Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I : modelling of incompressible charged porous media
Malakpoor, Kamyar ; Kaasschieter, Enrique F. ; Huyghe, Jacques M.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007), p. 661-678 / Harvested from Numdam

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.

Publié le : 2007-01-01
DOI : https://doi.org/10.1051/m2an:2007036
Classification:  76S05,  74B05,  74F10
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     author = {Malakpoor, Kamyar and Kaasschieter, Enrique F. and Huyghe, Jacques M.},
     title = {Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I : modelling of incompressible charged porous media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {41},
     year = {2007},
     pages = {661-678},
     doi = {10.1051/m2an:2007036},
     mrnumber = {2362910},
     zbl = {pre05289492},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2007__41_4_661_0}
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Malakpoor, Kamyar; Kaasschieter, Enrique F.; Huyghe, Jacques M. Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I : modelling of incompressible charged porous media. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 661-678. doi : 10.1051/m2an:2007036. http://gdmltest.u-ga.fr/item/M2AN_2007__41_4_661_0/

[1] R.M. Bowen, Theory of mixtures, in Continuum Physics, A.C. Eringen Ed., Vol. III, Academic Press, New York (1976) 1-127.

[2] R.M. Bowen, Incompressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 18 (1980) 1129-1148. | Zbl 0446.73005

[3] R.M. Bowen, Compressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 20 (1982) 697-735. | Zbl 0484.76102

[4] Y. Chen, X. Chen and T. Hisada, 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. | Zbl pre05155571

[5] B.D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13 (1963) 167-178. | Zbl 0113.17802

[6] 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. | Zbl 1062.76050

[7] A.J.H. Frijns, A four-component mixture theory applied to cartilaginous tissues. Ph.D. thesis, Eindhoven University of Technology (2001). | Zbl 0966.92002

[8] 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.

[9] W.Y. Gu, W.M. Lai and V.C. Mow, Transport of multi-electrolytes in charged hydrated biological soft tissues. Transport Porous Med. 34 (1999) 143-157.

[10] F. Helfferich, Ion exchange. McGraw-Hill, New York (1962).

[11] G.A. Holzapfel, Nonlinear Solid Mechanics. Wiley (2000). | MR 1827472 | Zbl 0980.74001

[12] J.M. Huyghe and J.D. Janssen, Quadriphasic mechanics of swelling incompressible porous media. Int. J. Engng. Sci. 35 (1997) 793-802. | Zbl 0903.73004

[13] 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.

[14] W.M. Lai, J.S. Houa and V.C. Mow, A triphasic theory for the swelling and deformation behaviours of articular cartilage. ASME J. Biomech. Eng. 113 (1991) 245-258.

[15] E.G. Richards, An introduction to physical properties of large molecules in solute. Cambridge University Press, Cambridge (1980).

[16] 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.