Journal of Elasticity (2004) 77: 123–138 © Springer 2005

DOI: 10.1007/s10659-005-4408-x

Constitutive Models for Compressible Nonlinearly

Elastic Materials with Limiting Chain Extensibility

CORNELIUS O. HORGAN1 and GIUSEPPE SACCOMANDI2

1 Structural and Solid Mechanics Program, Department of Civil Engineering, University of Virginia,
Charlottesville, VA 22904, USA. E-mail: coh8p@virginia.edu
2 Sezione di Ingegneria Industriale, Dipartimento di Ingegneria dell’Innovazione,
Università degli Studi di Lecce, 73100 Lecce, Italy. E-mail: giuseppe.saccomandi@unile.it

Received 20 July 2004; in revised form 15 December 2004

Abstract. Constitutive models are proposed for compressible isotropic hyperelastic materials that
reflect limiting chain extensibility. These are generalizations of the model proposed by Gent for in-
compressible materials. The goal is to understand the effects of limiting chain extensibility when the
compressibility of polymeric materials is taken into account. The basic homogeneous deformation
of simple tension is considered and simple closed-form relations for the deformation characteristics
are obtained for slightly compressible materials. An explicit first-order approximation is obtained
for the lateral contraction and for the Poisson function in terms of the axial extension which is
shown to be valid for each of two specific compressible versions of the Gent model. One of the
main results obtained is that the effect of limiting chain extensibility is to stiffen the material relative
to the neo-Hookean compressible case.

Mathematics Subject Classifications (2000): 74B20, 74G55.

Key words: limiting chain extensibility, compressible hyperelastic isotropic materials, constitutive
models.

1. Introduction
Constitutive models for rubber-like elastic materials and soft tissues are often pro-
posed with the added restriction of incompressibility. The incompressible con-
straint is clearly a theoretical idealization. For example, the bulk modulus of vul-
canized rubber is not infinitely large but simply much larger than the shear modulus
(usually the ratio of the latter to the former is of the order of 10−4 ). Thus, while the
assumption of incompressibility is generally a good approximation for vulcanized
rubber, this is not the case for foam rubber which can undergo considerable volume
change. From the mathematical point of view the advantage of the incompressibil-
ity assumption is clear: the nonlinear equations used to describe the mechanical
behavior of hyperelastic materials under the constraint of incompressibility sim-
plify considerably because of the geometric simplification of zero volume change.
Thus it has been possible to obtain analytic closed-form solutions for a large class
of interesting problems in this case. However, such solutions are relatively scarce
124 C.O. HORGAN AND G. SACCOMANDI

for compressible materials (see, e.g., the recent review article by Horgan [15]). It
is worth observing that, in finite element analyses, the incompressibility constraint
can cause numerical difficulties and in such cases nearly incompressible models are
often employed. Thus there is a need for continued research on the development of
robust constitutive models for compressible materials.
The aim of this paper is to consider a compressible analog of one of the phenom-
enological constitutive models for incompressible materials that has been success-
fully used to reflect limiting chain extensibility at the molecular level. This model
was first introduced by Gent [12] and has been investigated in detail by the present
authors and co-authors in a series of papers [16–29, 34]. The Gent model is a very
simple one that may be used instead of those formulated in terms of the inverse
Langevin function to capture the effect of finite chain extensibility in the thermo-
mechanical response of certain elastomers (see, for example, the review article [7]
for a discussion of the latter models and the papers [6, 23] for the relationship
between the Gent model and the Langevin function based ones).
In the present paper we introduce and investigate compressible versions of the
Gent model. Our goal is to understand the effects of limiting chain extensibility
when the compressibility of polymeric materials is taken into account. To this end
we consider the basic deformation of simple tension and are able to obtain simple
closed-form relations for the deformation characteristics by considering the case of
very small compressibility. Although this limit has been investigated previously by
many authors, to the best of our knowledge the results developed here are new even
for the standard case of a compressible neo-Hookean material. Our main finding is
that the effect of limiting chain extensibility is to stiffen the material relative to the
neo-Hookean compressible case. Explicit first-order approximations for the lateral
contraction and for the Poisson function in terms of the axial extension are obtained
which are shown to be valid for each of two specific compressible versions of the
Gent model.
The plan of the paper is as follows. In Section 2, we introduce the basic equa-
tions and in Section 3 we provide a discussion of the possible pressure–volume
responses that can be used to model compressibility in elastomers. In Section 4,
compressible versions of limiting chain extensibility models are described and
attention in focussed on two specific compressible versions of the Gent model. In
the limit as the limiting chain extensibility parameter tends to infinity, we recover
two particular neo-Hookean compressible models. The response of the models
proposed in Section 4 to a simple tension loading state is discussed in Sections 5–7.

