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Existence and uniqueness of a thermoelastic problem with variable parameters

Published online by Cambridge University Press:  04 May 2015

P. BARRAL ,
M. C. NAYA-RIVEIRO  and
P. QUINTELA
P. BARRAL
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain email: patricia.barral@usc.es, peregrina.quintela@usc.es
M. C. NAYA-RIVEIRO
Affiliation:
Department of Pedagogy and Didactics, Faculty of Educational Studies, Universidade da Coruña, 15071 A Coruña, Spain email: cristina.naya@udc.es
P. QUINTELA
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain email: patricia.barral@usc.es, peregrina.quintela@usc.es
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Abstract

The aim of this article is to study the existence and uniqueness of solution for a quasistatic fully coupled thermoelastic problem arising from some metallurgical processes. We consider mixed boundary conditions for both submodels, and a Robin boundary condition for the thermal one. Furthermore, the reference temperature, the thermal conductivity and the Lamé's parameters are assumed to depend on the material point.

Keywords

ThermoelasticityQuasistatic problemGalerkin's methodNon-homogeneous materials
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 4 , August 2015 , pp. 497 - 520
Copyright
Copyright © Cambridge University Press 2015 

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References

[1]Ames, K. A. & Payne, L. E. (1994) Uniqueness and continuous dependence of solutions to a multidimensional thermoelastic contact problem. J. Elast. 34, 139–148.CrossRefGoogle Scholar
[2]Barral, P., Naya-Riveiro, M. C. & Quintela, P. (2007) Mathematical analysis of a viscoelastic problem with temperature-dependent coefficients. Part I: Existence and uniqueness. Math. Methods Appl. Sci. 30 (13), 1545–1568.CrossRefGoogle Scholar
[3]Barral, P., Naya-Riveiro, M. C. & Quintela, P. (2007) Mathematical analysis of a viscoelastic problem with temperature-dependent coefficients. Part II: Regularity. Math. Meth. Appl. Sci. 30 (13), 1569–1592.CrossRefGoogle Scholar
[4]Barral, P. & Quintela, P. (1999) A numerical method for simulation of thermal stresses during casting of aluminium slabs. Comput. Methods Appl. Mech. Eng. 178 (1–2), 69–88.CrossRefGoogle Scholar
[5]Bermúdez, A., Muñiz, M. C. & Quintela, P. (1993) Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminum electrolytic cell. Comput. Methods Appl. Mech. Eng. 106 (1–2), 129–142.CrossRefGoogle Scholar
[6]Boley, B. A. & Weiner, J. H. (1960) Theory of Thermal Stresses, John Wiley & Sons Inc., New York.Google Scholar
[7]Brezis, H. (1983) Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maî trise, Masson, Paris.Google Scholar
[8]Carlson, D. E. (1972) Linear Thermoelasticity, Handbuch der Physik (Encyclopedia of Physics, Mechanics of Solids II), Vol. VIA/2, Springer, Berlin, pp. 297–345.Google Scholar
[9]Copetti, M. I. M. & Elliott, C. M. (1993) A one-dimensional quasi-static contact problem in linear thermoelasticity. Eur. J. Appl. Math. 4 (2), 151–174.CrossRefGoogle Scholar
[10]Dafermos, C. M. (1968) On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Ration. Mech. Anal. 29, 241–271.CrossRefGoogle Scholar
[11]Day, W. A. (1985) Heat Conduction within Linear Thermoelasticity, Springer-Verlag, New York–Berlin.CrossRefGoogle Scholar
[12]Duhamel, J. M. C. (1837) Second mémoire sur les phénomènes thermomécaniques. J. de l'École Polyt. 15, 1–57.Google Scholar
[13]Duvaut, G. & Lions, J. L. (1969) Nouvelles inéquations variationnelles rencontrées en thermique et en thermoélasticité. C. R. Acad. Sci. Paris Sér. A-B 269, A1198–A1201.Google Scholar
[14]Duvaut, G. & Lions, J. L. (1969) Sur de nouveaux problèmes d'inéquations variationnelles posés par la Mécanique. Le cas d'évolution. C. R. Acad. Sci. Paris Sér. A-B 269, A570–A572.Google Scholar
[15]Duvaut, G. & Lions, J. L. (1972) Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279.CrossRefGoogle Scholar
