Evaluating Plastic Deformation and Damage as Potential Mechanisms for Tendon Inelasticity Using a Reactive Modeling Framework

Abstract

Inelastic behaviors, such as softening, a progressive decrease in modulus before failure, occur in tendon and are important aspects in degeneration and tendinopathy. These inelastic behaviors are generally attributed to two potential mechanisms: plastic deformation and damage. However, it is not clear which is primarily responsible. In this study, we evaluated these potential mechanisms of tendon inelasticity by using a recently developed reactive inelasticity model (RIE), which is a structurally inspired continuum mechanics framework that models tissue inelasticity based on the molecular bond kinetics. Using RIE, we formulated two material models, one specific to plastic deformation and the other to damage. The models were independently fit to published macroscale experimental tensile tests of rat tail tendons. We quantified the inelastic effects and compared the performance of the two models in fitting the mechanical response during loading, relaxation, unloading, and reloading phases. Additionally, we validated the models by using the resulting fit parameters to predict an independent set of experimental stress–strain curves from ramp-to-failure tests. Overall, the models were both successful in fitting the experiments and predicting the validation data. However, the results did not strongly favor one mechanism over the other. As a result, to distinguish between plastic deformation and damage, different experimental protocols will be needed. Nevertheless, these findings suggest the potential of RIE as a comprehensive framework for studying tendon inelastic behaviors.

1. Introduction

Inelastic mechanical behaviors, where energy is dissipated during mechanical loading, occur during tendon's overload and in loading-induced pathological conditions such as degeneration and tendinopathy [1,2]. Inelastic behaviors in tissue include viscoelasticity, plastic deformation, and damage. In particular, plastic deformation and damage are associated with stress softening, commonly referred to as Mullin's effect in polymer mechanics [3,4]. In tissue, softening is observed as a progressive decrease in tensile modulus prior to failure [5–8] and can be described based on the concepts of plastic deformation and damage. In plastic deformation, there is a shift in the unloaded reference state, and in damage, the material loses its ability to absorb energy without a change to the unloaded reference state [3,9,10]. These inelastic behaviors can contribute to the development of clinical disorders, such as tendinopathy and rupture by changing the macroscale mechanical behavior [11] and can also affect tissue degeneration and regeneration, by changing the cell loading [12].

Tendon's mechanical and structural studies support both of the potential mechanisms of plastic deformation [13–15] and damage [16,17] for softening. However, plastic deformation and damage are hard to differentiate experimentally. Plastic deformation and damage cannot be differentiated during the commonly used ramp-to-failure tests due to the similar softening effect of these two mechanisms during loading. A repeated loading protocol, such as loading and unloading, is essential to quantify plastic deformation and damage. However, even when unloading is added, the low stiffness in the toe-region and viscoelastic behaviors can still limit differentiation between plastic deformation and damage [14]. Structural studies also do not resolve the question: nonrecoverable microscale sliding [18] supports plastic deformation, collagen denaturation due to mechanical loading [19,20] supports damage, and the inelastic tensile response of individual collagen fibrils [21,22] may support both. Despite numerous mechanical and structural studies, mechanisms for tendon's inelastic behaviors are not well described.

We recently developed the reactive inelasticity (RIE) theoretical framework to address the inelastic behaviors of soft tissue by using the kinetics of molecular bonds [23]. This structurally inspired continuum mechanics framework is an extension of the reactive viscoelasticity model by Ateshian [24] and prior work in polymer mechanics [3,25–27], which addresses all three of the aforementioned inelastic behaviors of tissue (i.e., viscoelasticity, plastic deformation, and damage) by using the same constitutive settings at different kinetic rates of bond breakage and reformation. This characteristic makes RIE distinct from other models for tissue's inelastic behaviors, which address viscoelasticity without softening [28–30], or include softening by focusing on either plastic deformation [14,31], damage [10,17,32–35], or their combination [36–39]. In the RIE framework, there are three bond types: formative bonds are used to model a transient viscoelastic behavior, permanent bonds are used for modeling the hyperelastic behavior, and sliding bonds are used for plastic deformation. Damage is added to each bond type by reducing the number fraction of the available bonds [23]. By combining different bond types, a wide variety of mechanical behaviors can be modeled. Hence, the RIE framework provides a comprehensive tool for investigating the inelastic behaviors of tissue by using a unified structurally inspired theory to model inelasticity.

Our long-term goal is to study tissue inelasticity, bridging the gap between experiments and modeling. The objectives of this study are to apply the RIE modeling framework to tendon's tensile behavior, to evaluate the model performance in fitting the macroscale experimental data for plastic deformation and damage mechanisms, and to validate the models by predicting an independent experimental dataset. Our previous study on rat tail tendons demonstrated that uniaxial mechanical loading can change both macroscale mechanical properties and microscale structure [18]. However, that experimental study was not designed to distinguish between plastic deformation and damage, and it did not provide numerical measures for inelastic effects. We will show that we can describe the inelastic behaviors in tendon by using RIE-based models specific to plastic deformation (combination of formative bonds and sliding bonds) or damage (combination of formative bonds and permanent bonds). Importantly, for validation, we then apply the resulting model parameters to predict results from a separate, independent, experiment. This study is significant in that it pursues a systematic approach to elucidate the mechanisms of tissue degeneration and failure, and it provides a path forward for understanding the connection between structural hallmarks of tendon pathology and mechanical behavior.

