1 Introduction

In a recent paper [1], a new four-parameter stretch-based constitutive model was proposed for incompressible isotropic hyperelastic soft materials. The model was deemed to be comprehensive in that several well-known strain-energies may be recovered for some particular and limiting values of some of the parameters. The rationale behind introduction of the four-parameter strain-energy (see (2.3) below) was described in detail in [1] where it was shown to generalize several related simpler models based on microstructural considerations that have been shown to match well with experimental data for a wide variety of elastomers, hydrogels and soft tissues. See [1] for an extensive list of references. As will be described in Section 2, the new model exhibits limited stretch extensibility and can be viewed as a generalization of the well-known Gent model. Furthermore, since the structure of the model is composed of powers of the principal stretches, it was shown in [1] that it is also a generalization of the one-term Ogden model [2]. While the Ogden model was originally developed to model the mechanics of rubber and was proposed as a k-term model involving 2 k material parameters, several recent investigations of the mechanical response of soft tissues make use of the one-term Ogden model. A catalog of some particular values of the strain-stiffening exponent that have been used in these studies is provided in [3] where it was remarked that soft tissues generally are modelled with exponents that are of much larger absolute value than those for elastomers. An extensive description of applications to biomaterials is given in [4]. Our purpose here is to demonstrate that the response of the new model in homogeneous deformations is very simply related to that for the one-term Ogden model. In particular, the recent results obtained in [3] on the Poynting and reverse Poynting effects in simple shear for the Ogden model will be shown to also hold for the new model.

In the next Section, we briefly discuss some relevant preliminaries from the stretch formulation of nonlinear hyperelasticity for isotropic incompressible materials and describe the new constitutive model and how it is a generalization of the classical one-term Ogden model. In Section 3, we show that the stress response of the new model for homogeneous deformations is largely dictated by the corresponding response for the Ogden one-term model and illustrate this result for the axial stress in uniaxial extension. In Section 4, we establish comparable results for all three stress-like quantities in the important homogeneous deformation of simple shear and show that recently obtained results [3] on the Poynting and reverse Poynting effects for the Ogden model apply equally well to the new model. Some conclusions are drawn in Section 5.

2 Preliminaries

Here we briefly summarize some relevant equations pertaining to the stretch formulation of the mechanical response of incompressible isotropic hyperelastic materials. On using the standard notation

$${i}_{1}={\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3},{i}_{2}={\lambda }_{1}{\lambda }_{2}+{\lambda }_{2}{\lambda }_{3}+{\lambda }_{3}{\lambda }_{1},{i}_{3}={\lambda }_{1}{\lambda }_{2}{\lambda }_{3}$$
(2.1)

for the three invariants of the stretch tensor in terms of the principal stretches \({\lambda }_{i}\), the incompressibility condition \({i}_{3}=1\) yields \({\lambda }_{1}{\lambda }_{2}{\lambda }_{3}=1\) so that only two independent stretches arise. The strain energy density per unit undeformed volume is given by \(W\left({\lambda }_{1},{\lambda }_{2},{\lambda }_{3}\right).\) We first recall the two-parameter one-term Ogden model [2]

$$\begin{array}{ccc}W=\frac{2\mu }{{\alpha }^{2}}\left({{\lambda }_{1}}^{\alpha }+{{\lambda }_{2}}^{\alpha }+{{\lambda }_{3}}^{\alpha }-3\right),& \alpha \ne 0,& {\lambda }_{1}{\lambda }_{2}{\lambda }_{3}=1,\end{array}$$
(2.2)

where \(\mu\) is the shear modulus for infinitesimal deformations and \(\alpha\) is a hardening parameter. This parameter may be positive or negative and need not be an integer. On setting \(\alpha =2\) and \(\alpha =1\) in (2.2) one recovers the classic neo-Hookean and Varga models respectively.

