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Fundamental solution in the linear theory of thermoviscoelastic mixtures

Published online by Cambridge University Press:  01 June 2007

MERAB SVANADZE
Affiliation:
Faculty of Physics and Mathematics, Ilia Chavchavadze State University, I. Chavchavadze Avenue 32, 0179 Tbilisi, Georgia email: [email protected]
GERARDO IOVANE
Affiliation:
Dipartimento di Ingegneria dell'Informazione e Matematica Applicata dell'Università di Salerno, Via Ponte don Melillo, Fisciano 84084, Italy email: [email protected]

Abstract

In this article the linear theory of thermoviscoelastic mixtures is considered. The fundamental solution of the system of linear-coupled partial differential equations of steady oscillations (steady vibrations) of the theory of thermoviscoelastic mixtures is constructed in terms of elementary functions and basic properties are established.

Type
Papers
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Truesdell, C. & Toupin, R. (1960) The classical field theories. In: Flügge, S. (editor), Handbuch der Physik, Vol. III/3, Springer-Verlag, Berlin, 226858.Google Scholar
[2]Eringen, A. C. & Ingram, J. D. (1965) A continuum theory of chemically reacting media – I. Int. J. Eng. Sci. 3, 197212.CrossRefGoogle Scholar
[3]Green, A. E. & Naghdi, P. M. (1965) A dynamic theory of interacting continua. Int. J. Eng. Sci. 3, 231241.CrossRefGoogle Scholar
[4]Müller, I. (1968) A thermodynamics theory of mixtures of fluids. Arch. Rational Mech. Anal. 28, 139.CrossRefGoogle Scholar
[5]Bowen, R. M. & Wiese, J. C. (1969) Diffusion in mixtures of elastic materials. Int. J. Eng. Sci. 7, 689722.CrossRefGoogle Scholar
[6]Green, A. E. & Steel, T. R. (1966) Constitutive equations for interacting continua. Int. J. Eng. Sci. 4, 483500.CrossRefGoogle Scholar
[7]Steel, T. R. (1967) Applications of a theory of interacting continua. Q. J. Mech. Appl. Math. 20, 5772.CrossRefGoogle Scholar
[8]Iesan, D. (1991) On the theory of mixtures of thermoelastic solids. J. Thermal Stresses 14, 389408.CrossRefGoogle Scholar
[9]Khoroshun, L. P. & Soltanov, N. S. (1984) Thermoelasicity of Two Component Mixtures, Naukova Dumka, Kiev. (Russian).Google Scholar
[10]Iesan, D. (1997) A theory of mixtures with different consituent temperatures. J. Thermal Stresses 20, 147167.CrossRefGoogle Scholar
[11]Bedford, A. & Stern, M. (1972) A multi-continuum theory for composite elastic materials. Acta Mech. 14, 85102.CrossRefGoogle Scholar
[12]Bedford, A. & Stern, M. (1972) Toward a diffusing continuum theory of composite elastic materials. J. Appl. Mech. 38, 814.CrossRefGoogle Scholar
[13]Tiersten, H. F. & Jahanmir, M. (1977) A theory of composite modeled as interpenetrating solid continua. Arch. Ratinal Mech. Anal. 65, 153192.CrossRefGoogle Scholar
[14]Iesan, D. (1994) On the theory of mixtures of elastic solids. J. Elasticity 35, 251268.CrossRefGoogle Scholar
[15]Nunziato, J. W. & Cowin, S. C. (1979) A non-linear theory of elastic materials with voids. Arch. Rational Mech. Anal. 72, 175201.CrossRefGoogle Scholar
[16]Goodman, M. A. & Cowin, S. C. (1972) A continuum theory for granular materials. Arch. Rational Mech. Anal. 44, 249266.CrossRefGoogle Scholar
[17]Drumheller, D. S. (1978) The theoretical treatment of a porous solid using a mixture theory. Int. J. Solids Struct. 14, 441456.CrossRefGoogle Scholar
[18]Iesan, D. (2004) On the theory of viscoelastic mixtures. J. Thermal Stresses 27, 11251148.CrossRefGoogle Scholar
[19]Quintanilla, R. (2005) Existence and exponential decay in the linear theory of viscoelastic mixtures. Eur. J. Mech. A / Solids 24, 311324.CrossRefGoogle Scholar
[20]Bowen, R. M. (1976) Theory of mixtures. In: Eringen, A. C. (editor). Continuum Physics, Vol. III, Academic Press, New York, 1127.Google Scholar
[21]Atkin, R. J. & Craine, R. E. (1976) Continuum theories of mixtures: Basic theory and historical development. Quart. J. Mech. Appl. Math. 29, 209243.CrossRefGoogle Scholar
[22]Bedford, A. & Drumheller, D. S. (1983) Theory of immiscible and structured mixtures. Int. J. Eng. Sci. 21, 863960.CrossRefGoogle Scholar
[23]Kupradze, V. D. (1965) Potential Methods in the Theory of Elasticity. Israel Program for Scientific Translations, Jerusalem.Google Scholar
[24]Kupradze, V. D., Gegelia, T. G., Basheleishvili, M. O. & Burchuladze, T. V. (1979) Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Amsterdam, New York, Oxford.Google Scholar
[25]Gurtin, M. E. (1972) The linear theory of elasticity. In: Truesdell, C. A. (editor), Handbuch der Physik, Vol. VI a/2, Springer-Verlag, Berlin, 1295.Google Scholar
[26]Nowacki, W. (1975) Dynamic Problems in Thermoelasicity, Noordhoff International Publishing, Leyden.Google Scholar
[27]Svanadze, M. (1988) The Fundamental matrix of the linearized equations of the theory elastic mixtures. Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 23, 133148.Google Scholar
[28]Svanadze, M. (1990) Fundamental solutions of equations of stable oscillation and pseudo-oscillation of a two-component elastic mixtures. Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 39, 227240.Google Scholar
[29]Svanadze, M. (1996) The fundamental solution of the oscillation equations of the thermoelasticity theory of mixture of two elastic solids. J. Thermal Stresses 19, 633648.CrossRefGoogle Scholar
[30]Lopatinsky, YA. B. (1951) Fundamental solution system of elliptic system of differential equations. Ukraine Math. J. 3, 338 (Russian).Google Scholar
[31]Lopatinsky, YA. B. (1951) Fundamental solutions of differential equations system of elliptic type. Ukraine Math. J. 3, 290316 (Russian).Google Scholar
[32]Hörmander, L. (1963) Linear Partial Differential Operators, Springer-Verlag, Berlin, Göttingen, Heidelberg.CrossRefGoogle Scholar
[33]Hörmander, L. (1983) The Analysis of Linear Partial Differential Operators II: Differential Operators With Constant Coefficients. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo.Google Scholar