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8/2/2019 kg07_11_608
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PRÜFEN UND MESSENTESTING AND MEASURING
608 KGK · November 2007
Thanks their special properties elastomers
are employed in many areas of the human
society [1, 2]. However, there is still a need
for describing exactly the nonlinear visco-
hyperelastic properties of these materials.
A step forward has been made in last
60 years [3] by taking the advantage of us-
ing computing tools such as the Finite ele-
ment method (FEM), which is well intro-
duced in rubber engineering today.
Our aim is to set up nonlinear material pa-
rameters of elastomers for numerical simu-
lations. In order to provide the engineering
constants for nonlinear viscohyperelastic
material models one needs to test material
in all deformation modes that will occur
during simulation. Usually three basic de-
formation modes are tested: uniaxial ten-
sion, equibiaxial tension and pure shear [4]
(Fig. 1). The uniaxial tension is easy to per-
form on standard testing machines [5].
However, special equipment is required for
the two other deformation modes [6]. One
of the suitable methods for characterizing
equibiaxial deformation of elastomers isthe bubble inflation technique in which an
elastomer plate is inflated to the shape of
bubble [7].
MethodsAn uniform circular specimen of elastomer
is clamped at the rim and inflated using
compressed air to one side. The specimen is
deformed to the shape of bubble (Fig. 2).
The inflation of the specimen results in a bi-
axial stretching near the pole of the bubble
and the planar tension near the rim.
Thanks to the spherical symmetry we canconsider
at the pole of the bubble.
Then we can write the Cauchy stress tensor
in spherical coordinates as:
rr
r z
0 0
0 0
0 0( , , )
(1)
The thickness of specimen is small and the
ratio between the thickness of the inflated
membrane t and the curvature radius r is
small enough, then the thin shell assumpti-
on allow us to neglect the radial stress rr infront of the stress
. In addition we equate
to the thickness-average hoop stress,
which leads to:
pr
t2(2)
where p is the differential inflation pressu-
re, r is curvature radius of specimen and t is
the specimen thickness.
With consideration of material incompress-
ibility we can express the thickness of in-
flated specimen as:
tt
0
(3)
where t0
is the initial thickness of specimen
(unloaded state). Further we have to measu-
re the stretch
at the pole of inflated ma-
terial. Generally stretch is the ratio
between the current length l and the initial
length l0:
l
l0
(4)
One can use an optical method for measu-
ring the elongation
and the curvature ra-
dius r (camera, video camera, laser etc.).
Substituting equation (3) into equation (2)one can compute the hoop stress
as fol-
lowing:
pr
t
2
2 0
(5)
ExperimentalThe schematic view of the testing equip-
ment is presented in Figure 3. The speci-
men (a) of 2mm sample plate is fixed be-
tween two rings with inner diameter
40 mm. The rings are clamped in a sup-
port (b). Next function of the support is
the distribution of the compressed air toone side of the specimen. The air pressure
is regulated with a pressure regulator (c)
and a regulating valve (g). The current
pressure value is recorded using a pressure
Elastomer · Hyperelasticity · Finite ele-ment method (FEM) · Equibiaxial ten-sion · Bubble inflation technique
The aim of this work is to set up nonlin-
ear viscohyperelastic material parame-ters of elastomers for numerical finiteelement simulation (FEM). The study isfocused on equibiaxial elongation byusing the “bubble inflation” technique.These data together with those fromuniaxial tests were used to create aFEM model.
Äquibiaxiale Prüfung vonElastomeren
Elastomer · Hyperelastizität · Finite-
elemente-Methode (FEM) · biaxialeDehnung · Aufblastest
Das Ziel dieser Arbeit ist die Bereitste-lung von Materialdaten zum nichtline-aren viskoelastischen Verhalten vonElastomeren, die für die numerische Si-mulation (FEM) erforderlich sind. DerSchwerpunkt dieses Beitrags ist dieäquibiaxialen Prüfung von Elastomerenmit Hilfe des „Aufblastests“
AuthorsJ.Javořik, Z.Dvořàk,Zlin (Czech Republic)
Corresponding author:Ing. Jakub Javořik, Rh.D.Tomas Bata University Fakulty of
TechnologyDept. of Manufacturing EngineeringNad Stranemi 45176272 Zlin, Czech RepublicE-mail: [email protected]
Equibiaxal Test of Elastomers
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609KGK · November 2007
sensor (d). The inflation of the specimen is
recorded using a high resolution CCD video
camera (f). A computer is used to control
the pressure valve.
