, Fonctions d'une variable réelle, Hermann, 1961. [5] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280. [6] H. Brézis, Problèmes unilatéraux, J. Math. Pures et Appli, 51 (1972), 1-168. [7] F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, in "Nonlinear Functional Analysis" (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), 1-308, Amer. Math. Soc., Providence, RI, 1976. [8] O. Chau, R. Oujja and M. Rochdi, A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity, Discrete and Continuous Dynamical Systems, Series S, Volume 1 (2008), 61-70. [9] O. Chau, D. Motreanu and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials, Applications of Mathematics, 47 (2002), 341-360. doi: 10.1023/A:1021753722771. [10] O. Chau and M. Rochdi, On a dynamic bilateral contact problem with friction for viscoelastic materials, Int. J. of Appli. Math. and Mech, 2 (2006), 41-52. [11] P. G. Ciarlet, "Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988. [12] M. Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact, Int. J. Engn. Sci, 34 (1996), 783-798. doi: 10.1016/0020-7225(95)00121-2. [13] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976. [14] C. Eck and J. Jarušek, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Mathematical models and Methods in Applied Sciences, 9 (1999), 11-34. doi: 10.1142/S0218202599000038. [15] H. Fraysse and J. M. Arnaudiès, "Cours de Mathématiques," Dunod, 1999. [16] D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics," Nonconvex Optimization and its Applications, 69, Kluwer Academic Publishers, Boston, MA, 2003. [17] I. Hlaváček, J. Haslinger, J. Necăs and J. Lovíšek, "Solution of Variational Inequalities in Mechanics," Applied Mathematical Sciences, 66, Springer-Verlag, New York, 1988. [18] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999. [19] J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Mathematical Journal, 46 (1996), 475-487. [20] N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, SIAM, Philadelphia, PA, 1988. [21] J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302. [22] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlin. Anal., 11 (1987), 407-428. doi: 10.1016/0362-546X(87)90055-1. [23] J. Nečas and I. Hlavaček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction," Elsevier, Amsterdam, 1981. [24] P. D. Panagiotopoulos, "Inequality Problems in Meechanical and Applications. Convex and Nonconvex Energy Functions," Birkhäuser Boston, Inc., Boston, MA, 1985. [25] P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering," Springer-Verlag, Berlin, 1993. [26] M. Raous, M. Jean and J. J. Moreau, eds., "Contact Mechanics," Plenum Press, New York, 1995. [27] M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity, 51 (1998), 105-126. doi: 10.1023/A:1007413119583. [28] E. Zeidler, "Nonlinear Functional Analysis and its Applications," Springer-Verlag, 1997.
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