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TECHNICAL PAPERS: Soft Tissue

A Bimodular Polyconvex Anisotropic Strain Energy Function for Articular Cartilage

[+] Author and Article Information
Stephen M. Klisch

Mechanical Engineering Department, California Polytechnic State University, San Luis Obispo, CA 93407sklisch@calpoly.edu

In this paper, the terms “Young’s modulus” and “Poisson’s ratio” will be used to refer to strain-dependent functions because a finite deformation theory is used.

The second-order model in terms of the first Piola-Kirchhoff stress developed in Ref. 23 was shown to satisfy stability criteria; however, the corresponding Cauchy stress was not.

Applications are presented in the “Discussion.”

See the “Discussion” for comments on “over-parameterization” in the context of the nonlinear regression analysis used here.

In particular, strong interaction terms that satisfy the bimodular stress–strain continuity conditions stated in Eq. 7 were not found for an orthotropic material.

See the comment below following Eq. 21.

If the secondary fibers are not bimodular and assumed mechanically equivalent, then the symmetry reduces to orthotropy as in (30,35).

These results are summarized in the “Discussion.”

This limitation is addressed in the “Discussion.”

The results of other models are summarized in the Discussion.

Quote taken from p. 798 of Ref. 44.

J Biomech Eng 129(2), 250-258 (Sep 15, 2006) (9 pages) doi:10.1115/1.2486225 History: Received May 01, 2006; Revised September 15, 2006

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex “strong interaction terms” for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson’s ratios, as well as axial stress–axial strain responses, in tension and compression for finite deformations.

FIGURES IN THIS ARTICLE
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Copyright © 2007 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic of the coordinate system and experimental specimen orientations in relation to anatomical directions. The unit vector E1 is parallel to the local split-line direction, the unit vector E3 is perpendicular to the articular surface, and the unit vector E2 is perpendicular to the split-line direction and parallel to the surface. The cylinders labeled P11, P22, and P33 represent specimens loaded in tension or compression along the E1, E2, and E3 directions, respectively.

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Figure 2

Schematic of the principal and secondary fiber orientations in relation to anatomical directions in the 1-2 plane. The two principal fiber directions are parallel to the unit vectors E1 and E2 and the two secondary fiber directions, denoted as E±12, are oriented at angles of ±ϕ12 to the E1 direction. The weights of the line elements represent the relative strength of the fiber directions as predicted by regression analysis; i.e., the principal fibers along the E1 direction are the strongest while the secondary fibers are the weakest.

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Figure 3

Predictions of the eight-parameter model (8-PAR) for the uniaxial tension (UT) response in the 1, 2, and 3 directions and the unconfined compression (UCC) response in the 1 direction. The theoretical UCC curves in the 2 and 3 directions are within 1% of the curve shown. UCC stress and strain values, although negative by definition, are plotted as positive numbers.

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Figure 4

Predictions of the four-parameter model (4-PAR) for the uniaxial tension (UT) response in the 1, 2, and 3 directions and the unconfined compression (UCC) response in the 1 direction. The theoretical UCC curves in the 2 and 3 directions are within 3% of the curve shown. UCC stress and strain values, although negative by definition, are plotted as positive numbers.

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Figure 5

Prediction of the eight-parameter model that does not include the traction-free boundary condition equations (8-PAR-B) for the uniaxial tension (UT) response in the 2 direction. The predictions for the UT response in the 1 and 3 directions and the UCC response in the 1 direction are similar to those of the 8-PAR model shown in Fig. 3.

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