A damage model for collagen fibres with an application to collagenous soft tissues

Abstract

We propose a mechanical model to account for progressive damage in collagen fibres within fibrous soft tissues. The model has a similar basis to the pseudoelastic model that describes the Mullins effect in rubber but it also accounts for the effect of cross-links between collagen fibres. We show that the model is able to capture experimental data obtained from rat tail tendon fibres, and the combined effect of damage and collagen cross-links is illustrated for a simple shear test. The proposed three-dimensional framework allows a straightforward implementation in finite-element codes, which are needed to analyse more complex boundary-value problems for soft tissues under supra-physiological loading or tissues weakened by disease.

1. Introduction

Fibrous soft tissues consist of distributions of collagen fibres which, for example, could be almost parallel, as is the case for tendons, dispersed, as for artery walls, or isotropic, as for the middle zone of cartilage. These fibres are embedded in an essentially isotropic extrafibrillar matrix consisting of elastic fibres (including elastin), proteoglycans, water, adhesion proteins and integrins, inter alia. Of particular interest are the mechanical properties of the collagen fibres, the main load-bearing constituents of soft tissues, and their contribution to the overall behaviour of the tissues [1]. There are several constitutive models available that capture the distribution of collagen fibres [2–5]. However, it is important to note that under certain supra-physiological loads and in certain tissue diseases collagen starts to soften and finally ruptures, as shown by the experimental data in Pins & Silver [6] on a single collagen fibre. But as yet such effects on the microscale have not been incorporated in constitutive models. For soft biological tissues, a number of damage models are available, as described in [7]. For example, one study [8] derives a fibre/fibril damage model based on failure once a critical tissue stretch is reached. The proposed model reproduces the typical nonlinear behaviour of ligaments, including the toe and linear regions, then damage and eventual failure. The computational work in [9] proposes a (rather complex) macroscopic tissue damage model by considering recruitment stretches for the fibre content and failure of fibres in a distribution at different stretches or strain energies. Decoupled damage mechanisms for the matrix and fibres are considered. On the basis of [8], the constitutive model in [10] considers fibre recruitment and damage distributions by using probability density functions. It also includes a constitutive model for unloading after damage. The work in [11] applies the constrained mixture theory of [12] to study the formation/dilatation of abdominal aortic aneurysms. In particular, the constitutive model accounts for continuous degradation and creation of collagen fibres (i.e. the disappearance of old collagen and appearance of new collagen). The purpose of the present paper is now to develop a basic model at the collagen/cross-link microscale level that can account for these softening and failure effects.

The influence of the concentration of collagen fibre cross-links on the anisotropic response of fibrous soft tissues such as arterial walls was first analysed with a fully three-dimensional model in [13]. However, that approach was solely based on the theory of hyperelasticity and no damage mechanism was included, which limits its applicability. Another aim of this paper is, therefore, to account for collagen fibre damage in the presence of undamaged cross-links, which is the subject of §2. The model has a similar structure to that of the pseudoelastic model of the Mullins effect of rubber published in [14] but takes account of damage during loading rather than unloading. In the present account, this damage model is used to reproduce the experimental behaviour of rat tail tendon fibres. In §3, with the inclusion of cross-links, we analyse the combined effect of damage and cross-links in the simple shear of a single family of parallel fibres embedded in an isotropic matrix. In particular, we demonstrate the influence of damage and the proportion of cross-links on the shear stress versus the amount of shear response. This illustrates that fibre damage leads to a softening behaviour and finally failure of the tissue. In §4, we provide concluding remarks and point to the needs for further experimental data at the microscopic level to inform the macroscopic tissue behaviour. This model approach can be implemented in a finite-element code to execute more realistic boundary-value problems, for which purpose we provide the elasticity tensor in appendix A.

2. Damage model considering cross-linking

a. Damage formulation

We start by introducing the deformation gradient F relative to a given reference configuration, and the related right and left Cauchy–Green tensors C = F^TF and b = FF^T, respectively. For further use, we define the isotropic invariant I₁ and the pseudo-invariant I₄ according to

$$I_1=\text{tr}\hspace{0.17em}\mathbf{\text{C}}\text{and}{I}_4=(\mathbf{\text{C}}\mathbf{\text{M}})\cdot \mathbf{\text{M}}={\lambda }^2,$$ 2.1

where M is the direction of aligned fibres in the stress-free reference configuration which are embedded in an isotropic matrix and λ is the fibre stretch. Now let us consider a fibre-reinforced material such as a collagenous soft tissue, which is subject to the incompressibility constraint det F = 1, with a strain-energy function of the form Ψ(I₁, I₄). The Cauchy stress tensor σ is then given by [15]

$$\mathit{\sigma }=2{\psi }_1\mathbf{\text{b}}+2{\psi }_4\mathbf{\text{m}}\otimes \mathbf{\text{m}}-p\mathbf{\text{I}},$$ 2.2

where p is a Lagrange multiplier and m = FM is the fibre direction in the deformed configuration. Here, for convenience, we have introduced the abbreviations

$${\psi }_1=\frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }{I}_1}\text{and}{\psi }_4=\frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }{I}_4}.$$ 2.3

