Foundations of multiphasic and porous materials

  • Wolfgang Ehlers
Chapter

Abstract

Miscible multiphasic materials like classical mixtures as well as immiscible materials like saturated and partially saturated porous media can be successfully described on the common basis of the well-founded Theory of Mixtures (TM) or the Theory of Porous Media (TPM). In particular, both the TM and the TPM provide an excellent frame for a macroscopic description of a broad variety of engineering applications and further problems in applied natural sciences. The present article portrays both the standard and the micropolar approaches to multiphasic materials reflecting their mechanical and their thermodynamical frameworks. Including some constitutive models and various illustrative numerical examples, the article can be understood as a reference paper to all the following articles of this volume on theoretical, experimental and numerical investigations in the Theory of Porous Media.

Keywords

Permeability Entropy Porosity Anisotropy Foam 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Biot, M. A.: Le problème de la consolidation de matières argileuses sous une charge. Ann. Soc. Sci. Bruxelles B 55 (1935), 110–113.Google Scholar
  2. 2.
    Biot, M. A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941), 155–164.MATHCrossRefGoogle Scholar
  3. 3.
    Biot, M. A.: Theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low frequency range. J. Acoust. Soc. Am. 28 (1956), 168–178.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bishop, A. W.: The effective stress principle. Teknisk Ukeblad 39 (1959), 859863.Google Scholar
  5. 5.
    Bluhm, J.: A consistent model for saturated and empty porous media. Forschungsberichte aus dem Fachbereich Bauwesen, Heft 74, Universität-GHEssen 1997.Google Scholar
  6. 6.
    de Boer, R.: Vektor-und Tensorrechnung für Ingenieure. Springer-Verlag, Berlin 1982.MATHCrossRefGoogle Scholar
  7. 7.
    de Boer, R.: Highlights in the historical development of porous media theory: toward a consistent macroscopic theory. Applied Mechanics Review 49 (1996), 201–262.CrossRefGoogle Scholar
  8. 8.
    de Boer, R.: Theory of Porous Media. Springer-Verlag, Berlin 2000.MATHCrossRefGoogle Scholar
  9. 9.
    de Boer, R., Ehlers, W.: Theorie der Mehrkomponentenkontinua mit Anwendung auf bodenmechanische Probleme. Forschungsberichte aus dem Fachbereich Bauwesen, Heft 40, Universität-GH-Essen 1986.Google Scholar
  10. 10.
    de Boer, R., Ehlers, W.: The development of the concept of effective stresses. Acta Mech. 83 (1990), 77–92.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    de Boer, R., Ehlers, W.: Uplift, friction and capillarity - three fundamental effects for liquid-saturated porous media. Int. J. Solids Structures 26 (1990), 43–57.CrossRefGoogle Scholar
  12. 12.
    de Boer, R., Ehlers, W., Kowalski, S., Plischka, J.: Porous media - a survey of different approaches. Forschungsberichte aus dem Fachbereich Bauwesen, Heft 54, Universität-GH-Essen 1991.Google Scholar
  13. 13.
    de Borst, R.: Simulation of strain localization: A reappraisal of the Cosserat continuum. Engineering Computations 8 (1991), 317–332.CrossRefGoogle Scholar
  14. 14.
    Bowen, R. M.: Theory of mixtures. In Eringen, A. C. (ed.): Continuum Physics, Vol. III, Academic Press, New York 1976, pp. 1–127.Google Scholar
  15. 15.
    Bowen, R. M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 18 (1980), 1129–1148.MATHCrossRefGoogle Scholar
  16. 16.
    Bowen, R. M.: Compressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 20 (1982), 697–735.MATHCrossRefGoogle Scholar
  17. 17.
    Coleman, B. D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13 (1963), 167–178.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et fils, Paris 1909. (Theory of Deformable Bodies, NASA TT F-11 561, 1968 ).Google Scholar
  19. 19.
    Coussy, O.: Mechanics of Porous Continua. Wiley, Chichester 1995.MATHGoogle Scholar
  20. 20.
