The Gent model for rubber-like materials: An appraisal foran ingenious and simple idea

Giuseppe Puglisi a, Giuseppe Saccomandi b,n

a Dipartimento di Scienze dell'Ingegneria Civile e dell'Architettura, Politecnico di Bari, 70126 Bari, Italyb Dipartimento di Ingegneria, Università degli Studi di Perugia, 06125 Perugia, Italy

a r t i c l e i n f o

Article history:Received 5 March 2014Accepted 7 May 2014Available online 4 June 2014

Keywords:Gent modelLimiting chain extensibilityDamage

a b s t r a c t

We review the main aspects of the celebrated Gent constitutive model for rubberlike materials.Emphasis is placed on the case of damageable materials describing possible damage and deformationlocalization.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

During 1996 Alan Gent publishes a short note [16] where heproposed a new constitutive equation for the non-linear elasticbehavior of rubberlike materials. Due to its formal simplicity, thismodel reached a great popularity in various scientific communitiesinterested in large elastic deformations of solids. The aim of thisnote is to present an appraisal of this simple constitutive model byshowing its effectiveness in describing the behavior of traditionaland many new elastomeric and biological materials. We remarkthat the Gent model [16] has been applied successfully in severaldifferent fields of material science and in the following we referonly to a small subset of the many possible applications of thesimple idea behind the Gent model. For a more detailed survey ofthe scientific literature related to the Gent model, we refer thereader to the paper by Horgan in this same volume [18].

Rubberlike materials are typically characterized by their abilityto undergo very large reversible deformations. As a consequencethey are modeled (at least in appropriate ranges of stretches,temperature and time scales) as hyperelastic materials. A generaltreatment of such an approach was given by Rivlin [32] and laterby many other authors (see, for example, [25]). This peculiarmacroscopic behavior arises from the structure of rubber-likematerials at the micro- (network) scale and we refer to theclassical book by Treloar [34] for an analysis of the amorphousproperties of such macromolecular materials at these scales.

To be more explicit, let us consider a deformation x̂ of a bodyΩ, Ω 3 X↦x¼ x̂ðXÞ and let F ¼Gradðx̂Þ be the deformation

gradient. Let then B¼ FFT be the left Cauchy–Green strain tensorand let

I1 ¼ trðBÞ; I2 ¼I21�trðB2Þ

2; I3 ¼ det B

be its principal invariants. If a material is hyperelastic and isotropic[25] we may introduce a strain-energy density function W ¼WðI1;I2; I3Þ. Moreover, because the bulk modulus of rubber is typicallysignificantly higher than the tensile modulus, rubber and rubber-like materials are often modeled as incompressible materials withan energy density depending on only the first two invariants:W ¼WðI1; I2Þ.

As a result, the Cauchy stress tensor T is given by

T ¼ �pIþ2∂W∂I1

B�2∂W∂I2

B�1; ð1:1Þ

where p is the Lagrange multiplier associated with the incompres-sibility constraint. Moreover, in the absence of body force, theequilibrium equations are

div T ¼ 0 ð1:2Þ

(here div is the usual divergence operator with respect to x).Schematically, we may consider three distinct classes of con-

stitutive approaches proposed in the framework of non-linearelasticity for rubberlike materials. The first class proposed phe-nomenological constitutive laws, relating them both to the experi-mental observations and to the known molecular structure of thematerial. In this class different simple microstructure based con-stitutive equations were taken into account to describe thethermo-mechanical behavior of rubber-like materials. The mostrepresentative strain energy density function in this class is for sure

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International Journal of Non-Linear Mechanics

http://dx.doi.org/10.1016/j.ijnonlinmec.2014.05.0070020-7462/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail addresses: giuseppe.puglisi@poliba.it (G. Puglisi),

giuseppe.saccomandi@unipg.it (G. Saccomandi).

