Cyril J.F. Kahn1 LEMTA, Cell and Tissue Engineering Group, Nancy-Université, 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandoeuvre-lès-Nancy Cedex, France e-mail: [email protected]

Xiong Wang LEMTA, UMR 7563, Cell and Tissue Engineering Group, Nancy-Université, CNRS, 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandoeuvre-lès-Nancy Cedex, France; Physiopatholgie, Pharmacologie et Ingénierie Articulaires, UMR 7561, Nancy-Université, CNRS, 9 Avenue de la Forêt de Haye, 54500 Vandoeuvre-Lès-Nancy, France e-mail: [email protected]

Rachid Rahouadj LEMTA, UMR 7563, Cell and Tissue Engineering Group, Nancy-Université, CNRS, 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandoeuvre-lès-Nancy Cedex, France e-mail: [email protected]

1

Nonlinear Model for Viscoelastic Behavior of Achilles Tendon Although the mechanical properties of ligament and tendon are well documented in research literature, very few unified mechanical formulations can describe a wide range of different loadings. The aim of this study was to propose a new model, which can describe tendon responses to various solicitations such as cycles of loading, unloading, and reloading or successive relaxations at different strain levels. In this work, experiments with cycles of loading and reloading at increasing strain level and sequences of relaxation were performed on white New Zealand rabbit Achilles tendons. We presented a local formulation of thermodynamic evolution outside equilibrium at a representative element volume scale to describe the tendon’s macroscopic behavior based on the notion of relaxed stress. It was shown that the model corresponds quite well to the experimental data. This work concludes with the complexity of tendons’ mechanical properties due to various microphysical mechanisms of deformation involved in loading such as the recruitment of collagen fibers, the rearrangement of the microstructure (i.e., collagens type I and III, proteoglycans, and water), and the evolution of relaxed stress linked to these mechanisms. 关DOI: 10.1115/1.4002552兴 Keywords: mechanical properties, tendon, Achilles tendon, loading cycles, relaxation, relaxed stress

Introduction

Ligaments and tendons are soft connective tissues whose roles are to transfer efforts from muscles to bones 共tendons兲 and guide normal joint motion 共ligaments兲. Thus, injuries to these tissues can lead to joint instability and greatly restrict the activity level of patients. A good understanding of the mechanical properties of ligaments and tendons is crucial to prevent injuries and to evaluate the quality of engineered biotissues or prostheses. Soft connective tissues have a multiscale hierarchical structure that induces diverse mechanical properties 关1,2兴. Ligaments and tendons are made up of water 共60–80% wet weight兲 and different collagens, such as collagen type I 共65–80% dry weight兲, III 共5–8% dry weight兲, and V 共low level兲 关2兴, which are the major collagen components. In addition, ligaments and tendons contain a certain amount of proteoglycans 共e.g., decorin兲, which ensure the extent of fibril cross-linking and a control of fibrillogenesis 关2,3兴. Many constitutive laws have been proposed to describe the mechanical properties of soft tissues 关4–7兴. The strain-stress behavior and viscoelasticity of ligaments and tendons have been greatly investigated with regard to tension, relaxation, and creep. Physi1 Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received March 21, 2010; final manuscript received September 9, 2010; accepted manuscript posted September 15, 2010; published online October 15, 2010. Assoc. Editor: John C. Criscione.

Journal of Biomechanical Engineering

ological uniaxial stress induces a typical stress-strain relationship, which is traditionally divided into three phases: a toe region with a very low stress, a heel, and a linear region 共Fig. 1兲 关2,8,9兴. To our knowledge, few models can describe the loading, relaxation, and unloading behavior of ligaments and tendons except the work of Peña et al. 关5兴, which proposed a three-dimensional finite strain anisotropic visco-hyperelastic model for finite element implementation. This lack must be remedied to model healthy native tissues and to evaluate the quality of regenerated tissues, scaffolds, and prostheses for tissue engineering of ligaments and tendons. Taking into account the literature and our data on tendons, we tried to model the mechanical behavior of tendons and ligaments with a nonlinear viscoelastic model based on irreversible theory. The aim of this study was to develop a uniaxial unified model to describe tendon 共or ligament兲 mechanical behavior.