The new model proposed in [1] is the four-parameter model

$$W=\frac{6\left(n-1\right)\mu N}{{\alpha }^{2}}\left(\frac{1-N}{1-nN}\right)\left[\frac{1}{3N(n-1)}\left({{\lambda }_{1}}^{\alpha }+{{\lambda }_{2}}^{\alpha }+{{\lambda }_{3}}^{\alpha }-3\right)-\mathrm{ln}\left(\frac{{{\lambda }_{1}}^{\alpha }+{{\lambda }_{2}}^{\alpha }+{{\lambda }_{3}}^{\alpha }-3N}{3-3N}\right)\right]$$
(2.3)

where \(N,n,\alpha \ne 0\) are dimensionless material parameters and \(\mu\) is the shear modulus introduced in (2.2) and \({\lambda }_{1}{\lambda }_{2}{\lambda }_{3}=1\) for incompressibility. For simplicity in what follows and to maintain conformity with the microstructural origins for (2.3), we assume that \(N>1\) and \(n>1.\) As described in [1], these restrictions may be weakened if necessary for fitting with experimental data. In order for the ln (■) function to be well defined, we require the constraint

$${{\lambda }_{1}}^{\alpha }+{{\lambda }_{2}}^{\alpha }+{{\lambda }_{3}}^{\alpha }<3N$$
(2.4)

to hold for all deformations under consideration and so, except in the limiting cases \(N\to \infty\) or as \(n\to 1\), the model (2.3) reflects limited extensibility. In [1], the model (2.3) is deemed to be comprehensive in that several well-known strain-energies may be recovered from some limiting values of the parameters. For example, as \(N\to \infty\) or as \(n\to 1\), one recovers the Ogden model (2.2). As \(n\to \infty\), one recovers models of the type suggested in [5] while as \(n\to \infty\) and \(\alpha =2\), it is shown in [1] that one recovers the celebrated Gent model [6] with limiting chain extensibility parameter in that model given by \({J}_{m}=3\left(N-1\right)\). When \(n=3\), one recovers a model introduced in [7]. The rationale behind introduction of the four-parameter strain-energy model (2.3) was described in detail in [1] where it was shown to generalize several related simpler models based on microstructural considerations that have been shown to match well with experimental data for a wide variety of elastomers, hydrogels and soft tissues. In particular, when \(n=3\) and \(\alpha =2\) in (2.3), one recovers the two-parameter generalized neo-Hookean model

$$\begin{array}{cc}W=\mu N\left(\frac{3-3N}{1-3N}\right)\left[\frac{1}{6N}\left({I}_{1}-3\right)-\mathrm{ln}\left(\frac{{I}_{1}-3N}{3-3N}\right)\right],& N>1,\end{array}$$
(2.5)

with the constraint:

$${I}_{1}<3N$$
(2.6)

to ensure that the ln (■) function is well defined. Here \({I}_{1}\) is the first principal invariant of the left Cauchy-Green deformation tensor B. The two-parameter model (2.5) was originally proposed by Anssari-Benam and Bucchi [8, 9] and has been shown to have widespread applications (see, e.g. [7, 10], and references cited therein) and to allow for analytic solution of boundary-value problems involving non-homogeneous deformations [11, 12]. An extension to model compressible materials is described in [13]. While the model (2.3) is expressed in terms of the principal stretches, the strain-energy (2.5) is written in terms of the first strain invariant and so is a generalized neo-Hookean model. Of course, the model (2.5) can also be written in terms of the principal stretches. The model (2.5) reflects limiting chain extensibility and is also simply related to the Gent model (see [14]).

3 Stress response for homogeneous deformations

For strain-energies of the form \(W\left({\lambda }_{1},{\lambda }_{2},{\lambda }_{3}\right),\) we recall that the principal Cauchy stresses in terms of the principal stretches are given by

$$\begin{array}{cc}{t}_{i}=-p+{\lambda }_{i}\frac{\partial W}{\partial {\lambda }_{i}}\text{,}& \left(\text{no sum}\right)\text{,}\end{array}$$
(3.1)

where p is the hydrostatic pressure arising due to the incompressible constraint. For homogeneous deformations, where the principal stretches are constants, the equilibrium equations are satisfied if and only if p is a constant and so the stresses are all constant.

