R. KEERTHIWANSA et al.: ELASTOMER TESTING: THE RISK OF USING ONLY UNIAXIAL DATA FOR FITTING ...
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ELASTOMER TESTING: THE RISK OF USING ONLY UNIAXIAL
DATA FOR FITTING THE MOONEY-RIVLIN
HYPERELASTIC-MATERIAL MODEL
TESTIRANJE ELASTOMEROV: RIZIKO UPORABE REZULTATOV
ENOOSNIH PREIZKUSOV ZA PRILAGAJANJE
MOONEY-RIVLINOVEMU MODELU ZA HIPERELASTI^NI
MATERIAL
Rohitha Keerthiwansa, Jakub Javorik, Jan Kledrowetz, Pavel Nekoksa
Tomas Bata University in Zlín, Faculty of Technology, Vavreckova 275, 760 01 Zlín, Czech Republic
keerthiwansa@utb.cz
Prejem rokopisa – received: 2017-06-27; sprejem za objavo – accepted for publication: 2017-10-20
doi:10.17222/mit.2017.085
The Mooney-Rivlin constitutive model is often used for the characterization of hyperelastic rubber-like materials. To obtain the
material constants for a model, only a uniaxial-tension-data set is usually used. Though it is regularly used for its easiness of
processing data in a simple and practical way, the method is considered to be insufficiently accurate. To analyse the shortcoming
of the method, a detailed examination was done with the Mooney-Rivlin two-parameter model. This paper discusses the
variations related to three basic load curves, i.e., uniaxial, equibiaxial and pure-shear curves. For a visual observation of the
fitted-data dispersion, two data-fitting cases were considered. The first one was the data fitting only through uniaxial data while
the second one was a combination of uniaxial and pure-shear experimental-data curve fitting. A detailed one-to-one comparison
of the curves was done to achieve an accurate estimation of the variations.
Keywords: uniaxial tension, equibiaxial loading, pure shear/planar shear loading, curve fitting, Mooney-Rivlin constitutive
model
Mooney-Rivlinov temeljni model se pogosto uporablja za karakterizacijo gumi podobnih hiperelasti~nih materialov. Za
materialne konstante se obi~ajno uporablja set podatkov, dobljenih z enoosnim nateznim preizkusom. ^eprav se ta na~in
uporablja zaradi enostavnosti in prakti~nosti metode pa ga lahko smatramo kot manj natan~nega. Zato, da bi avtorji tega
prispevka ugotovili neskladje, so izvedli natan~no analizo z Mooney-Rivlinovim dvoparametri~nim modelom. V ~lanku avtorji
obravnavajo variacije, ki se nana{ajo na tri osnovne krivulje obremenjevanja: enoosno, ekvivalentno dvoosno in ~isti strig.
Vizualno opazovanje raztrosa prilagojenih podatkov so izvedli na osnovi dveh na~inov prilagajanja podatkov. Prvo prilagajanje
podatkov so izvedli na osnovi rezultatov enoosnega nateznega preizkusa. V drugem primeru so uporabili kombinacijo obeh:
eksperimentalne rezultate enoosnega nateznega preizkusa in ~istega striga. Izvedli so natan~no primerjavo krivulj in ocenili
odstopanja.
Klju~ne besede: enoosni nateg, ekvivalentna biaksialna obremenitev, ~isti strig, ravninska stri`na obremenitev, prilagajanje
krivulje, Mooney-Rivlinov konstitutivni (temeljni) model
1 INTRODUCTION
Hyperelastic rubber-like materials are often used in
the industry for the construction of various machine
components due to their specific material properties.
During the design stage of these parts, in order to com-
pensate for the behaviour under real-time load condi-
tions, it is important to know the exact mechanical
characteristics of these materials. As they behave non-
linearly during large deformations, stress-strain curves
deviate from the typical pattern, therefore being difficult
to estimate. The most common method of finding these
characteristics at present is through several constitutive
models introduced by various scientists.1–4
Once an appropriate model is selected, it needs
material constants attached to it to complete the task of
material characterization. The method used to obtain
these material constants for a specific model is the fitting
of the data collected from stress-strain experiments. The
least squares approach of the numerical analysis is used
to mathematically solve this problem.5,6
1.1 Data fitting
Data fitting is an important element in the process.