International Journal of Non-Linear Mechanics 68 (2015) 17–24

the basic neo-Hookean model

W ¼ E6

ðI1�3Þ; ð1:3Þ

depending on one single material small strain tensile modulus Ethat has been related in various ways to the molecular materialproperties. This macroscopic constitutive law results as a homo-genized behavior of an isotropic network of ideal Gaussian chains.An introduction to this class of models can be found in the classicalwork of Treloar [34].

Successively the axiomatic theory of Continuum Mechanics[35] delivered a systematization of the so-called representationproblem by a systematic use of the methods of Linear Algebra (seefor example [32]). Thus, the second class of constitutive modelsfocused on the mathematical determination of specific forms of thegeneral representation formula (1.1). Several new interestingmodels resulted from the research in this direction among whichthe most widely adopted is the Mooney–Rivlin model consideringan energy function depending linearly also on the second invariant

W ¼ C1ðI1�3ÞþC2ðI2�3Þ: ð1:4Þ

Here C1 and C2 are two material moduli. Of course (1.3) is a specialcase of (1.4).

The third, more recent, class of constitutive theories resultedfrom a critical reassessment of all previous theories searching for adeeper connection between the macroscopic response of rubber-like materials and the behavior at the mesoscopic scale. The aim ofthese new models was both to cover the gap between thetheoretical predictions and the experimental behavior and togeneralize the elastic constitutive theories to more general mate-rial behaviors (e.g. residual strains and Mullin and Payne effects).In this case the main contribution was the introduction of modelsbased on the inverse Langevin function defined as LðxÞ ¼ cothðxÞ�1=x [3].

To understand the role of this function, we briefly recall thatrubberlike materials are amorphous materials consisting of verylong flexible chainlike molecules constituted by a backbone ofmany non-collinear single valence bonds around which, due tothermal agitation, barrier free rotations are possible. Rubberymaterials are composed of a network of thousands of these chainslinked through interchain bonds. These topological links deliverthe material a solid behavior while allowing the molecules tochange their shape easily and continuously at normal tempera-tures due to their Brownian motion. If a tensile force is applied tothe network then the chains assume a somewhat oriented con-figuration. Several interesting material behaviors descend fromthis microscale material structure. For example, if the macromo-lecules in the network are highly regular they are able to finelypack (for stereochemical reasons) so that they easily crystallizedue to van der Waals forces. In this case the polymeric materialbehaves like a crystalline solid exhibiting ductile and plasticbehavior. In other cases the monomers may hardly move due toviscous, inertial and entanglement effects under thermal agitationand therefore the polymer behaves in a glassy way.

The ability of the Gent model to describe important propertiesof rubberlike materials can be substantially noted by the followingobservation of the behavior of these materials at the networkscale. As described above, rubber materials are constituted by longchains energetically free to switch between a wide variety ofconformations through transitions' phenomena which are gov-erned mainly by the statistics of random processes. For this reasonwe say that the elasticity of rubber-like materials is mainlyentropic in nature, because their elasticity is regulated by theconfigurational entropy of the chains and not by enthalpic energycontributions. In particular, the statistics that governs the con-formation of the macromolecules over short chain lengths is

Gaussian, but, when we approach something like one-third ofthe fully stretched length of the chain (contour length), non-Gaussian effects become non-negligible. Macroscopically this isreflected in a large experimental discrepancy of the predictionsof a model such as (1.3) based on the Gaussian assumption, orits phenomenological generalization (1.4), with experimentalobservations.

To overcome this problem, two main methods have beenproposed. In one direction more complex phenomenological formsthan (1.4) have been proposed. Often the main drawback of theseconstitutive models is that they contain a large number ofempirical parameters that are not always easily determined byfitting procedures with experimental data. The advantage is thatthey are ready to use in the commercial finite element codes asgreatly appreciated by engineers.

