Rubber Modeling Using Uniaxial Test Data

Rubber Modeling Using Uniaxial Test Data

G. L. BRADLEY,1 P.C. CHANG,2 G. B. MCKENNA3

1 Alvi Associates, Inc., Towson, Maryland 21204

2 Department of Civil engineering, University of Maryland, College Park, Maryland 20742

3 Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409-3121

Received 8 March 2000; accepted 17 May 2000Published online 8 May 2001

ABSTRACT: Accurate modeling of large rubber deformations is now possible with finite-element codes. Many of these codes have certain strain-energy functions built-in, but itcan be difficult to get the relevant material parameters and the behavior of the differentbuilt-in functions have not been seriously evaluated. In this article, we show thebenefits of assuming a Valanis–Landel (VL) form for the strain-energy function anddemonstrate how this function can be used to enlarge the data set available to fit apolynomial expansion of the strain-energy function. Specifically, we show that in theABAQUS finite-element code the Ogden strain-energy density function, which is aspecial form of the VL function, can be used to provide a planar stress–strain data seteven though the underlying data used to determine the constants in the strain-energyfunction include only uniaxial data. Importantly, the polynomial strain-energy densityfunction, when fit to the uniaxial data set alone, does not give the same planarstress–strain behavior as that predicted from the VL or Ogden models. However, thepolynomial form does give the same planar response when the VL-generated planardata are added to the uniaxial data set and fit with the polynomial strain-energyfunction. This shows how the VL function can provide a reasonable means of estimatingthe three-dimensional strain-energy density function when only uniaxial data areavailable. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 81: 837–848, 2001

Key words: earthquake bearing; finite element analysis; mechanical properties;Mooney–Rivlin material; Ogden function; Rivlin expansion; rubber; strain energy func-tion; Valanis–Landel function

INTRODUCTION

Elastomeric bearings used for earthquake loadisolation consist of alternating layers of steel andrubber—the latter vulcanized under high temper-

ature—to form a composite bearing. Under axialloading, the rubber in the bearing expands out-ward. This deformation leads to the rupture ofsteel in the radial direction or debonding betweensteel and rubber. In prior work, the load-deflec-tion response of such a bearing was modeled us-ing the ABAQUS1 finite-element code.2 In thisarticle, we present the methodology used to de-termine the parameters employed in the consti-tutive model for the rubber used in that study.

Most finite-element codes today include a hy-perelastic constitutive model for analysis. In the

Correspondence to: G. B. McKenna ([email protected]).

Contract grant sponsor: Building and Fire Research Lab-oratory and Materials Science and Engineering Laboratory,National Institute of Standards and Technology, Gaithers-burg, MD.Journal of Applied Polymer Science, Vol. 81, 837–848 (2001)© 2001 John Wiley & Sons, Inc.

837

case of the ABAQUS version available at the timeof the current work, the hyperelastic models in-cluded were a multiterm Rivlin expansion and amultiterm version of the Ogden model. Both in-clude the possibility of material compressibility.A major difficulty in the use of such finite-elementcodes is the experimental determination of thematerial parameters to use in the model chosenfor calculation. For the polynomial expansion inthe invariants of the deformation tensor, the ex-perimental data needed include uniaxial, biaxial,and planar extension (pure shear) tests. Uniaxialdeformations include both tensile and compres-sive deformations. Yet, it is typical that only uni-axial tension tests are performed and this maylead to limited accuracy when attempting to de-termine the strain-energy density function. Inparticular, the exclusion of compression data mayresult in a strain-energy density function thatdoes not capture much of the strain energy’s de-pendence on I2. For this reason, tests were per-formed in both tension and compression.

