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Mathematical modelling and numerical solution of swelling of cartilaginous tissues.Part I: Modelling of incompressible charged porous media

Published online by Cambridge University Press:  04 October 2007

Kamyar Malakpoor ,
Enrique F. Kaasschieter  and
Jacques M. Huyghe
Kamyar Malakpoor
Affiliation:
Correspondence to: K. Malakpoor, Korteweg-de Vries Institute for Mathematics (Faculty NWI), University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands. k.malakpoor@uva.nl
Enrique F. Kaasschieter
Affiliation:
Departement of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. e.f.kaasschieter@tue.nl
Jacques M. Huyghe
Affiliation:
Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. j.m.r.huyghe@tue.nl
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Abstract

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task.

Keywords

Mixture theoryporous mediahydrated soft tissue.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 4 , July 2007 , pp. 661 - 678
Copyright
© EDP Sciences, SMAI, 2007

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References

R.M. Bowen, Theory of mixtures, in Continuum Physics, A.C. Eringen Ed., Vol. III, Academic Press, New York (1976) 1–127.
Bowen, R.M., Incompressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 18 (1980) 1129–1148. CrossRef
Bowen, R.M., Compressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 20 (1982) 697–735. CrossRef
Chen, Y., Chen, X. and Hisada, T., Non-linear finite element analysis of mechanical electrochemical phenomena in hydrated soft tissues based on triphasic theory. Int. J. Numer. Mech. Engng. 65 (2006) 147–173. CrossRef
Coleman, B.D. and Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13 (1963) 167–178. CrossRef
W. Ehlers, Foundations of multiphasic and porous materials, in Porous Media: Theory, Expriements and Numerical Applications, W. Ehlers and J. Blaum Eds., Springer-Verlag, Berlin (2002) 3–86.
A.J.H. Frijns, A four-component mixture theory applied to cartilaginous tissues. Ph.D. thesis, Eindhoven University of Technology (2001).
W.Y. Gu, W.M. Lai and V.C. Mow, A triphasic analysis of negative osmotic flows through charged hydrated soft tissues J. Biomechanics 30 (1997) 71–78.
Gu, W.Y., Lai, W.M. and Mow, V.C., Transport of multi-electrolytes in charged hydrated biological soft tissues. Transport Porous Med. 34 (1999) 143–157.
F. Helfferich, Ion exchange. McGraw-Hill, New York (1962).
G.A. Holzapfel, Nonlinear Solid Mechanics. Wiley (2000).
Huyghe, J.M. and Janssen, J.D., Quadriphasic mechanics of swelling incompressible porous media. Int. J. Engng. Sci. 35 (1997) 793–802. CrossRef
J.M. Huyghe, M.M. Molenaar and F.P.T. Baaijens, Poromechanics of compressible charged porous media using the theory of mixtures. J. Biomech. Eng. (2007) in press.
Lai, W.M., Houa, J.S. and Mow, V.C., A triphasic theory for the swelling and deformation behaviours of articular cartilage. ASME J. Biomech. Eng. 113 (1991) 245–258. CrossRef
E.G. Richards, An introduction to physical properties of large molecules in solute. Cambridge University Press, Cambridge (1980).
C. Truesdell and R.A. Toupin, The classical field theories, in Handbuch der Physik, Vol. III/1, S. Flügge Ed., Springer-Verlag, Berlin (1960) 226–902.