Journal of Elasticity

, Volume 74, Issue 2, pp 135–145 | Cite as

Some Properties of the Boundary Value Problem of Linear Elasticity in Terms of Stresses

  • V.A. Kucher
  • X. Markenscoff
  • M.V. Paukshto
Article

Abstract

The relation between the classical formulation of linear elastic problems in displacements and the stress formulation proposed by Pobedria is studied. It is shown that if the Navier and Pobedria differential operators are elliptic then corresponding boundary value problems are equivalent. The values of parameters for which Pobedria's boundary value problem has the Fredholm property are found. The homogeneous Pobedria's system is considered as a spectral problem with Poisson's ratio as a spectral parameter. The points of the essential spectrum are found and classified. The example of solving Pobedria's system for the Lamé problem for a spherical shell is presented.

spectral theory elliptic p.d.e.s Beltrami–Michell compatibility conditions 

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References

  1. 1.
    B.E. Pobedria, A new formulation of the problem of the mechanics of a deformable solid in stresses (in Russian). Dokl. Akad. Nauk SSSR 253(2) (1980) 295–297. English translation in Soviet Math. Dokl. 22(1) (1980) 88-91.Google Scholar
  2. 2.
    B.E. Pobedria, A problem in stresses for an anisotropic medium (in Russian). Prikl.Mat. Mekh. 58(1) (1994) 77–85. English translation in J. Appl. Math. Mech. 58(1) (1994) 81-89.Google Scholar
  3. 3.
    B.E. Pobedria, Some general theorems of the mechanics of a deformable solid (in Russian). Prikl. Mat. Mekh. 43(3) (1979) 531–541.Google Scholar
  4. 4.
    E. Cosserat and F. Cosserat, Sur les équations de la théorie de l'élasticité. C. R. Acad. Sci. Paris 126 (1898) 1089–1091.Google Scholar
  5. 5.
    E. Cosserat and F. Cosserat, Sur la solution des équations de l'élasticité dans le cas ou les valeurs des inconnues à la frontière sont données. C. R. Acad. Sci. Paris 133 (1901) 145–147.Google Scholar
  6. 6.
    S.G. Mikhlin, The spectrum of operator bundle of elasticity theory (in Russian). Uspekhi Mat. Nauk 28 (1973) 43–82. English translation in Russian Math. Surveys 28 (1973) 45-83.Google Scholar
  7. 7.
    J.L. Ericksen, On the Dirichlet problem for linear differential equations. Proc. Amer. Math. Soc. 8 (1957) 521–522.Google Scholar
  8. 8.
    H.C. Simpson and S.J. Spector, On failure of the complementing condition and nonuniqueness in linear elastostatics. J. Elasticity 15 (1985) 229–231.Google Scholar
  9. 9.
    S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17 (1964) 35–92.Google Scholar
  10. 10.
    I.S. Sokolnikoff, Mathematical Theory of Elasticity. McGraw-Hill, New York (1956).Google Scholar
  11. 11.
    R.J. Knops and L.E. Payne, Uniqueness Theorems in Linear Elasticity. Springer, New York (1971).Google Scholar
  12. 12.
    A.N. Kozhevnikov, The basic boundary value problems of the static elasticity theory. Math. Zeitschrift. 213 (1993) 241–274.Google Scholar
  13. 13.
    S.P. Timoshenko and J.N. Goodier, Theory of Elasticity. McGraw-Hill, New York (1970).Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • V.A. Kucher
    • 1
  • X. Markenscoff
    • 1
  • M.V. Paukshto
    • 2
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of California, San DiegoLa JollaU.S.A
  2. 2.Department of Mathematics and MechanicsSaint Petersburg State UniversitySt. PetersburgRussia

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