Fitting hyperelastic models to experimental dataR. W. Ogden, G. Saccomandi, I. Sgura

Abstract This paper is concerned with determiningmaterial parameters in incompressible isotropic elasticstrainenergy functions on the basis of a non-linear leastsquares optimization method by fitting data from theclassical experiments of Treloar and Jones and Treloar onnatural rubber. We consider three separate forms ofstrain-energy function, based respectively on use of theprincipal stretches, the usual principal invariants of theCauchy-Green deformation tensor and a certain set oforthogonal invariants of the logarithmic strain tensor.We highlight, in particular, (a) the relative errors gener-ated in the fitting process and (b) the occurrence of mul-tiple sets of optimal material parameters for the same datasets. This multiplicity can lead to very different numericalsolutions for a given boundary-value problem, and this isillustrated for a simple example.

Keywords Constitutive laws, Elasticity, Elastomers,Finite deformation, Isotropic

1IntroductionIn this paper, we carry out a systematic analysis of thefitting of incompressible isotropic hyperelastic constitutivelaws to experimental data, the determination of materialparameters and the corresponding relative errors. For thispurpose we use the original data of Treloar [1] and Jonesand Treloar [2] for natural rubber, a non-linear least

squares optimization tool from Matlab, and three specificmaterial models due to Ogden [3], based on stretches,Pucci and Saccomandi [4], based on the usual invariants I1and I2 of the Cauchy-Green deformation tensor, andCriscione et al. [5], based on a set of orthogonalinvariants of the logarithmic strain tensor.

An important aim of the paper is to show, by means ofnumerical examples, that the fitting of experimental datawithin the framework of the mechanics (or thermome-chanics) of elastomeric solids is a very delicate question.In particular, non-uniqueness of optimal materialparameters often occurs, and this non-uniqueness may bereflected very markedly in the solution of boundary-valueproblems. The difficulties arising in the fitting procedureare intrinsic to the considered problem and are not ingeneral dependent on the specific choice of constitutivelaw, although in some cases uniqueness may be achieved;one case in point is where the material constants occurlinearly in the stress-deformation equations; another thatof the PucciSaccomandi model discussed in Sect. 5.Moreover, the difficulties are not functions of the partic-ular formulation, whether in terms of stretches or invari-ants. If, for example, polynomial models are used (with thematerial constants appearing linearly) then to obtaingood fits to the data for an extensive range of deforma-tions it is necessary to employ high-order polynomials,which typically introduce numerical instabilities. On theother hand, if the material constants feature in a non-linear way then non-uniqueness of the optimal set ofconstants can be the rule unless some intrinsic property ofthe problem is used to avoid this.

We begin, in Sect. 2, by summarizing the relevantequations that describe the stresses in a homogeneouspure strain of an incompressible isotropic elastic solid asderivatives of the strain-energy function and the importantspecial cases of simple tension and equibiaxial tension. InSect. 3 the optimization method to be used in thesubsequent sections is described.

The Ogden model, with either six or eight materialparameters, is the focus of attention in Sect. 4. It is shownthat by fitting the Treloar data for simple tension (equi-biaxial tension) the prediction for equibiaxial tension(simple tension) is not good. This is not surprising. On theother hand, if both sets of data are fitted together thengood overall fits are obtained with reasonably low relativeerrors. Corresponding fits are performed for the full set ofthe JonesTreloar biaxial data. An important feature of theresults is that multiple sets of optimal values of thematerial parameters are generated by taking different

Computational Mechanics (2004) Springer-Verlag 2004DOI 10.1007/s00466-004-0593-y

Received: 16 January 2004 / Accepted: 4 May 2004Published online: 18 August 2004

R. W. Ogden (&)Department of Mathematics,University of Glasgow,Glasgow G12 8QW, Scotland, UKe-mail: rwo@maths.gla.ac.uk

G. SaccomandiDipartimento di Ingegneria dellInnovazione,Sezione di Ingegneria Industriale,Universita` degli Studi di Lecce, 73100 Lecce, Italye-mail: giuseppe.saccomandi@unile.it

I. SguraDipartimento di Matematica,Universita` degli Studi di Lecce, 73100 Lecce, Italye-mail: ivonne.sgura@unile.it

The work of G. S. was partially supported by GNFM, INDAM,Italy. The work of I. S. was partially supported by the ProgettoGiovani Ricercatori Universita` di Lecce MIUR 2001/2002.

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starting values and/or by setting different levels of accu-racy in the optimization procedure. This clearly hasimplications for the solution of boundary-value problems,and we emphasize this point by considering, in Sect. 7, asimple boundary-value problem in which significantlydifferent numerical predictions of the solution are ob-tained by using different optimal sets of parameters. It alsohas serious implications in respect of implementation ofmaterial models in commercial software, particularly sincenon-uniqueness is inherent in problems of non-linearelasticity. Thus, careful mathematical analysis is needed toensure confidence in the resulting numerical solutions.

In Sect. 5, for the Pucci and Saccomandi model, anextension of the Gent model [16] having three materialparameters, we find that there is no multiplicity of optimalsets of parameters. This model fits the Treloar data welland also the JonesTreloar data except in the compressionrange. We also report, in Sect. 7, the predictions of thedifferent parameter sets obtained for this model fordifferent sets of data in respect of the boundary-valueproblem mentioned above.

In Sect. 6, we examine the Criscione et al. [5] model onthe same basis as for the other two models. We find thatwhilst this model gives a good fit to the JonesTreloar datafor moderate stretches with just five material parameters itdoes not fit the Treloar simple tension data well for higherstretches unless a very much larger number of parametersis used.

2Basic equationsIn this section we summarize briefly the equations ofincompressible isotropic non-linear elasticity that arerequired for comparing the theory with experimental datain standard experimental protocols. The data that we shallconsider in this paper relate to experiments in which thedeformation is homogeneous. More specifically, we con-sider homogeneous deformations that can be classified aspure homogeneous strain, i.e. deformations of the form

x1 k1X1; x2 k2X2; x3 k3X3; 1where X1;X2;X3 are rectangular Cartesian coordinatesthat identify material particles in some unstressed refer-ence configuration, x1; x2; x3 are the correspondingcoordinates after deformation with respect to the sameaxes, and the coefficients k1; k2; k3 are positive constants,referred to as the principal stretches of the deformation.The principal axes of the deformation coincide with theCartesian coordinate directions and are fixed as the valuesof the stretches change.