2

Experimental

2.1 Materials and Methods. Animal care complied with the “Principle of Laboratory Animal Care and the Guide for the Care and Use of Laboratory Animals” 共National Institute of Health publication No. 80-23, revised 1978兲. Mechanical uniaxial tests in the longitudinal direction were carried out using an Adamel Lhomargy DY.22 testing machine 共MTS兲 关10兴 on a fresh white New Zealand rabbit’s Achilles tendon 共Fig. 1, n = 3 per group test兲. To measure the cross-section, two pictures of tendon were taken in two perpendicular directions 共X and Y兲. The mean diameters

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Fig. 1 Mean stress-strain curve of rabbit Achilles tendons „n = 10…. Stress response as a function of strain „ε˙ = 2 Ã 10−2 s−1…. „a… Muscle insertion and „b… bone insertion.

were measured on each picture, and the cross-section was calculated by approximating it as an ellipse 共Fig. 2兲. During mechanical tests, the tendons were immersed in 0.9% saline water maintained at 37 ° C. The risk that tendons slip in the grip was prevented by fixing the bony insertion with an orthopedic resin 共Palacos®R, Heraeus, Germany兲 and by peeling the muscle insertion in order to decrease its thickness; it was then placed between two sheets of emery cloth and held fast by a self-tightening grip. Finally, the first tendon was stretched at the same length as measured on the rabbit before taking off the Achilles tendon. This elongation corresponds to a force of 1 N. Thus, the initial length of the tendons was obtained by stretching them until 1 N was reached at the relaxation state. Because the tendons are sensitive to loading history, this procedure was preferred to the most often used preconditioning method, which consists of preconditioning the tendon with cycles of loading and unloading. Three sequences of mechanical loading were then applied to the tendons: loading, relaxation, and unloading. During relaxation tests, we assumed that relaxed stress was reached when the force

did not change significantly during an hour 共variations⬍ 1 N兲. Successive cycles of uniaxial loading and unloading were performed at strain ␧ = 2 ⫻ 10−2, 5 ⫻ 10−2, and 9 ⫻ 10−2 and stopped at 0.1, respectively. The contralateral Achilles tendon of each rabbit was used to perform the sequenced relaxation tests during loading at ␧ = 2 ⫻ 10−2, 4 ⫻ 10−2, 6 ⫻ 10−2, and 8 ⫻ 10−2. Contralateral tendons were then strained to 0.1, unloaded until 6 ⫻ 10−2 of deformation, and, finally, relaxed 共Fig. 3兲. All experiments were performed at a strain rate of 2 ⫻ 10−2 s−1. 2.2 Experimental Results. Our results confirmed the wellknown stress-strain response of tendons and ligaments reported for uniaxial strain at constant strain rate 关1,8,9兴, namely, the nonlinear properties of the relaxation tests as a function of strain 关11兴. The relaxed stress and the corresponding relaxation time depended on the strain level 共Fig. 3 and Table 1兲. The sequence of relaxations also revealed that relaxed stress was not reversible: A hysteresis loop of the relaxed stress can be seen in Fig. 3. During cycles of loading, unloading, and reloading, we observed that hysteresis loops depended on strain level and were amplified by the number of previously applied cycles 共Fig. 4兲. We also noted that the elastic modulus at the beginning of reloading was very low, i.e., the applied stress followed the unloading pathway 共Fig. 4共b兲兲 关9,12兴.

3

Thermodynamic Modeling

The results we obtained from our experiments on rabbit Achilles tendon and from the published data in the literature both show that the tendon behaves as an anisotropic viscoelastic material depending greatly on history of loading. Indeed, taking into account nonreversible macroscopic relaxed stress and the hysteresis loops during loadings and relaxations, the tendon evolves outside equilibrium state with nonlinearity depending on mechanical solicitations such as strain, strain rate 关6,7兴, and stress 关13兴. These observations led us to opt to use thermodynamic modeling based on outside equilibrium state describing material at a representative element volume 共REV兲 scale. This formulation has been used previously to study the mechanical properties of materials such as polymers 共such as high-density polyethylene 共HDPE兲 and melamine兲 关10,14,15兴, glass 关16兴, and metals.

Fig. 2 Mean section of Achilles tendon. Section of Achilles tendon obtained by MRI „solid line…, ellipse „dashed line…. Areas + and − are nearly equivalent.