The second, successful, direction of research was dedicated tothe refinement of the molecular description of the macromolecu-lar network and to the consideration of the non-Gaussian char-acter of the random processes of the chains conformations, takinginto account that the probability density function for the end-to-end distance of the chains has compact support, and not infinitetails as in the Gaussian case, since the macromolecules have finitecontour lengths. These studies generated several models, amongwhich the most popular is the Arruda–Boyce eight chain model [2]which is indeed based on the inverse Langevin function. Theadvantage of such models is that they usually contain fewconstitutive parameters and that such parameters are directlyconnected to the microscopic properties of the material. On theother hand the main disadvantage is the numerical and analyticalcomplexity of such models and the fact that the stress–strainrelationship is usually not expressed in closed explicit form.

The Gent constitutive equation has to be considered in suchframework (Paraphrasing Gent himself [16])

a simple, two-constant, constitutive relation, applicable over theentire range of strains. …,

Alan Gent, using the typical British understatement, justifies theintroduction of his model only on the basis of the simplemathematical feasibility. In reality, it was an ingenious idea thatallows, especially because of its mathematical simplicity, explora-tion of many mechanical issues associated with the elasticity ofrubber-like materials. These aspects are reviewed in detail in thepaper by Horgan in this issue [18] and we refer to this paper for amore detailed survey of the various results obtained in theliterature about the Gent model [18]. The aim of this paper is toshow the main features (and limits) of the Gent model byconsidering the simple deformation classes of simple shear andrectilinear shear. Then we show how the idea of Gent led thederivation of a new generation of models useful also in thedescription of important biomaterials like spider silk or proteinmacromolecular biomaterials.

2. The Gent model

The energy density function proposed by Gent in [16] forincompressible, isotropic, hyperelastic materials is, using his exactformulation,

W ¼ �E6

Jmln 1� I1Jm

� �: ð2:1Þ

Here E is the small strain tensile modulus that, for incompressiblematerials, is related to the infinitesimal shear modulus μ by therelation μ¼ E=3. Thus, since W depends on the only first invariantof B, the Gent model belongs to the class of the generalized neo-Hookean materials such that W ¼WðI1Þ.

G. Puglisi, G. Saccomandi / International Journal of Non-Linear Mechanics 68 (2015) 17–2418

The important parameter Jm represents the maximum value ofI1 with the energy that grows to infinity as I1-Jm. To betterunderstand its mechanical meaning we may recall a simple resultby Kearsley [22] that states that I1 “is equal to three times thesquare of the stretch ratio of an infinitesimal line elementaveraged over all possible orientations”. As a consequence Jm canbe seen as a fair (and natural) measure of the average contourlength of the chains composing the polymeric network.

Observe also that, as in the entropic models for molecularchains, showing that the non-Gaussian models converge to aGaussian one when the contour length grows to infinity, whenJm-1 (2.1) reduces to (1.3) corresponding to the averagedbehavior of an isotropic network of Gaussian chains. We may thensee the Gent model as a simple and direct generalization of theneo-Hookean strain energy density that takes into account thenon-Gaussian character of the macromolecular chains with finitecontour length.

The stress–strain relation for (2.1) is easily obtained as

T ¼ �pIþ E=31� I1=Jm

B; ð2:2Þ

and the generalized shear modulus function μ̂ is given by

μ̂ ¼ E=31� I1=Jm

ð2:3Þ

and is a function of the only first invariant.

2.1. Simple shear

To describe the behavior of the Gent model we begin by fixingthe ideas on the simple, analytically clear case of simple sheardeformations

x¼ XþκY ; y¼ Y ; z¼ Z: ð2:4ÞHere κ is the shear deformation parameter. This deformation isprobably the simplest example of finite deformation although it isa deformation which is difficult to produce experimentally due tothe required boundary conditions such as surface tractions.

In this case, given the energy W ¼WðI1; I2Þ, the components ofthe Cauchy stress are given by

T11 ¼ �pþ2ð1þκ2ÞW1�2W2; T12 ¼ 2κðW1þW2Þ;T13 ¼ T23 ¼ 0;

T22 ¼ �pþ2W1�2ð1þκ2ÞW2; T33 ¼ �pþ2W1�2W2; ð2:5Þwhere Wi ¼ ∂WðI1; I2Þ=∂Ii, i¼1,2.

