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

TL;DR: An orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor is derived as an alternative to traditional exponential type equations and suggests that the bimodular second- order models may be appealing for some applications with cartilaginous tissues.

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 compression. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equation, 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.

## 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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### Cites background or methods from "A Bimodular Theory for Finite Defor..."

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...(A3) For orthotropic, transversely isotropic, and isotropic materials, if F2e and T̂κ1(δFe) satisfy F2e = δFeF1e, T̂κ1(δFe) = T̂κ0(δFeF1e), (A4) then these are sufficient conditions for (A3) to hold Klisch (2006b)....

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...These two tools interactively solve the equilibrium growth boundary value problem (BVP) as proposed in (Klisch et al. 2001; Klisch 2006b), subject to the growth theories presented in the CGMM (Klisch et al. 2000, 2003; Klisch and Hoger 2003)....

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##### References

3,706 citations

### "A Bimodular Theory for Finite Defor..." refers background or methods in this paper

...2.2 Second-order orthotropic materials 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)....

[...]

...(40) In Ogden (1984) 7, this inequality is interpreted as stabil ity “under tractions which follow the material, i.e. rotate with the local rotation R . . . for an isotropic material.”...

[...]

...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)....

[...]

...Here, we follow the defini tion and interpretation of incremental stability for the conju gate pair of Biot stress and Biot strain as presented in Ogden (1984) and derive necessary and sufficient material stabil ity conditions for the reduced orthotropic bimodular secondorder model (23)....

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...…are the symmetric and skew parts of a tensor, and ∂Ŵ ∂2W ∂ I j ∂ Ik �1[E] = D (I)[E] = (II ) (I) (I) · E ∂U ∂ Ik ∂ I j ∂U ∂U ∂W ∂2 I j+ (II ) (I)[E], ∂ I j ∂U2 5 For example, see the discussion on molecular and phenomenologi cal models of rubber elasticity presented in Chapter 7 of Ogden (1984)....

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### "A Bimodular Theory for Finite Defor..." refers background in this paper

...2.2 Second-order orthotropic materials 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)....

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### "A Bimodular Theory for Finite Defor..." refers background in this paper

...The proteoglycans are negatively charged molecules that mainly resist compressive loads (Basser et al. 1998; Lai et al. 1991) while the collagen net work primarily resists tensile and shear loads (Mow and Ratcliffe 1997; Venn and Maroudas 1977)....

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...The proteoglycans are negatively charged molecules that mainly resist compressive loads (Basser et al. 1998; Lai et al. 1991) while the collagen net work primarily resists tensile and shear loads (Mow and Ratcliffe 1997; Venn and Maroudas 1977)....

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