To illustrate how the response for the new model (2.3) is related to that for the Ogden model (2.2) we begin with the simplest homogeneous deformation namely uniaxial extension. For uniaxial extension in the 1-direction so that \({t}_{1}\equiv T,{t}_{2}={t}_{3}=0\), we obtain

$$T={\lambda }_{1}\frac{\partial W}{\partial {\lambda }_{1}}-{\lambda }_{2}\frac{\partial W}{\partial {\lambda }_{2}}.$$
(3.2)

For the model (2.3), we have

$$\frac{\partial W}{\partial {\lambda }_{i}}=2\mu \left(\frac{1-N}{1-nN}\right){\left[\frac{{{\lambda }_{1}}^{\alpha }+{{\lambda }_{2}}^{\alpha }+{{\lambda }_{3}}^{\alpha }-3nN}{{{\lambda }_{1}}^{\alpha }+{{\lambda }_{2}}^{\alpha }+{{\lambda }_{3}}^{\alpha }-3N}\right]}\frac{{{\lambda }_{i}}^{\alpha -1}}{\alpha }$$
(3.3)

so that on using the relations between the stretches in uniaxial extension namely \({\lambda }_{1}=\lambda ,{\lambda }_{2}={\lambda }_{3}={\lambda }^{-1/2}\), we get from (3.2)

$$T=2\mu \left(\frac{1-N}{1-nN}\right)\left[\frac{{\lambda }^{\alpha }+2{\lambda }^{\frac{\alpha}{2}}-3nN}{{\lambda }^{\alpha }+2{\lambda }^{\frac{\alpha}{2}}-3N}\right]\frac{\left({\lambda }^{\alpha }-{\lambda }^{-\alpha /2}\right)}{\alpha }.$$
(3.4)

We write (3.4) as

$$T=\left(\frac{1-N}{1-nN}\right)\left[\frac{{\lambda }^{\alpha }+2{\lambda }^{\frac{\alpha}{2}}-3nN}{{\lambda }^{\alpha }+2{\lambda }^{\frac{\alpha}{2}}-3N}\right]{T}_{og}$$
(3.5)

where

$${T}_{og}=\frac{2\mu }{\alpha }\left({\lambda }^{\alpha }-{\lambda }^{-\alpha /2}\right).$$
(3.6)

As suggested by the notation and can be verified directly on using (3.2) with W given by (2.2), \({T}_{og}\) is the well-known axial stress response for the Ogden model. We write (3.5) as

$$T=F{T}_{og}$$
(3.7)

where

$$F\left(n,N,\alpha \right)=\frac{\left(1-N\right)}{1-nN}\left(\frac{{\lambda }^{\alpha }+2{\lambda }^{-\frac{\alpha }{2}}-3nN}{{\lambda }^{\alpha }+2{\lambda }^{-\frac{\alpha }{2}}-3N}\right).$$
(3.8)

The multiplicative decomposition (3.7) now shows that the stress response has two components namely the basic response \({T}_{og}\) which does not exhibit any limited extensibility modified by the factor F which is subject to the constraint (2.4) which for uniaxial extension reads

$${\lambda }^{\alpha }+{\lambda }^{-\alpha /2}<3N.$$
(3.9)

Thus the limited extensibility is reflected only in the factor F. It can be seen from (3.8) that as \(N\to \infty\) or as \(n\to 1\), one has \(F\to 1\) so that in this case the axial stress for the new model coincides with that for the Ogden model. This reflects the remark made after (2.4) regarding the form of the strain-energy (2.3) in these limits. In view of the restrictions \(N>1\) and \(n>1\) assumed in Section 2, it can be shown that

$$F(n,N,\alpha )\ge 1$$
(3.10)

with equality holding if and only if \(N\to \infty\) or \(n\to 1\). Thus we conclude that, for each fixed value of \(\alpha\), we have

$$T\ge {(T)}_{og}\text{.}$$
(3.11)

The relation (3.7) with (3.11) following as a consequence, is the key result established here. It might be deemed somewhat surprising given that the new model (2.3) is not of separable form in the principal stretches, as is the Ogden model (2.2), and that the strain-energy function in (2.3) involves a somewhat complex combination of the principal stretches.

