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TECHNICAL PAPERS: Soft Tissue

Poromechanics of Compressible Charged Porous Media Using the Theory of Mixtures

[+] Author and Article Information
J. M. Huyghe1

Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlandsj.m.r.huyghe@tue.nl

M. M. Molenaar

 Shell International Exploration and Production, B.V. Kessler Park 1, 2288GS Rijswijk, The Netherlandsmathieu.molenaar@shell.com

F. P. Baajens

Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlandsf.p.t.baajens@tue.nl

1

Corresponding author.

J Biomech Eng 129(5), 776-785 (Feb 20, 2007) (10 pages) doi:10.1115/1.2768379 History: Received September 25, 2006; Revised February 20, 2007

Osmotic, electrostatic, and/or hydrational swellings are essential mechanisms in the deformation behavior of porous media, such as biological tissues, synthetic hydrogels, and clay-rich rocks. Present theories are restricted to incompressible constituents. This assumption typically fails for bone, in which electrokinetic effects are closely coupled to deformation. An electrochemomechanical formulation of quasistatic finite deformation of compressible charged porous media is derived from the theory of mixtures. The model consists of a compressible charged porous solid saturated with a compressible ionic solution. Four constituents following different kinematic paths are identified: a charged solid and three streaming constituents carrying either a positive, negative, or no electrical charge, which are the cations, anions, and fluid, respectively. The finite deformation model is reduced to infinitesimal theory. In the limiting case without ionic effects, the presented model is consistent with Blot’s theory. Viscous drag compression is computed under closed circuit and open circuit conditions. Viscous drag compression is shown to be independent of the storage modulus. A compressible version of the electrochemomechanical theory is formulated. Using material parameter values for bone, the theory predicts a substantial influence of density changes on a viscous drag compression simulation. In the context of quasistatic deformations, conflicts between poromechanics and mixture theory are only semantic in nature.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Viscous drag compression: a pressure gradient is applied across a sample. The sample is supported by a porous filter. The hydraulic permeability of the porous filter is orders of magnitude larger than the hydraulic permeability of the sample.

Grahic Jump Location
Figure 2

Computed strain ε, fluid content ζf, cationic content ζ+, and anionic content ζ− as a function of depth x for a viscous drag experiment. Continuous line: storage modulus M=36.4GPa,  *****: storage modulus M=100GPa. The total viscous drag compression is independent of the storage modulus, while the compression of the fluid relative to the solid does depend on the storage modulus. The open circuit and closed circuit results coincide.

Grahic Jump Location
Figure 3

Computed electrical potential ξ as a function of depth x for a viscous drag experiment. Continuous line: closed circuit, M=36.4GPa; dashed line: open circuit, M=36.4GPa; and  *****: open circuit, M=100MPa. In closed circuit condition, the electrical potential is set to zero in the upstream and downstream reservoirs. The potentials calculated are Donnan potentials. The streaming current is F[(ζ+ν+s∕V¯+)−(ζ−ν−s∕V¯−)]=0.098μA∕mm2s. In the open circuit condition, the voltage of the upstream reservoir is set to zero. The resulting electrical potential profile is the sum of a Donnan potential and a streaming potential.

Grahic Jump Location
Figure 4

Computed strain ε, fluid content ζf, cationic content ζ+, and anionic content ζ− as a function of depth x for a viscous drag experiment. M=∞, α=1 (incompressible case). The open circuit and closed circuit results coincide for the quantities shown.

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