2. Methods

We formulated two material models using the RIE framework that address either plastic deformation or damage. These models account for the mechanics of tendon by simultaneously addressing: stress stiffening (toe-region), stress relaxation (viscoelasticity), unloading, and softening. The models were independently fit to the experimental data from Lee et al. [18]. We first evaluated the performance of these two models in fitting the experimental data. To validate the model prediction, we applied the resulting model parameters to a separate, independent experiment that measured the stress response of tendon fascicles in constant-rate tensile tests (Szczesny et al. [40]).

2.1. Reactive Inelastic Modeling.

In the following, we briefly describe the theoretical formulation of RIE and its specific application to this study; for further details the reader is referred to Ref. [23]. The RIE framework employs the kinetics of molecular bonds to simulate the inelastic behaviors. There are two levels for categorizations of bonds: “bond types” and “generations.” A bond type specifies its mechanical behavior (e.g., hyperelastic or plastic deformation). Bonds break when subjected to external loading, and reform to a new configuration. The reformed bonds initiate a new generation. The generations are bonds that were formed at the same time and have the same reference deformation gradient. Different generations can have different stress and relative deformations, but the sum of the number fraction of the generations is conserved at unity [24]. That is

$$\sum _{\alpha }{w}_{\gamma }^{\alpha }=1\hspace{0.17em}$$ (1)

where $w_{\gamma }^{\alpha }$ is the number fraction of the generation (α) from a bond type (γ), and is a positive scalar less than one. In this formulation, the number of generations is calculated based on the discretization of deformation into a finite number of time steps, which results in adequate numerical convergence.

By adding the response of generations, the overall response of a bond type is calculated, and by combining different bond types, an RIE material model is formulated. This constitutive setting allows for consistent modeling of the inelastic behaviors using the same set of equations [23]. We used a simplified one-dimensional version of RIE to model the rat tail tendon's axial tensile tests. We assumed that the stress in the material is the sum of all the bonds types ($T=\sum _{\gamma }{T}_{\gamma }$), and that the stress of a bond type is a weighted sum of the stresses from generations that depends on the overall stretch λ and the reference stretch of each generation ${\lambda }_{\gamma }^{\alpha }$ such that

$$T(\lambda ,w_{\gamma }^{\alpha },{\lambda }_{\gamma }^{\alpha })=\sum _{\gamma }\sum _{\alpha }{w}_{\gamma }^{\alpha }{\mathcal{T}}_{\gamma }(\lambda /{\lambda }_{\gamma }^{\alpha })$$ (2)

In this equation, ${\mathcal{T}}_{\gamma }$ is the intrinsic hyperelasticity stress function that provides a unique stress for each deformation for each generation of a bond type. As a result, the state variables that control stress are λ, $w_{\gamma }^{\alpha }$, and ${\lambda }_{\gamma }^{\alpha }$ [23].

We modeled the kinetics of molecular bonds to formulate the evolution of $w_{\gamma }^{\alpha }$, and ${\lambda }_{\gamma }^{\alpha }$ for the different bond types [23]. Three types of bonds are defined based on their kinetic rate: formative (γ = f), permanent (γ = p), and sliding (γ = s), which account for viscoelastic, hyperelastic, and plastic deformation behaviors, respectively [23]. The formative bonds have a finite rate of breakage and reformation and the reference stretch is determined by the deformation at which the generation was initiated. That is, ${\lambda }_f^{\alpha }=\lambda (t^{\alpha })$, and t^α is the time at which the generation was generated. This results in a viscoelastic behavior with a stress-free equilibrium condition (Figs. 1(a) and 1(d)) [24,27]. Permanent bonds do not break ($w_p^0=1$ and ${\lambda }_p^0=1$), which results in a hyperelastic behavior with no deformation history dependence. The permanent bonds are a special case of formative bonds at the limit of slow kinetics. Sliding bonds are the other limit case; this time the kinetics rate is fast and the bonds almost instantaneously break and reform. In this case, the plastic deformation behavior can be formulated using constitutive relations for ${\lambda }_s^{\alpha }$, where, for the 1D uniaxial tensile loading, ${\lambda }_s^{\alpha }\le \lambda$ is the sufficient condition [23].