The general expression (3.3) for \(\partial W/\partial {\lambda }_{i}\) is the key ingredient in establishing the foregoing result. Since this relationship is valid for all homogeneous deformations, it would be expected that results of the type (3.7) should hold in general for such deformations, with the particular form for F being different for each different deformation. The result (3.4) for uniaxial extension was obtained in [1] (see Eq. (28) there), without however making the direct connections (3.7) and (3.11) to its counterpart for the Ogden model. Analogous results to (3.4) for equi-biaxial deformation and for the shear stress in pure shear and simple shear were also given in [1]. A major focus in [1] is the demonstration that these results for the model (2.3) fit well with experimental data for a variety of soft materials. In particular, the axial stress (3.4) has been compared with the corresponding stress (3.6) for the Ogden model in Figs. 4 (a), (b) of [1] and fitting with human brain tissue experimental data is illustrated there. In the next section, we establish results of the type (3.7), (3.11) for three important stress quantities arising in simple shear and make use of recent findings obtained for the Ogden model in [3] to immediately infer analogous results on the Poynting effect for the model (2.3).

4 Simple shear

The simple shear deformation is described by

$${x}_{1}={X}_{1}+\kappa {X}_{2},{x}_{2}={X}_{2},{x}_{3}={X}_{3},$$
(4.1)

where \(\left({X}_{1},{X}_{2},{X}_{3}\right)\) and \(\left({x}_{1},{x}_{2},{x}_{3}\right)\) denote the Cartesian coordinates of a typical particle before and after deformation respectively and \(\kappa >0\) is an arbitrary dimensionless constant called the amount of shear. The angle of shear is \({\mathrm{tan}}^{-1}\kappa\). For the deformation (4.1), the formulation in terms of the principal stretches of the stretch tensor may be found, for example, in Ogden [15], Horgan and Murphy [16], Vitral [17] and Horgan and Vitral [18]. The deformation gradient tensor F and the left Cauchy-Green strain tensor B = \({\boldsymbol{FF}}^{T}\) for the deformation (4.1) are

$${{\boldsymbol{F}}}=\left[\begin{array}{ccc}1& \kappa & 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],{\boldsymbol{B}}=\left[\begin{array}{ccc}1+{\kappa }^{2}& \kappa & 0\\ \kappa & 1& 0\\ 0& 0& 1\end{array}\right].$$
(4.2)

The in-plane principal stretches \({\lambda }_{1}\) and \({\lambda }_{2}\) are given by (see, e.g., [16])

$${\lambda }_{1}=\frac{\kappa +\sqrt{4+{\kappa }^{2}}}{2}\text{ (>1), }{\lambda }_{2}=\frac{1}{{\lambda }_{1}}=\frac{-\kappa +\sqrt{4+{\kappa }^{2}}}{2},{\lambda }_{1}-\frac{1}{{\lambda }_{1}}=\kappa ,$$
(4.3)

while the out-of-plane stretch is \({\lambda }_{3}=1.\) These kinematic quantities are all constants and so the deformation is homogeneous.

We confine attention here primarily to the plane stress formulation for simple shear. See e.g., [16] for a detailed description of this approach and for a discussion of alternative methods to determine the hydrostatic pressure arising due to the incompressibility constraint. We first summarize relevant recent results obtained in [3]. It is shown there that the three stress-type components of primary interest in the plane stress formulation of simple shear are the shear stress

$${T}_{12}=\frac{{{\lambda }_{1}}^{2}}{1+{{\lambda }_{1}}^{2}}\frac{\partial W}{\partial {\lambda }_{1}}-\frac{{{\lambda }_{2}}^{2}}{1+{{\lambda }_{2}}^{2}}\frac{\partial W}{\partial {\lambda }_{2}},$$
(4.4)

