Let
\(\textbf{F}:=\nabla \varphi \) be the deformation gradient mapping infinitesimal vectorial elements in the reference configuration onto the current configuration. The mapping of infinitesimal volume elements is then described by
\(J:=\det \textbf{F}\). In order to automatically fulfill the principle of objectivity, we consider models formulated in the right Cauchy-Green deformation tensor
\({\textbf{C}}:={\textbf{F}}^T{\textbf{F}}\). For hyperelastic materials, a strain energy density function
\(\Psi \) as a function of the deformation tensor is defined. In order to guarantee a physically reasonable material response avoiding a loss of material stability [
40], here, polyconvex energy functions in the sense of [
3] will be considered. By evaluation of the second law of thermodynamics, the 2nd Piola-Kirchhoff stress tensor and the Cauchy stress tensor can be computed by
$$\begin{aligned} \textbf{S} = 2\frac{\partial \Psi }{\partial \textbf{C}} \quad \text{ and }\quad \varvec{\sigma } = J^{-1}\textbf{F}\textbf{S}\textbf{F}^T, \end{aligned}$$
respectively. Based on the Cauchy stress tensor, the von Mises stresses can be computed. For a convenient construction of suitable strain energy density functions, often the invariants of the functional arguments in the energy function are considered. For isotropic materials, the principle invariants of
\({\textbf{C}}\), that is,
$$\begin{aligned} I_1=\textrm{tr}\,{\textbf{C}}, \quad I_2=\textrm{tr}[\textrm{Cof}({\textbf{C}})], \quad I_3=\det {\textbf{C}}, \end{aligned}$$
with
\(\textrm{Cof}(\textbf{C})=\det (\textbf{C})\textbf{C}^{-T}\), are taken into account and thus,
\(\Psi :=\Psi (I_1,I_2,I_3)\). Note that in the context of soft biological tissues, often the dependence on the second invariant is omitted. Thereby, the material response is mainly governed by normal stretches described through
\(I_1\) and volume changes described by
\(I_3=(\det \textbf{F})^2\). As a result of embedded collagen fibers, arterial wall tissues behave anisotropically. The fibers are mainly wound crosswise helically around the artery and in healthy arteries symmetrically disposed with respect to the axial direction. Around these main fiber family directions, fibers are stochastically dispersed; cf., e.g., [
22]. Assuming a weak interaction between the fiber families, the resulting anisotropy can be modelled by superimposing two transversely isotropic models. For the individual fiber family and thus, the description of transverse isotropy, the structural tensor
\({\textbf{M}}=\textbf{a} \otimes \textbf{a}, \Vert \textbf{M}\Vert =1\) [
43] is considered as additional argument in the strain energy density function. Herein, the preferred direction vector
\(\textbf{a}\) describes the fiber orientation. Thus, for a coordinate-invariant formulation, the mixed invariants
$$\begin{aligned} J_4=\textrm{tr}[{\textbf{C}\textbf{M}}], \quad J_5=\textrm{tr}[{\textbf{C}^2\textbf{M}}], \end{aligned}$$
are taken into account. Whereas
\(J_4\) describes the square of the stretch in fiber direction
\(\textbf{a}\), the physical meaning of
\(J_5\) is unclear.
Table 2
Material parameter sets for the Neo-Hooke model
NH\(_1\) | 2500.0 | 50.4 |
NH\(_2\) | 3333.3 | 67.1 |
NH\(_3\) | 6333.3 | 127.52 |
NH\(_4\) | 7333.3 | 147.65 |
2.1 Simple, isotropic material model: Neo-Hooke
As a simple, isotropic nonlinear material model, we consider a standard compressible Neo-Hookean energy. It is often written in the form
$$\begin{aligned} \Psi _{\textrm{NH}} = \Psi _{\textrm{vol}} + \Psi _{\textrm{isoch}}, \end{aligned}$$
where the volumetric and isochoric parts are given by
$$\begin{aligned} \Psi _{\textrm{vol}} = \frac{\kappa }{2} ( \ln (I_3^{1/2}) )^2 \quad \text{ and }\quad \Psi _{\textrm{isoch}} = \frac{\mu }{2} ( I_3^{-1/3}I_1 - 3). \end{aligned}$$
The material parameters are
\(\kappa \) and
\(\mu \) and modulate an increase of energy related to a volume change and a change of isochoric deformations, respectively. Due to the fact that soft biological tissues are mostly assumed to be almost incompressible, here, the volumetric energy will be used as a penalty function to adjust for the quasi-incompressibility constraint. Since the stresses are obtained as derivatives of the energy functions with respect to the deformation tensor, an increase of
\(\mu \) thus corresponds to an increase in stiffness and for incompressibility mainly modulates the slope in stress–strain diagrams of uniaxial tension tests. Since the Neo-Hookean model is not able to catch the stiffening of the tissue caused by the collagen fibers, the parameters were fitted by hand to the experiments in [
28]; see also [
8]. There, uniaxial tension tests were performed in circumferential and axial direction of human artery segments. Whereas the parameter set NH
\(_1\) was fitted to approximate the slope of the experimental curves (media) in the range
\(0<\Delta l/l_o<0.2\), the set NH
\(_2\) approximately corresponds to the average slope of the experimental curve in the range
\(0<\Delta l/l_o<0.235\) and thus, results in a slightly stiffer behavior. NH
\(_3\) and NH
\(_4\) were chosen significantly stiffer in order to obtain a similar lumen area as the anisotropic models before the start of the heart beat; see also Sect.