In the case of Gent material (2.2) the shear stress is given by

T12 ¼ μκ

1�ðκ2þ3Þ=Jm: ð2:6Þ

To point out the effect of the parameter Jm we observe that

T12-1 when κ-ffiffiffiffiffiffiffiffiffiffiffiffiffiJm�3

pand

T12-μκ when Jm-1:

The behavior is reproduced in Fig. 1 where we show acomparison also with the 8-chain Arruda–Boyce model [3] forwhich after easy calculations one gets

T8c12 ¼

L�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þk2

3n

s0@

1A

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þk2

3n

s μk;

where n is the number of links in the chain.

We observe that the Gent model behaves as the neo-Hookeanmodel at low strain. Moreover the model, while keeping itsfundamental analytical simplicity, approximates very well themacroscopic 8-chain model. This last model is deduced from amolecular chain model of the inverse Langevin type as attainedunder a freely jointed chain behavior hypothesis [3].

It is important to observe that the generalized shear modulus forthe Gent material is a rational function. In this respect we observe thatin typical phenomenological models polynomial (Taylor series)approximations are considered for the constitutive laws. As a result,these models lose the possibility of describing well the asymptoticbehavior assigned by the contour length threshold. On the other hand,it is possible to introduce better approximations using rationalfunctions of given orders such as the Padé approximants [4] that areusually more accurate than Taylor expansions and are able to take intoaccount the presence of singularities (such as the limiting value of theadmissible amount of shear in (2.6)). The first to use a rationalapproximation for the inverse Langevin function was, to the authorsknowledge, Treloar [34]. Moreover, a Padé approximation was firstintroduced by Cohen in 1991 [5] and required the introduction ofsixth-order terms. Interestingly, later Horgan and Saccomandi in [20]realized that by fixing Jm (the maximum average contour length) thePadé approximant of the eight-chain model proposed by Arruda andBoyce [2] delivers exactly the Gent model. This result gives a naturalmolecular interpretation of the Gent model.

2.2. Limits of the Gent model

As typical of very simple models, the constitutive assumption(2.1) was an ingenious intuition, but because it uses only twoconstitutive parameters to describe many properties of a hugeclass of complex statistical models, it has some important limits. Inthis section we briefly analyze some of them. We remark, anyway,that the limits we describe in the following are indeed shared withall members of the class of generalized neo-Hookean models, i.e.materials with an energy density depending on the only firstinvariant: W ¼WðI1Þ. Therefore, the resulting unpleasant featuresthat will be discussed in the following are shared also with modelsof the Arruda–Boyce type [2,3].

To be specific, consider again the simple shear deformation andthe associated universal relations [33] that are assigned by the twotrivial conditions T13 ¼ T23 ¼ 0 plus the celebrated Rivlin universalrelation

T11�T22 ¼ κT12: ð2:7ÞAs a result, if experimentally (2.7) is not satisfied then one mayconclude that the material under investigation is not an isotropicCauchy elastic material.

0 1 2 3 40

5

10

15

20

25

30

k

neo−Hookean8−chain

Gent

T12ˆ

Fig. 1. Comparison of the shear stress/strain curves for the Gent model, the neo-Hookean model and the 8-chain model. Here the values Jm¼21 and n¼7 have beenchosen to let the Gent and 8-chain model attain the same limiting extensibility.

G. Puglisi, G. Saccomandi / International Journal of Non-Linear Mechanics 68 (2015) 17–24 19

Consider now a rectangular elastic body, under the sheardeformation (2.4) and suppose that the two faces perpendicularto the Z direction respect the boundary conditions T33 ¼ 0. Using(2.5) we obtain

p¼ 2ðW1�W2Þ:As a result we have

T11 ¼ 2κ2W1; T12 ¼ 2κðW1þW2Þ;

T22 ¼ �2κ2W2; T13 ¼ T23 ¼ T33 ¼ 0:

Now we have three trivial universal relations T13 ¼ T23 ¼ T33 ¼ 0plus (2.7). If then we suppose that W ¼WðI1Þ (generalized neo-Hookean material), we obtain

T11 ¼ 2κ2W1; T12 ¼ 2κW1;

T22 ¼ T13 ¼ T23 ¼ T33 ¼ 0:

This means that the new trivial universal relation T22 ¼ 0 arisesfrom the peculiar case of generalized neo-Hookean materials.