the lateral normal stress

$${T}_{22}=-\frac{\partial W}{\partial {\lambda }_{3}}+\frac{{\lambda }_{1}}{1+{{\lambda }_{1}}^{2}}\frac{\partial W}{\partial {\lambda }_{1}}+\frac{{\lambda }_{2}}{1+{{\lambda }_{2}}^{2}}\frac{\partial W}{\partial {\lambda }_{2}},$$
(4.5)

and the normal traction of the inclined lateral faces, denoted by \(\widehat{N}\), given by

$$\widehat{N}=-\frac{\partial W}{\partial {\lambda }_{3}}+\frac{{\lambda }_{1}}{1+{{\lambda }_{1}}^{6}}\frac{\partial W}{\partial {\lambda }_{1}}+\frac{{\lambda }_{2}}{1+{{\lambda }_{2}}^{6}}\frac{\partial W}{\partial {\lambda }_{2}}.$$
(4.6)

For the Ogden model (2.2), it is shown in [3] that

$${T}_{12}=\frac{2\mu }{\alpha }\left(\frac{{\lambda }^{\alpha +1}-{\lambda }^{-\alpha +1}}{1+{\lambda }^{2}}\right),$$
(4.7)

where we have introduced the notation \({\lambda }_{1}\equiv \lambda ,{\lambda }_{2}\equiv {\lambda }^{-1}\) which is used henceforth. The lateral normal stress is

$${T}_{22}=\frac{2\mu }{\alpha }\left(\frac{({\lambda }^{\alpha }-1)(1-{\lambda }^{2-\alpha })}{1+{\lambda }^{2}}\right),$$
(4.8)

while the normal traction \(\widehat{N}\) on the slanted face of the sheared specimen is given by

$$\widehat{N}=\frac{2\mu }{\alpha }\left(\frac{({\lambda }^{\alpha }-1)(1-{\lambda }^{6-\alpha })}{1+{\lambda }^{6}}\right).$$
(4.9)

As was pointed out in [3], an immediate consequence of (4.9) is that

$$\widehat{N}=0\text{ when }\alpha =6.$$
(4.10)

Thus since in experimental implementation of simple shear there is no traction applied on the slanted faces, it was proposed in [3] that \(\alpha =6\) might be considered in some sense to be an optimal parameter in the Ogden one-term model (2.2). It can be verified that for \(\alpha <6,\) the normal traction on the slanted faces is compressive while for \(\alpha >6,\) this traction is tensile. Thus there is a transition in the Poynting-type effect associated with this normal traction on the slanted faces.

An equally important aspect of simple shear is to consider the Poynting effect associated with the lateral normal stress perpendicular to the direction of shearing namely \({T}_{22}\) given in (4.8). As was established in [3] and can be seen directly from (4.8), we have

$$\begin{array}{ccc}{T}_{22}=0& \mathrm{when}& \alpha =2.\end{array}$$
(4.11)

Furthermore, when \(\alpha <2\), we see that \({T}_{22}<0\) and so this stress is compressive and we have the usual Poynting effect. For \(\alpha >2\), one has \({T}_{22}>0\) and so we have the reverse Poynting effect. In the former case, the sheared specimen tends to expand laterally while in the latter case, the specimen tends to contract laterally. The result (4.11) is the well-known fact that there is no Poynting effect in simple shear for neo-Hookean materials (see, e.g. [16, 19], and references cited therein for details). We now see from the preceding discussion that the neo-Hookean model is actually the transition model within the Ogden class at which the Poynting effect changes character. There has been considerable interest recently in investigation of the classic and reverse Poynting effect in soft materials (see, e.g., [19, 20] and references cited therein). The foregoing result provides a striking example where both effects can be demonstrated to occur for homogeneous isotropic one-term Ogden models depending on the stiffening exponent. This finding is in contrast to the conventional wisdom (based on the classical approach of Rivlin in terms of the strain invariants) that most isotropic materials only exhibit the classical Poynting effect.

It is instructive to provide explicit results for the special cases \(\alpha =2, \, {1}\) corresponding to the neo-Hookean and one-term Varga models respectively. These results will be expressed in terms of the principal stretches and also in terms of the amount of shear. We use an overbar to denote dimensionless quantities obtained by dividing by \(\mu\).