In the theory of hyperelasticity there exists a strain-energy function (or stored-energy function), defined perunit volume and denoted W . It depends symmetrically onk1; k2; k3, i.e.

Wk1; k2; k3 Wk1; k3; k2 Wk2; k1; k3; 2and for an incompressible material the principal stretchessatisfy the constraint

k1k2k3 1: 3The principal Cauchy stresses ri; i 2 f1; 2; 3g (defined

per unit cross-sectional area normal to the xi axis in the

deformed configuration), are related to the stretchesthrough W according to the equations

ri ki oWoki p; 4

where p is a Lagrange multiplier associated with the con-straint (3). Note that in (4) there is no summation over therepeated index i. The corresponding Biot (or nominal)stresses ti (defined per unit reference cross-sectional area)are the stresses that are often measured directly inexperiments and are given by

ti oWoki pk1i rik1i : 5

Since the material considered is incompressible, onlytwo stretches can be varied independently and hencebiaxial tests are sufficient to determine the form of W . Forthis reason it has been found convenient to use theincompressibility constraint (3) to express the strainenergy as a function of two independent stretches, andhere we use k1 and k2 and the notation defined by

W^k1; k2 Wk1; k2; k11 k12 : 6This enables p to be eliminated from (4) and leads to

r1 r3 k1 oW^ok1 ; r2 r3 k2oW^ok2

: 7

For the energy and stresses to vanish in the chosen ref-erence configuration and for consistency with the classicaltheory of incompressible isotropic elasticity we must have

W^1; 1 0; W^11; 1 W^21; 1 0;W^121; 1 2l; W^111; 1 W^221; 1 4l; 8where the subscripts 1 and 2 signify differentiation withrespect to k1 and k2, respectively, and l> 0 is the shearmodulus of the material in the reference configuration.

If the deformation (1) is applied to a thin sheet ofmaterial (a common situation experimentally) then a planestress condition applies and we may set r3 0, the stressnormal to the plane of the sheet. In terms of the nominalstresses we then have simply

t1 oW^ok1 ; t2 oW^ok2

; 9

which provides two equations relating k1; k2 and t1; t2 andtherefore a simple basis for characterizing W^ from mea-sured (homogeneous) biaxial data.

There are several special cases of the biaxial test whichare important from the experimental point of view. Theseare simple tension and equibiaxial tension, which we nowdiscuss separately and briefly, and pure shear, which we donot examine here. A subscript s or e will be used to labelquantities associated with, respectively, simple tension andequibiaxial tension.

(i) Simple tensionFor simple tension we set t2 0 and t1 t. Then, bysymmetry, the incompressibility constraint yields

k2 k3 k1=21 and we write k1 k. The strain energymay now be treated as a function of just k, and we define

Wsk W^k; k12; 10and (9) reduces to

t W 0sk; 11where the prime indicates differentiation with respect to k.

(ii) Equibiaxial tensionFor equibiaxial tension we have k1 k2 ( k, say),k3 k21 and t1 t2 ( t, say). Then, we defineWek W^k; k; 12and (9) reduces to

t 12

W 0ek: 13For discussion of the possible ranges of values of the

stretches in various tests we refer to Ogden [6].In Sect. 4 we shall focus on fitting members of a specific

class of energy functions to various sets of experimentaldata. The considered strain-energy functions have theform

W X1

i1

liaikai1 kai2 kai3 3; 14

where each li and ai is a material constant, the latter notnecessarily being integers (Ogden [3, 7, 8]). For practicalpurposes the sum in (14) is restricted to a finite number ofterms, while, for consistency with the classical theory, theconstants must satisfy the requirement

XM

i1liai 2l; 15

where M is a positive integer and l is again the shearmodulus of the material in the undeformed stress-free(natural) configuration, as identified in (8).

We shall also make use of the strain energy expressed asa function of invariants (symmetric functions of thestretches). In particular, we mention here the representa-tion of W in terms of the two independent invariants I1and I2 defined in terms of the stretches by

I1 k21 k22 k23; I2 k21 k22 k23 : 16Let W, regarded as a function of I1; I2, be written WI1; I2.Then, we have the identification

Wk1; k2; k3 WI1; I2: 17In terms of W the principal Cauchy stress differences

(7) become

r1r3 2k21k21 k22 o WoI1

2k21k22k21 o WoI2

;

r2r3 2k22k21 k22 o WoI1

2k21k22k22 o WoI2

: 18

Analogues of the conditions (8) may be written down inrespect of W but we omit the details.

Many specific forms of strain energy may be found inthe literature, expressed either in terms of the stretches orthe invariants I1 and I2 or other choices of independent

deformation variables. For representative reviews of manyof these we refer to, for example, Ogden [3, 7, 9], Treloar[10], Beatty [11] and Boyce and Arruda [12].

3Outline of the optimization methodThe experimental data used in this paper are the classicaldata of Jones and Treloar [2] for the biaxial tension dataand of Treloar [1] for the other tests, their numericalvalues having been obtained from the original experi-mental tables. These data have been used to identify theparameters in specific strain-energy functions by means ofthe least squares (LS) technique. It should, however, bepointed out that sometimes in the literature these data areobtained by digitization of the figures in [1] and [2], withconsequent loss of accuracy.

Let K K1;K2; . . . ;KmT be the vector of values of ameasure of stretch appropriate to the considered defor-mation and s s1; s2; . . . ; smT be the correspondingvalues of stress, which may be either nominal stress orCauchy stress, again depending on the deformation inquestion. Hence K; s are the given pairs of data values.The considered material model is represented by thestrain-energy function W , from which the stress is calcu-lated and written FK; p : R Rn ! R, where K takesvalues Ki; 1 1; . . . ;m, and p p1; p2; . . . ; pnT is a set ofn material parameters (constants) to be identified.

