Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Variational Continuum Multiphase Poroelasticity

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 67))

Abstract

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.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ateshian, G.A., Ricken, T.: Multigenerational interstitial growth of biological tissues. Biomech. Model. Mechanobiol. 9(6), 689–702 (2010)

    Article  Google Scholar 

  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. Bedford, A., Drumheller, D.: A variational theory of immiscible mixtures. Arch. Ration. Mech. Anal. 68(1), 37–51 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bedford, A., Drumheller, D.: A variational theory of porous media. Int. J. Solids Struct. 15(12), 967–980 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bedford, A., Drumheller, D.S.: Theories of immiscible and structured mixtures. Int. J. Eng. Sci. 21(8), 863–960 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berdichevsky, V.: Variational principles of continuum mechanics. Springer (2009)

    Google Scholar 

  10. Biot, M.: Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21(7), 597–620 (1972)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Biot, M., Willis, D.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957)

    MathSciNet  Google Scholar 

  13. Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182–185 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Bishop, A.: The effective stress principle. Teknisk Ukeblad 39, 859–863 (1959)

    Google Scholar 

  16. de Boer, R.: Theoretical poroelasticity – a new approach. Chaos, Solitons Fractals 25(4), 861–878 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. de Boer, R., Ehlers, W.: The development of the concept of effective stresses. Acta Mech. 83(1–2), 77–92 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bowen, R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20(6), 697–735 (1982)

    Article  MATH  Google Scholar 

  19. Coussy, O.: Mechanics of porous continua. Wiley (1995)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Drumheller, D.S.: The theoretical treatment of a porous solid using a mixture theory. Int. J. Solids Struct. 14(6), 441–456 (1978)

    Article  MATH  Google Scholar 

  28. Ehlers, W., Bluhm, J.: Porous media: theory, experiments and numerical applications. Springer Science & Business Media (2013)

    Google Scholar 

  29. Eremeyev, V., Pietraszkiewicz, W.: The nonlinear theory of elastic shells with phase transitions. J. Elast. 74(1), 67–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fillunger, P.: Erdbaumechanik? Selbstverl. d, Verf (1936)

    Google Scholar 

  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. Goodman, M., Cowin, S.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44(4), 249–266 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gouin, H., Gavrilyuk, S.: Hamilton’s principle and rankine-hugoniot conditions for general motions of mixtures. Meccanica 34(1), 39–47 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  35. Gray, W.G., Hassanizadeh, S.M.: Unsaturated flow theory including interfacial phenomena. Water Resour. Res. 27(8), 1855–1863 (1991)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Guo, Z.H.: Time derivatives of tensor fields in nonlinear continuum mechanics. Arch. Mech. 15, 131–163 (1963)

    MathSciNet  MATH  Google Scholar 

  39. Halphen, B., Nguyen, Q.S.: Sur les matériaux standard généralisés. J. de mécanique 14, 39–63 (1975)

    MATH  Google Scholar 

  40. Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 1. averaging procedure. Adv. Water Resour. 2, 131–144 (1979)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. Kenyon, D.E.: Thermostatics of solid-fluid mixtures. Arch. Ration. Mech. Anal. 62(2), 117–129 (1976)

    MathSciNet  MATH  Google Scholar 

  45. Kestin, J., Rice, J.R.: Paradoxes in the application of thermodynamics to strained solids. Citeseer (1969)

    Google Scholar 

  46. Kuhn, H.W., Tucker, A.W.: Proceedings of 2nd berkeley symposium (1951)

    Google Scholar 

  47. Lanczos, C.: The variational principles of mechanics, vol. 4. Courier Corporation (1970)

    Google Scholar 

  48. Landau, L., Lifshitz, E.: Mechanics. Course of theoretical physics, vol. 1 (1976)

    Google Scholar 

  49. Leech, C.: Hamilton’s principle applied to fluid mechanics. Q. J. Mech. Appl. Mech. 30(1), 107–130 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  50. Lopatnikov, S., Cheng, A.: Macroscopic Lagrangian formulation of poroelasticity with porosity dynamics. J. Mech. Phys. Solids 52(12), 2801–2839 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  53. Lopatnikov, S., Gillespie, J.: Poroelasticity-iii: conditions on the interfaces. Transp. Porous Media 93(3), 597–607 (2012)

    Article  MathSciNet  Google Scholar 

  54. Lubliner, J.: Plasticity theory. Courier Corporation (2008)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  58. Marsden, J., Hughes, T.: Mathematical foundations of elasticity. Courier Dover Publications (1994)

    Google Scholar 

  59. Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  60. Moiseiwitsch, B.L.: Variational principles. Courier Corporation (2013)

    Google Scholar 

  61. Moreau, J.: Sur les lois de frottement, de viscosité et de plasticité. CR Acad. Sci., Paris 271, 608–611 (1970)

    Google Scholar 

  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)

    Article  Google Scholar 

  63. Nguyen, Q., Germain, P., Suquet, P.: Continuum thermodynamics. J. Appl. Sci. 50, 1010–1020 (1983)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  67. Ogden, R.W.: Non-linear elastic deformations. Courier Corporation (1997)

    Google Scholar 

  68. Passman, S.: Mixtures of granular materials. Int. J. Eng. Sci. 15(2), 117–129 (1977)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  70. Schrefler, B.: Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions. Appl. Mech. Rev. 55(4), 351–388 (2002)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  78. Simo, J.C., Hughes, T.J.: Computational inelasticity, vol. 7. Springer Science & Business Media (2006)

    Google Scholar 

  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. 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. 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. 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. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. Handbuch der Physik, III/3. Springer, New York (1965)

    Google Scholar 

  84. Truesdell, C.: Thermodynamics of diffusion. In: Rational Thermodynamics, pp. 219–236. Springer (1984)

    Google Scholar 

  85. Truesdell, C., Toupin, R.: The classical field theories. Springer (1960)

    Google Scholar 

  86. Woods, L.: On the local form of the second law of thermodynamics in continuum mechanics. Q. Appl. Math. 39, 119–126 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  87. Woods, L.: Thermodynamic inequalities in continuum mechanics. IMA J. Appl. Math. 29(3), 221–246 (1982)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi del Sannio, Benevento, Italy

    Roberto Serpieri

  2. University of Miami, Biomechanics Research Laboratory, Coral Gables, FL, USA

    Francesco Travascio

Authors
  1. Roberto Serpieri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Travascio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Serpieri .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this chapter

Cite this chapter

Serpieri, R., Travascio, F. (2017). Variational Macroscopic Two-Phase Poroelasticity. Derivation of General Medium-Independent Equations and Stress Partitioning Laws. In: Variational Continuum Multiphase Poroelasticity. Advanced Structured Materials, vol 67. Springer, Singapore. https://doi.org/10.1007/978-981-10-3452-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-3452-7_2

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-3451-0

  • Online ISBN: 978-981-10-3452-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation