Access to this full-text is provided by Wiley.
Content available from Advances in Materials Science and Engineering
This content is subject to copyright. Terms and conditions apply.
Research Article
Study of Vertical Characteristics with Changes in Prepressure of
Rubber Pad Used by High-Speed EMU
Chuanbo Xu ,
1
,
2
Mao-Ru Chi ,
1
Liangcheng Dai ,
1
Yiping Jiang ,
1
and Zhaotuan Guo
1
1
State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, China
2
Henan Engineering Research Center of Rail Transit Intelligent Security, Zhengzhou Railway Vocational & Technical College,
Zhengzhou, China
Correspondence should be addressed to Mao-Ru Chi; cmr2000@163.com
Received 21 February 2020; Accepted 28 May 2020; Published 17 June 2020
Academic Editor: Dora Foti
Copyright ©2020 Chuanbo Xu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Rubber spring plays an important role in improving train performance, so the study of rubber spring is one of the focuses of train
dynamics. e vertical characteristic parameters of rubber spring are affected by prepressure significantly, as a result of varying
parameters of static stiffness, dynamic stiffness, periodic energy consumption, damping coefficient, and so on. In order to use the
theoretical method to calculate the precise static stiffness and predict the dynamic characteristics and to reduce the workload of the
rubber spring performance test, this paper takes the annular rubber pad as an example to study with different prepressures. In this
paper, the convexity coefficient correction formula (simply called the CCCF) for static stiffness calculation and the dynamic
fiducial conversion coefficient (simply called the DFCC) method based on different prepressures are proposed. rough further
analysis, the accuracy of CCCF and DFCC is proved both theoretically and experimentally. e results have shown precise
prediction of the variation of prepressure on rubber spring parameters by using CCCF and DFCC and can be used as the reference
of accurate vertical dynamic-static characteristics of the rubber spring.
1. Introduction
With the improvement of train speed rising, people are more
concerned about the riding comfort, reducing noise, and
increasing stability. Rubber components are being widely
used in rail transit vehicles, and they play a vital role for
improving the performance of the train. As a result, cal-
culation of the performance parameters has been one of the
key research areas in train dynamics. As a result, the accurate
calculation of the performance parameters of rubber com-
ponents has always been one of the focuses of train dynamics
research.
At present, there are many research studies on the static
stiffness calculation of rubber components under the con-
dition of small deformation at room temperature [1–7].
ere are many theoretical and empirical formulas for
specific shapes of rubber components, and many of these
formulas adopt linear stiffness or consider certain geometric
nonlinearity. A finite element method is also widely used in
the study of the rubber model; it can simulate the static test
more accurately. Moreover, this method has great advan-
tages for the rubber element with an irregular shape, but the
parameters calculated in the computer simulation still de-
pend on the test data. e performance of rail transit vehicles
is greatly affected by the dynamic mechanical performance
of rubber components. For this reason, the dynamic char-
acteristics of rubber components have always been an im-
portant research topic, and the research focuses on the
dynamic stiffness and dynamic damping. Most of the cal-
culations of dynamic characteristics adopt different com-
binations of springs, dampers, friction antibodies, and
fractional-order modules for the analysis, and the influ-
encing factors mainly can be given thought to frequency and
amplitude [8–16]. Sj¨
oberg [14–16] played a very important
role in the research of rubber spring combination model.
Recently, the fractional-order model has been studied
Hindawi
Advances in Materials Science and Engineering
Volume 2020, Article ID 8257286, 13 pages
https://doi.org/10.1155/2020/8257286
extensively and deeply, and many meaningful results have
been obtained. ere are many factors that affect the per-
formance of rubber springs, such as frequency, dynamic
amplitude, basic size, material, temperature, and pre-
pressure, but there are relatively few studies on temperature
and prepressure. Li et al. [17–22] studied the performance of
rubber components in different temperature states and have
proven that temperature has an influence on the perfor-
mance of rubber. Especially in the low-temperature state, the
rubber property is obviously changed, which is caused by
low-temperature crystallization. Kari [23, 24] proposed a
nonlinear temperature model of rubber pad based on shape
factor and studied the influence of temperature and pre-
pressure on material geometric parameters. e rectangular
approximation method was adopted for deformation of
rubber in this model. Cheng et al. [25] studied the nonlinear
relationship between stiffness and prepressure of different
types of rubber spring. Koh et al. [26, 27] derived the
theoretical formula for calculating the vertical stiffness of
vibration isolation rubber pad. Foti et al. [28] studied the
dynamic characteristics of the rolling isolation device from
the theoretical point of view and analyzed that its perfor-
mance was affected by many factors, including the load
factor. In addition, through further study, the research and
calculation method are provided for the optimal design of
rolling vibration isolation device [29].
