Continuum Mechanics and Thermodynamics

, Volume 28, Issue 5, pp 1553–1581 | Cite as

Saturated porous continua in the frame of hybrid description

  • Olga V. BrazginaEmail author
  • Elena A. Ivanova
  • Elena N. Vilchevskaya
Original Article


A method for modeling fluid–solid interactions in saturated porous media is proposed. The main challenge is the combination of the material and spatial descriptions. The deformation of the solid, which serves as a “container” to the fluid, is studied by following the motion of its material particles, i.e., in Lagrangian description. On the other hand, the motion of the fluid is described in spatial form, i.e., by using a Eulerian approach. However, the solid deforms and this implies a certain difference regarding the standard formulation used in spatial description of fluid mechanics where a fixed grid dissects space into elementary volumes. Here the grid is no longer fixed, and the elementary volumes will follow the deformation of the solid. Moreover, for the solid as well as for the fluid the balance equations are formulated in the current configuration, where interaction forces and couples are taken into account. By using Zhilin’s approach, entropy and temperature are incorporated in the system of equations. Constitutive equations are constructed for both elastic and inelastic components of force and couple stress tensors and interaction force and couple. The constitutive equations for elastic components are found on the basis of the energy balance equation; the constitutive equations for the inelastic components are proposed in accordance with the second law of thermodynamics. Particular emphasis is placed on the constitutive equations of the interaction force and couple, which result in a symmetric form only because of the “hybrid” approach combining the Lagrangian with the Eulerian description. Three possible examples of application of the theory have been presented. For each example, all required assumptions were first stated and discussed and then the complete set of the corresponding equations was presented.


