Open Access

Research articles

Switchless constitutive relation for arterial tissues: eliminating all discontinuities in mechanical response

Centre for Soft and Biological Matter, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India

Google Scholar

Find this author on PubMed

Search for more papers by this author

and

K. Kannan

http://orcid.org/0000-0002-1840-1086

Centre for Soft and Biological Matter, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India

[email protected]

Contribution: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Supervision, Writing - review & editing

Google Scholar

Find this author on PubMed

Search for more papers by this author

K. Arvind

http://orcid.org/0000-00002-7230-9141

Centre for Soft and Biological Matter, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India

Google Scholar

Find this author on PubMed

Search for more papers by this author

and

K. Kannan

http://orcid.org/0000-0002-1840-1086

Centre for Soft and Biological Matter, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India

[email protected]

Contribution: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Supervision, Writing - review & editing

Google Scholar

Find this author on PubMed

Search for more papers by this author

Published:16 April 2025https://doi.org/10.1098/rspa.2024.0634

Review history
- Review history is available via Web of Science at https://www.webofscience.com/api/gateway/wos/peer-review/10.1098/rspa.2024.0634

Abstract

The switching criterion in the constitutive relations for arterial tissues could introduce stress discontinuities and cause fibres to exhibit dual behaviour—simultaneously experiencing tension and compression—depending on the switching criteria. Furthermore, the resulting conditional constitutive relations do not distinguish between longitudinal and transverse shear in unidirectional composites, contrary to expectations. Consequently, the azimuthal and telescopic shear behaviours of a cylindrical annulus resemble that of their isotropic counterpart. To work though these concerns, we propose two classes of vanishing-matched-generalized invariants, based on the Green–Lagrange and Seth–Hill strains. For unidirectional composites, this invariant effectively nullifies fibre contributions in pure compression, thereby eliminating the need for switching criteria. Furthermore, for rotationally symmetric fibre distributions, the proposed relations produce the correct mechanical response for the entire range of dispersion, i.e. $0 \leq κ \leq 1 / 2$ . The resulting structural constitutive relation for the stress linearizes to the isotropic Hooke’s Law when considering all deformations simultaneously—a desirable feature of any constitutive relation. Compared to the existing models, both proposed classes—(i) the simple vanishing-matched invariants-based relations and (ii) polynomial forms—demonstrate significantly improved descriptive capability across 13 sets of planar biaxial data from various healthy and diseased human arterial tissues.

1. Introduction

An artery is a tri-layered, anisotropic, residually stressed and inhomogeneous material. The distribution of the latter three characteristics is patient-specific. For a detailed morphological and histological description of arteries, readers are referred to the review article by Gasser [1], which also includes a list of several constitutive relations used to describe their mechanical behaviour. Detailed aspects regarding the constitutive modelling of arteries can be found in the review articles by Holzapfel & Ogden [2] and Holzapfel et al. [3].

The wall of an artery is supplied with nutrients through the vaso vasoram, and its mechanical behaviour is essentially viscoelastic. This complex behaviour is further exemplified when an artery changes its diameter and remodels its structural elements due to sustained changes in blood flow within its lumen. Further complexities of modelling the deformation of arteries are discussed in [4]. This paper provides a contribution to approximating their mechanical behaviour as an anisotropic, nonlinearly elastic and incompressible material.

The switching criterion used in the existing models for arterial tissue mechanics to ‘turn off’ the contribution of compressed collagen fibres may lead to a stress discontinuity at the switch point (see §3c(i) and §3a of this article and also refer to [5,6]). A modified version of the switching criterion is implemented in the finite-element software ABAQUS [7] to eliminate the discontinuity in the stress. However, Latorre [6] noted that when the fibres along the mean direction are in compression, certain collagen fibres, away from the mean direction, may be in extension and hence contribute to the potential, which is not accounted for in the version implemented in ABAQUS. To address this, Latorre modified the Gasser–Ogden–Holzapfel (GOH) model [8] using suitable switching criteria to resolve the stress discontinuity and fibre contribution in compression. Nonetheless, if the material is assumed to be hyperelastic, the complete constitutive relation must be derived from a unified potential that encapsulates all the deformations, i.e. one should refrain from using switching criteria.

Latorre [6], and later Horgan [9], identified the ‘dual nature’ of the fibre stretch: the fibres predicted to be in compression according to the relations that describe anisotropy can be in tension after the switch is engaged. This prediction suggests that the criterion for switching and the constitutive relations may not be fully compatible.

Murphy [10] demonstrated that the absence of the $I_{5}$ and $I_{7}$ invariants in an anisotropic constitutive relation leads to the same predictions as its isotropic counterpart during axial and azimuthal shear of a cylindrical annulus, an undesirable outcome. He also drew attention to the other concerns regarding the number of parameters that a two-fibre family material should have upon linearization and the lack of three distinct shear responses during large deformation of unidirectional composites, i.e. transverse, longitudinal and perpendicular shear [11]. Furthermore, Destrade et al. [12] observed that strain energy functions of the form $W = W (I_{1}, I_{4})$ are inadequate for accurately describing off-axis simple tension. Feng et al. [13] proposed an anisotropic invariant combining $I_{4}$ and $I_{5}$ , that accounts solely for the fibre shear (see also [14]).

In the existing constitutive relations incorporating a switching criterion, the anisotropic contribution of collagen fibres to the stress is switched off whenever the mean direction of distributed collagen is subjected to compression [3,8]. As a result, the constitutive relations linearize to the generalized Hooke’s Law with isotropic material symmetry when the fibres are in compression, and with anisotropic material symmetry when the fibres are in tension. In other words, they do not linearize to the generalized Hooke’s Law with a fixed material symmetry when all the deformations are considered together, a desirable requirement for any constitutive relation. Mainly, since the material parameters of Hooke’s Law must be constants, the parameters related to the anisotropy of the linearized elastic relation cannot switch off in compression. Consequently, the relations must be modified so that the anisotropic contributions vanish faster than their isotropic counterpart within the ‘small’ deformation limit. A simple solution is to change the exponent on the generalized invariant from 2 to 3. It appears that Murphy [10,11] did not recognize this challenge on the linearization of constitutive relations as he focused on the number of parameters a linearized anisotropic material should have.

In contrast to the generalized structure tensor (GST)-based approach, where the constitutive relation depends on the averaged structure tensor obtained by integration over the unit sphere, the angular integration (AI) approach is based on the integration of constitutive relation for individual collagen fibres weighted by the orientation density [3,15–18] (the last two references [17,18] discuss AI-based approaches that exclude fibres under compression). In addition, some constitutive relations in the literature incorporate the GST and perform AI for certain other quantities [19,20]. For perfectly aligned fibres, the existing AI and GST approaches yield the same constitutive relation for the fibres that contribute to tension and vanish identically in compression (refer to [5,21]). Therefore, the challenges related to linearization also persist in the existing AI approaches. Furthermore, contrary to experimental data [22,23], the existing AI approaches do not distinguish between the transverse and longitudinal shears of unidirectional soft matter because the fibre contributions vanish in both cases; the observed difference is due to the varying amounts of reinforcement of fibres (filler effect) in the two shear modes.

Here, we develop a unitary constitutive relation encapsulating all deformations that incorporates the GST without employing a switching criterion. The total contribution of the fibres due to compression along the mean direction is automatically discounted to a varying degree depending on the amount of dispersion, by introducing a matched-generalized invariant and having the unified potential depend on it. For perfectly aligned fibres under uniaxial compression along their direction, this invariant vanishes. In this manner, the switching criteria and the dual nature of the fibre stretch are eliminated effectively. An added benefit of the use of the matched-generalized invariant is the natural emergence of the invariant $I_{5}$ , which results in the stress tensor that predicts distinct transverse, longitudinal and perpendicular shear responses of unidirectional composites. Moreover, the exponent 3 on the matched-generalized invariant ensures that the constitutive relation linearizes to Hooke’s Law with isotropic material symmetry for all deformations. While the proposed theory introduces some mathematical complexity compared to existing popular GST-based approaches, it effectively addresses several existing challenges, as demonstrated in the following sections.

2. Switchless structure tensor-based relation for the stress in polynomial form

(a) Vanishing-matched-invariant for unidirectional composites

Consider the following matched terms:

\begin{aligned} A \cdot (N \otimes N) + \sqrt{A^{2} \cdot (N \otimes N}) & = | | A N | | | | N | | \cos θ + | | A N | | \\ = | | A N | | (1 + \cos θ) \geq 0, \end{aligned}

2.1

where A is a second-order tensor, N is a unit vector along the perfectly aligned fibres of a transversely isotropic material,

| | \cdot | |

represents the Euclidean norm of a first-order tensor and

θ

denotes the angle between AN and N. It is widely accepted that the collagen in arteries does not support load during compression, which can be emulated by guiding the above expression to the null value. Note that equation (2.1) will not vanish if one uses the right Cauchy–Green stretch tensor C in place of A because the angle

θ

will always be acute. However, one can use any strain measure. In that case,

θ

180^{\circ}

for any extent of uniaxial compression along N, and the right-hand side of equation (2.1) will be identically zero. An obvious candidate is the Green–Lagrange strain tensor. We propose the use of Seth–Hill class of strain measures

E_{m}

, i.e.

E_{m} = {\begin{cases} \frac{1}{2 m} (U^{2 m} - I), & m > 0, \\ \log (U), & m = 0, \end{cases}

2.2

where $U^{2} = C$ and U is the right stretch tensor. The well-known strain tensors Hencky, Biot and Green–Lagrange strains are obtained by setting $m = 0, 1 / 2$ and 1, respectively. The matched-invariant based on the Seth–Hill strain measure for the first family of fibres is defined as follows:

E_{m, 1} = E_{m} \cdot (N \otimes N) + \sqrt{E_{m}^{2} \cdot (N \otimes N)} .

2.3

To elucidate the characteristics of the vanishing-matched-invariant (or matched-invariant for brevity), let us consider a uniaxial deformation along the fibres. From equations (2.2) and (2.3),

E_{m, 1} = \frac{1}{2 m} (I_{4} (C^{m}) - 1 + \sqrt{I_{5} (C^{m}) - 2 I_{4} (C^{m}) + 1}),

2.4

where

I_{4} (C^{m}) = C^{m} \cdot (N \otimes N)

I_{5} (C^{m}) = C^{2 m} \cdot (N \otimes N)

are the anisotropic invariants associated with the tensor

C^{m}

. Figure 1 portrays the effect of the parameter

m

on the invariant

E_{m, 1}

. The invariant is exactly zero when the fibres are under pure compression, and the value of the invariant increases monotonically at different rates depending on the value of

m

when the fibres are extended. Although the matched-invariant is continuous, it is not differentiable at

λ = 1

. To ensure differentiability, the theory with exponent

n > 1

on the matched-invariant in equation (2.11) must be considered. For tensile loading along the direction of fibres, the generalized invariant in the GOH relation (

E_{GOH} = I_{4} - 1

) and the matched-invariant

E_{1, 1}

(i.e.

m = 1

) assume the same value. However, it is well known that the former invariant cannot vanish in compression, which necessitates a switching criterion.

Figure 1. The plot emphasizes the self-annihilation of matched-invariant ( $E_{m, 1}$ ) during uniaxial compression along the direction of perfectly aligned fibres.

The constitutive relations containing only the $I_{4}$ invariant cannot describe distinct longitudinal and transverse shear responses [10,24]. On the other hand, the matched-invariant ( $E_{m, 1}$ ) containing $I_{5}$ results in non-zero longitudinal shear, i.e. it is able to distinguish between the various modes of shear for a transversely isotropic material. One can attribute such a response to the reinforcing effect of the fibres (filler effect) [22,23].

According to equation (2.3), it should be noted that the contribution of the invariant is zero only if N coincides with one of the principal direction of E and the eigenvalue along N is negative or zero. Equivalently, the value of the invariant is zero in the following cases:

—	pure compressive loading along the fibre direction (figure 1).
—	planar loading in the plane transverse to the fibre that leaves the fibre unextended. An example of this is simple shear transverse to the fibre direction.

For other deformations, the matched-invariant discounts the contribution from that fibre to a varying degree, i.e. depending on how close the state of loading is to the two cases mentioned above. Compared to the existing approaches in the literature, a key difference here is that the shear on the fibre contributes to the matched-invariant even if the fibres are moderately compressed (refer to section S1 of the electronic supplementary material for more details). In other words, under the current approach, obliquely oriented fibres that are compressed will contribute to the mechanical response.