Let us define the objective function as the squared2-norm

SFp : kFK; p sk22 Xm

i1FKi; p si2; 19

where FK; p : FK1; p; . . . ; FKm; pT.Hence, the minimization problem is given by

minp

SFp: 20

If FK; p is a non-linear function with respect to p, then anon-linear least squares (NLS) problem arises.

Several numerical algorithms have been used in theliterature to solve NLS problems (see, for example, Bjorck[13]) and they are usually based on some suitable modi-fication of the Newton method and require an initial guessfor the solution. Therefore, when some appropriate stop-ping criteria are satisfied, the iterative technique approx-imates numerically a minimum for the problem (20) andthen furnishes an optimal solution, p say, for the fittingproblem with residual S SFp.

For our purposes in the present paper we use the func-tion Lsqcurvefit in the Optimization Toolbox of MATLAB[14]. We choose the trust region (TR) method (see [14] fordetails) as the option for the descent algorithm and, ingeneral, we choose a randomly generated starting point.The iterations of the NLS algorithm can stop either whenthe Newton step becomes less than TolX 1e)8 or theinfinity norm of the estimated gradient r of the objectivefunction is less than TolFun 1e)12 or when a maximumof 3000 iterations have been performed. We remark that thevalue of r for each approximation obtained can beregarded as an indicator of the attainment of a localminimum for the NLS function.

We did not impose upper and lower bounds on theparameters p since we chose to use an unconstrainedoptimization algorithm and to discard non-physicalsolutions a posteriori.

Finally, to allow comparisons between the differentforms of FK; p, we report also the relative errors in thedata obtained for the corresponding optimal numericalfitting and calculated as

erri jFKi; p sij

maxf0:5; jsijg for i 1; . . . ;m:

This is a slight modification of the usual definition in thatthe 0:5 has been included in the denominator so as toavoid division by small values of jsij when the stretches areclose to 1.

Moreover, in order to estimate the predictive ability ofeach model, we propose a suitable modification of theabove MATLAB function so as to fit several different setsof data together. For example, if the simple tension Ks; ssand the equibiaxial tension data Ke; se are consideredtogether we choose to minimize the new objective function

minpkFsKs; p ssk22 kFeKe; p sek22; 21

where FsKs; p FsKs1; p; . . . ; FsKsm; pT, withKsi; i 1; . . . ;m, being the entries in the vector Ks and Fsthe appropriate form of F for simple tension. Similarly forFeKe; p, except that the number of data pairs (i.e. the valueof m) may be different. More details will be given in eachsection dedicated to specific models under examination.

We mention here that the units used in Treloar [1] are kgcm2 and those in Jones and Treloar [2] are N mm2. We usethese original units without further reference throughoutthe following sections, and we emphasize that the tests inthese two papers were carried out for different materials.

4The Ogden modelIn this section we focus on the Ogden model identified in(14). Considerations of physically realistic response andmaterial stability lead to the inequalities

liai > 0; 22for each i 1; . . . ;M (no sum over i) in addition to theconnection (15). Note, however, that, in general, if M 3it is not necessary that these hold for every i. This model ismost commonly implemented with M 3 or M 4, andthe theoretical predictions of such a model are generallyvery satisfactory, at least from a qualitative point of view.Based on the same set of data as used here, the paper byTwizell and Ogden [15] is dedicated to fitting (14) bymeans of the LevenbergMarquardt algorithm.

In the following subsections we shall examine thismodel in order to show that, given a set of experimentaldata, the following conclusions can be drawn:

it is possible to find a large number of optimal sets ofparameters;

the number of the optimal sets of parameters and thevalues of the parameters depend on M and also in a verysensitive way on the numerical accuracy required in the

NLS algorithm used (maximum number of iterationspermitted, tolerance errors, etc.);

the effect of this multiplicity of optimal parameters onthe solution of a boundary-value problem is not negli-gible from the quantitative point of view.

4.1Simple tensionIn [15] the set of optimal parameters obtained in order tofit Treloars simple tension experiment data [1] are

i ai li1 2:26 2:222 2:01 0:4483 10:01 3:9e)7

23

for M 3 andi ai li1 1:23 6:272 2:99 0:0543 4:44 0:0364 19:49 0:8e)15

24

for M 4. The analysis is based on equations equivalentto (11) with

Ws XM

i1

liai

kai 2kai=2 3

; 25

which is the appropriate specialization of (14) for simpletension. For the set of parameters (23) (M 3) theresidual S23 is found to be 13:449, and from Fig. 1, inwhich the relative errors are plotted as functions of thestretch k, it is clear that the relative error may be very largefor low values of the stretch and moderate around k 4.For the set of parameters (24) (M 4) the residual S24 islowered to 8.0129 and Fig. 1 shows that the relative error is

Fig. 1. Plots of the relative errors (%) against stretch k in thesimple tension data fits of the Twizell-Ogden parameter sets forM 3; 4

very low across the range of values of the stretch except forlimited ranges where it is moderate.

Let us focus on the case M 3 and use the set (23)together with 30 sets of random values as starting guessesfor the NLS algorithm. The algorithm converges to the setof parameters reported in Table 1, where for each optimalset p a1; l1; . . . ; aM; lMT we report also the residualS Sp and the measure of the gradient r, as explainedin Sect. 3.

By OT we identify the solution obtained from our algo-rithm by starting from (23) with at most four significantdigits and by OSSi; i 1; 2; 3, the other possible solutionsobtained by starting from the random guesses. By using alarger number of random guesses it is possible to find othersolutions. In this case, starting from the (23) guess, we find aset of parameters very similar to those in (23), but the newsolution OT improves the fit, as illustrated by the distribu-tion of the relative errors shown in Fig. 2, and, in particular,it should be noted that OSS2 gives the lowest relative error.

We repeat this procedure for the case M 4. The resultsare reported in Table 2 and the relative errors in Fig. 3. Notethat the OSS2 set of parameters gives theoretical predic-tions with all the relative errors under the 5% level.

As a general comment on the Ogden model, Tables 1and 2 give evidence of the multiple choices encountered.

