Abstract
Finite elastic or elastoplastic strains in fluid saturated soils are studied. Isothermal and quasi-static loading conditions are considered. The governing equations at the macroscopic level are derived in a spatial and a material setting. The constituents are assumed to be materially incompressible at the microscopic level. The elasto-plastic behaviour of the solid skeleton is described by the multiplicative decomposition of the deformation gradient into an elastic and a plastic part; the elasto-plastic evolution laws are developed in the spatial setting. The Kirchhoff effective stress tensor and logarithmic principal strains are used in conjunction with an hyperelastic free energy function. The effective stress state is limited by the von Mises or the Drucker-Prager yield surface with isotropic hardening. Algorithmically, a particular “apex formulation” is advocated for the latter case. The fluid is assumed to obey Darcy’s law. The consistent linearisation of the fully non-linear coupled system of equations is derived. A spatial finite element formulation is presented. Numerical examples highlight the developments.
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References
D.H.Advani, T.S. Lee, J.K. Lee, C.S. Kim: Hygrothermomechanical evaluation of porous media under finite deformation. Part. I - Finite element formulationsInt. J. Num. Meth. Eng.36, pp. 147–160, 1993.
F. Armero: Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditionsComput. Methods Appl. Mech. Eng.171, pp. 205–241, 1999.
J. Bluhm, R. De Boer: The volume fraction concept in the porous media theoryZAMM Z. Angew. Math. Mech.77, pp. 563–577, 1997.
R.I. Borja, E. Alarcon: A mathematical framework for finite strain elastoplastic consolidation. Part 1: balance laws, variational formulation and linearizationComputer Meth. Appl. Mech. Eng.122, pp. 145–171, 1995.
R.I. Borja, C. Tamagnini: Numerical implementation of a mathematical model for finite strain elastoplastic consolidation, in: D.R.J. OWEN, E. ONATE, E. HINTON (eds.):Computational Plasticity - Fundamentals and ApplicationsCIMNE, Barcelona, 1997, pp. 1631–1640.
P.G. Ciarlet:Mathematical Elasticity. Volume I: Three-Dimensional ElasticityElsevier Science Publishers, Amsterdam, 1988.
R. De Boer:Theory of Porous Media: Highlights in Historical Development and Current StateSpringer, Berlin, 2000.
J. Desrues, R. Chambon, W. Hammad, R. Charlier: Soil mod-elling with regard to consistency: Cloe, a new rate type constitutive model, in: C.S. DESAI, S. KREMPL (eds.):3rd Int. Conf. Constitutive Laws for Eng. Materials TucsonASME press, 1991, pp. 399–402.
S. Diebels, W. Ehlers: Dynamic analysis of a fully saturated porous medium accounting for geometrical and material non-linearitiesInt. J. Numer. Meth. Eng.39, pp. 81–97, 1996.
D.C. Drucker, W. Prager: Soil mechanics and plastic analysis or limit designQuart.Appl. Math.10, pp. 157–165, 1952.
W. Ehlers, G. Eipper: Finite elastic deformation in liquid-saturated and empty porous solidsTrans. Porous Media34, pp. 179–191, 1999.
D. Gawin, L. Sanavia, B.A. Scherfler: Cavitation modelling in saturated geomaterials with application to dynamic strain localisationInt. J. Num. Meth. Fluids27, pp. 109–125, 1998.
W.G. Gray, M. Hassanizadeh: Unsaturated flow theory including interfacial phenomenaWater Resour. Res.27, pp. 1855–1863, 1991.
M. Hassanizadeh, W.G. Gray: General conservation equations for multi-phase system: 1. Averaging technique; 2. Mass, Momenta, Energy and Entropy EquationsAdv. Water Res. 2, pp. 191–201/131–144, 1979.
M. Hassanizadeh, W.G. Gray: General conservation equations for multi-phase system: 3. Constitutive theory for porous media flowAdv. Water Res.3, pp. 25–40, 1980.