In this formulation, damage can be applied to all of the bond types by reducing the number fraction of the bonds able to carry load [23]. As a result

$$T(\lambda ,w_{\gamma }^{\alpha },{\lambda }_{\gamma }^{\alpha },D_{\gamma })=\sum _{\gamma }(1-D_{\gamma })\sum _{\alpha }{w}_{\gamma }^{\alpha }{\mathcal{T}}_{\gamma }(\lambda /{\lambda }_{\gamma }^{\alpha })$$ (3)

In the above relation, D_γ is the damage parameter for each bond type that is a scalar in the range [0,1] and it is defined as

$$D_{\gamma }=1-\frac{\sum _{\alpha }{\overline{w}}_{\gamma }^{\alpha }}{\sum _{\alpha }{w}_{\gamma }^{\alpha }}$$ (4)

Here, ${\overline{w}}_{\gamma }^{\alpha }$ is the number fraction of bonds with damage. According to Eq. (1) it reads as

$$D_{\gamma }=1-\sum _{\alpha }{\overline{w}}_{\gamma }^{\alpha }$$ (5)

Hence, if no bonds are damaged D_γ = 0, and if all of the bonds are damaged D_γ = 1 [23,35]. Both damage and the change of reference configuration in sliding bonds have a similar behavior during loading (Figs. 1(b) and 1(e)); however, during unloading, the difference between these two processes becomes evident. Unlike plastic deformation, damage causes the softening effect without a shift in the reference unloaded configuration (Figs. 1(c) and 1(f)) [23]. Note that for an example case with only permanent bonds (γ = p), because there is just one generation with nonzero number fraction (i.e., $w_p^{\alpha }=w_p^0=1$), Eq. (3) reduces to

$$T=(1-D_p){\mathcal{T}}_p(\lambda )$$ (6)

which is the common formulation used in classic continuum damage mechanics for isotropic damage [35,41–43].

2.2. Plastic Deformation and Damage Constitutive Relations.

In this study, two separate models were created using the RIE framework: (1) plastic deformation model and (2) damage model. For the plastic deformation model we used a combination of formative bonds (γ = fP) and sliding bonds (γ = sP), where

$$T_{\text{plastic}}=T_{\mathit{fP}}+T_{\mathit{sP}}$$ (7)

For the damage model, a combination of formative bonds (γ = fD) and permanent bonds (γ = pD) was used, where

$$T_{\text{damage}}=T_{\mathit{fD}}+T_{\mathit{pD}}$$ (8)

The details of the constitutive relations and parameters are explained in the following and are summarized in Table 1.

Table 1.

Summary of the bond types, constitutive relations, and the model parameters used for each of the inelastic models

	Bond type	Kinetics	Intrinsic hyperelasticity	Sliding	Damage	# of parameters
Plastic	Formative	Nth-order	Fiber-exponential	—	Weibull	10
		${[K({\mathrm{s}}^{-1}),N]}_{\mathit{fP}}$	${[C_1(\text{MPa}),C_2]}_{\mathit{fP}}$	—	${[k,l,r_0]}_{\mathit{fP}}$
	Sliding	Step	Fiber-exponential	Modified Weibull	—
	—	${[C_1(\text{MPa}),C_2]}_{\mathit{sP}}$	${[b,c,r_0]}_{\mathit{sP}}$	—
Damage	Formative	Nth-order	Fiber-exponential	—	Weibull	10
		${[K(s^{-1}),N]}_{\mathit{fD}}$	${[C_1(\text{MPa}),C_2]}_{\mathit{fD}}$	—	${[k,l,r_0]}_{\mathit{fD}}$
	Permanent	—	Fiber-exponential	—	Weibull
	—	${[C_1(\text{MPa}),C_2]}_{\mathit{pD}}$	—	${[k,l,r_0]}_{\mathit{pD}}$

2.2.1. Kinetics.

For formative bonds, an nth-order kinetics rate equation was used

$$\frac{d{w}_f^{\alpha }}{\mathit{dt}}=-K_f{\left(w_f^{\alpha }\right)}^{N_f}$$ (9)

In this relation, the model parameters are (K_f, N_f), where K_f is a positive number that scales the rate of the reaction and N_f is the order of the breakage reaction $(N_f\ge 1)$. For the special case of N_f = 1, this relation reduces to a first-order kinetics equation, where its time constant is τ_f = 1/K_f [23]. For the sliding bonds, a step kinetics relation is used, where for breaking generations $w_s^{\alpha }=0$, and for reforming generation $w_s^{\alpha +1}=1$ [23]. For the permanent bonds, since there is no bond breakage, ${\tau }_p=1/K_p\to \infty$.

