## Abstract

The phenomenological approach to the modelling of the mechanical response of arteries usually assumes a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strain-energy function in favour of a reduced set of only three invariants. It is shown that this reduced form is not consistent with the corresponding full linear theory based on infinitesimal strains. It is proposed that compatibility with the linear theory is an essential feature of any nonlinear model of arterial response. Two approaches towards ensuring such compatibility are proposed. The first is that the nonlinear theory reduces to the full six-constant linear theory, without any restrictions being imposed on the constants. An alternative modelling strategy whereby an anisotropic material is compatible with a simpler material in the linear limit is also proposed. In particular, necessary and sufficient conditions are obtained for a nonlinear anisotropic material to be compatible with an isotropic material for infinitesimal deformations. Materials that satisfy these conditions should be useful in the modelling of the crimped collagen fibres in the undeformed configuration.

## 1. Introduction

The phenomenological approach has been widely adopted in biomechanics in order to predict the mechanical response of soft tissue. Excellent introductions to the relevant mathematical theory and some bio-mechanical applications can be found, for example, in Holzapfel [1] and Ogden [2]. Almost universally, the constraint of incompressibility is adopted because of the high water-content of soft tissue and will be adopted here also. The choice of phenomenological model has been motivated by the morphology of the soft tissue under consideration and by experimental data. In particular, the nature of the reinforcement of soft tissue by collagen has proved an important structural component for the motivation of mathematical models. In ligaments, tendons and muscles, for example, the collagen has essentially a single preferred direction, which leads naturally to the choice of a transversely isotropic model; in arteries, the collagen is wound around the cylinder in two helices with opposite pitch, leading to the common assumption of an isotropic matrix reinforced with two families of mechanically equivalent fibres and this is the modelling problem of interest here. If one assumes that the material is hyperelastic, then the mathematical model is fully determined by the specification of the strain-energy function, a scalar function of seven well-defined combinations of a strain tensor and the two directions of the fibres, called invariants. The fundamental modelling problem is the specification of the strain-energy function in terms of the invariants and some material parameters, with the values of the material parameters obtained by fitting the model to experimental data. The model of Holzapfel *et al.* [3] is a particularly good example of this type of modelling and has been extremely influential. Ideally, one would want to choose as simple a model as possible that is consistent with experimental data. However, this choice is complicated by the lack of a benchmark set of data for any part of the cardiovascular system of any species. Despite this lack, many different models have been proposed, based primarily on mathematical convenience and on the need to capture the qualitative features that most commonly appear in the experimental literature, particularly the often noticed severe strain-hardening effect.

A fundamental question in any modelling endeavour is the number of material constants that are necessary to predict material response and one of the main arguments made here is that typically very few material parameters for modelling the mechanical response of arterial tissue are included in the models. It is suggested here that the corresponding linear theory involving infinitesimal deformations can, and should, play an important part in both motivating the form of specific models of nonlinear response and in assessing the validity of any simplifying assumption made outside this motivating framework. A modelling axiom could be that the number of material constants in a nonlinear model should be at least equal to the number of material constants in the corresponding linear theory. The nonlinear theory of a hyperelastic matrix reinforced with two families of mechanically equivalent fibres is of interest here and the corresponding linear theory is a six-constant theory. It seems unrealistic to expect that a material that requires six constants to describe infinitesimal deformations from an undeformed state could be described by a model with fewer constants over a range of nonlinear strain. For ease of exposition, the linearization procedure adopted here will be for two families of straight fibres and the physical constants used in the analysis will be obtainable from biaxial and simple shear experiments on cuboid specimens. This is not as unphysical as it may seem because the majority of material characterization testing of arteries involves precisely these experiments, as can be seen in §6.

It is also proposed here that, in addition to the central requirement of matching experimental data over the physiological strain range, a model of arterial response should also recover the linear theory on restriction to infinitesimal deformations, without restrictions being placed on the values of the corresponding material constants. This is not proposed as a necessary condition for a physically realistic model of arterial response (nor, of course, is it sufficient) given that physiological strains in the artery are never zero but it certainly seems to be a desirable feature. Satisfaction of this condition leads to a possible new modelling approach for arterial tissue whereby the albeit very limited experimental data for arterial material subjected to infinitesimal strains can be incorporated into the model before additional parameters in the model are determined from experimental data obtained under physiological conditions.