3.1 Basic Formulation of the Constitutive Equation. Our approach is based on a local formulation of the thermodynamic evolution of a REV, following the generalization of Gibbs’ and Gibbs–Duhem’s relations 关16,17兴, u共s, ␧,n兲 = Ts + ␴:␧ + ␮ · n

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共1兲

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Table 4 Parameters fitted for the Achilles tendon successive relaxation and cycles of loading „Figs. 3 and 4, n = 3… No. 1

Eu 共MPa兲 ␣c ␧thc 共⫻10−2兲 ␣d ␧thd 共⫻10−2兲 K0 共⫻10−3 m3 / mol兲 KA 共⫻10−3 m3 / mol兲

No. 2

No. 3

Mean⫾ SD

Cycles

Relax

Cycles

Relax

Cycles

Relax

Cycles

Relax

500 1.8 6 2.7 5.6 −2.8 −3.2

475 1.85 8 2.7 5.5 −1.5 −3.0

300 1.8 7 3 6.5 −3.5 −3.5

300 1.75 9 2.7 6.5 −1.5 −3.5

410 1.75 6 2.85 5 −3.4 −3.5

300 1.8 7.2 2.8 4.7 −1.5 −3.5

403⫾ 100 1.78⫾ 0.03 6.4⫾ 0.6 2.9⫾ 0.2 5.7⫾ 0.8 −3.2⫾ 0.4 −3.4⫾ 0.2

358⫾ 101 1.8⫾ 0.05 8.1⫾ 0.9 2.7⫾ 0.1 5.6⫾ 0.9 −1.4⫾ 0.2 −3.3⫾ 0.3

ing to the strain level shown in Table 1. This time must be reached when the stress is equal to the relaxed stress, i.e., when the relaxed r r,0 = ␶N 共␧ + ␧0兲2兲. Finally, the initial shift function is equal to 1 共␶N elastic modulus of the instantaneous stress 共Eu0兲 was measured with the initial tangent modulus of the tests. The last step was to define the last seven parameters according to different tests of successive cycles or relaxations 共Table 4兲. Using the proposed constitutive law herein, the various loadingunloading strains applied to rabbit Achilles tendons fitted quite well 共Fig. 7兲. The imposed and fitted parameters related to the identifications of experimental tests are summarized in Tables 2–4.

5

Discussion

This study presented a novel nonlinear viscoelastic model based on the formulation of thermodynamic properties at the REV level. The model is valid for large deformation and various uniaxial loadings such as successive relaxations and cyclic loadings. However, a transversely isotropic model would be suitable, but the lack of information on the transverse direction behavior and the difficulty to obtain them led us to use a uniaxial model. To our knowledge, in ligament and tendon biomechanics, few models describe the unloading behavior qualitatively and quantitatively. The extension of the quasi-linear viscoelastic modeling of Fung 关9兴 efficiently describes the properties in uniaxial traction and relaxation and also works well to describe the evolution of peak stress during cyclic loadings but is not able to adequately follow unloading properties and successive relaxations, as reported by Provenzano et al. 关11兴. Indeed, Provenzano et al. 关11兴 demonstrated that relaxation is a function of strain. Pioletti et al. 关19兴 and Woo et al. 关20兴 reported that Quasi-Linear Viscoelastic model 共QLV兲 gives an inaccurate description of strain rate effect and that the single integral finite strain viscoelastic model does not incorporate this effect 关19–21兴. Pioletti et al. 关2兴 and De Vita and Slaughter 关4兴 proposed a model to incorporate the effect of strain rate by explicitly using the strain rate as a variable. In the present study, we proposed a model that incorporates successive relaxations and cyclic loading uniaxial properties as discussed. The model is consistent: The differences in mean values between relaxation test fits and cycles test fits are not significant, taking into account the standard deviation 共Tables 3 and 4兲. Furthermore, the value obtained for characteristic physical parameters such as global elastic modulus and relaxation time spectrum are of the same order of magnitude to those found in research literature on the subject. Indeed, the global elastic modulus obtained on rabbit Achilles tendon 共350–575 MPa兲 was comparable to the elastic modulus reported in research literature 关22兴. This elastic modulus took into account the whole material at the scale of REV but not only collagen I, which explained the difference with the estimation of the elastic modulus of collagen type I 共1–10 GPa兲 关23–27兴. Indeed, tendons are complex multiscale architectural structures made up not just of collagens but also of a matrix made of water, proteoglycans, elastin, etc. 关1,3,26,27兴. Therefore,

Fig. 7 Typical curves of fitting experimental data: „a… and „b… fit of successive relaxations in function of strain and time, respectively, and „c… fit of cyclic loading. „쎲… is the experimental data, red solid line is the theoretical instantaneous stress, and green dashed line is the relaxed stress.