2.2.2. Intrinsic Hyperelasticity.

To account for nonlinear stiffening of tissue (toe region), since the rat tail tendon fascicles have a mostly parallel makeup [44], a typical nonlinear fiber-exponential constitutive relation was used as the intrinsic hyperelastic function of the bonds [45,46]

$${\mathcal{T}}_{\gamma }(\lambda /{\lambda }_{\gamma }^{\alpha })=C_1[\text{exp}(C_2(\lambda /{\lambda }_{\gamma }^{\alpha }-1))-1]u(\lambda /{\lambda }_{\gamma }^{\alpha }-1)$$ (10)

In this equation, C₁ and C₂ are positive-valued model parameters. In the plastic model, C₁ and C₂ are the same for formative and sliding bonds, and likewise for the damage model C₁ and C₂ are the same for formative and permanent bonds. This is a simplifying assumption that indicates the bonds from each model have the same load bearing capability, but their difference is due to the difference in their kinetics and inelastic parameters. In addition, u(.) is the Heaviside step function, which is included so that the constitutive relation does not result in compressive stiffness.

2.2.3. Sliding.

The axial stretch was used as the sliding variable (Ξ_s = λ(t)), which governs the sliding process in a nonreversible way [23]. That is, sliding stretch increases upon loading, and its value does not change during unloading or lower loading. Hence, the difference between the reference stretch of two consecutive generations of the sliding bonds (α – 1 and α) is calculated as [23]

$${\lambda }_s^{\alpha }-{\lambda }_s^{\alpha -1}=f_s({\Xi }_s(t^{\alpha }))-f_s({\Xi }_s(t^{\alpha -1}))$$ (11)

A modified Weibull's relation was used for f_s such that

$$f_s\left({\Xi }_s\right)=({\Xi }_s-1)\left(1-\text{exp}\left[-{\left(\frac{{\Xi }_s-{\left(r_0\right)}_s}{c-1}\right)}^b\right]\right)$$ (12)

The model parameters are $b\hspace{0.17em}(b\ge 1)$ the shape parameter, c the scale parameter, and (r₀)_s is the initial threshold of sliding $(c\ge 1\hspace{0.17em},{(r_0)}_s\ge 1)$. For a simple ramp loading with ${(r_0)}_s=1$, one can show that ${\lambda }_s^{\alpha }=f_s({\Xi }_s)+1$. This constitutive relation guarantees that for a tensile loading, the reference configuration of bonds is always a noncompressive configuration (i.e., $1\le {\lambda }_s^{\alpha }<\lambda$). For convenience, the sliding stretch of the last generation ${\lambda }_s^{\alpha }$ will be simply referred to as the sliding stretch λ_s.

2.2.4. Damage.

Similar to the sliding formulation, the axial stretch was used as the damage variable (Ξ_D = λ(t)) and the Weibull's cumulative distribution function was used for accumulation of damage [34,35]

$$f_D\left({\Xi }_D\right)=1-\text{exp}\left[-{\left(\frac{{\Xi }_D-{\left(r_0\right)}_D}{l-1}\right)}^k\right]$$ (13)

In this equation, (k, l) are the Weibull's shape and scale parameters, respectively, and (r₀)_D is stretch at the initial onset of damage $(k\ge 1,l\ge 1\hspace{0.17em},{(r_0)}_D\ge 1)$ [23]. The accumulation of damage is calculated by considering the history of deformation and ${\dot{D}}_{\gamma }=(\partial f_D/\partial {\Xi }_D){\dot{\Xi }}_D$ using a nonrecoverable scheme that only allows for increase in damage during loading [23]. Similar to ${\lambda }_s^{\alpha }$, it can be shown that if (r₀)_D = 1, for a simple ramp loading $D_{\gamma }=f_D({\Xi }_D)$. Damage is applied to permanent bonds for the damage model. Additionally, the formative bonds in both models are allowed to accumulate damage because otherwise post-yield failure would not be possible. This is also consistent with the creep rupture of tendon under force loading, because a viscoelastic response with no damage would converge to an “equilibrium” under that regime of loading, which further justifies our consideration for damage on formative bonds [47,48].

2.3. Experimental Microtensile Test Data.

Experimental data from Lee et al. [18] were used to fit the two models. Briefly, tendon fascicles were dissected from the tail of Sprague–Dawley rats and tested in a phosphate buffered saline bath. The mechanical tensile test consisted of ∼5 mN preload, five cycles of preconditioning to 4% strain, a ramp load at 1%/s to either 4%, 6%, or 8% grip-to-grip strains (n = 7/group) (loading phase), held for 15 min (relaxation phase). Following relaxation, the specimen was unloaded at 1%/s (unloading phase), and reloaded to failure (reloading phase) (Fig. 2). The initial cross-sectional area and grip-to-grip length were used to calculate stress and strain. We used the maximum optical strain (reported as the tissue strain in Ref. [18]) to scale the grip-to-grip strain to account for the gripping effects and to avoid the errors in estimation of the applied strain to the tissue. The stress data were smoothed using the moving average method and resampled to one-thousand data points for each loading, relaxation, unloading, and reloading phases, and were then used for curve fitting to calculate the model parameters.

