## Abstract

In this paper, we discuss the need for models that express the stretch (or strain) as a function of stress, or implicit constitutive models that relate the stretch (or strain) and stress, for describing the elastic response of some elastomers. We would like to provide an explanation for some experimental data for elastomeric materials that imply that the material moduli depend on pressure. Included in the class of models that are proposed are those which can explain limiting chain extensibility that is exhibited by some rubber-like solids. The models that are proposed stem from a completely different starting point from that for classical elastic bodies, so that these models cannot be obtained within the context of classical theory.

## 1. Introduction

There is clear experimental evidence that the material moduli of solids and fluids depend on the pressure. In the authoritative monograph titled the *Physics of high pressure*, Bridgman (1931) surveyed, with great care, the research work up to 1930 concerning the response of materials, both fluids and solids, that are subject to high pressure. Here, one can find a detailed exposition of the history of the experimental literature concerning the determination of the properties of materials when exposed to high pressure, and the subject matter that is relevant to this study is the elaborate discussion of the dependence of the viscosity of liquids on pressure. Experiments clearly suggest that viscosity varies exponentially with pressure, and that it is the relationship between the viscosity and the pressure that causes the tremendous change that has been observed in the fluidity of liquids in applications ranging from industrial lubrication to geology.

It is now well known and well accepted that the mechanical properties of many inhomogeneous materials with microstructure, such as geological or composite materials, are very sensitive to hydrostatic pressure (Shin & Pae 1992; Sahaphol & Miura 2005). A classical example is the shear modulus of confined soils.

The effect of hydrostatic pressure of up to 30 000 kg cm^{−2} on the tensile property of several metals and amorphous materials, such as melamine–formaldehyde resins, is presented in Bridgman (1953). Paterson (1964), based on his experiments, states that, in many rubbers, natural or synthetics, there is a clear dependence of Young’s modulus on the pressure. The results concerning polymeric materials have been confirmed in Jones Parry & Tabor (1974), where stress/strain curves for a polymer sample have been determined directly in a special device at various applied pressures. A review of the published data concerning this effect can be found in Jones Parry & Tabor (1973).

With regard to many liquids, while the viscosity can change by several orders of magnitude when subject to a large range of pressures, the changes in the density are of the order of a few per cent (see Hron *et al*. 2001; Rajagopal 2003), and thus the fluid can be considered incompressible and the viscosity can be assumed to depend on the pressure. Such a possibility is not allowed within the framework of classical theory. Only quite recently has a general theory, for fluids with pressure-dependent viscosity, been proposed and for a discussion of the relevant issues, we refer the reader to Hron *et al*. (2001). With regard to solids, early attempts, such as the model in Birch (1936), to investigate the effect of pressure upon the elastic parameters of isotropic solids, are partial and unsatisfactory. The model proposed by Birch is the set of equations of incremental elasticity, where a small deformation is superposed onto a finite strain state that is a consequence of an applied hydrostatic state of stress. There have been some studies that allow the material moduli of an elastic solid to depend upon the residual stress (Johnson & Hoger 1993; Hoger 1997). Man (1998) has proposed a model (eqn (11) of his paper) that provides a relation for the stress in terms of the residual stress and the current strain, but this, nor the other models in which the current stress depends on the strain and the residual stress, are models wherein the material moduli depend on the current value of the stress. Hartig’s law (see Bell 1973) presumes that the tangent modulus depends on the stress, and comes closest to a model wherein the material moduli depend on the stress. However, Hartig’s law is an ad hoc approximation in that the local tangent modulus depends on the stress within the context of a linearized theory. The material moduli of the class of models that we are interested in depend on the current values of the stress (in our case the mean normal stress), and not on some prescribed residual stress. When the current stress in the body depends on the deformation gradient and the residual stress, we have a model wherein one could view the residual stress as an internal variable. On the other hand, a model with the material moduli depending on the current stress is a totally different sort of constitutive relation. There is no quantity that comes into play that can be treated as an internal variable. Even if, instead of the stress, one only considers a part of the stress, say any single component of the stress, the equation is totally different from that in which the stress depends on the residual stress.