By comparing these theoretical results with experimental datathe authors in Pucci and Saccomandi [28] and in McKenna et al.[17], could conclude that

it is shown that at moderate ratios, the eight-chain model does notprovide even qualitative agreement with the experimental stress–strain data for both the dry and swollen states.

Similar comments can be extended also to the Gent model.Despite these limitations, the role played in the development of

the theory of rubber-like elasticity by the Gent model has beenfundamental due to its simplicity and its relation with the inverseLangevin function models. As a result the Gent model has been, in

the last decade, a true motor towards interesting models able toconnect the macroscopic scale behavior to the mesoscopic struc-ture of polymeric networks. Horgan and Saccomandi (see [18]) andother co-workers have shown that using the model (2.1) it ispossible to solve analytically a huge class of boundary valueproblems. Moreover, the simple structure of the Gent model maybe generalized to obtain interesting three-dimensional continuumversions of various molecular models such as the widely adoptedworm-like chain model for biological materials [26].

2.3. Rectilinear shear

The main effect of the parameter Jm on the solution of aboundary value problem is here exemplified by considering thesimple case of rectilinear shear deformations

x¼ Xþ f ðYÞ; y¼ Y ; z¼ Z;

where f(Y) is a function that must be determined from the balanceequations.

To be specific, consider the rectangular body schematized inFig. 2(a) under a constant load q with fixed boundaries at theedges Y¼0,1. The averaged total potential energy of the system is(G ¼ G=μ¼ 3G=EÞ

G ¼Z 1

0�12

Jmln 1�κ2þ3Jm

� ��q f

� �dY; ð2:8Þ

where q ¼ q=μ¼ 3q=E and the shear κ is now defined as κ ¼ df =dY .If we consider the Dirichlet boundary conditions

f ð0Þ ¼ 0; f ð1Þ ¼ 0; ð2:9Þ

q

X

Y1

1f(Y)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

Y

f

q 10

q 100

q

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

Y

k

q 10q 10

q 100 q 100

q

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

Y

G

q 100q 100

q 10 q 10

Fig. 2. Rectilinear shear for a rectangular body of a Gent material (here we assumed Jm¼21). (a) Scheme of the deformation class and assigned load q, (b) displacement,(c) shear strain, and (d) energy density fields at different values of the load q .

G. Puglisi, G. Saccomandi / International Journal of Non-Linear Mechanics 68 (2015) 17–2420

by integrating the Euler–Lagrange equilibrium equations weobtain the balance equation

κ1�ðκ2þ3Þ=Jm

þqY ¼ c1 ð2:10Þ

where c1 is an integration constant.Observe that, since for Jm43 the energy density (2.1) is a

strictly convex function of κ and the coercivity conditionWZ Jm lnðJm=ðJm�2ÞÞþ Jm=ðJm�2Þκ2 is satisfied, the solution tothe BVP we are considering exists and it is unique in a suitabledeformation space [1].

To search for the explicit solution, we first observe that sincethe problem is symmetric around Y¼1/2 the considered BVP canbe recast as the standard IVP

κ1�ðκ2þ3Þ=Jm

¼ q12�Y

� �; f̂ ð0Þ ¼ 0:

Thus, by using simple algebraic methods, we may obtain byquadrature the exact solution in terms of the function

ΦðYÞ ¼J log jY�1

2j

� �� J arcsinh

JffiffiffiffiffiffiffiffiffiffiJ�3

pqj2Y�1j

!