Therefore, it is essential to pay attention to the procedure
to do it in a proper manner. Because of the simplicity of
the method of data acquisition, quite often the data
collected through a uniaxial-tension test is used. Accord-
ing to M. Sasso et al.7 and others, the use of a single data
set related to one particular mode of load is insufficient
to obtained accurate results. The failure of the material
characterization occurring due to this inadequacy of the
data set during the data-fitting stage so far has not been
properly estimated. The objective of this paper is to
highlight the error imposed on the material characteriza-
tion when using only uniaxial data for the data-fitting
Materiali in tehnologije / Materials and technology 52 (2018) 1, 3–8 3
UDK 620.1:620.172.2:67.017 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 52(1)3(2018)
task compared to the combined, uniaxial and planar-
shear data fitting. Initially, in our effort, the Mooney-
Rivlin two-parameter model was used. Later, as an
improvement of the analysis, the Mooney three-para-
meter and Yeoh model were added.
1.2 Theory
The basis for any hyperelastic-material model is the
strain energy density function W. In the general case of
an anisotropic hyperelastic solid, the strain energy
density function must be a symmetrical function of the
stretch ratios 1, 2 and 3. Therefore, W can be defined
with three invariants I1, I2, I3 given in next Equations
(1–4):
I1 1
2
2
2
3
2= + +   (1)
I 2 1
2
2
2
2
2
3
2
1
2
3
2= + +      (2)
I 3 1
2
2
2
3
2=    (3)
Thus,
W f= ( )     2 3 (4)
In a hyperelastic-material research, various models
are used to define the strain energy function. The
Mooney-Rivlin model is one of the frequently used
models.
1.3 Mooney-Rivlin model
A suitable general form, a power series of the inva-
riants I1, I2 considering the incompressibility of the
material, is defined with Equation (5):
W I I C I C I( , ) ( ) ( )1 2 10 1 01 23 3= − + − (5)
In this equation, C10 and C01 are the material con-
stants.
Further extension of the Mooney-Rivlin model, a
three-parameter form, is defined as follows in Equation
(6):
W I I C I C I C I I( , ) ( ) ( ) ( )( )1 2 10 1 01 2 11 1 23 3 3 3= − + − + − − (6)
1.4 Yeoh model
In this model, the three-term function is considered
and written as given in Equation (7):
W I C I C I C I( ) ( ) ( ) ( )1 10 1 20 1
2
30 1
33 3 3= − + − + − (7)
2 EXPERIMENTAL PART
2.1 Material
Styrene-butadiene rubber (SBR), commonly used in
the tyre-manufacturing industry, was used as the test
material. For the experiments, all specimens were taken
from the same rubber sheet prepared under uniform
process conditions.
2.2 Experimental tests
In order to examine the mechanical behaviour of the
above-mentioned elastomer material, three different
deformation tests in the form of uniaxial, equibiaxial and
pure shear were done. After obtaining the data for these
tests separately, a regression analysis was done using the
least square approach. Brief descriptions of the three
tests are given below.
2.3 Uniaxial-tension test
This is a standard test. For this test, ISO 23529:2016
type-1 standard test pieces made of the martial men-
tioned above were used. Results were obtained at equal
time durations and were recorded as engineering stress
and engineering strain.
Equibiaxial-tension test
The use of an inflated rubber membrane for testing
biaxial tension has a long history. It was first introduced
by 8. Since then, this method has been frequently used by
scientists for testing rubber.9–11 As there is no standard
method regarding the test, several variants of the bubble
technique were used in different cases. However, the
basic elements of the testing method in all the cases were
the same. We adopted the method as described below.
A uniformly thin circular specimen shown in Fig-
ure 1, made of the test material, was held between two
metal rings (jaws) of the apparatus. Then, pressurised air
was introduced into the chamber where the top wall was
the specimen back side. With the increasing pressure
inside the chamber, the rubber membrane took the form
of a dome or bubble. The pole of the bubble underwent
equibiaxial tension and, therefore, a tiny area on the top
of the bubble was used as the reference for the calcula-
tion of the strain. While pressure values were recorded at
constant intervals, corresponding profile images of the
inflated membrane were captured with a digital camera.