2.4. Model Implementation and Parameter Identification.

The models were implemented using a custom written matlab program for reactive inelasticity (ReactiveBond v1.1 [49]). For each phase of loading, relaxation, unloading, and reloading, one-thousand time-steps were used that results in the same number of generations for loading, unloading, and reloading. Note that no generations are initiated at the relaxation phase. These time-steps were adequate for reaching numerical convergence. For parameter identification, we used a constrained multivariable nonlinear optimization method (fmincon, matlab). The root-mean-square error was used as the optimization cost function. To minimize the chance of convergence to local minima, we ran the optimization for each of the experimental stress curves using 48 randomly generated initial guesses and the solution with the minimum final residual value was selected as the optimal answer (MultiStart, matlab).

2.5. Data Analysis.

The optimized curve-fit results were individually plotted for each phase of the test, and the resulting model parameters were illustrated using parallel coordinate plots (PCP) to visualize the fit parameters in a compact way [50]. In addition to the individual fit parameters, the means, medians, and the interquartile range (IQR) were calculated and displayed in the PCP. The resulting fit parameters' similarity to normal distributions was tested using a Jarque–Bera test (jbtest. matlab) with a significance level set at (p < 0.05).

The goodness of the fits was assessed using the percent error between the fits and the experiments, calculated as $\%\text{err}=100(T_{\text{fit}}-T_{\hspace{0.17em}\text{exp}\hspace{0.17em}})/T_{\hspace{0.17em}\text{exp}\hspace{0.17em},\text{max}}$. The IQR of %err is evaluated for both of the models, and plotted for each phase of the deformation as a function of the normalized time of the phase defined as $t=(t-t_1)/(t_2-t_1)$, where t₁ and t₂ correspond to the time at the beginning and the end of the phase, respectively. Normalization of time was performed to visualize the patterns of the errors due to variability in strains reached at each phase.

The constitutive relations used for formative bonds were the same between the models, so we hypothesized that the formative bond parameters are not sensitive to the choice of other bond types. To test this hypothesis, we compared the pairs of formative bond model parameters by using Wilcoxon signed rank test (signrank matlab, p < 0.05).

2.6. Quantification of Inelastic Effects.

The accumulation of inelastic effects is plotted for each bond type in each model. The inelastic effect of damage is represented by the damage parameter D_γ for formative bonds and permanent bonds. To calculate the inelastic effect of sliding on stress, analogous to the damage parameter, we defined a normalized representation of the sliding effect (when the sliding is occurring) as

$${\pi }_{\mathit{sP}}=1-T_{\mathit{sP}}/T_{s0}$$ (14)

where $T_{s0}$ is the stress response of sliding bonds using the same intrinsic hyperelasticity relation with no sliding (equivalent to a permanent bond response). As a result, π_sP takes a value between zero and one, where ${\pi }_{\mathit{sP}}=0$ indicates that there is no sliding in the system, and ${\pi }_{\mathit{sP}}=1$ has a zero-stress response. To compare the inelastic effect between the models, the median stretches at the 50% inelastic effect ($D_{\mathit{fP}},{\pi }_{\mathit{sP}},D_{\mathit{fD}},D_{\mathit{pD}}=50\%$) were compared. Using paired comparison, median stretch of the formative bonds of the two models were compared to each other, and the median stretches for sliding bonds were compared to permanent bonds (signrank, p < 0.05).

2.7. Model Validation.

To validate the models, we compared the response of both models to the constant strain-rate ramp experimental tests by Szczesny et al. [40]. This type of mechanical testing is widely used for material characterization; thus, it is a suitable choice for validation [7,51]. It is important to note that the validation data were not included for the curve fitting. We used the nonstained constant-rate ramp data from Ref. [40] (n = 4, Fig. 6 in Ref. [40]). The median values from the fit results of the models (Table 2) are used for predicting the mechanical response to a 1%/s deformation curve, up to the peak stress, and the time-step between the initiation of the generations was set as 1.5 ms that results in adequate convergence. The modulus and the peak stress are compared between the experimental data and the model predictions.

3. Results

Both the plastic deformation and damage models resulted in excellent fits to the experimental data (Fig. 3). In particular, both of the models were capable of fitting the loading phase that showed stress stiffening (toe region) followed by the linear region (Figs. 3(a) and 3(e)). During the sustained loading of the relaxation phase, stress relaxation was closely fit as a result of using the nth-order kinetics relation for the formative bonds (Figs. 3(b) and 3(f)). The unloading phase had a different nonlinear response compared to the loading phase due to the inelastic behaviors (Figs. 3(c) and 3(g)). The reloading curves showed significant softening before reaching the peak stress that was captured by both models (Figs. 3(d) and 3(h)).

