Computational Geomechanics. Manuel Pastor

defined constitutive laws allow (almost) all geomechanical phenomena to be studied. In Chapter 3, we shall discuss in some detail approximation by the finite element method leading to their solution.

References

      1 Biot, M. A. (1941). General theory of three‐dimensional consolidation, J. Appl. Phys., 12, 155–164.

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

      3 Biot, M. A. (1956a). Theory of propagation of elastic waves in a fluid‐saturated porous solid, Part I: Low‐frequency range, J. Acoust. Soc. Am., 28, 2, 168–178.

      4 Biot, M. A. (1956b). Theory of propagation of elastic waves in a fluid‐saturated porous solid, Part II: Low‐frequency range, J. Acoust. Soc. Am., 28, 2, 179–191.

      5 Biot, M. A. (1962). Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys., 33, 4, 1482–1498.

      6 Biot, M. A. and Willis, P. G. (1957). The elastic coefficients of the theory consolidation, J. Appl. Mech., 24, 594–601.

      7 Bishop, A. W. (1959). The principle of effective stress, Teknisk Ukeblad, 39, 859–863.

      8 Bolzon, G., Schrefler, B. A. (1995). State surfaces of partially saturated soils: an effective pressure approach. Appl. Mech. Rev., 48 (10), 643–649.

      9 Borja, R. I. (2004). Cam‐Clay plasticity. Part V: A mathematical framework for three‐phase deformation and strain localization analyses of partially saturated porous media. Comp. Methods Appl. Mech. Eng., 193, 5301–5338.

      10 Bowen, R. M. (1976). Theory of Mixtures in Continuum Physics, Academic Press, New York.

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

      12 Bowen, R. M. (1982). Compressible porous media models by use of theories of mixtures, Int. J. Eng. Sci. 20, 697–735.

      13 Chan, A. H. C., Famiyesin, O. O., and Muir Wood, D. (1991). A fully explicit u‐w schemes for dynamic soil and pore fluid interaction. Asian Pacific Conference on Computational Mechanics, Hong Kong, Vol. 1, Balkema, Rotterdam, 881–887 (11–13 December 1991).

      14 Coussy, O. (1995). Mechanics of Porous Media, John Wiley & Sons, Chichester.

      15 Coussy, O. (2004). Poromechanics, John Wiley & Sons, Chichester.

      16 Craig, R. F. (1992). Soil Mechanics (5), Chapman & Hall, London.

      17 De Boer, R. (1996). Highlights in the historical development of the porous media theory, Appl. Mech. Rev., 49, 201–262.

      18 De Boer, R. and Kowalski, S. J. (1983). A plasticity theory for fluid saturated porous solids, Int. J. Eng. Sci., 21, 1343–1357.

      19 De Boer, R., Ehlers, W., Kowalski, S. and Plischka, J. (1991). Porous Media, a Survey of Different Approaches, Forschungsbericht aus dem Fachbereich Bauwesen, 54, Universitaet‐Gesamthochschule Essen.

      20 Derski, W. (1978). Equations of motion for a fluid saturated porous solid, Bull. Acad. Polish Sci. Tech., 26, 11–16.

      21 Desai, C. S. (1977a). Discussion—Finite element, residual schemes for unconfined flow, Int. J. Numer. Methods Eng., 11, 80–81.

      22 Desai, C. S. (1977b). Finite element, residual schemes for unconfined flow, Int. J. Numer. Methods Eng. 10, 1415–1418.

      23 Desai, C. S. and Li, G. C. (1983). A residual flow procedure and application for free surface flow in porous media, Adv. Water Resour., 6, 27–35.

      24 Fredlund, D. G., Morgenstern, N. R. (1977). Stress state variables for unsaturated soils, J. Geotech. Eng. Div. ASCE, 103, 447–466.

      25 Ghaboussi, J. and Wilson, E. L. (1972). Variational formulation of dynamics of fluid saturated porous elastic solids, ASCE EM, 98, EM4, 947–963.

      26 Gray, W. G. and Schrefler, B. A. (2001). Thermodynamic approach to effective stress in partially saturated porous media, Eur. J. Mech. A/Solids, 20, 521–538.

