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Asymptotic behavior of eigenfunctions and eigenfrequencies of oscillation boundary value problems of the linear theory of elastic mixtures

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Georgian Mathematical Journal

Abstract

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.

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References

  1. D. G. Natroshvili, A. J. Jagmaidze, and M. Zh. Svanadze, Some problems of the theory of elastic mixtures. (Russian)Tbilisi Univ. Press, Tbilisi, 1986.

    Google Scholar 

  2. L. P. Khoroshun and N. S. Soltanov, Thermoelasticity of two-component mixtures. (Russian)Naukova Dumka, Kiev, 1984.

    Google Scholar 

  3. Ya. Ya. Rushchitskii, Elements of mixture theory. (Russian)Naukova Dumka, Kiev, 1991.

    Google Scholar 

  4. I. G. Filippov, Dynamical theory of a relative flow of multicomponent media. (Russian)Prikl. Mekhanika 7(1971), No. 10, 92–99.

    Google Scholar 

  5. B. Lempriere, On practicability of analyzing waves in composites by the theory of mixtures.Lockheed Palo Alto Research Laboratory. Report No. LMSC-6-78-69-21 (1969), 76–90.

  6. H. D. McNiven and Y. A. Mengi, A mathematical model for the linear dynamic behavior of two-phase periodic materials.Int. J. Solids and Struct. 15(1979), No. 4, 571–580.

    Google Scholar 

  7. T. R. Steel, Applications of a theory of interacting continua.Quart. J. Mech. and Appl. Math. 20(1967), No. 1, 57–72.

    Google Scholar 

  8. M. Zh. Svanadze, Representation of a general solution of the equation of steady-state oscillations of two-component elastic mixtures. (Russian)Prikl. Mekhanika 29(1993), No. 12, 22–29.

    Google Scholar 

  9. M. Zh. Svanadze, The uniqueness of solutions of stable oscillation problems of the linear theory of a two-component elastic mixture. (Russian)Bull. Acad. Sci. Georgia 145(1992), No. 1, 51–54.

    Google Scholar 

  10. M. Zh. Svanadze, Investigation of boundary value problems of steady-state oscillations of the theory of elastic mixtures. 10th Conference of Problems and Methods in Math. Physics, September 13–17, 1993,Chemnitz, Germany, Abstracts, p. 60.

  11. M. Zh. Svanadze, Uniqueness theorem for solutions of the internal boundary value problems of steady-state oscillations of the linear theory of elastic mixtures.Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 46(1992), 179–190.

    Google Scholar 

  12. H. Weyl, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung.J. Reine und Angew. Math. 141(1912), 1–11.

    Google Scholar 

  13. H. Weyl, Über die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze.J. Reine und Angew. Math. 143(1913), 177–202.

    Google Scholar 

  14. R. Courant and D. Hilbert, Methoden der mathematischen Physik.Julius Springer, Berlin, B. 1, 2, 1931, 1937.

    Google Scholar 

  15. T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes.Comptes Rendus des Mathématiciens Scandinaves á Stockholm 14–18Aout 1934, 34–44,Lund, 1935.

  16. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers.Rendiconti del Circolo Matematico di Palermo 39(1915), 1–49.

    Google Scholar 

  17. A. Plejel, Propriétés asymptotiques des fonctions et valeurs propres de certains problems de vibrations.Arkiv för Math., Astr. och Fysik. 27A(1940), 1–100.

    Google Scholar 

  18. H. Niemeyer, Über die elastischen Eigenschwingungen endlicher Körper.Arch. Rat. Mech. Ann. 19(1965), 24–61.

    Google Scholar 

  19. T. V. Burchuladze, To the theory of boundary value problems of oscillation for an elastic body. (Russian)Proc. Tbilisi University, Mat. Mech. Astron. 64(1957), 215–240.

    Google Scholar 

  20. T. V. Burchuladze, On the asymptotic behavior of eigenfunctions of some boundary value problems of an anisotropic elastic body. (Russian)Bull. Acad. Sci. Georgian SSR 23(1959), No. 3, 265–272.

    Google Scholar 

  21. R. G. Dikhamindzhia, Asymptotic distribution of eigenfunctions and eigenvalues of some basic problems of vibration of the moment theory of elasticity. (Russian)Trudy Tbiliss. Matem. Inst. Razmadze 73(1983), 64–71.

    Google Scholar 

  22. C. Müller and H. Niemeyer, Greensche Tensoren und asymptotische Gesetze der elekromagnetischen Hohlraumschwingungen.Arch. Rat. Mech. Ann. 7(1961), 305–348.

    Article  Google Scholar 

  23. H. Niemeyer, Eine Verschärfung der asymptotischen Gesetze elektomagnetischen Hohlraumschwingungen.Arch. Rat. Mech. Ann. 7(1961), 412–433.

    Article  Google Scholar 

  24. W. Gromes, Über das asymptotische Verhalten der Spektralfunktion elliptischer Systeme.Math. Zeitschrift 118(1970), No. 4, 254–270.

    Article  Google Scholar 

  25. R. Brübach, Über die Spektralmatrix elliptischer Systeme.Math. Zeitschrift 140(1974), No. 3, 231–244.

    Article  Google Scholar 

  26. V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. (Translated from Russian)North-Holland series in applied mathematics and mechanics, v.25,North-Holland Publishing Company, Amsterdam-New York-Oxford, 1979; Russian original:Nauka, Moscow, 1976.

    Google Scholar 

  27. M. Zh. Svanadze, Fundamental solutions of equations of stable oscillation and pseodo-oscillation of a two-component elastic mixture. (Russian)Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 39(1990), 227–240.

    Google Scholar 

  28. I. G. Petrashen, Elastic wave propagation in anisotropic media. (Russian)Nauka, Leningrad, 1980.

    Google Scholar 

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Svanadze, M. Asymptotic behavior of eigenfunctions and eigenfrequencies of oscillation boundary value problems of the linear theory of elastic mixtures. Georgian Mathematical Journal 3, 177–200 (1996). https://doi.org/10.1007/BF02254739

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