×

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach. (English) Zbl 0936.74025

Summary: After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic-Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.

MSC:

74E05 Inhomogeneity in solid mechanics
74B99 Elastic materials
74A20 Theory of constitutive functions in solid mechanics
74M25 Micromechanics of solids
Full Text: DOI

References:

[1] Auriault, J.-L; Sanchez-Palencia, E., Etude du comportement macroscopique d’un milieu poreux saturé déformable, J. Méc., 16, n° 4, 575-603 (1977) · Zbl 0382.73013
[2] Ball, J. M., Convexity conditions and existence theorems in non-linear elasticity, Arch. Ration. Mech. Anal., 63, 337-403 (1977) · Zbl 0368.73040
[3] Biot, M. A., Theory of finite deformations of porous solids, Indiana Univ. Math. J., 21, 7, 597-620 (1972) · Zbl 0218.76090
[4] Biot, M. A., Variational lagrangian-thermodynamics of non isothermal finite strain mechanics of porous solids and thermomolecular diffusion, Int. J. Solids Structures, 13, 579-597 (1977) · Zbl 0361.73007
[5] Cheng, A. H.D, Material coefficients of anisotropic poroelasticity, Int. J. Rock Mech. Min. Sci., 34, n° 2, 199-205 (1997)
[6] Ciarlet, P. G., Elasticité tridimensionnelle (1986), Masson: Masson Paris · Zbl 0572.73027
[7] Cieszko, M.; Kubik, J., Constitutive relations and internal equilibrium condition for fluid-saturated porous solids-non linear theory, Arch. Mech., 48, 5, 893-910 (1996) · Zbl 0888.73002
[8] Coussy, O., Thermodynamics of saturated porous solids in finite deformation, Eur. J. Mech. A/Solids, 8, 1-14 (1989) · Zbl 0674.73005
[9] Coussy, O., Mechanics of Porous Continua (1995), John Wiley: John Wiley N.Y · Zbl 0838.73001
[10] Dangla, P.; Coussy, O., Drainage and drying of deformable porous materials: 1D case study, (Proc. IUTAM Symp., Mechanics of Granular and Porous Materials (1996))
[11] Dormieux, L.; Stolz, C., Variational approach for poroelastic medium, C. R. Acad. Sci. Paris, 315, 407-412 (1992), (in French, abridged English version) · Zbl 0749.73004
[12] Hill, R., On constitutive macro-variables for heterogeneous solids at finite strain, (Proc. R. Soc. A, 326 (1971)), 131-147 · Zbl 0229.73004
[13] Kowalczyk, P.; Kleiber, M., Modelling and numerical analysis of stresses and strains in the human lung including tissues-gas interaction, Eur. J. Mech. A/Solids, 13, n° 3, 367-393 (1994) · Zbl 0820.73061
[14] Ogden, R. W., On the overall moduli of non-linear elastic composite materials, J. Mech. Phys. Solids, 22, 541-553 (1974) · Zbl 0293.73003
[15] Thompson, M.; Willis, J. R., A reformulation of the equations of anisotropic poroelasticity, J. Appl. Mech. ASME, 58, 612-616 (1991) · Zbl 0754.73017
[16] Van Campen, D. H.; Huyghe, J. M.; Bovendeerd, P. H.M; Arts, T., Biomechanics of the heart muscle, Eur. J. Mech. A/Solids, 13, 19-41 (1994), special issue · Zbl 0815.73045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.