Porous media Continuum thermodynamics Interaction forces Interaction couples 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biot M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    de Boer R.: Trends in Continuum Mechanics of Porous Media. Springer, Netherlands (2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    de Boer R., Ehlers W.: On the problem of fluid- and gas-filled elasto-plastic solids. Int. J. Solids Struct. 22(11), 1231–1242 (1986)CrossRefzbMATHGoogle Scholar
  4. 4.
    Loret B., Simoes F.M.F.: A framework for deformation, generalized diffusion, mass transfer and growth in multi-specied multi-phase biological tissues. Eur. J. Mech. A Solid 24, 757–781 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wei C.H., Muraleetharan K.K.: A continuum theory of porous media saturated by multiply immiscible fluids: II. Lagrangian description description and variational structure. Int. J. Eng. Sci. 40, 1835–1854 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bowen R.M.: Compressive porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20(6), 697–735 (1982)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bowen R.M.: Incompressive porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18, 1129–1148 (1980)CrossRefzbMATHGoogle Scholar
  8. 8.
    Wilmanski K.: Continuum Thermodynamics. Part 1: Foundations. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Anferov S.D., Scul’skiy O.I.: Modeling of fluid filtration through plastically deformed porous medium in the process of extrusion (in Russian). PNRPU Mech. Bull. 2, 29–47 (2014)CrossRefGoogle Scholar
  10. 10.
    Hassanizaden M., Gray W.G.: General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv. Water Resour. 3, 25–40 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    Nigmatullin B.E.: Foundations of Heterogenious Continuum Mechanics (in Russian). Nauka, Moscow (1978)Google Scholar
  12. 12.
    Borja R.I.: On mechanical energy and effective stress in saturated and unsaturated porous continua. Int. J. Solids Struct. 43, 1764–1786 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    Chapelle D., Moireau P.: General coupling of porous flows and hyperelastic formulations—from thermodynamics principles to energy balance and compatible time schemes. Eur. J. Mech. B Fluid. 46, 82–96 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li C., Borja R.I., Regueiro R.A.: Dynamics of porous media at finite strain. Comput. Methods Appl. Mech. 193, 3837–3870 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Vuong A.-T., Yoshihara L., Wall W.A.: A general approach for modeling interacting flow through porous media under finite deformation. Comput. Methods Appl. Mech. 283, 1240–1259 (2015)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Arienti M., Hung P., Morano E., Shepherd J.E.: A level set approach to Eulerian–Lagrangian coupling. J. Comput. Phys. 185, 213–251 (2003)ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Donea J., Giuliani S., Halleux J.P.: An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid-structure interaction. Comput. Method. Appl. Mech. 33, 689–723 (1982)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hirt C.W., Amsden A.A., Cook J.L.: An arbitraty Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253 (1974)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    McGurn M.T., Ruggirello K.P., DesJardin P.E.: An Eulerian–Lagrangian moving immersed interface method for simulation burning solids. J. Comput. Phys. 241, 364–387 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Surana K.S., Blackwell B., Powell M., Reddy J.N.: Mathematical models for fluid–solid interaction and their numerical solutions. J. Fluid. Struct. 50, 184–216 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    Brown, D.L., Popov, P., Efendiev, Y.: Effective equations for fluid-structure interaction with application to poroelacticity: applicable analysis. Int. J. Eng. Sci. 93, 4 (2014) doi: 10.1080/00036811.2013.839780
  22. 22.
    Collins R.E.: Flow of Fluids Through Porous Materials. Reinolds Publishing Corporation, New York (1961)Google Scholar
  23. 23.
    Kunin I.A.: Kinematics of media with continuously changing topology. Int. J. Theor. Phys. 29(11), 1167–1176 (1990)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Gurtin M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)zbMATHGoogle Scholar
  25. 25.
    Hauke J.: An Intriduction to Fluid Mechanics and Transport Phenomena. Springer, Netherlands (2008)CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhilin P.A.: Rational Continuum Mechanics (in Russian). Polytechnic University Publishing House, St.Petersburg (2012)Google Scholar
  27. 27.
    Ivanova, E.A., Vilchevskaya, E.N., Müller, W.H.: Time derivatives in material and spatial description—What are the differences and why do they concern us?. In: Naumenko, K., Aßmus, M. (ed.) Advanced Methods of Continuum Mechanics for Materials and Structures, Springer, Berlin (2016)Google Scholar
  28. 28.
    Loicyanskii L.G.: Mechanics of Fluids (in Russian). Nauka, Moscow (1987)Google Scholar
  29. 29.