2. Basic Equations
The mechanical properties of elastomeric materials are usually represented in terms
of a strain-energy density function W (see, e.g., [3, 33]). The state of strain is
characterized by the principal stretches λ1 , λ2 , λ3 of the deformation or equiva-
lently by introducing a strain measure such as the right Cauchy–Green strain tensor
ELASTIC MATERIALS WITH LIMITING CHAIN EXTENSIBILITY 125

C = FT F. Here F is the gradient of the deformation. For an isotropic hyperelastic

material, W is a function of the strain invariants
1

I1 = tr C, I2 = (tr C)2 − tr C2 , I3 = det C = J 2 . (1)
2
Alternatively, one can use the left Cauchy–Green strain tensor B = FFT , which
has the same principal strain invariants (1) as C. For convenience it is usual to
require that the strain-energy function W should vanish in the reference configura-
tion where F = 1 so that I1 = I2 = 3 and I3 = 1. This assumption is expressed by
the normalization condition

W (3, 3, 1) = 0. (2)
√
Moreover it is standard to require that W should approach infinity as J = I3
tends to +∞ or 0+ , i.e.

lim W = +∞, lim W = +∞. (3)

J →+∞ J →0+

These conditions have the physical interpretation that an infinite amount of energy
is required in order to expand the body to infinite volume or to compress it to a
point with vanishing volume.
For isotropic materials the Cauchy stress may be represented as

T = β0 1 + β1 B + β−1 B−1 , (4)

where
2
β0 = (I2 W2 + I3 W3 ),
J
2 (5)
β1 = W1 ,
J
β−1 = −2J W2 ,
and the subscripts on W denote differentiation with respect to the corresponding
principal invariant. From (4) and (5) we obtain the representation equation for the
first Piola–Kirchhoff stress tensor S defined by

S = J TF−T , (6)

as


S = 2W1 FT +2W2 I1 1 − FT F FT + 2I3 W3 F−1 . (7)

It is usually assumed that the Cauchy stress vanishes in the undeformed state and
so we have the additional normalization condition

W 2 + W
1 + 2W 3 = 0, (8)
126 C.O. HORGAN AND G. SACCOMANDI

where the superposed hat notation on W signifies that the derivatives are evaluated
for I1 ≡ I2 = 3, I3 = 1.
For consistency with the classical infinitesimal theory, the strain-energy func-
tion must also satisfy


1 + W
W 2 ≡ − W 2 + W3 = µ ,
2 (9)
11 + 4W
W 12 + 4W 22 + 2W
13 + 4W23 + W 33 = κ + µ .
4 3
In (9) we have used the fact that in linearized isotropic elasticity there are two
fundamental parameters namely, the infinitesimal (or ground-state) shear mod-
ulus µ and the infinitesimal (or ground-state) bulk modulus κ. For vulcanized
rubber-like materials ε = µ/κ is usually a small parameter, but not zero. In the
nearly incompressible theory some corrections to the incompressible (zero-order)
approximation are introduced on considering ε  1.

