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Advances in Mathematical Physics
Volume 2016, Article ID 4792148, 9 pages
http://dx.doi.org/10.1155/2016/4792148
Research Article

Elastic Equilibrium of Porous Cosserat Media with Double Porosity

Ilia Vekua Institute of Applied Mathematics of Ivane Javakhishvili Tbilisi State University, 2 University Street, 0186 Tbilisi, Georgia

Received 11 May 2016; Accepted 30 June 2016

Academic Editor: John D. Clayton

Copyright © 2016 Roman Janjgava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. K. Wilson and E. C. Aifantis, “On the theory of consolidation with double porosity,” International Journal of Engineering Science, vol. 20, no. 9, pp. 1009–1035, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. D. E. Beskos and E. C. Aifantis, “On the theory of consolidation with double porosity-II,” International Journal of Engineering Science, vol. 24, no. 11, pp. 1697–1716, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M. Y. Khaled, D. E. Beskos, and E. C. Aifantis, “On the theory of consolidation with double porosity—III A finite element formulation,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 8, no. 2, pp. 101–123, 1984. View at Publisher · View at Google Scholar · View at Scopus
  4. G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, “Basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata),” Journal of Applied Mathematics and Mechanics, vol. 24, no. 5, pp. 1286–1303, 1960. View at Google Scholar
  5. M. A. Biot, “General theory of three-dimensional consolidation,” Journal of Applied Physics, vol. 12, no. 2, pp. 155–164, 1941. View at Publisher · View at Google Scholar · View at Scopus
  6. R. de Boer, Theory of Porous Media, Highlights in the Historical Development and Current State, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  7. N. Khalili and S. Valliappan, “Unified theory of flow and deformation in double porous media,” European Journal of Mechanics—A/Solids, vol. 15, no. 2, pp. 321–336, 1996. View at Google Scholar · View at Scopus
  8. N. Khalili and A. P. S. Selvadurai, “On the constitutive modelling of thermo-hydro-mechanical coupling in elastic media with double porosity,” Elsevier Geo-Engineering Book Series, vol. 2, pp. 559–564, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. J. G. Berryman and H. F. Wang, “The elastic coefficients of double-porosity models for fluid transport in jointed rock,” Journal of Geophysical Research, vol. 100, no. 12, pp. 24611–24627, 1995. View at Publisher · View at Google Scholar · View at Scopus
  10. J. G. Berryman and H. F. Wang, “Elastic wave propagation and attenuation in a double-porosity dual-permeability medium,” International Journal of Rock Mechanics and Mining Sciences, vol. 37, no. 1-2, pp. 63–78, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Svanadze, “Fundamental solution in the theory of consolidation with double porosity,” Journal of the Mechanical Behavior of Materials, vol. 16, no. 1-2, pp. 123–130, 2005. View at Publisher · View at Google Scholar
  12. I. Tsagareli and M. M. Svanadze, “Explicit solution of the boundary value problems of the theory of elasticity for solids with double porosity,” Proceedings in Applied Mathematics and Mechanics, vol. 10, no. 1, pp. 337–338, 2010. View at Publisher · View at Google Scholar
  13. M. Svanadze and S. De Cicco, “Fundamental solutions in the full coupled theory of elasticity for solids with double porosity,” Archives of Mechanics, vol. 65, no. 5, pp. 367–390, 2013. View at Google Scholar · View at MathSciNet
  14. M. Svanadze and A. Scalia, “Mathematical problems in the coupled linear theory of bone poroelasticity,” Computers & Mathematics with Applications, vol. 66, no. 9, pp. 1554–1566, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. I. Tsagareli and L. Bitsadze, “Explicit solution of one boundary value problem in the full coupled theory of elasticity for solids with double porosity,” Acta Mechanica, vol. 226, no. 5, pp. 1409–1418, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. E. Cosserat and F. Cosserat, Theorie des Corps Deformables, Hermann, Paris, France, 1909.
  17. C. Truesdell and R. A. Toupin, “The classical field theories,” in Handbuch der Physik, S. Flügge, Ed., vol. 1–3, Springer, Berlin, Germany, 1960. View at Google Scholar