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冋 册 ⌬E+ RT

␶jr = ␶jr,0 exp

共5兲

where the relaxation time of the mode j is determined by the variation in the activation energy ⌬E+, the ideal gas constant 共R兲, and the absolute temperature 共T兲 共see Appendix兲. To introduce the nonlinear effects of the loading history, an empirical form is adopted,

␶jr = ␶jr,0共␧ + ␧0兲2ar共␴, ␴r,T兲

冉

ar共␴, ␴r,T兲 = exp

Fig. 5 Schematic link between elastic stress, instantaneous stress, and relaxed stress

lus, ␶r,0 is the jth time relaxation at the initial state, ␴r is the j relaxed stress tensor, and E+ is the free energy of activation. This formulation describes the stress rate in two parts: elastic and dissipative. The elastic part is given by Eu, and the dissipative part is a function of the instantaneous time of relaxation + 共␶r,0 j exp关⌬E / RT兴兲 and the gap between the stress and the relaxed stress for each mode 共Fig. 5兲. The relaxation time spectrum 共␶r兲 changes during the loadings with the variation in the activation free energy 共⌬E+兲, which characterizes the material. It should be noted that the dissipative part will vanish when the stress is equal to the relaxed stress. This mechanical formulation has been widely used to describe the complex behavior of polymers such as HDPE and melamine 关10,14,15兴. 3.2 Application to Tendons. In the following section, we will discuss further the points mentioned in the previous section to model the behavior of rabbit Achilles tendon by adapting the formulation of nonlinear properties. To describe those tissues, we based our considerations on the reported mechanical properties in research literature and on our data concerning rabbit Achilles tendon. 3.2.1 Elastic Properties. The three phases of the typical uniaxial stress-strain curve for tendons 共i.e., toe, heel, and linear regions兲 are generally explained by a recruitment of fibers. To incorporate this phenomenon into our model, the proposition of De Vita and Slaughter 关4兴, who modeled this recruitment during the loading phases with a probability of taut collagen fibers as a function of strain by using a distribution of Weibull, was adapted to describe the variation in the elastic part of the extracellular matrix in loading and unloading conditions,

冋 冋 冉 冊 册册 再

Eu,eff = Eu0 + Eu 1 − exp −

␧loc ␧thi

␣i

,

i=

c loading d unloading

冎

共4兲 where Eu0 is the equivalent elastic modulus of the tissue taking into account the initial distribution of collagen fibers and the noncollagenous matrix 共unstrained state兲 and Eu is the equivalent elastic modulus of fibers and cross-link taut 共strained state兲. ␧loc is a variable determined as a function of ␧ and parameters of loading and unloading 共␧thc , ␣c , ␧thd , ␣d兲 and guarantees the continuity of the distribution function at the passage from loading to unloading and vice versa. 3.2.2 Relaxation Spectrum. Based on the developments reported by Cunat 关16兴, this spectrum of relaxation time is written as

K 0兩 ␴ − ␴ r兩 + K A共 ␴ − ␴ r兲 RT

冊

共6兲

where ar is the shift function of the relaxation time spectrum. The nonlinearities of the relaxation time spectrum are due to the strain and stress effects of the gap 共␴ − ␴r兲 during the loading, unloading, and reloading cycles. 3.2.3 Relaxed Stress. The relaxed state is presumed to be governed by the macromolecular fiber configuration related to the deformation state. Based on previous work by Mrabet et al. 关15兴 and Arieby 关14兴, we assumed that the relaxed stress evolution on the time scale of the experiment must follow this differential form,