It is also important to point out that many implicit constitutive models that qualify to describe elastic bodies, in the sense that they are incapable of dissipation in any process that they undergo, cannot be expressed as a finite number of piecewise explicit constitutive expression. For instance, Rajagopal & Srinivasa (2007) showed that a spiral relationship between the stress and the strain qualifies to be a thermodynamically acceptable constitutive relation for an elastic body. This constitutive relation cannot be expressed as a finite number of explicit constitutive functions. In fact, in this case, one would need infinity of explicit constitutive functions of both stress as a function of strain, and strain as a function of stress. Implicit constitutive theories for elasticity cast a far wider net for constitutive relations for elastic materials than explicit theories, whether they be explicit expressions for the stress in terms of the strain, or vice versa. To take the point of view that all material response can be modelled by explicit constitutive relations is short sighted and restricts one’s repertoire for modelling the response of bodies.

The point is that the classical mathematical theories for mechanical behaviour are structurally too restrictive to take into account the possibility of the various material moduli, such as the generalization of Young’s modulus or the shear modulus, to depend on the pressure. The aim of this study is to investigate the possibility of allowing the material moduli to depend on the pressure with a view towards generalizing the class of models used in the theory of elasticity of rubber-like materials, so that observed effects such as the dependence of the material moduli on the pressure can be accommodated.

In Truesdell & Noll (1992) (see §43), a material is called *elastic* if it is a *simple* material and if the stress at time *t* depends only on the local configuration at time *t*. This means that the constitutive equation must be expressed as
1.1
where **T** is the Cauchy stress tensor, **F** is the deformation gradient at the present time, taken with respect to a fixed but arbitrary local reference configuration, and is the response function of the elastic material.

Recently, in a series of papers, Rajagopal and co-workers (Rajagopal 2007; Rajagopal & Srinivasa 2007, 2009; Bustamante & Rajagopal in press) asserted that the usual interpretation of what one usually means by elasticity is much too restrictive and Rajagopal illustrated his thesis by introducing implicit constitutive theories that describe the non-dissipative response of solids. Indeed, it is not necessary to associate the original idea of elasticity considered by the *savants* of the eighteenth century with the more recent concept of Noll’s simple materials to describe an elastic material. Usually, what we require is only that the stress be determined by the strain and rotation from a single, fixed reference configuration. Therefore, there is no reason to not accept, as a mathematical model for an elastic material, any implicit constitutive equation of the form
1.2
rather than to restrict ourselves to equation (1.1).

In this paper, we show that an implicit theory of elasticity, or an explicit expression for the stretch in terms of the stretch, is necessary to describe the observations concerning the dependence of material moduli on pressure, as, in general, one cannot assume an invertible relation between the pressure and the volumetric deformation.

The plan of this paper is the following. In the next section, we review some basic equations and experimental facts. In §3, we develop a mathematical framework for small strains but large stresses to describe the non-dissipative (elastic) response of solids whose material moduli depend linearly on the pressure (i.e. mean normal stress). In §4, we consider a more general class of implicit models. The last section is devoted to concluding remarks.

## 2. Basic equations, models and experimental facts

The present paper is concerned with the interpretation of some data concerning rubber-like materials. For this reason, we start by reviewing some of the basic approaches we may use to develop a mathematical theory for the mechanical response of elastomeric materials. Within this framework, it is possible to consider a phenomenological approach or a molecular approach. The phenomenological point of view is based on the rational theory of continuum mechanics. We represent the motion of a body by a relation ** x**=

**χ**(

**,**

*X**t*), where

**denotes the current coordinates of a point occupied by the particle whose coordinates are**

*x***in the reference configuration, at the time**

*X**t*, and

**χ**is a one to one mapping. Also, we shall assume that it is sufficiently smooth to make all the derivatives meaningful. The deformation gradient

**(**

*F***,**

*X**t*) is defined as

**:=**

*F**∂*

**χ**/

*∂*

**and the left Cauchy–Green stretch tensor**

*X***:=**

*B*

*F**F*^{T}. In describing elastic materials, it is also usual to consider a strain-energy density

*W*(such elastic bodies are usually referred to as Green elastic or hyperelastic bodies) that, in the isotropic and incompressible case, must be a function of only the first two principal invariants of the left Cauchy–Green strain tensor, i.e.

*W*=

*W*(

*I*

_{1},

*I*

_{2}), where

*I*

_{1}=tr

**and**

*B**I*

_{2}=tr

*B*^{−1}. The explicit classical representation formula for the Cauchy stress tensor is given by 2.1 where

*p*is the indeterminate Lagrange multiplier associated with the internal constraint of incompressibility, .