Conversely, every model based on physiological strain data makes implicit predictions about the values of the material constants of the linear theory. It seems desirable that these predictions are at least of the same order as the experimentally determined values of these constants. The current method of constructing phenomenological models for the mechanical response of soft tissue is essentially pragmatic and is independent of any considerations based on the linear theory. One pairs a truncated Taylor series in some of the invariants, the choice motivated by mathematical convenience and experience, with functions of invariants that capture the above-mentioned severe strain-hardening effect, as is done, for example, in Holzapfel *et al.* [3]. The behaviour of these models for infinitesimal strains is ignored, with the reasonable expectation that a good fit with experimental data for moderate strains should result in a physically realistic model for the linear regime. However, it was shown in a recent paper by Murphy [4] that this is not necessarily the case for skeletal muscles and it will be shown here that it is not the case for an important model of arterial response proposed in Holzapfel *et al.* [3], the so-called HGO model. This is not a fatal flaw in this model as the unloaded state is not recoverable for arteries, but nonetheless even some qualitative agreement with observed material constants would be preferable, given that the HGO model is based on a physiological strain regime for which strains are only of the order of 10%.

It is shown here that the restrictions that need to be imposed on the strain-energy function of the nonlinear theory to recover the corresponding linear theory on restriction to infinitesimal strains motivate a quadratic function of the two strain invariants and the five pseudo-invariants as a simple model of nonlinear response, called here the basic linear model. It is also shown how the qualitative features that most commonly appear in the experimental literature, particularly the often noticed severe strain-hardening effect, can be easily accommodated using the basic linear model as a modelling basis.

Compatibility with the linear theory seems essential if one wishes a nonlinear anisotropic material to be anisotropic over the full range of strain. An alternative modelling perspective is based on the physics of the loading of the collagen fibrils. In the unloaded state, these fibrils have a sinusoidal character and only become fully engaged in the bearing of the applied load at moderate strains. This view is more fully motivated in, for example, Fung [5]. This perspective motivated Holzapfel *et al.* [3] to assume an additive decomposition of the strain-energy function, and thus the stress–strain constitutive law, into separate isotropic and anisotropic components, with the intention of having the isotropic component of the theory predict the mechanical response in some strain regime around the unloaded state and the anisotropic component functioning only at larger strains. A different approach is adopted here: it is assumed that there should be no anisotropic contribution to the constitutive law for infinitesimal strains. Materials which have this characteristic will be described as anisotropic materials which behave isotropically in the limit and will be termed NALI (‘nonlinear anisotropic, linear isotropic’) materials. Necessary and sufficient conditions for such behaviour will be given here and some simple examples of such materials are given. It is shown that the usual models of anisotropy, including the now classical HGO model, do not have this characteristic and some modifications to incorporate this behaviour are suggested. It is hoped that such materials could prove useful in the modelling of the crimped collagen fibrils in the undeformed configuration. Even though one can argue from a physics perspective that these models have some physical relevance, it must be emphasized that these models are motivated mostly on the basis of mathematical convenience and are not compatible with available experimental data which suggest that arterial tissue behaves as an anisotropic material even for infinitesimal strains. A particularly clear demonstration of this is given in fig. 4 of Zemanek *et al.* [6], where the axial Young's modulus is approximately twice the circumferential in equibiaxial experiments on porcine thoracic aortic specimens. Another excellent demonstration of anisotropy in the linear regime is fig. 7 of Holzapfel *et al.* [7], where the longitudinal and circumferential stress–strain plots are given for each of the three layers of human coronary arteries; a distinction between the infinitesimal moduli for the longitudinal and circumferential modes can clearly be seen. It remains to be seen what impact this simplifying assumption of isotropic behaviour in the limit has on the reliability of predictions of the mechanical response of arteries for physiological strains.

A framework to incorporate the linear theory within the nonlinear modelling of biological soft tissue is therefore provided here: one should recover either the unrestricted anisotropic linear theory or an explicitly chosen simpler type of anisotropy on restriction to infinitesimal deformations. This seems an advance on the current modelling practice wherein the corresponding linear theory is rarely considered.