册 兺冋 冋 冋 冉 冊 册册 N

␴˙ r = Er,eff␧˙ −

j=1

␴jr ␶jeq,0av共␧˙ 兲aeq共␴r兲

Er,eff = Er0 + Er 1 − exp −

冦

˙ ⬍ ␧˙0 if 兩␧兩

1

av共␧˙ 兲 = ␧˙0 ˙ 兩␧兩

␧ ␧thR

else

冉

aeq共␴r兲 = exp

Keq兩␴r兩 RT

冊

␶jeq,0 = ␶Neq,010−D共 N−1 兲 N−j

␣R

冧

共7a兲

共7b兲

共7c兲

共7d兲 共7e兲

The relaxed stress ␴ remains independent of the strain rate, thanks to the shift function av, which corresponds to the experimental observations. This formulation also takes into account the hysteresis behavior of the relaxed stress observed in Fig. 3 by introducing a dissipative part for the relaxed stress 共second part of the right hand, Eqs. 共7a兲–共7e兲兲. It should be noted that the characteristic time of dissipation for the relaxed stress needs to be long enough. Indeed, the time scale for a material to recover to relaxed state is shorter than the time required to achieve equilibrium. A summary of the whole model is given in the schematic algorithm for the stress computation 共Fig. 6兲. r

4

Identification

eq,0 Five parameters of the model, N, D, ␧0, ␶N , and ␧˙0, are fixed parameters 共Table 2兲. Cunat 关18兴 demonstrated that through 50 modes of dissipation over 6 decades, the time of relaxation spectrum remained quasi-continuous. We fixed the number of modes N and decades D at 30 and 3, respectively, because beyond these values the variations in N and D have almost no further effect on eq,0 , representing the time for the result. The time of relaxation ␶N the material to get from the relaxed state to the equilibrium state, is not a measurable parameter, and therefore its value was then fixed at 1.5⫻ 108 s. Finally, ␧0 and ␧˙0 have no physical signification. ␧0 is necessary to avoid a relaxation time null at the initial strain; its value was imposed at 8 ⫻ 10−4, which ensured a time of relaxation longer than the time of sound propagation across the tendon. ␧˙0 is usually fixed for this formulation at 10−7 s−1.

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Fig. 6 Schema of the algorithm used to compute instantaneous stress

The second step of the identification was to determine the five relaxed stress parameters and the two instantaneous stress parameters. Indeed, the relaxed stress is deduced from the stress observed at the end of each relaxation test of Fig. 3. All the relaxed stress parameters were identified with these records and were then fixed or re-adjusted to improve their conformity 共see Table 3兲. r,0 Then, the time of relaxation ␶N 共last mode of Eq. 共5兲, j = N兲 was determined by calculating the evolution of relaxation time accord-

Table 2 Fixed parameters

N

D

␧0

␶Neq,0 共s兲

␧˙0 共s−1兲

30

3

8 ⫻ 10−4

1.5⫻ 108

10−7

Table 3 Parameters predetermined with the rabbit Achilles tendon tests during successive relaxations and cycles of loading „Figs. 3 and 4, n = 3…. Prefitted parameters of the relaxed stress were re-adjusted for the cycles of loading tests fit to improve them. No. 1

共MPa兲 E 共MPa兲 ␣R ␧thr 共⫻10−2兲 Keq 共⫻10−3 m3 / mol兲 Eu0 共MPa兲 ␶Nr,0 共⫻106 s兲 Er0 r

No. 2

No. 3

Mean⫾ SD

Cycles

Relax

Cycles

Relax

Cycles

Relax

Cycles

Relax

22 180 1.9 8 −2.2 75 1.3

22 180 1.9 8 −2.2 80 1.3

20 110 2.3 9 −2.2 50 1.2

30 125 2.3 9 −2.6 83 1.2

22 135 2.3 9 −2.6 38 1.2

20 130 2.4 8.6 −2.75 80 1.1

21.3⫾ 1.2 142⫾ 35.5 2.2⫾ 0.2 8.7⫾ 0.6 −2.3⫾ 0.2 54.3⫾ 18.9 1.23⫾ 0.06

24⫾ 5.3 145⫾ 30.4 2.2⫾ 0.3 8.5⫾ 0.5 −2.5⫾ 0.3 81⫾ 1.7 1.2⫾ 0.1

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Table 4 Parameters fitted for the Achilles tendon successive relaxation and cycles of loading „Figs. 3 and 4, n = 3… No. 1