  1. 1.

    \(\alpha =2\): From (4.7)–(4.9) we find, on using (4.3), that

    $$\begin{array}{l}{\overline{T} }_{12}=\left(\frac{{\lambda }^{3}-{\lambda }^{-1}}{1+{\lambda }^{2}}\right)=\left(\lambda -\frac{1}{\lambda }\right)=\kappa ,{\overline{T} }_{22}=0\text{,}\\ \overline{\widehat{N} }=\left(\frac{({\lambda }^{2}-1)(1-{\lambda }^{4})}{1+{\lambda }^{6}}\right)=-\frac{{\kappa }^{2}}{1+{\kappa }^{2}}\end{array}.$$
    (4.12)

The first two of these are the well-known results for the neo-Hookean model and the last result in (4.12) was obtained in [16].

  1. 2.

    \(\alpha =1\):

    $${\overline{T} }_{12}=2\left(\frac{{\lambda }^{2}-1}{1+{\lambda }^{2}}\right)=\frac{2\kappa }{\sqrt{4+{\kappa }^{2}}},$$
    (4.13)
    $${\overline{T} }_{22}=-2\frac{{\left(\lambda -1\right)}^{2}}{1+{\lambda }^{2}}=-2\left[1+\frac{2}{\sqrt{4+{\kappa }^{2}}}\right],$$
    (4.14)
    $$\overline{\widehat{N} }=-2\left(\frac{(\lambda -1)({\lambda }^{5}-1)}{1+{\lambda }^{6}}\right)=-2\left[1+\frac{2+{\kappa }^{2}}{\left(1+{\kappa }^{2}\right)\sqrt{4+{\kappa }^{2}}}\right].$$
    (4.15)

As mentioned above and described in detail in [16], an alternative approach to obtaining the hydrostatic pressure in the constitutive law due to incompressibility is to assume that \(\widehat{N}=0\) at the outset rather than the plane stress assumption. This is motivated by the fact, as pointed out above, that such normal tractions on the slanted faces are not normally applied in experiments involving simple shear. One now finds that (see [3] for details)

$${\overline{T} }_{22}=\frac{2}{\alpha }\left(\frac{{\lambda }^{2-\alpha }({\lambda }^{2}-1)({\lambda }^{2\alpha }-1)}{1+{\lambda }^{6}}\right).$$
(4.16)

Thus \({\overline{T} }_{22}>0\) so that the reverse Poynting effect is predicted for all amounts of shear and all exponents \(\alpha\) in the model (2.2). As pointed out in [3], if an experiment implementing simple shear with no traction applied on the slanted face exhibits the classical Poynting effect for the lateral stress, then (4.16) shows that one-term Ogden models will not be a satisfactory simulation of the experiment. One such experiment demonstrating the classical Poynting effect is that of Destrade et al. [21] on shearing of brain tissue. On the other hand, the experiments of Sugerman et al. [22] on a blood thrombus mimic exhibited a reverse Poynting effect so that it is reasonable to use a one-term Ogden model in this case as was done in [22]. In [22] the range for \(\alpha\) found to match experimental data was \(11.23\le \alpha \le 16.38\), while in [4] it was suggested that the parameter \(\alpha\) be taken as \(\alpha =4.28\) to match with experimental data for pure shear. As shown in [3], there is also an out-of-plane normal stress \({T}_{33}\) which turns out to be identical to the negative of the right-hand side of (4.9). Thus this stress is zero for the special value \(\alpha =6\) and changes sign from tensile to compressive as the stiffening parameter \(\alpha\) transitions from \(\alpha <6\) to \(\alpha >6.\) The expression (4.7) for the shear stress remains unchanged.

