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Mathematical Analysis of Diffusion Models in Poro-Elastic Media

  • Hélène Barucq
  • Monique Madaune-Tort
  • Patrick Saint-Macary
Conference paper

Abstract

We consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity as well as a coupled quasi-static problem in thermoelasticity. The pioneering work of M. A. Biot [1, 2, 3, 4] has given rise to numerous extensions. Among them, R. E. Showalter [7] developed an existence, uniqueness and regularity theory for the degenerate quasi-static system using the theory of linear degenerate evolution equations in Hilbert space while C.M. Dafermos studied the classical coupled thermo-elasticity system which describes the flow of heat through an elastic structure [5]. Herein, we choose to consider the full dynamic coupled system which is a more general model including the previous ones and which describes a more complete phenomenon of consolidation. As far as applications are concerned, Biot’s theory is generally involved for the study of a soil under load but it also provides a good theory for the ultrasonic propagation in fluid-saturated porous media like cancellous bone. Our objective here is to develop the existence-uniqueness theory for the one dimensional systems both in the linear and nonlinear cases using classical functional arguments in the Sobolev background. In the linear case, we present a fixed point method based on J. L. Lions’ results [6] to establish the existence of a solution to the problem while in the nonlinear one, our approach involves Galerkin approximations coupled with a monotonicity method.

Keywords

Porous Media Hydraulic Conductivity Linear Case Nonlinear Case Regularity Theory 
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References

  1. 1.
    M. A. Biot: Le Problème de la Consolidation de Matières Argileuses sous une Charge, Ann. Soc. Sci. Bruxelles B55, (1935) pp. 110–113Google Scholar
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    M. A. Biot: General Theory of Three-Dimensional Consolidation, J. Appl. Phys. 12, (1941) pp. 155–164MATHCrossRefGoogle Scholar
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    M. A. Biot: Theory of Finite Deformations of Porous Solids, Indiana Univ. Math. J. 21 (1972), 597–620MathSciNetCrossRefGoogle Scholar
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    C. M. Dafermos: On the Existence and Asymptotic Stability of Solutions to the Equations of Linear Thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), pp. 241–271MathSciNetMATHCrossRefGoogle Scholar
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    R. Dautray, J. L. Lions: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. (Masson, Paris, 1988)Google Scholar
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    R. E. Showalter: Diffusion in Poro-Elastic Media, Jour. Math. Anal. Appl. 251, (2000) 310–340MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Hélène Barucq
    • 1
  • Monique Madaune-Tort
    • 1
  • Patrick Saint-Macary
    • 1
  1. 1.Laboratoire de Mathématiques Appliquées, IPRA, Avenue de l’UniversitéUniversité de Pau et des Pays de l’AdourPauFrance

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