The basic molecular theories for describing the response of polymers and rubber-like materials are based on a suitable idealization of the macromolecular chains composing the polymeric network and on the assumption that the elasticity of such materials is exclusively entropic. In the simplest theory, we start from a single ideal chain composed of *n* links of length *l*, such that the direction in space of any particular link is entirely random and independent of the neighbouring links. The problem is to find an answer to the following question, namely given one end of the chain being fixed at, say *A*, to determine the probability that the other end of the chain at *B*, is at a distance *r* from *A*. Once this probability density function is determined, the assumption that the elasticity of the chain is purely entropic allows one to determine the tension, *f*, of the single chain by differentiating the entropy with respect to *r*. To obtain the mechanical response of the complete polymeric network, we need some additional assumptions to compute the entropy of the full network and a sort of Cauchy–Born rule to relate the macroscopic deformation of the bulk material to the microscopic deformation of the chains.

In the simplest molecular theory, the Gaussian model, the probability density function for the end to end distance is described by the Gaussian distribution (Treloar 1949). Because the Gaussian probability density function does not have compact support, such a model is inadequate to describe the finite extensibility effects of real chains that are observed in the response of many polymers. Therefore, there is the need for more refined theories (see the review article by Horgan & Saccomandi (2006*b*) for a discussion of the relevant issues) that are usually referred to as non-Gaussian (Treloar 1949).

The early non-Gaussian theories of polymeric chains were the models proposed in 1942 by Kuhn & Grün and in 1943 by James & Guth (Treloar 1949). The model proposed by Kuhn and Grün is based on a probability density function expressed in terms of the Langevin function of , namely
2.2
For this model, the expression
2.3
holds, where *k* denotes the Boltzmann constant and *ϑ* represents the absolute temperature. The expression provides an explicit relationship for *r* (which is related to the macroscopic extension/stretch of the chain by making an appropriate assumption) in terms of the tension *f*. This explicit equation for the extension/stretch is recast as though it were an equation for the tension *f* by considering the *special function* known as the *inverse Langevin function*, and in this special instance, it takes the form
2.4
Sometimes this model is referred to as the *freely jointed chain* (FJC). Kratky & Porod (1949) proposed a more complex model for the polymeric chain. This model, referred to as the *worm-like chain* (WLC), envisions an idealization of the polymeric chain as an isotropic rod that is continuously flexible, in contrast with the previous model, which is flexible only at the links. This model has been used with some success to model the behaviour of many biopolymers in simple extension (the so-called *J-shaped* curves). This theory does not allow one to compute, in closed form, the formula for the tension *f* in terms of the end to end distance *r*. For this reason, in the literature, it is usual to use an interpolation formula for the WLC (see the discussion in Bouchiat *et al*. (1999)). In the stretching of DNA in a physiological buffer, *implicit generalizations* of the direct interpolation formulae are necessary, such as
2.5
where *P* is the persistence length, an additional parameter measuring the distance over which correlations among neighbouring links in the direction of the tangent are lost, and *K* is a stretch modulus that is used to account for the enthalpic compliance (Wang *et al*. 1997). (When, in equation (2.5), we let , we recover the classical explicit WLC model.)

The three-dimensional models that we obtain from the molecular theory must be compatible with the general representation formula (2.1). For example, using the Gaussian chain leads to a molecular model that is exactly the same as the phenomenological strain-energy characterizing the neo-Hookean model,
2.6
where *μ* is the infinitesimal shear modulus. The main difference between the molecular approach and the phenomenological approach is that, in the first approach, this parameter is given in terms of microscopic quantities, whereas in the latter, this is just a material modulus whose meaning within the context of mechanical response may be clarified a *posteriori*.

The three-dimensional model associated with the FJC and WLC is more complex, the simplest among the models that take into account non-Gaussian effect is the Gent model, where the strain-energy density, *W*, is given as
2.7
where *μ* is the infinitesimal shear modulus and *J*_{m} is the limiting chain parameter. It is clear that we must have *I*_{1}−3<*J*_{m}, and that when , we recover the classical neo-Hookean model (2.6) from the Gent model. The status of this model has been clarified in a series of papers by Horgan & Saccomandi (for a survey see Horgan & Saccomandi (2006*a*)).