For \(\alpha =2,\text{ 1,}\) one obtains

$${\overline{T} }_{22}=\left(\frac{({\lambda }^{2}-1)({\lambda }^{4}-1)}{1+{\lambda }^{6}}\right)=\frac{{\kappa }^{2}}{1+{\kappa }^{2}},$$
(4.17)

and

$${\overline{T} }_{22}=\frac{2\lambda {({\lambda }^{2}-1)}^{2}}{1+{\lambda }^{6}}=\frac{2{\kappa }^{2}}{\left(1+{\kappa }^{2}\right)\sqrt{4+{\kappa }^{2}}},$$
(4.18)

respectively, while the shear stresses are still given by (4.11) and (4.13). The positivity of the lateral normal stress \({\overline{T} }_{22}\) and so the occurrence of a reverse Poynting effect is evident.

We now proceed to obtain comparable results for the new model (2.3). For simplicity, we confine attention to the plane stress formulation. On substitution from (2.3) into (4.4) one finds that

$${T}_{12}=\frac{2\mu }{\alpha }\frac{(1-N)}{1-nN}\left(\frac{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3nN}{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3N}\right)\left(\frac{{\lambda }^{\alpha +1}-{\lambda }^{-\alpha +1}}{1+{\lambda }^{2}}\right).$$
(4.19)

It may be verified that this result is identical to that established in Eq. (34) of [1] when differences in notation are accounted for. The shear stress response (4.19) is examined in detail in [1] and shown to fit well with experimental data for a wide variety of soft materials. In particular, the shear stress (4.19) has been compared with the corresponding stress (4.7) for the Ogden model in Figs. 4 (a), (b) of [1] and fitting with human brain tissue experimental data is illustrated there.

In the case of simple shear, the constraint (2.4) reads

$${\lambda }^{\alpha }+{\lambda }^{-\alpha }+1<3N$$
(4.20)

Note that this constraint involves just the two material parameters \(\alpha\) and \(N\). It is important to observe that (4.20) imposes a restriction on the allowable maximum stretch (the “locking stretch”) and so, by virtue of (4.3), the amount of shear \(\kappa\) and the angle of shear \({\mathrm{tan}}^{-1}\kappa\) are also restricted. For example, when \(\alpha =2\), the constraint (4.20) can be written as

$${\left(\lambda -\frac{1}{\lambda }\right)}^{2}={\kappa }^{2}<3\left(N-1\right)$$
(4.21)

where we have made use of (4.3)3 in the second step. As we have remarked in Section 2, for the special case as \(n\to \infty\) and \(\alpha =2\), one recovers from (2.3) the celebrated Gent model [6] with limiting chain extensibility parameter in that model given by \({J}_{m}=3\left(N-1\right)\). Thus, for the Gent model, (4.21) shows that \({\kappa }_{\mathrm{max}}=\sqrt{{J}_{m}}\). This result was previously given and illustrated for some soft materials in [23]. For example, when \({J}_{m}=2.289\) (a value proposed for human arterial wall tissue [24]), one has \({\kappa }_{\mathrm{max}}=1.513\) and so the maximum angle of shear is approximately \({56.54}^{o}\).

We now turn to (4.5) and (4.6) respectively and find that the lateral normal stress is

$${T}_{22}=\frac{2\mu }{\alpha }\frac{(1-N)}{1-nN}\left(\frac{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3nN}{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3N}\right)\left(\frac{\left({\lambda }^{\alpha }-1\right)\left(1-{\lambda }^{2-\alpha }\right)}{1+{\lambda }^{2}}\right)$$
(4.22)

while the normal traction \(\widehat{N}\) on the slanted face of the sheared specimen is given by

$$\widehat{N}=\frac{2\mu }{\alpha }\frac{(1-N)}{1-nN}\left(\frac{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3nN}{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3N}\right)\left(\frac{({\lambda }^{\alpha }-1)(1-{\lambda }^{6-\alpha })}{1+{\lambda }^{6}}\right).$$
(4.23)

We now introduce the notation

$$G\left(n,N,\alpha \right)=\frac{\left(1-N\right)}{1-nN}\left(\frac{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3nN}{{\lambda }^{\alpha }+{\lambda }^{-\alpha }+1-3N}\right).$$
(4.24)