Eu 共MPa兲 ␣c ␧thc 共⫻10−2兲 ␣d ␧thd 共⫻10−2兲 K0 共⫻10−3 m3 / mol兲 KA 共⫻10−3 m3 / mol兲

No. 2

No. 3

Mean⫾ SD

Cycles

Relax

Cycles

Relax

Cycles

Relax

Cycles

Relax

500 1.8 6 2.7 5.6 −2.8 −3.2

475 1.85 8 2.7 5.5 −1.5 −3.0

300 1.8 7 3 6.5 −3.5 −3.5

300 1.75 9 2.7 6.5 −1.5 −3.5

410 1.75 6 2.85 5 −3.4 −3.5

300 1.8 7.2 2.8 4.7 −1.5 −3.5

403⫾ 100 1.78⫾ 0.03 6.4⫾ 0.6 2.9⫾ 0.2 5.7⫾ 0.8 −3.2⫾ 0.4 −3.4⫾ 0.2

358⫾ 101 1.8⫾ 0.05 8.1⫾ 0.9 2.7⫾ 0.1 5.6⫾ 0.9 −1.4⫾ 0.2 −3.3⫾ 0.3

ing to the strain level shown in Table 1. This time must be reached when the stress is equal to the relaxed stress, i.e., when the relaxed r r,0 = ␶N 共␧ + ␧0兲2兲. Finally, the initial shift function is equal to 1 共␶N elastic modulus of the instantaneous stress 共Eu0兲 was measured with the initial tangent modulus of the tests. The last step was to define the last seven parameters according to different tests of successive cycles or relaxations 共Table 4兲. Using the proposed constitutive law herein, the various loadingunloading strains applied to rabbit Achilles tendons fitted quite well 共Fig. 7兲. The imposed and fitted parameters related to the identifications of experimental tests are summarized in Tables 2–4.

5

Discussion

This study presented a novel nonlinear viscoelastic model based on the formulation of thermodynamic properties at the REV level. The model is valid for large deformation and various uniaxial loadings such as successive relaxations and cyclic loadings. However, a transversely isotropic model would be suitable, but the lack of information on the transverse direction behavior and the difficulty to obtain them led us to use a uniaxial model. To our knowledge, in ligament and tendon biomechanics, few models describe the unloading behavior qualitatively and quantitatively. The extension of the quasi-linear viscoelastic modeling of Fung 关9兴 efficiently describes the properties in uniaxial traction and relaxation and also works well to describe the evolution of peak stress during cyclic loadings but is not able to adequately follow unloading properties and successive relaxations, as reported by Provenzano et al. 关11兴. Indeed, Provenzano et al. 关11兴 demonstrated that relaxation is a function of strain. Pioletti et al. 关19兴 and Woo et al. 关20兴 reported that Quasi-Linear Viscoelastic model 共QLV兲 gives an inaccurate description of strain rate effect and that the single integral finite strain viscoelastic model does not incorporate this effect 关19–21兴. Pioletti et al. 关2兴 and De Vita and Slaughter 关4兴 proposed a model to incorporate the effect of strain rate by explicitly using the strain rate as a variable. In the present study, we proposed a model that incorporates successive relaxations and cyclic loading uniaxial properties as discussed. The model is consistent: The differences in mean values between relaxation test fits and cycles test fits are not significant, taking into account the standard deviation 共Tables 3 and 4兲. Furthermore, the value obtained for characteristic physical parameters such as global elastic modulus and relaxation time spectrum are of the same order of magnitude to those found in research literature on the subject. Indeed, the global elastic modulus obtained on rabbit Achilles tendon 共350–575 MPa兲 was comparable to the elastic modulus reported in research literature 关22兴. This elastic modulus took into account the whole material at the scale of REV but not only collagen I, which explained the difference with the estimation of the elastic modulus of collagen type I 共1–10 GPa兲 关23–27兴. Indeed, tendons are complex multiscale architectural structures made up not just of collagens but also of a matrix made of water, proteoglycans, elastin, etc. 关1,3,26,27兴. Therefore,

Fig. 7 Typical curves of fitting experimental data: „a… and „b… fit of successive relaxations in function of strain and time, respectively, and „c… fit of cyclic loading. „쎲… is the experimental data, red solid line is the theoretical instantaneous stress, and green dashed line is the relaxed stress.