(b) Generalization for dispersed fibres

To describe dispersed fibres, Gasser et al. [8] introduced a GST defined as

H = \int_{\partial Ω} ρ (N) N \otimes N d Ω,

2.5

where

\partial Ω

denotes the surface of the unit sphere, N is a unit vector along the direction of any fibre and

ρ (N)

is a normalized fibre-orientation distribution function. For rotationally symmetric fibre dispersions around a mean fibre direction M, equation (2.5) can be simplified to [8]

H_{rot} = κ I + (1 - 3 κ) M \otimes M,

2.6

where

κ

is the structural parameter that quantifies the amount of dispersion. Recent data on human arterial tissue have shown that collagen fibre dispersion is more pronounced in the circumferential–longitudinal plane than in the radial direction. Accordingly, a suitable GST for non-rotationally symmetric fibre dispersions was introduced by Holzapfel et al. [25] as follows:

H_{i} = A I + B M_{i} \otimes M_{i} + (1 - 3 A - B) M_{n} \otimes M_{n}, i = 1, 2,

2.7

where

M_{i}

is the mean direction of the

i

th fibre family in the circumferential–longitudinal plane and

M_{n}

is the out-of-plane unit vector in the radial direction. The quantities

A

and

B

are related to the in- and out-of-plane dispersion parameters

0 \leq κ_{i} \leq 1

and

0 \leq κ_{o} \leq 1 / 2

, respectively, through

A = 2 κ_{i} κ_{o}

and

B = 2 κ_{o} (1 - 2 κ_{i})

. The values

κ_{i} = 0

and

κ_{o} = 1 / 2

represent the perfect alignment of the fibres along the direction

M_{i}

. A planar-isotropic distribution corresponds to the values

κ_{i} = 1 / 2

and

κ_{o} = 1 / 2

. The values

κ_{i} = 1 / 2

and

κ_{o} = 1 / 3

correspond to the absence of preferred directions and the fibres are isotropically distributed.

For dispersed fibres, the averaged matched-invariant for the $i$ th family $E_{m, i}$ (or matched-generalized) is expressed as

E_{m, i} = \int_{\partial Ω} ρ_{i} (N) (E_{m} \cdot (N \otimes N) + \sqrt{E_{m}^{2} \cdot (N \otimes N)}) d Ω .

2.8

Since the exact integral is unknown, we look for an approximation. Consider a series expansion of the matched-invariant in equation (2.8) with respect to

N \otimes N

about

H_{i}

. The first term of the series is the evaluation of the matched-invariant at the mean, i.e.

H_{i}

. Using equation (2.5) along with the fact that

ρ (N)

is a normalized density function, equation (2.8) becomes

\begin{aligned} E_{m, i} & = E_{m} \cdot H_{i} + \sqrt{E_{m}^{2} \cdot H_{i}} - \frac{1}{8} {(E_{m}^{2} \cdot H_{i})}^{- 3 / 2} (E_{m}^{2} \otimes E_{m}^{2}) \cdot (H_{i} - H_{i} \otimes H_{i}) + \dots, \end{aligned}

2.9

where

H_{i} = \int_{\partial Ω} ρ_{i} (N) N \otimes N \otimes N \otimes N d Ω

is a fourth-order GST [21,26]. The first approximation

{\overset{˘}{E}}_{m, i}

for the

i

th fibre family is hence

\begin{aligned} {\overset{˘}{E}}_{m, i} & = E_{m} \cdot H_{i} + \sqrt{E_{m}^{2} \cdot H_{i}} \\ = A tr (E_{m}) + B E_{m} \cdot (M_{i} \otimes M_{i}) + C E_{m} \cdot (M_{n} \otimes M_{n}) \\ + {(A tr (E_{m}^{2}) + B E_{m}^{2} \cdot (M_{i} \otimes M_{i}) + C E_{m}^{2} \cdot (M_{n} \otimes M_{n}))}^{1 / 2}, \end{aligned}

2.10

where

C = 1 - 3 A - B

. The reader is referred to section S2 of the electronic supplementary material for proof that the matched-generalized invariant is always non-negative. For dispersed fibres, this invariant is strictly positive for non-zero strains. This aligns with the fact that some fibres will always be stretched, contributing to the matched invariant. The invariant will be zero only when the fibres are perfectly aligned, which is consistent with the findings of the previous section. In a similar manner, one can show that if

m = 1

, the matched-generalized invariant is an upper bound of the GOH generalized-invariant, i.e.

{\overset{˘}{E}}_{1, i} \geq E_{GOH, i}

, the equality holding for pure tension along the direction of a unidirectional composite. Note that since the matched-generalized invariant

{\overset{˘}{E}}_{m, i}

is always positive for dispersed fibres, the exponent on

{\overset{˘}{E}}_{m, i}

can be any real number.

Table 1 lists the expressions obtained from the matched-generalized invariant (equation (2.10)) for the Seth–Hill parameters $m = 0, 1$ and 2, for each of the unidirectional, planar-isotropic and isotropic distributions of the fibres, respectively. The basic isotropic invariants of C are $I_{1} = tr (C)$ , $I_{2} = tr (C^{2})$ and $I_{3} = tr (C^{3})$ , and they form an integrity basis for the representation of the strain energy of an isotropic material [27]. The invariants $I_{4} = C \cdot N \otimes N$ and $I_{5} = C^{2} \cdot N \otimes N$ are the contributions describing the perfectly aligned fibres. The various expressions involving Hencky strain-based invariants, i.e. $K_{1} = tr (\log U)$ , $K_{2} = \sqrt{tr {(\log U)}^{2}}$ , $K_{4} = \log U \cdot N \otimes N$ and $K_{5} = {(\log U)}^{2} \cdot N \otimes N$ can be recovered from the matched-invariant in the limit of $m$ approaching zero for different choices of material symmetry. For the planar-isotropic distribution, the fibres are assumed to be the in $X Y$ -plane. The matched-invariant for this case can be expressed in terms of the invariants $I_{1}, I_{2}$ and $I_{3}$ using equation (2.50); the invariants are to be ordered in the same manner as that of the principal stretches, and the respective expressions are chosen for $λ_{1}$ and $λ_{2}$ .

Table 1. Matched-generalized invariant subsumes the listed invariants. $^{b}$
Collapse
$m$	transverse isotropy	planar isotropy $^{a}$
0	$K_{4} + \sqrt{K_{5}}$	$(\log λ_{1} + \log λ_{2}) / 2 + {\sqrt{({(\log λ_{1})}^{2} + {(\log λ_{2})}^{2}) / 2}}^{}$
1	$\frac{1}{2} (I_{4} - 1 + \sqrt{I_{5} - 2 I_{4} + 1})$	$(λ_{1}^{2} + λ_{2}^{2} - 2) / 4 + \sqrt{({(λ_{1}^{2} - 1)}^{2} + {(λ_{2}^{2} - 1)}^{2}) / 8}$
2	$\frac{1}{4} (I_{5} - 1 + \sqrt{α_{0} + α_{1} I_{4} + α_{2} I_{5} + 3})$ , $α_{0} = \frac{1}{6} I_{1} (I_{1}^{3} - 3 I_{1} I_{2} + 2 I_{3}) = I_{1} det C$ , $α_{1} = \frac{1}{3} (I_{3} - I_{1}^{3})$ , $α_{2} = \frac{1}{2} (I_{1}^{2} + I_{2}) - 2$	$(λ_{1}^{4} + λ_{2}^{4} - 2) / 8 + \sqrt{({(λ_{1}^{4} - 1)}^{2} + {(λ_{2}^{4} - 1)}^{2}) / 32}$
$m$	isotropy
0	$K_{1} / 3 + K_{2} / {\sqrt{3}}^{}$
1	$\frac{1}{2} ((I_{1} - 3) / 3 + \sqrt{(I_{2} - 2 I_{1} + 3) / 3})$
2	$\frac{1}{4} ((I_{2} - 3) / 3 + \sqrt{(3 α_{0} + α_{1} I_{1} + α_{2} I_{2} + 3) / 3})$

$^{a}$ To express the principal stretches in terms of $I_{1}$ , $I_{2}$ and $I_{3}$ , one has to first arrange the principal stretches $λ_{1}$ , $λ_{2}$ and $λ_{3}$ in decreasing order and then substitute the solution of equation (2.50) in the same order.

$^{b}$ If the material is incompressible, $K_{1} = 0$ and $I_{3} = (1 / 2) (6 - I_{1}^{3} + 3 I_{1} I_{2})$ .

(c) Green–Lagrange strain-based switchless structural constitutive relation (vanGOH)

The collagen fibres are sheathed helically in the arterial wall and resist deformation during mechanical loading. Specifically, two families of collagen fibres with a positive–negative helix angle in the adventitia and the third family in the circumferential direction for the medial layer have been observed experimentally [28].

The proposed potential has the same general mathematical structure as the GOH model:

W = \frac{c}{2} (I_{1} - 3) + \sum_{i = 1}^{2} \frac{k_{1, i}}{3 k_{2, i}} (\exp (k_{2, i} {\overset{˘}{E}}_{1, i}^{3}) - 1),

2.11

where

c

is the shear modulus of the ground matrix,

k_{1, i}

and

k_{2, i}

are constants. There are two changes compared to the GOH model: (i) the generalized invariant is replaced with the vanishing-matched-generalized invariant

{\overset{˘}{E}}_{1, i}, i = 1, 2

defined in equation (2.10), and (ii) the exponent is changed from 2 to 3 for consilience with linearized elasticity. The two families in equation (2.11) represent the collagen fibres in the adventitia. The two families of adventitial collagen fibres are assumed to have the same strengths, i.e.

k_{1, 1} = k_{1, 2}

and

k_{2, 1} = k_{2, 2}

The matched-generalized invariant ${\overset{˘}{E}}_{1, i}$ for the vanishing-matched-invariant-based GOH (vanGOH) constitutive relation is based on the Green–Lagrange strain tensor $E_{1} = (C - I) / 2$ . Substituting $m = 1$ in equation (2.10) for dispersed fibres, one arrives at

\begin{aligned} {\overset{˘}{E}}_{1, i} & = E_{1} \cdot H_{i} + \sqrt{E_{1}^{2} \cdot H_{i}} \\ = \frac{1}{2} (A I_{1} + B I_{4, i} + (1 - 3 A - B) I_{4}^{⊥}) \\ + \underset{additional term}{\underset{⏟}{\frac{1}{2} {(A (I_{2} - 2 I_{1} + 3) + B (I_{5, i} - 2 I_{4, i} + 1) + (1 - 3 A - B) (I_{5}^{⊥} - 2 I_{4}^{⊥} + 1))}^{1 / 2}}}, \end{aligned}

2.12

where

I_{4, i} = C \cdot (M_{i} \otimes M_{i})

I_{5, i} = C^{2} \cdot (M_{i} \otimes M_{i})

I_{4}^{⊥} = C \cdot (M_{n} \otimes M_{n})

and

I_{5}^{⊥} = C^{2} \cdot (M_{n} \otimes M_{n})

. The resulting potential has two pairs of anisotropic invariants, namely,

I_{4}

and

I_{5}

, and

I_{4}^{⊥}

and

I_{5}^{⊥}

, thereby yielding three distinct shear responses [10]. The term

(I_{5, i} - 2 I_{4, i} + 1)

in equation (2.12) also appears in Murphy’s analysis of linear theory [10] and is linked to a coefficient that quantifies the distinct moduli in the longitudinal and transverse shear of unidirectional composites.

The corresponding Cauchy stress tensor is expressed as

T = - p I + c B + 2 F (\sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{1, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{1, i}^{3}) \frac{\partial {\overset{˘}{E}}_{1, i}}{\partial C}) F^{T},

2.13

where

F

is the deformation gradient and

B = F F^{T}

. From equation (2.12), we have

\frac{\partial {\overset{˘}{E}}_{1, i}}{\partial C} = \frac{1}{2} (H_{i} + \frac{1}{2 \sqrt{E_{1}^{2} \cdot H_{i}}} (E_{1} H_{i} + H_{i} E_{1})) = {\overset{˘}{H}}_{1, i}^{x^{*}} .

2.14

With the introduction of a unit strain tensor

{\hat{E}}_{1} = E_{1} / | | E_{1} | |

, equation (2.14) can be re-written as

{\overset{˘}{H}}_{1, i}^{x^{*}} = \frac{1}{2} (H_{i} + \frac{1}{2 \sqrt{{\hat{E}}_{1}^{2} \cdot H_{i}}} ({\hat{E}}_{1} H_{i} + H_{i} {\hat{E}}_{1})) .

2.15

With the above definition of

{\overset{˘}{H}}_{1, i}^{x^{*}}

, the matched-generalized invariant defined in equation (2.12) can be expressed as

{\overset{˘}{E}}_{1, i} = (C - I) \cdot {\overset{˘}{H}}_{1, i}^{x^{*}}

. Recall that for the GOH constitutive relation, the generalized invariant

E_{GOH, i} = (C - I) \cdot H_{i}

. Thus, here one needs to replace

H_{i}

with the modified tensor

{\overset{˘}{H}}_{1, i}^{x^{*}}

Substituting equation (2.14) in equation (2.13), the resulting Cauchy stress tensor for two families of fibres (V2F) is derived to be

T = - p I + c B + 2 \sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{i, 1}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{i, 1}^{3}) {\overset{˘}{h}}_{1, i}^{x^{*}},

2.16

where

{\overset{˘}{h}}_{1, i}^{x^{*}} = F {\overset{˘}{H}}_{1, i}^{x^{*}} F^{T}

and

i = 1, 2

represents the family number.

