Skip to main content
SpringerLink
Log in

Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

We study infinitesimal deformations of a porous linear elastic body saturated with an inviscid fluid and subjected to conservative surface tractions. The gradient of the mass density of the solid phase is also taken as an independent kinematic variable and the corresponding higher-order stresses are considered. Balance laws and constitutive relations for finite deformations are reduced to those for infinitesimal deformations, and expressions for partial surface tractions acting on the solid and the fluid phases are derived. A boundary-value problem for a long hollow porous solid cylinder filled with an ideal fluid is solved, and the stability of the stressed reference configuration with respect to variations in the values of the coefficient coupling deformations of the two phases is investigated. An example of the problem studied is a cylindrical cavity leached out in salt formations for storing hydrocarbons.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G.E. Andrews, R. Askey and R. Roy, Special functions. In: Encyclopedia of Mathematics and Its Applications. Cambridge Univ. Press., Cambridge (1999).

    Google Scholar 

  2. V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, translated by J. Szucs; English translation edited by M. Levi. Springer, New York (1983).

    MATH  Google Scholar 

  3. F.M. Arscott, Heun's equation. In: A. Ronveaux (ed.), Heun's Differential Equations. Oxford Univ. Press, Oxford (1995) Part A.

    Google Scholar 

  4. R.C. Batra, On nonclassical boundary conditions. Arch. Rational Mech. Anal. 48 (1972) 163–191.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. R.C. Batra, Thermodynamics of non-simple elastic materials. J. Elasticity 6 (1976) 451–456.

    Article  MATH  ADS  Google Scholar 

  6. R.C. Batra, The initiation and growth of, and the interaction among adiabatic shear bands in simple and dipolar materials. Internat. J. Plasticity 3 (1987) 75–89.

    Article  Google Scholar 

  7. R.C. Batra and L. Chen, Shear band spacing in gradient-dependent thermoviscoplastic materials, Comput. Mech. 23 (1999) 8–19.

    Article  MATH  Google Scholar 

  8. R.C. Batra and J. Hwang, Dynamic shear band development in dipolar thermoviscoplastic materials. Comput. Mech. 12 (1994) 354–369.

    Google Scholar 

  9. R.C. Batra and C.H. Kim, Adiabatic shear banding in elastic-viscoplastic nonpolar and dipolar materials. Internat. J. Plasticity 6 (1990) 127–141.

    Article  Google Scholar 

  10. P. Bérest, J. Bergues, B. Brouard, J.G. Durup and B. Guerber, A salt cavern abandonment test. Internat. J. Rock Mech. Min. 38 (2001) 357–368.

    Article  Google Scholar 

  11. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155–164.

    Article  MATH  ADS  Google Scholar 

  12. P. Blanchard and E. Bruning, Variational Methods in Mathematical Physics (a Unified Approach). Springer, Heidelberg (1992).

    MATH  Google Scholar 

  13. R.M. Bowen, Theory of mixtures. In: Continuum Physics, Vol. III (1976) pp. 2–127.

    Google Scholar 

  14. J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system. J. Chem. Phys. 31 (1959) 688–699.

    Article  ADS  Google Scholar 

  15. P. Casal, La théorie du second gradient et la capillarité. C. R. Acad. Sci. Paris Sér. A 274 (1972) 1571–1574.

    MATH  MathSciNet  Google Scholar 

  16. P. Casal and H. Gouin, Equations du movement des fluides thermocapillaires. C. R. Acad. Sci. Paris Sér. II 306 (1988) 99–104.

    MATH  MathSciNet  Google Scholar 

  17. P. Cosenza, M. Ghoreychi, B. Bazargan-Sabet and G. de Marsily, In situ rock salt permeability measurement for long term safety assessment of storage. Internat. J. RockMech. Min. 36 (1999) 509–526.

    Article  Google Scholar 

  18. R. de Boer, Theory of Porous Media. Springer, Berlin (2000).

    MATH  Google Scholar 

  19. F. dell'Isola, M. Guarascio and K. Hutter, A variational approach for the deformation of a saturated porous solid. A second gradient theory extending Terzaghi's effective stress principle. Archive Appl. Mech. 70 (2000) 323–337.

    Article  MATH  Google Scholar 

  20. F. dell'Isola and P. Seppecher, Edge contact forces and quasibalanced power. Meccanica 32 (1997) 33–52.

    Article  MATH  MathSciNet  Google Scholar 

  21. W. Elhers, Toward finite theories of liquid-saturated elasto-plastic porous media. Internat. J. Plasticity 7 (1991) 433–475.