The errors were generally small between the model fits and the experiments, where the median of errors did not exceed 5% (Fig. 4). However, the performance of the models was not equal in all of the phases. In particular, both of the models overestimated the loading toe region in the loading phase (Figs. 4(a) and 4(e)). The error during relaxation was maximum at the higher stress values—the start of relaxation—and it declined as the relaxation progressed (Figs. 4(b) and 4(f)). Both of the models had minimal overestimation during the unloading phase, where the plastic model had a slightly better performance toward the end of unloading (Figs. 4(c) and 4(g)). Upon reloading, the models underestimated the stress in the nonlinear stiffening region, and the error was less at the end of the reloading phase (Figs. 4(d) and 4(h)).

The fit parameters were visualized using PCP, where each piecewise line corresponds to the fit parameters of one sample (Fig. 5). Several model parameters had a different median and mean value, which is a sign of non-normality of the distributions; this was also confirmed by the normality test that indicates for an overall response the median is more appropriate compared to the mean value (Table 2). Both of the models showed a similar trend in fit parameters for formative bonds, which were modeled using the same parameter sets, but independently fit using each model (Fig. 5). However, the statistical comparisons indicated that formative bonds' parameters had some differences. In particular, for kinetics of bonds, K_f parameters were slightly different (p < 0.05), where for the plastic model $K_f=0.32\hspace{0.17em}{\mathrm{s}}^{-1}$, and for the damage model $K_f=0.34\hspace{0.17em}{\mathrm{s}}^{-1}$, but there was no difference between the order of the kinetics relation ($N_f\sim 1.5$). Additionally, contrary to our hypothesis, all of the pairs of intrinsic hyperelasticity ($C_{1f},C_{2f}$) and damage ($k_f,l_f,{(r_0)}_f$) parameters were different, except for only l_f that did not show a difference (p < 0.05) (Table 2). The sliding parameters of sliding bonds and the damage parameters of the permanent bonds were dissimilar, which was expected since they belong to different bond types (Fig. 5 and Table 2).

The inelastic effects on the bonds were calculated for each experiment and the stretch at which 50% inelastic effect for a bond type was reached (Fig. 6). In the plastic model, the formative bonds' damage reached D_fP = 0.50 at the stretch value of 1.07 [1.04, 1.09] (read as median [Q1, Q3], Fig. 6(a)). The sliding bonds in the plastic model, π_sP reached 50% at the stretch of 1.05 [1.04, 1.06] (Fig. 6(b)). For the damage model, the formative bonds reached D_fD = 0.50 at λ = 1.06 [1.05, 1.07], which had a slightly different distribution compared to that of formative bonds in the plastic model (p < 0.05) (Fig. 6(c)) and the permanent bonds D_pD = 0.5 was reached at λ = 1.05 [1.03, 1.05] which was similar to the sliding bonds (Fig. 6(d)). When looking at the individual samples, due to the nonrecoverable nature of the inelastic effects during reloading, higher values were reached for all of the inelastic parameters compared to their counterpart in the loading phase (Fig. 6). In addition to the inelastic effects, for sliding bonds, we plotted the reference stretch λ_s that reached the small value of 1.01 [1.0, 1.02] during loading, and it increased to 1.05 [1.04, 1.05] at the end of reloading phase (Fig. 6(b)).

To validate the model fits, the median parameter from both of the models was used to predict independent experiments [40], where both models produced similar stress-stretch curves that include the initial stiffening (toe-region) and subsequent increase in stress followed with softening (Fig. 7). Average experimental modulus and peak stress were 909±27 MPa and 58±4 MPa, respectively [40]. The predicted modulus of the plastic model was 832 MPa and the peak stress was 31 MPa. Respectively, these values were –8% and –46% different from the experiments. For the damage model, the modulus was 1036 MPa, and the peak stress was 64 MPa, which were 14% and 10% different from their experimental counterparts, respectively. Overall, both models produced similar stress-stretch curve shapes, and modulus values; however, the damage model resulted in a closer peak stress prediction when compared to the plastic model (Fig. 7).

4. Discussion

4.1. Comparison Between the Plastic Deformation and Damage Models.

In this study, we successfully applied the theoretical framework of reactive inelasticity to tendon experimental data [18]. We implemented two independent models, specific to plastic deformation or damage, and showed excellent fits to the experiments (Fig. 3), and visualized the outcome parameters using parallel coordinate plots, which enhances visualization of the complex relationships among them. The fits showed that the plastic model had slightly smaller errors toward the end of the unloading phase, which was due to inclusion of plastic deformation that agrees with previous studies [13,14]. However, when comparing the model predictions using the resultant fit parameters to independent validation experimental data, both of the models predict similar stress-stretch curves, with the damage model having more similar behavior (Fig. 7). These results do not strongly favor one approach over the other, and both of the inelastic mechanisms may contribute to the mechanical response; thus, further investigations and different experimental testing protocols are needed to differentiate between tendon plastic deformation and damage behaviors. Particularly, force-controlled tests that can enable unloading to zero-force rather than zero-deformation can be helpful to distinguish plastic deformation and damage.