It is clear that equations (2.6) and (2.7) are only two examples of possible strain-energy functions. Another possibility is the Mooney–Rivlin strain-energy density function
2.8
where *μ* is the infinitesimal shear modulus and *β* is a constant satisfying −1/2≤*β*≤1/2. (When *β*=1/2, we recover the neo-Hookean model (2.6).)

With any constitutive equation given by (2.1), it is possible to associate a *generalized* shear modulus. To illustrate this fact, let us consider the simple shear deformation
(Here, *γ* is the amount of shear.) In this case, *I*_{1}=*I*_{2}=3+*γ*^{2} and the corresponding shear stress is given by
and
2.9
is the generalized shear modulus. The infinitesimal shear modulus is defined as

For the neo-Hookean model (2.6), we find that *Q*≡*μ*, but for the Gent model (2.7), we find that
2.10
Therefore, we find that and . However, we note that , a result that it is pertinent to finite-chain extensibility.

In a nonlinear theory, it is also possible to introduce the tangent modulus while considering the deformation of simple extension. Young’s modulus is defined as the linear limit of the tangent modulus. If the nonlinear model is compressible, we may also consider the Poisson function, whose linear limit is the classical Poisson ratio (Beatty 1987).

In the literature that explores the effect of pressure on the material moduli of rubbers, the authors assume the shear modulus (or equivalently if the incompressibility constraint is in force, Young’s modulus) as a function of the pressure, *p*. We point out that it is not strictly correct to use the terminology shear modulus as a function of the pressure, as shear modulus is a term that refers to a constant that appears in the constitutive relation for a linearized elastic solid. For this reason, when we explore the possible dependence of the material moduli on pressure, we have to consider the generalized shear modulus.

Unlike the generalization of the classical incompressible Navier–Stokes fluid with a constant viscosity to one whose viscosity depends on pressure, one cannot generalize the classical model for incompressible elastic solids, such as the neo-Hookean or Mooney–Rivlin solid, to materials whose moduli depend on the Lagrange multiplier. Indeed, if we consider the modified neo-Hookean model whose Cauchy stress tensor takes the form
2.11
and we consider the simple shear deformation, we obtain the generalized shear modulus to be
Unlike the classical Navier–Stokes fluid theory, it is important to realize that *p* in equation (2.11) does not represent the mean normal stress, as the trace of ** B** is not zero. It is merely the Lagrange multiplier.

Now, suppose we subject an initially undeformed and unstressed material, whose constitutive equation is described by equation (2.11), to a simple shear defomation and a pressure *p*_{1}. At the end of the shear, let us increase the pressure to *p*_{2}. Let us reverse the shear and return the body to its initial configuration and finally let us reduce the pressure back to *p*_{1}. The body would have undergone a cycle in both deformation and stress, but the body would have done net work. This fact is in contradiction with elastic behaviour, as has been explained in detail, for example, in Rajagopal & Saccomandi (2006). For this reason, we have to relax the incompressibility constraint, allow the material to be a compressible material and consider a new model given by
where now λ and *μ* are constitutive functions of their argument. This model is not to be confused with equation (2.11), for what we have now is a compressible elastic material in which the material modulus depends on the mean normal stress. The new constitutive equation is not an explicit model for the stress in terms of the deformation gradient. On the other hand, the stretch ** B** can be expressed explicitly in terms of the stress. At this juncture, it is worth observing that one cannot replace the dependence of the modulus

*μ*on the mean normal stress to that on the volumetric deformation (volumetric strain), as the relationship between the mean normal stress and the volumetric deformation needs not be one to one. In fact, when dealing with implicit constitutive relations in general, and even with regard to the expression relating the stress and the Cauchy–Green stretch

**given earlier in this paragraph, we are primarily interested in the relationship between the mean normal stress and the volumetric strain not being one to one. It is worth remarking that the paper by Truesdell & Moon (1975) is devoted to delineating the class of models for which the classical relationship between the stress and the stretch is invertible. We are interested primarily in the relationship between the stress and the stretch being non-invertible. Of course, for the class of models where one does have invertibility of the relationship between the mean normal stress and the volumetric deformation, we do not have an implicit model, but an explicit model, for in this case, the stress belongs to the class of classical compressible Cauchy elastic materials.**

*B*In the paper by Jones Parry & Tabor (1974), the authors explained, from the molecular point of view, the reasons as to why the shear modulus must depend on the pressure that the specimen is subject to, and this translates to the mean normal stress within the body. Indeed, because in the entropic picture, the elasticity of rubber is associated with the mobility of the macromolecular chains, it is clear that the elasticity is strongly influenced by the available free volume, i.e. the interstitial volume between the chains. If the polymer is compressed, then the free volume will be reduced, the intermolecular forces between segments will increase and the shear modulus would be expected to increase. This means that we have to *relax* the incompressibility assumption and that a purely entropic description is no longer sufficient and an energetic contribution to the free energy must be considered.

