METHOD FOR MEASURING BIAXIAL DEFORMATION ON RUBBER AND
POLYPROPYLENE SPECIMENS
Ch. Feichter1, M. Jerabek1, Z. Major2 and R.W. Lang1, 2
Polymer Competence Center Leoben GmbH, Parkstraße 11, A-8700 Leoben
(2)
Institute of Materials Science and Testing of Plastics, University of Leoben,
Franz-Josef-Straße 13, A-8700 Leoben
[email protected]
(1)
ABSTRACT
Biaxial measurements are very often used in order to characterize the material behaviour in more constrained conditions
than in the usual tensile tests. For rubbers this is necessary for the determination of hyperelastic material laws [1]. But also
for detailed material characterization of polypropylene this loading condition is utilized. There exist several loading
methodologies for applying biaxial load. The basis of the method used in this paper is the inflation of a thin material sheet
to a bubble with the help of pressurized air [2]. This method demands for exact pressure measurement and for a
determination of the strain distribution around the pole of the resulting bubble. The strain measurement was done with a
full-field strain analysis system called Aramis (GOM mbH, Braunschweig, Germany) based on image correlation
techniques [3].
Introduction
Measurements of the biaxial deformation behaviour is an important procedure in material science. Such data is used
primarily in finite element calculations for the calibration of the constitutive model. It allows for a more reliable material
characterization and therefore for more reliable results. However, the measurement of such biaxial deformation is a
complicated matter. If a specimen is loaded in two perpendicular directions with a servohydraulic system the biaxial strain
state is only achieved in the middle of the specimen and due to the strain concentration in the edges of it, it is not possible
to load up to high strains, because the specimen starts to fail at this points.
A different method is necessary to receive stress-strain-curves. In this paper the deformation of a thin plate using
pressurized air is presented [4,5]. The plate is deformed like the skin of a balloon, which gives, aside from small amounts
of bending, perfectly biaxial deformation. With this approach different rubber and polypropylene types have been tested.
The results are presented along with the calculation method and compared to uniaxial or planar test results.
Experimental Setup
Uniaxial Measurements
Uniaxial Measurements are usually done on testing systems based on actuators moved with hydraulic oil or on
electromechanical movement via spindle drives. In this paper both systems have been used for the uniaxial
characterization. The mechanical properties of the polypropylene (PP) materials were characterized by tensile testing
using an electromechanical universal testing machine Instron 4504 (Instron Deutschland GmbH, Darmstadt, D). The strain
was measured by means of a mechanical extensometer type Instron 2630-112 with a gauge length of 50 mm. Injection
molded specimens were used according to ISO 3167. The measurements have been carried out at room temperature.
The strain rate was varied between 0.0087 and 0.0000087 1/s. The mechanical properties of the rubbery materials were
tested with a servo-hydraulic testing system called MTS 831.59 Polymer Test System (MTS Systems, Eden Prairie, MN,
USA) under displacement controlled loading conditions with a loading rate of 1 mm/s. Since the rubber specimens are
very soft compared to the polypropylene, no local strain measurement was used. Both testing systems are shown in
Figure 1.
Figure 1. Electro-mechanical test system of type Instron 4504 (left) and servo hydraulic test system MTS 831.59 (right)
with fixtures.
Biaxial Measurements
For the biaxial test proprietary equipment was developed. The rubber and the PP sheets were clamped into the
measurement system (Figure 2) and then the volume below the specimen was slowly inflated. This leads to the formation
of a bubble that increases its size with increasing preasure. The challengeing part is now the determination of stresses
and strains. Therefore the applied pressure was measured by a piezo-resistive pressure sensor of the type
RAG25A10BV1H (Kistler, Winterthur, CH).
Figure 2. Measurement system for the determination of biaxial stress-strain data. The shown equipment is used for
applying the load and measuring the pressure, from which the stresses are calculated. The strains are measured by using
the full-field strain analysis system (Aramis).
The resulting strains are measured with a full-field strain analysis system. For the full-field strain analysis, a 3D image
correlation photogrammetry system of the type Aramis (GOM, Braunschweig, D) was used. This system is based on
stereo viewing and therefore capable of measuring the strain distribution and topology of the specimen surface [13].