2q

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2þ4ðJ�3Þq2ðY�1ÞYþðJ�3Þq2

q2q

as

f ðY ; Jm; qÞ ¼ΦðY ; Jm; qÞ�Φð0; Jm; qÞ: ð2:11ÞThis solution is represented in Fig. 2.The peculiarity of this solution, first proposed in [27], is

obtained in the limit q-1 (see Fig. 2). Indeed, while for theneo-Hookean material the solution blows up as q-1, theasymptotic solution for (2.11) is given by

f ðY ; Jm; q-1Þ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiJm�3

p2

ð1�j2Y�1jÞ;kðY ; Jm; q-1Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiJm�3

p;

GðY ; Jm; q-1Þ-1: ð2:12ÞObserve that in the limit of growing load the system tends to a

solution that is uniform everywhere up to a point where theamount of shear has a jump. A clear phenomenon of localization ofthe strain, but what is more important is that everything happenswith finite values of strain and stress with possible importantapplications in fracture mechanics: a first attempt in this directionhas been provided in [19] and with more details and interestinginsights in [23]. The results contained in such papers confirm thatthe use of the idea of limiting chain extensibility may be a possiblesolution for some of the paradoxes of classical fracture mechanics.

3. Beyond elasticity

As remarked above the constitutive parameter Jm is connectedto a mesoscopic characteristic of the polymeric material (contourlength). As a consequence it is natural to extend the Gent model(2.1) to capture stress-softening and healing effects related to thevariations of this mesoscale parameter. Specifically, we point outthat damage in polymeric and many biological materials repre-sents macroscopic homogenized effects of complex phenomena ofbreaking [10] and re-crosslinking [6] at the network scales relatedto the evolution during the history of deformation of the averagelength of the macromolecular chains. The analysis of such behaviorat network scales suggested the introduction of microstructureinspired approaches such as the ones proposed in [29,10]. Thesemodels have been shown to be effective in describing the complexhistory dependent dissipative behavior of rubberlike materials also

in non-trivial inhomogeneous deformation histories [9], showingthe ability of describing experimentally verified damage localiza-tion effects [11,7] or instabilities observed in rubberlike balloons[8].

In this perspective and in relation to the Gent model, weremark that recent advances in experimental techniques, thatallow the analysis of the behavior of single macromoleculethermomechanical behavior [30,31], showed the fundamental rolein the macroscopic dissipation of the continuous variations of thechain contour length. These variations are accompanied byentropy variations revealed at the micro-scale by a sequence ofstress drops in the chains. Based on these observations anenergetic microstructure inspired approach [12] or statisticalmechanics multiscale approaches [13] has recently been proposedto describe the dissipative behavior of macromolecular materialsundergoing unfolding phenomena. Despite the ability of thesemodels to deliver a direct connection between the microstructureproperties and the macroscopic behavior of these materials,fundamental in particular in the field of the design of new bio-inspired materials, the possibility of models as simple as the Gentmodel, with a clear connection with the microstructure propertiesof the material, appears appealing both for numerical computa-tions and for the clearness of the resulting analytical interpreta-tions. This suggests extending the Gent model to non-elasticbehavior of biological and rubberlike materials [21].

The aim of this section is to show how it is possible to modifythe constitutive equation (2.1) to take into account damage. In theframework of the internal variable thermodynamics, under thehypothesis of isotropic damage, we may simply introduce aninternal damage variable α with a new history dependent energydensity function W ¼WðF ;αÞ. In this approach the main problemis the deduction of the dependence of the energy density on theinternal variable itself and of its evolution equation. In [7,13] theauthors showed the possibility of deducing such information byconsidering multiscale approaches.