To account for the additional contribution of the circumferentially oriented collagen fibres in the medial layer of an artery [29–31], we introduce a three-fibre-family potential:

W = \frac{c}{2} (I_{1} - 3) + \sum_{i = 1, 2} \frac{k_{1, i}}{3 k_{2, i}} (\exp (k_{2, i} {\overset{˘}{E}}_{1, i}^{3}) - 1) + \frac{k_{1, 3}}{3 k_{2, 3}} (\exp (k_{2, 3} {\overset{˘}{E}}_{1, 3}^{3}) - 1),

2.17

where

{\overset{˘}{E}}_{1, 3} = E_{1} \cdot H_{3} + \sqrt{E_{1}^{2} \cdot H_{3}}

is the matched-generalized invariant corresponding to the circumferentially oriented collagen fibres in the medial layer. Since the amount and the type of medial collagen is unique (contains a greater proportion of type-III collagen compared with the adventitial layer) [32], the parameters associated with it are distinct. Furthermore, the collagen fibres in the medial layer are less stiff than the ones in the adventitial layer [33]. The corresponding Cauchy stress tensor (V3F) is

\begin{aligned} T & = - p I + c B + 2 \sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{1, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{1, i}^{3}) {\overset{˘}{h}}_{1, i}^{x^{*}} + 2 k_{1, 3} {\overset{˘}{E}}_{1, 3}^{2} \exp (k_{2, 3} {\overset{˘}{E}}_{1, 3}^{3}) {\overset{˘}{h}}_{1, 3}^{x^{*}}, \end{aligned}

2.18

where

{\overset{˘}{h}}_{1, 3}^{x^{*}} = F {\overset{˘}{H}}_{1, 3}^{x^{*}} F^{T}

. The families of fibres are either assumed to be unidirectional (

κ_{i} = 0

κ_{o} = 1 / 2

) or dispersed whenever the in- and out-of-plane dispersion parameters are known from microstructural imaging. Hence, the constitutive relations V2F and V3F have four and six parameters, respectively. For small strains

ε

, on using the inequalities

ε \cdot H_{i} \leq | | ε | | | | H_{i} | |

and

| | ε^{2} | | \leq | | ε | |^{2}

, equation (2.16) simplifies to the following:

\begin{aligned} T + p I & \approx 2 c ε + 2 | | ε | |^{2} \sum_{i = 1}^{2} k_{1} {(| | H_{i} | | + \sqrt{| | H_{i} | |})}^{2} \times \frac{1}{2} (H_{i} + \frac{1}{2 \sqrt{{\hat{ε}}^{2} \cdot H_{i}}} (H_{i} \hat{ε} + \hat{ε} H_{i})), \end{aligned}

2.19

where

\hat{ε} = ε / | | ε | |

is the unit strain tensor. Hence, the linearized stress response is that of an isotropic incompressible material for all deformations.

While the GOH model features two response functions $\partial W / \partial I_{1}$ and $\partial W / \partial I_{4}$ , the vanGOH model includes four: $\partial W / \partial I_{1}$ , $\partial W / \partial I_{2}$ , $\partial W / \partial I_{4}$ and $\partial W / \partial I_{5}$ . However, these functions are interrelated: one can derive the following relation for the anisotropic part of the strain energy $W^{f, i}$ (see table 4 and refer to section S3 of the electronic supplementary material for more details):

\frac{W_{1}^{f, i}}{\partial {\overset{˘}{E}}_{1, i} / \partial I_{1}} = \frac{W_{2}^{f, i}}{\partial {\overset{˘}{E}}_{1, i} / \partial I_{2}} = \frac{W_{4}^{f, i}}{\partial {\overset{˘}{E}}_{1, i} / \partial I_{4, i}} = \frac{W_{5}^{f, i}}{\partial {\overset{˘}{E}}_{1, i} / \partial I_{5, i}} = W_{f, i}^{'} ({\overset{˘}{E}}_{1, i}),

2.20

where

W_{j}^{f, i}

denotes the partial derivative of

W^{f, i}

with respect to

I_{j}

for

j = 1, 2, 4, 5

for the

i

th family. Hence, there is only one independent response function for the vanGOH model (i.e.

W_{f, i}^{'} ({\overset{˘}{E}}_{1, i})

)—the same as that of GOH. Therefore, concerns of requiring additional datasets [34] and the lack of a global minimum [35] are not applicable here. Further, the number of parameters associated with the response function

W_{f, i}^{'} ({\overset{˘}{E}}_{1, i})

is identical to that of the GOH relation.

(i) Connection of the proposed structure tensor with current approaches

From the definition of the matched-generalized invariant in equation (2.8), it follows that

\begin{aligned} \frac{\partial E_{1, i}}{\partial C} & = \int_{\partial Ω} ρ (N) \frac{1}{2} (N \otimes N + \frac{1}{2 \sqrt{E_{1}^{2} \cdot N \otimes N}} (E_{1} (N \otimes N) + (N \otimes N) E_{1})) d Ω \\ = H_{1, i}^{x^{*}} \approx {\overset{˘}{H}}_{1, i}^{x^{*}}, \end{aligned}

2.21

where

{\overset{˘}{H}}_{1, i}^{x^{*}}

, defined in equation (2.15), is the first approximation of

H_{1, i}^{x^{*}}

A quantity $H_{i}^{x} := \int_{\partial Ω} ρ_{i} (N) H (I_{4} - 1) N \otimes N d Ω$ , where $H$ is the Heaviside function, was introduced by Melnik et al. [5] (iGST approach), and it can be interpreted as a GST constructed by excluding the fibres in compression, i.e. fibres having $I_{4} < 1$ . Note that for uniaxial tension along the fibre direction $M_{i}$ of the $i$ th family in a unidirectional composite,

H_{1, i}^{x^{*}} = {\overset{˘}{H}}_{1, i}^{x^{*}} = H_{i}^{x} = M_{i} \otimes M_{i} .

2.22

For uniaxial compression along the fibre direction of a unidirectional composite,

H_{1, i}^{x^{*}} = {\overset{˘}{H}}_{1, i}^{x^{*}} = H_{i}^{x} = 0 .

2.23

Further, for transverse shear approached from uniaxial extension and uniaxial compression along the fibre direction

M_{i}

H_{1, i}^{x^{*}} = {\overset{˘}{H}}_{1, i}^{x^{*}} = H_{i}^{x} = M_{i} \otimes M_{i} and H_{1, i}^{x^{*}} = {\overset{˘}{H}}_{1, i}^{x^{*}} = H_{i}^{x} = 0,

2.24

respectively. Note that

tr (H_{i}^{x}) = \int_{\partial Ω} ρ_{i} (N) H (I_{4} - 1) d Ω = χ_{i} .

2.25

Note also that while

tr (H_{i}) = 1

tr (H_{i}^{x}) \in [0, 1]

, and it is equal to the fraction of extended fibres within the

i

th family at a given state of stretch [5]. Melnik et al. [5], and more recently Breslavsky et al. [20], used the quantity

χ_{i}

in their constitutive relation where they appear as scaling factors for the anisotropic contribution to the strain energy.

It follows from equation (2.21) that

\begin{aligned} tr (H_{1, i}^{x^{*}}) & = \int_{\partial Ω} ρ_{i} (N) \frac{1}{2} (1 + \frac{I_{4} (N) - 1}{\sqrt{I_{5} (N) - 2 I_{4} (N) + 1}}) d Ω \\ = \int_{\partial Ω} ρ_{i} (N) \frac{1}{2} (1 + \cos (ϕ (N))) d Ω, \end{aligned}

2.26

where

ϕ (N)

is the angle between EN and N. From the last expression in equation (2.26), it follows that

0 \leq tr (H_{1, i}^{x^{*}}) \leq 1

. Unlike the artificial introduction of the Heaviside function in equation (2.25), a function involving the invariants

I_{4}

and

I_{5}

naturally emerges in equation (2.26) from equation (2.21). Figure 2a,b portrays the quantities

H (I_{4} (N) - 1)

and

\frac{1}{2} (1 + \cos (ϕ (N)))

, respectively, for three different fibres oriented at

0^{\circ}

45^{\circ}

and

90^{\circ}

to the uniaxial loading direction. For the fibres coinciding with the eigendirections of the strain tensor (here,

0^{\circ}

- and

90^{\circ}

-oriented fibres), the integrand

\frac{1}{2} (1 + \cos (ϕ (N)))

is identical to

H (I_{4} - 1)

. For all other fibres, there is a partial contribution (ranging between 0 and 1) that adjusts with deformation (figure 2b).

From equation (2.15),

tr ({\overset{˘}{H}}_{1, i}^{x^{*}}) = \frac{1}{2} + \frac{\hat{E} \cdot H_{i}}{2 \sqrt{{\hat{E}}^{2} \cdot H_{i}}} .

2.27

From the inequalities

- \sqrt{{\hat{E}}_{m}^{2} \cdot H_{i}} \leq {\hat{E}}_{m} \cdot H_{i} \leq \sqrt{{\hat{E}}_{m}^{2} \cdot H_{i}}

, it follows from equation (2.27) that

0 \leq tr ({\overset{˘}{H}}_{1, i}^{x^{*}}) \leq 1

, where the equalities are attained only for perfectly aligned fibres. For these cases in a unidirectional composite (zero-reinforcement scenarios),

tr (H_{1}^{x^{*}}) = tr ({\overset{˘}{H}}_{1}^{x^{*}}) = 0

: (i) pure compressive stretch along the fibre direction and (ii) planar loading in the plane transverse to the fibre that compresses the fibre. Hence, the tensor

H_{1, i}^{x^{*}}

can be interpreted as a GST constructed by excluding only the fibres in zero-reinforcement scenarios. All the other fibres contribute to

H_{1, i}^{x^{*}}

to varying degrees due to their state of combined tension/compression and shear.

Consider the uniaxial loading of a material reinforced with a single family of dispersed fibres. For simplicity, let the mean direction of the rotationally symmetric fibres $M$ coincide with the loading direction, say $e_{3}$ . The distribution follows the normalized density function given by

ρ (N) = ρ (Θ) = \frac{1}{π} \sqrt{\frac{b}{2 π}} \frac{\exp (2 b \cos^{2} Θ)}{erfi (\sqrt{2 b})},

2.28

where

erfi

is the imaginary error function. This density function leads to a rotationally symmetric structure tensor

H_{rot}

defined in equation (2.6). A uniaxial deformation described by

[F] = diag (λ_{Z}^{- 1 / 2}, λ_{Z}^{- 1 / 2}, λ_{Z})

can hence be sustained by this material. Using equation (2.27),

tr ({\overset{˘}{H}}_{1}^{x^{*}}) = \frac{1}{2} + \frac{(1 / 2) κ ((1 / λ_{Z}) - 1) + (1 / 4) (1 - 2 κ) (λ_{Z}^{2} - 1)}{\sqrt{(1 / 2) κ {((1 / λ_{Z}) - 1)}^{2} + (1 / 4) (1 - 2 κ) {(λ_{Z}^{2} - 1)}^{2}}} .

2.29

Figure 2c illustrates the quantities $χ$ , $tr (H_{1}^{x^{*}})$ and $tr ({\overset{˘}{H}}_{1}^{x^{*}})$ as functions of the uniaxial stretch $λ_{Z}$ . Recall that $χ$ denotes the fraction of extended fibres within the distribution [5,20]. As uniaxial extension increases, fibres tend to align with the loading direction and become extended (see the crosses in figure 2c for $λ_{Z} > 1$ ). Conversely, under greater compression, fibres tend to align orthogonally to the loading direction and become extended (see the crosses in figure 2c for $λ_{Z} < 1$ ). Note that $tr (H_{1, i}^{x^{*}})$ and $tr ({\overset{˘}{H}}_{1, i}^{x^{*}})$ exhibit a qualitative similarity to $χ_{i}$ (figure 2c).

(ii) Analysis of uniaxial stress response

ii.1 Perfectly aligned fibres

Consider a matrix embedded with two families of parallel fibres in the $X Y$ -plane oriented symmetrically with the $X$ -axis, i.e. $M_{1} = {(\cos (α), \sin (α), 0)}^{T}$ , $M_{2} = {(\cos (- α), \sin (- α), 0)}^{T}$ and subjected to a homogeneous uniaxial state of stress, i.e. $[T] = diag (T_{x x}, 0, 0)$ . It follows that the deformation gradient tensor has the form $[F] = diag (λ_{X}, λ_{Y}, λ_{Z})$ .

