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Micropolar mixture models on the basis of the Theory of Porous Media

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Porous Media

Abstract

The behaviour of porous media can be described in a continuum mechanical setting by the Theory of Porous Media, i. e. by a mixture theory extended by the concept of volume fractions. In addition to the volume fractions, micropolarity is taken into account to model the internal structure of porous media on the macroscopic scale. After a microscopic motivation of the approach, which shows that it is physically motivated to deal with micropolar mixture models, the kinematics, the balance relations, and the constitutive framing of such a theory are discussed. A set of model equations is formulated within the presented frame and applied to some boundary value problems showing the evidence of the theoretical approach.

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References

  1. Bauer, G.: Thermodynamische Betrachtung einer gesättigten Mischung. Dissertation, Technische Hochschule Darmstadt 1997.

    Google Scholar 

  2. Bluhm, J.: A consistent model for saturated and empty porous media. Forschungsberichte aus dem Fachbereich Bauwesen, 74, Universität-GH-Essen 1997.

    Google Scholar 

  3. Bluhm, J., de Boer, R.: Effective stress — a clarification. Arch. Appl. Mech. 66 (1996), 479–492.

    MATH  Google Scholar 

  4. de Boer, R.: Theory of porous media — highlights in the historical development and current state. Springer-Verlag, Berlin 2000.

    Google Scholar 

  5. de Boer, R., Ehlers, W.: Theorie der Mehrkomponentenkontinua mit Anwendung auf bodenmechanische Probleme, Teil I. Forschungsberichte aus dem Fachbereich Bauwesen, 40, Universität-GH-Essen 1986.

    Google Scholar 

  6. Bowen, R. M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 18 (1980), 1129–1148.

    Article  MATH  Google Scholar 

  7. Coleman, B. D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rat. Mech. Anal. 13 (1963), 167–178.

    Article  MathSciNet  MATH  Google Scholar 

  8. Cundall, P. A, Strack, O. D. L.: A discrete numerical model for granular assemblies. Géotechnique 29 (1979), 47–65.

    Article  Google Scholar 

  9. Diebels, S.: A micropolar theory of porous media: Constitutive modelling. Transport in Porous Media 34 (1999), 193–208.

    Article  Google Scholar 

  10. Diebels, S.: Mikropolare Zweiphasenmodelle: Formulierung auf der Basis der Theorie Poröser Medien. Habilitationsschrift, Bericht Nr. II-4, Institut für Mechanik (Bauwesen), Lehrstuhl II, Universität Stuttgart 2000.

    Google Scholar 

  11. Diebels, S., Ehlers, W.: On basic equations of multiphase micropolar materials. Technische Mechanik 16 (1996), 77–88.

    Google Scholar 

  12. Diebels, S., Ehlers, W., Michelitsch, T.: Particle simulations as a microscopic approach to a Cosserat continuum. J. Phys. IV France (2001), submitted.

    Google Scholar 

  13. Ehlers, W.: Poröse Medien — ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Forschungsberichte aus dem Fachbereich Bauwesen, 47, Universität-GH-Essen 1988.

    Google Scholar 

  14. Ehlers, W.: Compressible, incompressible and hybrid two-phase models in porous media theories. In Angel, Y. C. (ed.): Anisotropy and Inhomogeneity in Elasticity and Plasticity, AMD-Vol. 158, ASME 1993, pp. 25–38.

    Google Scholar 

  15. Ehlers, W.: Constitutive equations for granular materials in geomechanical context. In Hutter, K. (ed.): Continuum mechanics in environmental sciences and geophysics, CISM Courses and Lectures No. 337, Springer-Verlag, Wien 1993, pp. 313–402.

    Google Scholar 

  16. Ehlers, W., Diebels, S., Michelitsch, T.: Microscopic modeling of granular materials taking into account particle rotations. In Vermeer, P. A. et al. (eds.): Continuous and discontinuous modelling of cohesive frictional materials, Springer-Verlag, Berlin 2001, pp. 259–274.

    Chapter  Google Scholar 

  17. Ehlers, W., Ellsiepen, P.: Theoretical and numerical methods in environmental continuum mechanics based on the theorie of porous media. In Schrefler, B. A. (ed.): Environmental mechanics, CISM Courses and Lectures No. 417, Springer-Verlag, Wien 2001.

    Google Scholar 

  18. Eipper, G.: Theorie und Numerik finiter elastischer Deformationen in fluidgesättigten porösen Festkörpern. Dissertation, Bericht Nr. II-1, Institut für Mechanik (Bauwesen), Lehrstuhl II, Universität Stuttgart 1998.