Furthermore, measurements of pure shear ornonequal biaxial responses to obtain additionaldata, which would improve the performance ofthe polynomial model in the full range of defor-mation geometries, requires special mechanicaldevices that are expensive to make and not al-ways practicable to build. In this article, we showthat one possible and practical alternative to per-forming the pure shear (or other) tests is to usethe Valanis–Landel (VL) function3 to generatesuch data from experimental uniaxial tension andcompression data alone. An important point thatis made in the article is that, to the extent thatthe VL function is valid, it provides a means toestimate deformations other than uniaxial ten-sion and compression that differ from those esti-mated from the polynomial form of the strain-energy function when the latter is fitted to thesame set of uniaxial data. The polynomial form,when fitted to uniaxial tension, uniaxial compres-sion, and the VL-generated pure shear data, givesgood fits to the entire data set. This result sug-gests the need to adequately consider more thanjust uniaxial data when computing strain-energyfunctions for rubber. Finally, in the work here, wedescribe the use of a compressible strain-energyfunction. The use of such a function is importantin earthquake bearings because the high pres-sures involved in their loading can lead to asmuch as 10% vol change in the rubber.

CONSTITUTIVE MODELING OF RUBBER

General Considerations

Rubber is a nonlinear, nearly elastic materialeven at large strains. Such behavior is well char-acterized by a hyperelastic model. Here, we as-sume that the rubber is isotropic; hence, thestrain-energy density function can be written as afunction of the strain invariants I1; U 5 (I1, I2, I3),where I3 is the square of the volume ratio J (J5 =I3) and is equal to 1 for a perfectly incom-pressible material. It is common to assume thatrubber materials are incompressible when thematerial is not subjected to large hydrostaticloadings. However, in the case here, large hydro-static stresses arise in the compressive loading ofthe earthquake bearings and we need to considerboth the compressible and the incompressibleproblems.

Practically, solid rubber has a Poisson’s ratiothat ranges from 0.49 to 0.50.4 As described sub-sequently, the Poisson’s ratio for the rubber usedin this study was found by a volumetric compres-sion test to be 0.4994. Hence, the rubber is nearlyincompressible. If a material is incompressible,the hydrostatic stress cannot be found from thedisplacements, since the application of a hydro-static stress results in no deformation. “Mixed”formulations have successfully dealt with thisproblem.5,6 In a mixed formulation for a perfectlyincompressible material, the internal energy isaugmented by adding the term p(J 2 1), where pis a Lagrange multiplier (the hydrostatic stress)introduced to impose the constraint J 2 1 5 0 (J2 1 is the volumetric strain). This allows thehydrostatic stress to be approximated directly,independent of the displacements. The stress thatis derived from the displacements is the devia-toric stress. The sum of the hydrostatic and de-viatoric stress tensors gives the total stress ten-sor. We note that this is an approximation, asPenn7 and Fong and Penn8 showed deviationsfrom such a separability of deviatoric and hydro-static components in measurements of volumechanges in rubber under large uniaxial deforma-tions.

Following the development in ABAQUS,1 thedeviatoric portion of the strain energy can bewritten using revised invariants that remove anyeffect due to volume change. The strain energy iswritten as

U 5 U~I#1, I#2, I3! 5 U# ~I#1, I#2! 1 U~I3! (1)

838 BRADLEY, CHANG, AND MCKENNA

where the bars over the first two strain invariantsindicate removal of volume changes.

The decoupling of the deviatoric and volumet-ric strain energy can only be valid if the bulkmodulus is a constant (Sussman and Bathe,9 butsee also Penn7 and Fong and Penn,8 cited above).A constant bulk modulus implies that the pres-sure/volume ratio is constant. Results of volumet-ric testing, discussed later, show that the bulkmodulus is a constant, independent of the applieduniaxial compressive deformation. On this basis,the decoupling of the strain energy into deviatoricand volumetric portions is assumed to be a rea-sonable approximation for the rubber tested and,therefore, the use of the mixed approach is justi-fied.