For unidirectional fibres (table 1), the relation (equation (2.12)) simplifies to

\begin{aligned} {\overset{˘}{E}}_{1, i} & = E_{1} \cdot (M_{i} \otimes M_{i} + \frac{E_{1} (M_{i} \otimes M_{i}) + (M_{i} \otimes M_{i}) E_{1}}{2 \sqrt{E_{1}^{2} \cdot (M_{i} \otimes M_{i})}}) \\ = \frac{1}{2} (I_{4, i} - 1 + \underset{additional term}{\underset{⏟}{\sqrt{I_{5, i} - 2 I_{4, i} + 1}}}), \end{aligned}

2.30

for the

i

th family. The value of this invariant equals that of the GOH invariant for uniaxial extension along the unidirectional fibres and is automatically zero in the zero-reinforcement scenarios.

The associated V2F Cauchy stress tensor for unidirectional composites is obtained by setting $H_{i} = M_{i} \otimes M_{i}$ in equation (2.16):

\begin{aligned} T & = - p I + c B + 2 \sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{1, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{1, i}^{3}) \\ \times (\frac{1}{2} m_{i} \otimes m_{i} + \underset{additional term}{\underset{⏟}{\frac{1}{2 \sqrt{I_{5, i} - 2 I_{4, i} + 1}} (\frac{1}{2} (B - I) (m_{i} \otimes m_{i}) + (m_{i} \otimes m_{i}) \frac{1}{2} (B - I))}}), \end{aligned}

2.31

where

m_{i} = F M_{i}

. Figure 3 shows the stress–stretch curves corresponding to the vanGOH relation (equation (2.31)) and the GOH relation with a switching criterion (refer to section S6 of the electronic supplementary material for the constitutive relation) for three different fibre orientations: (i) both the families along the loading direction, i.e.

α = 0^{\circ}

(left-hand figure), (ii) both the families transverse to the loading direction, i.e.

α = 90^{\circ}

(middle figure), and (iii) the families at

\pm 40^{\circ}

to the loading direction (right-hand figure).

Figure 3. The uniaxial stresses for the vanGOH relation match those of the GOH relation with a switching criterion when fibres are perfectly aligned along any of the eigendirections of C (plots (a,b)). For other orientations, the vanGOH relation predicts higher stress due to the local shear experienced by the fibres (plot (c)). $c = 1$ kPa, $k_{1} = k_{1}^{GOH} = 5$ kPa and $k_{2} = k_{2}^{GOH} = 0.01$ . Fibre orientation (a) $α = 0^{\circ}$ , (b) $α = 90^{\circ}$ and (c) $α = 40^{\circ}$ .

For uniaxial loading along and transverse to the fibres, the vanGOH relation and the GOH relation, with a switching criterion, yield identical responses for the same values of the material parameters ( $k_{1} = k_{1}^{GOH}$ and $k_{2} = k_{2}^{GOH}$ as a result of equations (2.33) and (2.35) with $κ = 0$ ). With these same parameters, the uniaxial responses for a fibre orientation oblique to the loading direction ( $α = 40^{\circ}$ ) are shown in figure 3c. When the fibres are oriented obliquely to the loading direction, each of the fibres is in a state of combined tension/compression and shear. Consequently, the vanGOH relation predicts non-zero reinforcement during compression of oblique fibre families, which is not possible when one uses a switching criterion (figure 3c).

ii.2 Dispersed fibres

Here we compare the predictions of several GST-based models with that of the AI-based model of Lanir [36] considering exclusion of compressed fibres during uniaxial deformation for varying amounts of dispersion of fibres. The strain energy for the AI model with the exclusion of compressed fibres ( ${AI}^{x}$ , with $x$ denoting the exclusion of compressed fibres) is

\begin{aligned} W_{{AI}_{n}^{x}} = \int_{\partial Ω} ρ (N) H (I_{4} - 1) w (I_{4}) \sin Θ d Θ d Φ, \\ and & w (I_{4}) = \frac{c_{1}}{n c_{2}} (\exp (c_{2} {(I_{4} - 1)}^{n}) - 1), \end{aligned}}

2.32

where

\partial Ω

denotes the surface of the unit sphere

{(Θ, Φ) | Θ \in [0, π], Φ \in [0, 2 π]}

For comparison of the various models, we follow the procedure described by Holzapfel & Ogden [19]: prior to comparison of the nonlinear relations, for a given amount of fibre dispersion, the mechanical response in uniaxial extension of each of the linearized GST models are matched with the linearized ${AI}^{x}$ model to determine the appropriate material parameters for the GST-based models.¹ Whereas in [19], the comparison was made without excluding compressed fibres; here, we consider the exclusion of compressed fibres. The reader is referred to section S5 of the electronic supplementary material for the constitutive relations of the GOH, Variance and ${AI}_{n}^{x}$ models. For a small uniaxial strain $ε$ , the Green–Lagrange strain tensor $[E] \approx diag (- ε / 2, - ε / 2, ε)$ and $I_{4} (M) - 1 \approx 2 ε$ . One can derive the following:

\begin{aligned} T_{z z}^{GOH 2} & \approx 3 c ε + 4 k_{1}^{GOH 2} {(1 - 3 κ)}^{2} ε \times H (ε), \end{aligned}

2.33

\begin{aligned} T_{z z}^{var 2} & \approx 3 c ε + 4 k_{1}^{var 2} (1 - 6 κ + 18 \hat{κ}) ε \times H (E_{GOH}) \end{aligned}

2.34

\begin{aligned} and T_{z z}^{vanGOH 2} & \approx 3 c ε + \frac{1}{2} k_{1} (4 + 3 κ (- 5 + 6 κ) + 2 \sqrt{4 - 6 κ} (1 - 3 κ) sign (ε)) ε . \end{aligned}

2.35

For the GOH constitutive relation, the switching criterion based on

I_{4} (M)

is employed [37] while for the Variance relation [21], the switching criterion is based on mean-

I_{4}

. Since expressions analogous to equations (2.33)–(2.35) for the

{AI}_{n}^{x}

model (equation (2.32)) could not be analytically derived, each of the coefficient of the leading terms in equations (2.33)–(2.35) are equated to the value of the

{AI}_{2}^{x}

model to calculate the associated parameters

k_{1}^{GOH2}

k_{1}^{var2}

and

k_{1}

Figure 4 portrays the uniaxial responses of the GST-based relations in comparison to that of the corresponding ${AI}^{x}$ model for exponent 2 on the invariant. For clarity, only the stresses due to the fibres are plotted. For unidirectional fibres, the uniaxial responses of all the four constitutive relations coincide and hence they are not shown here. From figure 4, the following points can be observed:

—	Contribution from extended fibres during uniaxial compression: for compression along the mean direction of the dispersed fibres, the fibres will be in a state of combined tension and shear and combined compression and shear; hence it is reasonable to expect some level of reinforcement. In addition to having continuous stresses since it does not involve a switching criterion, the vanGOH relation is capable of describing the reinforcement when $λ_{Z} < 1$ (figure 4c).
—	Qualitative nature of the stresses during uniaxial tension for nearly isotropic fibre distribution: the tensile stresses of the GOH and Variance relations for well-dispersed fibres show limited correlation with those of the ${AI}_{2}^{x}$ relation. In addition, for a moderate level of dispersion, the stress–stretch curves of the GOH model exhibit qualitative trends that deviate from expected behaviour. This is due to the large values of $k_{1}^{GOH2}$ if the tensile stresses in the linear regime are to be matched. It can be seen from equation (2.33) that as $κ \to 1 / 3$ , $k_{1}^{GOH2} \to \infty$ so as to match the finite coefficient of ${AI}_{2}^{x}$ (also see eqn. (79) in [19]).
—	Stiffness for isotropic fibre distributions: from equation (2.33), when $κ = 1 / 3$ , the linearized stiffness in tension or compression for the GOH model is $3 c$ , irrespective of the stiffness of the embedded fibres. By contrast, the Variance and the vanGOH relations account for the fibre stiffness (linearized stiffnesses of $3 c + \frac{4}{5} k_{1}^{var2}$ and $3 c + \frac{1}{2} k_{1}$ , respectively).
—	Extended range of the dispersion parameter $1 / 3 < κ \leq 1 / 2$ : for negative values of the concentration parameter $b$ , the peak of the density function (equation (2.28)) shifts from $Θ = 0^{\circ}$ to $Θ = 90^{\circ}$ . The limiting case $b \to - \infty$ ( $κ \to 1 / 2$ ) corresponds to a planar isotropic distribution. Note that the vanGOH relation yields monotone stresses and has good agreement with ${AI}_{2}^{x}$ relation for both tension and compression.²

It must be noted that the mechanical responses of the GOH and Variance models corresponding to the dispersed fibres depend on the switching criteria, i.e. whether one uses the mean direction-based switch or the mean $I_{4}$ -based switch. At the same time, the vanGOH relation is switchless, so no such dependencies exist.

The vanGOH model (with exponent 2) shows good qualitative and quantitative agreement with the ${AI}_{2}^{x}$ relation for all levels of dispersion (the agreement is reasonably good for exponent 3 as well, but this is not shown). Further, the mathematical form of the vanGOH constitutive relation for dispersed fibres is simpler compared with that of the Variance model (compare equation (2.16) with eqns. (25), (26) and (33) in the electronic supplementary material). For a rotationally symmetric fibre distribution, the GST-based vanGOH relation appears to be a good approximation to the ${AI}_{n}^{x}$ relation, at least for uniaxial deformation along the mean direction.

In summary, the vanishing-matched-invariant-based GOH relation differs from the GOH relation only in the use of a modified structure tensor ${\overset{˘}{H}}_{1, i}^{x^{*}}$ in place of the term $H_{i}$ . The resulting switchless vanGOH relation effectively eliminates the various issues associated with the switching criterion while containing the same number of independent response functions and material parameters. It will be demonstrated in §3b that the vanGOH relation (equation (2.16)) has significantly improved descriptive capability, by testing using 13 sets of unequal-biaxial data, covering various modes of deformation.

(d) Seth–Hill strain-based switchless structural constitutive relation of the polynomial form

A majority of the existing constitutive relations describing anisotropy of arterial tissue involve only $I_{4}$ . Recently, Neff et al. [38] used the invariants based on the Hencky strain $\log U$ , expressly $K_{4}$ and $K_{5}$ , to describe the anisotropy of arterial tissue. Corresponding to the particular values of the Seth–Hill parameter $m$ , $I_{4}$ and $I_{5}$ , and the Hencky strain-based invariants describing anisotropy can be obtained from the matched-invariant, thereby addressing the concerns of Murphy regarding distinct shear responses in the three modes of shear [10,24]. The strain energy is obtained by replacing the Green–Lagrange strain tensor in the definition of the matched-generalized invariant in equation (2.11) with the Seth–Hill strain measure, i.e.

W = \frac{c}{2} (I_{1} - 3) + \sum_{i = 1, 2} \frac{k_{1, i}}{3 k_{2, i}} (\exp (k_{2, i} {\overset{˘}{E}}_{m, i}^{3}) - 1) .

2.36

The corresponding Cauchy stress tensor (P2F) is expressed as

T = - p I + c B + 2 F (\sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{m, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{m, i}^{3}) \frac{\partial {\overset{˘}{E}}_{m, i}}{\partial C}) F^{T}

2.37

and³

\begin{aligned} \frac{\partial {\overset{˘}{E}}_{m, i}}{\partial C} & = \frac{1}{m} {(\frac{\partial C^{m}}{\partial C})}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}}, where {\overset{˘}{H}}_{m, i}^{x^{*}} \\ = \frac{1}{2} (H_{i} + \frac{1}{2 \sqrt{{\hat{E}}_{m}^{2} \cdot H_{i}}} ({\hat{E}}_{m} H_{i} + H_{i} {\hat{E}}_{m})), \end{aligned}

2.38

with

{\hat{E}}_{m} := E_{m} / | | E_{m} | |

, a unit Seth–Hill strain measure.

There are four material parameters, namely, $c$ , $k_{1, 1} = k_{1, 2}$ , $k_{2, 1} = k_{2, 2}$ and $m$ , and three structural parameters, i.e. $κ_{i}$ , $κ_{o}$ and the mean angle of orientation $α$ of symmetrically oriented families with respect to the circumferential direction: therefore, P2F has five parameters including the angle $α$ .

Case 1: all eigenvalues of C are distinct

Consider the spectral decomposition of $C^{m}$ :

C^{m} = \sum_{j = 1}^{3} {(λ_{j}^{2})}^{m} P_{j},

2.39

where

λ_{j}^{2}

are the eigenvalues of the tensor C, and the corresponding eigen projections

P_{j}

are defined as

P_{j} = \prod_{\begin{matrix} k = 1 \\ k \neq j \end{matrix}}^{3} \frac{C - λ_{k}^{2} I}{λ_{j}^{2} - λ_{k}^{2}}, j = 1, 2, 3.