      27 Gray, W.G. and Schrefler, B. A. (2007). Analysis of the solid phase stress tensor in multiphase porous media, Int. J. Numer. Anal. Meth. Geomech., 31, 541–581.

      28 Green, A. E. (1969). On basic equations for mixtures, Quart. J. Mech. Appl. Math., 22, 428–438.

      29 Green, A. E. and Adkin, J. E. (1960). Large Elastic Deformations and Nonlinear Continuum Mechanics, Oxford University Press, London.

      30 Hassanizadeh, M. and Gray, W. G., (1979a). General conservation equations for multiphase systems: 1 Averaging procedure, Adv. Water Resour., 2, 131–144.

      31 Hassanizadeh, M. and Gray, W. G. (1979b). General conservation equations for multiphase systems: 2 Mass, momenta, energy and entropy equations, Adv. Water Resour., 2, 191–203.

      32 Hassanizadeh, M. and Gray, W. G. (1980). General conservation equations for multiphase systems: 3 Constitutive theory for porous media flow, Adv. Water Resour., 3, 25–40.

      33 Hassanizadeh, M. and Gray, W. G. (1990). Mechanics and thermodynamics of multiphase flow in porous media including interphase transport, Adv. Water Resour., 13, 169–186.

      34 Jaumann, G. (1905). Die Grundlagen der Bewegungslehre von einem modernen Standpunkte aus, Leipzig.

      35 Jommi, C. (2000). Remarks on the constitutive modeling of unsaturated soils, from experimental evidence and theoretical approaches in unsaturated soils. Proceedings of International Workshop on Unsaturated Soils, Trento. Italy (10–12 April 2000). Roterdam: Balkema, 139–153.

      36 Khalili, N, Khabbaz, M. H., Valliappan, S. (2000). An effective stress based numerical model for flow and deformation in unsaturated soils. Comput. Mech., 26 (2), 174–184.

      37 Kowalski, S. J. (1979). Comparison of Biot's equation of motion for a fluid saturated porous solid with those of Derski, Bull. Acad. Polish Sci. Tech., 27, 455–461.

      38 Lambe, T. W. and Whitman, R. V. (1969). Soil Mechanics, (SI Version), John Wiley & Sons, New York.

      39 Leliavsky, S. (1947). Experiments on effective area in gravity dams, Trans. Am. Soc. Civ. Eng., 112, 444.

      40 Lewis, R. W. and Schrefler, B. A. (1982). A finite element simulation of the subsidence of gas reservoirs undergoing a waterdrive in Finite Element in Fluids Vol. 4 R.H. Gallagher, D. H. Norrie, J. T. Oden, O. C. Zienkiewicz (Eds), John Wiley, 179–199.

      41 Lewis, R. W. and Schrefler, B. A. (1987). The Finite Element Method in the Deformation and Consolidation of Porous Media, John Wiley & Sons, Chichester.

      42 Lewis, R. W. and Schrefler, B. A. (1998). The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, John Wiley & Sons, Chichester.

      43 Li, X. K. and Zienkiewicz, O. C. (1992). Multiphase flow in deforming porous‐media and finite‐element solutions, Comp. Struct., 45, 2, 211–227.

      44 Li, X. K., Zienkiewicz, O. C. and Xie, Y. M. (1990). A numerical model for immiscible 2‐phase fluid‐flow in a porous medium and its time domain solution, Int. J. Numer. Methods Eng., 30, 6, 1195–1212.

      45 Liakopoulos, A. C. (1965). Transient flow through unsaturated porous media. D.Eng. dissertation. University of California, Berkeley, USA.

      46 Morland, L. W. (1972). A simple constitutive theory for fluid saturated porous solids, J. Geophys. Res., 77, 890–900.

      47 Nuth, M., Laloui, L. (2008). Effective stress concept in unsaturated soils: clarification and validation of a unified framework. Int. J. Numer. Anal. Methods Geomech., 32, 771–801.

      48 Runesson, K. (1978). On non‐linear consolidation of soft clay. Ph.D Thesis. Chalmers University of Technology, Goeteborg.

      49 Sandhu, R. S. and Wilson, E. L. (1969). Finite element analysis of flow in saturated porous elastic media, ASCE EM, 95, 641–652.

      50 Schrefler,