Two forms of the strain-energy density func-tion based on the mixed formulation were used.The first is the polynomial strain energy densityfunction, in which the strain energy is

U 5 Oi1j51

N

Cij~I#1 2 3!i~I#2 2 3!j 1 Oi51

N 1Di

~Jel 2 1!2i

(2)

The first summation is the contribution due todeviatoric effects, and the second summation isthe contribution due to volumetric effects. The Cijand Di are parameters that are found from therubber test data and I#1 and I#2 are the first andsecond invariants of the Cauchy–Green deforma-tion tensor with the volume change removed. Jelis the ratio of the current volume to the originalvolume excluding thermal effects. For N 5 1, thedeviatoric contribution to the strain energy iscalled the Mooney–Rivlin function. It is oftenwritten as

U 5 C10~I#1 2 3! 1 C01~I#2 2 3! (3)

In addition to the well-known invariant expan-sion described above, the strain-energy densityfunction of an isotropic elastic material can alsobe written in terms of the stretches l. Valanis andLandel3 postulated that the strain-energy densityfunction is a function that is separable in terms ofthe principal stretches, and the total strain en-ergy is

U 5 u~l1! 1 u~l2! 1 u~l3! (4)

We will refer to u( z ) as the Valanis–Landel or VLfunction.

In ABAQUS, a special form of the VL functionfor the strain-energy density is provided—it is theOgden strain-energy density function, which wealso consider in this study. The contribution dueto deviatoric effects is written as a function of thestretch ratios. In ABAQUS, a compressible formof the Ogden function is also provided and thecontribution due to volumetric effects is the sameas that used for the polynomial strain-energyfunction:

U 5 Oi51

N 2mi

ai2 ~l# 1

ai 1 l# 2ai 1 l# 3

ai 2 3! 1 Oi51

N 1Di

~Jel 2 1!2i

(5)

In eq. (5), mi, ai, and Di are parameters that arefound from the rubber test data. The principalstretch ratios l# k

ai have the volume change re-moved. To determine the deviatoric parametersCij in eq. (3) and mi and ai in eq. (5), experimentaldata can be fitted to one or more of the threedeformation modes: uniaxial (tension or compres-sion), equibiaxial (tension or compression), andplanar (tension or compression; this is pure shearwhen the material is incompressible).1 We re-mark further that the Cij and the mi and ai, areassumed to be independent of the volume.

Using a right-handed Cartesian coordinatesystem with axes x1, x2, and x3, let the uniaxialdeformation mode correspond to an application offorce along the longitudinal axis of the specimen.For the purpose of explanation here, we denotethis axis x1. Then, the specimen is free to expandor contract along axes x2 and x3. The equibiaxialdeformation mode corresponds to equal stretchesalong the x1 and x2 axes, with unrestrained move-ment along the x3 axis. Planar deformation corre-sponds to a stretch along the x1 axis, with nostretch allowed along the x2 axis, and unre-strained movement along the x3 axis. Each ofthese modes is an example of a deformation inwhich the directions of principal strain do notchange, that is, the deformations take place alongthe principal axes. Each mode can be written interms of a single stretch, which allows the stress–strain relationship to be easily measured. To de-termine the isotropic (or hydrostatic) parametersin Di in eqs. (2) and (4), constrained compressiontests in which volume change could be measuredwere performed.

RUBBER MODELING USING UNIAXIAL TEST DATA 839

For an incompressible material, uniaxial com-pression is equivalent to equibiaxial tension, uni-axial tension is equivalent to equibiaxial com-pression, and planar tension is equivalent to pla-nar compression. For the material analysisperformed here, experimental testing was per-formed in uniaxial tension and uniaxial compres-sion. A planar test could have been done had theappropriate equipment been available and mayhave captured dependence on the strain invari-ants that was missed by the uniaxial tests. Un-fortunately, this experiment requires more exten-sive preparation and tooling than was availablein our laboratories. In lieu of an actual planartest, the stress–strain response in planar tensionwas calculated using the VL function, which hasbeen shown to be an excellent descriptor of actualrubber behavior.10–13

Determination of the VL Strain-energy Function

As noted above, apart from the constrained com-pression, the mechanical testing of the rubberwas performed only in the uniaxial tensile andcompressive modes. The planar deformationmode is independent of either of the uniaxial ten-sion or uniaxial compression modes. No planardeformation tests were done, since a test appara-tus for this mode of deformation was neitheravailable nor practicable to build for small speci-mens. Planar deformation is also referred to as apure shear or strip biaxial deformation. It is dif-ferent from simple shear deformation in that theprincipal strain directions remain unchangedduring the deformation. Because of the lack oftest data and the known success of the VL func-tion in describing the behavior of rubber (see ref-erences cited above), we used the VL function togenerate stress data for planar deformation as ifthe tests had been performed.

