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Porous Media pp 199-225 | Cite as

Coupling between the evolution of a deformable porous medium and the motion of fluids in the connected porosity

  • Renato Lancellotta
Chapter

Abstract

This contribution is aimed at presenting a consistent mathematical description of porous media. Basic assumptions concerned with the representation of a porous medium as a continuum are discussed in detail, making a clear distinction between two geometric scales: at microscale each constituent occupies a specific domain, while at macroscale soil and fluid particles are superimposed at the same geometric point. At macroscale, a unified Lagrangean formulation is given, by assuming the soil skeleton as a material reference volume and by referring the fluid motion to the soil skeleton Finally, two problems are analysed, both of relevant interest in soil mechanics, i. e. the propagation of body waves in undrained conditions and the consolidation of a soft clay stratum.

Keywords

Porous Medium Pore Pressure Effective Stress Representative Elementary Volume Shear Wave Velocity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atkin, R. J., Craine, R. E.: Continuum theories of mixtures: basic theory and historical development. Q. J. Mech. Appl. Math. 29 (1976), 209–244.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bear, J.: Dynamics of fluids in porous media. Elsevier, 1972.Google Scholar
  3. 3.
    Bear, J., Bachmat, Y.: Introduction to modeling of transport phenomena in porous media. Kluwer, 1991.Google Scholar
  4. 4.
    Bedford, A., Drumheller, D. S.: Theory of immiscible and structured mixtures. Int. J. Eng. Sci. 21 (1983), 863–960.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Biot, M. A.: General theory of three-dimensional consolidation. J. Appl. Phy. 12 (1941), 155–165.MATHCrossRefGoogle Scholar
  6. 6.
    Biot, M. A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. Journal of the Acoustica Society of America 28 (2) (1956), 168–191.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Biot, M. A., Willisi, D. G.: The elastic coefficients of theory of consolidation. J. Appl. Mech. (1957), 594–601.Google Scholar
  8. 8.
    Biot, M. A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 23 (1962), 1482–1498.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Biot, M. A.: Variational Lagrangean thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion. Int. J. Solids Structures 13 (1977), 579–597.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    de Boer, R., Ehlers, W.: Theorie der Mehrkomponentenkontinua mit Anwendung auf bodenmechanische Probleme. Forschungsberichte aus dem Fachbereich Bauwesen, Heft 40, Universität GH Essen 1986.Google Scholar
  11. 11.
    de Boer, R.: Theory of Porous Media. Springer-Verlag, Berlin 2000.MATHCrossRefGoogle Scholar
  12. 12.
    Bourgeois, E., Dormieux, L.: Consolidation of a nonlinear poroelastic layer in finite deformations. Eur. J. Mech. A/Solids 15 (4) (1996), 575–598.MATHGoogle Scholar
  13. 13.
    Bowen, R. M.: Theory of mixtures. In Eringen, A. C. (ed.): Continuum physics III/1, Academic Press, 1976, p. 317.Google Scholar
  14. 14.
    Bowen, R. M.: Incompressible porous media models by the use of the theory of mixtures. Int. J. Eng. Sc. 18 (1980), 1129–1148.MATHCrossRefGoogle Scholar
  15. 15.
    Bowen, R. M.: Compressible porous media models by the use of the theory of mixtures. Int. J. Eng. Sc. 20 (1982), 697–735.MATHCrossRefGoogle Scholar
  16. 16.
    Carter, J. P., Booker, J. R., Small, J. C.: The analysis of finite elasto-plastic consolidation. Int. Journal for Num. An. Methods in Geomechanics 3 (1979), 107–129.MATHCrossRefGoogle Scholar
  17. 17.
    Coleman, B. D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rat. Mech. and Anal. 13 (1963), 167–178.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Coussy O.: Thermodynamics of saturated porous solids in finite deformations. Eur. J. Mech. A/Solids 8 (1989), 1–14.MathSciNetMATHGoogle Scholar
  19. 19.
    Coussy O.: Mechanics of porous continua. John Wiley and Sons, 1995.Google Scholar
  20. 20.
    Coussy, O., Dormieux, L., Detournay, E.: From mixture theory to Biot’s approach for porous media. Int. J. Solids Structures 35 (34–35) (1998), 4619–4635.MATHCrossRefGoogle Scholar
  21. 21.
    Drugan, W. J., Willis, J. R.: A micromechanics-based nonlocal constitutive equation and estimates of representative element size for elastic composites. J. Mech. Phys. Solids 44 (1996), 497–524.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Ehlers, W.: Constitutive equations for granular materials in geomechanical context. In Hutter, K. (ed.): Continuum mechanics in environmental sciences and geophysics, CISM Courses and Lectures No. 337, Springer-Verlag, 1993, pp. 313–402.Google Scholar