    Crawford, R. H., Anderson, D. C., Waggenspack, W. N.: Mesh rezoning of 2d isoparametric elements by inversion. International Journal for Numerical Methods in Engineering 28 (1989), 523–531.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Cross, J. J.: Mixtures of fluids and isotropic solids. Arch. Mech. 25 (1973), 1025–1039.MathSciNetMATHGoogle Scholar
  22. 22.
    Delesse, M.: Pour déterminer la composition des roches. Annales des mines, 4. séries 13 (1848), 379–388.Google Scholar
  23. 23.
    Diebels, S.: Mikropolare Zweiphasenmodelle: Formulierung auf der Basis der Theorie Poröser Medien. Habilitation, Bericht Nr. II-4 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 2000.Google Scholar
  24. 24.
    Diebels, S., Ehlers, W.: Dynamic analysis of a fully saturated porous medium accounting for geometrical and material non-linearities. Int. J. Numer. Methods Engng. 39 (1996), 81–97.MATHCrossRefGoogle Scholar
  25. 25.
    Diebels, S., Ehlers, W.: On fundamental concepts of multiphase micropolar materials. Technische Mechanik, 16: 77–88 (1996).Google Scholar
  26. 26.
    Diebels, S., Ellsiepen, P., Ehlers, W.: Error-controlled Runge-Kutta time integration of a viscoplastic hybrid two-phase model. Technische Mechanik 19 (1999), 19–27.Google Scholar
  27. 27.
    Ehlers, W.: On thermodynamics of elasto-plastic porous media. Arch. Mech. 41 (1989), 73–93.MathSciNetMATHGoogle Scholar
  28. 28.
    Ehlers, W.: Poröse Medien - ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Forschungsberichte aus dem Fachbereich Bauwesen, Heft 47, Universität-GH-Essen 1989.Google Scholar
  29. 29.
    Ehlers, W.: Toward finite theories of liquid-saturated elasto-plastic porous media. Int. J. Plasticity 7 (1991), 443–475.CrossRefGoogle Scholar
  30. 30.
    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. The American Society of Mechanical Engineers, New York 1993, pp. 25–38.Google Scholar
  31. 31.
    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
  32. 32.
    Ehlers, W.: Grundlegende Konzepte in der Theorie Poröser Medien. Technische Mechanik 16 (1996), 63–76.Google Scholar
  33. 33.
    Ehlers, W.: Continuum and numerical simulation of porous materials in science and technology. In Capriz, G., Ghionna, V. N., Giovine, G. (eds.): Modelling and Mechanics of Granular and Porous Materials, Modelling and Simulation in Science, Engineering and Technology Series, Birkhäuser Verlag, Basel 2002, pp. 243–289.Google Scholar
  34. 34.
    Ehlers, W., Ammann, M., Diebels, S.: h-Adaptive FE methods applied to single-and multiphase problems. International Journal for Numerical Methods in Engineering 54 (2002), 219–239.MATHCrossRefGoogle Scholar
  35. 35.
    Ehlers, W., Blome, P.: A triphasic model for unsaturated soils based on the theory of porous media. Mathematical and Computer Modelling,submitted.Google Scholar
  36. 36.
    Ehlers, W., Droste, A.: A continuum model for highly porous aluminium foam. Technische Mechanik 19 (1999), 341–350.Google Scholar
  37. 37.
    Ehlers, W., Droste, A.: FE simulations of metal foams based on the macroscopic approach of the Theory of Porous Media. In Banhart, J., Ashby, M., Fleck, N. (eds.): Proceedings of the International Conference of Metal Foams and Porous Metal Structures. Verlag MIT, Bremen 1999, pp. 299–302.Google Scholar
  38. 38.
    Ehlers, W., Ellsiepen, P.: Adaptive Zeitintegrations-Verfahren für ein elastisch-viskoplastisches Zweiphasenmodell. ZAMM 78 (1998), S361–S362.MATHGoogle Scholar
  39. 39.
