Variational Macroscopic Two-Phase Poroelasticity. Derivation of General Medium-Independent Equations and Stress Partitioning Laws

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


A macroscopic continuum theory of two-phase saturated porous media is derived by a purely variational deduction based on the least Action principle. The proposed theory proceeds from the consideration of a minimal set of kinematic descriptors and keeps a specific focus on the derivation of most general medium-independent governing equations, which have a form independent from the particular constitutive relations and thermodynamic constraints characterizing a specific medium. The kinematics of the microstructured continuum theory herein presented employs an intrinsic/extrinsic split of volumetric strains and adopts, as an additional descriptor, the intrinsic scalar volumetric strain which corresponds to the ratio between solid true densities before and after deformation. The present theory integrates the framework of the Variational Macroscopic Theory of Porous Media (VMTPM) which, in previous works, was limited to the variational treatment of the momentum balances of the solid phase alone. Herein, the derivation of the complete set momentum balances inclusive of the momentum balance of the fluid phase is attained on a purely variational basis. Attention is also focused on showing that the singular conditions, in which either the solid or the fluid phase are vanishing, are consistently addressed by the present theory, included conditions over free solid-fluid surfaces.


Fluid Phase Representative Volume Element Volumetric Strain Momentum Balance Reference Configuration 
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.
    Aizicovici, S., Aron, M.: A variational theorem in the linear theory of mixtures of two elastic solids. the quasi-static case. Acta Mech. 27(1), 275–280 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albers, B., Wilmański, K.: Influence of coupling through porosity changes on the propagation of acoustic waves in linear poroelastic materials. Arch. Mech. 58(4–5), 313–325 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andreaus, U., Giorgio, I., Lekszycki, T.: A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. Zeitschrift für Angewandte Mathematik und Mechanik 94(12), 978–1000 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ateshian, G.A., Ricken, T.: Multigenerational interstitial growth of biological tissues. Biomech. Model. Mechanobiol. 9(6), 689–702 (2010)CrossRefGoogle Scholar
  5. 5.
    Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics á la hamilton–piola least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids August 28, 1–44 (2013)Google Scholar
  6. 6.
    Bedford, A., Drumheller, D.: A variational theory of immiscible mixtures. Arch. Ration. Mech. Anal. 68(1), 37–51 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bedford, A., Drumheller, D.: A variational theory of porous media. Int. J. Solids Struct. 15(12), 967–980 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bedford, A., Drumheller, D.S.: Theories of immiscible and structured mixtures. Int. J. Eng. Sci. 21(8), 863–960 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berdichevsky, V.: Variational principles of continuum mechanics. Springer (2009)Google Scholar
  10. 10.
    Biot, M.: Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21(7), 597–620 (1972)CrossRefzbMATHGoogle Scholar
  11. 11.
    Biot, M.: Variational lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion. Int. J. Solids Struct. 13(6), 579–597 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Biot, M., Willis, D.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957)MathSciNetGoogle Scholar
  13. 13.
    Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182–185 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Bishop, A.: The effective stress principle. Teknisk Ukeblad 39, 859–863 (1959)Google Scholar
  16. 16.
    de Boer, R.: Theoretical poroelasticity – a new approach. Chaos, Solitons Fractals 25(4), 861–878 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    de Boer, R., Ehlers, W.: The development of the concept of effective stresses. Acta Mech. 83(1–2), 77–92 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bowen, R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20(6), 697–735 (1982)CrossRefzbMATHGoogle Scholar
  19. 19.
    Coussy, O.: Mechanics of porous continua. Wiley (1995)Google Scholar
  20. 20.
    Coussy, O., Dormieux, L., Detournay, E.: From mixture theory to Biot’s approach for porous media. Int. J. Solids Struct. 35(34), 4619–4635 (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Cowin, S., Goodman, M.: A variational principle for granular materials. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 56(7), 281–286 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    De Boer, R.: Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory. Appl. Mech. Rev. 49(4), 201–262 (1996)CrossRefGoogle Scholar
  23. 23.