On comparison of (4.19), (4.22), (4.23) with the results (4.7)-(4.9) for the Ogden model, we see that each of the quantities \({T}_{12},{T}_{22},\widehat{N}\) can be written as

$${T}_{12}=G{({T}_{12})}_{og},{T}_{22}=G{({T}_{22})}_{og},\widehat{N}=G{(\widehat{N})}_{og},$$
(4.25)

where the notation with subscript “og” refers to the expressions in (4.7)-(4.9). It is immediate that the conclusions made above regarding the Poynting effect for the special values of

$$\alpha =6 \, \text{and }\alpha ={2}$$
(4.26)

in the Ogden model remain valid for the new model (2.3). It can be seen from (4.24) that as \(N\to \infty\) or as \(n\to 1\), one has \(G\to 1\) so that in this case the results for the new model coincide with those for the Ogden model. This reflects the remark made after (2.4).

We see from (4.24), (4.25) that the stress and normal traction responses for the new model are just a multiple of the corresponding responses for the Ogden model. The parameters \(N\text{ and }n\) do not, of course, affect the Ogden model responses but rather the magnitude and character of the factor \(G.\) The role of the constraint (4.20) is now evident. As previously mentioned, this constraint on the principal stretch \(\lambda\) reflects the limiting chain extensibility character inherent in the model (2.3) and ensures that \(G\) is bounded. As pointed out in Section 3, the Ogden model itself does not involve any limited extensibility. In view of the restrictions \(N>1\) and \(n>1\) assumed in Section 2, it can be shown from (4.24) that

$$G(n,N,\alpha )\ge 1$$
(4.27)

with equality holding if and only if \(N\to \infty\) or \(n\to 1\). Thus, we conclude that, for each fixed value of \(\alpha\), we have

$${T}_{12}\ge {\left({T}_{12}\right)}_{og},{T}_{22}\ge {\left({T}_{22}\right)}_{og},\widehat{N}\ge {\left(\widehat{N}\right)}_{og}.$$
(4.28)

The relations (4.25) with (4.28) following as a consequence, are the analogs of (3.7) and (3.11) obtained for the axial stress in uniaxial extension. Here these relationships hold for all three stress-type quantities of interest in simple shear.

As an illustration of the foregoing results, we consider, for example, the lateral normal stress \({T}_{22}\). This dimensionless stress \({\overline{T} }_{22}\equiv {T}_{22}/\mu\) is shown in Fig. 1 where the response for the Ogden model is plotted versus the stretch \(\lambda\) for the fixed value of \(\alpha =4\) and compared with the response for the model (2.3) with the same value of \(\alpha\) and varying values of \(n\) and \(N.\)

Fig. 1
figure 1

The dimensionless lateral normal stress plotted versus the stretch for the one-term Ogden model (2.2) with \(\alpha =4\) (dashed curve) and for the model (2.3) for \(\alpha =4\) and varying values of \(n\) and \(N.\) The values of n and N are: Blue curve: n = 10; N = 5, Green curve: n = 5; N = 10, Red curve: n = 3; N = 15, Black curve: n = 2; N = 20

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5 Concluding Remarks

In this short paper, we have shown that the new four-parameter constitutive model (2.3) proposed in [1] has a very simple stress response for homogeneous deformations namely that the stresses are just a multiple of the corresponding response for the celebrated two-parameter one-term Ogden model. While a connection between these models might be anticipated in view of the structure of the model (2.3), it is not obvious a priori that the simple multiplicative character established here should hold. Indeed, in forthcoming work, it is shown that this is not the case for inhomogeneous deformations such as torsion. The multiplicative factor depends only on three parameters and reflects limited extensibility. This result was first illustrated for the axial stress in uniaxial extension and then for all three stress-like quantities of interest in simple shear. In particular, recently developed results [3] on the character of the Poynting and reverse Poynting effects for the one-term Ogden model in simple shear were shown to be equally valid for the new model. Further applications of the four-parameter model (2.3) have been described in [25] where the accuracy of fitting with experimental data for a variety of soft materials has been compared with that for the three-term Ogden model [2] with six material parameters.