Now we consider the possibility of damage occurring when the stretch λ in the fibre exceeds some critical value, say λ_c. To model the damage effect, we introduce a (dimensionless) damage variable η, which is an additional independent variable so that Ψ(I₁, I₄, η). In the damage phase, the Cauchy stress is again given by (2.2) with the optimization condition

$$\frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }\eta }=0,$$ 2.4

which gives η implicitly in terms of I₁ and I₄. Let $I_{4\text{c}}={\lambda }_{\text{c}}^2$ be the critical value of I₄. We take η = 1 whenever I₄ ≤ I_4c, so (2.2) applies with Ψ(I₁, I₄, 1) and (2.4) is not active. For definiteness, we now take

$$\mathit{\Psi }(I_1,I_4,\eta )={\mathit{\Psi }}_{\text{iso}}(I_1)+\eta {\mathit{\Psi }}_{\text{fib}}(I_4)+\varphi (\eta ),$$ 2.5

analogously to the model of the Mullins effect [14], where ϕ(η) is some measure of damage. Then (2.2) gives

$$\mathit{\sigma }=2{\psi }_{\text{i}}\mathbf{\text{b}}+2\eta {\psi }_{\text{f}}\mathbf{\text{m}}\otimes \mathbf{\text{m}}-p\mathbf{\text{I}},$$ 2.6

where we have introduced the abbreviations

$${\psi }_{\text{i}}=\frac{\mathrm{\partial }{\mathit{\Psi }}_{\text{iso}}}{\mathrm{\partial }{I}_1}\text{and}{\psi }_{\text{f}}=\frac{\mathrm{\partial }{\mathit{\Psi }}_{\text{fib}}}{\mathrm{\partial }{I}_4},$$ 2.7

and with (2.5) the optimization condition (2.4) gives

$${\varphi }^{\mathrm{\prime }}(\eta )=-{\mathit{\Psi }}_{\text{fib}}(I_4),$$ 2.8

which determines η in terms of I₄, i.e. damage is only related to the fibres. We require

$$\varphi (1)=0\text{and}{\varphi }^{\mathrm{\prime }}(1)=-{\mathit{\Psi }}_{\text{fib}}(I_{4\text{c}}).$$ 2.9

Note that if we use (2.4) to give η = η(I₄) and write, say,

$$\overline{\mathit{\Psi }}(I_4)=\eta {\mathit{\Psi }}_{\text{fib}}(I_4)+\varphi (\eta ),$$ 2.10

then by (2.8) we obtain ${\overline{\mathit{\Psi }}}^{\mathrm{\prime }}=\eta {\psi }_{\text{f}}$, where ψ_f is according to (2.7)₂. This leads to an alternative arrival at the second term on the right-hand side of (2.6).

A suitable choice of ϕ^′(η), which gives a decaying behaviour for η as I₄ increases beyond I_4c and damage progresses, is

$${\varphi }^{\mathrm{\prime }}(\eta )=m\log \eta -{\mathit{\Psi }}_{\text{fib}}(I_{4\text{c}}),$$ 2.11

and hence, by (2.8),

$$\eta =\exp (-\frac{{\mathit{\Psi }}_{\text{fib}}(I_4)-{\mathit{\Psi }}_{\text{fib}}(I_{4\text{c}})}{m}),$$ 2.12

where m > 0 is a parameter with the same dimension as Ψ. Figure 1 provides a schematic of the damage parameter η as a function of the stretch λ. It shows that η decreases from 1 when the stretch λ increases beyond the critical value λ_c down to the value η_f when λ reaches the failure value λ_f. This schematic is based on specific calculations of η for uniaxial extension and simple shear, which exhibit very similar behaviour.

Figure 1.

Damage parameter η versus stretch λ, with critical stretch λ_c and failure stretch λ_f.

b. Uniaxial extension

Let λ be the stretch in the fibre direction M and σ the related uniaxial Cauchy stress. Then, the components of (2.6) yield

$$\sigma =2{\psi }_{\text{i}}{\lambda }^2+2\eta {\lambda }^2{\psi }_{\text{f}}-p\text{and}0=2{\psi }_{\text{i}}{\lambda }^{-1}-p,$$ 2.13

and hence, on elimination of p,

$$\sigma =2{\psi }_{\text{i}}({\lambda }^2-{\lambda }^{-1})+2\eta {\lambda }^2{\psi }_{\text{f}},$$ 2.14

where ψ_i and ψ_f are defined in (2.7). Now choose

$${\mathit{\Psi }}_{\text{iso}}=\frac{\mu }{2}(I_1-3)\text{and}{\mathit{\Psi }}_{\text{fib}}=\frac{k_1}{2k_2}\{\exp [k_2{(I_4-1)}^2]-1\},$$ 2.15

for the matrix and the fibre properties, respectively, where μ and k₁ are stress-like parameters, while k₂ is dimensionless. Hence, according to (2.7), 2ψ_i = μ and 2ψ_f = 2k₁(I₄ − 1)exp [k₂(I₄ − 1)²], so that (2.12), gives

$$\eta =\exp [-\frac{k_1}{2m{k}_2}\{\exp [k_2{(I_4-1)}^2]-\exp [k_2{(I_{4\text{c}}-1)}^2]\}],$$ 2.16

with I₄ = λ² and $I_{4\text{c}}={\lambda }_{\text{c}}^2$.

From (2.14), we then obtain

$$\sigma =\mu ({\lambda }^2-{\lambda }^{-1})+2k_1{\lambda }^2({\lambda }^2-1)\exp [k_2{({\lambda }^2-1)}^2],$$ 2.17

when no damage occurs (λ ≤ λ_c), and

$$\sigma =\mu ({\lambda }^2-{\lambda }^{-1})+2\eta k_1{\lambda }^2({\lambda }^2-1)\exp [k_2{({\lambda }^2-1)}^2],$$ 2.18

with (2.16), when λ ≥ λ_c.

The nominal stress P = σ/λ is plotted in figure 2 as a fit to the uniaxial test data from a rat tail tendon fibre shown in [6], using the parameter values μ = 0, k₁ = 115 MPa, k₂ = 7.7, m = 6 MPa and λ_c = 1.05. Note that, since we are modelling a single fibre here, we emphasize that there is no need to include the isotropic term.

Figure 2.

Fit to experimental data from a rat tail tendon fibre taken from [6], whereby the dots represent the data extracted digitally from the curve in [6]. The solid curve shows the nominal stress P = σ/λ versus stretch λ according to (2.18) with μ = 0 and k₁ = 115 MPa, k₂ = 7.7, m = 6 MPa, λ_c = 1.05.