## 2. Preliminaries

Incompressible, homogeneous, nonlinearly elastic materials with two preferred directions ** M** and

**′ in the undeformed configuration are of interest here. The preferred directions are usually physically induced by the presence of fibres embedded in an elastic matrix and this will be assumed here also. Let**

*M***denote the deformation gradient tensor. Then the incompressibility constraint takes the form 2.1where**

*F**λ*

_{i}are the principal stretches. The tensors

**and**

*B***are the left and right Cauchy–Green strain tensors, respectively, and therefore are positive definite symmetric tensors with strain invariants defined by 2.2For incompressible materials, all deformations are isochoric and therefore det(**

*C***)=det(**

*B***)=1. The so-called pseudo-invariants (e.g. [1]) are defined by 2.3These are positive definite quadratic forms, with the exception of**

*C**I*

_{8}. Note that

*M.M*^{′}is a constant for given preferred directions and it can therefore be set equal to 1, without loss of generality, with the exception of initially orthogonal directions for which

*I*

_{8}=0. In this special case, there is therefore no constitutive dependence on this invariant. The preferred directions

**,**

*M***′ in the undeformed configuration are transformed into the vectors**

*M***,**

*FM***′, respectively, in the deformed configuration. As 2.4**

*FM**I*

_{4}and

*I*

_{6}are the squared stretch of line elements aligned in the original directions of anisotropy.

The constitutive law for the Cauchy stress tensor ** σ** for these materials has the form [2]
2.5where

*p*is an arbitrary scalar field. Here,

*W*=

*W*(

*I*

_{1},

*I*

_{2},

*I*

_{4},

*I*

_{5},

*I*

_{6},

*I*

_{7},

*I*

_{8}) is the strain-energy function per unit undeformed volume and attached subscripts denote partial differentiation with respect to the appropriate principal strain invariant or pseudo-invariant. To ensure that the stress is identically zero in the undeformed configuration, it will be required that 2.6where the 0 superscript denotes evaluation in the reference configuration, for which

*I*

_{1}=

*I*

_{2}=3,

*I*

_{j}=1,

*j*=4,5,6,7,

*I*

_{8}=

*M.M*^{′}and

*p*

^{0}is the value of the pressure term in the reference configuration. It will also be assumed that the strain energy vanishes in the undeformed configuration, i.e. that

*W*

^{0}=0.

A major difficulty in the analysis of anisotropic materials is the complexity of this constitutive law. Consequently, physically realistic constitutive assumptions that can reduce this complexity are highly desirable. The search for such assumptions is hindered by the lack of a simple physical interpretation of the pseudo-invariants *I*_{5}, *I*_{7} and *I*_{8}, in contrast to the identification of *I*_{4}, *I*_{6} as the square of the fibre stretch in the directions along ** FM**,

*FM*^{′}, respectively. This lack and the realization that a major simplification of the constitutive law occurs for strain energies independent of these invariants has resulted in soft tissue reinforced with two families of mechanically equivalent fibres often modelled by strain-energies independent of

*I*

_{5},

*I*

_{7}and

*I*

_{8}. Additionally, to reduce the algebraic complexity even further, dependence on

*I*

_{2}is usually dispensed with and strain energies of the form 2.7are the norm for soft tissue reinforced with two families of fibres. The most important application of such strain energies is the modelling of the mechanical response of large elastic arteries. One particularly influential model is the so-called HGO strain-energy function 2.8where

*c*,

*k*

_{1}>0 are stress-like material parameters and

*k*

_{2}>0 is a dimensionless parameter, proposed by Holzapfel

*et al.*[3]. A generalization of this model that accounts for fibre dispersion was proposed by Gasser

*et al.*[8] with 2.9where

*κ*denotes the dispersion factor. Although some predictive capability of the mechanical response of arteries must inevitably be lost when the reduced forms (2.7) of the strain-energy function are used, the hope must be that the most important features of the mechanical response can still be captured by some specification of the reduced form. It will be shown later here that this is not necessarily the case.

## 3. Linearized constitutive law

The linearization procedure adopted here follows that of Atkin & Fox [9]. Let ** u**=

**−**

*x***denote the displacement of a typical particle from the reference configuration to the current. Let**

*X***≡∂**

*H***/∂**

*u***and**

*X***≡(1/2)(**

*ϵ***+**

*H*

*H*^{T}). Assume now that 3.1where ∥⋅∥ denotes the

*sup*norm. The constraint of perfect incompressibility (2.1) then takes the form 3.2where here, and in what follows, only the zero- and first-order terms in

*ϵ*are considered. As

**=**

*F***+**

*I***, it follows that 3.3and therefore, for infinitesimal deformations, 3.4because incompressibility condition (3.2) holds.**

*H*Infinitesimal deformations of a nonlinearly elastic matrix reinforced with two families of straight, mechanically equivalent fibres will be considered. If the fibres have directions ** M** and

*M*^{′}in the reference configuration, then Cartesian axes are chosen so that the fibres have the following orientations: 3.5with

*e*_{z}normal to the plane of the fibres. There are two important special cases: the case of transverse isotropy, corresponding to both

*Θ*=0

^{°}and

*Θ*=90

^{°}, and orthogonal fibres, with

*Θ*=45

^{°}.

