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Journal of Elasticity 68: 167–176, 2002. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity CORNELIUS O. HORGAN1 and GIUSEPPE SACCOMANDI2

1 Department of Civil Engineering, P.O. Box 400742, University of Virginia, Charlottesville, VA 22904, U.S.A. E-mail: [email protected] 2 Sezione di Ingegneria Industriale, Dipartimento di Ingegneria dell’Innovazione, Università degli Studi di Lecce, 73100 Lecce, Italy. E-mail: [email protected]

Received 6 May 2003 Abstract. Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strainenergy density in the Gent model depends only on the first invariant I1 of the Cauchy–Green strain tensor, is a simple logarithmic function of I1 and involves just two material parameters, the shear modulus µ and a parameter Jm which measures a limiting value for I1 − 3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Padè approximant for this function. The constants µ and Jm in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closedform solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function. Mathematics Subject Classifications (2000): 74A25, 74B20. Key words: Gent constitutive model, strain hardening phenomena, incompressible rubber.

1. Introduction Mechanical properties of elastomeric materials are usually represented in terms of a strain-energy density function W (see, e.g., [3, 25]). The state of strain is characterized by the principal stretches λ1 , λ2 , λ3 of the deformation or equivalently by introducing a strain measure such as the left Cauchy–Green strain tensor B = FFT . Here F is the gradient of the deformation. For an isotropic hyperelastic material, W is a function of the strain invariants I1 = trB,

I2 =

1 2 trB − (trB)2 , 2

I3 = det B.

(1)

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C.O. HORGAN AND G. SACCOMANDI

Because rubber can be considered to behave in an incompressible manner as long as the hydrostatic stress does not become too large, it is usual to adopt the internal constraint of incompressibility so that all the admissible deformations must be √ isochoric, i.e. det F = I3 = 1. Modern design practices in the rubber industry are largely based on finite element simulations and the accuracy of these analyses relies on the ability of the constitutive model used to predict the mechanical response of the elastomeric material. For this reason the choice of the functional form of the strain-energy density is of fundamental importance. Constitutive models have been developed mainly using two approaches: the molecular and continuum mechanics viewpoints. A recent review on this subject may be found in [6]. See also [9]. The basic strain-energy density for rubber elasticity is the Mooney–Rivlin strainenergy 1 1 1 1 + γ µ(I1 − 3) + − γ µ(I2 − 3), (2) W = 2 2 2 2 where µ > 0 is the infinitesimal constant shear modulus and γ is a dimensionless constant in the range − 12 γ 12 . When γ = 12 we obtain the neo-Hookean strain-energy. The theoretical predictions based on (2) do not adequately describe experimental data especially at high values of strain. To model the typical hardening at large deformations, a number of alternative models have been proposed. Some of these models involve a strain-energy density of the form W = W (I1 ), and so are called generalized neo-Hookean models. In the molecular theory of elasticity (see, e.g., [8]) these models involve the introduction of a distribution function for the end-to-end distance r of the polymeric chain which is not Gaussian. The most widely used and simplest non-Gaussian probability distribution P (r) with compact support, i.e. P (r) = 0 if r > rmax , is due to Kuhn and Grün [24]. In the framework of the phenomenological theory similar models have been obtained on considering the idea of limiting chain extensibility, i.e. by considering strain-energy density functions that have a singularity when the first invariant I1 reaches a finite value I1∗ . The simplest model with limiting chain extensibility is due to Gent [10, 11] who proposed the strain-energy density µ I1 − 3 , (3) WI = − Jm ln 1 − 2 Jm where µ is the shear modulus for infinitesimal deformations and Jm is the constant limiting value for I1 − 3, taking into account limiting polymeric chain extensibility. On taking the formal limit as the polymeric chain extensibility parameter tends to infinity (Jm → ∞), (3) reduces to the classical neo-Hookean form µ (4) W = (I1 − 3). 2 Limiting chain extensibility from a phenomenological point of view may be introduced in many ways and a detailed review of some of the possibilities may be

GENT CONSTITUTIVE MODEL OF RUBBER ELASTICITY

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found in the paper by Horgan and Saccomandi [18]. The model (3) is a basic one because it is associated with the simplest rational approximation of the response functions in the Cauchy stress representation formula for isotropic incompressible elastic materials [27, 19]. The Cauchy stress associated with (3) is given by TI = −pI + µ

Jm B, Jm − (I1 − 3)

(5)

so that the stress has a singularity as I1 → Jm + 3, reflecting the rapid strain hardening observed in experiments. In a recent paper [4], Beatty has considered a stretch averaged full-network model of rubber elasticity. On considering a non-Gaussian network of perfectly flexible chains and using the approximate expression for the probability distribution function for the end-to-end distance introduced by Kuhn and Grün [24], it is shown in [4] that the macroscopic constitutive equation for the Cauchy stress tensor obtained by averaging in a suitable way is given by TII = −pI + X(I1 )B,

(6)

where X(I1 ) ≡

µ0 β(λˆ r ) . 3λˆ r

(7)