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the apparent elastic modulus should take into account this architecture and the mechanical properties of the different components using a homogenization model such as the works of Fesel et al. 关28兴 or Redaelli et al. 关29兴. Very few papers deal with relaxed stress: Generally, relaxation tests are done with a recovery time fixed between 30 min and 60 min for each strain 关7,30兴. Our study demonstrates that this assumption only works for low strain levels under 4%. Stopping relaxation tests before reaching the relaxed stress can distort the results on relaxation time because of a lack of information at the end of the relaxation test. We saw that, for example, at 6% of strain the relaxation time is about 4700 s. However, our results and model are comparable to reported data at the same strain level. Indeed, Abramowitch et al. 关30兴 found a relaxation time spectrum from 0.54⫾ 0.15 s to 1602⫾ 581 s for a relaxed test at 2.76% of strain on goat medial collateral ligament 共MCL兲. The model presented herein and adjusted through tests carried out on rabbit Achilles tendons led to a relaxation time spectrum from 1 s to 1000 s for the same strain and strain rate as tests performed by Abramowitch et al. Moreover, our model follows the strain dependent relation of relaxation behavior, a phenomenon also reported by Provenzano et al. 关11兴, who worked with the MCL of Sprague Dawley rats. Contrary to usual research habits, a single relaxation test does not efficiently characterize and evaluate soft tissues such as tendons 关4,31兴. To the authors’ knowledge, it is the first time that a model can be used for such diverse type of loadings, as shown in this paper. Quantifying the properties of this range of loadings will allow us to obtain a deeper understanding of the complex mechanical properties of Achilles tendons. This model can be used in the future to evaluate and compare scaffolds and regenerated biotissues to native tendon in tissue engineering of ligament and tendon by using it in more relevant tests.

Acknowledgment This study was supported by the French CNRS 共ATIP/SPI兲 and ANR 共TELiTeR兲 and the French Ministry of Education and Research. We would like to thank Pr. Delagoutte for providing orthopedic resin and Alain Gérard for his technical assistance with the mechanical testing machine.

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

Symbol Scalar u ⫽ s ⫽ T ⫽ ⌬E+ ⫽

Considering a homogeneous continuous medium, Cunat 关32,33兴 showed that the evolution of intensive variables Y 共e.g., stress tensor ␴ and temperature T兲 and intensive generalized forces A may be associated with the dual extensive variables y 共e.g., volume weighted strain tensor V␧, entropy s兲 and extensive internal variables z 共describing the microstructure兲 by the rate form of the constitutive equations,

冉 冊 冉 冊冉 冊

⳵u ˙ ⳵y Y ⇒ ⳵u ˙ −A −A= ⳵z

Y=

tensor of first and second scalar parameter at unrelaxed state parameter at relaxed state parameter at equilibrium state time derivation of the variable A AikBkj AijBji specific internal energy specific entropy absolute temperature free energy of activation

Symbol of Vector ␮ ⫽ chemical potential vector n ⫽ amount of substance vector pi,0 ⫽ weight vector of the dissipative modes at the i state ␶i,0 ⫽ relaxation time vector to get to the i state Symbol of Second Order Tensor ␴ ⫽ Cauchy stress tensor ␧ ⫽ true strain tensor a ⫽ Tisza matrix g ⫽ matrix of dissipation Journal of Biomechanical Engineering

=

b

y˙

bT g

z˙

a

共A1兲

where a = ⳵2u / ⳵y2 is the Tisza matrix, b = ⳵2u / ⳵y ⳵ z is the coupling matrix, and g = ⳵2u / ⳵z2 is the matrix of dissipation, defined as the matrices of second order derivatives of the generalized internal energy potential u共y , z兲. For the sake of simplicity, these matrices are assumed to be constant in the vicinity of the thermodynamical equilibrium. It should be noted that for a rotation of the VER, the time derivation must be calculated toward an objective derivation such as Jaumann’s or Xiao’s one 关34兴. For this kind of derivation, the symbol ⵜ can be used to differentiate it from the ⵜ ˙ + W : ␴-␴ : W兲. time derivation 共␴ = ␴ Equation 共A1兲 may be reasonably completed 关32,33兴 by a linear kinetic evolution of the internal variables, 共A2兲