Linearizing the strain measures in the constitutive law (2.5) yields the following linearized stress–strain relation for an incompressible elastic matrix reinforced with two initially straight families of mechanically equivalent fibres:
3.6where
3.7and
3.8In the special case of orthogonal fibres, there is no dependence on the *I*_{8} invariant, and therefore
3.9This is therefore a six-constant theory with material parameters , , , , , *d*_{xy} (for orthogonal fibres there are only four constants, as discussed above). This result was obtained by Spencer [10] using an alternative approach. Destrade *et al.* [11] have given a rigorous treatment of the influence of the incompressibility constraint on the form of the linear theory that is in agreement with the results obtained above. To help the constitutive modelling process, it seems desirable to interpret these parameters in terms of experimental constants determined by material characterization tests. This will be considered next.

## 4. Material characterization tests

There are an infinite number of ways to relate the six material constants of the linearized theory to six physical constants obtained from material characterization tests. In view of the structure of constitutive law (3.6), it seems sensible to include the shear moduli in the suite of physical constants to be used. Define the three shear infinitesimal moduli as follows:
4.1Then it follows from (3.8) and (3.6) that
4.2If the fibres are not initially orthogonal, then *C*^{2}≠*S*^{2} and these relations can be inverted to yield
4.3For the special case of orthogonal fibres, with ,
4.4

Assume now a stress-controlled, in-plane biaxial testing regime, with therefore *σ*_{zz}=*σ*_{xz}=*σ*_{yz}=*σ*_{xy}=0. This type of testing for nonlinear, anisotropic materials is discussed in more detail in, for example, Ogden [2]. It follows from (3.6) that *ϵ*_{xz}=*ϵ*_{yz}=*ϵ*_{xy}=0 and that the in-plane normal stresses are given by
4.5noting that *ϵ*_{zz}=−*ϵ*_{xx}−*ϵ*_{yy} by incompressibility and using (4.2)_{2,3}. For simple tension in the *x*-direction, *σ*_{yy}=0, and therefore
4.6where *ν*_{x} can be viewed as a Poisson's ratio, and then *σ*_{xx}=*E*_{x}*ϵ*_{xx}, where
4.7is Young's modulus in the *x*-direction. Similarly, for simple tension in the *y*-direction, the corresponding Poisson's ratio *ν*_{y} and Young's modulus *E*_{y} are defined as
4.8so that *σ*_{yy}=*E*_{y}*ϵ*_{yy}. The four physical constants *E*_{x}, *ν*_{x}, *E*_{y}, *ν*_{y} are not independent with *E*_{x}*ν*_{y}=*E*_{y}*ν*_{x}.

In what follows, the six physical constants used in the analysis will be the three shear moduli and *E*_{x}, *E*_{y} and *ν*_{x}. Inverting the above relations yields
4.9assuming that .

## 5. Restrictions on the strain-energy function

Both the Young's moduli and the Poisson ratio *ν*_{x} will be used together with the three shear moduli to characterize the mechanical response for infinitesimal strains. The restrictions imposed on the strain-energy function of the nonlinear theory by imposing compatibility with the linear theory obtained in the last section will now be summarized here. Assume first that the material constants *μ*_{xy}, *μ*_{xz}, *μ*_{yz}, *E*_{x}, *ν*_{x} and *E*_{y} have been obtained experimentally and that the fibre orientation is known. Then, recalling initial conditions (2.6) and definitions (3.7), the strain-energy function of an elastic matrix reinforced with the two families of straight, mechanically equivalent fibres must satisfy the following conditions:
5.1Thus, there are 11 conditions to be satisfied, and therefore it seems that an 11-constant strain-energy function is necessary to ensure compatibility with the linear theory. Any strain-energy function with a smaller number of constants will either violate the initial conditions or else impose a relationship between the material constants. In the absence of experimental data indicating that such a relationship exists, the strain-energy function is therefore overprescriptive.