    Ehlers, W., Ellsiepen, P.: PANDAS: Ein FE-System zur Simulation von Sonderproblemen der Bodenmechanik. In Wriggers, P., Meißner, U., Stein, E., Wunderlich, W. (eds.): Finite Elemente in der Baupraxis: Modellierung, Berechnung und Konstruktion. Ernst & Sohn, Berlin 1998, pp. 391–400. Beiträge zur Tagung FEM ‘88 an der TU Darmstadt am 5. und 6. März 1998.Google Scholar
  40. 40.
    Ehlers, W., Ellsiepen, P.: Zeit-und ortsadaptive Verfahren zur Berechnung von Scherbändern in porösen Materialien. In Mahnken, R. (ed.): Theoretische und Numerische Methoden in der Angewandten Mechanik mit Praxisbeispielen. Forschungs-und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover, Bericht-Nr. F98/4, 1998, pp. 99–106. Festschrift anläßlich der Emeritierung von Herrn Prof. Dr.-Ing. Dr.-Ing. E. h. E. Stein.Google Scholar
  41. 41.
    Ehlers, W., Ellsiepen, P.: Theoretical and numerical methods in environmental continuum mechanics based on the theory of porous media. In Schrefler, B. A. (ed.): Environmental Geomechanics, CISM Courses and Lectures No. 417. Springer-Verlag, Wien 2001, pp. 1–81.Google Scholar
  42. 42.
    Ehlers, W., Ellsiepen, P., Ammann, M.: Time-and space-adaptive methods applied to localization phenomena in empty and saturated micropolar and standard porous materials. International Journal for Numerical Methods in Engineering 52 (2001), 503–526.MATHCrossRefGoogle Scholar
  43. 43.
    Ehlers, W., Ellsiepen, P., Blome, P., Mahnkopf, D., Markert, B.: Theoretische und numerische Studien zur Lösung von Rand-und Anfangswertproblemen in der Theorie Poröser Medien, Abschlußbericht zum DFG-Forschungsvorhaben Eh 107/6–2. Bericht aus dem Institut für Mechanik (Bauwesen), Nr. 99–II-1, Universität Stuttgart 1999.Google Scholar
  44. 44.
    Ehlers, W., Markert, B.: On the viscoelastic behaviour of fluid-saturated materials. Granular Matter 2 (2000), 153–161.CrossRefGoogle Scholar
  45. 45.
    Ehlers, W., Markert, B.: A linear viscoelastic biphasic model for soft tissues based on the theory of porous media. ASME Journal of Biomechanical Engineering 123 (2001), 418–424.CrossRefGoogle Scholar
  46. 46.
    Ehlers, W., Markert, B.: A visco-elastic two-phase model for articular cartilage tissues. In Ehlers, W. (ed.): IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials. Kluwer, Dordrecht 2001, pp. 87–92.Google Scholar
  47. 47.
    Ehlers, W., Markert, B.: A macroscopic finite strain model for cellular polymers. International Journal of Plasticity,in press.Google Scholar
  48. 48.
    Ehlers, W., Markert, B., Klar, O.: Biphasic description of viscoelastic foams by use of an extended Ogden-type formulation. In Ehlers, W., Bluhm, J. (eds.): Porous Media: Theory, Experiments and Numerical Applications. Springer-Verlag, Berlin 2002, pp. 275–294.Google Scholar
  49. 49.
    Ehlers, W., Müllerschön, H.: Stress-strain behaviour of cohesionless soils: Experiments, theory and numerical computations. In Cividini, A. (ed.): Application of Numerical Methods to Geotechnical Problems, CISM Courses and Lectures No. 397. Springer-Verlag, Wien 1998, pp. 675–684.Google Scholar
  50. 50.
    Ehlers, W., Volk, W.: On shear band localization phenomena induced by elasto-plastic consolidation of fluid-saturated soils. In Owen, D. R. J., Oíiate, E., Hinton, E. (eds.): Computational Plasticity - Fundamentals and Applications. CIMNE, Barcelona 1997, pp. 1657–1664.Google Scholar
  51. 51.
    Ehlers, W., Volk, W.: On shear band localization phenomena of liquid-saturated granular elasto-plastic porous solid materials accounting for fluid viscosity and micropolar solid rotations. Mech. Cohes.-Frict. Mater. 2 (1997), 301–320.CrossRefGoogle Scholar
  52. 52.