To emphasize that application of the NLS method to theOgden function (14) can be very sensitive to the numericalsettings used in the implementation of the algorithm, wereport the following considerations. In the numericalexperiments above, we found that the NLS optimizationoften stops when the Newton step becomes less thanTolX 1e)8 even if neither the estimated gradient r northe residual S is sufficiently small. Moreover, if we setTolX 1e)15 up to the machine precision and we restartthe algorithm from one solution p obtained whenTolX 1e)8 is used, sometimes a completely different setof parameters with a lower residual can be obtained. For

example, if we choose p OT in Table 1 as a new startingguess, then the new set

i ai li1 2:7908 0:811072 1:6207 3:96473 10:95 5:0211e)8

26

with a residual S 9:57949 and r 0:504 is obtained.This behavior of the Newton iterations usually indicates

that the problem becomes ill-conditioned, that is low andflat local minima can be present in the objective functionof the optimization problem.

Fig. 2. Plots of the relative errors (%) against stretch k in respectof the Ogden strain energy with M 3 in simple tension for theparameter sets OT, OSS1, OSS2, OSS3 given in Table 1

Table 2. Parameter values for the Ogden model with M 4: simple tensioni OT OSS1 OSS2 OSS3

ai li ai li ai li ai li

1 1.23 6.27 10.376 1.6988e)7 2.4536 0.15853 22.89 6.1302e)192 )1.99 )0.054 )1.605 )1.4396 )2.0354 )3.8145 )12.161 )9.8765e)43 4.44 0.036 )1.6087 )2.0189 13.945 9.5425e)11 )0.039294 )43.2394 19.49 7.6559e)16 2.654 1.0229 3.2503 0.33457 )4.3008 )2.0949

S 6.97 10.386 8.1035 5.7977r 1175.1 0.68402 1.6933 38416

Table 1. Parameter values for the Ogden model with M 3: simple tensioni OT OSS1 OSS2 OSS3

ai li ai li ai li ai li

1 2.2519 2.2118 )3.3288 )2.7776 10.666 9.1866e)8 )4.6141 )0.923542 )2.054 )0.61139 3.2481 0.25031 2.845 0.63604 )4.5926 )1.20373 10.01 3.9204e)7 )22.522 )2.5625e)8 )2.7138 )2.7391 )20.594 )2.1212e)7

S 12.876 9.3318 9.9655 11.1r 5.82 0.24963 0.56965 0.03575

4.2Sensitivity of the Ogden strain energyIn the fittings discussed above we have noted that therelative errors are always higher in the region of smallstretches. It is therefore interesting to perform a sensitivityanalysis for the Ogden strain-energy function (14) in thecase of simple tension, in which case, withk1 k; k2 k3 k1=2, we may write the energy as afunction of k, as in (25). The corresponding nominal stressts is then

ts XM

i1li k

ai1 kai=21

Fsk; p: 27

For each i, the absolute sensitivities of the parameters ai; lican be assessed by considering

otsoli

kai1 kai=21

;

otsoai

li ln k kai1 1

2kai=21

:

However, such quantities can be inappropriate becausethey are not invariant with respect to changes in themagnitude of either ts or of one parameter. Relativechanges in ts with respect to a relative change in ai or liare provided, respectively, by

lit0

otsoli

;ait0

otsoai

;

where t0 is the value of ts at some specified set ofparameter values. For example, by considering ts to beevaluated at the set (23) for M 3, we obtain, fori 1; . . . ; 3,

lit0

otsoli

li k

ai1 kai=21

jt0j 28

and

ait0

otsoai

li ln k kai1 12 kai=21

ai

t0j j ; 29

where

t0 2:22k1:26 k2:13 0:448k3:01 k0:005 3:9e 7k9:01 k6:005:

Since these functions are both linear in li, we fix li 1and we plot them in Fig. 4 as functions of ai and k. Weobserve that all parameters are sensitive for stretches inthe region 0:5 k 1:5 in each case. This could alsoexplain why bigger relative errors are generally concen-trated in this region. A similar analysis can be carried outfor the other deformations considered in the followingsubsections.

4.3Equibiaxial deformationsWe now turn attention to equibiaxial deformations. Theparameters reported in [15] that provide fits to the Treloardata are

i ai li1 1:46 5:392 2:03 0:5313 9:68 0:19e)5

30

for M 3, andi ai li1 1:26 6:172 2:01 0:00913 4:26 0:0464 19:49 1e)14

31

for M 4, and the relevant relationship between k and t isgiven by (13) with (12), specialized in respect of (14).

Fig. 3. Plots of the relative errors (%) against stretch k in respectof the Ogden strain energy with M 4 in simple tension for theparameter sets OT, OSS1, OSS2, OSS3 given in Table 2

Fig. 4. Relative sensitivity of the Ogden parameters: plot of (28)(left) and (29) (right) against k and ai

The set of parameters reported for M 3 is not con-sidered since in this case the total residual is 4158:9, whilefor M 4 it is 3:6391. The results obtained from oursimulations are summarized in Tables 3 and 4, and thecorresponding distributions of relative errors are shown inFigs. 5 and 6, respectively. Note that if the NLS techniquestarts with (30) then the solution obtained does not satisfythe condition (22) for every i.

4.4Simple and equibiaxial tension comparedIn this section we compare the fits for simple and equi-biaxial tension by selecting those parameters in the pre-vious tables that give the lowest relative error in theinfinity norm for both sets of Treloar data.

Consider the case M 3. Among the optimal minimaobtained for simple tension, the OT set in Table 1produces the best fit (in the sense just specified) for theequibiaxial data. On the other hand, among the optimalminima obtained for equibiaxial tension, the OSS2 set inTable 3 produces the best fit for the simple tension data.These behaviours are reported in Figs. 7 and 8, respec-tively, and it easy to see that in each case no good agree-ment is obtained for both simple tension and equibiaxialtension simultaneously. The situation is only slightlyimproved if we take M 4, so we do not show the ana-logues of Figs. 7 and 8 for this case.

