Mécanique & Industries 3 (2002) 245–252

Strain energy density function for carbon black filled rubber vulcanizedfor industrial applications

Énergie volumique de déformation du caoutchouc vulcanisé et renforcéau noir de carbone pour des applications industrielles

Hocine Bechira,∗, Khaled Boufalaa, Yvon Chevalierb

a Laboratoire de Technologie des Matériaux et de Génie des Procédés, Université de Béjaïa, Route Targua Ouzemour, 06000, Béjaïa, Algérieb ISMCM-CESTI, Laboratoire d’ingénierie des Structures Mécanique et des Matériaux (LISMMA), Groupe Vibroacoustique,

3, Rue Fernand-Hainaut, F-93407 Saint-Ouen cedex, France

Received 20 September 2000; accepted 1 February 2002

Abstract

The hyperelastic behavior of rubber-like materials can be derived from the strain energy density functionW which is determined fromexperimental results obtained in homogeneous strain, such as uniaxial tension, pure shear and equibiaxial tension. In the particular case ofcarbon filled reinforced rubber, the shape of theW function can be identified from a minimum number of experimental tests. The form of theW function has been proposed in the literature only for large strain domain. We present in this paper a new strain energy density functionW

for carbon black reinforced rubber which can predict the mechanical behavior of the material at small and large strain. 2002 Éditionsscientifiques et médicales Elsevier SAS. All rights reserved.

Résumé

Les lois de comportement élastique non-linéaire des élastomères, peuvent être déduites de l’énergie volumique de déformationW ; on ditque ces matériaux sont hyper-élastiques ou élastiques de Green. L’énergie volumique de déformationW est identifiée à partir des résultatsexpérimentaux obtenus en déformations homogènes : par exemple, essais de tension uni-axiale, de cisaillement pur et de tension bi-axiale.Cependant, on peut déterminer l’énergie volumique de déformationW par un minimum d’essais. En effet, l’expression analytique de l’énergievolumique de déformationW a été proposée pour ce matériau, donnant de bons résultats seulement dans le domaine des grandes déformations.On présente dans cet article, une nouvelle énergie volumique de déformationW qui peut prédire le comportement mécanique du matériau,aussi bien dans le domaine des petites que des grandes déformations. 2002 Éditions scientifiques et médicales Elsevier SAS. All rightsreserved.

Keywords:Rubbers; Large strain; Strain energy density

Mots-clés :Caoutchoucs ; Grandes déformations ; Énergie volumique de déformation

1. Introduction

The use of rubber materials for engineering applicationsis very wide, including engine mounts, building and bridgebearings, vehicle door seals, tires, off-shore structure flexjoints. These applications: anti-vibration mounts, impact

* Correspondence and reprints.E-mail addresses:Bech_dz@yahoo.fr (H. Bechir),

Yvon.Chevalier@ismcm-cesti.fr (Y. Chevalier).

absorbers, and flexible couplings, are still major uses ofrubber materials. To design these structures efficiently it isimportant to know about the behavior of rubber material, itis normally classified as hyperelastic. Rivlin’s polynomialfunction [1] is one of most popular strain energy densityfunctionsW for rubber due its generality. This polynomialfunction is normally truncated to Mooney–Rivlin functionalforms for industrial applications [2]. The major disadvantageof using Mooney–Rivlin model is that the coefficients ofthe strain energy density functionW curve fitted from

1296-2139/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.PII: S1296-2139(02 )01166-1

246 H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252

Nomenclature

I1, I2 andI3 strain invariantsI identity tensorF deformation gradient tensorC right Cauchy strain tensorB left Cauchy strain tensorE Green–Lagrange strain tensorS second Piola–Kirchhoff stress tensor . . . N·m−2

σ Cauchy stress tensor . . . . . . . . . . . . . . . . . N·m−2

W strain energy density function . . . . . . . . MJ·m−3

λi (i = 1,2,3) extension ratio in thei directionλ extension ratio in direction of the applied forceγ simple strain shearφ reduced stress in uniaxial tension or

compression . . . . . . . . . . . . . . . . . . . . . . . . . N·m−2

ψ reduced stress in simple shear . . . . . . . . . N·m−2

uniaxial tensile test data cannot be applied to shear or morecomplicated deformation especially for carbon black filledengineering rubber. Another words, the material parametersdepend on the state of strain. Analysts who want to usethe described material law for designing rubber bodies haveto know values of the Rivlin constants. Rubber compoundsare different in their molecular synthesis and processing,consequently, their mechanical properties vary significantly.In fact, even material of the same chemical compositionshows different behavior due to that different industrialcharge. To use the model some more research has to be donein the direction to obtain more sets of these constants. Ourobjective is to search a strain energy density functionW withthe material parameters independent of the deformation, andwith the shape of the strain energy density function that canbe identified from a minimum number of experimental tests,like the uniaxial tension and the biaxial tension.

