kg07_11_608

8/2/2019 kg07_11_608

http://slidepdf.com/reader/full/kg0711608 1/3

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

8/2/2019 kg07_11_608


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

8/2/2019 kg07_11_608


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

kg07_11_608

Documents