Abstract
In this paper, we present a synthesis of different results obtained recently in the papers (Chau and Awbi, Appl Anal 83:635–648, 2004; Addi et al. Discret Contin Dyn Syst 31:1039–1051, 2011; Adly et al. Numer Algebra Control Optim 2:89–101, 2012; Adly and chau, to appear in Mathematical Programming; Chau et al., Int J Appl Math Mech, 2012). It concerns the study of contact problems for viscoelastic materials with possible thermal effects. We first describe a general thermo-viscoelastic model involving a thermo-viscoelastic Kelvin–Voigt constitutive law, a temperature field governed by the heat equation and a subdifferential surface contact condition. Then, we study a model which describes the frictional contact between a short memory thermo-viscoelastic body and a given rigid foundation. The free boundary contact problem for a long memory viscoelastic material is also considered. Finally, we provide numerical simulations for different fundamental examples of thermal contact problems.
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References
Addi, K., Chau, O., Goeleven, D.: On some frictional contact problems with velocity condition for elastic and visco-elastic materials. Discret. Contin. Dyn. Syst. 31(4), 1039–1051 (2011)
Adly, S., Chau, O.: On some dynamical thermal non clamped contact problems. To appear in Mathematical Programming, series B
Adly, S., Chau, O., Rochdi, M.: Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numer. Algebra Control Optim. (NACO) 2(1), 89–101 (2012)
Bermudez, A., Moreno, C.: Duality methods for solving variational ineq ualities. Comput. Math. Appl. 7, 43–58 (1981)
Brézis, H.: Problèmes unilatéraux. J. Math. Pure Appl. 51, 1–168 (1972)
Brézis, H.: Operateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert. North-Holland, Amterdam (1973)
Brézis, H.: Analyse fonctionnelle, Théorie et Application. Masson, Paris (1987)
Chau, O., Awbi, B.: Quasistatic thermovisoelastic frictional contact problem with damped response. Appl. Anal. 83(6), 635–648 (2004)
Chau, O., Motreanu, D., Sofonea, M.: Quasistatic frictional problems for elastic and viscoelastic materials. Appl. Math. 47(4), 341–360 (2002)
Chau, O., Goeleven, D., Oujja, R.: A numerical treatment of a class variational inequalities arising in the study of viscoelastic materials. Submitted to Int. J. Appl. Math. Mech. (May 2012)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
Ciarlet, P.G.: Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. North Holland, Amsterdam (1988)
Duvaut, G., Lions, J.L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972)
Ekeland, I., Teman, R.: Analyse convexe et problèmes variationelles. Gauthier-Villars, Paris (1984)
Glowinski, R.: Numerical Methods for Nonlinear Variational problems. Springer-Verlag, New York (1984)
Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics. Kluwer Academic, Dordrecht (2003)
Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer-Verlag, New York (1999)
Jeantet, R., Croguennec, T., Schuck, P., Brulé, G.: Science des Aliments. Lavoisier, Paris (2006)
Kikuchi, N.N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod et Gauthier-Villars, Paris (1969)
Ne\v cas, J., Hlava\v cek, I.: Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction. Elsevier, Amsterdam (1981)
Panagiotopoulos, P.D.: Inequality Problems in Mechanical and Applications. Birkhauser, Basel (1985)
Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer-Verlag, Berlin (1993)
Pazy, A.: Semigroups of non-linear contractions in Hilbert Spaces. In: Prodi, G. (ed.) Problems in Nonlinear Analysis (C.I.M.E. Ed.). Cremonese, Roma (1971)
Raous, M., Jean, M., Moreau, J.J. (eds.): Contact Mechanics. Plenum Press, New York (1995)
Zeidler, E.: Nonlinear Functional Analysis and its Applications. Springer, New York (1997)
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Chau, O., Goeleven, D., Oujja, R. (2014). Variational Inequality Models Arising in the Study of Viscoelastic Materials. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_4
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DOI: https://doi.org/10.1007/978-1-4939-1106-6_4
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