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Biot's poroelasticity equations by homogenization

  • Robert Burridge
  • Joseph B. Keller
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 154)

Abstract

Equations are derived which'govern the linear macroscopic mechanical behavior of a porous elastic solid saturated with a compressible viscous fluid. The derivation is based on the equations of linear elasticity in the solid, the linearized Navier-Stokes equations in the fluid, and appropriate conditions at the solid-fluid boundary. The scale of the pores is assumed to be small compared to the macroscopic scale, so that the two-space method of homogenization can be used to deduce the macroscopic equations. When the dimensionless viscosity of the fluid is small, the resulting equations are those of Biot, who obtained them by hypothesizing the form of the macroscopic constitutive relations. The present derivation verifies those relations, and shows how the coefficients in them can be calculated, in principle, from the microstructure. When the dimensionless viscosity is of order one, a different equation is obtained, which is that of a viscoelastic solid.

Keywords

Porous Medium Macroscopic Scale Solid Region Field Quantity Fourth Rank Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M.A. Biot, General theory of three-dimensional consolidation, J. App. Phys. 12, 155–164 (1941).Google Scholar
  2. 2.
    , Theory. of elasticity and consolidation for a porous anisotropic solid, J. App. Phys. 26, 182–185 (1955).Google Scholar
  3. 3.
    , General solutions of the equations of elasticity and consolidation for a porous material, J. App. Mech. 23, 91–95 (1956)Google Scholar
  4. 4.
    , The theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low frequency range, II. Higher frequency range, J. Acoust. Soc. Am. 28, 168–178, 179–191 (1956).Google Scholar
  5. 5.
    , Mechanics of deformation and acoustic propagation in porous media, J. App. Phys. 33, 1482–1498 (1962).Google Scholar
  6. 6.
    , Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Am. 34, 1256–1264 (1962).Google Scholar
  7. 7.
    J.R. Rice and M.P. Cleary, Some basic stress-diffusion solutions for fluid saturated elastic porous media with compressible constituents, Rev. Geophys. Space Phys. 14, 227–241 (1976).Google Scholar
  8. 8.
    R. Burridge and C.A. Vargas, The fundamental solution in dynamic poroelasticity, Geophys. J. Roy. Astro. Soc. 58, 61–90 (1978).Google Scholar
  9. 9.
    M.P. Cleary, Fundamental solutions for fluid-saturated porous media and applications to localized rupture phenomena, Ph.D. Thesis, Brown University, Division of Engineering, 1975.Google Scholar
  10. 10.
    , Elastic and dynamic response regimes of fluid-impregnated solids with diverse microstructures, Int. J. Solids Structures 14, 795–819 (1978).Google Scholar
  11. 11.
    E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, 1980.Google Scholar
  12. 12.
    J.B. Keller, Effective behavior of heterogeneous media, in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed., Plenum, New York, 1977, pp. 631–644.Google Scholar
  13. 13.
    A. Bensoussan, J.-L. Lions, and G.C. Papanicolaou, Asymptotic analysis for periodic structures, in Studies in Mathematics and Its Applications, Vol. 5, North-Holland, 1978.Google Scholar
  14. 14.
    E.W. Larsen, Neutron transport and diffusion in inhomogeneous media, II, Nucl. Sci. Eng. 60, 357–368 (1976).Google Scholar
  15. 15.
    J.B. Keller, Darcy's law for flow in porous media and the two-space method, in Nonlinear Partial Differential Equations in Engineering and Applied Science, R.L. Sternberg, A.J. Kalinowski and J.S. Papadakis, eds., Marcel Dekker, New York, 1980, 429–443.Google Scholar
  16. 16.
    F.K. Lehner, A derivation of the field equations for slow viscous flow through a porous medium, I & EC Fundamentals 18, 41–45 (1979).Google Scholar
  17. 17.
    R. Burridge and J.B. Keller, Poroelasticity equations derived from microstructure, J. Acoust. Soc. Amer. 70, 1140–1146 (1981).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Robert Burridge
    • 1
  • Joseph B. Keller
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York University New York
  2. 2.Departments of Mathematics and Mechanical Engineering

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