    Ehlers, W., Volk, W.: On theoretical and numerical methods in the theory of porous media based on polar and non-polar elasto-plastic solid materials. Int. J. Solids Structures 35 (1998), 4597–4617.MATHCrossRefGoogle Scholar
  53. 53.
    Ehlers, W., Volk, W.: Localization phenomena in liquid-saturated and empty porous solids. Transport in Porous Media 34 (1999), 159–177.MathSciNetCrossRefGoogle Scholar
  54. 54.
    Eipper, G.: Theorie und Numerik finiter elastischer Deformationen in fluidgesättigten porösen Medien. Dissertation, Bericht Nr. II-1 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 1998.Google Scholar
  55. 55.
    Ellsiepen, P.: Zeit-und ortsadaptive Verfahren angewandt auf Mehrphasenprobleme poröser Medien. Dissertation, Bericht Nr. II-3 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 1999.Google Scholar
  56. 56.
    Eringen, A. C., Kafadar, C. B.: Polar field theories. In Eringen, A. C. (ed.): Continuum Physics, Vol. VI. Academic Press, New York 1976, pp. 1–73.Google Scholar
  57. 57.
    Gallimard, L., Ladevéze, P., Pelle, J. P.: Error estimation and adaptivity in elastoplasticity. Int. J. Numer. Methods Engng. 39 (1996), 129–217.CrossRefGoogle Scholar
  58. 58.
    Günther, W.: Zur Statik und Kinematik des Cosseratschen Kontinuums. Abh. Braunschweig. Wiss. Ges. 10 (1958), 195–213.MATHGoogle Scholar
  59. 59.
    Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods. Springer-Verlag, Berlin 1989.Google Scholar
  60. 60.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations, Vol. 2: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin 1991.Google Scholar
  61. 61.
    Hassanizadeh, S. M., Gray, W. G.: General conservation equations for multiphase systems: 1. Averaging procedure. Adv. Water Resources 2 (1979), 131144.Google Scholar
  62. 62.
    Hassanizadeh, S. M., Gray, W. G.: General conservation equations for multiphase systems: 2. Mass, momentum, energy and entropy equations. Adv. Water Resources 2 (1979), 191–203.Google Scholar
  63. 63.
    Haupt, P.: Foundation of continuum mechanics. In Hutter, K. (ed.): Continuum Mechanics in Environmental Sciences and Geophysics, CISM Courses and Lectures No. 337. Springer-Verlag, Wien 1993, pp. 1–77.Google Scholar
  64. 64.
    Haupt, P.: Continuum Mechanics and Theory of Materials. Springer-Verlag, Berlin 2000.MATHGoogle Scholar
  65. 65.
    Heinrich, G., Desoyer, K.: Hydromechanische Grundlagen für die Behandlung von stationären und instationären Grundwasserströmungen. Ing.-Archiv 23 (1955), 182–185.MathSciNetCrossRefGoogle Scholar
  66. 66.
    Heinrich, G., Desoyer, K.: Hydromechanische Grundlagen für die Behandlung von stationären und instationären Grundwasserströmungen, II. Mitteilung. Ing.-Archiv 24 (1956), 81–84.MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Heinrich, G., Desoyer, K.: Theorie dreidimensionaler Setzungsvorgänge in Tonschichten. Ing.-Archiv 30 (1961), 225–253.MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Huyghe, J. M., Jansson, C. F., Lanier, Y., von Donkelaar, C. C., Maroudas, A., van Campen, D. H.: Experimental measurement of electrical conductivity and electro-osmotic permeability of ionised porous media. In Ehlers, W., Bluhm, J. (eds.): Porous Media: Theory, Experiments and Numerical Applications. Springer-Verlag, Berlin 2002, pp. 295–313.Google Scholar
  69. 69.
    Kafadar, C. B., Eringen, A. C.: Micropolar media - I: the classical theory. Int. J. Engng. Sci. 9 (1971), 271–305.MATHCrossRefGoogle Scholar
  70. 70.
    Kossaczkÿ, I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comp. Appl. Math. 55 (1994), 275–288.MATHCrossRefGoogle Scholar
  71. 71.
