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Research Papers

# Anisotropic Hydraulic Permeability Under Finite Deformation

[+] Author and Article Information
Gerard A. Ateshian

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

Jeffrey A. Weiss

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

$Df(x)[u]=d/dε∣ε=0f(x+εu)$ for any function $f(x)$.

The tensor product $⊗̱$ between second-order tensors $A$ and $B$ is defined by $(A⊗̱B)ijkl=AikBjl$ in a Cartesian basis (15).

The tensor products $⊗$ and $⊗¯̱$ between second-order tensors $A$ and $B$ are, respectively, defined by $(A⊗B)ijkl=AijBkl$ and $(A⊗¯̱B)ijkl=(AikBjl+AilBjk)/2$ in a Cartesian basis (15).

Conventionally, in a transversely isotropic formulation, the strain invariants $M3:C$ and $M3:C2$ are denoted by $I4$ and $I5$, respectively. They appear as $I6$ and $I9$ in this treatment to maintain a consistent notation between orthotropic and transversely isotropic formulations.

A second-order tensor $T$ is isotropic when $T=αI$ for any scalar $α$.

J Biomech Eng 132(11), 111004 (Oct 15, 2010) (7 pages) doi:10.1115/1.4002588 History: Received March 25, 2010; Revised September 09, 2010; Posted September 21, 2010; Published October 15, 2010; Online October 15, 2010

## Abstract

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.

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## Figures

Figure 1

Finite element results for permeation through a cylinder subjected to finite torsion: (a) cylinder geometry in reference configuration, (b) cylinder geometry after one half-turn (180 deg twist), (c) vector plot of relative volume flux of solvent in deformed configuration using k=k0I, which does not account for strain-induced anisotropy, and (d) vector plot of flux in deformed configuration using k=k1b, which accounts for strain-induced anisotropy

Figure 2

Close-up of relative volume flux of solvent, in relation to mesh deformation: (a) reference configuration, (b) k=k0I, (c) k=k1b, and (d) k=2k2b2. Case (c) confirms that this constitutive relation produces strain-induced anisotropy that follows an affine transformation of preferred material directions since the flux realigns exactly with the deforming mesh. Cases (b) and (d) represent nonaffine transformations.

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