From (2.9)₁ and (2.11), we have

$$\varphi (\eta )=m\eta \log \eta +(1-\eta )[m+{\mathit{\Psi }}_{\text{fib}}(I_{4\text{c}})],$$ 2.19

and from (2.12)

$$m\log \eta ={\mathit{\Psi }}_{\text{fib}}(I_{4\text{c}})-{\mathit{\Psi }}_{\text{fib}}(I_4).$$ 2.20

There is no energy loss for I₄ ≤ I_4c. Energy loss for I₄ ≥ I_4c is given by

$${\mathit{\Psi }}_{\text{fib}}-[\eta {\mathit{\Psi }}_{\text{fib}}+\varphi (\eta )]=m(\eta -\log \eta -1)>0\text{for}\hspace{0.17em}\eta <1.$$ 2.21

Although there are no data available for the unloading phase, it is worthwhile illustrating the stress softening effect induced by η during unloading prior to failure. For uniaxial extension, this is shown by the schematic in figure 3 with three unloading curves from different points on the loading path. This parallels figure 2 with the nominal stress versus stretch.

Figure 3.

Schematic of the nominal stress P versus stretch λ during loading in uniaxial extension (continuous curve), with unloading curves (dashed) from three different points on the loading curve prior to failure.

c. Inclusion of collagen fibre cross-links

Let us use the unit vector M, which identifies the collagen fibre direction in the reference configuration, and introduce N, which is an arbitrary unit vector orthogonal to M. Now we consider two families of cross-links around the collagen fibre direction M with the unit vectors L⁺ and L⁻, which are rotationally symmetric about M, and with the action of F on them defined by

$${\mathbf{\text{L}}}^{\pm }=\pm \cos {\alpha }_0\mathbf{\text{M}}+\sin {\alpha }_0\mathbf{\text{N}}\text{and}\mathbf{\text{F}}{\mathbf{\text{L}}}^{\pm }=\pm c_0\mathbf{\text{F}}\mathbf{\text{M}}+s_0\mathbf{\text{F}}\mathbf{\text{N}},$$ 2.22

where α₀ defines their orientation relative to the direction M (figure 4). For conciseness, we have written s₀ = sinα₀ and c₀ = cosα₀.

Figure 4.

(a) Parallel fibres in the M direction with two families of parallel cross-links described by the vectors L⁺ and L⁻ making an angle α₀ with M. (b) Detail of a pair of cross-links, showing rotational symmetry about the M direction with the orthogonal vector N (modified from [13]).

The invariant I₄ associated with the fibre direction is given in (2.1)₂, and the invariants I^±, which are the squares of the stretches in the cross-link directions, and the quantities $I_8^{\pm }$ describing the coupling between the collagen fibres and cross-links are defined by Holzapfel & Ogden [13]

$$I^{\pm }=c_0^2{I}_4\pm 2s_0{c}_0(\mathbf{\text{C}}\mathbf{\text{M}})\cdot \mathbf{\text{N}}+s_0^2(\mathbf{\text{C}}\mathbf{\text{N}})\cdot \mathbf{\text{N}}\text{and}{I}_8^{\pm }=\pm c_0{I}_4+s_0(\mathbf{\text{C}}\mathbf{\text{M}})\cdot \mathbf{\text{N}}.$$ 2.23

Note that, in general, I⁺ ≠ I⁻ and $I_8^{+}\ne -I_8^{-}$.

From the derivatives of I₄, I^± and $I_8^{\pm }$ with respect to the right Cauchy–Green tensor C given in [13] applied to a strain-energy function $\mathit{\Psi }(I_1,I_4,I^{+},I^{-},I_8^{+},I_8^{-}),$ we obtain the general expression of the Cauchy stress tensor as

$$\begin{array}{rl}\mathit{\sigma }& =-p\mathbf{\text{I}}+2{\psi }_1\mathbf{\text{b}}+2{\psi }_4\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{M}}\\ & +2{\psi }_{I^{+}}[c_0^2\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{M}}+s_0{c}_0(\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{N}}+\mathbf{\text{F}}\mathbf{\text{N}}\otimes \mathbf{\text{F}}\mathbf{\text{M}})+s_0^2\mathbf{\text{F}}\mathbf{\text{N}}\otimes \mathbf{\text{F}}\mathbf{\text{N}}]\\ & +2{\psi }_{I^{-}}[c_0^2\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{M}}-s_0{c}_0(\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{N}}+\mathbf{\text{F}}\mathbf{\text{N}}\otimes \mathbf{\text{F}}\mathbf{\text{M}})+s_0^2\mathbf{\text{F}}\mathbf{\text{N}}\otimes \mathbf{\text{F}}\mathbf{\text{N}}]\\ & +{\psi }_{8^{+}}[2c_0\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{M}}+s_0(\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{N}}+\mathbf{\text{F}}\mathbf{\text{N}}\otimes \mathbf{\text{F}}\mathbf{\text{M}})]\\ & +{\psi }_{8^{-}}[-2c_0\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{M}}+s_0(\mathbf{\text{F}}\mathbf{\text{M}}\otimes \mathbf{\text{F}}\mathbf{\text{N}}+\mathbf{\text{F}}\mathbf{\text{N}}\otimes \mathbf{\text{F}}\mathbf{\text{M}})],\end{array}$$ 2.24

as in [13], but with a slightly different notation, where we have used the abbreviations

$${\psi }_{I^{\pm }}=\frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }{I}^{\pm }}\text{and}{\psi }_{8^{\pm }}=\frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }{I}_8^{\pm }}$$ 2.25

in addition to ψ₁ and ψ₄ defined in (2.3).