3. Pressure–Volume Response and Compressibility

There are two methods that are commonly used to formulate constitutive equations
for compressible hyperelastic materials (see, e.g., [5, 8, 14, 31–33]). Both methods
are based on the addition of an extra volumetric term to a basic incompressible
strain-energy density function.
In the first method the strain-energy function for an incompressible material
WINC is viewed as the restriction to the subspace (I1 , I2 , 1) of a strain-energy W 
defined in the full space (I1 , I2 , I3 ). The strain-energy density for a compressible
hyperelastic material is then written as
W =W  (I1 , I2 , I3 ) + WVOL (J ), (10)
where the second term (the pure volumetric part) is included to capture the pres-
sure–volume response. For example, the classical incompressible neo-Hookean
form
µ
WINC = (I1 − 3) (11)
2
may be associated with the full-space strain-energy
 = µ (I1 − 3 − 2 ln J ),
W (12)
2
and this provides a possible functional form to be used in (10). This form was
proposed by Simo and Pister [36]. Clearly, for a given WINC , the choice of W  is not
unique. (It is interesting to note that the choice (12) is also recovered on using the
classical molecular theory. See, for example, [9].)
The second method is to use deformation measures that decompose the pure
isochoric part of the deformation from the pure volumetric part of the deformation.
For example, this may be achieved on using the deviatoric strain invariants
I1 I2
I 1 = 2/3 , I 2 = 4/3 , (13)
J J
ELASTIC MATERIALS WITH LIMITING CHAIN EXTENSIBILITY 127

and then requiring that

W = WDEV (I 1 , I 2 ) + WVOL (J ). (14)

Here WDEV (I 1 , I 2 ) = WINC (I1 , I2 ) when I3 = 1.

Obviously many other approaches are possible but the two described here are
the most popular and this is because the above mentioned decomposition arises
directly from the theory of nearly incompressible materials.
In the literature many empirical laws for the pure volumetric part (or bulk term)
of the strain-energy have been proposed. For example, in [31, 32], the following
expression has been proposed
I
WVOL (J ) = cβ −2 (β ln J + J −β − 1), (15)

where c is a parameter to be related to the bulk modulus by the requirements (9)

and β is an empirical parameter. Good agreement with experimental data is demon-
strated in [31]. For the special case when β = −2, the expression (15) reduces to a
model proposed in [35]. On the other hand the law
II
WVOL (J ) = c1 ln J + c2 (ln J )2 + c3 (J 2 − 1), (16)

has been proposed by Flory [11] in the case c1 = c3 = 0 and c2 = c/2 and in [35]
when c1 = −c/2, c2 = 0, c3 = c/4 in which case (16) is identical to (15) when
β = −2. In a recent paper by Bischoff et al. [5] the authors have proposed the term
c
III
WVOL (J ) = {cosh(β(J − 1)) − 1}. (17)
β2
The volumetric terms (15), (16) and (17) contain non-algebraic terms that allow
one to satisfy the conditions (3).
Another volumetric term has been proposed in [30], namely
µ 
IV
WVOL (J ) = d(J 2 − 1) − 2(d + 1)(J − 1) + 2 ln J , (18)
2
where d is a parameter to be related to the bulk modulus by the conditions (9). This
model is appropriate only for slightly compressible materials.

4. Compressible Versions of Limiting Chain Extensibility Models

In a series of papers [16–29] the present authors have shown that one of the most
tractable phenomenological constitutive models that can be used to study limiting
chain extensibility for incompressible materials is the model first introduced by
Gent in [12, 13], i.e.,
 