A better fit is obtained if we apply the NLS algorithm toboth sets of Treloar data simultaneously, i.e. Ks; ss forsimple tension and Ke; se for equibiaxial tension. In this

Table 4. Parameter values for the Ogden model with M 4: equibiaxial tensioni OT OSS1 OSS2 OSS3

ai li ai li ai li ai li

1 1.2619 6.1718 1.6918 0.89887 )0.017504 )125.91 2.8434 0.888982 )2.0115 )0.092621 1.6918 1.6841 )0.017524 )124.9 )0.49624 )3.98193 4.2615 0.047581 5.4929 0.013381 )0.017577 )125.22 )0.49625 )3.81574 19.49 1.0484e)12 1.6918 2.31 )2.1483 )0.13082 )7.9792 )8.5091e)10

S 0.43686 0.30387 0.80212 3.5249r 2.3394 1.0259e)7 0.027966 0.78817

Fig. 5. Plots of the relative errors (%) against stretch k in respectof the Ogden strain energy with M 3 in equibiaxial tension forthe parameter sets OSS1, OSS2, OSS3 given in Table 3

Fig. 6. Plots of the relative errors (%) against stretch k in respectof the Ogden strain energy with M 4 in equibiaxial tension forthe parameter sets OT, OSS1, OSS2, OSS3 given in Table 4

Table 3. Parameter values forthe Ogden model with M 3:equibiaxial tension

i OSS1 OSS2 OSS3

ai li ai li ai li

1 )6.7054 )4.0441e)9 1.6918 4.4184 4.259 0.138732 1.58 5.1733 5.4929 0.013381 )0.022671 )144.693 )2.4591 )0.03445 1.6919 0.47456 )0.022671 )142.01

S 0.31666 0.30387 0.70829r 0.020419 2.821e)5 0.0088361

case we optimize the objective function (21), wherep a1; l1; . . . ; aM; lMT.

Consider M 3. By starting the algorithm with, forexample, the optimal set OT Table 1 and also from 10random choices, the optimal set obtained is, in eachcase,

i ai li1 8:3952 1:2069e)52 1:8821 3:77293 2:2453 0:052171

32

with S 20:013, r 0:0015682, Ss 17:233, Se 2:7801,where Ss and Se are the residuals for the simple tension

Fig. 7. Fit to the simple tension and equ-ibiaxial tension data (circles) using the OT Table 1 parameter set obtained from thesimple tension data, and the relative errors:M 3. In the left-hand (right-hand) figuresnominal stresses (relative errors) are plot-ted against stretch

Fig. 8. Fit to the simple tension and equ-ibiaxial tension data (circles) using theOSS2 Table 3 parameter set obtained fromthe equibiaxial tension data, and therelative errors: M 3. In the left-hand(right-hand) figures nominal stresses(relative errors) are plotted against stretch

and equibiaxial tension data separately. The correspond-ing fits and relative errors are shown in Fig. 9. Note thatrelative errors around the 10% level are obtained.

If M 4, the solution obtained isi ai li1 2:7971 0:778172 2:7188 0:0112293 10:505 1:269e)74 0:33382 16:169

33

and in this case we have S 10:904, r 1:3192,Ss 10:427, Se 0:4765, and the simple tension andequibiaxial tension fits are both shown in Fig. 10 alongwith the relative errors.

4.5Biaxial dataAs discussed in Ogden [6] as well as in the book [9], the setpog given by

Fig. 9. Fits using both simple andequibiaxial tension data (circles) and thecorresponding relative errors: M 3. In theleft-hand (right-hand) figures nominalstresses (relative errors) are plotted againststretch

Fig. 10. Fits using both simple and equi-biaxial tension data (circles) and the cor-responding relative errors: M 4. In theleft-hand (right-hand) figures nominalstresses (relative errors) are plotted againststretch

i ai li1 1:3 0:692 4 0:013 2 0:0122

34

provides a good prediction of the biaxial data for thematerial tested by Jones and Treloar [2], which is a dif-ferent material from that used by Treloar [1] so that cross-correlation between the two sets of data is not admissible.The basis for the fitting in this case is the stress differencer1 r2, calculated from (7) and specialized for (14),plotted against k1 for a series of fixed values of k2. Therelevant function Fk; p is given by

Fk1; p XM

i1likai1 kai2 ; 35

with p l1; a1; . . . ; lM; aMT, M 3 and k2 fixed. Weestimate that pog produces a minimum residual

Smin 0:026728 and a maximum residual Smax 0:035938on the biaxial data amongst the different values of k2.

In our simulation, the NLS algorithm is applied foreach set of data with k2 fixed and then predictions for thesets of data with other values of k2 are derived using theoptimal set of parameters obtained. Also, for the biaxialdata, we show that the Ogden strain-energy functiongives good fits with several different optimal sets ofparameters. In fact, by starting the NLS procedure frompog, we find the set reported as OG in Table 5, whilestarting with 30 random samples different local minimaare obtained. Some of these sets of parameters are shownin Table 5 together with the corresponding minimum andmaximum residuals between the different values of k2. InFig. 11 we show the fits obtained using the OSS1 set(which has the lowest residuals) and the correspondingrelative errors.

If the Ogden strain energy with M 4 is considered,then the optimal sets in Table 6 are obtained, but the

Fig. 11. Fit of the biaxial data (circles)using the Ogden strain energy for theoptimal set OSS1 in Table 5, and the cor-responding relative errors: M 3. In theleft-hand figure r1 r2 is plotted againstk1 for the fixed values of k2 indicated

Table 5. Parameter values forthe Ogden model with M 3:biaxial deformation

i OG OSS1 OSS2

ai li ai li ai li

1 1.2921 0.7332 6.8269 3.8409e)4 4.5686 0.00489522 4.5537 0.0049951 1.3193 0.73974 1.2935 0.732753 )2.4381 )0.0052363 )2.6059 )0.0040178 )2.4363 )0.0052583

Smin 0.0064745 0.004571 0.0064744Smax 0.061694 0.051234 0.06157r 2.0927e)6 3.6893e)6 5.63e)7

relative errors are not improved compared with the case ofM 3.