Note that the second Piola–Kirchhoff stress, which is important for finite-element implementations, here denoted by S, is related to σ by S = F⁻¹σF^−T for an incompressible material. Consequently, we can determine the total differential

$$\text{d}\mathbf{\text{S}}=\mathbb{C}:\frac{1}{2}\text{d}\mathbf{\text{C}},$$ 2.26

where the colon denotes the standard double contraction and $\mathbb{C}$ is the elasticity tensor in the material description required for finite-element analysis. For a general explicit expression for $\mathbb{C}$, we refer to appendix A.

i. (Uniaxial extension with cross-linking)

For uniaxial extension with stretch λ in the fibre direction, we have

$$\mathbf{\text{F}}\mathbf{\text{M}}=\lambda \mathbf{\text{M}}\text{and}\mathbf{\text{F}}\mathbf{\text{N}}={\lambda }^{-1/2}\mathbf{\text{N}}.$$ 2.27

Hence, with these two equations we can deduce from (2.22)₂

$$\mathbf{\text{F}}{\mathbf{\text{L}}}^{\pm }=\pm c_0\lambda \mathbf{\text{M}}+s_0{\lambda }^{-1/2}\mathbf{\text{N}},$$ 2.28

and (2.23) specializes to

$$I\equiv I^{\pm }=c_0^2{\lambda }^2+s_0^2{\lambda }^{-1},I_8=I_8^{+}=c_0{\lambda }^2\text{and}{I}_8^{-}=-I_8^{+}.$$ 2.29

Then ${\psi }_I={\psi }_{I^{+}}={\psi }_{I^{-}},{\psi }_{8^{+}}=-{\psi }_{8^{-}}={\psi }_8,$ and (2.24) specializes to

$$\begin{array}{rl}\mathit{\sigma }& =-p\mathbf{\text{I}}+2{\psi }_1({\lambda }^2\mathbf{\text{M}}\otimes \mathbf{\text{M}}+{\lambda }^{-1}\mathbf{\text{N}}\otimes \mathbf{\text{N}})+2{\psi }_4{\lambda }^2\mathbf{\text{M}}\otimes \mathbf{\text{M}}\\ & +4{\psi }_I(c_0^2{\lambda }^2\mathbf{\text{M}}\otimes \mathbf{\text{M}}+s_0^2{\lambda }^{-1}\mathbf{\text{N}}\otimes \mathbf{\text{N}})+4{\psi }_8{c}_0{\lambda }^2\mathbf{\text{M}}\otimes \mathbf{\text{M}},\end{array}$$ 2.30

the relevant components of which are

$$\sigma =-p+2{\psi }_1{\lambda }^2+2{\psi }_4{\lambda }^2+4{\psi }_I{c}_0^2{\lambda }^2+4{\psi }_8{c}_0{\lambda }^2$$ 2.31

and

$$0=-p+2{\psi }_1{\lambda }^{-1}+4{\psi }_I{s}_0^2{\lambda }^{-1}.$$ 2.32

By eliminating the Lagrange multiplier p we obtain

$$\sigma =2{\psi }_1({\lambda }^2-{\lambda }^{-1})+2{\psi }_4{\lambda }^2+4{\psi }_I(c_0^2{\lambda }^2-s_0^2{\lambda }^{-1})+4{\psi }_8{c}_0{\lambda }^2.$$ 2.33

Now let us use the specific strain-energy functions (2.15) supplemented by quadratic energy functions associated with the cross-links and fibre/cross-link interactions so that

$$\mathit{\Psi }=\frac{1}{2}\mu (I_1-3)+\eta \frac{k_1}{2k_2}\{\exp [k_2{(I_4-1)}^2]-1\}+\frac{1}{2}\nu {(I-1)}^2+\frac{1}{2}\kappa {(I_8-c_0)}^2,$$ 2.34

where ν and κ are parameters with dimension of stress associated with the cross-links and interactions, respectively. In particular, a larger ν corresponds to a larger density of cross-links, while κ is a measure of the interaction energy. Here, we have adopted very simple models for the energy in the cross-links and the interaction energy since there are no data available to justify more sophisticated forms of energy. From (2.12) with (2.15)₂, η is given by

$$\eta =\exp [-\frac{k_1}{2m{k}_2}\{\exp [k_2{({\lambda }^2-1)}^2]-\exp [k_2{({\lambda }_{\text{c}}^2-1)}^2]\}].$$ 2.35

From (2.33) to (2.34), with the help of (2.3) and (2.25), the Cauchy stress σ then becomes

$$\begin{array}{rl}\sigma & =\mu ({\lambda }^2-{\lambda }^{-1})+2k_1\eta {\lambda }^2({\lambda }^2-1)\exp [k_2{({\lambda }^2-1)}^2]\\ & +4\nu (I-1)(c_0^2{\lambda }^2-s_0^2{\lambda }^{-1})+4\kappa (I_8-c_0)c_0{\lambda }^2.\end{array}$$ 2.36

The fit to the experimental data of [6] is of similar agreement to that shown in figure 2. Specific parameters are, for example, k₁ = 120 MPa, ν = 15 MPa, k₂ = 6.4, m = 6 MPa, α₀ = π/4, λ_c = 1.02, κ = 8 MPa and μ = 0.

3. Application to planar deformation of soft tissues

Next, we consider the situation in which the collagen fibres and cross-links are restricted to the (E₁, E₂) plane and we define by M the direction of the family of aligned fibres and its normal N as

$$\mathbf{\text{M}}=\cos \alpha {\mathbf{\text{E}}}_1+\sin \alpha {\mathbf{\text{E}}}_2\text{and}\mathbf{\text{N}}=-\sin \alpha {\mathbf{\text{E}}}_1+\cos \alpha {\mathbf{\text{E}}}_2,$$ 3.1

where α is the angle between the fibre direction and the E₁ axis (figure 5). With respect to M and N the cross-link directions L^± between members of the family and the action of F thereon are again given by (2.22). The invariant I₄ = (CM) · M is as in (2.1)₂, but with M now defined by (3.1)₁, while the invariants I^± and the quantities $I_8^{\pm }$ are again given by (2.23). The Cauchy stress tensor σ has the same form (2.24) as in three dimensions but is now restricted to two dimensions.