2.40

The action of the fourth-order tensor acting on the second-order tensor in equation (2.38) can be expressed as [39]⁴

\begin{aligned} {(\frac{\partial C^{m}}{\partial C})}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}} & = {(S (\sum_{j = 1}^{3} m {(λ_{j}^{2})}^{m - 1} P_{j} ⊠ P_{j} + \sum_{\begin{matrix} j, k = 1 \\ k \neq j \end{matrix}}^{3} \frac{λ_{j}^{2 m} - λ_{k}^{2 m}}{λ_{j}^{2} - λ_{k}^{2}} P_{j} ⊠ P_{k}) S)}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}}, \end{aligned}

2.41

where

i = 1, 2

, and

S

is the symmetrizer. By using the properties of the transpose of the fourth-order tensors⁵ and the fact that

S^{T} = S

S {\overset{˘}{H}}_{m, i}^{x^{*}} = {\overset{˘}{H}}_{m, i}^{x^{*}}

and

{(P_{j} ⊠ P_{k})}^{T} = P_{j}^{T} ⊠ P_{k}^{T} = P_{j} ⊠ P_{k}

, one arrives at

\begin{aligned} {(\frac{\partial C^{m}}{\partial C})}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}} & = S (\sum_{j = 1}^{3} m {(λ_{j}^{2})}^{m - 1} P_{j} ⊠ P_{j} + \sum_{\begin{matrix} j, k = 1 \\ k \neq j \end{matrix}}^{3} \frac{λ_{j}^{2 m} - λ_{k}^{2 m}}{λ_{j}^{2} - λ_{k}^{2}} P_{j} ⊠ P_{k}) {\overset{˘}{H}}_{m, i}^{x^{*}} . \end{aligned}

2.42

On using the action of a fourth-order tensor on a second-order tensor (see footnote 3) and the subsequent action by the symmetrizer

S

(see footnote 4), one arrives at the following relation:

\begin{aligned} {(\frac{\partial C^{m}}{\partial C})}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}} & = \sum_{j = 1}^{3} m {(λ_{j}^{2})}^{m - 1} P_{j} {\overset{˘}{H}}_{m, i}^{x^{*}} P_{j} \\ + \sum_{\begin{matrix} j, k = 1 \\ k \neq j \end{matrix}}^{3} \frac{λ_{j}^{2 m} - λ_{k}^{2 m}}{λ_{j}^{2} - λ_{k}^{2}} P_{j} {\overset{˘}{H}}_{m, i}^{x^{*}} P_{k}, i = 1, 2. \end{aligned}

2.43

By using equations (2.43), (2.38) and (2.40) in equation (2.37), and collecting the eigenvalues, the Cauchy stress tensor can be shown to be

\begin{aligned} T = - p I + c B + \frac{2}{m} \sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{m, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{m, i}^{3}) \\ \times (a_{1} B^{2} {\overset{˘}{h}}_{m, i}^{x^{*}} B^{2} + a_{2} (B^{2} {\overset{˘}{h}}_{m, i}^{x^{*}} B + B {\overset{˘}{h}}_{m, i}^{x^{*}} B^{2}) + a_{3} B {\overset{˘}{h}}_{m, i}^{x^{*}} B \\ + a_{4} (B^{2} {\overset{˘}{h}}_{m, i}^{x^{*}} + {\overset{˘}{h}}_{m, i}^{x^{*}} B^{2}) + a_{5} (B {\overset{˘}{h}}_{m, i}^{x^{*}} + {\overset{˘}{h}}_{m, i}^{x^{*}} B) + a_{6} {\overset{˘}{h}}_{m, i}^{x^{*}}), \end{aligned}

2.44

where

{\overset{˘}{h}}_{m, i}^{x^{*}} = F {\overset{˘}{H}}_{m, i}^{x^{*}} F^{T}

with

{\overset{˘}{H}}_{m, i}^{x^{*}}

being defined in equation (2.38). The Cauchy stress tensor is expressed using a tensor basis involving the tensors I, B and

{\overset{˘}{h}}_{m, i}^{x^{*}}

in the polynomial form (P2F). The coefficients

a_{i}

are expressed in terms of the principal values of C, and they are independent of the order of the stretches:

\begin{aligned} a_{1} & = \frac{m}{Δ^{2}} \sum_{τ \in π} {(λ_{i}^{2} - λ_{j}^{2})}^{2} {(λ_{k}^{2})}^{m - 1} + \frac{2}{Δ} \sum_{τ \in π} \frac{λ_{i}^{2 m} - λ_{j}^{2 m}}{{(λ_{i}^{2} - λ_{j}^{2})}^{2}}, \\ a_{2} & = - \frac{m}{Δ^{2}} \sum_{τ \in π} ({(λ_{i}^{2} - λ_{j}^{2})}^{2} (λ_{i}^{2} + λ_{j}^{2}) {(λ_{k}^{2})}^{m - 1}) \\ - \frac{1}{Δ} \sum_{τ \in π} (\frac{(λ_{i}^{2} + λ_{j}^{2} + 2 λ_{k}^{2}) (λ_{i}^{2 m} - λ_{j}^{2 m})}{{(λ_{i}^{2} - λ_{j}^{2})}^{2}}), \\ a_{3} & = \frac{m}{Δ^{2}} \sum_{τ \in π} ({(λ_{i}^{2} - λ_{j}^{2})}^{2} {(λ_{i}^{2} + λ_{j}^{2})}^{2} {(λ_{k}^{2})}^{m - 1}) \\ + \frac{2}{Δ} \sum_{τ \in π} (\frac{(λ_{i}^{2} + λ_{k}^{2}) (λ_{j}^{2} + λ_{k}^{2}) (λ_{i}^{2 m} - λ_{j}^{2 m})}{{(λ_{i}^{2} - λ_{j}^{2})}^{2}}), \\ a_{4} & = \frac{m}{Δ^{2}} \sum_{τ \in π} (λ_{i}^{2} λ_{j}^{2} {(λ_{i}^{2} - λ_{j}^{2})}^{2} {(λ_{k}^{2})}^{m - 1}) \\ + \frac{1}{Δ} \sum_{τ \in π} (\frac{(λ_{i}^{2} λ_{k}^{2} + λ_{j}^{2} λ_{k}^{2}) (λ_{i}^{2 m} - λ_{j}^{2 m})}{{(λ_{i}^{2} - λ_{j}^{2})}^{2}}), \\ a_{5} & = - \frac{m}{Δ^{2}} \sum_{τ \in π} (λ_{i}^{2} λ_{j}^{2} {(λ_{i}^{2} - λ_{j}^{2})}^{2} (λ_{i}^{2} + λ_{j}^{2}) λ_{k}^{2}) \\ - \frac{1}{Δ} \sum_{τ \in π} (\frac{((λ_{i}^{2} + λ_{k}^{2}) λ_{j}^{2} λ_{k}^{2} + (λ_{j}^{2} + λ_{k}^{2}) λ_{i}^{2} λ_{k}^{2}) (λ_{i}^{2 m} - λ_{j}^{2 m})}{{(λ_{i}^{2} - λ_{j}^{2})}^{2}}) \\ and a_{6} & = \frac{m}{Δ^{2}} \sum_{τ \in π} (λ_{i}^{4} λ_{j}^{4} {(λ_{i}^{2} - λ_{j}^{2})}^{2} {(λ_{k}^{2})}^{m - 1}) + \frac{2}{Δ} \sum_{τ \in π} (\frac{λ_{i}^{2} λ_{j}^{2} λ_{k}^{4} (λ_{i}^{2 m} - λ_{j}^{2 m})}{{(λ_{i}^{2} - λ_{j}^{2})}^{2}}), \end{aligned}}

2.45

where

τ

is a permutation

i, j, k

of the numbers 1,2,3 and

π

is the set of even permutations. These coefficients can also be represented in the closed-form involving C, and a trace-less tensor G that is orthogonal to devC [27], the deviatoric part of C. The trace-less tensor is defined as

G (C) = \frac{1}{3} ((3 I_{2} - I_{1}^{2}) C^{2} + (I_{1} I_{2} - 3 I_{3}) C + (I_{1} I_{3} - I_{2}^{2}) I) .

2.46

The sets of eigenvalues associated with C, G(C) and cof(C) are

{λ_{1}^{2}, λ_{2}^{2}, λ_{3}^{2}}

- Δ / 3 {λ_{1}^{2} - λ_{2}^{2}, λ_{2}^{2} - λ_{3}^{2}, λ_{3}^{2} - λ_{1}^{2}}

and

{λ_{1}^{2} λ_{2}^{2}, λ_{2}^{2} λ_{3}^{2}, λ_{3}^{2} λ_{1}^{2}}

, respectively, where

cof (C)

denotes the cofactor of C and

Δ = (λ_{1}^{2} - λ_{2}^{2}) (λ_{2}^{2} - λ_{3}^{2}) (λ_{3}^{2} - λ_{1}^{2})

. Note that the first line of the expression

a_{1}

involves the trace of the product of the tensors

C^{m - 1}

and

G^{2} (C)

with the suitable scalar multiplying the expression. In a similar manner, one can represent every coefficient in equation (2.45). Accordingly, we define the additional tensors

X = G (C^{m}) G {(C)}^{- 1} G {(C)}^{- 1} and Y = \frac{9 m}{Δ^{4}} G^{2} (C) C^{m - 1},

2.47

and concomitantly, the coefficients

a_{i}

are expressed in the closed-form as follows:

\begin{aligned} a_{1} = tr (Y - \frac{2 Δ}{3 Δ_{m}} X), \\ a_{2} = tr (Y (C - I_{1} I) + \frac{Δ}{3 Δ_{m}} (C + I_{1} I) X), \\ a_{3} = tr (Y {(C - I_{1} I)}^{2} - \frac{Δ}{3 Δ_{m}} (2 C^{2} + (I_{1}^{2} - I_{2}) I) X), \\ a_{4} = tr (Y cof C - \frac{Δ}{3 Δ_{m}} (\frac{I_{1}^{2} - I_{2}}{2} I - cof C) X), \\ a_{5} = tr (Y cof C (C - I_{1} I) + \frac{Δ}{3 Δ_{m}} (det (C) I + \frac{I_{1}^{2} - I_{2}}{2} C) X) \\ and & a_{6} = tr (Y {(cof C)}^{2} - det (C) \frac{2 Δ}{3 Δ_{m}} X C), \end{aligned}}

2.48

where

Δ_{m} = (λ_{1}^{2 m} - λ_{2}^{2 m}) (λ_{2}^{2 m} - λ_{3}^{2 m}) (λ_{3}^{2 m} - λ_{1}^{2 m}) = - {(- 27 det (G (C^{m})))}^{1 / 4}

. If

m = 0

, the tensors X and Y, and

Δ_{m}

in equation (2.48) are replaced with

X_{0}

Y_{0}

and

Δ_{0}

, respectively, where

Δ_{0} = (\log (λ_{1}^{2}) - \log (λ_{2}^{2})) (\log (λ_{2}^{2}) - \log (λ_{3}^{2})) (\log (λ_{3}^{2}) - \log (λ_{1}^{2})) = - {(- 27 det (G (\log C)))}^{1 / 4}

X_{0} = G (\log C) G {(C)}^{- 1} G {(C)}^{- 1} and Y_{0} = \frac{9}{Δ^{4}} G^{2} (C) C^{- 1} .

2.49

Alternatively, the coefficients

a_{i}

in equation (2.45) can be expressed in terms of the invariants

I_{1}

I_{2}

and

I_{3}

using the following relations:

\begin{aligned} {\hat{λ}}_{1}^{2} = \frac{I_{1}}{3} + \frac{1}{3} \sqrt{6 I_{2} - 2 I_{1}^{2}} \cos (\frac{1}{3} \cos^{- 1} (Z)), \\ {\hat{λ}}_{2}^{2} = \frac{I_{1}}{3} - \frac{1}{3} \sqrt{6 I_{2} - 2 I_{1}^{2}} \cos (\frac{1}{3} (\cos^{- 1} (Z) + π)) \\ and & {\hat{λ}}_{3}^{2} = \frac{I_{1}}{3} - \frac{1}{3} \sqrt{6 I_{2} - 2 I_{1}^{2}} \cos (\frac{1}{3} (\cos^{- 1} (Z) - π)), \\ Z = \frac{\sqrt{2} (2 I_{1}^{3} - 9 I_{1} I_{2} + 9 I_{3})}{{(3 I_{2} - I_{1}^{2})}^{3 / 2}}, \end{aligned}}

2.50

where the square of the stretches are ordered as

{\hat{λ}}_{1}^{2} \geq {\hat{λ}}_{2}^{2} \geq {\hat{λ}}_{3}^{2}

. It should be noted that

m

can take any non-negative real value in these equations. When

m

is a positive integer, equation (2.38) can be expressed in a relatively simple form [40]:

{(\frac{\partial C^{m}}{\partial C})}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}} = \sum_{r = 0}^{m - 1} C^{m - 1 - r} {\overset{˘}{H}}_{m, i}^{x^{*}} C^{r}, i = 1, 2.

