The Linear Isotropic Variational Theory and the Recovery of Biot’s Equations

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


In this chapter, the general framework presented in Chap.  2 is specialized to address linear isotropic two-phase poroelasticity. The elastic moduli of the resulting isotropic theory are derived with the special forms achieved by the governing PDEs for hyperbolic and parabolic problems. Next, the hyperbolic system is deployed to analyze the propagation of purely elastic waves. The chapter is concluded with a section dedicated to a comparison between the hyperbolic isotropic equations resulting from the present theory and their counterparts in Biot’s theory. This comparison shows the recovery by the medium-independent VMTPM framework of the essential structure of Biot’s PDEs. This recovery is herein deductively achieved in absence of heuristic statements, proceeding from the consideration of individual strain energies of the solid and fluid phases and from the minimal kinematic hypotheses of Chap.  2. This study is complemented by an analysis of the bounds of the elastic moduli of the isotropic theory, which is undertaken deploying a generalization to the present two-phase context of the Composite Sphere Assemblage homogenization technique by Hashin.


Strain Energy Density Volumetric Strain Solid Volume Fraction Isotropic Theory Rotational Wave 
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.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)CrossRefzbMATHGoogle Scholar
  2. 2.
    Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. low-frequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biot, M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33(4), 1482–1498 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corapcioglu, M.Y., Tuncay, K.: Propagation of waves in porous media. In: Corapcioglu, M.Y. (ed.) Advances in Porous Media, vol. 3, pp. 361–440. Elsevier, Amsterdam (1996)Google Scholar
  5. 5.
    Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech. 29(1), 143–150 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hashin, Z.: Analysis of composite materials-a survey. J. Appl. Mech. 50(3), 481–505 (1983)CrossRefzbMATHGoogle Scholar
  7. 7.
    Lee, K., Westmann, R.: Elastic properties of hollow-sphere-reinforced composites. J. Compos. Mater. 4(2), 242–252 (1970)CrossRefGoogle Scholar
  8. 8.
    Markert, B.: A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua. Transp. Porous Media 70(3), 427–450 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity. Courier Dover Publications, Chelmsford (1994)zbMATHGoogle Scholar
  10. 10.
    Serpieri, R., Travascio, F.: A purely-variational purely-macroscopic theory of two-phase porous media–part I: derivation of medium-independent governing equations and stress partitioning laws. SubmittedGoogle 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.
    Timoshenko, S., Goodier, J., Abramson, H.: Theory of elasticity. J. Appl. Mech. 37, 888 (1970)CrossRefGoogle Scholar
  15. 15.
    Wilmański, K.: A few remarks on Biot’s model and linear acoustics of poroelastic saturated materials. Soil Dyn. Earthquake Eng. 26(6), 509–536 (2006)CrossRefGoogle 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