    Krause, R., Rank, E.: A fast algorithm for point-location in a finite element mesh. Computing 57 (1996), 49–62.MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Lade, P., de Boer, R.: The concept of effective stress for soil, concrete and rock. Géotechnique 47 (1997), 61–78.CrossRefGoogle Scholar
  73. 73.
    Ladevèse, P., Pelle, J. P., Rougeot, P.: Error estimation and mesh optimization for classic finite elements. Eng. Comp. 8 (1991), 69–80.Google Scholar
  74. 74.
    Lewis, R. W., Schrefler, B. A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd Edition. Wiley, Chichester 1998.MATHGoogle Scholar
  75. 75.
    Liu, I-S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46 (1972), 131–148.MATHGoogle Scholar
  76. 76.
    Liu, I-S., Müller, I.: Thermodynamics of mixtures of fluids. In Truesdell, C. (ed.): Rational Thermodynamics, 2nd Edition. Springer-Verlag, New York 1984, pp. 264–285.CrossRefGoogle Scholar
  77. 77.
    Mahnkopf, D.: Lokalisierung fluidgesättigter poröser Festkörper bei finiten elastoplastischen Deformationen. Dissertation, Bericht Nr. II-5 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 2000.Google Scholar
  78. 78.
    Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering 171 (1999), 387–418.MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Mitchell, W. F.: Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. J. Comp. Appl. Math. 36 (1991), 65–78.MATHCrossRefGoogle Scholar
  80. 80.
    Mow, V. C., Ateshian, G. A., Lai, W. M., Gu, W. Y.: Effects on fixed charges on the stress-relaxation behavior of hydrated soft tissues in a confined compression problem. Int. J. Solids Structures 35 (1998), 4945–4962.MATHCrossRefGoogle Scholar
  81. 81.
    Mow, V. C., Gibbs, M. C., Lai, W. M., Zhu, W. B., Athanasiou, K. A.: Biphasic indentation of articular cartilage - II. J. Biomechanics 22 (1989), 853–861.CrossRefGoogle Scholar
  82. 82.
    Mow, V. C., Sun, D. D., Guo, X. E., Likhitpanichkul, M., Lai, W. M.: Fixed negative charges modulate mechanical behaviours and electrical signals in articular cartilage under confined compression. In Ehlers, W., Bluhm, J. (eds.): Porous Media: Theory, Experiments and Numerical Applications. Springer-Verlag, Berlin 2002, pp. 227–247.Google Scholar
  83. 83.
    Müllerschön, H.: Spannungs-Verzerrungsverhalten granularer Materialien am Beispiel von Berliner Sand. Dissertation, Bericht Nr. II-6 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 2000.Google Scholar
  84. 84.
    Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon Press, Oxford 1986.MATHGoogle Scholar
  85. 85.
    Perzyna, P.: Fundamental problems in viscoplasticity. Adv. Appl. Mech. Eng. 9 (1966), 243–377.CrossRefGoogle Scholar
  86. 86.
    Plischka, J.: Die Bedeutung der Durchschnittsbildungstheorie für die Theorie poröser Medien. Dissertation, Fachbereich Bauwesen, Universität-GH-Essen 1992.Google Scholar
  87. 87.
    Schrefler, B. A.: Modelling of subsidence due to water or hydrocarbon withdraw from the subsoil. In Schrefler, B. A. (ed.): Environmental Geomechanics,CISM Courses and Lectures No. 417. Springer-Verlag, Wien 2001, pp. 235301.Google Scholar
  88. 88.
    Schrefler, B. A., Scotta, R.: A fully coupled model for two-pase flow in de-formable porous media. Comp. Methods Appl. Mech. Engrg. 190 (2001), 32233246.Google Scholar
  89. 89.
    Schrefler, B. A., Simoni, L., Xikui, L., Zienkiewicz, O. C.: Mechanics of partially saturated porous media. In Desai, C. S., Gioda, G. (eds.): Numerical Methods and Constitutive Modelling in Geomechanics, CISM Courses and Lectures No. 311. Springer-Verlag, Wien 1990, pp. 169–209.Google Scholar
  90. 90.