2.51

Table 2 lists the coefficients

a_{i}

for a few integral values of

m

. Particularly, when

m = 1

, the special constitutive relation is termed vanGOH, which has a relatively simple mathematical structure, and it has been discussed in §2c.

Table 2. Expressions for the coefficients of the Cauchy stress tensor for a few integer values of $m$ . If the material is incompressible, replace $I_{3}$ with $(1 / 2) (6 - I_{1}^{3} + 3 I_{1} I_{2})$ .
Collapse
$m$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$
1	0	0	0	0	0	1
2	0	0	0	0	1	0
3	0	0	1	1	0	0
4	0	1	0	$I_{1}$	$- \frac{1}{2} (I_{1}^{2} - I_{2})$	$\frac{1}{3} (I_{1}^{3} - 3 I_{1} I_{2} + 2 I_{3})$
5	1	$I_{1}$	$- (I_{1}^{2} - I_{2})$	$\frac{1}{2} (I_{1}^{2} + I_{2})$	$- \frac{1}{6} (I_{1}^{3} + 3 I_{1} I_{2} - 4 I_{3})$	$\frac{1}{3} I_{1} (I_{1}^{3} - 3 I_{1} I_{2} + 2 I_{3})$
6	$2 I_{1}$	$I_{2}$	$- \frac{2}{3} (I_{1}^{3} - I_{3})$	$\frac{1}{3} (I_{1}^{3} + 2 I_{3})$	$\frac{1}{12} (I_{1}^{4} - 12 I_{1}^{2} I_{2} + 8 I_{1} I_{3} + 3 I_{2}^{2})$	$\frac{1}{6} (I_{1}^{2} + I_{2}) \times (I_{1}^{3} - 3 I_{1} I_{2} + 2 I_{3})$

Case 2: any two eigenvalues coincide

Without loss of generality, let $λ_{1} \neq λ_{2} = λ_{3}$ . Consequently [39],

\begin{aligned} {(\frac{\partial C^{m}}{\partial C})}^{T} {\overset{˘}{H}}_{m, i}^{x^{*}} & = m {(λ_{1}^{2})}^{m - 1} P_{1} {\overset{˘}{H}}_{m, i}^{x^{*}} P_{1} + m {(λ_{2}^{2})}^{m - 1} {\bar{P}}_{1} {\overset{˘}{H}}_{m, i}^{x^{*}} {\bar{P}}_{1} + \frac{λ_{1}^{2 m} - λ_{2}^{2 m}}{λ_{1}^{2} - λ_{2}^{2}} (P_{1} {\overset{˘}{H}}_{m, i}^{x^{*}} {\bar{P}}_{1} + {\bar{P}}_{1} {\overset{˘}{H}}_{m, i}^{x^{*}} P_{1}), \end{aligned}

2.52

where

i = 1, 2

and

P_{1} = (C - λ_{2}^{2} I) / (λ_{1}^{2} - λ_{2}^{2})

{\bar{P}}_{1} = I - P_{1}

. Accordingly, the Cauchy stress is represented in the following reduced form:

\begin{aligned} T & = - p I + c B + \frac{2}{m} \sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{m, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{m, i}^{3}) \times (b_{1} B {\overset{˘}{h}}_{m, i}^{x^{*}} B + b_{2} (B {\overset{˘}{h}}_{m, i}^{x^{*}} + {\overset{˘}{h}}_{m, i}^{x^{*}} B) + b_{3} {\overset{˘}{h}}_{m, i}^{x^{*}}), \end{aligned}

2.53

where

\begin{aligned} b_{1} & = z^{2} (\frac{m}{3} (2 tr (C^{m - 1}) + z_{m - 1}^{- 1}) - \frac{2 z}{z_{m}}), \\ b_{2} & = z^{2} (- \frac{m}{9} (tr (C^{m - 1}) (2 I_{1} + z^{- 1}) + z_{m - 1}^{- 1} (I_{1} - 4 z^{- 1})) + \frac{2 I_{1} + z^{- 1}}{3} \frac{z}{z_{m}}), \\ b_{3} & = z^{2} (\frac{m}{27} (tr (C^{m - 1}) (2 I_{1}^{2} + 2 I_{1} z^{- 1} + 5 z^{- 2})) + z_{m - 1}^{- 1} (I_{1}^{2} - 8 I_{1} z^{- 1} - 2 z^{- 2}) \\ - \frac{2}{9} (I_{1} - z^{- 1}) (I_{1} + 2 z^{- 1}) \frac{z}{z_{m}}) \\ and z & = (- 2 / 9) tr [{(dev C)}^{- 1}] and z_{m} = (- 2 / 9) tr [{(dev (C^{m}))}^{- 1}] .^{6} \end{aligned}}

2.54

Alternatively, the Cauchy stress tensor can be derived by taking the limit of equation (2.44) by letting

λ_{3} \to λ_{2}

For three families of collagen fibres, the corresponding Cauchy stress tensor (P3F) is

\begin{aligned} T & = - p I + c B + \frac{2}{m} \sum_{i = 1}^{2} k_{1, i} {\overset{˘}{E}}_{m, i}^{2} \exp (k_{2, i} {\overset{˘}{E}}_{m, i}^{3}) \\ \times (a_{1} B^{2} {\overset{˘}{h}}_{m, i}^{x^{*}} B^{2} + a_{2} (B^{2} {\overset{˘}{h}}_{m, i}^{x^{*}} B + B {\overset{˘}{h}}_{m, i}^{x^{*}} B^{2}) + a_{3} B {\overset{˘}{h}}_{m, i}^{x^{*}} B \\ + a_{4} (B^{2} {\overset{˘}{h}}_{m, i}^{x^{*}} + {\overset{˘}{h}}_{m, i}^{x^{*}} B^{2}) + a_{5} (B {\overset{˘}{h}}_{m, i}^{x^{*}} + {\overset{˘}{h}}_{m, i}^{x^{*}} B) + a_{6} {\overset{˘}{h}}_{m, i}^{x^{*}}) \\ + \frac{2}{m_{1}} k_{1, 3} {\overset{˘}{E}}_{m_{1}, 3}^{2} \exp (k_{2, 3} {\overset{˘}{E}}_{m_{1}, 3}^{3}) ({\hat{a}}_{1} B^{2} {\overset{˘}{h}}_{m_{1}, 3}^{x} B^{2} \\ + {\hat{a}}_{2} (B^{2} {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} B + B {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} B^{2}) + {\hat{a}}_{3} B {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} B \\ + {\hat{a}}_{4} (B^{2} {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} + {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} B^{2}) + {\hat{a}}_{5} (B {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} + {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}} B) + {\hat{a}}_{6} {\overset{˘}{h}}_{m_{1}, 3}^{x^{*}}), \end{aligned}

2.55

where

{\overset{˘}{E}}_{m_{1}, 3} = E_{m_{1}} \cdot H_{3} + \sqrt{E_{m_{1}}^{2} \cdot H_{3}}

and the coefficients

{\hat{a}}_{i}

differ from

a_{i}

only in

m

, and it has the same mathematical form given in equation (2.45).

The Cauchy stress tensor for unidirectional fibres of two and three families can be obtained by setting $κ_{i} = 0$ and $κ_{o} = 1 / 2$ in equations (2.44) and (2.55), respectively.

3. Effectiveness of switchless constitutive relations

(a) Uniaxial response of human thoracic aortic tissue

Breslavsky et al. [20] performed uniaxial tensile tests on circumferential and longitudinal strips of human thoracic aortic tissue. The in-plane lateral strains were also recorded during the test.

Since the microstructure of the collagen fibres was not quantified in [20], we assumed the collagen fibres were dispersed only in the $Θ - Z$ plane, so $κ_{o}$ was fixed at the value 1/2 and $κ_{i}$ included in the parameter space during optimization. In addition, the state of uniaxial stress was assumed to be homogeneous. The optimization was performed using the genetic algorithm (ga) in $MATLAB$ to determine the set of five parameters $c$ , $k_{1}$ , $k_{2}$ , $α$ and $κ_{i}$ for the vanGOH (equation (2.16)) as well as the GOH relation that fits the stresses. For the vanGOH relation, the material parameters were $c = 26.37$ kPa, $k_{1} = 25.58$ kPa, $k_{2} = 13.53$ , $α = {34.73}^{\circ}$ and $κ_{i} = 0.121968$ . For the GOH relation, the material parameters were $c = 12.53$ kPa, $k_{1} = 203.67$ kPa, $k_{2} = 40.00$ , $α = {40.22}^{\circ}$ and $κ_{i} = 0.135923$ . Figure 5a shows the fitted stress–strain curves of the two relations, and figure 5b shows the predictions of the lateral strains.

Since the mean fibre directions of the two fibre families for the GOH model are determined to be $\pm {40.22}^{\circ}$ with respect to the circumferential axis, the mean direction M initially has $I_{4} (M) < 1$ for tensile loading of the axial strip. Only after some load (corresponding to approx. 0.1 strain in this case) does the mean direction begin to extend, thereby activating the tension–compression switch that includes the anisotropic contribution to the mechanical response. Since the switch point is not at the reference state, one encounters an abrupt change, i.e. a discontinuity in the stress component at the switching point (see also [5]⁷) (see the inset in figure 5a).

The vanGOH relation, being switchless, always yields continuous stresses. In addition, it provides a better description of the stresses (figure 5a) and better predictions of the lateral strains (figure 5b), while having the same number of material parameters as the GOH relation.

(b) Biaxial response of arterial tissue

In this section, we evaluate the effectiveness of the proposed switchless relations—vanGOH (V2F and V3F) and polynomial types (P2F and P3F)—by comparing their predictions with load- and displacement-controlled biaxial data. We also assess the well-known GST-based relations, such as the GOH [8], Variance [21] and four-fibre-family (FFF) [42] (with exponent 3 on the generalized-invariant corresponding to the collagen fibres) (see section S6 in the electronic supplementary material). Thirteen comprehensive unequal-biaxial datasets from normal and diseased arteries are analysed, considering various locations within the human body. To avoid shear stresses, experiments are conducted in the circumferential and longitudinal directions, resulting in biaxial deformation under purely normal forces in the X and Y directions (which correspond to the $Θ - Z$ directions of the excised tissue). Unlike uniaxial tensile tests, where fibres can experience compression, these biaxial tests are tensile, meaning fibres seldom enter a compressive state. Consequently, the anisotropic contribution is rarely turned off in the GOH, Variance and FFF relations.

In this section, only certain fits are portrayed in figures 6 and 7. However, the tables in section S5 of the electronic supplementary material list the coefficient of determination $R^{2}$ and the associated material parameters for each of the proposed and well-known GST-based models.

Figure 5. The left-hand panel shows the uniaxial stress–strain response of aortic strips oriented circumferentially ( $Θ$ axis) and axially ( $Z$ -axis). The right-hand panel displays the model lateral strain predictions. Accurately capturing both stress and lateral strains remains a challenging task (see [41] for more details). (a) Fitted uniaxial stresses and (b) predicted lateral strains.

Figure 6. Multiple models are fitted to the load-controlled biaxial data of a diseased coronary artery [43]. The associated material parameters are listed in table 3.

Figure 7. Multiple models are fitted to the biaxial data of a human common carotid artery [44]. The associated constitutive parameters are listed in table 3.