4.1. The parameter sets are summarized in Table
2.
The resulting stress-stretch curves are compared with the experiments in Fig.
1. Note that the experimental data shows a slight hysteresis resulting from a negligible visco-elastic response. We will see in the FSI simulations in Sect.
4 that, for NH
\(_1\) and NH
\(_2\), the resulting material behavior is significantly softer compared to the sophisticated, anisotropic models; see Fig.
4.
For the sets NH\(_3\) and NH\(_4\) the in- and outflow lumen in the simulations presented later is, at the end of the ramp, similar to the lumen areas of the sophisticated models. These parameter sets correspond well for a specific loading scenario in a structural problem, but due to their artificially stiff response, the associated stress–strain curves will not match well the experimental curves on average. For all parameter sets, the compression modulus \(\kappa \) was chosen such that the volume change of the model response was kept below 1% in the uniaxial tension tests.
Note that, in the benchmark computations presented in [
7, Figure 21], based on the material model
\(\Psi _A\), the increase of the arterial circumference during the heartbeat was below
\(20\,\%\).
2.2 Anisotropic material models
Following the analysis in [
10], we consider different anisotropic and quasi-incompressible material models for arterial walls. They are of the form
$$\begin{aligned} \Psi _{\textrm{X}} = \Psi _{\textrm{X, iso}} (I_1,I_3) + \sum _{a=1}^2 \Psi _{\textrm{X}, (a)}^{\textrm{ti}} (I_1,I_3,J_4^{(a)},J_5^{(a)}) \end{aligned}$$
where
\(X\in \{A,B,E\}\). Herein, the mixed invariants are considered separately for the two fiber family directions
\(\textbf{a}^{(a)},a = 1,2\), i.e.,
\(J_4^{(a)} = \text{ tr }(\textbf{C}\textbf{M}^{(a)})\) and
\(J_5^{(a)} = \text{ tr }(\textbf{C}^2\textbf{M}^{(a)})\) with
\(\textbf{M}^{(a)} = \textbf{a}^{(a)}\otimes \textbf{a}^{(a)}\). In detail, the individual functions are
Table 3
Material parameter sets for nonlinear, anisotropic models
\(\Psi _A\) | 17.5 | 499.8 | 2.4 | 30,001.9 | 5.1 | – | – |
\(\Psi _B\) | 10.7 | 207.1 | 9.7 | – | – | 1018.8 | 20.0 |
\(\Psi _E\) | 9.7 | 95.3 | 3.8 | – | – | 687.6 | 20.0 |
Model \(\Psi _A\) [
8]
$$\begin{aligned}&\Psi _{\textrm{A, iso}} (I_1,I_3) = \epsilon _1 (I_3^{\epsilon _2} + I_3^{-\epsilon _2} -2) + c_1 (I_1 I_3^{-1/3}-3), \\&\Psi _{\textrm{A}, (a)}^{\textrm{ti}}(I_1,J_4^{(a)},J_5^{(a)}) = \alpha _1 \left\langle I_1 J_4^{(a)} - J_5^{(a)} -2 \right\rangle ^{\alpha _2}, \end{aligned}$$
Model \(\Psi _B\) (
isochoric and anisotropic part from [
29])
$$\begin{aligned}&\Psi _{\textrm{B, iso}} (I_1,I_3) = \epsilon _1 (I_3^{\epsilon _2} + I_3^{-\epsilon _2} -2) + c_1 (I_1 I_3^{-1/3}-3), \\&\Psi _{\textrm{B}, (a)}^{\textrm{ti}} (I_3,J_4^{(a)}) = \frac{k_1}{2k_2} \left( \exp \left( k_2 \left\langle J_4^{(a)} I_3^{-1/3} - 1 \right\rangle ^{2} \right) - 1 \right) , \end{aligned}$$
Model \(\Psi _E\) (
anisotropic part from [
30])
$$\begin{aligned}&\Psi _{\textrm{E, iso}} (I_1,I_3) = \epsilon _1 (I_3^{\epsilon _2} + I_3^{-\epsilon _2} -2) + c_1 (I_1 - 3 - \ln (I_3)) ,\\&\Psi _{\textrm{E}, (a)}^{\textrm{ti}} (J_4^{(a)}) = \frac{k_1}{2k_2} \left( \exp \left( k_2 \left\langle J_4^{(a)} -1 \right\rangle ^2 \right) - 1 \right) . \end{aligned}$$
Herein, the Macauley brackets
\(\langle (\bullet )\rangle := ((\bullet ) - |(\bullet )|)/2\) filter out negative values. Note that
\(\Psi _{\textrm{A, iso}} = \Psi _{\textrm{B, iso}}\). The models
\(\Psi _B\) and
\(\Psi _E\) are based on the well-known Holzapfel, Gasser, and Ogden model, where the transversely isotropic parts do not include
\(I_1\) and