2. Basic equations

We consider the rubber-like materials as perfectly elas-tic; this assumption is valid only if the time dependent ef-fects (such as creep, relaxation and Mullins) are negligibleduring the experiment procedures. If we adopt a phenom-enological point of view, isotropic hyperelastic materials areconveniently represented in terms of a strain energy densityfunctionW [1,3].

The state of strain is characterized by the principalstretches:λ1, λ2 andλ3, or equivalently by the strain invari-antsI1, I2 and I3. A rubber can be considered to behavewithout any appreciable volume change and the assumptionof incompressibility is adopted. This condition requires thatλ1λ2λ3 = 1, and therefore,I3 = 1. The basic problem is toconstruct a strain energy density functionW , dependent onI1 andI2 only, consistent with the observed behavior of car-bon black reinforced rubber. As Rivlin [1] showed, the strainenergy density functionW of a homogeneous isotropic ma-

terial depends on the strain invariantsI1, I2 andI3 definedas

I1 = tr(C)= λ21 + λ2

2 + λ23

I2 = 1

2

[(tr(C)

)2 − tr(C)2]

= (λ1λ2)2 + (λ2λ3)

2 + (λ1λ3)2

I3 = det(C)= λ21λ

22λ

23

(1)

where tr(C) and tr(B) stand for the trace of the secondorder tensorsC andB, respectively.C = FTF is the rightCauchy strain tensor andB = FFT is the left Cauchy straintensor.F is the deformation gradient tensor,λi the principalratio, and the notationT indicates the transpose of a tensor.The stress–strain relation derives fromW and may be givenin terms of the second Piola–Kirchhoff stress tensor,S =det(F )FTσF , σ being the Cauchy stress:

S = ∂W

∂E= 2

∂W

∂C(2)

E = (1/2)(C − I) is the Green–Lagrange tensor, andI theidentity tensor.

S = ∂W

∂E= 2

∂W

∂C= 2

[∂W

∂I1

∂I1

∂C+ ∂W

∂I2

∂I2

∂C+ ∂W

∂I3

∂I3

∂C

](3)

or equivalently,

S =[(∂W

∂I1+ I1∂W

∂I2

)I − ∂W

∂I2C + ∂W

∂I3C

−1]

(4)

The general constitutive form is

σ = 2√I3

[(I2∂W

∂I2+ I3∂W

∂I3

)I − I3∂W

∂I2B−1 + ∂W

∂I1B

](5)

Volume changes in rubber-like materials are very smalland the assumption of incompressibility is a good approxi-mation if one considers the problem of plane stress: uniaxialtension, pure shear or biaxial tension are studied, the con-straint I3 = 1 is true identically through the material. Thestrain energy density functionW is then considered as afunction ofI1 andI2 only. The Cauchy stress tensor reducesto a function of the partial derivatives∂W/∂I1 and∂W/∂I2up to an arbitrary hydrostatic pressurep:

σ = −pI + 2

[∂W

∂I1B − ∂W

∂I2B−1

](6)

H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252 247

Rivlin [1] pointed out that the strain energy densityW(per unit original volume) for an isotropic material can beexpressed as a function of the three strain invariantsI1, I2andI3 = 1. The general form of a Rivlin-type strain energydensity functionW [1] is, therefore,

W =∞∑i,j=0

Cij (I1 − 3)i(I2 − 3)j (7)

whereC00 is set equal to zero to reflect zero strain energyin the unstrained state. Eq. (7) is a power series, which isusually truncated to the first few terms. Taking only the firsttwo terms yields the Mooney–Rivlin [4] material model:

W = C10(I1 − 3)+C01(I2 − 3) (8)

It appears that the obtained partial derivatives∂W/∂I1and∂W/∂I2 are constants. However, Kawabata and Kawai[5] stated that the derivatives, in general, are functions ofI1and I2 for carbon black reinforced rubber. Tschoegl [6]expressed the opinion that the failure of the Mooney–Rivlinequation to describe the behavior of rubbers must be theresult of neglecting higher-order terms in the expression (7).James and Green [7] developed the strain energy densityfunction W that includes five to nine terms from theseries (7). Haines and Wilson [8] selected the terms of thepower series after inspecting the stress–strain data. However,developing the strain energy at higher order does notguaranty obtaining the best predictions for stress–strain data.