z˙ = L · A

where L represents a matrix of kinetic coefficients satisfying the Onsager properties 关35兴. To obtain the corresponding constitutive equations, Cunat proposed to introduce a state of reference, called “the relaxed state” defined by the condition −A˙ r = 0 共the superscript “r” refers to this state兲. Using Eqs. 共A1兲 and 共A2兲, it can be demonstrated that the kinetic evolution equation of the N microstructural reactions advancement is given by z˙j = −

Nomenclature Notation A, a A, a A u, a u A r, a r Aeq, aeq ˙ A A·B A:B

Appendix Constitutive Law Within the Framework of the Thermodynamics of Relaxation

zj − zjr , ␶j

j = 1, . . . ,N

共A3兲

where zrj is the value of the jth reaction advancement at the relaxed state, N is the number of chemical process 共or dissipative modes兲, and ␶j is the corresponding relaxation time. The time integration of the difference of Eq. 共A1兲 between the unrelaxed and relaxed states gives a relation between the internal variable and the intensive variable unrelaxed and relaxed variables as follows: Yj − Yjr = bj共zj − zjr兲

共A4兲

Relation 共A1兲 incorporating Eqs. 共A3兲 and 共A4兲 gives

兺冉 N

˙ = Euy˙ − Y

j=1

N

A˙ i = − bi␧˙ +

兺g j=1

ij

冉

冊

共A5a兲

i = 1, . . . ,N

共A5b兲

Yj − Yjr ␶j

冊

Yj − Yjr , b j␶ j

Let us now consider the application to the inelastic behavior for isothermal uniaxial loading paths. The controlled variables 共y兲 correspond to the uniaxial strain tensor ␧, associated with the intensive variable, which is the Cauchy stress tensor ␴. Equation 共A5兲 gives 关33,34兴 N

␴˙ = Eu␧˙ −

兺 j=1

冉

␴j − ␴jr ␶j

冊

共A6a兲

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N

兺

A˙ i = − bi␧˙ +

gij

j=1

冉

冊

␴j − ␴jr , b j␶ j

i = 1, . . . ,N

共A6b兲

where the constants Eu 共the superscript “u” means “unrelaxed” or instantaneous兲, bj, and gij are defined as the following second order derivatives of u:

⳵u ⳵ ␧2

关11兴 关12兴 关13兴

2

Eu =

共A7a兲

关14兴

关15兴

bj =

⳵ 2u , ⳵ ␧ ⳵ zj

j = 1, . . . ,N

共A7b兲

gij =

⳵ 2u , ⳵ zi ⳵ zj

i,j = 1, . . . ,N

共A7c兲

among which we recognize the instantaneous Young modulus Eu. Cunat showed that the relaxation times ␶j may be modeled as functions of the temperature T only, according to

冉

冉 冊

冊

⌬Ej+ h ⌬Hj − T⌬Sj exp ␶j共T兲 = = ␶jr,0 exp k BT RT RT

共A8兲

where h is the Planck constant, kB is the Boltzmann constant, R is the ideal gas constant, and ␶r,0 j is the relaxation time reference of the jth mode at initial time. The physical parameter ⌬E+j corresponds to the apparent variation of activation free energy of the jth mode. The activation free energies are presumed to be the same for each mode 共⌬E+j = ⌬E+兲, indicating that the material is considered “thermodynamically simple.” Replacing Eq. 共A8兲 in Eq. 共A6兲 gives the following relations: N

␴˙ = Eu␧˙ −

兺 j=1

N

A˙ i = − bi␧˙ +

兺g j=1

ij

冢

冢

冋 册冣

␴j − ␴jr

␶jr,0

⌬E+ exp RT

冋 册冣

␴j − ␴jr bj␶jr,0 exp

⌬E+ RT

共A9a兲

关16兴

关17兴 关18兴 关19兴 关20兴 关21兴 关22兴 关23兴

关24兴

关25兴

,

i = 1, . . . ,N 关26兴

共A9b兲

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