Therefore, a stochastic dot pattern was applied on the specimen by using an aerosol can. The dots had to be small but
large enough that the camera could detect the pattern and not only a grey surface. Figure 3 shows a typical dot pattern
used for this analysis. The three white square boxes symbolize three parts of the pattern that can be detected. The center
of the pattern is the recognized data point. Figure 4 shows the setup of the two cameras of the full-field strain analysis
system and the calibration laser between the cameras, focused on the center of the measuring area.
Figure 3. Dot pattern used for the full-field strain analysis system type Aramis. The small dots (white) can be seen on the
black surface of the rubber specimen. The size of the image is approximately 7 x 5 mm.
Object
α
Cameras
Figure 4. Camera setup of the full-field strain analysis system with laser between cameras for calibration purposes.
Before the measurement can be done, the system has to be calibrated to the lenses and the angle between the two
cameras. This procedure was done by using special calibration plates, which had to be measured with the system in
different positions. This calibration procedure allowed to correct for distortions of the lenses and to calibrate the distance of
measured points to the cameras. After that step, the pattern was applied on the surface that should be measured, in that
case the area surrounding the crack tip, and the measurement could be started. The analysis software detected the
pattern in the taken images (Figures 5a) and the pattern movement was tracked. The three-dimensional surface
coordinates were calculated by correlating pictures at the same time of the left and right camera, the strains were
calculated from the pictures, which had been taken at different times [13]. Figure 5b demonstrates the true biaxial
character of this type of measurement.
(a)
(b)
28
26
24
22
20
Strain, %
18
16
Identical values proof
the biaxial deformation
14
12
10
8
6
Strain εx
4
Strain εy
Strain εmax
2
0
0
20
40
60
80
100
Distance, mm
Figure 5. Strain distribution on the inflated rubber bubble measured with the Aramis full-field strain analysis system. It can
be seen that the deformation at the pole is higher than at the rest of the surface. The average value at the pole was taken
for the material law calculation.
Data Analysis
The strains are analyzed by the Aramis full-field strain analysis system. They can be exported as text files. The measured
pressure data has to be analyzed separately. Here the first step is the calculation of the stress. For demonstration
purposes the situation on the bubble is shown in Figure 6. This figure is a horizontal projection of the middle of the
specimen surface.
sy
s
sx
β
∆p
rB
M
2 rL
Figure 6. Explanation of the situation on the bubble.
Since the initial thickness t0 is reduced by the deformation, the effective thickness t has to be calculated prior to the
calculation of the stresses with the help of λ, which is the extension ratio and can be calculated by adding 1 to the stress
(λ = 1 + ε).
t=
t0
λ2
(1)
At the beginning we set up the following equation:
∆p ⋅ rL ⋅ π = s y ⋅ 2 ⋅ rL ⋅ π
2
(2)
where rL is the effective diameter of the specimen, ∆p is the pressure within the bubble and sy is the line force (force per
length) in y-direction. With
s y = s ⋅ sin α
(3)
this leads to
σ=
∆p ⋅ rL ⋅ λ2
2 ⋅ t0 ⋅ sin α
(4)
and with
rL
= rB
sin α
(5)
together with the true strain, exported from the strain analysis system, the resulting true stress can be achieved. It is
calculated using Equation 6.
σ=
∆p ⋅ rB ⋅ λ2
2 ⋅ t0
Here rB is the bubble radius measured with the full-field strain analysis system.
(6)
Results
The experimental results are shown in this section. Initially the strain rate of the biaxial results for the polypropylene
materials is presented (Figure 7). This is a very important fact since the strain rate has a big influence on the material
stiffness. Therefore, also the control mode of the testing system has an influence on the results, since force controlled
loading in non-linear materials results in non-constant loading rates. The uniaxial materials have been measured in a
displacement controlled way, the biaxial measurements have been tested in a pressure controlled way.