4.2. Interpretation of Inelastic Effects.

The J-shaped form for the accumulation of inelastic behaviors in our study (Fig. 6) is consistent with the findings of a recent study that reported the increase in denatured collagen during tensile loading by using collagen hybridizing peptides [20]. This indicates that the accumulation of inelastic effects may be correlated with collagen denaturation due to mechanical loading. However, the collagen hybridizing peptides study suggested onset of damage at higher strains (∼8% strain) [20] compared to our study (${(r_0)}_{\gamma }<5\%$, Table 2) and other studies [13,14,18]. This suggests that in addition to collagen denaturation, other molecular mechanisms and proteins in extra-cellular matrix (perhaps elastin or decorin) may play a role in the inelastic mechanical response of tendon [52–54]. Further, part of the inelastic response can also arise from multiscale “sliding” between tendon's fibrous components [40,55] which is not intended to be represented by the sliding bonds, the bonds specifically address the macroscale response in a continuum mechanics scheme. While identification of the relationships between tendon's inelastic mechanics and molecular disruptions require further investigations, the inelastic effects can be quantified by using mechanical modeling for understanding the mechanical properties of tendon, and assessing its altered mechanical properties in various stages of disease, injury, and healing [48,56,57].

4.3. Remarks About Reactive Inelasticity Modeling.

We hypothesized that the formative bond parameters would be the same for both models regardless of the choice of sliding or permanent bonds for the equilibrium response. While they were similar, there were some unexpected and statistically significant differences (Table 1). This may be due to the simplifying assumption we made that the formative bonds had the same intrinsic hyperelasticity parameters as the sliding or permanent bonds (Table 1). We made this assumption in order to use the minimum number of variables to model the tendon response. It is unclear if full independence of bond parameters would provide a benefit over the current modeling framework. Nonetheless, it is likely that including interaction terms between bonds or addition of more bonds and parameters, as a spectrum [58,59], could be beneficial for accurate modeling of tissue's inelastic behaviors.

4.4. Sources of Error.

The percent error (%err) of the models were quite small, but both of the models had relatively large errors in the toe-regions and the beginning of the relaxation phase (Fig. 4). Here, our continuum modeling scheme is intended to study the macroscale inelastic behaviors, and it does not account for the microstructure. Smoother fits could be achieved with more elaborate constitutive relations and microstructural considerations, such as fiber-recruitment for intrinsic hyperelasticity [60–62] or normalized energy and stress for sliding and damage [35,63,64], and by adding derivatives of the stress response to the cost function. For the relaxation phase, despite some deviations at the beginning, this phase was well fit by using a generalized nth-order kinetics for formative bonds (Figs. 4(b) and 4(f)). The error in the relaxation phase may be related to fluid flow-dependent viscoelasticity [46,65], which was not included in this study. We expect this to be a small effect for tail tendon; however, it can be added by using a biphasic mixture with RIE used to model the solid phase [30,66]. Additionally, we assumed that the kinetics rate of bond breakage and reformation is not dependent on the level of strain; however, adding strain dependence to kinetics parameters might be necessary for modeling more complicated loading scenarios such as incremental stress relaxation [24,67].

4.5. Conclusion and Future Direction.

In conclusion, we applied the theoretical framework of RIE to model viscoelasticity, plastic deformation, and damage to determine tendon's inelastic mechanical response. This study is novel in that (1) it investigated tendon inelastic effects without a prior assumption about the softening mechanism and compared the two modeling approaches for inelasticity, (2) it demonstrated numerical values for inelastic effects on tendon mechanical response, and (3) it provides a path forward to relate the molecular structure of tendon to its mechanical response. We applied the models to the experimental data, and validated the model by comparing the predictions to an independent set of experimental data. Both models were successful in fitting and predicting the experimental data, although the plastic model had slightly better fits during the unloading phase, and the damage model had better predictions of the independent validation data set. However, the results largely indicate that deformation-driven experiments can be equally described with both plastic deformation and damage perspectives. Thus, distinguishing plastic deformation and damage mechanisms will require further experimental studies using different loading protocols.

Appendix: Fit Parameters

The detailed descriptive statistics of the fit parameters (Table 2). The normality tests are marked on the mean values and the comparison between the formative bond parameters are marked on the median values.

Fig. 1.

Schematic difference between the plastic deformation and damage models. In the plastic model (a–c), a combination of formative bonds with damage and sliding bonds is used. For the damage model (d–f), a combination of formative bonds with damage and permanent bonds with damage is used. Both of the models have similar behaviors during loading, and relaxation ((a and d) and (b and e)); however, when looking closer at the boxed region in (b) and (e) during unloading (c and f), the plastic model shows a shift in reference unloaded configuration, where there is none for the damage model.

Fig. 2.

Microtensile mechanical testing profile: (a) The samples were ramped to a target grip strain (4%, 6%, or 8%) at a 1%/s rate (loading phase), held for 15 min (relaxation phase), unloaded at 1%/s (unloading phase), and reloaded to failure (reloading phase). The profile shown corresponds to a 6% target strain test.