    Article  Google Scholar 

  22. P. Fillunger, Erdbaumechanik. Selbstverlag des Verfassers, Wien (1936).

    Google Scholar 

  23. P. Germain, La méthode des puissances virtuelles en mécanique des milieux continus. J. Mécanique 12(2) (1973) 235–274.

    MATH  MathSciNet  Google Scholar 

  24. P. Germain, The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J. Appl. Mech. 25(3) (1973) 556–575.

    Article  MATH  MathSciNet  Google Scholar 

  25. M.A. Goodman and S.C. Cowin, A continuum theory for granular materials. Arch. Rational Mech. Anal. 44 (1972) 249–266.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. H. Gouin, Tension superficielle dynamique et effet Marangoni pour les interfaces liquidevapeur en théorie de la capillarité interne. C. R. Acad. Sci. Paris Sér. II 303(1) (1986).

  27. S. Krishnaswamy and R.C. Batra, A thermomechanical theory of solid-fluid mixtures. Math. Mech. Solids 2 (1997) 143–151.

    MATH  Google Scholar 

  28. G.A. Maugin, The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech. 35 (1980) 1–70.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. L.W. Morland, A simple constitutive theory for a fluid saturated porous solid. J. Geoph. Res. 77 (1972) 890–900.

    Article  ADS  Google Scholar 

  30. I. Müller, A thermodynamic theory of mixtures of fluids. Arch. Rational Mech. Anal. 28 (1968) 1–39.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. I. Müller, Thermodynamics. Pittman, Boston (1985).

    MATH  Google Scholar 

  32. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Noordorf, Groningen (1953).

  33. K.R. Rajagopal and L. Tao, Mechanics of Mixtures. World Scientific, Singapore (1995).

    MATH  Google Scholar 

  34. G. Sciarra, F. dell'Isola and K. Hutter, A solid-fluid mixture model allowing for solid dilatation under external pressure. Continuum Mech. Thermodyn. 13 (2001) 287–306.

    MATH  MathSciNet  ADS  Google Scholar 

  35. G. Sciarra, K. Hutter and G.A. Maugin, A variational approach to a micro-structured theory of solid-fluid mixtures, in preparation.

  36. P.Seppecher, Equilibrium of a Cahn-Hilliard fluid on a wall: Influence of the wetting properties of the fluid upon the stability of a thin liquid film. European J. Mech. B Fluids 12(1) (1993) 69–84.

    MATH  MathSciNet  Google Scholar 

  37. S.M.R.I. Solution, Mining Research Institute, Technical class guidelines for safety assessment of salt caverns, Fall Meeting, Rome, Italy (1998).

  38. B. Svendsen and K. Hutter, On the thermodynamics of a mixture of isotropic materials with constraints. Internat. J. Engrg. Sci. 33 (1995) 2021–2054.

    Article  MATH  MathSciNet  Google Scholar 

  39. R. Thom, Stabilité structurelle et morphogénèse: Essai d'une Théorie Générale des Modèles. Benjamin, New York (1972).

    Google Scholar 

  40. C.A. Truesdell, Sulle basi della termomeccanica. Lincei Rend. Sc. Fis.Mat. Nat. XXII (Gennaio 1957).

  41. C.A. Truesdell, Thermodynamics of diffusion, Lecture 5. In: C.A. Truesdell (ed.), Rational Thermodynamics. Springer, Berlin (1984) pp. 216–219.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dip. Ingegneria Strutturale e Geotecnica, Universita degli Studi di Roma “La Sapienza”, via Eudossiana 18, 00184, Roma, Italy

    F. dell'Isola

  2. Dip. Ingegneria Chimica, dei Materiali, delle Materie Prime e Metallurgia, Università degli Studi di Roma “La Sapienza”, via Eudossiana 18, 00184, Roma, Italy

    G. Sciarra

  3. Department of Engineering Science and Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, U.S.A.

    R.C. Batra

Authors
  1. F. dell'Isola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Sciarra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R.C. Batra
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

dell'Isola, F., Sciarra, G. & Batra, R. Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid. Journal of Elasticity 72, 99–120 (2003). https://doi.org/10.1023/B:ELAS.0000018765.68432.bb

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:ELAS.0000018765.68432.bb

Search

Navigation