Figure 5.

M represents the direction of a family of aligned fibres with unit normal N with respect to background axes E₁ and E₂, and M makes an angle α with respect to the E₁ direction. L^± represents the directions of two families of cross-links, and L^± make an angle α₀ with respect to the ±M direction (modified from Holzapfel & Ogden [13]).

a. Simple shear

For simple shear in the E₁ direction in the considered plane, the deformation gradient is given by F = I + γE₁ ⊗ E₂, where γ is the amount of shear. It follows that

$$\mathbf{\text{F}}\mathbf{\text{M}}=\mathbf{\text{M}}+\gamma \sin \alpha {\mathbf{\text{E}}}_1\text{and}\mathbf{\text{F}}\mathbf{\text{N}}=\mathbf{\text{N}}+\gamma \cos \alpha {\mathbf{\text{E}}}_1.$$ 3.2

The invariant I₄ = (CM) · M is

$$I_4=1+\gamma \sin 2\alpha +{\gamma }^2{\sin }^2\alpha ,$$ 3.3

while the required expressions (CN) · N and (CM) · N are given by

$$(\mathbf{\text{C}}\mathbf{\text{N}})\cdot \mathbf{\text{N}}=1-\gamma \sin 2\alpha +{\gamma }^2{\cos }^2\alpha \text{and}(\mathbf{\text{C}}\mathbf{\text{M}})\cdot \mathbf{\text{N}}=\gamma \cos 2\alpha +{\gamma }^2\sin \alpha \cos \alpha .$$ 3.4

On substitution of (3.4) into (2.23), we obtain

$$I^{\pm }=1+\gamma \sin 2(\alpha \pm {\alpha }_0)+{\gamma }^2{\sin }^2(\alpha \pm {\alpha }_0)$$ 3.5

and

$$I_8^{\pm }=\pm c_0+\gamma \sin ({\alpha }_0\pm 2\alpha )+{\gamma }^2\sin \alpha \sin ({\alpha }_0\pm \alpha ).$$ 3.6

From (2.24), the components of the Cauchy stress can be obtained, but we only need here the shear component σ₁₂, i.e.

$$\begin{array}{rl}{\sigma }_{12}& =2{\psi }_1\gamma +2[{\psi }_4+c_0^2({\psi }_{I^{+}}+{\psi }_{I^{-}})+c_0({\psi }_{8^{+}}-{\psi }_{8^{-}})]s(c+\gamma s)\\ & s_0[2c_0({\psi }_{I^{+}}-{\psi }_{I^{-}})+{\psi }_{8^{+}}+{\psi }_{8^{-}}](c^2-s^2+2\gamma sc)\\ & +2s_0^2({\psi }_{I^{+}}+{\psi }_{I^{-}})c(\gamma c-s)\equiv \frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }\gamma },\end{array}$$ 3.7

where for conciseness we have written s = sinα and c = cosα.

For illustrative purposes, we now consider the model strain-energy function [13]

$$\begin{array}{rl}\mathit{\Psi }& =\frac{1}{2}\mu (I_1-3)+\eta \frac{k_1}{2k_2}\{\exp [k_2{(I_4-1)}^2]-1\}+\frac{1}{2}\nu {(I^{+}-1)}^2+\frac{1}{2}\nu {(I^{-}-1)}^2\\ & +\frac{1}{2}\kappa {(I_8^{+}-c_0)}^2+\frac{1}{2}\kappa {(I_8^{-}+c_0)}^2,\end{array}$$ 3.8

which generalizes equation (2.34) to the case in which I⁺ ≠ I⁻ and $I_8^{+}\ne -I_8^{-}$. With (2.25) it follows that

$$\begin{array}{rl}& {\psi }_{I^{+}}+{\psi }_{I^{-}}=2\nu \gamma [2sc(c_0^2-s_0^2)+\gamma (s_0^2{c}^2+s^2{c}_0^2)],\end{array}$$ 3.9

$$\begin{array}{rl}& {\psi }_{I^{+}}-{\psi }_{I^{-}}=4\nu \gamma s_0{c}_0(c^2-s^2+\gamma sc),\end{array}$$ 3.10

$$\begin{array}{rl}& {\psi }_{8^{+}}+{\psi }_{8^{-}}=2\kappa \gamma s_0(c^2-s^2+\gamma sc)\end{array}$$ 3.11

$$\begin{array}{rl}\text{and}& {\psi }_{8^{+}}-{\psi }_{8^{-}}=2\kappa \gamma s{c}_0(2c+\gamma s).\end{array}$$ 3.12

In this case, from (2.12), we obtain, with the help of (2.15)₂ and (3.3),

$$\eta =\exp [-\frac{k_1}{2m{k}_2}\{\exp [k_2{\gamma }^2{s}^2{(\gamma s+2c)}^2]-\exp [k_2{\gamma }_{\text{c}}^2{s}^2{({\gamma }_{\text{c}}s+2c)}^2]\}],$$ 3.13

where γ_c is the critical value of γ at which damage is initiated.