Fig. 3.

Fit results for individual experiments. The fits are overlaid (black dashed line) on top of individual experimental stress responses, for (a and e) loading, (b and f) relaxation, (c and g) unloading, and (d and h) reloading phases of the experiment. (a–d) are fits to the plastic model and (e–f) to the damage model.

Fig. 4.

The IQR of %err of the model fits of (a and e) loading, (b and f) relaxation, (c and g) unloading, and (d and h) reloading phases as a function of the normalized time of each phase defined as $\tilde{t}=(t-t_1)/(t_2-t_1)$, where t₁ is the beginning of the phases and t₂ corresponds to end of the phase. (a–d) are %err for the plastic model and (e–f) for the damage model (median = dashed line, Q1, Q3 = dotted lines).

Fig. 5.

Parallel coordinate plots of the fit parameters for (a) plastic deformation model and (b) damage models, color-coded based on the nominal grip strain (red = 4%, orange = 6%, and blue = 8%). The IQR is shaded, where the median value is plotted using a solid black line. The mean value is marked with “×”. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.

Fig. 6.

Accumulation of damage and plastic deformation. The overall response for accumulation of damage (D_γ), and normalized plastic deformation (π_s) are plotted for each bond type using the median of the fit parameters (solid line). The 50% inelastic effect lines are marked using horizontal straight lines and the median stretch of all samples' intersected with this 50% inelastic effect line is marked with a vertical line. Additionally, the maximum damage and normalized plastic deformation reached during individual fits in loading (open circle), and reloading (star) phases for the individual fits. For the sliding bonds, the sliding stretch (λ_s) is also plotted (dashed line relative to right y-axis) and the maximum plastic deformation stretch during loading (open square) and reloading (cross) phases is plotted with respect to the right y-axis (brown).

Fig. 7.

Validation plots for prediction of plastic and damage models, showing the comparison of the experimental data from Ref. [40] to the predictions of the plastic model (red) and damage model (blue) to a ramp loading. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.

Table 2.

Fit parameter results

	Bond type	Kinetics	Intrinsic hyperelasticity	Sliding	Damage
Plastic model	Formative	${[K({\mathrm{s}}^{-1}),N]}_{\mathit{fD}}$	${[C_1(\text{MPa}),C_2]}_{\mathit{fP}}$	—	${[k,l,r_0]}_{\mathit{fP}}$
	Mean	[0.43*, 1.44*]	[2.98, 74.34*]	N/A	[1.74*, 1.04*, 1.05]
	Median	[0.32a, 1.49]	[2.62b, 47.47c]	N/A	[1.50d, 1.03, 1.04e]
	Q1	[0.23, 1.42]	[0.67, 41.76]	N/A	[1.34, 1.03, 1.02]
	Q3	[0.55, 1.56]	[4.76, 113.87]	N/A	[1.70, 1.06, 1.06]
	Sliding	—	${[C_1(\mathit{MPa}),C_2]}_{\mathit{sP}}$	${[b,c,r_0]}_{\mathit{sP}}$	—`
	Mean	N/A	Same as fP	[1.18*, 1.07*, 1.03]	N/A
	Median	N/A		[1.04, 1.06, 1.03]	N/A
	Q1	N/A		[1.00, 1.06, 1.02]	N/A
Q3	N/A		[1.22, 1.08, 1.03]	N/A
Damage model	Formative	${[K(s^{-1}),N]}_{\mathit{fD}}$	${[C_1(\text{MPa}),C_2]}_{\mathit{fD}}$	—	${[k,l,r_0]}_{\mathit{fD}}$
	Mean	[0.52*, 1.49*]	[2.09, 81.48]	N/A	[2.27, 1.05, 1.02]
	Median	[0.34a, 1.51]	[1.97b, 61.53c]	N/A	[2.10d, 1.05, 1.03e]
	Q1	[0.25, 1.38]	[0.61, 50.01]	N/A	[1.26, 1.03, 1.02]
	Q3	[0.67, 1.58]	[3.37, 123.12]	N/A	[2.72, 1.06, 1.03]
	Permanent	—	${[C_1(\mathit{MPa}),C_2]}_{\mathit{pD}}$	—	${[k,l,r_0]}_{\mathit{pD}}$
	Mean	N/A	Same as fD	N/A	[1.40*, 1.02, 1.03]
	Median	N/A		N/A	[1.16, 1.02, 1.03]
	Q1	N/A		N/A	[1.06, 1.02, 1.02]
Q3	N/A		N/A	[1.43, 1.03, 1.03]

^*The distribution is non-normal (p < 0.05).

^a–eThe medians of the parameter distributions are different (p < 0.05).

Funding Data

• National Institutes of Health (Grant No.: R01EB002425; Funder ID: 10.13039/100000002).