µ I I1 − 3
WINC = − Jm ln 1 −
I
, (19)
2 JmI
128 C.O. HORGAN AND G. SACCOMANDI

where µ is the shear modulus for infinitesimal deformations and JmI is the limiting
value for I1 − 3, taking into account limiting polymeric chain extensibility. Thus
I1 is constrained to lie in the range 3
I1 < Jm + 3. In the limit as the polymeric
chain extensibility parameter tends to infinity (JmI → ∞), (19) reduces to the
classical neo-Hookean form (11). Since WINC does not depend on the second in-
variant I2 , the Gent model is of generalized neo-Hookean type. The Cauchy stress
for the model (19) has a singularity as I1 → JmI + 3, reflecting the rapid strain
hardening observed in experiments.
The model (19) is a basic one because it is associated with the simplest rational
approximation of the response functions in the Cauchy stress representation for-
mula for isotropic incompressible elastic materials [34, 24]. A molecular-statistical
basis for (19) has been given in [23]. Another molecular based model for in-
compressible materials, also of generalized neo-Hookean type, is the eight-chain
model of [1] which involves the inverse Langevin function. See, e.g., [6, 7] for a
comparison between the eight-chain model and (19).
On the other hand, the authors have recently shown [27] that it is possible to
formulate a model similar to (19) of the form
 II 3 
µ II (Jm ) − (JmII )2 I1 + JmII I2 − 1
WINC = − Jm ln
II
, (20)
2 (JmII − 1)3
where JmII is also a limiting chain extensibility parameter. As JmII → ∞, one again
recovers the neo-Hookean material (11). The differences between (19) and (20) are
II
the following. First of all, we see that WINC does depend on the second invariant I2 .
More importantly, for the Gent model, the parameter JmI is the maximum value for
I1 − 3, whereas it is shown in [27] that, for the model (20), the parameter JmII is the
maximum of the squared principal stretches, i.e.,
max(λ21 , λ22 , λ23 ) = JmII .
The response of (20) to basic homogeneous deformations is discussed in [27, 29].
The method we use in this paper to generalize the Gent model to take compress-
ibility into account may be also applied to (20) or any other strain-energy density.
Here, for simplicity, we consider only the strain-energy (19).
The extension of (19) to the whole space (I1 , I2 , I3 ) may be written as
 
 µ I I1 − 3
W =−
I
J ln 1 − + 2 ln J , (21)
2 m JmI
where the term ln J arises from the usual considerations of the molecular theory
of elasticity using Gaussian statistics to reflect changes in the volume element
containing the chain end (see [5]). In the limit as JmI → ∞, (21) reduces to (12),
i.e. a compressible neo-Hookean model.
In [5], generalizations of the eight-chain model to include compressibility are
given and the results compared with experimental data. Here our aim is more re-
stricted. We are interested in investigation of the joint effect of compressibility and
ELASTIC MATERIALS WITH LIMITING CHAIN EXTENSIBILITY 129

limiting chain extensibility. Therefore we restrict our attention to a strain-energy

density function for compressible materials defined as
 (I1 , I2 , I3 ) + cβ −2 (β ln J + J −β − 1),
W =W (22)
or
 (I1 , I2 , I3 ) + µ 
W =W d(J 2 − 1) − 2(d + 1)(J − 1) + 2 ln J , (23)
2
where W  is given by (21) or its limiting case (12). Thus we are considering the
approach (10) and for the sake of definiteness we use the volumetric term (15) or
the slightly compressible model (18). Our methods of investigation may be easily
generalized to other specific functional forms for the volumetric part of the strain-
energy density.

5. Simple Tension
Let us consider a simple tensile loading. In this case the principal Cauchy stress
components tk are assumed to be
t3 = T , t1 = t2 = 0, T > 0. (24)
Thus, provided that the usual empirical inequalities hold (see [3]), that is,
β0
0, β1 > 0, β−1
0, (25)
where the response functions βK (K = 0, 1, −1) are defined in (5), it was shown
by Batra [2] that (24) produces a corresponding extensional deformation as
x1 = λ 1 X 1 , x2 = λ1 X2 , x3 = λX3 , (26)
and we have
I1 = 2λ21 + λ2 , I2 = 2λ2 λ21 + λ41 , I3 ≡ J 2 = λ2 λ41 . (27)
When we use (26) and (27) in (4) we find that in order for t1 = t2 = 0 one must
have
(λ + λ−1 λ21 )W2 + λλ21 W3 + λ−1 W1 = 0, (28)
where the derivatives of the strain-energy are evaluated at the values of the invari-
ants given in (27).
In a simple tension test, equation (28) is interpreted as a restriction on the strain-
energy density that defines a relation between the longitudinal extension λ  1 and
the lateral contraction λ1 . The empirical inequalities ensure that we do indeed have
contraction so that λ1
1. It is usual (see [4]) to define the ratio
λ1 (λ)
α(λ) = , (29)
λ
130 C.O. HORGAN AND G. SACCOMANDI

as the lateral contraction function. In addition, one may define the Poisson func-
tion [4] as
1 − λ1 (λ)
ν(λ) = . (30)
λ−1
The value of the function (30) in the undistorted natural state where λ = 1, i.e.,
dλ1 (λ)
ν0 = lim ν(λ) = − , (31)
λ→1 dλ λ=1

is the infinitesimal Poisson’s ratio for general homogeneous and isotropic elastic
solids (here we are assuming that λ1 (1) = 1).
 is inde-
In this paper we confine attention to the special case of (10) where W
pendent of I2 so that
 (I1 , I3 ) + WVOL (J ).
W =W (32)