The white strips were drawn in the centralarea of specimen for stretch measurement.
It is important to measure the elongation
and the curvature radius only in the area
near to pole (between the strips) of the in-
flated specimen and not on the entire bub-
ble contour because only on the pole the
equibiaxial state of stress occurs (Fig. 4).
A common SBR compound for tire manufac-
turing was tested. The material was loaded
until failure. The necessary values for the
stretch ratios
and the curvature radii r
were obtained from image analysis of the
video record. In order to obtain the stress-strain diagram the stretch values
were
converted into strain values ( 1) and
the Cauchy
stress into the engineering
stress by taking into account of the assump-
tion that the material is incompressible:
(6)
ResultsThe equibiaxial stress-strain diagram of
tested material is shown in Figure 5. Also
the uniaxial stress data are presented in this
diagram for comparison. One can observethe generally known fact [3], that the equib-
iaxial stress values are 1.5 2 times larger
than the uniaxial ones.
The common hyperelastic material models
(3rd order Yeoh and 5-terms Mooney-Rivlin)
and the experimental results were com-
pared. The importance of equibiaxial test is
demonstrated in this comparison. In thefirst case (Fig. 6 a), only the data for uniaxial
tension were used for the determination of
the material constants. In the second case
(Fig. 6 b), both the data of the uniaxial and
the equibiaxial tension were used to deter-
mine the material constants.
The FEM model of specimen inflation (based
on 5-terms Mooney-Rivlin hyperelastic
model) was created and compared with ex-
periment. The comparison of real stretch of
material
with stretch of FEM model is
shown in Figure 7.
It is clear form Figure6 that we were not ableto predict the biaxial behaviour of the elas-
tomer from the uniaxial data only. One can
see from fig.6a that both models used close-
ly follow the uniaxial experimental data but
that the biaxial prediction is very inaccurate
(especially for Mooney-Rivlin model). Whilein Fig.6b (where both the uniaxial and the
biaxial data were used) the material models
closely follow both the uniaxial and the bi-
axial experimental data. The inaccuracy of
biaxial Yeoh model is due to its simplicity
and unsuitability for large stretch ratios, but
in this case the model is still more accurate
then in case a). One can see the necessity of
equibiaxial test for prediction of hyperelas-
tic behaviour of elastomers from this com-
parison. In addition, one can see the differ-
ence between the stretch of FEM model and
the experiment at large deformation of thematerial in Fig7. For more accurate results
Deformation modes11
1
The bubble inflation technique22
2
Schematic view of the testing equip-ment
333
Stress-strain diagram of tested material
555
Inflated specimen in the equibiaxialtest
44
4
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PRÜFEN UND MESSENTESTING AND MEASURING
610 KGK · November 2007
one would need also data from pure shear
test. Even with this inaccuracy one can say
that results are very close to the experiment
up to a deformation of 2, that is seems
to be a sufficient large range for most appli-
cations. Still we have to be aware that with-
out equibiaxial data the FEM simulation
would not be possible at all.
The tests of all three modes of deformation
(uniaxial tension, equibiaxial tension, pure
shear) are needed for next development of
this work. Future improvement of equibiax-
ial test device is planned too.
AcknowledgementThis work was prepared under support of
project MSM 7088352102 (provider: Minis-
try of Education, Youth and Sports of Czech
Republic).
References
[1] A.B. Davey, A. R.Payne, Rubber in Engineering
Practice. London, Maclaren & sons Ltd., 1966,
501.
[2] A. N.Gent, Engineering with Rubber, Munich,
Hanser, 2001, 365.
[3] R.W. Ogden, Non-linear Elastic Deformations,
Dover Publications, Mineola, NY (1997).
[4] P.Kohnke, ANSYS – Theory reference, Canons-
burg, PA, USA, ANSYS, Inc. 1998, 965.
[5] L. P.Smith, The language of Rubber, Oxford, But-
terworth-Heinemann Ltd., 1993, 257.
[6] MSC.Software Corporation: Nonlinear finiteelement analysis of elastomers, http://www.
mscsoftware.com/assets/103_elast_paper.
pdf,MSC. Software Corporation, 2000, 64.
[7] N.Reugen, F.M. Schmidt, Y.le Maoulty, M.Ra-
chik, F.Abbé, Polymer Engineering and Science.
Society of Plastics Engineers 41 (2001) 522.
Comparison betweenexperiment and hyper-elastic material models
666
Comparison of realstretch of material
with stretch of FEM
model
777