\(J_5\). They are formulated such that a specifically stiff response is purely generated in the fiber directions. In contrast to this, the model
\(\Psi _A\) includes
\(J_5\) and even a coupling of
\(I_1\) with
\(J_4\). Although
\(J_5\) may not directly have a physical meaning, the term
\(I_1 J_4 - J_5\) as part of
\(\Psi _A\) was found in [
39] to describe the change of infinitesimal area elements with normal vectors perpendicular to the fiber direction. Due to the coupling term
\(I_1 J_4\) in this model, a somewhat dispersed stiffness around the fiber direction is also included. Note that more sophisticated models for the description of fiber dispersion are available (cf., e.g., [
22]), which allow for a more independent and less phenomenological quantification of dispersion intensity. Only the model
\(\Psi _B\) is formulated in a volumetric-isochoric split. Whereas this may enable a more direct quantitative interpretation of the material parameters, it also renders the model response questionable under purely volumetric loading since the stress response will then be purely isotropic. In addition to that, the uncontrolled volume change in the isochoric energy may be problematic in finite element formulations where, for example, the volume change is only considered as volume average of each finite element, or associated terms are used for reduced integration. Summarizing, all models are quasi-incompressible, provided that sufficiently large parameters in the volumetric penalty functions are considered, they are highly nonlinear, anisotropic, polyconvex, and widely used, but each of them has certain advantages and disadvantages. We use the parameters for the media fitted in [
10, Figure 2] to the experiments performed in [
28] which are given in Table
3. Note that the parameters used here for
\(\Psi _A\) correspond to
\(\Psi _A\) Set 2 in [
10] and are identical to the parameters used for
\(\Psi _A\) in [
7]. Although the models
\(\Psi _A\) and
\(\Psi _B\) have the same isotropic part
\(\Psi _{\mathrm{*, iso}}\), the parameters for
\(\Psi _{\mathrm{*, iso}}\) are not the same.
Note that all parameters for the simple isotropic and the sophisticated anisotropic models were obtained from adjusting to uniaxial tension tests which correspond to extreme values of stress ratios of circumferential to axial stresses ranging from 0 to infinity, which are often found difficult, already for engineering materials [
34]. In real arteries, however, a uniaxial stress scenario cannot be expected and rather biaxial stress scenarios appear with stress ratios of moderate intensity in between the extreme values. Therefore, in principle, it would be advantageous to also include biaxial test data, but these are in turn technically difficult to obtain for soft biological tissues and their accuracy should be considered critically, especially for experiments performed on individual layers like the media and adventitia of small arteries. Uniaxial tests may therefore be considered more reliable, but the level of information regarding the mechanical response is rather restricted compared to biaxial tests; cf., e.g., [
21]. Therefore, in general, it is recommended to calibrate a model to the average of many test results including uniaxial and biaxial tests with varying load ratios and not to overaccelerate the use of isolated test results. However, here we are rather interested in a qualitatively realistic, characteristic response and thus, a perfect match of the models with the experimental data should not be overrated. Therefore, the somewhat better agreement of model
\(\Psi _A\) with the nonlinear experimental stress–strain response (cf. Fig.
1) should not be given too much importance.