Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The dynamical problems of the theory of elasticity and thermoelasticity

  • 92 Accesses

  • 3 Citations

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    T. V. Burchuladze, “The asymptotic behavior of the fundamental oscillation functions of an elastic body,” Author's Abstract of Candidate's Dissertation in the Physicomathematical Sciences, Tbilis. Univ., Tbilisi (1956).

  2. 2.

    T. V. Burchuladze and R. V. Rukhadze, “On the Green tensors in the theory of elasticity,” Differents. Uravnen.,10, No. 10, 1849–1865 (1974).

  3. 3.

    M. I. Vishik and O. A. Ladyzhenskaya, “Boundary-value problems for partial differential equations and certain classes of operator equations,” Usp. Mat. Nauk,11, No. 6, 41–97 (1956).

  4. 4.

    N. M. Gyunter (Günter), Potential Theory and Its Applications to Basic Problems of Mathematical Physics, F. Ungar Publ. Co., New York (1967).

  5. 5.

    V. A. Il'in, “On the solvability of mixed problems for hyperbolic and parabolic equations,” Usp. Mat. Nauk,15, No. 2, 97–154 (1960).

  6. 6.

    N. S. Kakhniashvili, “The fundamental existence theorems in thermoelasticity,” Proc. Tbilisi Univ.,A8, No. 153, 79–91 (1974).

  7. 7.

    V. D. Kupradze, “The boundary-value problems in the theory of stationary elastic oscillations,” Usp. Mat. Nauk,8, No. 3(55), 21–74 (1953).

  8. 8.

    V. D. Kupradze, The Boundary-Value Problems of the Theory of Oscillations and Integral Equations [in Russian], Gostekhizdat, Moscow (1950).

  9. 9.

    V. D. Kupradze, The Method of Potentials in the Theory of Elasticity [in Russian], Fizmatgiz, Moscow (1963).

  10. 10.

    V. D. Kupradze, “On solving the three-dimensional mixed boundary-value problem in the theory of elasticity,” in: The Mechanics of the Continuous Medium and Related Problems of Analysis. For the 80th anniversary of Acad. N. I. Muskhelishvili, Moscow (1972).

  11. 11.

    V. D. Kupradze and T. V. Burchuladze, “Boundary-value problems in thermoelasticity,” Differents. Uravnen.,5, No. 1, 3–43 (1969).

  12. 12.

    V. D. Kupradze and T. V. Burchuladze, “The solving of dynamical problems of the theory of elasticity,” Annotatsii Dokl. Semin. Inst. Prikl. Matem. Tbilis. Univ.,3, 66–77 (1970).

  13. 13.

    V. D. Kupradze and T. V. Burchuladze, “The proof of the existence and the computation of the solutions of the fundamental mixed problems of the dynamics of a three-dimensional elastic body of an arbitrary form,” Tbilisis Matem. Inst. Shrom. Sakart. SSR Metsn. Akad. Tr. Tbilis. Matem. Inst. Akad. Nauk GruzSSR,39, 23–42 (1971).

  14. 14.

    V. D. Kupradze, T. G. Gegeliya, M. O. Basheleishvili and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity [in Russian], Tbilisi Univ., Tbilisi (1968).

  15. 15.

    V. D. Kupradze, T. G. Gegeliya, M. O. Basheleishvili, and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity [in Russian], Nauka, Moscow (1975).

  16. 16.

    O. A. Ladyzhenskaya, The Mixed Problem for Hyperbolic Equations [in Russian], Gotekhizdat, Moscow (1953).

  17. 17.

    O. A. Ladyzhenskaya, “On nonstationary operator equations and their applications to the linear problems of mathematical physics,” Matem. Sb.,45, No. 2, 123–158 (1958).

  18. 18.

    O. A. Ladyzhenskaya, The Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Leningrad (1973).

  19. 19.

    A. M. Lyapunov, Papers on the Theory of Potential [in Russian], GITTL, Moscow (1949).

  20. 20.

    L. G. Magnaradze, “On the general representations of certain classes of solutions of linear nonstationary partial differential equations and a new method of solving the Cauchy-Dirichlet problems and their analogs,” Tbilisis Univ. Gamokkheneb. Matem. Inst. Shrom., Tr. Inst. Prikl. Matem. Tbilis. Univ., No. 2, 72–92 (1969).

  21. 21.

    C. Miranda, Partial Differential Equations of Elliptic Type, Springer, New York (1970).

  22. 22.

    S. G. Mikhlin, Direct Methods in Mathematical Physics [in Russian], Gostekhizdat, Moscow (1950).

  23. 23.

    S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, Holden-Day, San Francisco (1965).

  24. 24.

    S. G. Mikhlin, Mathematical Physics, An Advanced Course, North-Holland, Amsterdam (1970).

  25. 25.

    H. Hochstadt (editor), Linear Equations of Mathematical Physics, Holt, Rinehart, and Winston, New York (1967).

  26. 26.

    K. Maurin, Methods of Hilbert Spaces, Polish Scientific Publishers, Warsaw (1972).

  27. 27.

    N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1966).

  28. 28.