    De Buhan, P., Dormieux, L.: On the validity of the effective stress concept for assessing the strength of saturated porous materials: a homogenization approach. J. Mech. Phys. Solids 44(10), 1649–1667 (1996)CrossRefGoogle Scholar
  24. 24.
    dell’Isola, F., Madeo, A., Seppecher, P.: Boundary conditions at fluid-permeable interfaces in porous media: a variational approach. Int. J. Solids Struct. 46(17), 3150–3164 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    dell’Isola, F., Placidi, L.: Variational principles are a powerful tool also for formulating field theories. CISM Courses and Lectures, vol. 535. Springer (2012)Google Scholar
  26. 26.
    dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Drumheller, D.S.: The theoretical treatment of a porous solid using a mixture theory. Int. J. Solids Struct. 14(6), 441–456 (1978)CrossRefzbMATHGoogle Scholar
  28. 28.
    Ehlers, W., Bluhm, J.: Porous media: theory, experiments and numerical applications. Springer Science & Business Media (2013)Google Scholar
  29. 29.
    Eremeyev, V., Pietraszkiewicz, W.: The nonlinear theory of elastic shells with phase transitions. J. Elast. 74(1), 67–86 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Fillunger, P.: Erdbaumechanik? Selbstverl. d, Verf (1936)Google Scholar
  31. 31.
    Gajo, A.: A general approach to isothermal hyperelastic modelling of saturated porous media at finite strains with compressible solid constituents. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society (2010)Google Scholar
  32. 32.
    Goodman, M., Cowin, S.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44(4), 249–266 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Gouin, H., Gavrilyuk, S.: Hamilton’s principle and rankine-hugoniot conditions for general motions of mixtures. Meccanica 34(1), 39–47 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gouin, H., Ruggeri, T.: Hamiltonian principle in binary mixtures of Euler fluids with applications to the second sound phenomena. Rendiconti Matematici dell’Accademia dei Lincei 14(9), 69–83 (2003)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Gray, W.G., Hassanizadeh, S.M.: Unsaturated flow theory including interfacial phenomena. Water Resour. Res. 27(8), 1855–1863 (1991)CrossRefGoogle Scholar
  36. 36.
    Gray, W.G., Miller, C.T., Schrefler, B.A.: Averaging theory for description of environmental problems: what have we learned? Adv. Water Resour. 51, 123–138 (2013)CrossRefGoogle Scholar
  37. 37.
    Gray, W.G., Schrefler, B.A., Pesavento, F.: The solid phase stress tensor in porous media mechanics and the hill-mandel condition. J. Mech. Phys. Solids 57(3), 539–554 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Guo, Z.H.: Time derivatives of tensor fields in nonlinear continuum mechanics. Arch. Mech. 15, 131–163 (1963)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Halphen, B., Nguyen, Q.S.: Sur les matériaux standard généralisés. J. de mécanique 14, 39–63 (1975)zbMATHGoogle Scholar
  40. 40.
    Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 1. averaging procedure. Adv. Water Resour. 2, 131–144 (1979)CrossRefGoogle Scholar
  41. 41.
    Hassanizadeh, S.M., Gray, W.G.: Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Res. 13(4), 169–186 (1990)CrossRefGoogle Scholar
  42. 42.
    Jardine, R., Gens, A., Hight, D., Coop, M.: Developments in understanding soil behaviour. In: Advances in Geotechnical Engineering: The Skempton Conference, p. 103. Thomas Telford (2004)Google Scholar
  43. 43.
    Karush, W.: Minima of functions of several variables with inequalities as side constraints. Ph.D. thesis, Master’s thesis, Department of Mathematics, University of Chicago (1939)Google Scholar
  44. 44.
    Kenyon, D.E.: Thermostatics of solid-fluid mixtures. Arch. Ration. Mech. Anal. 62(2), 117–129 (1976)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Kestin, J., Rice, J.R.: Paradoxes in the application of thermodynamics to strained solids. Citeseer (1969)Google Scholar
  46. 46.
    Kuhn, H.W., Tucker, A.W.: Proceedings of 2nd berkeley symposium (1951)Google Scholar
  47. 47.
    Lanczos, C.: The variational principles of mechanics, vol. 4. Courier Corporation (1970)Google Scholar
  48. 48.
    Landau, L., Lifshitz, E.: Mechanics. Course of theoretical physics, vol. 1 (1976)Google Scholar
  49. 49.
    Leech, C.: Hamilton’s principle applied to fluid mechanics. Q. J. Mech. Appl. Mech. 30(1), 107–130 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lopatnikov, S., Cheng, A.: Macroscopic Lagrangian formulation of poroelasticity with porosity dynamics. J. Mech. Phys. Solids 52(12), 2801–2839 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Lopatnikov, S., Gillespie, J.: Poroelasticity-i: governing equations of the mechanics of fluid-saturated porous materials. Transp. Porous Media 84(2), 471–492 (2010)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Lopatnikov, S., Gillespie, J.: Poroelasticity-ii: on the equilibrium state of the fluid-filled penetrable poroelastic body. Transp. Porous Media 89(3), 475–486 (2011)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Lopatnikov, S., Gillespie, J.: Poroelasticity-iii: conditions on the interfaces. Transp. Porous Media 93(3), 597–607 (2012)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Lubliner, J.: Plasticity theory. Courier Corporation (2008)Google Scholar