This energetic part is associated with the variation of volume, and therefore the incompressibility constraint is no longer compatible with the experimental data. We point out that the experimental data allow for pressures up to 6000 atm, and rubber may be considered incompressible up to around 200 atm.

To develop a model that takes into account all these features, we consider an implicit constitutive equation as in equation (1.2). It is convenient to follow such an approach from the very beginning. If we require that in equation (1.2) is an isotropic function, it immediately follows that
2.12
where the material functions *α*_{i},*i*=0,1,…,8, depend upon
2.13
(By the use of the Cayley–Hamilton theorem, *B*^{2} may be replaced by *B*^{−1}.)

Of course, representations such as equation (2.12) are too general to be useful, as it is nearly impossible to fashion an experimental programme within which all the material functions can be determined. Hence, we have to pick a simpler subclass of equation (2.12). Several strategies are possible. Let us consider the class of models represented by
2.14
where *γ*_{i},*i*=0,1,2, are yet dependant on the quantities indicated in equation (2.13). Notice that equation (2.14) is an implicit model in virtue of the material functions *γ*_{i}.

For a hydrostatic state of stress, a dependence of the shear modulus on the pressure means a dependence of the shear modulus with respect to the invariant *Θ*≡tr(** T**)/3 because, for an hydrostatic state of stress,

*T*

_{11}=

*T*

_{22}=

*T*

_{33}=−

*p*.

Elastomers may experience high stresses but low strains, and for this reason, if we consider the classical linear measure of strain
2.15
where **u** is the displacement vector, we can simplify equation (2.14).

First of all, we note that to obtain the classical linear and affine constitutive equations (i.e. not only linearity in ** ϵ** but also in

**) of isotropic linearized elasticity, it is sufficient to consider a special member of equation (2.14), 2.16 and to require where as usual λ and**

*T**μ*are the

*Lamé constants*. Now, if we consider a simple shear state of stress

**=**

*T**T*

_{12}

*e*_{1}⊗

*e*_{2}, we must have

*Θ*=0 and hence

*α*

_{1}=0, therefore from equation (2.16), we obtain and clearly

*μ*is the infinitesimal shear modulus.

Now, assume that the only non-zero stress component is *T*_{11}=*T*, where *T* is constant, then from the constitutive equation, we have *ϵ*_{11}=*T*/*E*, where
is Young’s modulus. This definition may be generalized to a nonlinear theory. Indeed, in the classical nonlinear theory as *T*=*T*(*ϵ*_{11}), Young’s modulus is obtained as the limit , where the derivative is usually denoted as the *tangent modulus*. In an implicit theory, it is still possible to compute the derivative d*T*/*d**ϵ*_{11}. When the model is such that , we may recover Young’s modulus considering the limit .

The last elastic modulus of interest in classical experiments is Poisson’s function, *ν*(*T*), a generalization of the classical linear Poisson’s ratio *ν*_{0}=λ/2×(λ+*μ*). In the problem under consideration, we obtain
2.17
This function measures the lateral strain component accompanying the extension in a given direction.