  18. G. Grioli, Elasticita Asimetria, vol. 50 of Annali di Matematica Pura ed Applicata, 1960.
  19. E. V. Kuvshinskii and E. L. Aero, “Continuum theory of asymmetric elasticity. Equilibrium of an isotropic body,” Fizika Tverdogo Tela, vol. 6, no. 9, pp. 2689–2699, 1964 (Russian). View at Google Scholar
  20. R. D. Mindlin, “Influence of couple-stresses on stress concentrations,” Experimental Mechanics, vol. 3, no. 1, pp. 1–7, 1963. View at Publisher · View at Google Scholar
  21. V. A. Palmov, “Fundamental equations of the theory of asymmetric elasticity,” Journal of Applied Mathematics and Mechanics, vol. 28, no. 3, pp. 496–505, 1964 (Russian). View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. A. C. Eringen and E. S. Suhubi, “Nonlinear theory of simple micro-elastic solids-I,” International Journal of Engineering Science, vol. 2, no. 2, pp. 189–203, 1964. View at Publisher · View at Google Scholar · View at Scopus
  23. W. Novacki, “Couple stresses in the theoty of thermoelasticity,” Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 14, article 8, 1966. View at Google Scholar
  24. A. E. Green and P. M. Naghdi, “The linear theory of an elastic Cosserat plate,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 63, no. 2, pp. 537–550, 1967. View at Publisher · View at Google Scholar
  25. W. Novacki, “On the completeness of stress functions in asymmetric elasticity,” Bulletin of the Polish Academy of Sciences: Technical Sciences, vol. 14, no. 7, 1968. View at 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, North-Holland Publishing, Amsterdam, The Netherlands, 1979. View at MathSciNet
  27. W. Novacki, Theory of Asymmetric Elasticity, Polish Scientific, Warszawa, Poland, 1986.
  28. R. D. Gauthier and W. E. Jahsman, “A quest for micropolar elastic constants,” Journal of Applied Mechanics, vol. 42, no. 2, pp. 369–374, 1975. View at Publisher · View at Google Scholar · View at Scopus
  29. É. A. Bulanov, “On the moment theory of elasticity. Plane deformation. Part 3,” Strength of Materials, vol. 30, no. 5, pp. 516–521, 1998. View at Publisher · View at Google Scholar · View at Scopus
  30. N. Khomasuridze, “Some problems of thermoelastic equilibrium of a rectangular parallelepiped in terms of asymmetric elasticity,” Georgian Mathematical Journal, vol. 8, no. 4, pp. 767–784, 2001. View at Google Scholar · View at MathSciNet
  31. М. А. Kulesh, V. P. Мatveenko, and I. N. Shardakov, “Construction and analysis of exact analytical solution of kirsch problem within the cosserat continuum and pseudo-continuum,” Journal of Applied Mechanics and Technical Physics, vol. 42, no. 4, pp. 145–154, 2001 (Russian). View at Google Scholar
  32. E. Providas and M. A. Kattis, “Finite element method in plane Cosserat elasticity,” Computers and Structures, vol. 80, no. 27–30, pp. 2059–2069, 2002. View at Publisher · View at Google Scholar · View at Scopus
  33. M. A. Kulesh, V. P. Matveenko, and I. N. Shardakov, “Parametric analysis of analytical solutions to one- and two-dimensional problems in couple-stress theory of elasticity,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 83, no. 4, pp. 238–248, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. R. Janjgava, “The approximate solution of some plane boundary value problems of the moment theory of elasticity,” Advances in Mathematical Physics, vol. 2016, Article ID 3845362, 12 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  35. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, The Netherlands, 1953. View at MathSciNet
  36. I. N. Vekua, Shell Theory: General Methods of Construction, Pitman Advanced Publishing Program, Boston, Mass, USA, 1985. View at MathSciNet
  37. T. V. Meunargia, Development of a Method of I. N. Vekua for Problems of the Three-Dimensional Moment Theory Elasticity, TSU Tbilisi, 1987 (Russian).
  38. R. Janjgava, “Derivation of a two-dimensional equation for shallow shells by means of the method of I. Vekua in the case of linear theory of elastic mixtures,” Journal of Mathematical Sciences, vol. 157, no. 1, pp. 70–78, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. R. Janjgava and M. Narmania, “The solution of some two-dimensional problems of thermoelasticity taking into account the microtemperature,” Journal of Thermal Stresses, vol. 39, no. 1, pp. 57–64, 2016. View at Publisher · View at Google Scholar · View at Scopus