R. KEERTHIWANSA et al.: ELASTOMER TESTING: THE RISK OF USING ONLY UNIAXIAL DATA FOR FITTING ...
4 Materiali in tehnologije / Materials and technology 52 (2018) 1, 3–8
Figure 1: Equibiaxial specimen with related measurements
The pressure inside the bubble was increased up to the
bursting point of the bubble. A complete set of data was
collected covering the full deformation of the specimen
membrane.
The stretch ratio lambda () was calculated by com-
paring the length of the curvature between two marked
positions (Figure 1) in each image and the undeformed
specimen length (l0) corresponding to these points in
Equation (8):
 
l
l 0
(8)
By considering the material incompressibility, the
hoop stress at the pole of the bubble () can be ex-
pressed as follows in Equation (9):
 

	

	p r
t
⋅ ⋅
0
(9)
In Equation (9), p is the applied pressure, to is the
initial thickness of the specimen, r is the radius of the
curvature and  is the stretch ratio at the pole.
2.5 Pure-shear test
The pure-shear test was done with a uniformly thin,
rectangular 240 mm × 24 mm × 2 mm rubber sample.
The centre line across the larger side was inked with two
cross-hair marks (20 mm apart) for the reference pur-
pose.
Planar-shear experiments are normally done with one
dimension of the sample restricted with respect to defor-
mation.12,13 Here, the specimen dimensions were selected
in such a way that the deformation along one direction
was very large compared to the deformation of the
direction perpendicular to that. Thereby, the small
deformation could be considered zero compared to the
larger deformation. During the test, the specimen was
held between the jaws of the apparatus along the larger
edge. The load and the elongation were recorded at
several stages and the stress and the strain were
calculated from the load and elongation data. Finally, the
results were tabulated accordingly.
3 RESULTS AND DISCUSSION
The scientific work describes deformation curves ob-
tained for three models through two different data-fitting
methods. The uniaxial, equibiaxial and pure-shear curves
provide for notable variations in these two separate
instances and the following section describes the men-
tioned variations in detail. For the purpose of an easy
identification, only uniaxial (OU) data fitting and
combined, uniaxial and pure-shear (CUS) data fitting
were used.
3.1 Experimentally obtained data dispersion
Regarding the dispersion of the three experimental-
data sets in the stress-strain domain, the following
observations were noted. As evident from Figures 2 to 7,
the biaxial-data set is located at a considerable distance
from the other two data sets. The set of pure-shear data
points is between the uniaxial- and biaxial-data sets,
more inclined towards the uniaxial-data set. The uni-
axial-data set is the closest to the strain axis.
Hence, at any given strain value, biaxial data shows
the highest stress while uniaxial data shows the lowest
stress. The pure-shear data for the same strain gives an
intermediate value. This pattern can be observed in all
the graphs.
3.2 Mooney-Rivlin two-parameter model
When referring to Figure 2, the first notable feature
is a drastic deviation of the biaxial and pure-shear
theoretical curves from the respective experimental-data
R. KEERTHIWANSA et al.: ELASTOMER TESTING: THE RISK OF USING ONLY UNIAXIAL DATA FOR FITTING ...
Materiali in tehnologije / Materials and technology 52 (2018) 1, 3–8 5
Figure 2: Uniaxial data fitting the Mooney two-parameter model
graphs Figure 3: Combined data fitting the Mooney two-parameter model
sets. This discrepancy may have been caused because the
data fitting was done with only one data set.
The second important feature is the near compatibi-
lity of the theoretical uniaxial curve with the data set.
This is because the data fitting too was done with the
uniaxial-data set. Furthermore, if we discuss the values
of the residual sum of squares (RSS) given in Table 1,
high values of the first two curves indicate the visible
discrepancy mentioned above. However, the situation
seems to be visibly improved with the combined data
fitting (Figure 3). Comparatively low values obtained for
RSS for both biaxial and pure-shear cases further
establish the improvement with respect to the theoretical
curves of the combined data fitting (Table 1). On the
other hand, the theoretical uniaxial curve seems to be a
bit degraded with the combined data fitting. A mode-
rately higher value obtained for RSS in the case of the
combined data fitting confirms this observation.
