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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...(14)P̃−(E) if gE (E) < 0 PE− In Curnier et al. (1995), a theorem that established necessary and sufficient conditions for the stress–strain equation to be continuous across the surface of discontinuity was proved....
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...Here, that theorem is slightly modified, because the major symmetry of the elasticity tensor for linear elastic materi als was invoked in Curnier et al. (1995) whereas the elas ticity tensor for finitely elastic materials need not possess major symmetry....
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...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....
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...…constants {λ11, λ22, λ33, λ, μ, γ1, γ2, γ3}, which are defined in terms of the strain energy function W in Appendix B. 2.3 Bimodular elastic materials Curnier et al. (1995) developed a bimodular theory for lin ear elastic materials in terms of the second Piola–Kirchhoff stress and Lagrange…...
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...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....
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180 citations
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...A justification for not allowing the first-order constants to jump is that the experi mental stress–strain response is continuous through the ori gin in strain space, as demonstrated for the annulus fibrosus (Wagner and Lotz 2004) and articular cartilage (Chahine et al. 2004)....
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...Lotz and colleagues developed an orthotropic finite deformation model for the annulus fibrosus using an exponential strain energy function; however, maximum errors between the the oretical and experimental stresses in uniaxial tension (UT) were approximately 50% (Klisch and Lotz 1999; Wagner and Lotz 2004)....
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...In contrast to earlier studies with exponential strain energy functions (Klisch and Lotz 1999; Wagner and Lotz 2004), in this study an exact solution to UT can be obtained for the second-order models....
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...(26) In (25), the isotropic part (i.e., the term dependent on I7) was based on the orthotropic model of Wagner and Lotz (2004) while the anisotropic part was based on the study of Holzapfel et al. (2004)....
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...…colleagues developed an orthotropic finite deformation model for the annulus fibrosus using an exponential strain energy function; however, maximum errors between the the oretical and experimental stresses in uniaxial tension (UT) were approximately 50% (Klisch and Lotz 1999; Wagner and Lotz 2004)....
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"A Bimodular Theory for Finite Defor..." refers background or methods in this paper
...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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168 citations
"A Bimodular Theory for Finite Defor..." refers background or methods in this paper
...For infinitesimal strains, Ateshian and colleagues (Soltz and Ateshian 2000; Wang et al. 2003) have employed elastic and biphasic models with a bimodular stress constitutive equation that allows for different mechanical proper ties in tension and compression....
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...Due to this molecular structure, articular cartilage typically exhibits a mechanical response with marked anisotropy and tension–compression asymmetry (Akizuki et al. 1986; Laasanen et al. 2003; Soltz and Ateshian 2000; Wang et al. 2003; Woo et al. 1976, 1979), and likely experiences finite, multi-dimensional strains due to typical in vitro and in vivo loads....
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...…exhibits a mechanical response with marked anisotropy and tension–compression asymmetry (Akizuki et al. 1986; Laasanen et al. 2003; Soltz and Ateshian 2000; Wang et al. 2003; Woo et al. 1976, 1979), and likely experiences finite, multi-dimensional strains due to typical in vitro and in vivo loads....
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...As mentioned before, Ateshian and colleagues (Soltz and Ateshian 2000; Wang et al. 2003) used a bimodular model for infinitesi mal strains....
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...The rationale for this requirement is that previous analyses suggested that material stability is difficult to ensure if the first-order constants jump, because eight stiffness matrices must be positive-definite (see, Klisch et al. 2004; Wang et al. 2003)....
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167 citations
"A Bimodular Theory for Finite Defor..." refers background or methods in this paper
...A justification for not allowing the first-order constants to jump is that the experi mental stress–strain response is continuous through the ori gin in strain space, as demonstrated for the annulus fibrosus (Wagner and Lotz 2004) and articular cartilage (Chahine et al. 2004)....
[...]
...A justification for not allowing the first-order constants to jump is that the experi mental stress–strain response is continuous through the ori gin in strain space, as demonstrated for the annulus fibrosus (Wagner and Lotz 2004) and articular cartilage (Chahine et al. 2004)....
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...For UT in the 3 direction, the parameters from the UT in the 1-direction were scaled down using a ratio of the infinitesimal Young’s moduli reported in Chahine et al. (2004) for bovine cartilage....
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