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TECHNICAL PAPERS: Soft Tissue

# Anisotropy of Fibrous Tissues in Relation to the Distribution of Tensed and Buckled Fibers

[+] Author and Article Information
Gerard A. Ateshian

Department of Mechanical Engineering, Columbia University, 500 West 120th Street, MC4703, 220 S.W. Mudd, New York, NY 10027ateshian@columbia.edu

In contrast, in the classical continuum approach where the definition of the material points comprising a body does not change, there can only be one material symmetry associated with the reference configuration $κ0(B)$ of the tissue.

Note that $0≤ϕ0≤π∕2$.

The strain energy density $Wt$ of a real fibrous tissue could be represented by $Wt=W+Wg$, where $Wg$ is the strain energy density of the ground substance.

Similarly, it can be shown that such a material cannot sustain uniaxial unconfined compression loading, by repeating this analysis for the $CT$ case. The corresponding solution yields $ϕ0=π∕2$, indicating that none of the fibers would be resisting load and implying that the material would collapse axially.

J Biomech Eng 129(2), 240-249 (Sep 29, 2006) (10 pages) doi:10.1115/1.2486179 History: Received April 18, 2006; Revised September 29, 2006

## Abstract

Fibrous tissues are characterized by a much higher stiffness in tension than compression. This study uses microstructural modeling to analyze the material symmetry of fibrous tissues undergoing tension and compression, to better understand how material symmetry relates to the distribution of tensed and buckled fibers. The analysis is also used to determine whether the behavior predicted from a microstructural model can be identically described by phenomenological continuum models. The analysis confirms that in the case when all the fibers are in tension in the current configuration, the material symmetry of a fibrous tissue in the corresponding reference configuration is dictated by the symmetry of its fiber angular distribution in that configuration. However, if the strain field exhibits a mix of tensile and compressive principal normal strains, the fibrous tissue is represented by a material body which consists only of those fibers which are in tension; the material symmetry of this body may be deduced from the superposition of the planes of symmetry of the strain and the planes of symmetry of the angular fiber distribution. Thus the material symmetry is dictated by the symmetry of the angular distribution of only those fibers which are in tension. Examples are provided for various fiber angular distribution symmetries. In particular, it is found that a fibrous tissue with isotropic fiber angular distribution exhibits orthotropic symmetry when subjected to a mix of tensile and compressive principal normal strains, with the planes of symmetry normal to the principal directions of the strain. This anisotropy occurs even under infinitesimal strains and is distinct from the anisotropy induced from the finite rotation of fibers. It is also noted that fibrous materials are not stable under all strain states due to the inability of fibers to sustain compression along their axis; this instability can be overcome by the incorporation of a ground matrix. It is concluded that the material response predicted using a microstructural model of the fibers cannot be described exactly by phenomenological continuum models. These results are also applicable to nonbiological fiber–composite materials.

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## Figures

Figure 1

Double elliptical cone of normal strain, shown with its three planes of symmetry

Figure 2

The three planes of symmetry of the elliptical cone of normal strain in relation to planes of symmetry (not shown) of a transversely isotropic fiber angular distribution represented by a cone: (a) example when all three planes coincide; (b) alternate example when all three planes coincide; (c) example of only one coincident plane (normal to m3); and (d) example of no coincident planes

Figure 3

The three planes of symmetry of the elliptical cone of normal strain (with unit normals {m1,m2,m3}) in relation to the three planes of symmetry of an orthotropic fiber angular distribution (with unit normals {a1,a2,a3}): (a) example when all three planes coincide; (b) example of only one coincident plane (normal to a3 and m3); and (c) example of no coincident planes

Figure 4

Conical angle of fiber recruitment, ϕ0, versus λg∕λf, for uniaxial loading of a bar of fibrous material with isotropic fiber angular distribution

Figure 5

Tensile Young’s modulus for the example shown in Fig. 4, normalized to EY∞=(9−23)λf∕45

Figure 6

Apparent Poisson’s ratio ν=−E1∕E3=1∕tan2ϕ0 corresponding to Fig. 4

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