3. Experiments

To determine the partial derivatives ofW(I1, I2) we mayuse experimental data to carbon black reinforced rubberwhich undergoes homogeneous deformations. We refer tothe experiments of Yeoh [9] which exhibit a vulcanizedrubber material containing 40 phr of carbon black undersimple tension, simple shear and uniaxial compression.Let us express the theoretical principal stresses for theseexperiments. The invariantsI1 and I2 are calculated withthe given stretch ratiosλ1, λ2 andλ3 for each experiment.The indeterminacy introduced in (6) by the hydrostaticpressurep is eliminated by the combination of two of theprincipal stresses.

3.1. Uniaxial tension and compression

We consider a rectangular block of rubber stretched byλ

in the direction of elongation, and letσ be the correspondingprincipal stress of the Cauchy stress tensor, the two otherprincipal stresses being zero:

σ = 2

(λ− 1

λ2

)(∂W

∂I1+ 1

λ

∂W

∂I2

)

λ1 = λ, λ2 = λ3 = 1√λ

(9)

I1 = λ2 + 2

λ, I2 = 2λ+ 1

λ2

3.2. Simple shear

In simple shear, one of the principal stretch ratios isheld fixed, i.e.,λ3 = 1. Thenλ1 = λ givesλ2 = 1/λ. Theprincipal stress associated to the direction of stretching, i.e.,the direction associated toλ1, is

σ = 2

(λ− 1

λ

)(∂W

∂I1+ ∂W

∂I2

)with

λ1 = λ, λ2 = 1

λ, λ3 = 1 (10)

I1 = I2 = 1+ λ2 + 1

λ2

If the corresponding shear strainγ is defined as

γ = λ− 1

λ(11)

then from Eq. (10) we obtain the equivalent relation:

σ = 2γ

(∂W

∂I1+ ∂W

∂I2

)

I1 = I2 = 3+ γ 2(12)

Also, when deformations are homogeneously applied, theprincipal stresses may be given in terms of the followingreduced stresses:

φ = σ

λ− 1/λ2 = 2

(∂W

∂I1+ 1

λ

∂W

∂I2

)

ψ = σ

γ= 2

(∂W

∂I1+ ∂W

∂I2

) (13)

3.3. Equibiaxial tension

A rubber is equally stretched in two orthogonal direc-tions, for instance, 1 and 2, andσ = σ1 = σ2 are the non-zeroprincipal stresses:

σ = 2

(λ− 1

λ5

)(∂W

∂ I1+ λ2 ∂W

∂ I2

)

λ1 = λ2 = λ, λ3 = 1

λ2

I1 = 2λ2 + λ−4, I2 = λ4 + 2λ−2

(14)

3.4. Biaxial tension

The rubber sheet is still stretched in the two directions 1and 2, but not by the same amounts; the associated principalstressesσ1 andσ2 are different and their theoretical formsare given by

σ1 = 2

{∂W

∂I1

(λ1 − λ−3

1 λ−22

) + ∂W

∂I2

(λ1λ

22 − λ−3

1

)}

σ2 = 2

{∂W

∂I1

(λ2 − λ−3

2 λ−21

) + ∂W

∂I2

(λ2λ

21 − λ−3

2

)}

I1 = λ21 + λ2

2 + λ−21 λ−2

2 , I2 = λ−21 + λ−2

2 + λ21λ

22

(15)

248 H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252

4. The shape of the functionW for carbon reinforcedrubber

Gregory [10] noted that a simple relationship existsbetween stress–strain data obtained in uniaxial tension,uniaxial compression and simple shear. Such a relationshipexists if:

(a) ∂W/∂I1 � ∂W/∂I2, and(b) ∂W/∂I1 is independent ofI2.