Kearsley and Zapas13 showed how to obtain theVL function from uniaxial tensile and compres-sive data, which we did obtain. In terms of thetrue stress t and the stretch l, the following re-cursive expression can be used to obtain the de-rivative u9(l) of the VL function:

limn3`

Ok50

n21

@t~l~1/4!k! 1 t~l2~1/2!~1/4!k

!# 5 lu9~l! (6)

The right-hand side of eq. (6) is an infinite sum ofthe true stresses at the stretches specified in thebracketed expression. The rapid convergence ofthis infinite series allows one to obtain the VL

function with a small number of terms. Note thatthe first and second terms are on opposite sides ofthe undeformed state (l 5 1), so that both exten-sion and compression data are required, and theterms of the series converge rapidly from bothsides of the undeformed state. A computer pro-gram was written to find the values m9(l) at thedesired stretch values. The formula for uniaxialengineering stresses, written in terms of the de-rivative of the VL function, is

s 5 u9~l! 2 l23/2u9~l21/2! (7)

EXPERIMENTAL

Materials

The material used in this study was a carbonblack-reinforced rubber compound of the samecomposition as that used in the manufacture ofthe elastomeric bearings. It was provided in theform of sheets provided by the bearing manufac-turer and prepared according to ASTM D3182-89.14 Nominal rubber mechanical properties pro-vided by the manufacturer are presented in Ta-ble I.

Mechanical Testing Procedures

All rubber testing was performed on an Instron1125 test machine. The accuracy of this machinewas established through a calibration test usingknown dead loads and was calibrated according toASTM E4, which requires that the machine read-ing be within 1% of the true reading. The machinemet these requirements. This is equivalent to arelative expanded uncertainty (k 5 2) of 1% and issmall compared to specimen variability for therubber uniaxial testing.

The uniaxial tension tests were performed upto engineering tensile strains of 6.00, very nearthe ultimate elongation of the rubber, to ensurethat the strain-energy density function would ad-equately represent the severe engineering tensilestresses and strains expected in the finite-ele-ment analysis of the elastomeric bearings. Thetesting was performed on 14 dumbbell specimens.The variation about the mean of these tests wasapproximately 610%, so the variability can beattributed to factors outside the resolution of thetesting machine. Sources of variation include (1)differences in the properties of the rubber and (2)measurement error. Displacements were re-corded manually using a hand-held caliper, and


force readings were manually taken at the time ofeach displacement reading. Measurement errorin the cross-sectional area can also affect the re-sults. Based on this limited testing sample andmanual testing methodology, it is difficult to sayhow much of the variability is due strictly to dif-ferences in the properties of the rubber, but it islikely that the largest portion of the variability isdue to the manual measurement process. Testingof the rubber was performed at strain rates rang-ing from 5 3 1024 to 4.4 3 1022 s21. There was notrend of the data with the strain rate and the testresults reported here are for the strain rate of 53 1024 s21 (see Bradley2).

The uniaxial compressive tests were carried toa stress level exceeding 100 MPa. At this stresslevel, the engineering compressive strain ex-ceeded 0.85. To get accurate displacement read-ings for the compression test, machine compli-ance was removed from the displacement data.The uniaxial compression test was made to sim-ulate homogeneous compression, in whichstraight lines before deformation remain straightafter deformation. This test requires the loadedends of the specimen to slide freely along theloaded surface, so there is no bowing of the un-loaded surface. Attaining perfect homogeneouscompression requires a frictionless contact be-tween the platen and the sample surface. Thiscondition was approximated by applying a lubri-cant to the specimen’s plane faces.

