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Diffusion in poro-plastic media. (English) Zbl 1095.74011
Summary: A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial-boundary value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto-viscoplastic type. The variational form of this problem in Hilbert space is a nonlinear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi-static momentum equation. The essential sufficient conditions for the well-posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid.

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S05 Finite element methods applied to problems in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
76R50 Diffusion
Full Text: DOI
[1] Sobolev spaces. Pure and Applied Mathematics, vol. 65. Academic Press: New York, 1975.
[2] Auriault, Journal de Mécanique 16 pp 575– (1977)
[3] Babu?ka, RAIRO Modélisation Mathématique et Analyse Numérique 32 pp 521– (1998)
[4] Biot, Journal of Applied Physics 12 pp 155– (1941)
[5] Biot, Journal of Applied Physics 26 pp 182– (1955)
[6] Biot, Indiana University Mathematics Journal 21 pp 597– (1971)
[7] Burridge, Journal of Acoustic Society of America 70 pp 1140– (1981)
[8] Linear thermoelasticity. In Handbuch der Physik, vol. VIa/2. Springer: New York, 1972.
[9] Singular and Degenerate Cauchy Problems. Academic Press: New York, 1976.
[10] Rock mechanics. vol. 1, Theoretical Fundamentals, Editions Technip: Paris, 1991.
[11] A numerical algorithm for single phase fluid flow in elastic porous media. In Lecture Notes in Physics, vol. 552. Springer: Berlin, 2000; 80-92. · Zbl 1018.74037
[12] Coussy, Transport in Porous Media 4 pp 281– (1989) · Zbl 0674.73005
[13] Dafermos, Archives of Rational Mechanics and Analysis 29 pp 241– (1968)
[14] Heat Conduction within Linear Thermoelasticity. Springer: New York, 1985. · Zbl 0577.73009 · doi:10.1007/978-1-4613-9555-3
[15] Inequalities in mechanics and physics. Grundlehren der Mathematishen Wissenshaften, vol. 219. Springer-Verlag: Berlin, 1976.
[16] Fichera, Archives of Mechanics 26 pp 903– (1974)
[17] Plasticity: mathematical theory and numerical analysis. Interdisciplinary Applied Mathematics, vol. 9. Springer-Verlag: New York, 1999. · Zbl 0926.74001
[18] Hollingsworth, Discrete and Continuous Dynamical Systems 1 pp 59– (1995)
[19] Computational Methods in Subsurface Flow. Academic Press: New York, 1983. · Zbl 0577.76001
[20] Johnson, Journal de Mathematiques Pures et Appliquees (9) 55 pp 431– (1976)
[21] Johnson, Journal de Mathematiques Pures et Appliquees 62 pp 325– (1978)
[22] Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakkot?sho: Tokyo, 1996.
[23] Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland: Amsterdam, 1979.
[24] Lewis, International Journal of Numerical and Analytical Methods in Geomechanics 17 pp 577– (1993)
[25] Li, SIAM Journal on Numerical Analysis 33 pp 809– (1996)
[26] Non-homogeneous boundary value problems and applications. die Grundlehren der Mathematishen Wissenshaften, Band 181, vol. 1. Springer-Verlag: New York, Heidelberg, 1972.
[27] Coupled geomechanics and flow simulation for time-lapse seismic modeling. In Proceedings of the 69th Annual International Meeting of the Society of Exploration Geophysicists, Houston, TX, 1999; 1667-1670.
[28] Coupling of geomechanics and reservoir simulation models. In Computer Methods and Advances in Geomechanics, 1994; 2151-2158.
[29] Murad, International Journal of Engineering Science 34 pp 313– (1996)
[30] Rice, Reviews in Geophysics and Space Physics 14 pp 227– (1976)
[31] Rockafellar, Pacific Journal of Mathematics 24 pp 525– (1968) · Zbl 0159.43804 · doi:10.2140/pjm.1968.24.525
[32] Rockafellar, Journal of Mathematical Analysis in Applications 28 pp 4– (1969)
[33] Convex integral functionals and duality. In Contributions to Nonlinear Functional Analysis. (ed.). Academic Press: New York, 1971; 215-236. · doi:10.1016/B978-0-12-775850-3.50012-1
[34] (ed.). Mechanics of poroelastic media. Solid Mechanics and its Applications, vol. 35. Kluwer Acad. Pub.: Dordrecht, 1996.
[35] Monotone operators in banach space and nonlinear differential equations. Mathematical Surveys and Monographs, vol. 49. American Mathematical Society: Providence, 1997.
[36] Showalter, Journal of Mathematical Analysis and Applications 251 pp 310– (2000)
[37] Showalter, Dynamics of Continuous, Discrete and Impulsive Systems 10 pp 661– (2003)
[38] Showalter, Mathematical Methods in Applied Science 25 pp 115– (2002)
[39] Showalter, Journal of Mathematical Analysis and Applications 216 pp 218– (1997)
[40] Showalter, Discrete and Continuous Dynamical Systems (Ser. B: Applications and Algorithms) 1 pp 403– (2001)
[41] Partially saturated flow in a composite poroelastic medium. In Poromechanics II, et al. (eds.). Balkema: Lisse, 2002; 549-554.
[42] Computational inelasticity. Interdisciplinary Applied Mathematics, vol. 7. Springer-Verlag: New York, 1998.
[43] Suquet, Quarterly Applied Mathematics 38 pp 391– (1980)
[44] Erdbaumechanic auf Bodenphysikalischer Grundlage. Franz Deuticke: Leipzig, 1925.
[45] Differential models of hysteresis. Applied Mathematical Sciences, vol. 111. Springer: Berlin, 1994. · Zbl 0820.35004
[46] ?eni?ek, RAIRO Modélisation Mathématique et Analyse Numérique 18 pp 183– (1984)
[47] Computational Geomechanics. Wiley: Chichester, 1999.
[48] Zienkiewicz, Geotechnique 30 pp 385– (1980)
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