Hence, from (3.7), we obtain with (2.3) and (3.9)–(3.12)

$$\begin{array}{rl}{\sigma }_{12}& =\mu \gamma +2\eta k_1\exp [k_2{\gamma }^2{s}^2{(2c+\gamma s)}^2]s^2(c+\gamma s)(2c+\gamma s)\gamma \\ & +4\nu \gamma \{2s^2{c}^2{(c_0^2-s_0^2)}^2+2s_0^2{c}_0^2{(c^2-s^2)}^2\\ & +3sc\gamma [(c_0^2-s_0^2)(c_0^2{s}^2+s_0^2{c}^2)+2s_0^2{c}_0^2(c^2-s^2)]+{\gamma }^2[{(c_0^2{s}^2+s_0^2{c}^2)}^2+4s_0^2{c}_0^2{s}^2{c}^2]\}\\ & +2\kappa \gamma \{s_0^2+4s^2{c}^2(c_0^2-s_0^2)+3sc\gamma [(s_0^2{c}^2+s^2{c}_0^2)+s^2(c_0^2-s_0^2)]+2{\gamma }^2{s}^2(s_0^2{c}^2+s^2{c}_0^2)\}.\end{array}$$ 3.14

In figure 6, we plot the dimensionless shear stress ${\overline{\sigma }}_{12}={\sigma }_{12}/\mu$ from (3.14) against the amount of shear γ in order to illustrate the dependence on the various parameters. The specific parameter values used are ${\overline{k}}_1=k_1/\mu =2$, α = π/3, k₂ = 0.1, γ_c = 0.7, m/μ = 10, α₀ = π/5, and $\overline{\nu }=\nu /\mu =0.5,2,\overline{\kappa }=\kappa /\mu =0.6$. Without cross-links $\overline{\nu }=\overline{\kappa }=0$. Clearly, the shear stress response stiffens with an increase in the cross-link parameter $\overline{\nu }$ without damage, while when damage is included the shear stress is reduced after a critical value of γ and can reach a maximum as γ increases, as evidenced in the case when there are no cross-links. For the related elasticity tensor of model (3.8), which is relatively simple, we refer to appendix A.

Figure 6.

Plots of the dimensionless shear stress ${\overline{\sigma }}_{12}$ versus the amount of shear γ with and without fibre damage (dashed and solid curves, respectively), with ($\overline{\nu }=\nu /\mu =0.5,2,\overline{\kappa }=\kappa /\mu =0.6$) and without ($\overline{\nu }=\overline{\kappa }=0$) cross-links. Parameter values: ${\overline{k}}_1=k_1/\mu =2$, α = π/3, k₂ = 0.1, γ_c = 0.7, m/μ = 10, α₀ = π/5.

4. Discussion and concluding remarks

This study proposes a simple mechanical model for the damage progression of stretched collagen fibres based on a pseudoelastic approach. The model is used to fit the limited data that are available on the stretching of an individual collagen fibre, and the agreement with the data is very satisfactory. The model has then been used for the construction of a constitutive model for fibre-reinforced soft tissues in which the collagen fibres are supported by cross-links. The predictions of the model have been illustrated by an application to a simple shear test in which both damage and cross-links are accounted for. The related elasticity tensor is also provided with a view to analysing more complex boundary-value problems requiring a finite-element implementation.

To inform the further development of models that incorporate damage and cross-linking more data are needed on the response and damage of stretched single fibres, their influence on aggregates of collagen fibres embedded in tissues and also the mechanical properties of the cross-links. It is well known that the proportion of cross-links increases with age and causes a stiffening of the tissue [16]. In addition, several studies have shown that the stiffening of fibrous tissues is related to the concentration of cross-links (e.g.[17,18]). Such a relationship is captured by our model.

The effect of proteoglycans is essentially incorporated into the isotropic part of the tissue model, partly because it is still unclear what their mechanical contribution is to the overall response of the tissue. However, there is evidence that proteoglycans can support forces in the piconewton range when stretched [19], but it is not clear if the level of stresses they can support is relevant for a constitutive model of the type proposed in this paper. The review article by Scott [20] has described a mechanism between the collagen fibrils (as distinct from fibres) governed mainly by proteoglycans, which are essentially orthogonal to the fibrils (note that the author calls this complex an ‘elastic shape module’). However, it seems that there is no quantification yet available that shows how the force is transmitted between the individual fibrils. In the present paper, we focus on the collagen fibre level without accounting for the structure of fibrils and proteoglycans. In addition, because of the orthogonal arrangement of the proteoglycans with respect to the fibrils (see [21]), the force transition would only be relevant for rather large deformations. It is worth pointing out, however, that force transition between proteoglycans and fibrils was accounted for in a mechanical model in [22], in which a collagen fibre is represented as a bundle of collagen fibrils cross-linked by proteoglycans.

More advanced multi-scale models are needed that capture the behaviours of the individual constituents such as proteoglycans, cross-linking proteins and their interaction with collagen molecules, fibrils and fibres and their aggregated contributions to the tissue. There is hope that current imaging modalities will allow a better understanding of the structure down to the nanoscale, but there is also a need for mechanical information at the same level. In order to tackle organ-level simulations the proposed model allows for a straightforward implementation within the finite-element method, which is a powerful tool for analysing more clinically relevant problems in health and disease.

Appendix A

Here, we explicitly present the elasticity tensor $\mathbb{C}$ in the material description, which is defined by

$$\mathbb{C}=4\frac{{\mathrm{\partial }}^2\mathit{\Psi }}{\mathrm{\partial }\mathbf{\text{C}}\mathrm{\partial }\mathbf{\text{C}}}.$$ A 1