The conditions (9) then simplify to

1 ≡ −W
W 3 = µ ,
2 (33)
11 + 2W
W 33 = κ + µ ,
13 + W
4 3
where we recall that the superposed hat notation indicates that the derivatives of
the strain-energy are evaluated at the ground state.

5.1. NEO - HOOKEAN COMPRESSIBLE MATERIAL

It is of interest to begin by considering the model (12) in (22), that is, a com-
pressible neo-Hookean material augmented by the volumetric term (15). For this
material it can be easily shown that (33) are satisfied provided that c =
κ − 2µ/3 ≡ , where  is the infinitesimal Lamé constant of the infinitesimal
theory. Equation (28) now reduces to
 
2ε

1− 1 − (λ21 λ)−β + βε(λ21 − 1) = 0, (34)
3
where we recall that ε = µ/κ. Since µ  κ for vulcanized rubber-like materials
so that ε  1, it is convenient to seek the solution of (34) for nearly incompressible
materials by considering a Taylor expansion of the form
(β)
λ1 (λ) ≈ λ1,0 + λ1,1 ε + O(ε2 ). (35)

This is the standard method of solution used to study slightly compressible ma-
terials used by many authors (see, for example, [32, 33] and the references cited
ELASTIC MATERIALS WITH LIMITING CHAIN EXTENSIBILITY 131

therein). We thus obtain

   
(β) 1 1 (6β + 11)λ − 6β − 15 2
λ1 (λ) ≈ √ + (λ − 1) ε+ ε
λ 2λ3/2 24λ5/2

+ O(ε3 ) . (36)

The asymptotic result (36) is a very good approximation for λ  1 and shows that
for slightly compressible materials the effect of β on the solution is of second-
order.
Consider now the Levinson–Burgess model
µ 
W = I1 − 3 + d(I3 − 1) − 2(d + 1)(J − 1) , (37)
2
which contains purely algebraic terms. It can be readily verified that (33) are
satisfied if
1 1
d= + . (38)
ε 3
On using (37) in (28), we obtain an equation whose unique real positive root is
given by
√
(4ε + 3)λ − 3ε
λ1 (λ) = √ . (39)
λ ε+3
This result is valid in the range [3ε/(4ε + 3), ∞]. It is worth noting that the first-
order approximation arising from (39), namely
 
1 1
λ1 (λ) = √ + (λ − 1) ε + O(ε2 ), (40)
λ 2λ3/2
is identical to that obtained in (36) for the model (12) augmented by the Ogden
volumetric term (15).
The foregoing results show that the rather simple Levinson–Burgess model (37),
which has been widely used in the literature, does accurately capture small com-
pressibility effects in simple tension. Furthermore the first-order approximation
(40) has been shown to be valid for each of two well-known compressible neo-
Hookean models.

5.2. GENT COMPRESSIBLE MATERIAL

We now consider use of the model (21) in (22). We have

1I = µ JmI 3I = − µ ,

W , W (41)
2 JmI − (I1 − 3) 2I3
132 C.O. HORGAN AND G. SACCOMANDI

and for WVOL (J ) given by (15), we have

∂WVOL 1 ∂(WVOL ) c 1 − J −β
= = . (42)
∂I3 2J ∂J 2β I3
Thus (33)1 is satisfied. Since W13 = 0 and
µ JmI µ c (2 + β)J −β − 2
W11 + W33 = + + , (43)
2 (JmI − I1 + 3)2 2I32 4β J4
we see that (33)2 is satisfied if
 