From equation (2.16), it is clear that the dependence of the shear modulus, Young’s modulus or Poisson’s ratio on *Θ* is possible by considering an explicit expression for the stretch (strain) in terms of the stress or an implicit model. Paterson (1964) carried out experiments in a pressure vessel into which a piston is introduced, to determine stress–strain curves for tensile loading. The apparatus is not described in detail, but Paterson measured Young’s modulus. Typically, for natural rubber, he measured a sort of linear increase from about 1.5×10^{7} to 2×10^{7} dyn^{2} cm^{−2} up to 4000 atm and a rapid 1000-fold increase between 4000 and 6000 atm, after which the modulus becomes relatively insensitive to pressure changes. Therefore, the plot of Young’s modulus versus pressure is *S*-shaped. Clearly, in this range of pressure, rubber may not be considered incompressible (the volume variations are more than 10%). In Jones Parry & Tabor (1973), the shear modulus is measured and in a range of pressure that goes up to 18×10^{−3} psi a linear dependence of the shear modulus on pressure is confirmed. In this case, the deformation involved is shear. What happens during the 1000-fold increase is clearly related to the glass transition of rubber. Indeed, the final Young’s modulus is comparable with the modulus of ebonite. It is not possible to model the full glass transition via the theory of elasticity, as a permanent set in the deformation is observed. For this reason, we shall concentrate our attention mainly on the initial phase to describe the pressure rise (the first half of the *S*-shaped curve).

In the following sections, we shall consider a simple method to obtain mathematical models of this mechanical phenomenon. First, we shall derive elastic models that are capable of describing a linear dependence of the various generalized moduli on pressure. Then, using the ideas of limiting chain extensibility, we shall discuss a simple model for the sharp increase in the material moduli that is observed experimentally.

## 3. Models with linear dependence on the pressure

The aim of this section is to derive a specific subclass of the class of models (2.16) where the shear and Young’s moduli depend linearly on the pressure, and to describe the experiments in Jones Parry & Tabor (1973) and Paterson (1964) up to around 4000 atm. To derive such a model, we first consider a state of shear superimposed on a hydrostatic pressure and then a simple tension superimposed on a hydrostatic pressure.

We start with the following stress field:
where *p* and τ are constants. We then obtain that *Θ*=−*p*,
3.1
and
3.2
By using the usual notation *α*_{2}=(2*μ*(*Θ*))^{−1}, if we choose
3.3
we find that *μ* depends linearly on the pressure *p*, as predicted by the experiments of Jones Parry & Tabor (1973). We point out that this result is valid, in the approximation of small strains, in the case of both small and large stresses. Unluckily, we cannot deduce from the data in Jones Parry & Tabor (1973) the amount of stress and strain that are attained.

The situation with regard to the experiments described in Paterson (1964) is more complex. Here, where we have to consider a state of simple tension superimposed on a pure hydrostatic pressure, i.e.
where *p* is the hydrostatic pressure and *T* the tensile stress. In this case,
3.4
It then follows from our definition of *α*_{2}(*Θ*) directly after equation (3.2) that
3.5

We shall assume that the experiments are performed for very large values of the pressure *p*, but for small *T*. Paterson (1964) remarked that the measured values of Young’s modulus are at low strains. Then, because *Θ*=*T*/3−*p*, if we restrict our attention to a linear setting, it is possible to express *ϵ*_{11} as
3.6
What is observed in the experiments of Patterson (1964) is that1
3.7
A possible choice of *α*_{1}=*α*_{1}(*Θ*) compatible with the requirement in equation (3.7) is given by
3.8
where *C* is an arbitrary constant that may be fixed by considering what happens when *T*=0. The description of the modulus of compression before the simple tension experiments start seems to have been completely omitted in the data reported in Paterson (1964) and Jones Parry & Tabor (1973), where Poisson’s ratio is hypothesized to be constant; but this assumption is clearly at odds with the proposed aim of the experiment. (In the range of pressures that it is considered, rubber is clearly a compressible material.)

The constitutive equation that we propose as a possible alternative is 3.9

If we consider a state of simple tension such that *T*_{11}=*T* with the remaining stress components being zero, it is possible to compute the tangent modulus for such a model. In this case, *Θ*=*T* and
3.10
Clearly, in this case, the tangent modulus does not depend linearly on *T*. If we wish to develop such a model, we start again from
3.11
and we have that
and by requiring
we obtain
3.12
where *C* is a constant to be determined; in this case, considering that for *E*_{1}, , we have to recover the classical results corresponding to the linearized theory of elasticity. Therefore, we would need
It then follows that the constitutive relation that we are considering takes the form
3.13

It is clear that the determination of the parameters *E*_{0} and *E*_{1} is quite simple using the first part of the data plotted in figure 1. To determine the other parameters, we need more experiments.