Another, more phenomenological approach is to considerevolution equations reflecting the macroscopic hysteretic anddamage behavior of the system [15] or to deduce, knowing theenergy density function, the evolution equation of the damagevariable based on a hypothesis of maximization of the dissipationrate (see [29]). To be more explicit, consider the usual definition ofthe dissipation rate

ξ¼ T � D� _W ¼ T�∂W∂F

FT� �

� Lþg _α ; ð3:1Þ

where L¼ _FF �1, the stretching tensor is defined as D¼ ðLþLT Þ=2and g≔�∂W=∂α is the generalized force working for the damageincrement [36]. To maximize the dissipation rate subjected to theisochoricity condition

L � I ¼ 0

and to the requirement

ξZ0; ð3:2Þ

representing the Clausius–Duhem dissipation inequality [36], weintroduce the Lagrangean function

L¼ ξþη1 T�∂W∂F

TT� �

� Lþg _α�ξ� �

þη2L � I:

The Euler–Lagrange necessary condition (modulo a relabeling ofthe Lagrange multipliers η1;η2) delivers the usual formula for theCauchy stress tensor of incompressible materials

T ¼ �pIþ∂W∂F

FT ; ð3:3Þ

G. Puglisi, G. Saccomandi / International Journal of Non-Linear Mechanics 68 (2015) 17–24 21

where p¼ �η2=η1 and the relationship

η∂ξ∂ _α

¼ g

where η¼ ð1þη1Þ=η1. As a result we have

ξ¼ g _α ¼ η∂ξ∂ _α

_α : ð3:4Þ

This equation defines the rate of damage depending on the rate ofdissipation of the material.

Since here we are interested in the description of materialsshowing a rate-independent behavior we assume that the drivingforce g¼ gðαÞ is independent of _α . Moreover, since with theprogression of the damage the average contour length of themacromolecular network increases we assume that

JmðαÞ ¼ J0mþαJ1m; αA ð0;1Þ ð3:5Þ

so that in this context the damage of the material coincides with avariation of the parameter Jm. Here Jm ¼ J0m represents the value ofJm for the undamaged material, whereas Jm ¼ J0mþ J1m is the value ofJm at damage saturation. As a result we have

g¼ gðI1;αÞ ¼I1þðJm� I1Þ log ð1� I1=JmÞ

6ðJm� I1ÞEJ1m40; ð3:6Þ

so that, using (3.1)–(3.3), we obtain the reduced dissipationinequality

ξ¼ g _αZ0; ð3:7Þ

showing that in this simple setting the Clausius–Duhem inequalityfor this simple model is respected iff damage grows

_αZ0

or, using (3.5), equivalently

_JmZ0:

Regarding the choice of the evolution law, it must be fixedthrough a phenomenological approach or deduced by an analysisof the behavior at the network scale [13]. Here we are interested indescribing deterioration effects in materials such as rubber show-ing a Mullins type damage effect [24] whose memory is restrictedto the only maximum values of the attained stretches. Then, basedon the Gent idea, on the microstructure considerations reportedabove and on the results in [13], we assume that the damagevariable α depends only on the maximum attained value of thefirst invariant, I1max. In particular we may simply choose a growingfunction of I1max (see again [13] for a microstructure justification ofthis choice or [21] for the application of this simple idea todescribe the Mullins effect). As a result the system undergoesdamage only when I1 ¼ Imax

1 , with a dissipation potential

D¼ gðα; Imax1 Þ _α :

The maximization of the entropy production associated with thistype of potential delivers the conditions (3.3), (3.4) and theconsistency condition

ðgðα; Imax1 Þ�gðα; I1ÞÞ _α ¼ 0

that, due to the fact that g grows with I1, ensures that damage isattained only when I1 ¼ Imax

1 . As a result we obtain the historydependent elastic domain IAð3; Imax

1 Þ.Now, if we fix again our attention on the simple shear

deformation class, we may for the sake of algebraic simplicityassume the following monotonic damage law, depending only on

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

k0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

50

k

T12

k0

0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

k0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

k

T12

k0

α

α

Fig. 3. Shear stress–shear strain diagrams for a damageable Gent material under simple shear. Here we assumed μ¼ 1, J0m ¼ 3:5; J1m ¼ 1:0. In (b) we show the behavior for amaterial with a damage function represented in (a) assigned by δ¼ 1 and n¼2. In (d) we show the non-monotone shear stress–strain corresponding to the damage functionin (c) corresponding to δ¼ 4 and n¼6.

