Physical response of hyperelastic models for composite materials and soft tissues
- Minh Tuan Duong^{1, 2},
- Nhu Huynh Nguyen^{1, 3} and
- Manfred Staat^{1}Email authorView ORCID ID profile
DOI: 10.1186/s40540-015-0015-x
© Duong et al. 2015
Received: 16 April 2015
Accepted: 9 November 2015
Published: 8 December 2015
Abstract
A hyperelastic model must not only characterize the mechanical response of a composite material such as soft tissue, but also ensure numerical stability by a feasible set of material parameters. Apart from the well-known ill-conditioning problem caused by the incompressibility constraint, the paper indicates another ill-conditioning occurring in any general fibre-reinforced material model for tubular organs when unbalance between the fibre strain energy and the matrix strain energy becomes too large. Specifically, although the Holzapfel model is polyconvex, this problem can be observed as an unphysical behaviour in a physiological deformation range of a tissue such as arterial wall and intestine by thickening in the thickness direction associated with a volume growth of a specimen in a tension test. Particularly, the same problem for a polyconvex modified Fung-type model with the matrix characterized by the neo-Hookean model has been discussed for the first time. By investigating the influence of the shear modulus in these two models, we not only prove the cause of the ill-conditioning but also propose a solution to control the unbalance in the strain energy. The numerical results show significant enhancement of the model stability in overcoming the unphysical deformation.
Keywords
Physical behaviour Ill-conditioned Numerical stability Constitutive model Soft tissuesBackground
Composite materials such as biological soft tissues can be modelled using a phenomenological approach such as an exponential function by Fung et al. [1] and later fully completed in a more general form by Humphrey [2]. This classical Fung-type model has been widely used in biomechanics such as rabbit skin (Tong and Fung [3]), rabbit arteries (Chuong and Fung [4]) and a right coronary artery (Takamizawa [5]). On the contrary, a micro/structural approach models the microstructural constituents of tissues such as modelling fibres of connective tissue as long sinusoidal beams by Comninou and Yannas [6] and flat collagenous fibres (Lanir [7]). Particularly, Holzapfel et al. [8] postulated a strain–energy function (called the Holzapfel model) for arteries based on microstructures of the tissues by considering each layer of the artery as a composite material whose non-collagenous tissues, e.g. elastin, are considered as isotropic matrix substance. The collagen, which is assumed to be composed of two families of collagenous fibres, plays the role of a reinforcing material, and is assumed to be perfectly embedded in the isotropic matrix. The fibres and the matrix are characterized by different strain–energy functions and the material coefficients have clear physical meaning.
Besides the capability of representing the material behaviour of biological soft tissue accurately, constitutive equations must also ensure numerical stability in computer simulation by satisfying the Legendre–Hadamard or the ellipticity condition equivalent to a convexity or polyconvexity condition (Marsden et al. [9]).
Although the Holzapfel model is polyconvex (Schröder et al. [10]), a numerical instability known as unphysical behaviour can occur for an improper set of material coefficients as described in tension tests by Gasser et al. [11]. Considering the tests, the structural arrangement of the stiffened fibres in combination with the soft ground matrix activates a load-carrying mechanism in which the collagen fibres need to rotate towards the loading direction until they are able to carry significant load. Consequently, there is an unrealistic thickening of the sample strip in tension tests and it violates the incompressibility condition with a volume growth. To prevent this non-physical behaviour, Gasser et al. [11] took into account an effect of the dispersion of the collagen fibres whose orientations are not uniform. Admittedly, this proposal helps the model come closer to the real behaviour. In fact, by investigating terms of the strain–energy function, it is obvious that the incorporated dispersion is able to control the non-physical thickening by decreasing the difference between the orthotropic energy and the isotropic energy.