Whether or not bowing of the unloaded sur-faces occurred was very difficult to determine vi-sually because the loaded specimens became verythin and the lubricant squeezing out from theedges of the loaded surfaces obscured the view.After unloading, the circle of lubricant on theplaten showed that the loaded surface had slippedconsiderably. To assure ourselves that, in fact,the specimens were not bowing very much, wetested two specimens in compression, one with ashape factor of 0.55 and the other with a shapefactor of 1.10. The shape factor, S, is a measure of

the relative thickness of a specimen. It is definedas the ratio of the area of one loaded surface to thearea of the unloaded surface. For cylindrical spec-imens, S 5 r/2T, where r is the radius and t is thethickness of the specimen. The responses werenearly identical after removal of the machinecompliance. Since the stress–strain response is astrong function of the shape factor for bondedspecimens16,17 and is independent of shape factorfor homogeneous compression, this result gives usconfidence that the compression test was a fairlygood representation of homogeneous compres-sion. For the two compression tests done, the vari-ation of the test values about the mean of the twotests was approximately 62.8%.

Three volumetric tests were performed, eachusing a cylindrical compression-set rubber speci-men, using a device manufactured for this exper-iment. It consists of two steel plates joined to-gether by cap screws, with a hole cut into theupper plate. The diameter of the hole is 1.100 in.,with tolerances of 20.000 and 10.001 in. A pis-ton, to which compressive load was applied, wascut to a diameter of 1.100 in. with tolerances of10.000 and 20.001 in. The tight tolerances werenecessary because of the extreme importance ofproviding no room for the rubber to extrude dur-ing the test. The piston slid freely through thehole when the bottom plate was removed, butwith the bottom plate attached, entrapped airretarded its movement considerably, showingthat the device and sample diameters are identi-cal. After removal of the machine compliance, theraw data were converted into a pressure and vol-ume ratio.

RESULTS

Figure 1 depicts the experimentally determineduniaxial tensile and compressive stress–straindata (crosses). These data were used to determinethe deviatoric constants in the strain-energy den-

Table I Nominal Mechanical Properties of the Rubber Supplied by the Elastomeric BearingManufacturer and Tested for This Study

Material

Shear Modulus[ASTM D-4014-89,

Annex A1] MPa(psi)

Tensile Strength[ASTM D-412-92,Method A] MPa

(psi)

Ultimate Elongation[ASTM D-412-92,

Method A] (%)

Compression Set[ASTM D-395-

89, Method B15](%)

Manufacturer suppliedcomposite batch 0.883 (128) 16.59 (2406) 621 20.75


sity functions and to determine the VL function(solid line), both of which were used in the finite-element modeling [see eqs. (2), (4), (5), and (7)].Figure 2 shows the results of the constrainedcompression test. These data confirm the con-stancy of the bulk modulus and justify the ap-proximation involved in decomposing the strain-energy density function into a deviatoric and ahydrostatic component [see eqs. (1), (2), and (4)].Additionally, these data were used to determinethe hydrostatic (or volumetric) constants in thestrain-energy density functions, which were then

used in the finite-element modeling. For the threevolumetric tests, the slopes of each graph, whichcorrespond to the bulk modulus, differ by only60.86% from the mean value. This difference can-not be attributed to any specific errors in testingor differences in the bulk modulus from sample tosample because they are within the resolution ofthe testing machine.

The data of Figures 1 and 2 permit the deter-mination of the compressible VL function param-eters. The solid line in Figure 1 is the VL functionfit to the uniaxial tension and compression data[crosses in Fig. 1; also see eq. (7)]. Clearly, the VLfunction provides an excellent fit to the uniaxialstress–strain data.

As another check on the stresses generated bythe VL function, we compare equibiaxial stressesgenerated in two different ways, as follows: First,uniaxial compression and equibiaxial tensionare equivalent for an incompressible material.Through a simple transformation, one can showthat the equibiaxial stress–strain, written as afunction of uniaxial compression and the stretch,is

sequibiaxial 5 2scompressivel3/2 (8)

and, second, in terms of the VL function, theequibiaxial stress–strain relationship can be ex-pressed as follows:

sequibiaxial 5 u9~l! 21l3 u9S 1

l2D (9)

Figure 1 Plot of the uniaxial stress–strain behaviorof the filled rubber: (3) experimental data; (solid line)VL fit to the data.

Figure 2 Comparison of volumetric compression testresults and finite-element results for filled rubber.

