The solid phase stress tensor in porous media mechanics and the Hill-Mandel condition. (English) Zbl 1170.74324

Gray, William G.; Schrefler, Bernhard A.; Pesavento, Francesco

J. Mech. Phys. Solids 57, No. 3, 539-554 (2009).

Summary: An assessment of the stress tensors used currently for the modeling of partially saturated porous media is made which includes concepts like total stress, solid phase stress, and solid pressure. Thermodynamically constrained averaging theory is used to derive the solid phase stress tensor. It is shown that in the upscaling procedure the Hill conditions are satisfied, which is not trivial. The stress tensor is then compared to traditional stress measures. The physical meaning of two forms of solid pressure and of the Biot coefficient is clarified. Finally, a Bishop-Skempton like form of the stress tensor is obtained and a form of the total stress tensor that does not make use of the effective stress concept.

MSC:

74F10

Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)

Keywords:

porous media; Hill-Mandel condition; elasticity; bishop parameters; skempton stress

References:

[1]	Alonso, E. E.; Gens, A.; Josa, A., A constitutive model for partially saturated soils, Géotechnique, 40, 3, 403-430 (1990)
[2]	Bachmat, Y., Spatial macroscopization of processes in heterogeneous systems, Israel Journal of Technology, 10, 391-403 (1972)
[4]	Biot, M.; Willis, D. G., The elastic coefficients of the theory of consolidation, Journal of Applied Mechanics, 24, 594-601 (1957)
[5]	Biot, M. A., Theory of elasticity and consolidation for a porous anisotropic solid, Journal of Applied Physics, 26, 2, 182-185 (1955) · Zbl 0067.23603
[6]	Bishop, A. W., The principle of effective stress, Teknisk Ukeblad, 39, 859-863 (1959)
[7]	Bishop, A. W., The influence of undrained change in stress on the pore pressure in porous media of low compressibility, Géotechnique, 23, 435-442 (1973)
[8]	Bolzon, G.; Schrefler, B. A., State surfaces of partially saturated soils: an effective pressure approach, Applied Mechanics Reviews, 48, 10, 643-649 (1995)
[9]	Bolzon, G.; Schrefler, B. A.; Zienkiewicz, O. C., Elastoplastic soil constitutive laws generalized to partially saturated states, Géotechnique, 46, 2, 279-289 (1996)
[10]	Borja, R. I., Cam-clay plasticity. Part V: a mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media, Computer Methods in Applied Mechanics and Engineering, 193, 5301-5338 (2004) · Zbl 1112.74430
[12]	Callen, H. B., Thermodynamics and an Introduction to Thermostatistics (1985), Wiley: Wiley New York · Zbl 0989.80500
[13]	Chateau, X.; Dormieux, L., Micromechanics of saturated and unsaturated porous media, International Journal for Numerical and Analytical Methods in Geomechanics, 26, 831-844 (2002) · Zbl 1016.74021
[14]	Coussy, O., Mechanics of Porous Continua (1995), Wiley: Wiley Chichester
[15]	Coussy, O., PoroMechanics (2004), Wiley: Wiley Chichester
[17]	Ehlers, W.; Graf, T.; Ammann, M., Deformation and localization analysis of partially saturated soil, Computer Methods in Applied Mechanics and Engineering, 193, 2885-2910 (2004) · Zbl 1067.74543
[18]	Eringen, A. C., Mechanics of Continua (1980), Krieger: Krieger New York · Zbl 0181.53802
[19]	Fredlund, D. G.; Morgenstern, N. R., Stress state variables for unsaturated soils, ASCE Journal of the Geotechnical Engineering Division, 103, 447-466 (1977)
[20]	Gawin, D.; Pesavento, F.; Schrefler, B. A., Modelling creep and shrinkage of concrete by means of effective stress, Materials and Structures, 40, 579-591 (2007)
[21]	Gray, W. G.; Hassanizadeh, S. M., Unsaturated flow theory including interfacial phenomena, Water Resources Research, 29, 1855-1863 (1991)
[22]	Gray, W. G.; Lee, P. C.Y., On the theorems for local volume averaging of multiphase systems, International Journal of Multiphase Flow, 3, 333-340 (1977) · Zbl 0375.76081
[23]	Gray, W. G.; Miller, C. T., Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview, Advances in Water Resources, 28, 161-180 (2005)
[24]	Gray, W. G.; Miller, C. T., Consistent thermodynamic formulations for multiscale hydrologic systems: fluid pressures, Water Resources Research, 43 (2007), (W09408), doi:10.1029/2006WR005811.
[25]	Gray, W. G.; Schrefler, B. A., Thermodynamic approach to effective stress in partially saturated porous media, European Journal of Mechanics, A/Solids, 20, 521-538 (2001) · Zbl 1034.74019
[26]	Gray, W. G.; Schrefler, B. A., Analysis of the solid phase stress tensor in multiphase porous media, International Journal for Numerical and Analytical Methods in Geomechanics, 31, 4, 541-581 (2007) · Zbl 1196.74040
[27]	Gray, W. G.; Leijnse, A.; Kolar, R. L.; Blain, C. A., Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems (1993), CRC Press: CRC Press Boca Raton · Zbl 0859.35097