A set of material constants, which is identified by curve fitting and satisfies the goodness of fit, is not unique in general. For example, by fixing the shear modulus, Helfenstein et al. [12] identified two sets of material constants of the Holzapfel model with and without a volumetric–isochoric split. This split was also known as another source of the non-physical response (Ehlers and Eipper [13]). Shear moduli differing by a factor of 10 can be observed for two different arterial layers (Holzapfel et al. [8]). Obviously, a too soft ground matrix characterized by the small shear modulus of the neo-Hookean model may cause the constitutive matrix of the overall model to be ill-conditioned because the entries of this matrix are summations of both the small stiffness of the ground matrix and the large stiffness of the fibres, leading to a high condition number. Thus, this composite material model shows strongly directional behaviour and would behave non-physically in a direction along which the strain–energy dissipation is minimal, e.g. in directions orthogonal to the fibre orientations (Duong et al. [14]). In tubular organs this is observed in the radial direction.
In this paper, we show that the above thickening effect is concerned with the ill-conditioning problem and no recourse to the “fibre rotation”, as given by Gasser et al. [11], is needed. Observations of the ill-conditioning for nonlinear elastic models for certain classes of soft biological tissue have been mentioned earlier, see, e.g. Sun and Sacks [15]. Besides, the formulation of biological models is not similar, and hence mechanisms causing the ill-conditioning of each model are different. Herein, we would prove that the unrealistic phenomenon of the Holzapfel model is resulting from the ill-conditioned constitutive matrix caused by large differences of the magnitudes of the shear terms in the elasticity tensor. Moreover, the modified Fung-type model, abbreviated as MFH model (Holzapfel [16]), has been proven to be affected by this ill-conditioning problem in the physiological deformation range for the first time (Duong [17]). However, no indication for relation between this numerical instability and the mentioned “fibre rotation” or the source of the thickening effect has been provided in literature [11–13]. In addition, no suggestion for treating the ill-conditioning issue as well as for obtaining a lower condition number has been proposed because solving the condition number equation through matrix manipulation leads to a very complicated formulation. Thus, in this paper the numerical condition number of the elasticity tensor (hereafter it is shortly named the condition number), whose value changes as a function of material parameters and deformation, is also investigated. Numerical results show the significant influence of the shear moduli in the two considered models on the unphysical behaviour. Consequently, this suggests how to treat the unphysical mechanical response of the material law by controlling the unbalance of the energy with the highest possible shear modulus, leading to significant improvements in the numerical results.
Constitutive Equations
Models for soft biological tissues
For the sake of comparison and investigation of the ill-conditioning, in this section two well-known material models as well as their modifications are recalled; the Fung model [1] and the Holzapfel model [8].
General Fung model
Holzapfel model
Some modified material laws
Non-physical behaviour
To obtain the unique solution of a boundary-value problem, the strain–energy function is required to satisfy the quasi-convexity condition. Ball [23] proposed a polyconvexity concept implying quasi-convexity and strong ellipticity. This guarantees material stability. For isotropic materials, the neo-Hookean model satisfies the concept of polyconvexity (Schröder and Neff [25]). Similarly, the Holzapfel model is also a polyconvex function with positive material parameters (Schröder and Neff [25]). In addition, the MFH model is also polyconvex (Balzani [24]). However, polyconvexity is not sufficient to ensure physical responses of the Holzapfel model and of the MFH as described in the following.
Holzapfel model with two fibre families and one fibre family
The simulation of a half of the specimen in Fig. 2 with Poisson’s ratio v = 0.49996 shows clearly the thickening of the strip at the middle, and this effect becomes more severe if the load continuously increases. Specifically, the thickness in Or direction (O1, displacement 1) orthogonal to the fibre plane \(O\theta z\) (O23) for the two fibre families increases which is inconsistent with the physically expected transverse contraction. Interestingly, for the case of one fibre family, the thickness increases in all directions in a plane orthogonal to the fibre direction O3, see Fig. 2. Although this set of material parameters ensures the polyconvexity of the Holzapfel model, the numerical results are, however, incorrect. The MFH model is subjected to this problem in the same manner.