Figure 3 Comparison of the equibiaxial stress–strainbehavior of the filled rubber determined from eqs. (8)and (9).


Figure 3 shows excellent agreement between theequibiaxial stresses calculated from eqs. (8) and (9).

In terms of the VL function, the planar stress–strain relationship can be expressed as follows:

splanar 5 u9~l! 21l2 u9S1

lD (10)

Figure 4 is a comparison between the planar ten-sile (pure shear) stresses calculated from the VLfunction and the experimental uniaxial tensilestresses. Note that the planar stresses areslightly larger than are the uniaxial stresses at agiven level of strain. The relative values are con-sistent with experimental data reported by Tre-loar.18 This result is expected since the force re-quired to deform a specimen in planar tensionshould be at least as large as is the force requiredin uniaxial tension because additional constraintsare imposed on the specimen along the axis per-pendicular to the direction of loading.

As noted above, the uniaxial engineeringstress–strain data points were used to obtain thedeviatoric constants for the rubber model. Whenonly these experimental data points are used tofind the constants in the polynomial and Ogdenstrain-energy density functions [see eqs. (2) and(5)], the model is referred to as experimental axialdata (EAD). Additionally, a second data set was

generated to determine the deviatoric constantsand included the planar tension data calculatedfrom the VL function and the uniaxial engineer-ing stress–strain data points. We refer to thismodel as generated planar data (GPD). The hy-drostatic portion of the polynomial and Ogdenmodels remains the same, and the constants werefound from the experimental pressure–volumedata shown in Figure 2.

COMPARISON OF THE RUBBER MODELSIN ABAQUS

One way to verify if the generated planar tensiondata improve the computer model is to comparethe finite-element results of the GPD and EADmodels. This comparison was performed for fourmodes of deformation: uniaxial tension, uniaxialcompression, equibiaxial tension, and planar ten-sion.

First, we determine the number of termsneeded to approximate the strain-energy densityfunction. It is helpful to look at the reducedstress19,20:

tR 5t

Sl2 21lD

5 2SUI1

11l

UI2

D (11)

Figure 5 Comparison of the reduced stress as calcu-lated from the different rubber models with that deter-mined from the experimental data.

Figure 4 Comparison of the planar tensile stress–strain response calculated from the VL function for therubber with the measured uniaxial tensile stress–strain response.


The reduced stress provides a sensitive measureof the deviations of the elastic properties fromconstant values.18,20 By plotting 1/l as the ab-scissa, the compression domain is the regiongreater than 1 and the tension domain is com-pressed between 0 and 1.

The polynomial strain-energy density functionis shown in eq. (2). For N 5 1, we get the Mooney–Rivlin model [see eq. (3)]. Taking the appropriatederivatives of eq. (3) and inserting them into eq.(11), the reduced stress–strain relation for uniax-ial tension and compression is

tR 5 2C10 11l

2C01 (12)

Equation (12) shows that the Mooney–Rivlinmodel results in the reduced stress being linear in1/l. Figure 5 shows the experimental uniaxialtensile and compressive data expressed as re-

duced stresses and plotted versus 1/l. Also shownare the results from the finite-element analysis.

For our present discussion, we look only at theexperimental data. There are several points to benoted from the data. First, the behavior is un-usual in two respects: At small deformations,there appears to be a large upswing in the re-duced stress values as l approaches the unde-formed state. This is sometimes thought to occurbecause of experimental uncertainties at smallerdeformations,19 but do seem to be outside of theexperimental errors here. It is possible that theresult is due to the specifics of the unknown rub-ber compound and filler. We address this pointfurther, subsequently. The second unusual resultis the concave curvature of the behavior at highstretches and large compressions (ignoring thesmall-strain region). Generally, unfilled rubberdoes not exhibit such behavior21 and this may bedue to limited chain extensibility that is exagger-ated by the presence of the reinforcing filler. The

Figure 6 Comparison of the experimental uniaxialstress–strain response of the rubber with finite-ele-ment calculations for different strain-energy functionsdetermined without using the VL function-generateddata (planar tension).