3.3 Mooney-Rivlin three-parameter model
Graphs in Figure 4 illustrate a relative dispersion of
the theoretical curves against the experimental-data sets
for the Mooney three-parameter model fitted only with
the uniaxial data. Positions of the theoretical curves of
the biaxial and pure shear seem to have further dete-
riorated as these two curves bend towards the negative
stress region. However, the uniaxial curve does not
appear to be affected by this.
At the same time, high values of RSS obtained for
the initial two cases of the Mooney three-parameter
model give further indication of the complete mismatch.
The RSS value of the near unity achieved for the uniaxial
curve means that the theoretical curve in this case nearly
R. KEERTHIWANSA et al.: ELASTOMER TESTING: THE RISK OF USING ONLY UNIAXIAL DATA FOR FITTING ...
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Figure 4: Uniaxial data fitting the Mooney three-parameter model
graphs
Figure 5: Combined uniaxial and pure-shear data fitting the Mooney
three-parameter model graphs
Figure 7: Combined uniaxial and pure-shear data fitting the Yeoh
model graphs
Figure 6: Uniaxial data fitting the Yeoh model graphs
coincides with the actual data. When we talk about the
combined fitting of this model, a certain improvement
can be seen from all the results. However, the biaxial
curve seems to deviate from the data values with the
increased strain. A high RSS value visible in this parti-
cular situation might have been a result of this deviation.
3.4 Yeoh model
When we consider the Yeoh model, the curves fitted
with a single data set, as given in Figure 6, provide
results similar to those in the previous two cases. The
only prominent feature here is a large deviation of the
theoretical biaxial curve from the actual data. Even with
the combined fitting effort, the theoretical biaxial curve
does not seem to improve. Finally, the material constants
and RSS values obtained as a result of the data-fitting
efforts related to this work are presented below in Table 1.
With respect to Table 1, the residual sum of squares
(RSS)* is defined as follows:
RSS
m d
di
n
i i
i
=
−⎛
⎝
⎜
⎞
⎠
⎟
=
∑
 

2
1
(10)
where n is the number of data. Considering an arbitrary
ith strain value, the theoretical stress value according to
the model is given by m (a point on the curve) and the
experimental stress data for the same strain is given by
d (the actual data value).
4 CONCLUSIONS
In order to discuss the risk of using only uniaxial data
for fitting hyperelastic-material models in general and
the Mooney-Rivlin model in specific, an in-depth com-
parison was done. All three models show some improve-
ment with combined-data fitting compared to only
uniaxial-data fitting. Out of the three models, the
Mooney two-parameter model allows the best improve-
ment. The Yeoh model seems to be the least responsive
to the change of the data-fitting method. From the results
of this work, it is evident that in order to obtain para-
meters for hyperelastic-material models through data
fitting, the use of only one set of data, i.e., the uniaxial
data is not sufficient.
Acknowledgment
This work and the project were realised with the
financial support of an internal grant of the TBU in Zlin,
No. IGA/FT/2017/002, funded from the resources for the
specific university research.
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Materiali in tehnologije / Materials and technology 52 (2018) 1, 3–8 7
Table 1: Values of material parameters predicted using only uniaxial data and using combined data sets (both uniaxial and pure shear)
Model Constant Value
Residual sum of squares (RSS)*
Uniaxial Biaxial Pure shear
Mooney two-parameter model
(Only uniaxial fitting)
C10 0.481348
1.7155 32.9896 7.99229
C01 -0.188224
Mooney two-parameter model
(Combined uniaxial and pure-shear fitting)
C10 0.30894
3.419414 0.856019 1.026144
C01 0.105854
Mooney three-parameter model
(Only uniaxial fitting)
C10 1.06424
1.005845 967.2367 89.36178C01 -0.838632
C11 -0.127043
Mooney three-parameter model
(Combined uniaxial and pure-shear fitting)
C10 0.28328
0.730736 82.77204 1.194405C01 0.107756
C11 0.0154692
Yeoh
(Only uniaxial fitting)
C10 0.316354
1.910457 16.5195 3.979336C20 0.0491813
C30 -0.0114659
Yeoh
(Combined uniaxial and pure-shear fitting)
C10 0.368486
2.688881 7.415695 2.007579C20 0.0207127
C30 -0.00394144
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