A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage
Summary (2 min read)
Introduction
- The work in this paper is motivated by the difficulty of, and the need for, developing accurate stress constitutive equa tions for fiber-reinforced cartilaginous tissues.
- For articular cartilage, there has been no finite deformation model presented that accurately describes its orthotropic response for multiple experimental protocols including tension and compression.
- Those models were based on a bimodular theory for infinitesimal strains (Curnier et al. 1995) in which the material constants may be discon tinuous (or jump) across a surface of discontinuity in strain space, provided that the stress continuity conditions are sat isfied at the surface.
2 Methods
- The authors outline the derivation of a second-order stress constitutive equation for orthotropic materials.
- Then, the authors propose a general theory for bimodular elastic materials and, consequently, derive a bimodular second-order stress constitutive equation.
- Finally, the authors study the abilities of bi modular second-order and exponential models to describe the mechanical response of articular cartilage in uniaxial tension (UT) and confined compression (CC).
- The deformation gradient tensor F is uniquely decomposed by the polar decomposition theorem as F = RU, (1) where R is a proper-orthogonal tensor and the right stretch tensor U is a symmetric positive-definite tensor.
T = RT̂(U)RT = RT̃(E)RT , P = RP̂(U) = RP̃(E), (6)
- For isotropic elastic materials, various second-order theo ries for Green-elastic materials have been proposed using different strain tensors (Hoger 1999; Murnaghan 1937, 1951; Rivlin 1953) which differ depending on which strain tensor is used (Ogden 1984).
- First, the authors require that the first-order constants {λ11, λ22, λ33} be continuous across the surfaces of discontinuity.
- The bimodular stress constitutive equation corresponding to the general second-order orthotropic material may have a total of 61 mate rial constants.
- The Poisson’s ratio at 16% strain was specified using the data reported in Huang et al. (1999).
- Four second-order models were studied, with 6, 7, 8, and 9 parameters (Table 2), and one exponential model was studied, with 7 parameters.
3 Results
- Qualitatively, the theoretical predictions of the UT re sponses were the same for all models, while the theoretical predictions of the CC response and Poisson’s ratios were different (Figs. 2–6 and Table 4).
- For the exponential model, the CC responses were nonlinear and equal in all 3 directions while the Poisson’s functions were nonlinear and equal in all 3 directions.
4 Discussion
- A bimodular theory for finite deformations was developed with the aim of accurately modeling the orthotrop ic and asymmetric mechanical response of cartilage.
- For articular cartilage, a stress constit utive equation for finite deformations that can accurately de scribe the orthotropic and asymmetric mechanical response for multiple experimental protocols has not been proposed.
- As mentioned before, Ateshian and colleagues (Soltz and Ateshian 2000; Wang et al. 2003) used a bimodular model for infinitesi mal strains.
- The bimodular second-order stress constitutive equation has several features that may make it desirable for some appli rial constants.
- The results of the present study revealed that the numerical regression algorithm did not initially con verge to a positive-definite stiffness matrix for the secondorder models.
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References
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...Although MRI mea surements of in situ and in vivo joints have predicted that cartilage is subject to average strains of less than 10% under physiologic loading conditions (Eckstein et al. 2000; Herberhold et al. 1999), local strains may be much higher due to nonhomogeneous mechanical properties that depend on both anatomic location (Laasanen et al....
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...For physiologic loading conditions, FEM contact analyses (Donzelli et al. 1999; Krishnan et al. 2003) suggest that in situ cartilage may experience local strains up to 26%, suggest ing that the tissue is in the nonlinear range of its stress–strain relationship (Huang et al. 1999)....
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...For physiologic loading conditions, FEM contact analyses (Donzelli et al. 1999; Krishnan et al. 2003) suggest that in situ cartilage may experience local strains up to 26%, suggest ing that the tissue is in the nonlinear range of its stress–strain relationship (Huang et al....
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...They may be used in FEMs of in vivo joints; the re sults of Donzelli et al. (1999) and Krishnan et al. (2003) sug gest that more accurate stress constitutive equations for large deformations may lead to an improved understanding of car tilage degeneration and failure....
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...2003) and depth from the articular surface (Schinagl et al. 1997; Wang et al. 2001)....
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...2001) and a 6-parameter exponential model with material constants specified to be consistent with published material properties (Baer et al. 2004)....
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