Figure 7 Comparison of the experimental uniaxialstress–strain response of the rubber with finite-ele-ment calculations for different strain-energy functionsdetermined using the VL function-generated data (pla-nar tension) (GPD).

Table II Average RMS Error of Calibration Runs for Uniaxial Tension

Rubber Model Ogden (N 5 6) Ogden (N 5 3) Polynomial (N 5 2)

EAD Compressible 0.0195 0.0159 0.0177GPD Compressible 0.0119 0.0160 0.0195


origins of the behavior are beyond the scope of thecurrent work and we do not address this pointfurther.

The net outcome of the observations, however,is that it becomes clear that the simplest form ofpolynomial expression cannot describe the behav-ior (i.e., the Mooney–Rivlin representation). Forthis reason, we chose to use N 5 2 for the poly-nomial strain-energy density function. This wasthe highest order for this function available in theversion of the finite-element code used and hasseven constants Cij.

For the Ogden strain-energy density function,determination of the order needed (N 5 1, . . . ,6)to capture the nonlinearity in the experimentaldata is not as easy to see because the function iswritten in terms of the stretches. In this case, thederivative with respect to the strain invariants isnot readily evaluated. We believe that using moreterms should help capture the nonlinearity in thereduced stress versus the stretch curve. We usedN 5 6, which is the largest number of termsavailable in the finite-element code used. We alsoused N 5 3 for comparison.

To compare the finite-element results againstthe experimental and calculated data for thethree models, we calculated the average root-mean-square (rms) error. The average rms erroris calculated from the formula shown in eq. (13)and measures the relative error in stress:

avg rms error 51M ÎO

i51

N Ssexp 2 sFE

sexpD (13)

M is the number of points at which the stress wascalculated by the finite-element analysis; sFE, thestress calculated by the finite-element analysis;

and sexp, the stress obtained from the experimen-tal data.

Table II shows the average rms error for eachof the calibration runs for uniaxial tension. Theperformance of each model is good, with slightlybetter results from the model based on GPD. Fig-ures 6 and 7 are plots of the finite-element resultsversus the experimental uniaxial data (tensionand compression). This plot graphically depictsthe closeness of the finite-element approxima-tions to the tensile data, using both experimentaldata and data generated by the VL function.

Table III shows the average rms error for eachof the calibration runs for uniaxial compression.Each finite-element result provides a good ap-proximation to the experimental compressiondata, with EAD being slightly better. Also, notethat the Ogden model with six terms (N 5 6)results in the lowest average rms error for eachrubber model. In Figures 6 and 7, the compres-sion data corresponds to strains less than zero.

Table IV shows the average rms error for eachof the calibration runs for equibiaxial tension,and Figures 8 and 9 show plots of each run. Thecomparisons are between the data generated bythe VL function and the ABAQUS results usingthe two rubber models. The results for each rub-ber model are comparable, but the experimentalrubber model EAD does a slightly better job thandoes GPD when using the Ogden, N 5 6, and thepolynomial, N 5 2, strain-energy density func-tions, with the Ogden, N 5 6, strain-energy den-sity function providing the best fit to the equibi-axial data generated by the VL function. It isimportant to note that rubber models EAD andGPD do not include the equibiaxial data set, yetthe finite-element models closely approximate theequibiaxial deformation mode.

Table III Average RMS Error of Calibration Runs for Uniaxial Compression



Table IV Average RMS Error of Calibration Runs for Equibiaxial Tension




Table V shows the average rms error for each ofthe calibration runs for planar tension, and Fig-ures 10 and 11 show plots of each run. An impor-tant observation is that the Ogden strain-energyfunction, using the rubber model EAD (whichdoes not include the planar data), does a muchbetter job of fitting the planar data than does thepolynomial strain-energy function. This is proba-bly because the Ogden model is a special form ofthe VL function. A second observation is thatwhen these calculated data are included in thedata set used to determine the strain-energy den-sity function (GPD) the polynomial strain-energydensity function performs nearly as well as doesthe Ogden function.