The special cases of isotropy, transverse isotropy and orthogonal fibres can be obtained from the previous analysis. The trivial case of isotropy is recovered from (5.1) by requiring that all partial derivatives with respect to the anisotropic pseudo-invariants are identically zero. This yields 5.2with .

Without loss of generality, only the case of transverse isotropy with *Θ*=0^{°}, corresponding to fibres aligned along the *x*-axis, will be considered. With this fibre angle and setting all derivatives with respect to *I*_{6},*I*_{7},*I*_{8} identically zero, (5.1) reduce to
5.3together with
5.4where the , terms have been rescaled by the factor 2 because there is now only one family of fibres. These last three conditions were previously obtained by Murphy [4] and, in an equivalent form, by Merodio & Ogden [12].

The special case of orthogonal fibres cannot be obtained directly from (5.1) because these conditions were obtained assuming that *C*^{2}≠*S*^{2}. The relationships between the shear moduli and the strain-energy function are given in (4.4). It follows from (3.9) and (3.6) that the normal stresses have the form
5.5Therefore, *E*_{x}=*E*_{y} with
5.6

Excluding these special cases from consideration in what follows, restrictions (5.1) can be immediately employed as an effective constitutive modelling tool for the nonlinear regime. It follows from the third and fourth set of restrictions above that the nonlinear model must include the anisotropic pseudo-invariants *I*_{4}, *I*_{5}, *I*_{6} and *I*_{7}, because if one of these is not included in the constitutive mix it follows from the third and fourth set of restrictions that
5.7This seems to be physically unrealistic, with the exception for orthogonal fibres discussed above, and certainly there is no experimental evidence in the literature to suggest that this assumption is appropriate for any type of soft tissue. The *I*_{8} invariant must also be included in the set of modelling invariants; if it is not, then the last three restrictions require that
5.8This certainly would not be a typical initial constitutive assumption for arterial tissue but it is a necessary condition for the exclusion of *I*_{8} from the corresponding mathematical model and it is highly improbable that these conditions would be even approximately satisfied for any real material. The author is unaware of any elastic material reinforced by two families of mechanically equivalent fibres for which these conditions are satisfied. Indeed, the author is unaware of any such material for which the physical constants, or even a significant subset, have even been measured for a specific material. This need for an *I*_{8} dependence seems at first glance to contradict the conclusions of Merodio & Ogden [13], whose analysis suggested that inclusion of *I*_{8} weakens the stability of the model. However, their analysis was based on strain energies of the form *W*=*W*(*I*_{1},*I*_{8}), which are not compatible with the linear theory.

So an important almost trivial consequence of requiring that the nonlinear theory be compatible with the linear theory is that *the nonlinear strain-energy function must contain all the anisotropic pseudo-invariants* *I*_{4}, *I*_{5}, *I*_{6}, *I*_{7} and *I*_{8}. It also seems that at least a quadratic dependence on some of these invariants is necessary, with a quadratic dependence on *I*_{8} seemingly essential. It is also evident that product terms in the invariants *I*_{4}, *I*_{6} and *I*_{5}, *I*_{7} should be included. Finally, a product term containing *I*_{8} and some of the other pseudo-invariants needs inclusion. These observations and the restrictions themselves will now be used to motivate a simple strain-energy function, and the emphasis will be on simplicity of form, that is fully compatible with the linear theory. Consider, therefore,
5.9where (5.1)_{1,5} have been satisfied identically. It is easy to check that the model parameters *c*_{1},…,*c*_{7} can be written uniquely in terms of the six material constants; seven model parameters are necessary because the initial condition *W*_{4}+2*W*_{5}=0 must also be satisfied. Enforcing this initial condition explicitly yields
5.10making an obvious change in the notation for the last constant. This strain-energy function can be viewed as a prototype for all anisotropic materials that are fully compatible with the linear theory and will be called here the ‘basic linear model’. For applications later, the following identities for this material are useful:
5.11together with and .