  55. 55.
    Madeo, A., dell’Isola, F., Darve, F.: A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J. Mech. Phys. Solids 61(11), 2196–2211 (2013)MathSciNetCrossRefGoogle Scholar
  56. 56.
    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
  57. 57.
    Markert, B.: A biphasic continuum approach for viscoelastic high-porosity foams: comprehensive theory, numerics, and application. Arch. Comput. Methods Eng. 15(4), 371–446 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Marsden, J., Hughes, T.: Mathematical foundations of elasticity. Courier Dover Publications (1994)Google Scholar
  59. 59.
    Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Moiseiwitsch, B.L.: Variational principles. Courier Corporation (2013)Google Scholar
  61. 61.
    Moreau, J.: Sur les lois de frottement, de viscosité et de plasticité. CR Acad. Sci., Paris 271, 608–611 (1970)Google Scholar
  62. 62.
    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
  63. 63.
    Nguyen, Q., Germain, P., Suquet, P.: Continuum thermodynamics. J. Appl. Sci. 50, 1010–1020 (1983)zbMATHGoogle Scholar
  64. 64.
    Nunziato, J.W., Walsh, E.K.: On ideal multiphase mixtures with chemical reactions and diffusion. Arch. Ration. Mech. Anal. 73(4), 285–311 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Nur, A., Byerlee, J.: An exact effective stress law for elastic deformation of rock with fluids. J. Geophys. Res. 76(26), 6414–6419 (1971)CrossRefGoogle Scholar
  66. 66.
    Nuth, M., Laloui, L.: Effective stress concept in unsaturated soils: clarification and validation of a unified framework. Int. J. Numer. Anal. Meth. Geomech. 32(7), 771–801 (2008)CrossRefzbMATHGoogle Scholar
  67. 67.
    Ogden, R.W.: Non-linear elastic deformations. Courier Corporation (1997)Google Scholar
  68. 68.
    Passman, S.: Mixtures of granular materials. Int. J. Eng. Sci. 15(2), 117–129 (1977)CrossRefzbMATHGoogle Scholar
  69. 69.
    Pietraszkiewicz, W., Eremeyev, V., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 87(2), 150–159 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Schrefler, B.: Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions. Appl. Mech. Rev. 55(4), 351–388 (2002)CrossRefGoogle Scholar
  71. 71.
    Sciarra, G., dell’Isola, F., Hutter, K.: Dilatancy and compaction around a cylindrical cavern leached-out in a fluid saturated salt rock, pp. 681–687 (2005)Google Scholar
  72. 72.
    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
  73. 73.
    Serpieri, R., Della Corte, A., Travascio, F., Rosati, L.: Variational theories of two-phase continuum poroelastic mixtures: a short survey. In: Altenbach, H., Forest, S. (eds.) Generalized Continua as Models for Classical and Advanced Materials, pp. 377–394. Springer, Cham (2016)CrossRefGoogle Scholar
  74. 74.
    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
  75. 75.
    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
  76. 76.
    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, pp. 1142–1153 (2013)Google Scholar
  77. 77.
    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
  78. 78.
    Simo, J.C., Hughes, T.J.: Computational inelasticity, vol. 7. Springer Science & Business Media (2006)Google Scholar
  79. 79.
    Skempton, A.: Terzaghi’s discovery of effective stress. From Theory to Practice in Soil Mechanics: Selections from the Writings of Karl Terzaghi, pp. 42–53 (1960)Google Scholar
  80. 80.
    Terzaghi, K.: The shearing resistance of saturated soils and the angle between the planes of shear. In: International Conference on Soil Mechanics and Foundation Engineering, Cambridge (1936)Google Scholar
  81. 81.
    Travascio, F., Asfour, S., Serpieri, R., Rosati, L.: Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach. Mathematics and Mechanics of Solids (2015). doi: 10.1177/1081286515616049
  82. 82.
    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
  83. 83.
    Truesdell, C., Noll, W.: The non-linear field theories of mechanics. Handbuch der Physik, III/3. Springer, New York (1965)Google Scholar
  84. 84.
    Truesdell, C.: Thermodynamics of diffusion. In: Rational Thermodynamics, pp. 219–236. Springer (1984)Google Scholar
  85. 85.
    Truesdell, C., Toupin, R.: The classical field theories. Springer (1960)Google Scholar
  86. 86.
    Woods, L.: On the local form of the second law of thermodynamics in continuum mechanics. Q. Appl. Math. 39, 119–126 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  87. 87.
    Woods, L.: Thermodynamic inequalities in continuum mechanics. IMA J. Appl. Math. 29(3), 221–246 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

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

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