Ill-conditioning and condition number of constitutive matrix
Results
Numerical tests: physical behaviour of submucosa layer
In this section, we indicate the relation between the ill-conditioning and the non-physical behaviour of the Holzapfel model and the MFH model. Obviously, for a 3D boundary-value problem, it is impossible to compute analytically the condition number because of complicated matrix manipulations. Therefore, the condition number is calculated numerically for our tension tests.
Equibiaxial test of submucosa
Uniaxial tension test of submucosa
Material parameters for the submucosa layer of large intestine (μ* = 1.58 kPa)
μ/μ* | μ (kPa) | k _{1p } (kPa) | k _{2p } = k _{3p } | α (°) |
---|---|---|---|---|
1.0 | 1.580* | 0.9095* | 12.1000* | 60* |
0.1 | 0.158 | 0.9134 | 12.0927 | 60 |
10.0 | 15.800 | 0.8710 | 12.1752 | 60 |
Unrealistic response of the MFH model with two fibre families
Discussion
For the Holzapfel model, the difference amongst the components of \({\mathbb{C}}\) can be adjusted through the shear modulus \(\mu\). If the magnitudes of the components of \({\mathbb{C}}_{\text{iso}}\) are more comparable to the corresponding ones of \({\mathbb{C}}_{\text{ani}}\), e.g. using a larger shear modulus \(\mu\), then the relative difference amongst the components of \({\mathbb{C}}\) are reduced, see Fig. 11. This helps decrease the condition number of the elasticity tensor in the equibiaxial simulation and hence the critical stretch is significantly increased, see Fig. 8.
For the case of the smallest shear modulus \(\mu = \mu^{*} /10\), the unphysical deformation begins from the smallest critical stretch \(\lambda = \lambda_{\text{c}}^{\mu^*/10}\), see Fig. 5. When larger values of the shear modulus \(\mu = \mu^{*}\) and \(\mu = 10\mu^{*}\) were used, the critical stretches increases significantly, denoted by \(\lambda_{\text{c}}^{\mu^* }\) and \(\lambda_{\text{c}}^{10\mu^* }\), respectively. Consequently, we have \(\lambda_{\text{c}}^{{10\mu^{*} }} > \lambda_{\text{c}}^{{\mu^{*} }} > \lambda_{\text{c}}^{{\mu^{*} /10}}\). Correspondingly, Fig. 8 shows the relative differences of the condition numbers for the three cases. These differences always start from the corresponding lower critical stretch, at which the unphysical deformation begins to take place. For example, if \(\mu = 10\mu^{*}\) and \(\mu = 100\mu^{*}\) the condition number is decreased by 55 and 75 %, respectively. These moderate reductions already ensure the physical behaviour of the material model. When the augmented Lagrange method was not utilized, the smallest critical stretch was decreased, \(\lambda_{\text{c}}^{{\mu^{*} /10}} = 1.2\), since the other ill-conditioning is caused by the nearly incompressible constraint. Therefore, the augmented Lagrange method can only mitigate the ill-conditioning problem but cannot prevent this in the deformation range of interest as an alternative solution as suggested by Helfenstein et al. [12]. Thus, apart from the source of the thickening effect proposed by Helfenstein et al. [12], we demonstrated that it is caused by the large difference between the two energy terms. Figure 2 shows that the unphysical deformation is observed in a plane or in directions orthogonal to the fibre direction. This is also supported by the equibiaxial test with the unrealistic displacement perpendicular to the fibre plane.