If it is desired to combine compatibility with the linear theory for infinitesimal strains with the severe strain hardening seen in tension tests of arterial tissue at moderate strains, then an additive model of the form
5.12seems an obvious model, with given by (5.9) and , a symmetric function of *I*_{4},*I*_{6} to account for the mechanical equivalence of the fibres, satisfying the initial conditions
5.13and with
5.14These conditions ensure that restrictions (5.1) only affect the component of the strain-energy function and that the strain-hardening effect is separately accounted for by the term. An alternative strategy would be to replace, say, the quadratic term in *I*_{4}−1,*I*_{6}−1 in (5.9) with a symmetric function such that
5.15This mimics the contribution of the *c*_{3}([*I*_{4}−1]^{2}+[*I*_{6}−1]^{2}) term to identities (5.1), while modelling the desired strain hardening.

## 6. The linear theory and *I*_{1},*I*_{4},*I*_{6} models

Almost all arterial tissue is modelled by strain energies that depend on a restricted set of invariants and the strain energies of choice have the form given by (2.7), i.e. *W*=*W*(*I*_{1},*I*_{4},*I*_{6}). It was already shown in the last section that these strain-energy functions are *not* compatible with the linear form because with no *I*_{5},*I*_{7} and *I*_{8} dependence, *μ*_{xz}=*μ*_{yz}(≡*μ*_{⊥}) and relations (5.8) must hold. The other restrictions imposed by the linear theory on this form will now be considered. For this class of materials, (5.1) yields
6.1where the last two of (5.1) have been eliminated because (5.8) has been assumed to hold. If, as is commonly further assumed, the *I*_{4}, *I*_{6} terms are separable, then and yet another restriction must be identically satisfied, i.e.
6.2If any of these relations is not satisfied for a given set of physical constants, then *W*=*W*(*I*_{1},*I*_{4},*I*_{6}) is not an appropriate strain-energy function. It seems highly unlikely that all these complex identities could be satisfied simultaneously for a given material, which suggests that it is highly probable that *W*=*W*(*I*_{1},*I*_{4},*I*_{6}) is not a physically realistic choice for infinitesimal strains.

The HGO model (2.8) is of the form (2.7) with separate *I*_{4},*I*_{6} terms, and therefore (5.7), (5.8) and (6.2) must be all identically satisfied before it can be used. *If* these conditions are satisfied, and there is no experimental evidence to suggest that this is a possibility, then (6.1) yield
6.3noting that initial conditions (6.1)_{1,3} are satisfied identically and that there is no identification of the model parameter *k*_{2} with a material constant of the linear theory. Using experimental data obtained by Chuong & Fung [14], Holzapfel *et al.* [3] used the material parameters
6.4to predict the mechanical response of the adventitia of carotid arteries from rabbits. As already noted in the Introduction, *if* (2.7) is a valid model for this material, then all the physical constants can in principle be determined from these model parameters. Working to one decimal place, it follows from (6.3) that the corresponding shear material parameters are predicted to be
6.5The author is unaware of any reported values for the shear moduli for the adventitia of carotid arteries of rabbits. Indeed, it is difficult to locate any reliable measurements of any shear moduli. One exceptional set of measurements was obtained by Lu *et al.* [15], who measured the equivalent of *μ*_{xy} of porcine coronary arteries using a tension–torsion machine. Their data suggest that *μ*_{xy}≈20 kPa, and therefore it seems that the prediction of the HGO model is an order of magnitude too small.

Identities (5.8) and (6.2) further yield
6.6These values furnish a means to test the validity of (2.8), with the prescribed material parameters (6.4), for infinitesimal strains: if the results from biaxial testing of carotid adventitia from rabbits yield physical parameters different from (6.6), or the in-plane shear moduli are found to be distinct in shear wave tests, then the model is not appropriate. As is almost always the case for soft tissue, there is a lack of useful experimental data. There is some indirect evidence, however, to suggest that these calculated values of the material constants are not physically realistic. Patel & Janicki [16] measured the following respective circumferential and longitudinal moduli for the common carotid artery in dogs:
6.7One would expect the Young's moduli, *E*_{x}, *E*_{y}, based on the testing of rectangular specimens, to be very similar to the respective Young's moduli, *E*_{θ},*E*_{z} obtained from cylindrical sections, as was done by Patel & Janicki [16]. A comparison of the derived moduli (6.6) with the experimental moduli (6.7) reveals differences of several orders of magnitude between the respective moduli, mirroring the differences between the derived and experimental values for the shear moduli, and a qualitative difference, in that the experimental circumferential modulus is much larger than the longitudinal, whereas the predicted longitudinal modulus is larger. These large differences between predicted and experimental values of the various material constants are not fatal flaws in the HGO model, as, understandably, Holzapfel *et al.* [3] were concerned with modelling the physiological strain range but the discrepancies seem curious. Ideally, one would prefer some correspondence between models for the physiological strain regime and their linear counterparts.