Consider an energy function of the form $\mathit{\Psi }(I_1,I_4,I^{\pm },I_8^{\pm },\eta )$, where the four invariants are defined according to (2.1) and (2.23), η = η(I₄) is a damage variable accommodating damage only in the fibres, which is not specified explicitly at this point, and where I₄ is the square of the stretch in the fibre direction. Then the derivatives of the invariants and η with respect to C are

$$\frac{\mathrm{\partial }{I}_1}{\mathrm{\partial }\mathbf{\text{C}}}=\mathbf{\text{I}},\frac{\mathrm{\partial }{I}_4}{\mathrm{\partial }\mathbf{\text{C}}}={\mathbf{\text{A}}}_{\text{M}},\frac{\mathrm{\partial }{I}^{\pm }}{\mathrm{\partial }\mathbf{\text{C}}}=c_0^2{\mathbf{\text{A}}}_{\text{M}}\pm 2s_0{c}_0{\mathbf{\text{A}}}_{\text{MN}}+s_0^2{\mathbf{\text{A}}}_{\text{N}},\frac{\mathrm{\partial }{I}_8^{\pm }}{\mathrm{\partial }\mathbf{\text{C}}}=\pm c_0{\mathbf{\text{A}}}_{\text{M}}+s_0{\mathbf{\text{A}}}_{\text{MN}}$$

and

$$\frac{\mathrm{\partial }\eta }{\mathrm{\partial }\mathbf{\text{C}}}={\eta }^{\prime }(I_4){\mathbf{\text{A}}}_{\text{M}},$$

where we have introduced the notations

$${\mathbf{\text{A}}}_{\text{M}}=\mathbf{\text{M}}\otimes \mathbf{\text{M}},{\mathbf{\text{A}}}_{\text{MN}}=\frac{1}{2}(\mathbf{\text{M}}\otimes \mathbf{\text{N}}+\mathbf{\text{N}}\otimes \mathbf{\text{M}})={\mathbf{\text{A}}}_{\text{NM}},{\mathbf{\text{A}}}_{\text{N}}=\mathbf{\text{N}}\otimes \mathbf{\text{N}}.$$

It follows that

$$\begin{array}{rl}\frac{\mathrm{\partial }\mathit{\Psi }}{\mathrm{\partial }\mathbf{\text{C}}}& ={\psi }_1\mathbf{\text{I}}+[{\psi }_4+c_0^2({\psi }_{I^{+}}+{\psi }_{I^{-}})+c_0({\psi }_{I_8^{+}}-{\psi }_{I_8^{-}})+{\psi }_{\eta }{\eta }^{\prime }]{\mathbf{\text{A}}}_{\text{M}}\\ & +[2s_0{c}_0({\psi }_{I^{+}}-{\psi }_{I^{-}})+s_0({\psi }_{I_8^{+}}+{\psi }_{I_8^{-}})]{\mathbf{\text{A}}}_{\text{MN}}+s_0^2({\psi }_{I^{+}}+{\psi }_{I^{-}}){\mathbf{\text{A}}}_{\text{N}},\end{array}$$

where we have used the abbreviations (2.3), (2.25) and ${\psi }_{\eta }=\mathrm{\partial }\mathit{\Psi }/\mathrm{\partial }\eta$. Then, we can write

$$\begin{array}{rl}\frac{{\mathrm{\partial }}^2\mathit{\Psi }}{\mathrm{\partial }\mathbf{\text{C}}\mathrm{\partial }\mathbf{\text{C}}}& =a_1\mathbf{\text{I}}\otimes \mathbf{\text{I}}+a_2(\mathbf{\text{I}}\otimes {\mathbf{\text{A}}}_{\text{M}}+{\mathbf{\text{A}}}_{\text{M}}\otimes \mathbf{\text{I}})+a_3{\mathbf{\text{A}}}_{\text{M}}\otimes {\mathbf{\text{A}}}_{\text{M}}+a_4(\mathbf{\text{I}}\otimes {\mathbf{\text{A}}}_{\text{N}}+{\mathbf{\text{A}}}_{\text{N}}\otimes \mathbf{\text{I}})\\ & +a_5(\mathbf{\text{I}}\otimes {\mathbf{\text{A}}}_{\text{MN}}+{\mathbf{\text{A}}}_{\text{MN}}\otimes \mathbf{\text{I}})+a_6({\mathbf{\text{A}}}_{\text{M}}\otimes {\mathbf{\text{A}}}_{\text{N}}+{\mathbf{\text{A}}}_{\text{N}}\otimes {\mathbf{\text{A}}}_{\text{M}})+a_7{\mathbf{\text{A}}}_{\text{N}}\otimes {\mathbf{\text{A}}}_{\text{N}}\\ & +a_8({\mathbf{\text{A}}}_{\text{M}}\otimes {\mathbf{\text{A}}}_{\text{MN}}+{\mathbf{\text{A}}}_{\text{MN}}\otimes {\mathbf{\text{A}}}_{\text{M}})+a_9({\mathbf{\text{A}}}_{\text{N}}\otimes {\mathbf{\text{A}}}_{\text{MN}}+{\mathbf{\text{A}}}_{\text{MN}}\otimes {\mathbf{\text{A}}}_{\text{N}})+a_{10}{\mathbf{\text{A}}}_{\text{MN}}\otimes {\mathbf{\text{A}}}_{\text{MN}},\end{array}$$