Moreover, the model of Weiss et al. [32] is constructed by an isotropic strain–energy term as a polynomial function in terms of the invariants \(I_{1}\) and \(I_{2}\), whereas the anisotropic term is characterized as an exponential function for one embedded fibre family. Therefore, this model would have a large difference in the mechanical contribution between the isotropic energy and the anisotropic energy, leading to the same instability problem as reported in Helfenstein et al. [12]. The thickening effect was also similarly observed for pure shear tests (Duong [17]). More importantly, the same is found for simulation of tension tests using of the MFH model. To this end, the influence of the shear modulus was also investigated for the MFH model as shown in Fig. 10. The model shows the bad effect in the physiological deformation range of the tension test with \(\mu = \mu^{*}\) and \(\mu = 0.1\mu^{*}\). The higher the shear modulus (\(\mu = 10\mu^{*}\)) is chosen, the more stable the MFH model becomes. Thus, the critical stretch increases with the increase of the shear modulus which should be chosen as large as possible for numerical stability. For the case of an incompressible Fung model in-plane stress, Sun and Sacks [15] imposed both the upper bound for the condition number and the convexity for the model to achieve numerical stability. However, this approach is limited to a specific case of plane stress. As shown in Fig. 8, the condition number is a function of deformation and the material parameters. Thus, preventing the ill-conditioning by fixing a predicted upper bound for the condition number is not always plausible.
Figure 5 shows an equibiaxial test simulated by the neo-Hookean model with only a linear polynomial function. The corresponding condition number curve of the neo-Hookean elasticity tensor is regular in Fig. 8. Moreover, Helfenstein et al. [12] have even utilized the real measure of fibre stretch (\(I_{4}\) and \(I_{6}\)) instead of (\(\overline{{I_{4} }}\) and \(\overline{{I_{6} }}\)) for the model by Rubin and Bodner [33], but there is of course negligible effect observed. This is due to the fact that there are only exponential terms in the strain–energy formulation. Thus, the instability problem originates from the unbalance of strain energy. Moreover, a model composed of the neo-Hookean and an isotropic version of the Fung-type potential (Nguyen et al. [20]; Duong et al. [21]) [see (5)] is also immune from the ill-conditioning even at very high stretch up to 2.0 due to the absence of an anisotropic term. A model proposed by Peng et al. [37] has both the fibre and the matrix terms described by polynomial functions and a shear interaction is modelled by a product of a weak exponential and a polynomial function in terms of all principal invariants. Therefore, the difference between the anisotropic and isotropic strain energy is equilibrated, leading to a stable model. In addition, Gasser et al. [11] have modified the structure tensor characterizing dispersion of fibre orientations to overcome the unphysical deformation. As discussed before, the fibre dispersion helps reduce the large difference between the isotropic strain energy and the anisotropic strain energy and therefore ensure stability of their model. However, the fibre dispersion is not always easily obtained due to limits of experiments
Conclusions
Generally, the larger difference in mechanical contributions apparently gives rise to the larger differences amongst the components of the material tangent stiffness matrix and therefore leads to the ill-conditioned stiffness matrix. The similar ill-conditioning is known from the simulation of composite structures comprising two main constituents with a great difference in mechanical stiffness. Typical examples of this are the organic-coated metals. Throughout the investigation of the ill-conditioning problem, the magnitude of \(\mu\) strongly affects the simulation accuracy of the Holzapfel model but has no influence on the accuracy of the neo-Hookean model. Therefore, the thickening effect does not originate from the “fibre rotation” as discussed by Gasser et al. [11]. Moreover, the shear modulus also significantly influences the numerical stability of the MFH model but has no effect on any isotropic Fung-type model (Duong [17]). It accounts for the ill-conditioning problem, which results from the large condition numbers of the elasticity tensors of the Holzapfel model and the MFH model. Since the shear modulus \(\mu\) of soft tissue usually varies in a bounded set, the shear modulus \(\mu\) should be chosen closer to the physically upper bound (its natural value). Alternatively, Shore hardness indentation tests could be used to measure the shear modulus of the matrix directly (Duong [17]). These measured values of the shear modulus help ensure the numerical stability of the material law (Duong [17]).
Declarations
Authors’ contributions
NHN was first to observe the nonphysical behavior originating from the ill-conditioned constitutive matrix discussed in the article, MS has connected it with the overall ill conditioning and MTD has proposed some improved material models. NHN and MTD have performed the FEM analyses and created the overall codes. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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