1 1
c = κ − 2µ + I . (44)
3 Jm
In the limit as JmI → ∞, we recover from (44) the result c = κ − 2µ/3 for the
neo-Hookean compressible material.
Turning now to the condition (28), we see that it reduces to
λ2 λ21 W3 + W1 = 0. (45)
On using J = λ21 λ and recalling that ε = µ/κ (κ = 0), we thus obtain
   
βJmI λ21 1 1
2 −β

ε I −β −2 + 1 − (λ1 λ)
Jm − (2λ21 + λ2 − 3) 3 JmI


+ 1 − (λ21 λ)−β = 0. (46)
On taking the limit as JmI → ∞ in (46) we recover (34).
For slightly compressible materials, we again seek an approximate solution
(β)
of (46) for simple tension by considering the standard Taylor expansion λ1 (λ) =
λ10 (λ) + λ11 (λ)ε + · · · . In this way it is possible to find that, to first-order in ε,

(β) 1 1 (λ − 1)(λ2 + λ − JmI − 2)

λ1 (λ) ≈ √ + √ ε. (47)
λ 2 λ[λ3 − (JmI + 3)λ + 2]
The first important remark is that the parameter β does not appear in (47) and
so, just as in (36), the effect of β is of second-order. Secondly, (47) allows one to
determine the effect of the limiting chain extensibility parameter JmI . From (47), we
see that the coefficient λ11 (λ, JmI ) of ε is a monotone increasing function of JmI and
so the maximum first-order correction is in the compressible neo-Hookean case
where
λ−1
lim λ11 (λ, JmI ) = . (48)
JmI →∞ 2λ3/2
Therefore, in simple tension, we have established the rather intuitive result that
the effect of limiting chain extensibility is to stiffen the material relative to the
neo-Hookean compressible model.
ELASTIC MATERIALS WITH LIMITING CHAIN EXTENSIBILITY 133

Instead of (22), we may use the Levinson–Burgess volumetric term as in (23)

and therefore consider
 
µ I I1 − 3 µ 
W = − Jm ln 1 − I
+ d(J 2 − 1) − 2(d + 1)(J − 1) . (49)
2 Jm 2
It may again be verified that (33)1 is satisfied. The requirement (33)2 now gives
1 1 2
d= + − . (50)
ε 3 JmI
In the limit as JmI → ∞, we recover from (50) the result (38). Equation (45) can
be rewritten as a quadratic equation in λ21 as
λ41 (λ) + ω1 λ21 (λ) + ω0 = 0, (51)
where
[(ε + 3)JmI − 6ε][λ3 − (JmI + 3)λ] − 2[(4ε + 3)JmI − 6ε]
ω1 = ,
2[(JmI − 6)ε + 3JmI ]λ
(52)
[(4ε + 3)JmI − 6ε][λ3 − (JmI + 3)λ] − 3ε(JmI )2
ω0 = − .
2[(JmI − 6)ε + 3JmI ]λ2
On carrying out a long but straightforward computation, it is possible to verify that
in (51) we have λ1 (1) = 1 and so we obtain
1 1
λ1 (λ) = − ω1 + (ω12 − 4ω0 ). (53)
2 2
If we let JmI → ∞ in (52), (53) we recover the simple result (39) obtained for the
classical Levinson–Burgess strain-energy (37). The approximate solution of (53)
to first-order is given by
1 1 (λ − 1)(λ2 + λ − JmI − 2)
λ1 (λ) ≈ √ + √ ε + O(ε2 ), (54)
λ 2 λ[λ3 − (JmI + 3)λ + 2]
which is identical to that obtained in (47) for the Gent compressible model (21)
augmented with the volumetric term (15). Thus we have shown that the basic
result (48) and the conclusion stated following that equation are also valid for
the material model (49).