Data published by Yamashita and Kawabata [11] indicatethat∂W/∂I1 is much greater than∂W/∂I2 for carbon blackfilled rubber, and data from Kawabata and Kawai [5] showedthe same results for rubbers. These results showed that∂W/∂I1 and∂W/∂I2 decreased rapidly with increasingI1and had a minimum value in the region of small invariantI1, followed by an increase with increasingI1. In contrast,∂W/∂I2 increased rapidly, then decreased with increasingI2, after passing through a maximum at a low value ofI2.Recently, Yeoh [9] made the simplifying assumption that∂W/∂I2 is equal to zero. He experimentally showed thatthe reduced stress in uniaxial and simple shear decreased,then passed through a minimum and finally increased withincreasing (I1 − 3). He, thus, proposed the cubic strainenergy density functionW :

W = C10(I1 − 3)+C20(I1 − 3)2 +C30(I1 − 3)3 (16)

However, the assumption∂W/∂I2 = 0 is not valid at smallstrain. Treolar [12] and Killiam et al. [13] have demonstratedthat the Mooney–Rivlin model is the most general first-order equation givingW in terms of the invariants. Later,Lambert-Diani and Rey [14] assumed that∂W/∂I1 = f (I1)and∂W/∂I2 = g(I2) at large strain.

Yamashita and Kawabata [11] proposed the followingstrategy to build a strain energy density functionW forcarbon black reinforced rubber:

W = C1(I1 − 3)+C2(I2 − 3)+ γ (I1) (17)

where the first two terms on the right in Eq. (16) representthe Mooney–Rivlin model, Eq. (8), and the third termγ (I1)is a noninteger power function of(I1 −3)N(N = 3.3). In thefirst approximation of the strain energy density functionW ,the stress–strain curves show a poor approximation forequibiaxial extension. In order to obtain a best fit ofstress–strain curves, these authors suggest a second-orderapproximation forW , the termγ (I1, I2) depends on theproduct of(I1 − 3)n+1(I2 − 3)−m for which n andm arenot integers(n = 4.6, m = 1.3). With this type of strainenergy density functionW , Yamashita and Kawabata [11]demonstrated the ability of their law to represent dataaccurately for equibiaxial extension and uniaxial tension.However, in practice, analyzing the strain and stress usingfinite element analysis (FEM), the Rivlin model strainenergy density functionW , Eq. (7), is usually chosen forthe rubber components employed as structural material. This

model has been implemented in many commercial finiteelement codes. On the ground of the above models, weintroduce the functions:

∂W

∂I1= f (I1)=

i=∞∑i=0

ai(I1 − 3)i (18)

∂W

∂I2= f (I2)=

i=∞∑i=0

bj (I2 − 3)j (19)

Our problem is to search the good approximation of thesefunctionsf (I1) andg(I2). The response of material dependson the state of strain. Another words, different experimentsmust be taken into account to fit the functionsf andg. Butfrom experiment point of view, sets of experimental data areavailable, like uniaxial tension, biaxial tension and simpleshear. Our strategy requires to determine these functionsby only less data experiments; Lambert-Diani and Rey [14]advanced that we need only two groups of tests: the one-direction stretch like uniaxial tension and the two-directionstretch experiments like equibiaxial tension. According tothese authors, if we consider the family of one-directionstretching experiments, it appears that theI2 contributionto the stress is largely below theI1 contribution at largedeformations. We can assume that when the material isuniaxially stretched, the contribution ofg to the stress isnegligible; the functionf is then fitted using the dataexperiments of uniaxial tension.

4.1. Identification of the function∂W/∂I1 = f (I1) fromexperiment

The function f is determined from uniaxial tensionexperiment data. We assume that the reduced stress functionEq. (13) depends only on the function∂W/∂I1:

φ = ∂W

∂I1+ 1

λ

∂W

∂I2(20)

Setting∂W/∂I2 = 0, (19) gives direct expression off ,

φ = ∂W

∂I1= f (I1) (21)

Yeoh [9] shows that the tension and compression resultsdo approximate to a single curve. It is seen that the plot ofreduced stressφ (or f ) versus(I1 − 3) can be approximatedby a polynomial form function of(I1 − 3):

f (I1)=n=3∑n=0

an(I1 − 3)n (22)

where parametersa are calculated by a standard least-squares fit. We now integrate the derivative Eq. (21) inorder to obtain the dependence of the strain energy densityfunctionW on (I1 − 3) which is written as a third-orderapproximation:

W1 = C10(I1 − 3)+C20(I1 − 3)2 +C30(I1 − 3)3 (23)

H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252 249

Table 1The constants for different models

Model ParametersCij (MPa)

Mooney–Rivlina C10 = 0.13,C01 = 0.095Yeohb C10 = 0.23,C20 = −0.0028,C30 = 0.0055Present modelc C10 = 0.23,C01 = −0.0028,C20 = −0.0028,

C02 = 0.0448,C30 = 0.0055

a Solved by using Eq. (8).b Solved by using Eq. (16).c Solved by using Eq. (29).