The generalization of the HGO model that includes fibre dispersion is given by (2.9). This model with the parameters
6.8was proposed by Gasser *et al.* [8] to predict the mechanical response of the iliac adventitia. Restrictions (6.1) yield for general model (2.9)
6.9and so, in particular, for material parameters (6.8),
6.10The remaining constants are determined as before and are given by
6.11There is a lack of reliable data on the mechanical response of iliac arteries and a critical evaluation of the above predicted values for the material constants cannot therefore be done. Nonetheless, it again seems that the predicted values for the Young's moduli, in particular, are much smaller than that one would expect for arterial tissue (cf. (6.7)).

It has been shown therefore that the restrictions imposed by the linear theory on the models of arterial tissue of the form (2.7) are quite prescriptive and, in particular, make unrealistic predictions about the relationships between the different physical constants. The question naturally arises as to whether these strain energies, and in particular the functional forms promoted by Gasser, Holzapfel and Ogden in (2.8) and (2.9), can be generalized to account for physically realistic behaviour for infinitesimal strains. Two strategies are proposed in the last section to achieve this. The first is to simply add the given *I*_{1},*I*_{4} and *I*_{6} strain energy to , given by (5.9), provided the given strain energy satisfies the conditions (5.13). Unfortunately, it is easy to check that functional forms (2.8) and (2.9) are *not* compatible with initial conditions (5.13) because the condition is not satisfied. If, however, an exponent of the invariants greater than two is used so that, for example, instead of (2.8),
6.12then all conditions (5.13) are met. The GOH model (2.9) can be modified in a similar way. The second proposed strategy is, perhaps, a better fit with the original HGO model, (2.8) (this strategy is not appropriate for the GOH model). Setting *k*_{1}=*c*_{3}, the model
6.13simplifies to (2.8) when *c*_{2}=*c*_{4}=*c*_{5}=*c*_{6}=*c*_{7}=0. This model recovers the linear model on restriction to infinitesimal deformations and has the same qualitative strain-hardening characteristics as the HGO model.

## 7. Anisotropic materials that are isotropic in the limit

The linear theory for an elastic medium reinforced with two families of fibres is a six-constant theory. Less complex material models are obviously described by fewer constants. It could be the case that, for a given artery, only a subset of the values of the six material constants is available. One would then like to use this information in a rational and coherent way. Alternatively, one could adopt the constitutive assumption that the two fibre family of materials behaves more simply for infinitesimal strains. The restrictions imposed on the strain-energy function of the elastic matrix reinforced with two families of mechanically equivalent fibres so that it behaves as if it were a simpler material for infinitesimal strains are now obtained. The simplest behaviour is where anisotropic materials are isotropic in the limit of infinitesimal deformations.

The linear theory of incompressible isotropic materials is characterized by the value of the shear modulus, *μ*. If it is desired that an anisotropic material behaves as if it were isotropic for infinitesimal strains, then *μ*_{xy}=*μ*_{xz}=*μ*_{yz}=*μ*, *E*_{x}=*E*_{y}=3*μ*, *ν*_{x}=1/2. Identities (5.1) then yield
7.1

These then are the necessary and sufficient conditions for an incompressible anisotropic material with two families of mechanically equivalent fibres to be isotropic for infinitesimal strains. Materials that satisfy this set of conditions will be termed NALI materials. It is not possible to determine the general form of the strain-energy function that satisfies these conditions; all that is possible is to determine the conditions, if any, that must be imposed on a given strain-energy function so that it is compatible with (7.1). For example, the only form of the basic linear material (5.10) that is compatible with this set of restrictions is the isotropic, neo-Hookean material, *W*=*μ*/2(*I*_{1}−3). This assumption of isotropy in the limit has the appeal of significantly reducing the number of constants needed to model small strains. It also has the appeal of being a simple method to incorporate the observed crimping of the collagen fibres in arteries on unloading, without recourse to an arbitrary stretch at which the collagen becomes engaged in load bearing. In addition, in contrast to the material symmetries discussed in the last section, there is no explicit dependence on fibre angle in this set of conditions. For all these reasons, it is anticipated that this reduction in anisotropy will be the model of choice, if some explicit, rational accounting for the unloaded state is desired in a model of arterial response. If this approach is adopted, then one could view anisotropy as evolving out of isotropy in the reference configuration with increasing strain.

