Analysis of the Quasi-static Consolidation Problem of a Compressible Porous Medium

  • Roberto SerpieriEmail author
  • Francesco Travascio
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 67)


Hereby, we present an analysis of the stress partitioning mechanism for fluid saturated poroelastic media in the transition from drained (e.g., slow deformations) to undrained (e.g., fast deformation) flow conditions. Our objective is to derive fundamental solutions for the general consolidation problem and to show how Terzaghi’s law is recovered as the limit undrained flow condition is approached. Accordingly, we present the linearized form of VMTPM in a u-p dimensionless form. Subsequently, we investigate the behavior of the poroelastic system as a function of governing dimensionless numbers for the case of a displacement controlled compression test. The results of this analysis confirm that, in the limit of undrained flow, the solutions of the consolidation problem recover Terzaghi’s law. Also, it is shown that a dimensionless parameter (\(P_{I}\)), which solely depends on mixture porosity and Poisson ratio of the solid phase, governs the consolidation of the poroelastic system.


Fluid Pressure Deborah Number Poroelastic Medium Porous Plug Solid Stress 
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.


  1. 1.
    Ateshian, G., Lai, W., Zhu, W., Mow, V.: An asymptotic solution for the contact of two biphasic cartilage layers. J. Biomech. 27(11), 1347–1360 (1994)CrossRefGoogle Scholar
  2. 2.
    De Hoog, F.R., Knight, J., Stokes, A.: An improved method for numerical inversion of laplace transforms. SIAM J. Sci. Stat. Comput. 3(3), 357–366 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ehlers, W., Markert, B.: A linear viscoelastic biphasic model for soft tissues based on the theory of porous media. J. Biomech. Eng. 123(5), 418–424 (2001)CrossRefGoogle Scholar
  4. 4.
    Marshall, R., Metzner, A.: Flow of viscoelastic fluids through porous media. Ind. Eng. Chem. Fundam. 6(3), 393–400 (1967)CrossRefGoogle Scholar
  5. 5.
    Mow, V., Kuei, S., Lai, W., Armstrong, C.: Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J. Biomech. Eng. 102(1), 73–84 (1980)CrossRefGoogle Scholar
  6. 6.
    Reiner, M.: The deborah number. Phys. Today 17(1), 62 (1964)CrossRefGoogle Scholar
  7. 7.
    Rice, J.R., Cleary, M.P.: Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. Space Phys. 14(2), 227–241 (1976)CrossRefGoogle Scholar
  8. 8.
    Runesson, K., Perić, D., Sture, S.: Effect of pore fluid compressibility on localization in elastic-plastic porous solids under undrained conditions. Int. J. Solids Struct. 33(10), 1501–1518 (1996)CrossRefzbMATHGoogle Scholar
  9. 9.
    Serpieri, R.: A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents. Transp. Porous Media 90(2), 479–508 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Serpieri, R., Rosati, L.: Formulation of a finite deformation model for the dynamic response of open cell biphasic media. J. Mech. Phys. Solids 59(4), 841–862 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Serpieri, R., Travascio, F.: General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach. Continuum Mech. Thermodyn. 28(1–2), 235–261 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Serpieri, R., Travascio, F., Asfour, S.: Fundamental solutions for a coupled formulation of porous biphasic media with compressible solid and fluid phases. In: Computational Methods for Coupled Problems in Science and Engineering V—A Conference Celebrating the 60th Birthday of Eugenio Onate, Coupled Problems 2013, pp. 1142–1153 (2013)Google Scholar
  13. 13.
    Serpieri, R., Travascio, F., Asfour, S., Rosati, L.: Variationally consistent derivation of the stress partitioning law in saturated porous media. Int. J. Solids Struct. 56–57, 235–247 (2015)CrossRefGoogle Scholar
  14. 14.
    Travascio, F., Asfour, S., Serpieri, R., Rosati, L.: Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach. Math. Mech. Solids (2015). doi: 10.1177/1081286515616049
  15. 15.
    Travascio, F., Serpieri, R., Asfour, S.: Articular cartilage biomechanics modeled via an intrinsically compressible biphasic model: Implications and deviations from an incompressible biphasic approach. In: ASME 2013 Summer Bioengineering Conference, pp. V01BT55A004–V01BT55A004. American Society of Mechanical Engineers (2013)Google Scholar
  16. 16.
    Verruijt, A.: Theory and problems of poroelasticity. Delft University of Technology (2013)Google Scholar
  17. 17.
    Yoon, Y.J., Cowin, S.C.: The elastic moduli estimation of the solid-water mixture. Int. J. Solids Struct. 46(3), 527–533 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Zhang, H., Schrefler, B.: Uniqueness and localization analysis of elastic-plastic saturated porous media. Int. J. Numer. Anal. Methods Geomech. 25(1), 29–48 (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Zienkiewicz, O.C., Chan, A., Pastor, M., Schrefler, B., Shiomi, T.: Computational Geomechanics. Wiley, Chichester (1999)zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Dipartimento di IngegneriaUniversità degli Studi del SannioBeneventoItaly
  2. 2.University of MiamiBiomechanics Research LaboratoryCoral GablesUSA

Personalised recommendations