Table 3. Material parameters of the GOH, V2F and P2F constitutive relations for the coronary artery [43] and common carotid artery [44]. The values of the quantities $c$ , $k_{1, i}$ and $ϵ$ are in kPa.
Collapse
coronary artery [43]
model	$c$	$k_{1}^{GOH}$	$k_{2}^{GOH}$		$α$	$ϵ$	$R^{2}$
GOH	320	1.72e4	3061.4		${56.6}^{\circ}$	12.8	0.95
	$c$	$k_{1, i}$	$k_{2, i}$	$m$	$α$	$ϵ$	$R^{2}$
V2F	103	2.29e4	1665.5	1	${54.9}^{\circ}$	9.45	0.97
P2F	58.8	2.63e4	1930.3	0	${54.5}^{\circ}$	8.02	0.98
common carotid artery [30]
model	$c$	$k_{1}^{GOH}$	$k_{2}^{GOH}$		$α$	$ϵ$	$R^{2}$
GOH	20.3	6.589	1.7629		${35.4}^{\circ}$	7.44	0.82
	$c$	$k_{1, i}$	$k_{2, i}$	$m$	$α$	$ϵ$	$R^{2}$
V2F	10.1	27.59	0.4968	1	${47.4}^{\circ}$	5.41	0.94
P2F	19.6	2.525	0.1096	2.7409	${54.3}^{\circ}$	2.73	0.99

Table 4. Characteristics and predictive accuracy ( $R^{2}$ ) of various constitutive models, with new models in bold; median values reflect extensive fitting across 13 unequal biaxial datasets.
Collapse
model	no. of fibre families	no. of independent response functions	no. of constitutive parameters	no. of structural parameters	median $R^{2}$ value
GOH-2F	2	1	3	3	0.83
Variance-2F			3		0.83
V2F			3		0.94
P2F			4		0.98
GOH-3F	3	2	5	3	—
Variance-3F			5		—
V3F			5		0.98
P3F			7		0.98
GOH-4F	4	3	7	3	0.99
V4F			7		—

The mean fibre directions of two families representing collagen in the adventitia are assumed to lie in the $X Y$ -plane: $M_{1} = {(\cos (α), \sin (α), 0)}^{T}$ and $M_{2} = {(\cos (- α), \sin (- α), 0)}^{T}$ . The components of the unit vectors are relative to the Cartesian basis with $e_{1}$ along the circumferential, $e_{2}$ along the longitudinal and $e_{3}$ along the radial directions of an artery. The third mean-direction connected with the unidirectional circumferential fibres in the medial layer is assumed to be along $e_{1}$ . The two adventitial fibre families are primarily dispersed in the $X Y$ -plane, with a small fraction oriented radially. A suitable structure tensor introduced by Holzapfel et al. [25] with in- and out-of-plane dispersion parameters is used. The values of the dispersion parameters $κ_{i}$ and $κ_{o}$ corresponding to the medial collagen were fixed at 0 and 1/2, respectively, prior to the optimization. It follows that, for biaxial deformation along the circumferential-longitudinal directions, i.e. with $[F] = diag (λ_{1}, λ_{2}, {(λ_{1} λ_{2})}^{- 1})$ , ${\overset{˘}{E}}_{m, 1} = {\overset{˘}{E}}_{m, 2} \equiv {\overset{˘}{E}}_{m}$ . Using equation (2.55), the non-zero components of the Cauchy stress tensor for the P3F relation with dispersion are

\begin{aligned} T_{x x} & = - p + c λ_{1}^{2} + \frac{k_{1} {\overset{˘}{E}}_{m}^{2} e^{k_{2} {\overset{˘}{E}}_{m}^{3}} λ_{1}^{2 m}}{m \sqrt{E_{m}^{2} \cdot H_{i}}} (λ_{1}^{2 m} - 1 + 2 m \sqrt{E_{m}^{2} \cdot H_{i}}) (A + B \cos^{2} α) \\ + \frac{k_{1, 3} {\overset{˘}{E}}_{m_{1}, 3}^{2} e^{k_{2, 3} {\overset{˘}{E}}_{m_{1}, 3}^{3}} λ_{1}^{2 m_{1}} (λ_{1}^{2 m_{1}} - 1 + \sqrt{{(λ_{1}^{2 m_{1}} - 1)}^{2}})}{\sqrt{{(λ_{1}^{2 m_{1}} - 1)}^{2}}}, \\ T_{y y} & = - p + c λ_{2}^{2} + \frac{k_{1} {\overset{˘}{E}}_{m}^{2} e^{k_{2} {\overset{˘}{E}}_{m}^{3}} λ_{2}^{2 m}}{m \sqrt{E_{m}^{2} \cdot H_{i}}} (λ_{2}^{2 m} - 1 + 2 m \sqrt{E_{m}^{2} \cdot H_{i}}) (A + B \sin^{2} α), \\ p & = c {(λ_{1} λ_{2})}^{- 2} + k_{1} {\overset{˘}{E}}_{m}^{2} e^{k_{2} {\overset{˘}{E}}_{m}^{3}} {(λ_{1} λ_{2})}^{- 2 m} \\ \times \frac{({(λ_{1} λ_{2})}^{- 2 m} - 1 + 2 m \sqrt{E_{m}^{2} \cdot H_{i}}) (1 - 2 A - B)}{m \sqrt{E_{m}^{2} \cdot H_{i}}}, \\ {\overset{˘}{E}}_{m} & = ((λ_{2}^{2 m} - 1) (A + B \sin^{2} α) + (λ_{1}^{2 m} - 1) (A + B \cos^{2} α) \\ + (1 - 2 A - B) ({(λ_{1} λ_{2})}^{- 2 m} - 1) \\ + ({(λ_{2}^{2 m} - 1)}^{2} (A + B \sin^{2} α) + {(λ_{1}^{2 m} - 1)}^{2} (A + B \cos^{2} α) \\ + (1 - 2 A - B) {{({(λ_{1} λ_{2})}^{- 2 m} - 1)}^{2})}^{1 / 2}) / (2 m) \\ and E_{m}^{2} \cdot H_{i} & = \frac{1}{4 m^{2}} {((λ_{1}^{2 m} - 1)}^{2} (A + B \cos^{2} α) + {(λ_{2}^{2 m} - 1)}^{2} (A + B \sin^{2} α) \\ + (1 - 2 A - B) {({(λ_{1} λ_{2})}^{- 2 m} - 1)}^{2}), \end{aligned}}

3.1

for

i = 1 or 2

. In the case of perfectly aligned fibres, the expression

E_{m}^{2} \cdot H_{i}

reduces to

E_{m}^{2} \cdot H_{i} = \frac{1}{4 m^{2}} ({(λ_{1}^{2 m} - 1)}^{2} \cos^{2} α + {(λ_{2}^{2 m} - 1)}^{2} \sin^{2} α) .

3.2

Accordingly, the components of the Cauchy stress are obtained by setting

A = 0

and

B = 1

Similarly, the components associated with P2F are obtained by setting $k_{3} = 0$ in equation (3.1). The components of the stress associated with the dispersed and perfectly aligned vanGOH constitutive relation (V2F and V3F) can be obtained by setting $m = m_{1} = 1$ in the corresponding polynomial-type relations. For the GOH relation and its variant, i.e. GOH3, mean-direction-based switching is employed [37], whereas for the Variance relation and the FFF relations, mean- $I_{4}$ -based switching is used. We compare the predictions of the existing GST-based models—GOH, GOH3, FFF and the Variance—with those of the proposed constitutive relations.

The biaxial data for the femoral artery obtained from the tissues harvested from the patients with ages 25, 47 and 70 (diseased), together with the structural measurements (mean direction, $κ_{i}$ and $κ_{o}$ ) are available in Jadidi et al. [30]. The constitutive relations with in- and out-of-plane dispersions are employed for these three data. The rest of the biaxial data totaling 10 does not have structural information on the arteries. Accordingly, all the constitutive relations are assumed to possess perfect alignment. For the two families of fibres, they were assumed to be perfectly aligned at angles $\pm α$ from the circumferential direction. For the dispersed fibres, the mean directions are assumed to be $\pm α$ degrees for the adventitia.

The material parameters for the perfectly aligned P3F, i.e. $c, k_{1, 1} = k_{1, 2}, k_{2, 1} = k_{2, 2}, m, k_{1, 3}, k_{2, 3}, m_{1}$ and the structural parameter $α$ are determined by using the differential evolution [45] algorithm in $MATHEMATICA$ . Whenever histological information ( $α$ , $κ_{i}$ and $κ_{o}$ ) was available, the values were substituted in equation (3.1) prior to optimization. The objective function for the minimization is the weighted root-mean-square (RMS) error:

\begin{aligned} ϵ & = \sum_{i = 1}^{P} w_{x x, i} {(\frac{1}{M (i)} \sum_{j = 1}^{M (i)} {(T_{x x}^{j} - T_{x x}^{e, j})}^{2})}^{1 / 2} + w_{y y, i} {(\frac{1}{N (i)} \sum_{j = 1}^{N (i)} {(T_{y y}^{j} - T_{y y}^{e, j})}^{2})}^{1 / 2} in kPa, \end{aligned}

3.3

where

T_{x x}^{e}

T_{y y}^{e}

are the measured stresses,

w_{x x, i}

and

w_{y y, i}

denote the weights associated with the RMS of

T_{x x}

and

T_{y y}

, respectively,

P

is the total number of testing protocols, and

M (i)

and

N (i)

are the number of pairs of data points in the

i

T_{x x}

λ_{x}

and

T_{y y}

λ_{y}

data, respectively. The complementary weights

w_{x x, i}

and

w_{y y, i}

are calculated as

\begin{aligned} w_{x x, i} = (1 - \frac{{(T_{x x}^{e, i})}_{max}}{\sum_{j = 1}^{P} {(T_{x x}^{e, j})}_{max} + {(T_{y y}^{e, j})}_{max}}) \frac{1}{2 P - 1} \\ and & w_{y y, i} = (1 - \frac{{(T_{y y}^{e, i})}_{max}}{\sum_{j = 1}^{P} {(T_{x x}^{e, j})}_{max} + {(T_{y y}^{e, j})}_{max}}) \frac{1}{2 P - 1}, \end{aligned}}

3.4

such that

\sum_{i = 1}^{P} (w_{x x, i} + w_{y y, i}) = 1

and the protocol with the lower values of the maximum stress are allotted higher weights. An overall coefficient of determination

R^{2}

is calculated as

\begin{aligned} R^{2} & = 1 - \frac{\sum_{i = 1}^{n_{x x}} {(T_{x x}^{e, i} - T_{x x}^{i})}^{2} + \sum_{j = 1}^{n_{y y}} {(T_{y y}^{e, j} - T_{y y}^{j})}^{2}}{\sum_{i = 1}^{n_{x x}} {(T_{x x}^{e, i} - {\bar{T}}^{e})}^{2} + \sum_{j = 1}^{n_{y y}} {(T_{y y}^{e, j} - {\bar{T}}^{e})}^{2}} \\ and n_{x x} & = \sum_{i = 1}^{P} M (i), n_{y y} = \sum_{j = 1}^{P} N (j), \end{aligned}}

3.5

where

{\bar{T}}^{e}

is the mean value of all the measured stresses, both circumferential and longitudinal. In other words, a single

R^{2}

is calculated for each model for each of the 13 tissue samples.

(i) Diseased human arterial tissue

The data for the following diseased tissues are considered: abdominal aortic aneurysmal tissue [46,47], coronary arterial tissue [43], a moderately diseased common iliac arterial tissue [44] and a severely calcified human femoral arterial tissue [30]. While the material parameters and $R^{2}$ are determined for all the said diseased tissues (refer to section S5 of the electronic supplementary material), only the predictions and data for the coronary artery are portrayed in figure 6.

Kural et al. [43] performed biaxial experiments on four coronary and three carotid arteries using force-controlled testing protocols, i.e. the force ratios in the $X$ and $Y$ directions were 1 : 1, 1 : 0.7, 1 : 0.5, 0.7 : 1 and 0.5 : 1. The predictions of GOH, V2F and P2F relations corresponding to the coronary artery of an 81-year-old donor with chronic obstructive pulmonary disease are illustrated in figure 6. The material parameters allied with the constitutive relations are listed in table 3. Note that the proposed relations provide a qualitatively better description of the 0.7 : 1 and 0.5 : 1 protocols. The addition of the third family of fibres did not improve the fit for the proposed constitutive relations. Further, the parameter $m$ approaches zero during minimization: the best fit is attained when one uses the logarithmic strain tensor-based constitutive relation with two families of fibres. The stiffening is somewhat gentle compared to healthy arterial tissues. Note that the mechanical response during 1:0.5 and 0.5:1 is quite knotty, but the proposed relations emulate the data well.

(ii) Healthy human arterial tissue

The following data for healthy arterial tissues are studied: abdominal aortic (AA) [44,47], thoracic, subclavian, renal, common carotid [44] and two femoral arterial [30] tissues. The material parameters associated with each specimen are listed in the electronic supplementary material.

Kamenskiy et al. [44] performed planar biaxial tests on 21 common carotid arteries. Seven protocols of force-controlled tests were conducted, that is, force ratios of 1 : 1, 1 : 2, 1 : 4, 1 : 10, 2 : 1, 4 : 1 and 10 : 1. The model fits and material parameters associated with a representative dataset are illustrated in figure 7 and listed in table 3, respectively. The $R^{2}$ value for the vanGOH relation is 0.94 as compared to 0.82 for the GOH relation-a significant improvement. The polynomial-type relation P2F with $m \approx 2.7$ provides a better description of the data ( $R^{2} \approx 0.99$ ) than the vanGOH relation. This highlights the importance of using polynomial-type relations for an accurate description of the stiffening response.

(iii) Summary

The number of material parameters and independent response functions associated with each constitutive relation is listed in table 4.

The statistics of the coefficients of determination $R^{2}$ associated with the proposed and the well-known constitutive relations, calculated using equation (3.5), are illustrated in a box-and-whisker plot in figure 8. The boxes represent the interquartile range where 50% of the actual data lie ( $R^{2}$ values).

Figure 8. Box-and-whisker plot based on the fitting of biaxial data for 13 healthy and diseased human arterial tissues. The number in the brackets are the number of parameters associated with the constitutive model. The P3F, P2F, V3F and V2F are the proposed, and the remaining are the existing GST-based relations.

