We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a deformable obstacle, the so-called foundation. The material’s behaviour is modelled with a viscoelastic constitutive law with long memory. The contact is frictionless and is defined using a multivalued normal compliance condition. We present a regularization method in the study of a class of variational inequalities involving history-dependent operators. Finally, we apply the abstract results to analyse the contact problem.
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
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F. Pătrulescu, A regularization method for a viscoelastic contact problem, Math. Mech. Solids, vol. 23, no. 2 (2018), pp. 181-194
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 C. Baiocchi, A. Capelo, Variational and quasivariational inequalities: applications to free-boundary problems, John Wiley, Chichester (1984).
 V. Barbu, Optimal control of variational inequalities, Pitman, Boston (1984).
 H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualite, Ann. Inst.Fourier, 18 (1968), 115-175.
 H. Brezis, Problemes unilateraux, J. Math. Pures Appl., 51 (1972), 1-168.
 G. Duvaut, J.L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin (1976).
 C. Eck, J. Jarusek, Existence results for the semicoercive static contact problem with Coulomb friction, Nonlinear Anal., 42 (2000), 961-976.
 R. Glowinski, J.L. Lions, R. Tremolieres, Numerical analysis of variational inequalities, North-Holland Publishing Company, Amsterdam (1981).
 R. Glowinski, Numerical methods for nonlinear variational problems, Springer-Verlag, New York (1984).
 W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, 30, American Mathematical Society–International Press, Sommerville, MA (2002).
 J. Haslinger, I. Hlavacek, J.Necas, Numerical methods for unilateral problems in solid mechanics, In Handbook of Numerical Analysis, vol. IV, P.G. Ciarlet and J.-L. Lions (eds.), North-Holland, Amsterdam (1996).
 I. Hlavacek, J. Haslinger, J. Necas et al., Solution of variational inequalities in mechanics, Springer-Verlag, New-York, 1988.
 N. Kikuchi, J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988.
 N. Kikuchi, J.T. Oden, Theory of variational inequalities with applications to problems of flow through porous media, Int. J. Engng. Sci., 18 (1980), 1173-1284.
 D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, SIAM, Philadelphia, 2000.
 J.T. Oden, J.A.C. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527-634.
 M. Shillor, M. Sofonea, J.J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, 655, Springer, Berlin (2004).
 M. Sofonea, A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge (2012).
 M. Sofonea, A. Matei. History-dependent quasivariational inequalities arising in Contact Mechanics, Eur. J. Appl. Math., 22 (2011), 471-491.
 P. Wriggers, Computational Contact Mechanics, Wiley, Chichester (2002).
 M. Sofonea, W. Han, M. Barboteu, Analysis of a viscoelastic contact problem with multivalued normal compliance and unilateral constraint, Comput. Methods Appl. Mech. Engrg., 26 (2013), 12-22.
 M. Sofonea, F. Patrulescu, Penalization of history-dependent variational inequalities, Eur. J. Appl. Math., 25 (2014), 155-176.
 A. Klarbring, A. Mikelic, M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811-832.
 A. Klarbring, A. Mikelic, M. Shillor, On friction problems with normal compliance, Nonlinear Analysis, 13 (1989), 935-955.
 P.D. Panagiotopoulos, Inequality problems in mechanics and applications, Birkhauser, Boston (1985).
 F. Patrulescu, A. Ramadan, Convergence results for contact problems with memory term, Math. Rep., 17 (67) (2015), 24-41.