  23. 23.
    Gibson, R. E., England, G. L., Hussey, M. J. L.: The theory of one dimensional consolidation of saturated clays. Géotechnique 17 (1967), 261–273.CrossRefGoogle Scholar
  24. 24.
    Gu, W. Y., Lai, W. M., Mow, V. C.: Transport of multi-electrolytes in charged hydrated biological soft tissues. In de Boer, R. (ed.): Porous media: Theory and Experiments, Kluwer Academic Press, 1999, pp. 143–157.Google Scholar
  25. 25.
    Hutter, K.: The foundations of thermodynamics, its basic postulates and implications. A review of modern thermodynamics. Acta Mechanica 27 (1977), 1–54.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lancellotta, R.: Geotechnical Engineering. Balkema, 1993.Google Scholar
  27. 27.
    Lancellotta, R., Preziosi, L.: A general nonlinear mathematical model for soil consolidation problems. Journal of Engineering Sciences 35 (1997), 1045–1063.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Lewis, R. W., Schrefler, B. A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. 2nd ed., John Wiley and Sons, Chichester 1998.Google Scholar
  29. 29.
    Liu, I-S.: Method of Lagrange multipliers for exploitation of entropy principle. Arch. Rat. Mech. and Anal. 46 (1972), 131–148.MATHGoogle Scholar
  30. 30.
    Lorentz, H. A.: The theory of electrons. Teubner, Leipzig 1909, reprint Dover 1952.Google Scholar
  31. 31.
    Markov, K. Z.: Elementary micromechanics of heterogeneous media. In Markov, Preziosi (eds.): Heterogeneous Media, Birkhäuser Verlag, 2000, pp. 1–162.Google Scholar
  32. 32.
    Müller, I.: A thermodynamic theory of mixture of fluids. Arch. Rat. Mech. Anal. 28 (1968), 1–39.MATHCrossRefGoogle Scholar
  33. 33.
    Müller, I.: Die Kältefunktion, eine universelle Funktion in der Thermodynamik viskoser warmeleitender Flüssigkeiten. Arch. Rat. Mech. and Anal. 40 (1971), 1–36.MATHCrossRefGoogle Scholar
  34. 34.
    MURST: Analisi geotecnica della vulnerabilità sismica dei monumenti storici. Unità operativa di Torino: Caratterizzazione in sito mediante misure della propagazione di onde sismiche superficiali, 2000.Google Scholar
  35. 35.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall properties of heterogeneous solids. Elsevier, 1993.Google Scholar
  36. 36.
    Neuman, S. P.: Theoretical derivation of Darcy’s law. Acta Mechanica 25 (1977), 153–170.MATHCrossRefGoogle Scholar
  37. 37.
    Pane, V.: Sedimentation and consolidation of clays. Ph. D. Thesis, University of Colorado, Boulder 1985.Google Scholar
  38. 38.
    Rajagopal, K. R., Tao, L.: Mechanics of Mixtures. World Scientific, 1995.Google Scholar
  39. 39.
    Slattery, J. C.: Flow of viscoelastic fluids through porous media. Am. Inst. Chem. Eng. J. 13 (1967), 1066.Google Scholar
  40. 40.
    von Terzaghi, K.: Die Berechnung der Durchlässigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. Sitz. Akad. Wissen. Wien, Math.-Naturw. Kl. Abt. IIa 132 (1923), 125–138.Google Scholar
  41. 41.
    von Terzaghi, K.: Erdbaumechanik auf bodenphysikalischer Grundlage. Deuticke, Leipzig 1960. See also: From Theory to Practice, John Wiley and Sons, 1925, 146–148.Google Scholar
  42. 42.
    Truesdell, C.: Thermodynamics of diffusion. In Truesdell, C. (ed.): Rational Thermodynamics, Springer-Verlag, 1984, pp. 219–236.Google Scholar
  43. 43.
    Truesdell, C., Toupin, R. A.: The classical field theories. In Flügge, S. (ed.): Handbuch der Physik, III/1, Springer-Verlag, 1960, pp. 226–902.Google Scholar
  44. 44.
    Wang, Y., Hutter, K.: Comparison of two entropy principles and their applications in granular flows with/without fluid. Arch. Mech. (1999), 1–18.Google Scholar
  45. 45.
    Whitaker, S.: The equation of motion in porous media. Chem. Eng. Sc. 21 (1966), 291.CrossRefGoogle Scholar
  46. 46.
    Wilmanski, K.: Porous media at finite strains: The new model with balance equation for porosity. Arch. Mech. 48 (4) (1996), 591–628.MATHGoogle Scholar
  47. 47.
    Wilmanski, K.: Thermomechanics of Continua. Springer-Verlag, Berlin 1998.MATHCrossRefGoogle Scholar
  48. 48.
    Wilmanski, K.: Mathematical theory of porous media - Lectures notes. WIAS preprint 602, Berlin 2000.Google Scholar
  49. 49.
    Wilmanski, K.: Note on the notion of incompressibility in theories of porous and granular materials. WIAS preprint 465, Berlin 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Renato Lancellotta
    • 1
  1. 1.Department of Structural and Geotechnical EngineeringTechnical University of Torino (Politecnico)TorinoItaly

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