A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

Abstract

Cartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with highly asymmetric tensile-compressive responses. Due to this complex behavior, it is difficult to develop accurate stress constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compres sion. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain ten sor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equa tion, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.

A nonlinear finite element model of cartilage growth

“…This limitation does not reflect the true nature of native cartilage which has been found to display anisotropic behavior (Wang et al 2003;Ficklin et al 2006). Future work will extend the bimodular stressstrain equation for the collagen constituent to include anisotropic behavior (Klisch 2006a). Also, it has been suggested that to accurately model specific growth experiments a collagen remodeling parameter (related to collagen-specific crosslink or fibril binding molecules) should be included that changes the mechanical properties of the collagen constituent (Klisch et al 2005a).…”

“…Those models were based on a bimodular theory for infinitesimal strains (Curnier et al 1995) in which the material constants may be discontinuous (or jump) across a surface of discontinuity in strain space, provided that stress continuity conditions are satisfied at the surface. Recently, the theory of Curnier et al (1995) was generalized to second-order elasticity by Klisch (2006a). Here, we briefly describe the extension of these theories to the present application.…”

“…Several different models of this type have been investigated and found to provide reasonably accurate predictions of cartilage in confined compression, unconfined compression, and tension (Klisch 2006a). The current model is a simplified version of those studied previously with the collagen modeled as a compressible Neo-Hookean material (Schroder and Neff 2003) (see Appendix B for more details on bimodular materials).…”

“…In order to ensure that the stress-strain equation will be continuous across the surface of discontinuity, we must satisfy the following (necessary and sufficient) conditions (Curnier et al 1995;Klisch 2006a):…”

Special Issue on Cartilage (Part II)

“…The fibrillar nature of the collagen matrix of cartilage produces a modulus which is significantly higher in tension than compression. The formulation of constitutive models which can describe this behavior has posed a challenge in the cartilage literature; Klisch (2006) presents a novel bimodular theory of elastic materials which extends the analysis to the range of finite deformations. Complementing these analyses of the solid matrix behavior is the successful theory of mixtures which can account for interstitial fluid flow and diffusion of charged ions through the charged collagen-proteoglycan matrix of cartilage, as reviewed by Yao and Gu (2006).…”