The functionW1 is determined from the uniaxial tensionexperiment data. The experimental strain energy densityfunction Wexp per unit volume at initial configuration iscalculated by integrating the Cauchy stressσ with smallincrements given byε = d lnλ:

Wexp=∫ ε

0σ dε (24)

The method of least squares is employed to minimizethe sum of the squares of deviations between the measuredstrain energy density function and that described by Eq. (22)or (23). The sum of the squares of the deviations is presentedby

ξ =N∑k=1

{W

[(I1(k)− 3

),(I2(k)− 3

)] −Wexp}2 (25)

where N is the number of data points used, and theparametersCij are determined from

∂ξ

∂Cij= 0 (26)

Developing this last equation, we are lead to solving asystem of linear algebraic equations to obtain the numericalvalues of the parametersCij (Table 1).

4.2. Identification of the function∂W/∂I2 = g(I2) fromexperiment

Experimental results of Kawabata et al. [15] showed that∂W/∂I2 = g(I2) is negative in the domain of small strains(I2 < 5), and is a linear function versus(I2 − 3) at largestrains(I2 ≥ 5); we can assume that the expression ofg is

g = ∂W

∂I2=n=1∑n=0

bn(I2 − 3)n (27)

where parametersb are calculated by a standard least-squares fit. In order to obtain the dependence of strain energydensity functionW on (I2 − 3), we integrate Eq. (27):

W2 = C01(I2 − 3)+C02(I2 − 3)2 (28)

Hence, the strain energy density functionW is

W(I1, I2) = C10(I1 − 3)+C01(I2 − 3)+C20(I1 − 3)2

+C02(I2 − 3)2 +C30(I1 − 3)3 (29)

Fig. 1. Calculatedg(I2) from equibiaxial tension experimental of Jamesand Green [7] for natural rubber vulcanizate containing 40 phr of carbonblack.

Fig. 1. Détermination de la fonctiong(I2) à partir des résultats expérimen-taux de l’essai équi-biaxial de James et Green [7].

Two methods are developed for determining the constantsC01 andC02 (Eq. (28)) in the literature. The first one is givenby Yamashita and Kawabata [11] using the one-directionexperiment like pure shear for valuating the functiong.However, in our case, this method gives a poor correlationbetween the model and experimental results of equibiaxialtension; for identifying these constants, we applied thesecond method proposed by Lambert-Diani and Rey [14],using the results of equibiaxial tension. By substituting (21)for (14) the principal nonzero stress component becomes

τ = 2{f (I1)

(λ− λ−5) + g(I2)

(λ3 − λ−3)} (30)

where τ is the first Piola–Kirchhoff stress tensor,τ =(detF)σF−T, where σ is the Cauchy stress tensor. Thefunction is defined in (23) and its parameters, calculated onthe uniaxial tension data, are given in Table 1.

Therefore, functiong directly derives from (30):

g = τ

2(λ3 − λ−3)− f

λ2(31)

Considering the case of James and Green data [7], valuesof g are calculated replacing(λ, τ ) with the correspondingexperimental data, and are plotted versus(I2 − 3) in Fig. 1.The first point associated to relatively small deformations isstill neglected; the linear function is

g(I2)= b1(I2 − 3)+ b0 (32)

where parametersb are calculated using a standard least-squares fit, 4C02 and 2C01 are found from the inclination andintercept, respectively. The results of the different constantsfound with the uniaxial tension and equibiaxial deformationare shown in Table 1. These constants are substituted intoEq. (9) for uniaxial tension (the experimental results were re-arranged using the true stressσ and true strainε = lnλ). The

250 H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252

Fig. 2. Uniaxial stress–strain curve for natural rubber vulcanizate containing40 phr of carbon black. These experimental results are compared withcalculated results based on Eqs. (8), (16) and (29).