    Google Scholar 

  19. Ellsiepen, P. (ed.): PANDAS — Benutzer- und Referenzhandbuch. Bericht Nr. 97-II-9, Institut für Mechanik (Bauwesen), Lehrstuhl II, Universität Stuttgart 1997.

    Google Scholar 

  20. Eringen, A. C.: Simple microfluids. Int. J. Engng. Sci. 2 (1964), 205–217.

    Article  MathSciNet  MATH  Google Scholar 

  21. Eringen, A. C., Kafadar, C. B.: Polar field theories. In Eringen, A. C. (ed.): Continuum Physics, Vol IV — Polar and nonlocal field theories, Academic Press, New York 1976, pp. 1–73.

    Google Scholar 

  22. Gibson, L. J., Ashby, M. F.: Cellular solids — structure and properties. 2nd ed., Cambridge University Press, Cambridge 1997.

    Google Scholar 

  23. Hassanizadeh, S. M., Gray, W. G.: General conservation equations for multiphase systems: 2. Mass, momentena, energy, and entropy equations. Advances in Water Resources 2 (1979), 191–208.

    Article  Google Scholar 

  24. Hutter, K.: The foundations of thermodynamics, its basic postulates and implications. A review of modern thermodynamics. Acta Mech. 27 (1977), 1–54.

    Article  MathSciNet  Google Scholar 

  25. Lade, P., de Boer, R.: The concept of effective stress for soil, concrete and rock. Géotechnique 47 (1997), 61–78.

    Article  Google Scholar 

  26. Lewis, R. W., Schrefier, B. A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. 2nd ed., John Wiley Sons, Chichester 1998.

    MATH  Google Scholar 

  27. Liu, I-S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rat. Mech. Anal. 46 (1972), 131–148.

    MATH  Google Scholar 

  28. Liu, I-S., Müller, I.: Thermodynamics of mixtures of fluids. In Truesdell, C. (ed.): Rational Thermodynamics, 2nd ed., Springer-Verlag, New York 1984, pp. 264–285.

    Chapter  Google Scholar 

  29. Müller, I.: Thermodynamik — Grundlagen der Materialtheorie. Bertelsmann Universitätsverlag, Düsseldorf 1973.

    Google Scholar 

  30. Onck, P. O.: Notch-strengthening in two-dimensional foams. J. Phys IV France (2001), submitted.

    Google Scholar 

  31. Simio, J. C., Taylor, R. L.: Penalty function formulations for incompressible nonlinear elastostatics. Comp. Meth. Appl. Mech. Engng. 35 (1982), 107–118.

    Article  Google Scholar 

  32. Steinmann, P.: Lokalisierungsprobleme in der Plasto-Mechanik. Dissertation, Universität Karlsruhe 1992.

    Google Scholar 

  33. Steinmann, P.: A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int. J. Solids Struct. 31 (1994), 1063–1084.

    Article  MathSciNet  MATH  Google Scholar 

  34. Suklje, L.: Rheological Aspects of Soil Mechanics. Wiley Interscience, London 1969.

    Google Scholar 

  35. Svendsen, B., Hutter, K.: On the thermodynamics of a mixture of isotropic materials with constraints. Int. J. Engng. Sci. 33 (1995), 2021–2054.

    Article  MathSciNet  MATH  Google Scholar 

  36. von Terzaghi, K.: Zur Berechnung der Durchlässigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. Sitzungsber. Akad. Wiss. Wien 132 (1923), 125–138.

    Google Scholar 

  37. Truesdell, C.: Thermodynamics of diffusion. In Truesdell, C. (ed.): Rational Thermodynamics, 2nd ed., Springer-Verlag, New York 1984, pp. 219–236.

    Chapter  Google Scholar 

  38. Truesdell, C., Toupin, R. A.: The classical field theories. In Flügge, S. (ed.): Handbuch der Physik, III/1, Springer-Verlag, Berlin 1960, pp. 226–793.

    Google Scholar 

  39. Volk, W.: Untersuchung des Lokalisierungsverhaltens mikropolarer poröser Medien. Dissertation, Bericht Nr. II-2, Institut für Mechanik (Bauwesen), Lehrstuhl II, Universität Stuttgart 1999.

    Google Scholar 

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Diebels, S. (2002). Micropolar mixture models on the basis of the Theory of Porous Media. In: Ehlers, W., Bluhm, J. (eds) Porous Media. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04999-0_3

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  • DOI: https://doi.org/10.1007/978-3-662-04999-0_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07843-9

  • Online ISBN: 978-3-662-04999-0

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