where

$$\begin{array}{rl}{a}_1& ={\psi }_{11},\hspace{0.17em}{a}_2={\psi }_{14}+{\psi }_{1\eta }{\eta }^{\prime }+c_0^2({\psi }_{1I^{+}}+{\psi }_{1I^{-}})+c_0({\psi }_{1I_8^{+}}-{\psi }_{1I_8^{-}}),\\ a_3& ={\psi }_{44}+2{\psi }_{4\eta }{\eta }^{\prime }+{\psi }_{\eta \eta }{{\eta }^{\prime }}^2+{\psi }_{\eta }{\eta }^{″}+2c_0^2[{\psi }_{4I^{+}}+{\psi }_{4I^{-}}+{\eta }^{\prime }({\psi }_{\eta I^{+}}+{\psi }_{\eta I^{-}})]\\ & +2c_0[{\psi }_{4I_8^{+}}-{\psi }_{4I_8^{-}}+{\eta }^{\prime }({\psi }_{\eta I_8^{+}}-{\psi }_{\eta I_8^{-}})]+c_0^4({\psi }_{I^{+}{I}^{+}}+{\psi }_{I^{-}{I}^{-}}+2{\psi }_{I^{+}{I}^{-}})\\ & +c_0^2({\psi }_{I_8^{+}{I}_8^{+}}+{\psi }_{I_8^{-}{I}_8^{-}}-2{\psi }_{I_8^{+}{I}_8^{-}})+2c_0^3({\psi }_{I^{+}{I}_8^{+}}-{\psi }_{I^{+}{I}_8^{-}}+{\psi }_{I^{-}{I}_8^{+}}-{\psi }_{I^{-}{I}_8^{-}}),\\ a_4& =s_0^2({\psi }_{1I^{+}}+{\psi }_{1I^{-}}),\hspace{0.17em}{a}_5=2s_0{c}_0({\psi }_{1I^{+}}-{\psi }_{1I^{-}})+s_0({\psi }_{1I_8^{+}}+{\psi }_{1I_8^{-}}),\\ a_6& =s_0^2[{\psi }_{4I^{+}}+{\psi }_{4I^{-}}+{\eta }^{\prime }({\psi }_{\eta I^{+}}+{\psi }_{\eta I^{-}})]+s_0^2{c}_0^2({\psi }_{I^{+}{I}^{+}}+{\psi }_{I^{-}{I}^{-}}+2{\psi }_{I^{+}{I}^{-}})\\ & s_0^2{c}_0({\psi }_{I^{+}{I}_8^{+}}-{\psi }_{I^{+}{I}_8^{-}}+{\psi }_{I^{-}{I}_8^{+}}-{\psi }_{I^{-}{I}_8^{-}}),\\ a_7& =s_0^4({\psi }_{I^{+}{I}^{+}}+{\psi }_{I^{-}{I}^{-}}+2{\psi }_{I^{+}{I}^{-}}),\\ a_8& =2s_0{c}_0[{\psi }_{4I^{+}}-{\psi }_{4I^{-}}+{\eta }^{\prime }({\psi }_{\eta I^{+}}-{\psi }_{\eta I^{-}})]+s_0[{\psi }_{4I_8^{+}}+{\psi }_{4I_8^{-}}+{\eta }^{\prime }({\psi }_{\eta I_8^{+}}+{\psi }_{\eta I_8^{-}})]\\ & +2s_0{c}_0^3({\psi }_{I^{+}{I}^{+}}-{\psi }_{I^{-}{I}^{-}})+s_0{c}_0({\psi }_{I_8^{+}{I}_8^{+}}-{\psi }_{I_8^{-}{I}_8^{-}})\\ & +s_0{c}_0^2(3{\psi }_{I^{+}{I}_8^{+}}-{\psi }_{I^{+}{I}_8^{-}}-{\psi }_{I^{-}{I}_8^{+}}+3{\psi }_{I^{-}{I}_8^{-}}),\\ a_9& =2s_0^3{c}_0({\psi }_{I^{+}{I}^{+}}-{\psi }_{I^{-}{I}^{-}})+s_0^3({\psi }_{I^{+}{I}_8^{+}}+{\psi }_{I^{+}{I}_8^{-}}+{\psi }_{I^{-}{I}_8^{+}}+{\psi }_{I^{-}{I}_8^{-}})\\ \text{and}{a}_{10}& =4s_0^2{c}_0^2({\psi }_{I^{+}{I}^{+}}+{\psi }_{I^{-}{I}^{-}}-2{\psi }_{I^{+}{I}^{-}})+s_0^2({\psi }_{I_8^{+}{I}_8^{+}}+{\psi }_{I_8^{-}{I}_8^{-}}+2{\psi }_{I_8^{+}{I}_8^{-}})\\ & +4s_0^2{c}_0({\psi }_{I^{+}{I}_8^{+}}+{\psi }_{I^{+}{I}_8^{-}}-{\psi }_{I^{-}{I}_8^{+}}-{\psi }_{I^{-}{I}_8^{-}}).\end{array}$$

For notational simplicity, the subscripts 1, 4, 8⁺ and 8⁻ on ψ stand for I₁, I₄, $I_8^{+}$ and $I_8^{-}$, respectively, and we have introduced the abbreviation ${\psi }_{\bullet \star }={\mathrm{\partial }}^2\mathit{\Psi }/\mathrm{\partial }(\bullet )\mathrm{\partial }(\star )$. Note that in the expression for a₂ in eqn (78) of [13] the sign before ${\psi }_{1I_8^{-}}$ was incorrectly written as +.

For the model (3.8), these expressions simplify considerably, giving

$$a_1=a_2=a_4=a_5=a_8=a_9=0$$

and

$$a_3=\eta ({\psi }_{\text{ff}}-{\psi }_{\text{f}}^2/m)+2\nu c_0^4+2\kappa c_0^2,\hspace{0.17em}{a}_6=2\nu s_0^2{c}_0^2,\hspace{0.17em}{a}_7=2\nu s_0^4,\hspace{0.17em}{a}_{10}=8\nu s_0^2{c}_0^2+2\kappa s_0^2,$$

where ψ_f is defined according to (2.7)₂ and ${\psi }_{\text{ff}}={\mathrm{\partial }}^2{\mathit{\Psi }}_{\text{fib}}/\mathrm{\partial }{I}_4^2$. Note that the explicit expressions for η and Ψ_fib are provided in (2.12) and (2.15)₂, respectively.

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Authors' contributions

G.A.H. and R.W.O. conceived and designed the research, interpreted the results and drafted the manuscript. Both authors edited and revised the manuscript, and gave final approval for publication.

Competing interests

We declare we have no competing interest.

Funding

The work of R.W.O. was funded in part by UK EPSRCgrant no. EP/N014642/1.