Fig. 2. Courbes contrainte-déformation en traction uniaxiale pour lecaoutchouc vulcanisé contenant 40 phr de noir de carbone. Ces résultatsexpérimentaux sont comparés avec ceux obtenus par les Éqs. (8), (16)et (29).

Fig. 3. Reduced stress curve of Mooney–Rivlin plot against inverse elon-gation (1/λ) in uniaxial tension–compression tests of Yeoh [9] for naturalrubber vulcanizate containing 40 phr of carbon black. These experimentalresults are compared with calculated results based on Eqs. (13) and (19).

Fig. 3. Courbe expérimentale de la contrainte réduite de Mooney–Rivlinen fonction de l’inverse de l’élongation(1/λ) en traction–compressionuniaxiale de Yeoh [9] pour le caoutchouc naturel vulcanisé contenant 40 phrdu noir de carbone. Ces résultats sont comparés avec ceux obtenus par lesÉqs. (13) et (19).

present model is compared with that of Yeoh [9] in Fig. 2.Both models give a very good correlation with the exper-imental results for uniaxial tension. Fig. 3 rearranges thedata using the reduced stressφ and the inverse elongationratio 1/λ to determine the coefficients of the strain energydensity function. Clearly, the Mooney–Rivlin model gives

Fig. 4. Shear stress curve against(I1 − 3) for natural rubber vulcanizatecontaining 40 phr of carbon black. These experimental results are comparedwith calculated results based on Eqs. (12), (23) and (28).

Fig. 4. Courbes des contraintes en cisaillement simple en fonction de(I1 − 3) pour le caoutchouc vulcanisé contenant 40 phr de noir de carbone.Ces résultats expérimentaux sont comparés avec ceux obtenus par lesÉqs. (12), (23) et (28).

straight lines in theφ versus 1/λ plots. Fig. 3 shows a typ-ical stress–strain plot of carbon black filled rubber vulcan-izate under uniaxial tension/compression. The upturn occursat relatively low strain, one can observe that the Mooney–Rivlin model, which represents a linear relation between re-duced stress and 1/λ in uniaxial deformation, is not capableof capturing the upturn in the small strain region, and thenonlinearity beyond∼200 % tension and∼20 % compres-sion. The Yeoh [9] model approximates the test data suc-cessfully at high range forλ in uniaxial compression with theexception for moderate tensile extension from the test data.Our model, the third-order approximation, gives a good fitfor extension ratio up toλ= 0.74 for uniaxial compression.

Substituting respectivelyf andg for Eq. (12) gives a verygood correlation between theoretical results (circles) and thecorresponding Yeoh data (solid lines), for pure shear in thedomain of relatively large deformations including relativelysmall elongation where data points were neglected. A poorcorrelation is observed for the third-order model at largestrains. Final results have been plotted in Fig. 4, and, asexpected, when the functiong is introduced in equibiaxialtension Eq. (14), theoretical results remain very good forour model and significant deviation is observed for Yeoh [9]model in Fig. 5.

Stress values for biaxial deformation from calculationsbased on the third-order approximation Eq. (15) (circles),Yeoh [9] model (dashed lines) and measured stress values(solid lines) of James and Green [7] for different combina-tions of λ1 andλ2 are compared in Fig. 6. The degree ofapproximation is good forλ1 = 1.2 andλ2 = 1.25, and athigh ratios errors increase. It is evident that errors are pro-duced with biaxial deformation for all models. The origin of

H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252 251

Fig. 5. Lines are predictions from equibiaxial data of James and Green [7];stress values are calculated from Eqs. (14), (29) (circles); Yeoh [9] model,Eqs. (14), (23) (+ symbols).

Fig. 5. Courbe expérimentale de traction équibiaxiale de James etGreen [7] ; ces résultats sont comparés avec ceux obtenus par les Éqs. (14),(29) ; le model de Yeoh [9] est obtenu à partir des Éqs. (14), (23).

Fig. 6. General biaxial data of James and Green [7]; stress values arecalculated from Eqs. (15), (29), Yeoh [9] model, Eqs. (15), (23), andmeasured stress values for different combinations ofλ1 = Elongation 1 andλ2 = Elongation 2.