[28]	Hassanizadeh, S. M.; Gray, W. G., Mechanics and thermodynamics of multiphase flow in porous media including interphase transport, Advances in Water Resources, 13, 169-186 (1990)
[29]	Haupt, P.; Tsakmakis, C., On the application of dual variables in continuum mechanics, Journal of Continuum Mechanics and Thermodynamics, 1, 165-196 (1989) · Zbl 0771.73003
[30]	Hill, R., Elastic properties of reinforced solids: some theoretical principles, Journal of the Mechanics and Physics of Solids, 11, 357-372 (1963) · Zbl 0114.15804
[31]	Hill, R., The essential structure of constitutive laws for metal composites and polycrystals, Journal of the Mechanics and Physics of Solids, 15, 79-95 (1967)
[32]	Houlsby, G. T., The work input to an unsaturated granular material, Géotechnique, 47, 1, 193-196 (1997)
[33]	Hutter, K.; Jöhnk, K., Methods of Physical Modeling: Continuum Mechanics, Dimensional Analysis, Turbulence (2004), Springer: Springer Berlin · Zbl 1058.76001
[34]	Hutter, K.; Laloui, L.; Vulliet, L., Thermodynamically based mixture models of saturated and unsaturated soils, Mechanics of Cohesive-Frictional Materials, 4, 295-338 (1999)
[36]	Khalili, N.; Khabbaz, M. H.; Villiappan, S., An effective stress based numerical model for flow and deformation in unsaturated soils, Computational Mechanics, 26, 2, 174-184 (2000) · Zbl 0991.74071
[37]	Kohgo, Y.; Nakano, M.; Myazaki, T., Theoretical aspects of constitutive modelling for unsaturated soils, Soils and Foundations, 33, 4, 49-63 (1993)
[38]	Kondepudi, D.; Prigogine, I., Modern Thermodynamics (1998), Wiley: Wiley Chichester · Zbl 0902.00007
[39]	Korobeinikov, S. N., Strictly conjugate stress and strain tensors, Journal of Applied Mechanics and Technical Physics, 41, 3, 513-518 (2000)
[40]	Lewis, R. W.; Schrefler, B. A., A finite element simulation of the subsidence of gas reservoirs undergoing a water drive, (Gallagher, R. H.; Norrie, D. H.; Oden, J. T.; Zienkiewicz, O. C., Finite Elements in Fluids, vol. 4. (1982), Wiley: Wiley Chichester), 179-199
[41]	Lewis, R. W.; Schrefler, B. A., The Finite Element Method in the Deformation and Consolidation of Porous Media (1998), Wiley: Wiley Chichester · Zbl 0935.74004
[43]	Miller, C. T.; Gray, W. G., Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 2. Foundation, Advances in Water Resources, 28, 181-202 (2005)
[45]	Narasimhan, T. N.; Witherspoon, P. A., Numerical model for saturated-unsaturated fluid flow in deformable porous media 1. Theory, Water Resources Research, 13, 3, 657-664 (1977)
[46]	Noorishad, J.; Mehran, M.; Narasimhan, T. N., On the formulation of saturated-unsaturated fluid flow in deformable porous media, Advances in Water Resources, 5, 61-62 (1982)
[47]	Nur, A.; Byerlee, J. D., An exact effective stress law for elastic deformation of rock with fluids, Journal of Geophysical Research, 76, 26, 6414-6419 (1971)
[48]	Nuth, M.; Laloui, L., Effective stress concept in unsaturated soils: clarification and validation of a unified framework, International Journal for Numerical and Analytical Methods in Geomechanics, 32, 771-801 (2008), DOI: 10.1002/nag.645. · Zbl 1273.74293
[49]	Safai, N. M.; Pinder, G. F., Vertical and horizontal land deformation in a desaturating porous medium, Advances in Water Resources, 2, 19-25 (1979)
[52]	Schrefler, B. A.; Gawin, D., The effective stress principle: incremental or finite form?, International Journal for Numerical and Analytical Methods in Geomechanics, 20, 785-814 (1996)
[53]	Schrefler, B. A.; Scotta, R., A fully coupled dynamic model for two-phase fluid flow in deformable porous media, Computer Methods in Applied Mechanics and Engineering, 190, 3223-3246 (2001) · Zbl 0977.74019
[55]	Sheng, D.; Sloan, S. W.; Gens, A., A constitutive model for unsaturated soils: thermomechanical and computational aspects, Computational Mechanics, 33, 6, 453-465 (2004) · Zbl 1115.74347
[57]	Slattery, J. C., Momentum, Energy, and Mass Transfer in Continua (1972), McGraw-Hill: McGraw-Hill New York
[59]	Wheeler, S. J.; Sivakumar, V., An elasto-plastic critical state framework for unsaturated soil, Géotechnique, 41, 1, 35-53 (1995)
[60]	Whitaker, S., The Method of Volume Averaging (Theory and Applications of Transport in Porous Media) (1998), Springer: Springer Berlin
[61]	Williams, W. O., Constitutive equations for flow of an incompressible viscous fluid through a porous medium, Quarterly of Applied Mathematics, 10, 255-267 (1978) · Zbl 0392.76086
[62]	Wilmanski, K., Thermomechanics of Continua (1998), Springer: Springer Berlin · Zbl 0917.73001
[63]	Zienkiewicz, O. C.; Chan, A.; Pastor, M.; Schrefler, B. A.; Shiomi, T., Computational Geomechanics with Special Reference to Earthquake Engineering (1999), Wiley: Wiley Chichester · Zbl 0932.74003