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Fundamental Solutions in Elasticity

  • Merab Svanadze
Chapter
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 51)

Abstract

This chapter is concerned with the fundamental solutions of the systems of equations in the linear theory of elasticity for materials with quadruple porosity.

References

  1. 18.
    Augustin, M.A.: A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs. Springer, Cham (2015)CrossRefGoogle Scholar
  2. 27.
    Basheleishvili, M.O.: On fundamental solutions of differential equations of an anisotropic elastic body (Russian). Bull. Acad. Sci. Georgian SSR 19, 393–400 (1957)Google Scholar
  3. 66.
    Burridge, R., Vargas, C.A.: The fundamental solution in dynamic poroelasticity. Geophys. J. R. Astr. Soc. 58, 61–90 (1979)CrossRefGoogle Scholar
  4. 101.
    Cleary, M.P.: Fundamental solutions for a fluid-saturated porous solid. Int. J. Solids Struct. 13, 785–806 (1977)MathSciNetCrossRefGoogle Scholar
  5. 119.
    d’Alembert, J.R.: Recherches sur la courbe que forme une corde tendue mise en vibration. Mém. Acad. Roy. Sci. Belles-Lett. de Berlin 3, 214–219 (1747/1749)Google Scholar
  6. 129.
    de Boer, R., Svanadze, M.: Fundamental solution of the system of equations of steady oscillations in the theory of fluid-saturated porous media. Transp. Porous Media 56, 39–50 (2004)MathSciNetCrossRefGoogle Scholar
  7. 134.
    Dragos, L.: Fundamental solutions in micropolar elasticity. Int. J. Eng. Sci. 22, 265–275 (1984)MathSciNetCrossRefGoogle Scholar
  8. 136.
    Ehrenpreis, L.: Solution of some problems of division. Part I. Division by a polynomial of derivation. Am. J. Math. 76, 883–903 (1954)zbMATHGoogle Scholar
  9. 143.
    Fredholm, I.: Sur les équations de l’équilibre d’un corps solide élastique. Acta Math. 23, 1–42 (1900)MathSciNetCrossRefGoogle Scholar
  10. 161.
    Günther, N.M.: Potential Theory and its Applications to Basic Problems of Mathematical Physics. Ungar, New York (1967)Google Scholar
  11. 162.
    Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2, pp. 1–296. Springer, Berlin (1972)Google Scholar
  12. 165.
    Hetnarski, R.B., Ignaczak, J.: Mathematical Theory of Elasticity, 2nd edn. Taylor and Francis, Abingdon (2011)zbMATHGoogle Scholar
  13. 167.
    Hörmander, L.: Local and global properties of fundamental solutions. Math. Scand. 5, 27–39 (1957)MathSciNetCrossRefGoogle Scholar
  14. 168.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols. I–IV. Springer, Berlin (1983/1985)Google Scholar
  15. 194.
    Kaynia, A.M., Banerjee, P.K.: Fundamental solutions of Biot’s equations of dynamic poroelasticity. Int. J. Eng. Sci. 31, 817–830 (1993)CrossRefGoogle Scholar
  16. 215.
    Kupradze, V.D.: Boundary Value Problems of the Oscillation Theory and Integral Equations. M.-L., State Publishing House of technical and theoretical Literature (1950) (Russian). German translation: Kupradze, V.D.: Randwertaufgaben der Schwingungstheorie und Integralgleichongen. Veb Deutscher verlag der Wissenschaften, Berlin (1956)Google Scholar
  17. 217.
    Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program for Scientific Translations, Jerusalem (1965)Google Scholar
  18. 218.
    Kupradze, V.D., Burchuladze, T.V.: Boundary value problems of thermoelasticity. Diff. Equ. 5, 3–43 (1969)MathSciNetGoogle Scholar
  19. 219.
    Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979)Google Scholar
  20. 220.
    Kythe, P.K.: Fundamental Solutions for Differential Operators and Applications. Birkhäuser, Boston (1996)CrossRefGoogle Scholar
  21. 221.
    Laplace, P.S.: Mémoire sur la théorie de l’anneau de Saturne. Mém. Acad. Roy. Sci. Paris 201–234 (1787/1789)Google Scholar
  22. 238.
    Malgrange, B.: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier 6, 271–355 (1955/1956)MathSciNetCrossRefGoogle Scholar
  23. 263.
    Nowacki, W.: Thermoelasticity. Pergamon Press, Oxford (1962)zbMATHGoogle Scholar
  24. 268.
    Ortner, N., Wagner, P.: On the fundamental solution of the operator of dynamic linear thermoelasticity. J. Math. Anal. Appl. 170, 524–550 (1992)MathSciNetCrossRefGoogle Scholar
  25. 269.
    Ortner, N., Wagner, P.: Fundamental Solutions of Linear Partial Differential Operators: Theory and Practice. Springer, Basel (2015)CrossRefGoogle Scholar
  26. 276.
    Poisson, S.D.: M’em. Acad. Sci. Paris 3, 131–176 (1818)Google Scholar
  27. 293.
    Sandru, N.: On some problems of the linear theory of asymmetric elasticity. Int. J. Eng. Sci. 4, 81–96 (1966)CrossRefGoogle Scholar
  28. 316.
    Somigliana, C.: Sopra l’equilibrio di un corpo elastico isotropo. Nuovo Cimento, ser 3, 17–20 (1885)Google Scholar
  29. 331.
    Svanadze, M.: Fundamental solution in the theory of consolidation with double porosity. J. Mech. Behav. Mater. 16(1–2), 123–130 (2005)Google Scholar
  30. 336.
    Svanadze, M.: Fundamental solution in the linear theory of consolidation for elastic solids with double porosity. J. Math. Sci. 195, 258–268 (2013)MathSciNetCrossRefGoogle Scholar
  31. 345.
    Svanadze, M.: Fundamental solutions in the theory of elasticity for triple porosity materials. Meccanica 51, 1825–1837 (2016)MathSciNetCrossRefGoogle Scholar
  32. 357.
    Svanadze, M., de Boer, R.: On the representations of solutions in the theory of fluid-saturated porous media. Quart. J. Mech. Appl. Math. 58, 551–562 (2005)MathSciNetCrossRefGoogle Scholar
  33. 358.
    Svanadze, M., De Cicco, S.: Fundamental solutions in the full coupled linear theory of elasticity for solid with double porosity. Arch. Mech. 65, 367–390 (2013)MathSciNetGoogle Scholar
  34. 368.
    Svanadze, M.M.: Fundamental solution and uniqueness theorems in the linear theory of thermoviscoelasticity for solids with double porosity. J. Therm. Stresses 40, 1339–1352 (2017)CrossRefGoogle Scholar
  35. 370.
    Svanadze, M.M.: Fundamental solutions and uniqueness theorems in the theory of viscoelasticity for materials with double porosity. Trans. A. Razmadze Math. Inst. 172, 276–292 (2018)MathSciNetCrossRefGoogle Scholar
  36. 371.
    Thomson, W. (Lord Kelvin): On the equations of equilibrium of an elastic solid. Camb. Dubl. Math. J. 3, 87–89 (1848)Google Scholar
  37. 391.
    Volterra, V.: Sur les vibrations des corps élastiques isotropes. Acta Math. 18, 161–232 (1894)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Merab Svanadze
    • 1
  1. 1.Ilia State UniversityTbilisiGeorgia

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