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Rubber modeling using uniaxial test data

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Rubber Modeling Using Uniaxial Test Data
G. L. BRADLEY,
1
P.C. CHANG,
2
G. B. MCKENNA
3
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 2000
Publishedonline8May2
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 the ABAQUS 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 estimating the 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 of steel 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 ABAQUS
1
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 (greg.mckenna@ coe.ttu.edu).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
I
2
. 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 of the 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) function
3
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 than just 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
I
1
;
U
(
I
1
,
I
2
,
I
3
),where
I
3
is the square of the volume ratio
J
(
J
I
3
) 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 of the 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
1), where
p
is a Lagrange multiplier (the hydrostatic stress)introduced to impose the constraint
J
1
0 (
J
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, asPenn
7
and Fong and Penn
8
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
U
I
1
,
I
2
,
I
3
U
I
1
,
I
2
U ˜
I
3
(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 Penn
7
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
i
j
1
N
C
ij
I
1
3
i
I
2
3
j
i
1
N
1
D
i
J
el
1
2
i
(2)The ﬁrst summation is the contribution due todeviatoric effects, and the second summation isthe contribution due to volumetric effects. The
C
ij
and
D
i
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.
J
el
is the ratio of the current volume to the srcinal volume excluding thermal effects. For
N
1, thedeviatoric contribution to the strain energy iscalled the Mooney–Rivlin function. It is oftenwritten as
U
C
10
I
1
3
C
01
I
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
. Valanis andLandel
3
postulated that the strain-energy densityfunction is a function that is separable in terms of the principal stretches, and the total strain en-ergy is
U
u
1
u
2
u
3
(4)We will refer to
u
(
) 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
i
1
N
2
i
i
2
1
i
2
i
3
i
3
i
1
N
1
D
i
J
el
1
2
i
(5)In eq. (5),
i
,
i
, and
D
i
are parameters that arefound from the rubber test data. The principalstretch ratios
k
i
have the volume change re-moved. To determine the deviatoric parameters
C
ij
in eq. (3) and
i
and
i
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
C
ij
and the
i
and
i
, areassumed to be independent of the volume.Using a right-handed Cartesian coordinatesystem with axes
x
1
,
x
2
, and
x
3
, let the uniaxialdeformation mode correspond to an application of force along the longitudinal axis of the specimen.For the purpose of explanation here, we denotethis axis
x
1
. Then, the specimen is free to expandor contract along axes
x
2
and
x
3
. The equibiaxialdeformation mode corresponds to equal stretchesalong the
x
1
and
x
2
axes, with unrestrained move-ment along the
x
3
axis. Planar deformation corre-sponds to a stretch along the
x
1
axis, with nostretch allowed along the
x
2
axis, and unre-strained movement along the
x
3
axis. Each of these modes is an example of a deformation inwhich the directions of principal strain do notchange, that is, the deformations take place along the 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
D
i
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 of test 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 if the tests had been performed.KearsleyandZapas
13
showedhowtoobtainthe VL function from uniaxial tensile and compres-sive data, which we did obtain. In terms of thetrue stress
t
and the stretch
, the following re-cursive expression can be used to obtain the de-rivative
u
(
) of the VL function:lim
n
3
k
0
n
1
t
1/4
k
t
1/2
1/4
k
u
(6)The right-hand side of eq. (6) is an inﬁnite sum of the true stresses at the stretches speciﬁed in thebracketed expression. The rapid convergence of this inﬁnite series allows one to obtain the VLfunction with a small number of terms. Note thatthe ﬁrst and second terms are on opposite sides of the undeformed state (
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
(
) at thedesired stretch values. The formula for uniaxialengineering stresses, written in terms of the de-rivative of the VL function, is
u
3/2
u
1/2
(7)
EXPERIMENTAL
Materials
The material used in this study was a carbonblack-reinforced rubber compound of the samecomposition as that used in the manufacture of the 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 using known dead loads and was calibrated according to ASTM 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
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
10%, 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
840
BRADLEY, CHANG, AND MCKENNA

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