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Research Papers

Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation

[+] Author and Article Information
Gerard A. Ateshian, Michael B. Albro

Department of Mechanical Engineering,  Columbia University, New York, NY 10027

Steve Maas, Jeffrey A. Weiss

Department of Bioengineering,  University of Utah, Salt Lake City, UT 84112

J Biomech Eng 133(8), 081005 (Sep 06, 2011) (12 pages) doi:10.1115/1.4004810 History: Received July 21, 2011; Revised August 01, 2011; Posted August 02, 2011; Published September 06, 2011; Online September 06, 2011

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http://mrl.sci.utah.edu/software).

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Finite element mesh for 2D axisymmetric Fickian diffusion

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Figure 2

2D axisymmetric Fickian diffusion, showing finite element results (symbol) and analytical solution (surface), at two representative time steps: (a) t=4.64 s; (b) t=72.08 s.

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Figure 5

Concentration enhancement under a step, infinitesimal, compression of a disk. The radial distribution of the solute concentration, normalized to the bath concentration, is displayed at selected time points. Finite element results (thick black curves) are compared to a finite difference solution (thinner gray curves).

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Figure 6

Concentration enhancement under a step, finite, compression of a disk. The temporal response of the average solute concentration in the disk (normalized to the bath concentration c*), shows an initial rise before slowly subsiding back to the initial, ambient concentration.

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Figure 3

Osmotic loading of a spherical gel, under infinitesimal strain conditions, comparing the finite element solution (symbol) to the finite difference solution (solid curve) at representative time points. (a) Finite element geometry and mesh. (b) Radial displacement of the solid matrix. (c) Solute concentration.

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Figure 4

Relative volume V/V0 and average effective solute concentration c̃avg during osmotic loading of a spherical gel, under finite strain conditions. V/V0 initially decreases as a result of solvent outflux; as the effective solute concentration in the gel slowly increases by diffusion from the bath, solvent flows back into the gel and V/V0 recovers to a steady-state value less than unity (since κ̃<1).

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