# American Institute of Mathematical Sciences

December  2011, 15(3): 687-705. doi: 10.3934/dcdsb.2011.15.687

## Analysis of a frictional contact problem for viscoelastic materials with long memory

 1 Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow, Poland, Poland 2 Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan

Received  July 2009 Revised  November 2010 Published  February 2011

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is time-dependent, the material behavior is described with a viscoelastic constitutive law with long memory and the contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of a hemivariational inequality with Volterra integral term for the displacement field. Then we prove existence and uniqueness results in the study of abstract inclusions as well as in the study of abstract hemivariational inequalities with Volterra integral term. The proofs are based on arguments on pseudomonotone operators, compactness and fixed point. We use the abstract results to prove the unique solvability of the frictional contact problem. Finally, we present examples of contact and frictional boundary conditions for which our results work.
Citation: Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
##### References:
 [1] F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, Interscience, New York, 1983. [2] Z. Denkowski, S. Migórski and N. S. Papageorgiou, "An Introduction to Nonlinear Analysis: Theory," Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [3] Z. Denkowski, S. Migórski and N. S. Papageorgiou, "An Introduction to Nonlinear Analysis: Applications," Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [4] A. D. Drozdov, "Finite Elasticity and Viscoelasticity - A Course in the Nonlinear Mechanics of Solids," World Scientific, Singapore, 1996. [5] G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976. [6] C. Eck, J. Jarušek and M. Krbec, "Unilateral Contact Problems: Variational Methods and Existence Theorems," Pure and Applied Mathematics, 270, Chapman/CRC Press, New York, 2005. [7] W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," American Mathematical Society, International Press, 2002. [8] I. R. Ionescu, C. Dascalu and M. Campillo, Slip-weakening friction on a periodic system of faults: Spectral analysis, Z. Angew. Math. Phys. (ZAMP), 53 (2002), 980-995. doi: 10.1007/PL00012624. [9] I. R. Ionescu, Q.-L. Nguyen and S. Wolf, Slip displacement dependent friction in dynamic elasticity, Nonlinear Analysis, 53 (2003), 375-390. doi: 10.1016/S0362-546X(02)00302-4. [10] S. Migórski, Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity, Discrete Continuous Dynam. Syst. Ser. B, 6 (2006), 1339-1356. doi: 10.3934/dcdsb.2006.6.1339. [11] S. Migórski, A. Ochal and M. Sofonea, Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact, Mathematical Models and Methods in Applied Sciences, 18 (2008), 271-290. doi: 10.1142/S021820250800267X. [12] Z. Naniewicz and P. D. Panagiotopoulos, "Mathematical Theory of Hemivariational Inequalities and Applications," Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995. [13] P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering," Springer-Verlag, Berlin, 1993. [14] A. D. Rodríguez-Aros, M. Sofonea and J. M. Viaño, A class of evolutionary variational inequalities with Volterra-type integral term, Mathematical Models and Methods in Applied Sciences, 14 (2004), 555-577. [15] M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact," Lecture Notes Physics, 655, Springer, Berlin, Heidelberg, 2004. [16] M. Sofonea and A. Matei, "Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems," Advances in Mechanics and Mathematics, 18, Springer, New York, 2009. [17] M. Sofonea, A. D. Rodríguez-Aros and J. M. Viaño, A class of integro-differential variational inequalities with applications to viscoelastic contact, Mathematical and Computer Modelling, 41 (2005), 1355-1369. doi: 10.1016/j.mcm.2004.01.011. [18] E. Zeidler, "Nonlinear Functional Analysis and Applications II A/B," Springer, New York, 1990.

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##### References:
 [1] F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, Interscience, New York, 1983. [2] Z. Denkowski, S. Migórski and N. S. Papageorgiou, "An Introduction to Nonlinear Analysis: Theory," Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [3] Z. Denkowski, S. Migórski and N. S. Papageorgiou, "An Introduction to Nonlinear Analysis: Applications," Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [4] A. D. Drozdov, "Finite Elasticity and Viscoelasticity - A Course in the Nonlinear Mechanics of Solids," World Scientific, Singapore, 1996. [5] G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976. [6] C. Eck, J. Jarušek and M. Krbec, "Unilateral Contact Problems: Variational Methods and Existence Theorems," Pure and Applied Mathematics, 270, Chapman/CRC Press, New York, 2005. [7] W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," American Mathematical Society, International Press, 2002. [8] I. R. Ionescu, C. Dascalu and M. Campillo, Slip-weakening friction on a periodic system of faults: Spectral analysis, Z. Angew. Math. Phys. (ZAMP), 53 (2002), 980-995. doi: 10.1007/PL00012624. [9] I. R. Ionescu, Q.-L. Nguyen and S. Wolf, Slip displacement dependent friction in dynamic elasticity, Nonlinear Analysis, 53 (2003), 375-390. doi: 10.1016/S0362-546X(02)00302-4. [10] S. Migórski, Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity, Discrete Continuous Dynam. Syst. Ser. B, 6 (2006), 1339-1356. doi: 10.3934/dcdsb.2006.6.1339. [11] S. Migórski, A. Ochal and M. Sofonea, Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact, Mathematical Models and Methods in Applied Sciences, 18 (2008), 271-290. doi: 10.1142/S021820250800267X. [12] Z. Naniewicz and P. D. Panagiotopoulos, "Mathematical Theory of Hemivariational Inequalities and Applications," Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995. [13] P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering," Springer-Verlag, Berlin, 1993. [14] A. D. Rodríguez-Aros, M. Sofonea and J. M. Viaño, A class of evolutionary variational inequalities with Volterra-type integral term, Mathematical Models and Methods in Applied Sciences, 14 (2004), 555-577. [15] M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact," Lecture Notes Physics, 655, Springer, Berlin, Heidelberg, 2004. [16] M. Sofonea and A. Matei, "Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems," Advances in Mechanics and Mathematics, 18, Springer, New York, 2009. [17] M. Sofonea, A. D. Rodríguez-Aros and J. M. Viaño, A class of integro-differential variational inequalities with applications to viscoelastic contact, Mathematical and Computer Modelling, 41 (2005), 1355-1369. doi: 10.1016/j.mcm.2004.01.011. [18] E. Zeidler, "Nonlinear Functional Analysis and Applications II A/B," Springer, New York, 1990.
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