    Falkovich G.: Fluid Mechanics. A Short Course for Physicists. Cambridge University Press, New York (2011)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ivanova E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 215, 261–286 (2010)CrossRefzbMATHGoogle Scholar
  31. 31.
    Ivanova, E.A.: On one model of generalized continuum and its thermodynamical interpretation. In: Altenbach, H., Maugin, G.A., Erofeev, V. Mechanics of Generalized Continua., pp. 151–174. Springer, Berlin (2011)Google Scholar
  32. 32.
    Ivanova E.A.: Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum. Tech. Mech. 32, 273–286 (2012)MathSciNetGoogle Scholar
  33. 33.
    Ivanova E.A.: Description of mechanism of thermal conduction and internal damping by means of two component Cosserat continuum. Acta Mech. 225, 757–795 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ivanova E.A.: A new model of a micropolar continuum and some electromagnetic analogies. Acta Mech. 226, 697–721 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Treugolov I.G.: Moment theory of electromagnetic effects in anisotropic solids. Appl. Math. Mech. 53(6), 992–997 (1989)Google Scholar
  36. 36.
    Zhilin L.G.: Advanced Problems in Mechanics, vol. 1, 2. Institute for Problems in Mechanical Engineering, St. Petersburg (2006)Google Scholar
  37. 37.
    Tiersten H.F.: Coupled magnetomechanical equations for magnetically saturated insulators. J. Math. Phys. 5(9), 1298–1318 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Maugin G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier, Oxford (1988)zbMATHGoogle Scholar
  39. 39.
    Eringen A.C., Maugin G.A.: Electrodynamics of Continua. Springer, New York (1990)CrossRefGoogle Scholar
  40. 40.
    Fomethe A., Maugin G.A.: Material forces in thermoelastic ferromagnets. Contin. Mech. Thermodyn. 8, 275–292 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Shliomis M.I., Stepanov V.I.: Rotational viscosity of magnetic fluids: contribution of the Brownian and Neel relaxational processes. J. Magn. Magn. Mater. 122, 196–199 (1993)ADSCrossRefGoogle Scholar
  42. 42.
    Ivanova, E.A., Vilchevskaya, E.N.: Description of thermal and micro-structural processes in generalized continua: Zhilin’s method and its modifications. In: Altenbach, H., Forest, S., Krivtsov, A.M. (eds.) Generalized Continua as Models for Materials with Multi-scale Effects or Under Multi-field Actions, pp. 179–197. Springer, Berlin (2013)Google Scholar
  43. 43.
    Vilchevskaya E.N., Ivanova E.A., Altenbach H.: Description of liquid–gas phase transition in the frame of continuum mechanics. Contin. Mech. Thermodyn. 26(2), 221–245 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Zhilin P.A.: Rigid body oscillator: a general model and some results. Acta Mech. 142, 149–193 (2000)CrossRefzbMATHGoogle Scholar
  45. 45.
    Truesdell C.: A First Course in Rational Continuum Mechanics. The John Hopkins University, Maryland, Baltimore (1972)Google Scholar
  46. 46.
    Altenbach H., Naumenko K., Zhilin P.: A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Contin. Mech. Thermodyn. 15(6), 539–570 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Zolotukhin A.B., Ursin J.-R.: Introduction to Petroleum Reservoir Engineering. Norwegian Academic Press, Kristiansand (2000)Google Scholar
  48. 48.
    Lurie A.I.: Nonlinear Theory of Elasticity. Elsevier, Amsterdam (1991)Google Scholar
  49. 49.
    Babichev A.P.: Physical Quantities: Handbook (in Russian). Energoatomizdat, Moscow (1991)Google Scholar
  50. 50.
    DeGroot C.H.T., Straatman A.G.: Towards a porous media model of the human lung. Int. J. Eng. Sci. AIP Conf. Proc. 1453, 69–74 (2012)ADSCrossRefGoogle Scholar
  51. 51.
    Miguel A.F.: Lungs as a natural porous media: architecture, airflow characteristics and transport of suspended particles. In: Delgado, J. M. P. Q. (ed.) Heat and Mass Transfer in Porous Media, pp. 115–138. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  52. 52.
    Klaas M., Koch E., Schröder W.: Fundamental Medical and Engineering Investigations on Protective Artificial Respiration. Springer, Berlin (2011)CrossRefGoogle Scholar
  53. 53.
    Bazarov I.P.: Thermodynamics. Pergamon Press, New York (1964)Google Scholar
  54. 54.
    Bower A.F.: Applied Mechanics of Solids. CRC Press, Boca Raton (2010)Google Scholar
  55. 55.
    Suganthy J.: Plastination using standard S10 technique—our experience in Christian Medical College. J. Anat. Soc. India 61(1), 44–47 (2012)CrossRefGoogle Scholar
  56. 56.
    Landau L.D., Lifshitz E.M.: Theory of Elasticity. Vol. 7. 1st edn. Pergamon Press, Oxford (1959)Google Scholar
  57. 57.
    Nowacky W.: Thermoelasticity. Pergamon Press, Warsaw (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Olga V. Brazgina
    • 1
    Email author
  • Elena A. Ivanova
    • 1
    • 2
  • Elena N. Vilchevskaya
    • 1
    • 3
  1. 1.Departament of Theoretical MechanicsPeter the Great St.Petersburg Polytechnic UniversitySt.-PetersburgRussia
  2. 2.Laboratory of Mechatronics, Institute for Problems in Mechanical Engineering of Russian Academy of Sciences (IPME RUS)St.-PetersburgRussia
  3. 3.Laboratory of Mathematical Methods in Mechanics of Materials, Institute for Problems in Mechanical Engineering of Russian Academy of Sciences (IPME RUS)St.-PetersburgRussia

Personalised recommendations