0.0012
PP
PP with Glass
0.0010
Strain Rate, 1/s
0.0008
0.0006
0.0004
0.0002
0.0000
0
20
40
60
80
100
120
Testing Time, s
Figure 7. Strain rate as a function of testing time is shown in this diagram for both polypropylene types. The presented
data has been treated with an smoothing filter.
The pure polypropylene shows a nearly constant strain rate, whereas the glass filled polypropylene shows an increase.
As can be seen in the diagram, the strain rate is nearly constant for the pure polypropylene at a value of about 0.0004 1/s.
The value of the glass filled polypropylene increases with the testing time. The reason for that lies in the method of
applying the strains. Since the material is inflated with pressurized air instead of applying the desired strains it cannot be
constant. The data analysis that delivers the strains takes about half an hour on a dual-processor computer and therefore
cannot be used for controlling the loading.
The presented biaxial data is therefore shown together with uniaxial stress-strain curves of different loading rates. The
loading rates are within a region of 0.0087 1/s and 0.0000087 1/s. They cover the strain rates of the biaxial tests. Figures
8 and 9 show the measured data.
35
30
Stress, MPa
25
20
15
PP
uniaxial, 0.0087 1/s
uniaxial, 0.00087 1/s
uniaxial, 0.000087 1/s
uniaxial, 0.0000087 1/s
biaxial, 0.0004-0.0005 1/s
10
5
0
0.00
0.01
0.02
0.03
0.04
0.05
Strain
Figure 8. True stress vs. true strain diagram for the polypropylene. Uniaxial as well as the biaxial results are shown. The
uniaxial results are shown for different strain rates.
The biaxial results lie perfectly between the curves with slightly lower and higher strain rates.
30
True Stress, MPa
25
20
15
PP with Glass
uniaxial, 0,0087 1/s
uniaxial, 0.00087 1/s
uniaxial, 0.000087 1/s
uniaxial, 0.0000087 1/s
biaxial, 0.0004-0.001 1/s
10
5
0
0.00
0.01
0.02
0.03
0.04
0.05
True Strain
Figure 9. True stress vs. true strain diagram for the polypropylene with glass. The material has a glass bead content of 7
volume percents. Uniaxial as well as the biaxial results are shown. The uniaxial results are shown for different strain rates.
Here the results are also in the expected region. However, the polypropylene filled with glass beads can only be loaded to
about half of the strain of the unfilled material. The overall stiffness was higher than for the unfilled material.
The results for the rubber materials are shown in Figures 10 and 11. Due to the lower material stiffness the FKM could be
loaded up to 100 % and the HNBR was loaded up to 40 % strain. As expected the biaxial material behavior is stiffer than
for the uniaxial measurement. This is a typical result for rubber and caused by the higher geometrical constrained loading
conditions.
25
FKM
uniaxial
biaxial
True Stress, MPa
20
15
10
5
0
0.0
0.5
1.0
1.5
2.0
2.5
True Strain
Figure 10. True stress vs. true strain diagram for the FKM rubber. The higher stiffness of the biaxial measurement can
clearly be observed.
25
HNBR
uniaxial
biaxial
True Stress, MPa
20
15
10
5
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
True Strain
Figure 11. True stress vs. true strain diagram for the HNBR rubber. The higher stiffness of the biaxial measurement can
clearly be observed.
Conclusions
The presented apparatus for biaxial measurements was found to be suitable for biaxial measurements on rubbers and
softer thermoplastics. The specimens have to be thin (up to 2 mm thickness) plates with a size of 150 x 150 mm. The
results are in the expected region, which proofs that the loading concept is applicable. The accuracy of the measurements
was found to be good. Limits for the usability are on the one hand the stiffness of the material, because if it is too high the
resolution of the strain measurement is too small, and on the other hand the maximum deformation, since the surface
moves nearer to the lenses of the strain analysis system and therefore blurs and the resulting blurry images cannot be
analyzed afterwards.
Acknowledgments
The research work for this paper was performed at the Polymer Competence Center Leoben GmbH (PCCL, Austria) within
the framework of the Kplus-program of the Austrian Ministry of Traffic, Innovation and Technology with contributions of the
University of Leoben. The PCCL is funded by the Austrian Government and the State Governments of Styria and Upper
Austria.
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