5Limiting chain extensibility modelsIn order to contrast with the results in Sect. 4 we nowconsider the strain energy introduced by Pucci andSaccomandi [4], which we referred to in Sect. 1 as thePucciSaccomandi model. The Gent model itself reflects,at the phenomenological level, the behaviour of molecularmodels based on Langevin statistics. In particular, it rep-resents very well the stiffening of the material at largestrains. It has the advantage of mathematical simplicityand allows detailed analysis and explicit solution of par-ticular boundary-value problems. Since the PucciSacco-mandi model is a modification of the Gent model [16] withthe addition of a logarithmic term similar to that in theGent model we also refer to it as the Gent Gent model.

We use the abbreviation GG to identify the model, whichhas the form

WI1; I2 l2

Jm log 1 I1 3Jm

C2 log I23

;

36

where Jm is a positive material constant that identifies theupper limit of I1 (i.e. I1 < 3 Jm) associated with limitingchain extensibility in the molecular-based models, C2 isanother material constant (associated with the contribu-tion of I2) and we recall the expressions (16) for theinvariants I1 and I2, which are subject to (3). Note thatl 2C2=3 is the infinitesimal shear modulus and that theoriginal Gent model corresponds to C2 0. The associatedprincipal stress differences may be calculated from theformulas (18), but we do not give them explicitly here.

By fitting the GG model for the simple tension andequibiaxial tension data we obtain the sets of parametersin Table 7. In Figs. 12 and 13 we show that each setexhibits a valuable predictive property since relative errorsbelow 20% are generated when the parameters obtained bythe simple tension (equibiaxial tension) fit are used tocompare the theory with the data for equibiaxial tension

Table 6. Parameter values for the Ogden model with M 4:biaxial deformation

i OSS1 OSS2

ai li ai li

1 )0.0067956 )15.013 )0.0079652 )17.8242 )3.3132 )8.0266e)4 )3.4693 4.5919e)43 1.7426 0.43689 2.0035 0.33014 )0.0067961 )15.284 )0.0079625 )18.026

Smin 0.0058526 0.00948Smax 0.53747 0.12028r 1.1627e)5 3.2675e)4

Fig. 12. Fit of the GG model to simpletension and equibiaxial tension data (cir-cles) and the corresponding relative errorswith the optimal parameter set based onuse of the simple tension data. In the left-hand (right-hand) figures nominal stresses(relative errors) are plotted against stretch

Table 7. Parameter values from the fit of the GG model to simpletension and equibiaxial tension data

l Jm C2 S rSimple tension 2.4195 77.931 1.0233 7.6082 4.4123e)5Equib. tension 3.1674 88.13 0.43659 0.36221 8.1682e)8

(simple tension). We remark that the GG model does notproduce multiple local minima in the NLS optimizationapproximation. This is associated with the form of the logfunction and the fact that only three material constants areinvolved.

If the two data sets are fitted together according to (21)then the NLS yields the optimal set

l 2:5343; Jm 79:793; C2 1:0368;

with residual S 38:754, r 2:58e)5, Ss 12:574,Se 26:179. The fits and the corresponding relative errorsare shown in Fig. 14. Note that the coupled data do notlead to an improvement in the fit. This is because thelimited number of constants does not allow enough flexi-bility to accommodate the additional data and the con-straints associated with the location of the limiting-chainextensibility asymptotes.

Fig. 13. Fit of the GG model to simpletension and equibiaxial tension data (cir-cles) and the corresponding relative errorswith the optimal parameter set based onuse of the equibiaxial tension data. In theleft-hand (right-hand) figures nominalstresses (relative errors) are plotted againststretch

Fig. 14. Fit of the GG model to simpletension and equibiaxial tension data(circles) and the corresponding relativeerrors with the optimal parameter setbased on use of both the simple tensionand equibiaxial tension data together.In the left-hand (right-hand) figuresnominal stresses (relative errors) areplotted against stretch

If the GG model is used to fit the biaxial tension data, wefind that each data set for a fixed value of k2 yields a goodfit, with the values of l and C2 very similar in each case.For example, the set

l 0:34676; Jm 85:79; C2 0:05471is obtained for k2 1 and then used to fit the data for theother values of k2. The resulting fits are shown in Fig. 15.The minimum and the maximum residuals with respect toall the biaxial data are Smin 0:039846, Smax 0:9132.Note, however, that the predictive property of the biaxialtension fitting is not good for compressive strains, irre-spective of the value of Jm.

Let us now apply the GG model to the biaxial case byfitting separately the data in the compression and exten-sion ranges. Consider the data for k1 < 1. The solution

l 0:3313; Jm 32:823; C2 0:050567is obtained with residuals Smin 0:00045909, Smax 0:37933. The fits and relative errors are shown as the uppertwo figures in Fig. 16. Consider next the data for k1 1.Then, the solution

l 0:34112; Jm 133:52; C2 0:10485is obtained with residuals Smin 0:0010215, Smax 0:016084. The corresponding fits and relative errors areshown in the lower two figures in Fig. 16.

This difference between the two estimated values of Jmmay be clarified by considering simple tension. For thiscase I1 k2 2=k. Therefore, for fixed Jm I1max 3, theresulting cubic equation k3 Jm 3k 2 0 for k hasthree real roots. One of these roots is negative and need

not be considered; of the other roots one is less than andone greater than 1. The roots are important because theyfix the locations of the asymptotes between which thesimple tension curve lies. Clearly, when the stresses for thedata in tension and in compression are considered theasymptote in the compressive region and that in tensionzone will not in general correspond to the same value ofJm. This suggests that the assumption of isotropy should beviewed with caution since it is clear that we do not havereflectional symmetry with respect to the undeformedconfiguration.

To avoid this limitation of the GG model, one approachis to modify it by incorporating an element of anisotropy.A possible procedure for incorporating anisotropy in aslightly different context, but which could be adapted here,is provided by Horgan et al. [17]. In the present paper,however, we restrict attention to isotropy.