For all the constitutive relations with two families of fibres, i.e. GOH, GOH3, Var2F, V2F and P2F, the bottom whisker or the outlier represent $R^{2} < 0.75$ , signifying a poor fit. This corresponds to a femoral artery specimen that is stiffer in the circumferential direction, even though the collagen fibres are aligned at $\pm 49^{\circ}$ to the circumferential direction. Consequently, any relation with two families of fibres cannot capture the mechanical response appropriately. Adding a third family along the circumferential direction dramatically improves the fit. Although the Variance model has higher $R^{2}$ values than the GOH model for the three tissues with known microstructural information, the median $R^{2}$ values remain the same. This is due to the fact that, out of the 13 tissues studied, 10 are assumed to have unidirectional collagen fibres because their microstructural information is not available. In these cases, the Variance relation is identical to the GOH relation. Compared to the GOH, GOH3 and Var2F with four material parameters, the proposed V2F with the same number of material parameters has a significantly improved interquartile range, suggesting that V2F is likely to emulate any new biaxial data very well. This significant improvement (median $R^{2}$ value for V2F is 0.94 as compared to 0.83 for GOH) is owing to the difference in the treatment of oblique fibres (see §2c(i) and 2c(ii)). Despite P3F having three families of fibres, the plots for FFF (four families) and P3F indicate a similar ability to mimic the data.

To test the assertions made in the preceding paragraph, a paired t-test is conducted on the mean-values of $R^{2}$ related to a pair of constitutive relations in question. The $R^{2}$ values of any two constitutive relations are paired for the same material; there are 13 pairs. The null hypothesis is whether the pair of mean-vales are the same. A threshold of $p < 0.05$ indicates that the null hypothesis is false. The findings are as follows:

(i)	The mean-value of the vanGOH relation with two families of fibres (V2F) is significantly better than that of the GOH3 relation ( $p = 5 \times 10^{- 5}$ ).
(ii)	The mean-value of P3F is statistically equal to that of FFF ( $p = 0.10$ ).
(iii)	The mean-value of vanGOH with three families of fibres (V3F) is significantly better than that of the V2F relation ( $p = 0.004$ ).
(iv)	The mean-value of P3F is significantly better than that of V3F ( $p = 0.005$ ).

4. Conclusion

The proposed constitutive relations, with a mild increase in mathematical complexity compared with the existing GST-based approaches, addresses several existing challenges in the hyperelastic modelling of arterial tissues:

—	Discontinuity in stress at the switch point: the proposed matched-invariant-based relations are switchless, ensuring that the resulting stresses remain continuous across the entire range of deformation. Moreover, the constitutive relation is unitary, encapsulating the full deformation spectrum.
—	Ambiguity in fibre stretch: the ‘dual’ nature of fibre stretch is not a concern in this model.
—	Lack of distinct shear responses in a transversely isotropic material: the matched-invariant-based approach naturally incorporates two anisotropic invariants, leading to distinct responses in longitudinal and transverse shear.
—	For rotationally symmetric distribution of fibres, the proposed relations provide realistic results in the entire range $0 \leq κ \leq 1 / 2$ .

In addition:

—	The linearized-stress-matched uniaxial response of a rotationally symmetric fibre distribution, derived from the proposed relation, is quantitatively similar to that of the AI relation by Lanir [36] with exclusion of compressed fibres.
—	A structure tensor-like quantity, ${\overset{˘}{H}}_{1, i}^{x^{}}$ , emerges in the theory for which the trace is qualitatively similar to the fraction of extended fibres $χ_{i}$ in the literature [5,20]. Instead of artificially introducing a Heaviside-based function of $I_{4}$ as in the traditional approaches, our method naturally derives a function of both $I_{4}$ and $I_{5}$ as a consequence of the matched-invariant, yielding an approximate closed-form expression for the trace. Notably, $tr ({\overset{˘}{H}}_{1, i}^{x^{}})$ and $χ_{i}$ are identical for certain special fibre orientations.
—	The vanGOH relation, a switchless alternative to the GOH relation, provides good description of the planar biaxial mechanics of arterial tissue with the same number of material parameters.

Further, the challenges that arise during the finite-element implementation of constitutive relations with a switching criterion are not present for the proposed relations—deriving the elasticity tensor for the vanGOH relation is straightforward. The modified tensor

{\overset{˘}{H}}_{m, i}^{x^{*}}

preserves all the benefits of the pre-integrated GST-based formulations and, at the same time, automatically discounts the mechanical contributions from the fibres in zero-reinforcement scenarios. Therefore, it is highly advantageous to use

{\overset{˘}{H}}_{m, i}^{x^{*}}

in formulating new switchless constitutive relations.

We have proposed a relatively simple vanGOH potential (V2F/V3F) and the polynomial-type P2F and P3F potentials. Even though V3F has a relatively simple mathematical form compared to that of P3F, the efficacy of both relations are excellent. Most of the biaxial experiments on soft tissues are in the circumferential–longitudinal directions, where only the normal stresses are present. Simultaneous measurements of normal and shear stresses in the plane of loading are possible using the biaxial testing system developed by Sacks et al. [48,49] and Potter et al. [50]. Accurately emulating both the normal and shear stresses is quite tricky. We believe that in such a scenario, P3F will outperform V3F because the Seth–Hill parameter $m$ can control the relative magnitude of normal and shear stresses.

To model more realistic passive behaviour of arteries, the effects of the presence of fluid [51,52], inhomogeneity [53] and residual stress need to be accounted for [54–56].

Data accessibility

The data used in this work is from various publications of other authors.

The data are provided in electronic supplementary material [57].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Conflict of interest declaration

We declare we have no competing interests.

Authors' contributions

A.K.: formal analysis, investigation, methodology, writing—original draft; K.K.: conceptualization, formal analysis, funding acquisition, investigation, methodology, supervision, writing—review and editing.

Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Funding

The authors acknowledge the generous financial support of the Ministry of Human Resource Development (MHRD) for the Centre for Soft and Biological Matter, bearing the number SP22231234CPETWOCSMHOC.

Acknowledgements

The authors thank the anonymous reviewers for their constructive comments, which have contributed to improving the quality of the manuscript.

Footnotes

1 When $κ > 1 / 3$ , for the Variance and vanGOH relations, the linearized stresses in compression were matched with ${AI}_{n}^{x}$ . For the GOH relation, the linearized stresses in tension was used, since the switching criterion $H (I_{4} (M) - 1)$ nullifies all of the anisotropic contribution in uniaxial compression.

2 The proposed relations yield monotone pressure-radius curves during inflation of a thin-walled cylinder for $1 / 3 < κ \leq 1 / 2$ (not shown here) [2].

3 The operation $A A = B$ in the component form is $A_{i j k l} A_{k l} = B_{i j}$ for any fourth-order tensor $A$ and second order tensors $A$ and $B$ . As a special case, $(A ⊠ B) C = A C B^{T}$ where the operator ‘ $⊠$ ’ denotes the square tensor product.

4 The fourth-order tensor $S$ is the symmetrizer defined through $S A = \frac{1}{2} (A + A^{T})$ for any second-order tensor A. In the component form, ${(S A)}_{i j} = \frac{1}{2} (A_{i j} + A_{j i})$ .

5 The transpose of a fourth-order tensor $A$ is defined using the inner product as $A \cdot A B = B \cdot A^{T} A$ , for any second-order tensors A and B. In the component form, ${(A^{T})}_{i j k l} = A_{k l i j}$ . Consequently, ${(A_{1} + A_{2} + \dots + A_{n})}^{T} = A_{1}^{T} + A_{2}^{T} + \dots + A_{n}^{T}$ and ${(A B C)}^{T} = C^{T} B^{T} A^{T}$ .

6 Note that $dev C$ and $dev (C^{m})$ are always invertible when not all eigenvalues of $C$ are equal.

7 Note that the switching criterion used by Melnik et al. [5] for the GOH model is different from the one used in this work. The latter is based on Holzapfel & Ogden [37]. However, both the switching criteria could lead to discontinuities in the stress.

Appendix A. Terminology

$N$ —unit vector along an arbitrary fibre direction.

$M_{i}$ —unit vector along the mean fibre direction of the $i$ th family of fibres.

$H$ —GST defined in equation (2.5).

$H_{rot}$ —GST for rotationally symmetric fibre dispersion (equation (2.6)).

$H_{i}$ —GST for non-rotationally symmetric fibre dispersion of the $i$ th family (equation (2.7)).

$E_{GOH, i}$ —generalized invariant introduced by Gasser, Ogden and Holzapfel [8] corresponding to the $i$ th family.

$H_{i}^{x}$ —GST corresponding to the $i$ th family excluding the compressed fibres [5].

$χ_{i}$ —fraction of extended fibres in the distribution of the $i$ th family at a given stretch state ([5,20]).

$E_{m, i}$ —vanishing-matched-generalized invariant or matched-generalized invariant for the $i$ th family of fibres corresponding to Seth–Hill parameter $m$ ; defined in equation (2.8).

${\overset{˘}{E}}_{m, i}$ —first approximation to the matched-generalized invariant for the $i$ th family of fibres corresponding to Seth–Hill parameter $m$ ; defined in equation (2.10).

$H_{m, i}^{x^{*}}$ —modified tensor for the $i$ th family of fibres with exclusion of fibres in zero-reinforcement scenarios and corresponding to Seth–Hill parameter $m$ (defined in equation (2.21)).

${\overset{˘}{H}}_{m, i}^{x^{*}}$ —first approximation to the modified tensor $H_{m, i}^{x^{*}}$ (equation (2.38)).

GOH—Cauchy stress tensor derived by Gasser, Ogden and Holzapfel [8].

GOH3—Cauchy stress tensor corresponding to exponent 3 on the generalized invariant $E_{GOH}$ ; the exponent 2 on $E_{GOH}$ in the GOH stored energy is replaced with 3 (see the electronic supplementary material).

vanGOH—Cauchy stress tensor corresponding to the exponent 3 on the matched-generalized invariant with $m = 1$ , i.e. ${\overset{˘}{E}}_{i, 1}$ ; switchless analogue of the classical GOH that linearizes correctly for all deformations (defined in equation (2.16)).

P3F—switchless Cauchy stress tensor of the polynomial form with three families of fibres incorporating the vanishing matched-generalized invariant with an arbitrary $m$ ; most general constitutive relation introduced for three families of fibres (defined in equation (2.55)).

P2F—switchless Cauchy stress tensor of the polynomial form with two families of fibres incorporating the vanishing matched-generalized invariant with an arbitrary $m$ ; most general constitutive relation introduced for two families of fibres (defined in equation (2.44)).

V3F—vanGOH incorporating three families of fibres; P3F with $m = m_{1} = 1$ (defined in equation (2.18)).

V2F—vanGOH incorporating two families of fibres; P2F with $m = 1$ (defined in equation (2.16)).

FFF—Cauchy stress tensor for the Four-fibre-family derived by Baek et al. [42] (with exponent 3 on the generalized invariant $E_{GOH}$ ).

Var2F—Cauchy stress tensor derived by Pandolfi et al. [21] for the Variance-based constitutive relation incorporating two families of fibres with exponent 3 (refer to the electronic supplementary material).

$k_{1, i}$ , $k_{2, i}$ —stress-like scaling and dimensionless parameters for the $i$ th family, $i = 1, 2, 3$ (equations (2.37) and (2.55)).

$c_{1}$ , $c_{2}$ —stress-like scaling and dimensionless parameters for the AI relation (equation (2.32)).

This paper is dedicated to the memory of Prof. K. R. Rajagopal, a distinguished scholar, mentor and teacher.

Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.7741391.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

VIEW FULL TEXT DOWNLOAD PDF

This Issue

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

April 2025

Volume 481Issue 2312

Article Information

DOI:https://doi.org/10.1098/rspa.2024.0634
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received23/08/2024
Manuscript accepted17/03/2025
Published online16/04/2025

License:

Citations and impact

Keywords

Subjects

Your password must have 8 characters or more and contain 3 of the following:

Switchless constitutive relation for arterial tissues: eliminating all discontinuities in mechanical response

Abstract

1. Introduction

2. Switchless structure tensor-based relation for the stress in polynomial form

(a) Vanishing-matched-invariant for unidirectional composites

(b) Generalization for dispersed fibres

(c) Green–Lagrange strain-based switchless structural constitutive relation (vanGOH)

(i) Connection of the proposed structure tensor with current approaches

(ii) Analysis of uniaxial stress response

ii.1 Perfectly aligned fibres

ii.2 Dispersed fibres

(d) Seth–Hill strain-based switchless structural constitutive relation of the polynomial form

Case 1: all eigenvalues of C are distinct

Case 2: any two eigenvalues coincide

3. Effectiveness of switchless constitutive relations

(a) Uniaxial response of human thoracic aortic tissue

(b) Biaxial response of arterial tissue

(i) Diseased human arterial tissue

(ii) Healthy human arterial tissue

(iii) Summary

4. Conclusion

Data accessibility

Declaration of AI use

Conflict of interest declaration

Authors' contributions

Funding

Acknowledgements

Footnotes

Appendix A. Terminology

This Issue