    N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Functions Theory and Their Applications to Mathematical Physics [in Russian], Nauka, Moscow (1968).

  29. 29.

    V. Novatskii, “A survey of the dynamical problems of thermoelasticity,” Period. Sb. Perev.,6, No. 100, 101–142 (1966).

  30. 30.

    I. G. Petrovskii, Lectures on Partial Differential Equations, Interscience, New York (1954).

  31. 31.

    I. G. Petrovskii, “On some problems in the theory of partial differential equations,” Usp. Mat. Nauk,1, Nos. 3–4, 44–70 (1946).

  32. 32.

    I. G. Petrovskii, On the Theory of Partial Differential Equations. Jubilee volume dedicated to the thirtieth anniversary of the Great October Socialist Revolution, I, Moscow (1947), pp. 214–230.

  33. 33.

    R. V. Rukhadze, “On the solvability of the first fundamental mixed problem of the dynamics of a three-dimensional elastic body,” Sakart. SSR Metsn. Akad. Moambe. Soobshch. Akad. Nauk GruzSSR,73, No. 2, 289–292 (1974).

  34. 34.

    R. V. Rukhadze, “On the solvability of the second fundamental and some other mixed problems in the dynamics of the theory of elasticity,” Sakart. SSR Metsn. Akad. Moambe. Soobshch. Akad. Nauk GruzSSR,74, No. 1, 45–48 (1974).

  35. 35.

    V. I. Smirnov, A Course of Higher Mathematics, Vol. 4, Pergamon Press, Oxford (1964).

  36. 36.

    Kh. L. Smolitskii, “The estimate of the derivatives of the Neumann functions,” Dokl. Akad. Nauk SSSR,106, No. 5, 785–788 (1956).

  37. 37.

    S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, American Mathematical Society, Providence (1963).

  38. 38.

    S. L. Sobolev, Partial Differential Equations of Mathematical Physics, Pergamon Press, Oxford (1964).

  39. 39.

    D. M. Éidus, “Estimates for the derivatives of the Green functions,” Dokl. Akad. Nauk SSSR,106, No. 2, 207–209 (1956).

  40. 40.

    D. M. Éidus, “Inequalities for the Green functions,” Matem. Sb.,45, No. 4, 455–470 (1958).

  41. 41.

    P. Chadwick, Thermoelasticity. The Dynamical Theory, Progress in Solid Mechanics, Vol. 1, North-Holland, Amsterdam (1960), pp. 263–328.

  42. 42.

    C. M. Dafermos, “On the existence and the asymptotic stability of solutions of the equations of linear thermoelasticity,” Arch. Rational Mech. Anal.,29, No. 4, 241–271 (1968).

  43. 43.

    G. Fichera, “Sull'esistenza e sul calcolo delle soluzioni dei problemi al contorno, relativi all'equilibrio di un corpo elastico,” Ann. Scuola Norm. Sup. Pisa, Ser. 3,4, 35–99 (1950).

  44. 44.

    G. Fichera, Existence Theorems in Elasticity. Handbuch der Physik, Vol. Y1a/2, 347–389 (1972).

  45. 45.

    K. O. Friedrichs, “On the boundary-value problems of the theory of elasticity and Korn's inequality,” Ann. Math.,48, 441–471 (1947).

  46. 46.

    R. Furuhashi, “On Fourier's method in the mixed problems of elastodynamics,” Mem. Inst. Sci. and Technol. Meiji Univ.,11, 79–82 (1972).

  47. 47.

    G. Giraud, “Équations à intégrales principales. Étude suivie d'une application,” Ann. Sci. Ecole Norm. Super.,51 251–372 (1934).

  48. 48.

    O. D. Kellogg, Foundations of Potential Theory, Springer, Berlin (1929); F. Ungar, New York (1946).

  49. 49.

    A. Korn, “Über die Lösung des Grundproblemes der Elastizitätstheorie,” Math. Ann.,75, 497–544 (1914).

  50. 50.

    V. D. Kupradze and T. V. Burchuladze, On dynamic problems of the theory of elasticity. Trends in elasticity and thermoelasticity (Witold Nowacki Anniversary Volume). Wolters-Noordhoff (1971), pp. 137–149.

  51. 51.

    R. Leis, Zur Theorie Elastischer Schwingungen, Ber. Ges. Math. und Datenverar. Bonn, BMFT-GMD-72, 4–54 (1973).

  52. 52.

    A. E. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, New York (1927).

  53. 53.

    W. Nowacki, Thermoelasticity, Pergamon Press, Oxford (1962).

  54. 54.

    W. Nowacki, Thermoelasticity, Pergamon (1963).

  55. 55.

    H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers,” Rend. Circ. Mat. Palermo,39, 1–50 (1915).

Download references

Additional information

Translated from Itogi Nauki i Tekhniki (Sovremennye Problemy Matematiki), Vol. 7, pp. 163–294, 1975.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kupradze, V.D., Burchuladze, T.V. The dynamical problems of the theory of elasticity and thermoelasticity. J Math Sci 7, 415–500 (1977). https://doi.org/10.1007/BF01091837

Download citation

Keywords

  • Dynamical Problem