6Logarithmic strain based modelsIn Criscione et al. [5], a new constitutive formulationbased on certain invariants, denoted K1;K2;K3, of thenatural or Hencky strain is considered. The latter hasprincipal values ln k1; ln k2; ln k3. These invariants pos-sess the property that they form an orthogonal invariantbasis, so that the associated stress response terms aremutually orthogonal [18]. The authors assert that this isadvantageous since the orthogonal entities exhibit a min-imum covariance and therefore maximal independence. Inparticular, for incompressible materials, K1 lnk1k2k3 0, and, in terms of the stretches, the othertwo invariants are given as

Fig. 15. Fit of the GG model to biaxialdata (circles) and the relative errors. Inthe left-hand figure the stress differencer1 r2 is plotted against k1 for thefixed values of k2 indicated

K2 ln k12 ln k22 ln k32

q;

K3 36

p ln k1 ln k2 ln k3K32

; 37

and the strain energy of an isotropic material may beregarded as a function of K2 and K3: W WK2;K3. In[5] a specific form of energy depending linearly on K3 is

Fig. 16. Fit of the GG modeland the relative errors sepa-rately for biaxial compression(upper figures) and biaxialtension (lower figures); thecircles correspond to the data.In the left-hand figures thestress difference r1 r2 isplotted against k1 for the fixedvalues of k2 indicated

considered for purposes of comparison with theJonesTreloar data. Explicitly, Criscione et al. [5] consider

WK2;K3 X1

i2

1

ici1K

i2 K3

X1

j3njK

j2; 38

where ci, i 1; 2; . . ., and ni; i 3; 4; . . ., are materialconstants and l c1=2 is the infinitesimal shear modulus.

The model may be approximated by a truncated N-termseries, so that, in particular,

oWoK2

XN

i1ciK

i2 K3

XN1

j1nj2j 2Kj12 :

Then, 2N 1 parameters p c1; . . . ; cN ; n3; . . . ; nN1Thave to be determined.

In this case application of the LS method yields alinear problem. In fact, if m is the size of the data set,an over-determined linear system Ap s has to besolved, where A : R2N1 ! Rm. In the following wedenote by Ss kAsp ssk22 the residual for simpletension.

It is easy to show that for the simple and equibiaxialtension tests K3 1;K2

6

p ln k=2 and K3 1,K2

6

pln k, respectively. For biaxial tension, we have

Fig. 17. Fits of the Criscionestrain energy (N 3) for sim-ple tension data (circles, upperleft figure) and relative errors(upper right figure), and com-parison of the predictive capa-bility on all the data (circles)with respect to the MooneyRivlin and Ogden M 1models (lower figure)

K2 2

p ln k12 ln k22 ln k1 ln k2

q;

K3 36

p ln k1 ln k2ln k1 ln k2K32

;

and the Cauchy stress difference

r1 r2 k1 oWK2;K3ok1 k2oWK2;K3

ok2may be calculated accordingly.

In the analysis of the JonesTreloar data conducted in[5] the value N 3 is suggested. Here, we begin the fittingof the Treloar simple tension data by taking N 3. Wefind that not only is the fit poor quantitatively, but it is alsopoor qualitatively and the main features of the data curveare not captured (see Fig. 17). To obtain an acceptablefitting we have determined that it is necessary to increasethe value of N up to 8 (which involves 15 materialparameters).

If we fix N 3 it is possible to obtain a good fit only byrestricting attention to moderate stretches (k < 3). (In-deed, this is the range of stretches for all the examplesreported in [5].) The parameters identified for this reducedrange of stretches are

pred fc1 9:7841; c2 0:4506; c3 0:3789;n3 1:3517; n4 1:5157g;

with residual Ss 0:0424. The optimal set obtained for thewhole range of data is

pall fc1 58:073; c2 11:351; c3 3:2796;n3 34:053; n4 13:118g;

with residual Ss 985:59.In Fig. 17 (upper) we show both fits for the Criscione

model with N 3 and the corresponding relative errors.We note, however, that for k < 3 very good agreement isalso obtained using, for example, the Ogden model forM 1 a1 2:1474; l1 1:2487; Ss 0:0556, or theclassical Mooney-Rivlin model c1 1:0505; c2 1:0878;Ss 0:0716. In Fig. 17 (lower), we show the predictivecapability of these two models compared with that of theN 3 Criscione model on the whole range of simpletension data.

In Sect. 4 and 5 we have seen that the main differencebetween the Ogden and GG models is that the first oneadmits multiple NLS solutions (for M 3) while thesecond one admits a unique NLS solution for each dataset. Concerning the linear least squares (LLS) approxi-mation of the Criscione model, it is easy to verify thatwhen simple, equibiaxial and biaxial tension data areconsidered the coefficient matrix A does not have fullrank, which means that infinitely many solutions withthe same residual are present. This is due to the choiceof the particular polynomial form of WK2;K3. In thiscase, we have calculated the solution with minimum2-norm by means of the singular value decomposition(SVD) of A according to well known formulae, given, forexample, in [19].

Criscione et al. [5] did not encounter the problem ofmultiple solutions. In fact, they took advantage of theproperties of the K invariants in order to identify theparameters c c1; c2; c3T and n n3; n4T by means oftwo different fits with third order (N 3) polynomials,that is by solving two different LLS problems, namely

min jjA1c b1jj22; min jjA2n b2jj22;where A1 2 Rm3;A2 2 Rm2 are in the class of Vander-monde matrices and b1 and b2 contain the Cauchy stressdata or linear combinations of these data. In [5] theseproblems are solved only for pure shear (k2 1) and theparameters identified are

pcr fc1 0:96; c2 0:16; c3 0:35; n3 0:065;n4 0:039g:

The authors declare a good predictive capability for this setof parameters on the simple, equibiaxial and biaxial tensiondata. However, we find that for simple tension the set pcrleads to the residuals Ss 110:6067 with k < 3 andSs 20975 if all the values of k are included. Moreover, theirequibiaxial data are extracted from the biaxial ones and aredifferent from those considered in the present paper.

It is worth noting that to obtain good fits for all thestretches it would be necessary to consider, for example,A1 2 Rm8. But, it is well known that the Vandermondematrices become ill-conditioned as the dimensionsincrease.