G. Hofstetter, R.L. Taylor: Non-associative Drucker-Prager plasticity at finite strainsComm. Appl. Num. Meth. Engrg.6, pp. 583–589, 1990.
E.H. LEE: Elastic-plastic deformation at finite strainsJ. Appl. Mech.36, pp. 1–6, 1969.
R.W. Lewis, B.A. Schrefler:The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous MediaJohn Wiley and Sons, New York, 1998.
J.E. Marsden, T.J.R. Hughes:Mathematical Foundations of ElasticityPrentice Hall Inc., Upper Saddle River, New Jersey, 1983.
E. Meroi, B.A. Schrefler, O.C. Zienkiewicz: Large strain static and dynamic semi-saturated soil behaviourInt. J. Num. Anal. Meth. Geomech.19, pp. 81–106, 1995.
C. Miehe: Computation of isotropic tensor functionsComm. Num. Meth. Eng.9, pp. 889–896, 1993.
M. Mokni, J. Desrues: Strain localisation measurements in undrained plane-strain biaxial tests on Hostun RF sandMech. Coh.-Frict. Mater.4, pp. 419–441, 1998.
S. Nemat-Nasser: On finite plastic flow of crystalline solids and geomaterialsTrans. ASME50, pp. 1114–1126, 1983.
J.F. Peters, P.V. Lade, A. Bro: Shear band formation in triaxial and plane strain tests, in: R.T. DONAGHE, R.C. CHANEY, M.L. SILVER (eds.):AdvancedTriaxialTesting of Soil and Rock, Am. Soc. Testing and Materials 977, Philadelphia, 1988, pp. 604–615.
S. Reese: Elastopastic material behaviour with large elastic and large plastic deformationZAMM Z. Angew. Math. Mech. 77, pp. S277–S278, 1997.
B.A. Schrefler, L. Sanavia, C.E. Majorana: A multiphase medium model for localisation and postlocalisation simulation in geomaterialsMech. Coh.-Frict. Mater.1, pp. 95–114, 1996.
J.C. SIMO: Numerical analysis and simulation of plasticity, in: P.G. CIARLET, J.L. LIONS (eds.): Numerical methods for solids (Part 3)Handbook of Numerical Analysis 6North-Holland, Amsterdam, 1998.
J.C. Simo, T.J.R. Hughes:Computational InelasticitySpringer, Berlin, 1998.
J.C. Simo, R. Taylor: Consistent tangent operators for rate-independent elastoplasticityComp. Meth. Applied Mech. Eng.48, pp. 101–118, 1985.
P. Steinmann: A finite element formulation for strong discontinuities in fluid-saturated porous mediaMech. Coh.-Frict. Mater. 4, pp. 133–152, 1999.
I. VardoulakisJ.SULEM: Bifurcation Analysis in Geomechanics, Blakie Academic and Professional, London, 1995.
P. Wriggers: Continuum mechanics, non-linear finite element techniques and computational stability, in: E. STEIN (ed.):Progress in Computational Analysis of Inelastic Structures CISM 321, Springer-Verlag, Wien-New York, 1993.
H.W. Zhang, L. Sanavia, B.A. Schrefler: An internal length scale in strain localisation of multiphase porous mediaMech. Coh.Frict. Mater.4, pp. 433–460, 1999.
H.W. Zhang, L. Sanavia, B.A. Schrefler: Numerical analysis of dynamic strain localisation in initially water saturated dense sand with a modified generalised plasticity modelComp. and Struct.79, pp. 441–459, 2001.
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Sanavia, L., Schrefler, B.A., Steinmann, P. (2002). A Mathematical and Numerical Model for Finite Elastoplastic Deformations in Fluid Saturated Porous Media. In: Capriz, G., Ghionna, V.N., Giovine, P. (eds) Modeling and Mechanics of Granular and Porous Materials. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0079-6_10
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DOI: https://doi.org/10.1007/978-1-4612-0079-6_10
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