## 4. Implicit theories and limiting chain extensibility

In this section, we wish to model the behaviour observed in experiments at the transition pressure. This phenomenon has been well described in Paterson (1964), at least qualitatively. This phenomenon is similar to the one we encounter in the case of limiting chain extensibility, there is a critical pressure at which the free volume reduces, leading to a marked increase of the tangent modulus. When this critical pressure is reached, we observe that the elastic range is small, a yield stress arises accompanied by a permanent set. The transition pressure is defined as the inflection point in the curve of the tangent modulus versus pressure. The aim of our model is to describe these curves up to the transition pressure, by postulating that the transition point is a vertical asymptote. We are not interested in what happens after the transition pressure because a model for glass transition needs to be put into place, and the resulting model is not that of an elastic body.

It has been pointed out that the basic ingredient of the phenomenological models corresponding to non-Gaussian molecular models is the use of rational functions instead of polynomial functions. This is confirmed by the Gent model that was introduced earlier. Therefore, the basic idea is to start again from the stress corresponding to a pure shear superimposed on a hydrostatic pressure and to require that the shear modulus is of the form
4.1
where *Θ** is the maximum pressure, i.e. the transition pressure. Now, as before, the true problem is the modelling of the tangent modulus. In order to illustrate the phenomenon of limited extensibility easily and with clarity, we choose a small strain response. In this case,
4.2
and what we want to obtain is
4.3
A possible choice of *α*_{1}=*α*_{1}(*Θ*) compatible with this requirement is given by
4.4
where *C* is the constant that must be tuned with the data for the modulus of compression. It follows that the constitutive model with the desired features is given by
4.5

Our method is quite general, and it allows us to also develop a model that is able to reproduce (qualitatively) the behaviour we wish to model in the range of pressures of interest. We postulate that (we do not have experimental data for the generalized shear modulus)
4.6
We suppose that
and we require that the tangent modulus is given by a *kink*,
4.7
To use this formula to fit the data in figure 1, first of all it is convenient to shift the data such that they are symmetrically arranged with respect to the independent variable axis, then via *E*_{2}, we have to arrange the step of the kink; with *E*_{1}, we have to centre the kink and with *E*_{0}, we arrange the thickness of the step of the kink.

The choice *α*_{1}=*α*_{1}(*Θ*) compatible with equation (4.7) cannot be computed in closed form, but it may be indicated as

Clearly, the fact that this model cannot be expressed in closed form does not detract from its applicability. Also, from the mechanical point of view, the fact that we are able to reproduce the experimental curve beyond the glass transition with an elastic model is only metaphoric, as a real material during and beyond glass transition will produce entropy and cannot be well approximated by an elastic model. On the other hand, this metaphoric exercise has allowed us to develop a model that describes behaviour which is similar to that exhibited by materials with limiting chain extensibility. A horizontal asymptote for the stress corresponds to a vertical asymptote for the strain. Therefore, this picture is clearly compatible with the strain hardening phenomena in natural rubber, which is due to the crystallization of the material.

## 5. Concluding remarks

In this paper, we have examined the dependence of the material moduli with respect to the mean normal stress with a view towards explaining the experimental data that are avalaible for elastomeric materials. It is clear that the basic theoretical ideas that we have explored, concerning fully implicit models or for that matter models wherein one has an explicit expression for the Cauchy–Green stretch in terms of the stress, have relevance to other classes of materials. We hope that we have made a compelling argument that it is only via explicit nonlinear models for the strain in terms of the stress or truly implicit models would it be possible to explain the experimental data. On the other hand, we have also shown that the data reported in the literature are not suffcient to determine, in a unique way, the mathematical models that could be used to describe the observed response. Indeed, it is necessary that a careful characterization of the response of the material to a hydrostatic pressure field be matched with the direct measurement of the generalized Young’s or shear modulus. We feel that explicit nonlinear models for the strain in terms of the stress or implicit models relating the stress and the strain warrant further scrutiny.

## Footnotes

↵1 At first sight, equation (3.7) may be confused with Hartig’s law; however, it should not be so confused. As may be checked (see Bell 1973), Hartig’s law takes the form d

*T*_{11}/d*ϵ*_{11}=*E*_{0}−*b**T*_{11}, which is clearly different from equation (3.7). Moreover, here, we are within a nonlinear three-dimensional framework and we are not considering a material modulus, but a constitutive relation.- Received August 10, 2009.
- Accepted August 24, 2009.

- © 2009 The Royal Society