G. Puglisi, G. Saccomandi / International Journal of Non-Linear Mechanics 68 (2015) 17–2422

the maximum previously attained value of the shear strain kmax:

α¼ 1�exp �βffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκ2max�κ20

q� �γ� �; ð3:8Þ

where β and γ are positive material parameters, and κ0offiffiffiffiffiffiffiffiffiffiffiffiffiJ0m�3

qis the activation shear threshold. As a result we deduce the shearstress–shear strain relation

T12 ¼

κμ

1�κ2þ3

J0m

if kmaxok0

κμ

1� κ2þ3

J0mþαJ1m

if kmaxZk0:

8>>>>>>>><>>>>>>>>:

ð3:9Þ

The corresponding stress–strain curves are represented inFig. 3(b, d) for the two different damage functions representedin Fig. 3(a, c), respectively. Interestingly, according to the choice ofthis function, two different scenarios of damage evolution can beobtained. If the contour length grows slowly as I1

max grows, thesystem shows a continuous damage evolution up to damagesaturation as shown in Fig. 3(c). If instead the contour lengthgrows fast the corresponding softening can prevail with a non-monotone stress–strain diagram. This possibility was evidenced inthe microstructure inspired damage model proposed in [11] wherethe authors showed the resulting possibility of damage anddeformation localization. The analytical variational approach inthis case (non-convex energy densities) can be found in [7] wheredamage localization for antiplane shear deformations wasdescribed.

4. Concluding remarks

In the theory of polymer mechanics many constitutive forms ofthe strain-energy density have been proposed, but only few ofthese constitutive equations delivered true breakthroughs. Anexample is the Mooney–Rivlin strain energy density function atthe basis of the Rivlin research on finite elasticity. Anotherexample is the Ogden strain energy density function, the firstfunctional form that allowed to fit in a careful way the experi-mental data for a variety of deformations and a significant range ofstrains. Such constitutive equations were at the base of clearadvancements in rubber constitutive modeling in various direc-tions. On the same pathway the Gent model, because of itsmathematical feasibility, was at the base of a new impetus ofresearch analyzing the multi-scale behavior rubber-like materials.

The use of models based on the inverse Langevin function isclearly restricted by the fact that even for very simple deformationfields it is not possible to obtain results in closed form. Onthe other hand, the Gent model which has the same accuracy infitting experimental data as the Arruda–Boyce model basedon the inverse Langevin function, allows closed form solutionsfor a huge class of deformations with a much clearer mechanicalinterpretation.

Moreover, the simple and clear mathematical structure of theGent model allows us to understand how it is possible to general-ize constitutive theories beyond classical Taylor expansions. Inter-estingly the damage extension of the Gent model here proposed,considering a history dependent parameter Jm, let us describecomplex phenomena such as the transition from homogeneous tolocalized damage configurations, relating them to microstructureproperties such as the rate of contour length variations of thechains. The clear interpretation of the material moduli involved inthe Gent model is the key ingredients for such an intuitive way ofextending this model.

We are sure that the Gent model will continue to stimulate inthe next decades many interesting researches in the field ofmacromolecular materials and, as we have tried to show in thispaper, not only in the framework of non-linear elasticity, but alsoin its extension to dissipative behaviors.

All people engaged in the mechanics of elastomeric materialshave to be grateful to the simple idea of Alan Gent a truegentleman who has always communicated his findings profession-ally, with enthusiasm, humility and an open mind.

Acknowledgments

This paper is dedicated to the memory of Alan Gent. G.P and G.S. have been partially supported by GNFM of Istituto Nazionale diAlta Matematica. G.S. is grateful to C.O. Horgan and E. Pucci forsharing many good ideas about the Gent model during the last15 years. We thank Ray Ogden for helpful criticisms.

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