7A test boundary-value problemIn this section we solve the boundary-value problem(BVP) arising in the modelling of rectilinear shear for thestrain energies examined in Sects. 4 and 5. In particular,for the Ogden model we show how the presence of mul-tiple solutions in the NLS approximation carries over tothe approximation of the BVP solution. For this purpose,for each table in Sect. 4, we illustrate the results graphi-cally and compare the numerical solutions correspondingto the different sets of parameters.

The differential equation and boundary conditionsmodelling the deformation yx in a slab x 2 0; 1 are

Cy02 k^ x 12

; y0 y1 0; 39

where

Cy02 2 W1y02 W2y02;k^ is a constant and Wiy02 is the value of o W=oIi forI1 I2 3 y02. By differentiation, we transform theproblem into a second-order two-point BVP and we usethe function bvp4c in MATLAB 6.5 as the numerical solver.Since we are interested in comparing the solutionsindependently of the values of k^, we fix it as k^ 0:5 andnote that if higher values of k^ are chosen the code is notalways able to approximate the solution since boundarylayers appear, in which case a different numerical method

coupled with a more suitable stepsize variation strategywould be needed.

7.1The Ogden strain energyLet us solve the problem (39) using the Ogden strainenergy for the cases M 3 and M 4 with the parametersreported in Tables 1 and 2 and (23) and (24) (simpletension) and in Tables 3 and 4 and (30) and (31) (equi-biaxial tension). The numerical solutions are shown inFigs. 18 and 19, respectively. Included, additionally, arethe results, labelled OSSse, based on the parameter sets (32)and (33) for M 3 and M 4, respectively. We note thatthe presence of multiple optimal sets of parameters fromthe NLS fittings produces numerical BVP approximationsthat are similar qualitatively but often very differentquantitatively. In Fig. 18 the labels of the solutions cor-responding to the smallest residuals in the parameter fitsare shown in bold type. They are OSSs1 for M 3 and OSSs3for M 4 and we remark that in each case they corre-

spond to the uppermost solution. The curves labelled set(23) and set (24) are based on the parameters in (23) and(24), respectively. In Fig. 19 the picture is similar exceptthat the solution corresponding to the lowest residual isOSSe1 for both M 3 and M 4. In the case of M 4,however, the data set OT (from Table 4), which also hassmall, but a slightly larger, residual, yields the uppermostsolution.

7.2The GG strain energyThe BVP has also been solved for the GG strain energy forthe parameters determined by the fits for all the defor-mations examined in Sect. 5. The optimal sets were givenin Sect. 5 and are collected together here for conveniencein Table 8, in which the subscripts indicate the associateddeformation (s: simple tension; e: equibiaxial tension; se: sand e combined; biax: biaxial data; biax-c: biaxial data incompression; biax-e: biaxial data in extension). Thecorresponding solutions of the BVP are shown in Fig. 20.Note, however, that the first three sets of parameterscorrespond to the Treloar data and the last three to theJonesTreloar data, which, we mention again, are for adifferent material.

8Concluding remarksIn this paper we have carried out a systematic study of thefitting of stress-stretch equations for incompressible iso-tropic hyperelastic constitutive laws to experimental datafor rubber in simple tension, equibiaxial tension and

Table 8. Parameters for the GG strain energy

GG l Jm C2

GGs 2.4195 77.9310 1.0233GGe 3.1674 88.1300 0.4366GGse 2.5343 79.793 1.0368GGbiax 0.3468 85.7900 0.0547GGbiax-c 0.3313 32.8230 0.0506GGbiax-e 0.3411 133.5200 0.1049

Fig. 18. Solution of the BVP (39) using the Ogden strain energywith different parameter sets obtained from the simple tensionfits: M 3 (left figure) and M 4 (right figure)

general biaxial tension tests. We have focused on thedetermination of material parameters and the corre-sponding relative errors. We have highlighted, in partic-ular, the occurrence of multiple sets of optimal materialparameters in the Ogden strain-energy function, which isformulated in terms of principal stretches, and the con-sequences of this for the solution of boundary-valueproblems. It appears that a study of this kind has not beenconducted previously, although the paper by Gendy andSaleeb [20] should be mentioned since it considers certain

aspects of the optimization problem in respect of theOgden strain-energy function. However, in that paper thequestion of multiple optimal sets of parameters was notencountered.

For comparison, we have applied a similar analysis andfitting procedure to the PucciSaccomandi model, which isbased on the invariants I1 and I2, and to the Criscionemodel, formulated in terms of the invariants K2 and K3.We have found that the PucciSaccomandi model, con-sidering that it involves only three material constants,

Fig. 20. Solution of the BVP (39) using the GGstrain energy with parameters obtained from all thedata fits with parameters given in Table 8

Fig. 19. Solution of the BVP(39) using the Ogden strainenergy with parameters ob-tained from the equibiaxialtension fits: M 3 (left figure)and M 4 (right figure)

gives a good fit to the range of data except for compressivevalues of the stretch in biaxial tests.

The Criscione model, while fitting the JonesTreloardata well with five material constants for moderate stret-ches, does not provide an adequate representation of thedata at larger stretches, such as those achieved in theTreloar simple tension tests, although this can beimproved at the expense of introducing significantly morematerial parameters. Thus, the feature of the specialinvariants introduced in the Criscione model that theyminimize the covariance between the various tensorialquantities in the Cauchy stress representation formuladoes not appear to be advantageous in helping to findeasily and directly improved functional forms for possiblestrain energies. In fact, since the considered space is finitedimensional all norms are equivalent and it cannot beexpected that there is a privileged set of invariants forpurposes of fitting the data.

We hope that the results in this paper will be usefulfor improving understanding of problems encounteredin fitting theory to experimental data and their practicaleffect on the solution of boundary-value problems. Onepossible outcome of this study is that, at least in thecase of elastomeric materials, it is advisable to examinetheir behaviour first via mathematical models ratherthan by simulation models since the former can bedesigned to capture in a clear way the main qualitativefeatures of the experimental data while the latter oftenaim to produce quantitative predictions by introducinglarge numbers of material parameters whose physicalsignificance is unclear.

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