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Quasi-Static Linear Thermo-Viscoelastic Process with Irregular Viscous Dissipation

Part of: Equations and systems of special type Coupling of solid mechanics with other effects Materials of strain-rate type and history type, other materials with memory

Published online by Cambridge University Press:  18 January 2017

Farid Messelmi ,
Abdelbaki Merouani  and
Hicham Abdelaziz
Farid Messelmi*
Affiliation:
Department of Mathematics and LDMM Laboratory, University Ziane Achour de Djelfa, Djelfa 17000, Algeria
Abdelbaki Merouani*
Affiliation:
Department of Mathematics, University of Bordj Bou Arreridj, Bordj Bou Arreridj 34000, Algeria
Hicham Abdelaziz*
Affiliation:
Department of Mathematics, University of Bordj Bou Arreridj, Bordj Bou Arreridj 34000, Algeria
*
*Corresponding author. Email:foudimath@yahoo.fr (F. Messelmi), badri_merouani@yahoo.fr (A. Merouani), hichem.abd21@gmail.com (H. Abdelaziz)
*Corresponding author. Email:foudimath@yahoo.fr (F. Messelmi), badri_merouani@yahoo.fr (A. Merouani), hichem.abd21@gmail.com (H. Abdelaziz)
*Corresponding author. Email:foudimath@yahoo.fr (F. Messelmi), badri_merouani@yahoo.fr (A. Merouani), hichem.abd21@gmail.com (H. Abdelaziz)
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Abstract

We consider a mathematical model which describes the quasi-static evolution of a thermo-viscoelastic linear body with taking into account the effects of internal forces which generate a non linear viscous dissipative function. We derive a variational formulation of the system of equilibrium equation and energy equation. An existence result of weak solutions was obtained in an appropriate function space.

Keywords

Thermo-viscoelasticityviscous dissipation

MSC classification

Secondary: 35M10: Equations of mixed type 74D05: Linear constitutive equations 74F05: Thermal effects
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 924 - 943
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Amassad, A., Fabre, C. and Sofonea, M., A quasistatic viscoplastic contact problem with normal compliance and friction, IMA J. Appl. Math., 69 (2004), pp. 463–482.Google Scholar
[2] Boccardo, L., Dall’Aglio, A., Gallouët, T. and Orsina, L., Quasi-linear parabolic equations with measure data, In Proceedings of the International Conference on Nonlinear Differential Equations, Kiev, (1995).Google Scholar
[3] Bonetti, E. and Bonfanti, G., Existence and uniqueness of the solution to 3D thermoviscoelastic system, Electron. J. Differential Equations, 50 (2003), pp. 1–15.Google Scholar
[4] Brezis, H., Equations et inéquations non linéaires dans les espaces en dualité, Ann. Inst. Fourier, 18(1) (1968), pp. 115–175.CrossRefGoogle Scholar
[5] Consiglieri, L., Stationary solution for a Bingham flow with nonlocal frictions, In Mathematical Topics in Fluid Mechanics, Rodrigues, J. F. and Sequeira, A. (eds), Pitman Res. Notes in Math. Longman (1992), pp. 237–252.Google Scholar
[6] Duvaut, G. et Lions, J. L., Les Inéquations en Mécanique et en Physique, Dunod, (1976).Google Scholar
[7] Duvaut, G. et Lions, J. L., Transfert de la Chaleur dans un fluide de Bingham dont la viscosité Dépend de la température. J. Funct. Anal., 11 (1972), pp. 85–104.Google Scholar
[8] Ky, F., Fixed point and min-max theorems in locally convex topological linear spaces, Proc. Natl. Acad. Sci. USA, 38(2) (1952), pp. 121–126.Google Scholar
[9] Fernández-García, J. R., Sofonea, M. and Viaňo, J. M., A frictionless contact problem for Elastic-Viscoplastic materials with normal compliance, Numer. Math., 90 (2002), pp. 689–719.Google Scholar
[10] Germain, P., Cours de Mécanique des Milieux Continus, Masson et Cie, Paris, (1973).Google Scholar
[11] Kobayashi, S. and Robelo, N., A coupled analysis of viscoplastic deformation and heat transfer: i theoretical consideration, II applications, Int, J. Mech. Sci., 22 (1980), pp. 699–705, pp. 707–718.Google Scholar
[12] Lions, J. L., Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, (1969).Google Scholar
[13] Lions, J. L. et Magenes, E., Problèmes aux Limites non Homogènes et Applications, Volume I, Dunod, (1968).Google Scholar
[14] Merouani, A. and Messelmi, F., Dynamic evolution of damage in elastic-thermo-viscoplastic materials, Electron. J. Differential Equations, 2010(129) (2010), pp. 1–15.Google Scholar
[15] Merouani, B., Messelmi, F. and Drabla, S., Dynamical flow of a Bingham fluid with subdifferential boundary condition, An. Univ. Oradea Fasc. Mat, Tome, XVI (2009), pp. 5–30.Google Scholar
[16] Messelmi, F., Merouani, B. and Bouzeghaya, F., Steady-state thermal Herschel-Bulkley flow with Tresca's friction law, Electron. J. Differential Equations, 2010(46) (2010), pp. 1–14.Google Scholar
[17] Nečas, J. and Kratochvil, J., On existence of the solution boundary value problems for elastic-inelastic solids, Comment. Math. Univ. Carolinea, 14 (1973), pp. 755–760.Google Scholar
[18] Simon, J., Compact sets in the space Lp (0, T, B), Ann. Mt. Pura Appl., 146 (1987), pp. 64–96.Google Scholar
[19] Sofonea, M., Quasistatic processes for elastic-viscoplastic materials with internal state variables, Annales Scientifiques de l’Université Clermont-Ferrand 2, Tome 94, Série Mathématiques, n° 25 (1989), pp. 47–60.Google Scholar