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General theorems and fundamental solutions in the dynamical theory of mixtures

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Abstract

This paper is concerned with the dynamic theory of a binary mixture consisting of a gas and an elastic solid. First, we derive general theorems (reciprocal theorem, uniqueness theorem, minimum principle). The, we establish a Galerkin representation and the fundamental solutions in the case of steady vibrations.

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References

  1. C. Truesdell and R. Toupin, The Classical Field Theories. In Handbuch der Physik (edited by S. Flügge), vol III/3. Berlin: Springer-Verlag (1960).

    Google Scholar 

  2. A. C. Eringen and J. D. Ingram, A continuum theory of chemically reacting media—I. Int. J. Engng. Sci. 3 (1965) 197–212.

    Article  Google Scholar 

  3. J. D. Ingram and A. C. Eringen, A continuum theory of chemically reacting media—II. Int. J. Engng. Sci. 5 (1967) 289–322.

    Article  Google Scholar 

  4. A. E. Green and P. M. Naghdi, A dynamical theory of interacting continua. Int. J. Engng. Sci. 3 (1965) 231–241.

    Google Scholar 

  5. I. Müller, A thermodynamic theory of mixtures of fluids. Arch. Rational Mech. Anal. 28 (1968) 1–39.

    Article  Google Scholar 

  6. R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials. Int. J. Engng. Sci. 7 (1969) 689–722.

    Article  Google Scholar 

  7. R. M. Bowen, Theory of mixtures. In: A. C. Eringen (ed.), Continuum Physics, vol. 3. New York: Academic Press (1976).

    Google Scholar 

  8. R. J. Atkin and R. E. Craine, Continuum theories of mixtures: basic theory and historical development. Q.J. Mech. Appl. Math. 29 (1976) 209–245.

    Google Scholar 

  9. A. Bedford and D. S. Drumheller, Theory of immiscible and structured mixtures. Int. J. Engng. Sci. 21 (1983) 863–960.

    Article  Google Scholar 

  10. K. R. Rajagopal and A. S. Wineman, Developments in the mechanics of interactions between a fluid and a highly elastic solid. In: D. De Kee and P. N. Kalany (eds), Recent Developments in Structured Continua, vol. II. Research Notes in Mathematics Series. Essex: Longman (1990) pp. 249–292.

    Google Scholar 

  11. D. Iesan, Reciprocity, uniqueness and minimum principles in the linear theory of piezoelectricity. Int. J. Engng. Sci. 28 (1990) 1139–1149.

    Article  Google Scholar 

  12. Y.M. Lee and C. J. Martin, A reciprocal theorem for a mixture of interacting continuous media. Int. J. Engng. Sci. 13 (1975) 979–991.

    Article  Google Scholar 

  13. R. J. Knops and T. R. Steel, Uniqueness in the linear theory of a mixture of two elastic solids. Int. J. Engng. Sci. 7 (1969) 571–577.

    Article  Google Scholar 

  14. R. J. Atkin, P. Chadwick and T. R. Steel, Uniqueness theorems for linearized theories of interacting continua. Mathematika 14 (1967) 27–42.

    Google Scholar 

  15. A. E. Green and P. M. Naghdi, The flow of fluid through an elastic solid. Acta Mech. 9 (1970) 329–340.

    Google Scholar 

  16. R. Reiss, Minimum principles for linear elastodynamics. J. Elasticity 8 (1978) 35–45.

    Google Scholar 

  17. R. Reiss and E. J. Haug, Extremum principles for linear initial value problems of mathematical physics. Int. J. Engng. Sci. 16 (1978) 231–251.

    Article  Google Scholar 

  18. R. M. Bowen, Green's functions for consolidation problems. Int. J. Engng. Sci. 19 (1981) 455–466.

    Article  Google Scholar 

  19. A. E. Green and T. R. Steel, Constitutive equations for interacting continua. Int. J. Engng. Sci. 4 (1966) 483–500.

    Article  Google Scholar 

  20. R. J. Atkin, Conservative theory for a mixture of an isotropic elastic solid and a non-Newtonian fluid. ZAMP 18 (1967) 803–825.

    Google Scholar 

  21. D. Graffi, Sui teoremi de reciprocità nei fenomeni non stazionari. Atti Accad. Sci: Ist. Bologna, Serie 11, 10 (1963) 33–40.

    Google Scholar 

  22. J. Ignaczak, A completeness problem for the stress equation of motion in the linear theory of elasticity. Arch. Mech. Stosow. 15 (1963) 225–234.

    Google Scholar 

  23. M. E. Gurtin, The Linear Theory of Elasticity, pp. 1–295 of Flügge's Handbuch der Physik, vol. VIa/2 (edited by C. Truesdell). Berlin: Springer (1972).

    Google Scholar 

  24. D. Ieşan, On some reciprocity theorems and variational theorems in linear dynamic theories of continuum mechanics. Memorie Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., Serie 4, 17 (1974).

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Authors and Affiliations

  1. Dipartimento di Ingegneria dell' Informazione e Matematica Applicata, Università di Salerno, 84100, Salerno, Italy

    Michele Ciarletta

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  1. Michele Ciarletta
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Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.), with the grant 60% M.U.R.S.T. (ITALY).

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Ciarletta, M. General theorems and fundamental solutions in the dynamical theory of mixtures. J Elasticity 39, 229–246 (1995). https://doi.org/10.1007/BF00041839

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  • DOI: https://doi.org/10.1007/BF00041839

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