6. Poisson Functions
Neo-Hookean compressible models. On using (40) in (30) we obtain the first-
order approximation for the Poisson function, valid for each of the two neo-Hoo-
kean models considered in Section 5.1, as
1 ε
ν(λ) = √ √ − 3/2 . (55)
λ( λ + 1) 2λ
A plot of ν(λ) versus λ for ε = 0.1 is given in Figure 1(a).
134 C.O. HORGAN AND G. SACCOMANDI

(a)

(b)
Figure 1. (a) Plot of the Poisson function (55) for the neo-Hookean compressible models with
I = 97 (solid line),
ε = 0.1. (b) Plots of the Poisson function (57) for the Gent models for Jm
I I
Jm = 30 (dashed line) and ε = 0.1. The curves for both values of Jm are virtually coincident.

On letting λ → 1 in (55), we obtain the first-order approximation for the

infinitesimal Poisson ratio as
1 ε
ν0 = − . (56)
2 2

Gent compressible models. On insertion of the first-order approximations (47)

or (54) in (30), we obtain the approximate Poisson function
ELASTIC MATERIALS WITH LIMITING CHAIN EXTENSIBILITY 135

1 (λ2 + λ − JmI − 2)ε

ν(λ, JmI ) = √ √ − √ , (57)
λ( λ + 1) 2 λ[λ3 − (JmI + 3)λ + 2]

which is valid for both of the Gent compressible models under consideration.
The coefficient of ε in (57) is a monotone increasing function of JmI and so the
maximum first-order correction is in the compressible neo-Hookean case where
ν(λ) = lim ν(λ, JmI ) as JmI → ∞ is given by (55). On letting λ → 1 in (57) we
find that the first-order approximation for the infinitesimal Poisson ratio is identical
to (56) obtained for the compressible neo-Hookean material and so the limiting
chain parameter does not affect this first-order approximation.
A plot of ν(λ, JmI ) versus λ for ε = 0.1 and JmI = 97 and JmI = 30, respectively,
is given in Figure 1(b).

7. Stress Response
We conclude by considering the stress response in simple tension. For simplicity,
we shall consider only the volumetric term given by (23). From (4) and (5) when
W = W (I1 , I3 ) we have
T = 2J W3 1 + 2J −1 W1 B, (58)
and so in simple tension the only nonzero stress is t3 given by
T = 2(λ21 W3 + λ−2
1 W1 )λ. (59)
For the compressible neo-Hookean model (37) we obtain, on using the exact
result (39),
   
T −1 1 (4ε + 3)λ − 3ε λ3 (ε + 3) −1 4
= ε + + − ε + . (60)
µ 3 λ(ε + 3) (4ε + 3)λ − 3ε 3
If instead we use the approximation (40), then
   
T −1 1 (2λ + ελ − ε)2 4λ4 −1 4
= ε + + − ε + . (61)
µ 3 4λ2 (2λ + ελ − ε)2 3
In Figure 2 we plot (60) and (61) when ε = 0.1. Both curves are virtually identical
and are depicted in Figure 2 by a single dotted curve.
On the other hand, for the Gent compressible model (49), we find that
 
T 1 2 JmI
= ε + − I λ21 λ + λ−2
−1
λ
µ 3 Jm 1
JmI − (2λ21 + λ2 ) + 3
 
4 2
− ε−1 + − I , (62)
3 Jm
where λ1 (λ) is given by (54). In Figure 2, for ε = 0.1, we display a comparison
between the stress response (62) for the model (49) with JmI = 97 and (61) for the
136 C.O. HORGAN AND G. SACCOMANDI

Figure 2. Plots of the axial stress (60), (61) for the compressible neo-Hookean model (37)
for ε = 0.1. The plots for the exact and approximate stresses are virtually identical and are
depicted by a single dotted curve. The axial stress (62) for the Gent compressible model (49)
with JmI = 97 and ε = 0.1 is also shown (solid curve).

neo-Hookean compressible model (37). The significant strain-hardening for the

Gent model is evident in Figure 2.

Acknowledgements
The research of C.O.H. was supported by the U.S. National Science Foundation
under DMS 0202834. The work of G.S. was partially supported by GNFM of Ital-
ian INDAM and by PRIN2003 Problemi Matematici Non Lineari di Propagazione
e Stabilità nei Modelli del Continuo. The constructive helpful comments by a
reviewer on an earlier version of the manuscript are greatly appreciated.

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