As mentioned previously, the volumetric testruns were only performed using the compressibil-ity data for rubber models EAD and GPD. Usingan incompressible model would result in no defor-mation, since the rubber is perfectly confined.Figure 2 presents the finite-element results com-

pared to the experimental data. The finite-ele-ment program accurately reproduces the experi-mental data.

Returning now to Figure 5 and the plot of thereduced stress versus for the EAD rubber model,the experimental data exhibit a rapid increase inthe modulus as the unstrained state is ap-proached. The results using the GPD data areidentical. This was observed by McKenna andZapas19,22 and other authors. The finite-elementresults using the polynomial or Ogden forms ofthe strain-energy function do not capture thischange in reduced stress. McKenna and Zapas22

also noted that the VL function did not capturethis behavior at small strains. McKenna and Za-pas19 reported that the compression modulus isgreater than is the tension modulus near the un-deformed state. This observation agrees with ourexperimental data. The reasons for this phenom-enon are not clear. It is interesting to note thatMcKenna and Zapas19,22 used bonded cylinders.

Figure 9 Comparison of the experimental equibiaxialstress–strain response of the rubber with finite-ele-ment calculations for different strain-energy functionsdetermined using the VL function-generated data (pla-nar tension).

Figure 8 Comparison of the experimental equibiaxialstress–strain response of the rubber with finite-ele-ment calculations for different strain-energy functionsdetermined without using the VL function-generateddata (planar tension).

Table V Average RMS Error of Calibration Runs for Planar Tension




They corrected the results to homogeneous com-pression using factors determined in a finite-dif-ference analysis and conjectured that this phe-nomenon might be due to their experimental ap-proach. However, in the current experiments, thetests were carried out on unbonded cylinders thathad been lubricated to reduce friction along theloaded surfaces.

CONCLUSIONS

The strain-energy density function for an elas-tomeric material that is used in rubber bearingsfor earthquake isolation of buildings was deter-mined in order that the load-deformation re-sponse of the bearing could be calculated fromthe hyperelastic models in the ABAQUS1 finite-element program. Because only uniaxial testingcould be performed, we used the VL function toextend the experimental rubber stress–straindata into the planar tensile (pure shear) mode.The polynomial and Ogden forms of the strain-energy function that were available in ABAQUSwere then fitted to the tension and compressiondata and the VL expanded data set that in-cludes planar tension. Using only uniaxial datato determine the elastic constants, the Ogden

strain-energy density function, which is a spe-cial case of the VL function, was able to predictthe full and expanded data set. The polynomialstrain-energy density function with second-or-der terms (N 5 2, corresponding to seven terms)significantly overestimates the planar tensilestresses calculated from the VL or Ogden func-tions. This suggests that the polynomial strain-energy density function has a different sensitiv-ity to the test geometries used for the data inputthan does the VL function (or the Ogden func-tion).

It should be noted that most work in rubberusing the VL function and Mooney–Rivlin plotswas done with unfilled rubber. However, therubber used in this testing program is filledwith carbon black, typical of most structuraluses of rubber (tires, elastomeric bearings, etc.).The Mooney–Rivlin plots of unfilled rubbers ex-hibit different behavior from that observedhere.21 With unfilled rubbers, the responsestarts at a low value, goes toward a maximumin the vicinity of 1/l 5 1, and then decreasesmildly before, possibly, increasing again. Here,the response at values of 1/l increases at arapid rate.

The authors gratefully acknowledge partial support ofthis work by the Building and Fire Research Labora-

Figure 10 Comparison of the VL-calculated planarstress–strain response of the rubber with finite-ele-ment calculations for different strain-energy functionsdetermined without using the VL function-generateddata (planar tension). Note that the polynomial expan-sion provides a poor fit to this case.

Figure 11 Comparison of the VL-calculated stress–strain response of the rubber with finite-element calcu-lations for different strain-energy functions determinedusing the VL function-generated data (planar tension).Note: Compare the improvement of the polynomial ex-pansion fit to the data.


tory and the Materials Science and Engineering Labo-ratory of the National Institute of Standards and Tech-nology in Gaithersburg, Maryland.

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Rubber Modeling Using Uniaxial Test Data

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