Some indications as to the forms of the strain energies that reduce to isotropy in the limit can be gleaned from these conditions. It seems that strain energies independent of *I*_{8} are especially appropriate as the last two of (7.1) are then identically satisfied. Popular models for soft tissue rarely contain either of *I*_{5},*I*_{7}, mainly because the corresponding terms in the constitutive law (2.5) are complex. If one therefore assumes a model of the form (2.7), then these conditions simplify to
7.2These are the conditions that are most likely to be applied when testing whether a given anisotropic model for soft tissue is isotropic for infinitesimal deformations. If polynomial models of the general form (2.7) are required, then it is immediate that quadratic models compatible with (7.2) are not possible, and therefore the standard reinforcing model is not isotropic in the limit. The simplest polynomial must be cubic in the pseudo-invariants and has the form
7.3with more general power-law models of the form
7.4also possible.

The HGO model (2.8) also does not satisfy (7.2). Some modifications of the form of this model that behave as isotropic materials in the limit are now sought because of its popularity. It is easy to check, for example, that models of the form
7.5do satisfy these conditions, and therefore behave isotropically for infinitesimal strains. If one wishes to simply add a component to the strain energy (2.8) to ensure infinitesimal strain isotropic behaviour, in keeping with the additive form of the HGO strain energy itself, then an obvious candidate is given by
7.6The exponential term is presumably a function of a quadratic term in *I*_{4}−1 in the original HGO model to ensure zero stress in the undeformed configuration. Adopting the approach of an additive component to ensure isotropic behaviour in the limit, then this quadratic dependence can be modified. If a simple linear dependence is adopted instead, then a modified HGO model takes the form
7.7

The recovery of more complex behaviour can also be accommodated. Two obvious, particular cases are considered below for completeness.

### (a) Orthogonal fibres

Suppose it is desired to model a material with two families of fibres that are not orthogonal as a material with orthogonal fibres in the linear regime, where now knowledge of only four material constants is necessary as opposed to the original six. For materials with orthogonal fibres, 7.8The first of these restrictions yields, using (4.2), 7.9However, as the fibres have been assumed mechanically equivalent, the last condition becomes 7.10which in turn yield 7.11as a result of imposing the initial conditions (2.6).

If the second of (7.8) holds, then it follows from the relation *E*_{x}*ν*_{y}=*E*_{y}*ν*_{x} that *ν*_{x}=*ν*_{y}, and therefore from (4.8),
7.12where , are defined in (3.7). Restrictions (7.9)_{2} and (7.12) are the two conditions that ensure that the six-constant linear theory of circular orthotropy is reduced to the four-constant linear theory for orthogonal fibres.

### (b) Transverse isotropy

Assume that one now wants transversely isotropic behaviour for infinitesimal strains, with the axis of symmetry along the *x*-axis. Then one condition that needs to be satisfied is that *μ*_{xy}=*μ*_{xz}(≡*μ*_{L}), which yields
7.13The other two conditions that must be enforced are that
7.14The first of these is equivalent to
7.15The terms in the second of these identities can be written as
7.16The three equations (7.13), (7.14)_{2} and (7.15) reduce the number of constants from the six of circular orthotropy theory to the three associated with the linear theory of transversely isotropic materials.

## 8. Summary

An explicit, rational accounting for the existence of the reference configuration is proposed here for anisotropic materials reinforced with two families of fibres. A choice is proposed: either anisotropic models of nonlinear elasticity should recover the corresponding linear theory on restriction to infinitesimal deformations, in which case models of incompressible nonlinearly elastic matrices reinforced by two families of mechanically equivalent fibres must contain at least six constants, or the anisotropic material should lose some anisotropy in the limit of infinitesimal strains, the simplest model being the anisotropic material that behaves isotropically in this limit. It is suggested that this model could prove useful in the modelling of the crimping of collagen fibres in the unloaded state of arteries.

## Acknowledgements

The author thanks Prof. Ray Ogden for suggesting this problem and Prof. Michel Destrade, as always, for stimulating discussions on this and other topics. The author is particularly grateful to Prof. Alain Goriely for his insightful comments and his encouragement. The constructive comments and suggestions of the anonymous referees and Board Member have been incorporated into this final, much improved version of the original manuscript.

- Received August 12, 2013.
- Accepted October 21, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.