The constant µ0 = nkT where k is the Boltzmann constant, T is the absolute temperature and n is the chain density. Here the mean relative chain stretch λˆ r√(the ratio of the current chain vector length rchain to its fully extended length rL = N) is defined by I1 , (8) λˆ r = 3N where N is the number of rigid links composing a single chain, and β ≡ L−1 (λˆ r )

(9)

is the inverse of the Langevin function L(β). Therefore λˆ r = L(β) ≡ coth(β) −

1 . β

(10)

√ The definition of λˆ r implies that (9) is defined only in the range (1/ N , 1) and thus from (8) we see that I1 is defined in the range (3, 3N). As is pointed out by Beatty [4], the general result (6) contains some specific network models, e.g., the Wang–Guth 4-chain model and the Arruda–Boyce 8-chain model as special cases. The aim of this note is to investigate the connections between the models (5) and (6). We show that the Gent constitutive law (5) is a very accurate approximation of (6) that retains all the qualitative features of this molecular based model

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and with similar quantitative predictions. Thus it is possible to give a molecular interpretation to the phenomenological constitutive constants in (3). 2. The Comparison between (5) and (6) On using (8) one obtains I1 = 3N λˆ 2r , and so (5) can be rewritten as TI = −pI + µ

N −1 1 B, N 1 − λˆ 2r

(11)

where we have used the fact that if the maximum value for I1 is 3N then Jm = 3(N − 1).

(12)

From (11) it is clear that Gent model cannot be obtained by a power series approximation of the molecular models based on the inverse Langevin function. The use of a power series to approximate the inverse Langevin function is somewhat misleading because (7) has a singularity as λˆ r → 1, and functions with singularities cannot be approximated globally by polynomial expressions. For this reason the comparison carried out in [5] between the eight chain molecular model of [1] and the Gent model is not complete. In order to obtain a useful approximation of the inverse Langevin function it is necessary to use rational functions. In the framework of rubber-like materials this type of approximation was apparently introduced for the first time by Cohen [7] and exploited by Perrin [26]. The idea in [7] is to use Padè approximants [2], a rational function type approximation. The [L, M] Padè approximant of a given real function f (x), at x = 0, is an approximation of O(x L+M+1 ). Considering a polynomial PL (x) of degree L and a polynomial QM (x) of degree M, define L ai x i PL (x) = Mi=0 . (13) [L, M] = QM (x) ˜k xk k=0 a Then [L, M] is the Padè approximant of f (x) at x = 0 if f (x) − PL (x) = O x L+M+1 as x → 0. QM (x)

(14)

j Therefore if one considers the Taylor expansion of f (x) = ∞ j =0 fj x and introduce this expression in (14) the coefficients ai and a˜ k may be easily obtained on solving a linear algebraic system. In [7] it is shown that the [3, 2] Padè approximant for the inverse Langevin function is A + B λˆ 2r + O λˆ 6r , L−1 (λˆ r ) ≈ λˆ r 2 1 + C λˆ r

(15)

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where A = 3,

B=−

36 , 35

C=−

33 . 35

(16)

Note that the first term on the right in (15) does not have a singularity at λˆ r = 1. Cohen [7] proposed rounding off the fraction C to C = −1 to ensure that such a singularity does occur. A further simplification is proposed in [7] by rounding off B to B ≈ −1 so that L−1 (λˆ r ) ≈ λˆ r

3 − λˆ 2r + O λˆ 6r . 1 − λˆ 2r

(17)

The key feature of (17) is that the rational approximation and the inverse Langevin function have a vertical asymptote at the same location λˆ r = 1. The approximation (17) has been employed by Haward [12] in the modeling of thermoplastic elastomers. On observing that the infinitesimal shear moduli for (5) and (6) are µ and µ0 , respectively, the first step in our comparison is to set µ = µ0 , and so it remains to compare the function β(λˆ r ), i.e. L−1 (λˆ r ), with N − 1 3λˆ r . N 1 − λˆ 2r

(18)

The function (18) is the ratio of two polynomials, and is simpler than (17). If we compute the [1, 2] Padè approximant of the inverse Langevin function we obtain L−1 (λˆ r ) ≈

3λˆ r 1 − (3/5)λˆ 2r

+ O λˆ 4r .

(19)

The approximation (19) has the same structure as (18), but the location of the singularity is not the same. Therefore the [1, 2] Padè approximation introduces too large an error in the location of the vertical asymptote and this is the reason for the use of a [3, 2] approximation in [7]. The foregoing remarks are quantified in Figure 1 where we plot (17) (solid curve), (18) for large N (dot-dash curve) and (19) (dotted curve). Note that the dotted curve (19) does not have the same asymptote as the other two curves. We also observe that (18) approaches the vertical asymptote at λˆ r = 1 slightly faster than (17). We note that Treloar in his celebrated book [29] (page 178 of the third edition) has provided an empirical approximation for the inverse Langevin function, namely L−1 (λˆ r ) ≈

3λˆ r 1 − [(3/5)λˆ 2r + (1/5)λˆ 4r + (1/5)λˆ 6r ]

.

(20)

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Figure 1. Plots of (17), (18) for large N, (19) versus λˆ r .