[16]Duvaut, G. & Lions, J. L. (1972) Les inéquations en mécanique et en physique. Travaux et Recherches Mathématiques, Vol. 21, Dunod, Paris.Google Scholar
[17]Figueiredo, I. & Trabucho, L. (1995) A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity. Int. J. Eng. Sci. 33 (1), 45–66.CrossRefGoogle Scholar
[18]Figueiredo, I. & Trabucho, L. (1995) Some existence results for contact and friction problems in thermoelasticity and in thermoviscoelasticity. In: Asymptotic Methods for Elastic Structures (Lisbon, 1993), Gruyter, Berlin, pp. 223–235.Google Scholar
[19]Flint, G., Usmani, A., Lamont, S., Torero, J. & Lane, B. (2006) Effect of fire on composite long span truss floor systems. J. Constr. Steel. Res. 62, 303–315.CrossRefGoogle Scholar
[20]Gawinecki, J. (1981) Uniqueness and regularity of the solution of the first boundary-initial value problem for thermal stresses equations of classical and generalized thermomechanics. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 29 (11–12), 231–238.Google Scholar
[21]Gawinecki, J. (1982) Existence and uniqueness of the solution of the third boundary-initial value problem for thermal stresses equations of classical and generalized thermomechanics. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 30 (3–4), 155–163.Google Scholar
[22]Gawinecki, J. (1983) Existence, uniqueness and regularity of the first boundary-initial value problem for thermal stresses equations of classical and generalized thermomechanics. J. Tech. Phys. 24 (4), 467–479.Google Scholar
[23]Gawinecki, J. (1986) Existence, uniqueness and regularity of the solution of the first boundary-initial value problem for the equations of linear thermomicroelasticity. Bull. Pol. Acad. Sci. Tech. Sci. 34 (7–8), 447–460.Google Scholar
[24]Gawinecki, J. (1987) Existence, uniqueness and regularity of the first boundary-initial value problem for hyperbolic equations system of the thermal stresses theory for temperature-rate-dependent solids. Bull. Pol. Acad. Sci. Tech. Sci. 35 (7–8), 411–419.Google Scholar
[25]Gawinecki, J. (1987) The Faedo–Galerkin method in thermal stresses theory. Comment. Math. Prace Mat. 27 (1), 83–107.Google Scholar
[26]Gawinecki, J., Kowalski, T. & Litewska, K. (1982) Existence and uniqueness of the solution of the mixed boundary-initial value problem in linear thermoelasticity. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 30 (11–12), 173–178.Google Scholar
[27]Ieşan, D. (1989) On some theorems in thermoelastodynamics. Rev. Roum. Sci. Tech. Sér. Méc. Appl. 34 (2), 101–111.Google Scholar
[28]Ionescu-Cazimir, V. (1964) Problem of linear coupled thermoelasticity. III. Uniqueness theorem. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 12 (12), 565–573.Google Scholar
[29]Jiang, S. & Racke, R. (2000) Evolution Equations in Thermoelasticity. Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[30]Knops, R. & Payne, L. (1970) On uniqueness and continuous dependence in dynamical problems of linear thermoelasticity. Int. J. Solids Struct. 6 (8), 1173–1184.CrossRefGoogle Scholar
[31]Martins, J. A. C. & Oden, J. T. (1987) Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal.-Theory Methods Appl. 11 (3), 407–428.CrossRefGoogle Scholar
[32]Muñoz Rivera, J. E. & Racke, R. (1998) Multidimensional contact problems in thermoelasticity. SIAM J. Appl. Math. 58 (4), 1307–1337.CrossRefGoogle Scholar
[33]Naya-Riveiro, M. C. & Quintela, P. (2008) Modelling of materials with long memory. Int. J. Solids Struct. 45 (24), 6133–6156.CrossRefGoogle Scholar
[34]Shi, P. & Shillor., M. (1992) Existence of a solution to the N-dimensional problem of thermoelastic contact. Commun. Partial Differ. Equ. 17 (9–10), 1597–1618.Google Scholar
[35]Viaño Rey, J. M. (1981) Existencia y aproximación de soluciones en termoelasticidad y elastoplasticidad. Phd thesis, Department of Applied Mathematics, Universidade de Santiago de Compostela.Google Scholar
[36]Weiner, J. (1957) A uniqueness theorem for the coupled thermoelastic problem. Q. Appl. Math. 15, 102–105.CrossRefGoogle Scholar