    Schrefler, B. A., Zhan, X.: A fully coupled model for water flow and air flow in deformable porous media. Water Res. Research 29 (1993), 155–167.CrossRefGoogle Scholar
  91. 91.
    Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen. Habilitation, Bericht Nr. 1–7 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 2000.Google Scholar
  92. 92.
    Shewchuk, J. R.: Triangle: A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator School of Computer Science, Carnegie Mellon University, Pittsburgh, Pensilvania 1996. http://vim.cs.cmu.edu/“quake/triangel.html.Google Scholar
  93. 93.
    Skempton, A. W.: Significance of Terzaghi’s concept of effective stress (Terzaghi’s discovery of effective stress). In Bjerrum, L., Casagrande, A., Peck, R. B., Skempton, A. W. (eds.): From Theory to Practice in Soil Mechanics. Wiley, New York 1960, pp. 42–53.Google Scholar
  94. 94.
    Steinmann, P.: A micropolar theory of finite deformation and finite rotation multiplicative elasto-plasticity. Int. J. Solids Structures 31 (1994), 1063–1084.MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    Suquet, P. M.: Elements of homogenization for inelastic solid mechanics. In Sanches-Palencia, E., Zaoui, A. (eds.): Homogenization techniques for composite media Lecture Notes in Physics, Springer-Verlag, Berlin 1987, pp. 193277.Google Scholar
  96. 96.
    Svendsen, B., Hutter, K.: On the thermodynamics of a mixture of isotropic materials with constraints. Int. J. Engng Sci. 33 (1995), 2021–2054.MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Terzaghi, K.: Die Berechnung der Durchlässigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. IIa 132 (1923), 125–138.Google Scholar
  98. 98.
    Terzaghi, K.: Erdbaumechanik auf bodenphysikalischer Grundlage. Franz Deuticke, Leipzig 1925.MATHGoogle Scholar
  99. 99.
    Terzaghi, K., Jelinek, R.: Theoretische Bodenmechanik. Springer-Verlag, Berlin 1954.CrossRefGoogle Scholar
  100. 100.
    Truesdell, C.: Sulle basi delle termomeccanica. Rend. Lincei 22 (1957), 158166.Google Scholar
  101. 101.
    Truesdell, C.: Rational Thermodynamics, 2nd Edition. Springer-Verlag, New York 1984.MATHCrossRefGoogle Scholar
  102. 102.
    Truesdell, C.: Thermodynamics of diffusion. In Truesdell, C. (ed.): Rational Thermodynamics. 2nd Edition, Springer-Verlag, New York 1984, pp. 219–236.CrossRefGoogle Scholar
  103. 103.
    Truesdell, C., Toupin, R. A.: The classical field theories. In Flügge, S. (ed.): Handbuch der Physik, Vol. III/1, Springer-Verlag, Berlin 1960, pp. 226–902.Google Scholar
  104. 104.
    Ulm, F.-J., Coussy, O.: Environmental chemomechanics of concrete. In Schrefler, B. A. (ed.): Environmental Geomechanics, CISM Courses and Lectures No. 417. Springer-Verlag, Wien 2001, pp. 301–350.Google Scholar
  105. 105.
    van Genuchten, M. T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44 (1980), 892–898.CrossRefGoogle Scholar
  106. 106.
    Volk, W.: Untersuchung des Lokalisierungsverhaltens mikropolarer poröser Medien mit Hilfe der Cosserat-Theorie. Dissertation, Bericht Nr. II-2 aus dem Institut für Mechanik ( Bauwesen ), Universität Stuttgart 1999.Google Scholar
  107. 107.
    Woltman, R.: Beyträge zur Hydraulischen Architektur. Dritter Band, Johann Christian Dietrich, Göttingen 1794.Google Scholar
  108. 108.
    Zienkiewicz, O. C., Zhu, J. Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Engng. 24 (1987), 337–357.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Wolfgang Ehlers
    • 1
  1. 1.Institute of Applied Mechanics (CE)University of StuttgartStuttgartGermany

Personalised recommendations