The expression (20), first developed by Treloar in [28], is a rational approximation which is closely related to the [1, 6] Padè approximation of the inverse Langevin function. The latter is readily calculated to be given by L−1 (λˆ r ) ≈

3λˆ r 1 − [(3/5)λˆ 2r + (36/175)λˆ 4r + (108/875)λˆ 6r ]

+ O λˆ 7r .

(21)

We note that the first term on the right in (21) does not have a singularity at λˆ r = 1. Since 36/175 = 0.20571 ≈ 1/5 and 108/875 = 0.12343, the Treloar approximation (20) may be considered as a modification of the [1, 6] Padè approximant, modified to locate the pole at λˆ r = 1. Indeed in (20) and (21) the coefficients of the λˆ 2r terms are identical and those of the λˆ 4r terms are virtually the same, whereas the coefficient of λˆ 6r in (20) has been determined by the desired location of the singularity. The qualitative and quantitative predictions of (20) and (17) are very similar (a plot of the former in Figure 1 would be indistinguishable from the solid curve corresponding to (17)), but clearly (20) is a much more complicated algebraic expression than (17) or (18).


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Figure 2. Plots of the Langevin function (10), (22) for N = 26, (23), and (28) versus β.

To compare (18) directly with the inverse Langevin function it is more convenient to consider their respective inverses. Thus we compare L(β) with the positive branch of the inverse function to (18), i.e. 2

1 N − 1 N − 1 . (22) 9 + 4β 2 − 3 λˆ Ir = 2β N N When N is greater than 20 (as is usual for macromolecules) we take (N −1)/N ≈ 1 and (22) simplifies to λˆ Ir ≈

1 9 + 4β 2 − 3 . 2β

(23)

In Figure 2 we have plotted the Langevin function (10) (solid curve), the righthand side of (22) with N = 26 (dot-dashed curve) and the right-hand side of (23) (dotted curve). It is clear that the function (18) arising in the Gent model is a very good approximation to the inverse Langevin function. By considering the difference between L(β) and the right-hand side of (23), i.e. coth(β) −

1 1 − 9 + 4β 2 − 3 , β 2β

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it is possible to show that the maximum error is ≈0.057. Therefore from a quantitative point of view the approximation is also accurate. 3. A Van der Waals Model Another type of limiting chain extensibility model is based on the Van der Waals strain-energy which has been studied in detail by Kilian [23]. On reconsidering the well known analogy between rubber and gases, Kilian, on using the Van der Waals equation of state, has proposed the following strain-energy density

I1 − 3 I1 − 3 WIII = −µJm ln 1 − + , (24) Jm Jm where µ is the shear modulus, and Jm is the constant limiting value for I1 − 3. In this case, the counterpart of (5) is √ Jm √ B, (25) TIII = −pI + µ √ Jm − I1 − 3 which is also singular as I1 → Jm + 3. If we compare (25) with (6), we are led to a comparison between L−1 (λˆ r ) and √ 3λˆ r N − 1 (26) √ N − 1 − N λˆ 2r − 1 or equivalently between L(β) and the positive branch of the inverse of (26), i.e. 2 2 ˆλIIIr = β(3 − 3N + 9 − 9N + β N ) . (27) 9 − 9N + β 2 N For large N, equation (27) reduces simply to λˆ IIIr ≈

β . β +3

(28)

In Figure 2, we have plotted the right-hand side of (28) (dashed curve). It is clear that the approximation (28) is inferior to (23) and so the strain-energy WI provides a better agreement with the molecular model than WIII . It is also worth noting that the algebraic structure of (26) shows that it cannot be related to a Padè approximant of the inverse Langevin function. 4. Concluding Remarks In this note we have compared the phenomenological constitutive model for incompressible rubber proposed by Gent [10] with the model that is obtained by


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molecular arguments using the Kuhn and Grün non-Gaussian probability distribution function for the end-to-end distance. We have shown that the Gent model provides a very good qualitative and quantitative approximation of such models. We have also seen that the Gent model is closely related to Padè approximants for the inverse Langevin function that arises in the non-Gaussian molecular models. The Gent model is very simple, is amenable to closed-form analytic solutions for many fundamental boundary-value problems (see [13–20, 22]) and is also applicable to the modeling of soft biological tissues [21]. We have also shown that the constitutive coefficients in the Gent model are simply related to microscopic quantities namely the infinitesimal shear modulus is µ = nkT as is usual in the molecular models and Jm = 3(N − 1). Therefore the Gent model is simple, with similar predictions to the molecular models and with a clear microscopic meaning for the constitutive coefficients. Thus the Gent model is an attractive alternative to the comparatively complicated molecular models for incompressible rubber and warrants inclusion in the libraries of large scale commercial computer codes (e.g., ABACUS or ANSYS) especially since analytic solutions to benchmark problems are now available for this model.

Acknowledgements The research of C.O.H. was supported by the U.S. National Science Foundation under DMS 0202834. The work of G.S. was partially supported by GNFM of the Italian INDAM. We are grateful to Professor M.F. Beatty for providing us a preprint of his work [4] prior to publication and to Professor A.N. Gent for his helpful comments on an earlier version of the manuscript.

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