Fig. 6. Résultats expérimentaux de traction biaxiale de James et Green [7] ;ces résultats sont comparés avec ceux obtenus par les Éqs. (15), (29) et lemodel de Yeoh [9] est obtenu à partir des Éqs. (15), (23).

errors may be due to the fact that the reinforcing effect of thecharge is more complex than shown by Eq. (23), so that thefunctionf is not only a function of(I1 − 3), but includes tosome extent an effect of (I2 − 3). In fact, the functiong isnot affected by the reinforcement at large strains [15]. Thesum of ∂W/∂I1 and ∂W/∂I2 is almost constant in the re-gion of small strains, however,∂W/∂I1 and∂W/∂I2 rapidlychange in this region. Another words, the Young and shearmodulus in this region are constants, respectively, and in cal-

culating stresses large errors are not produced by using ourapproximation function in this region.

5. Conclusion

A strain energy density function has been presented forthe representation of the stress–strain behavior of carbonblack reinforced rubber. We designed this function to coverthe small and moderate extension ratios. The present modelaccurately predicts the response of the equibiaxial tensionproblem, which could not be explained by the model ofYeoh [9].

The identification of parameters of our strain energy den-sity function is estimated by experimentation on uniaxialstretch and equibiaxial modes, it gives high precision in thedomain of relatively large moderate deformations for differ-ent deformation modes; we have presented a simple and fastmethod for the identification of materials parameters.

One can observe that the biaxial tensile testing requiresa complex apparatus, it is preferable to use only pure sheardeformations and uniaxial stretch characteristics, which arecomparatively easy to test and also require simple apparatus.However, identifying the constants from the one-directiongroup (uniaxial tension and simple shear), we lost in termsof high precision in the stress–strain behavior for all modesof deformations at large strains.

Our strain energy density function is deduced fromthe general Rivlin strain energy density function and itcan obtained from the popular third-order deformationapproximation of James and Green [7] (settingC02 =C11(I1 − 3)). Thus, it is already available in many finite-element analysis programs.

References

[1] R.S. Rivlin, Large elastic deformation of isotropic materials: I. Funda-mental concepts, II. Some uniqueness theories for pure homogeneousdeformations, Philos. Trans. Roy. Soc. London Ser. A 240 (1948) 459–508.

[2] H. Peeken, R. Döpper, B. Orschall, A 3-D rubber material modelverified in user-supplied subroutine, Computers Structures 26 (1/2)(1987) 181–189.

[3] R.W. Ogden, Nonlinear Elastic Deformation, Ellis Horwood Edition,1984.

[4] R. Mooney, A theory of large elastic deformation, J. Appl. Phys. 11(1940) 582–592.

[5] S. Kawabata, H. Kawai, Strain energy density function of rubbervulcanizate from bi-axial extension, Adv. Polymer. Sci. 24 (1977) 90–124.

[6] N.W. Tschoegl, Constitutive equations for elastomers, J. Appl. Poly-mer Sci. A 1 (1971) 1959–1970.

[7] A.G. James, A. Green, Strain energy function of rubber. II. Thecharacterization of filled vulcanizates, J. Appl. Polymer Sci. 19 (1975)2319–2330.

[8] D.W. Haines, W.D. Wilson, Strain energy density function for rubber-like materials, J. Mech. Phys. Solids 27 (1979) 345–360.

[9] O.H. Yeoh, Characterization of elastic properties of carbon black filledrubber vulcanizates, Rubber Chemistry Technol. 33 (1990) 792–805.

252 H. Bechir et al. / Mécanique & Industries 3 (2002) 245–252

[10] M.J. Gregory, The stress–strain behavior of filled rubbers at moderatestrains, Plastics and Rubbers Materials and Applications (1979).

[11] Y. Yamashita, S. Kawabata, Approximated form of the strain energy-density function of carbon black filled rubber for industrial applica-tions, Internat. Polymer Sci. Technol. 20 (2) (1993) T/52–T/64.

[12] L.R.G. Treolar, The Physics of Rubber Elasticity, 3rd ed., OxfordUniversity Press, 1975.

[13] H.G. Killiam, H. Schenk, S. Wolff, Large deformation in fillednetworks of different architecture and its interpretation in terms of the

Van der Waals network model, Colloid Polymer Sci. 265 (1987) 410–423.

[14] J. Lambert-Diani, C. Rey, New phenomenological behavior laws forrubbers and thermoplastic elastomers, Eur. J. Mech. A/Solids 18(1999) 1027–1043.

[15] S. Kawabata, M. Matsuda, K. Tei, H. Kawai, Experimental surveyof strain energy density function of isoprene rubber vulcanizate,Macromolecules 14 (1981) 154–162.