As the performance of high-speed trains is more in-
tensified with the load conditions variation, it is necessary to
conduct a study of the dynamic and static performance of
rubber springs under different prepressures. In this study,
with annular rubber pad of the high-speed train bogie as an
example, the dynamic-static performances under different
prepressures were studied. Based on the study, we proposed
a convexity coefficient correction formula (simply called the
CCCF) for static stiffness calculation and the dynamic fi-
ducial conversion coefficient (simply called the DFCC)
method for dynamic performances calculation, and, fur-
thermore, the theoretical rationality is proved and the
conformity of the test results is analyzed. e results show
that CCCF and DFCC can well calculate and describe the
dynamic-static performances of annular rubber pad.
2. Parameter Definition and Test
2.1. Parameter Definition. Rubber spring has nonlinear
dynamic-static characteristics, so some characteristic pa-
rameters have different definition methods under different
research focuses and calculation conditions. In order to
facilitate the research, some characteristic parameters are
defined in this study.
As shown in Figure 1, the stiffness of the rubber spring is
defined as
k�Fmax −Fmin
xmax −xmin
,(1)
where xmax and xmin represent the maximum and minimum
displacements and Fmax and Fmin represent the corre-
sponding maximum and minimum forces; take the pre-
loading position as the displacement point 0, and set the
corresponding force value as 0. e equivalent damping
coefficient cis defined as
c�W
πωx2
0
�S
πωx2
0
,(2)
where Wis the energy dissipated during a period of vi-
bration, Sis the area enclosed by the hysteresis curve, and ω
and x0represent the angular frequency and amplitude of
vibration, respectively. Loss factor ηrepresents the rubber
spring’s ability to dissipate energy in one cycle. e hys-
teresis curve is assumed to be a central symmetric graph with
a center point of 0; ηis defined as
η�W
2kx2
0+1/2W�S
2kx2
0+1/2S�cπωx2
0
2kx2
0+1/2cπωx2
0
,(3)
for the rubber spring, 2kx2
0+1/2Sis the maximum work
done by the external force in one stable period.
With regard to the static stiffness, it is impossible to
measure stiffness in an absolute static condition, so the quasi-
static stiffness at extremely low frequencies is used. e static
stiffness is also calculated in the same way as the dynamic
stiffness, namely, formula (1). In this study, static stiffness and
quasi-static stiffness are considered to be the same. Owing to
the equivalent damping coefficient due to internal friction in
the rubber spring, the hysteretic curve is always present no
matter how low the frequency of the quasi-static test is, but
the hysteretic curve is closer to the linear form.
2.2. Test Equipment and Data Processing. e annular rubber
pad used for the bogie of the Standard EUM in China is
selected for the test analysis. e first series rubber pad has a
metal cover plate at both ends, as shown in Figure 2. e
hardness is 55 HS, and Young’s modulus Eis 2.28 MPa.
Vertical dynamic and static tests were performed on the
rubber pad in Figure 2 using the test equipment shown in
Figure 3. In the test, the room temperature was set to 25 ±2°C,
and the input was sinusoidal x�x0sin ωtx �x0sin ωt,
taking preloading position as displacement 0 mm. e static
test was performed by using the quasi-static test method with
a very low frequency, the frequency less than 0.005 Hz, and a
cycle lasts more than 5 min at quasi-static condition.
In the quasi-static conditions, the amplitudes were set as
0.2 mm, 0.5mm, 0.7mm, 1mm, and 2mm; the prepressures
F
X
0
xmin
xmax
W = S
Fmax
Fmin
k
Figure 1: Force-displacement hysteresis curve of rubber spring.
2Advances in Materials Science and Engineering
were set as 45 kN, 57 kN, 65 kN, 75 kN, and 85 kN. In the
dynamic conditions, the frequencies were 0.5 Hz, 1 Hz, 2Hz,
5 Hz, 7 Hz, and 10 Hz, and the amplitude was 0.2 mm,
0.5 mm, 1 mm, 1.5 mm, and 2 mm; the prepressures were
45 kN, 57 kN, 65 kN, 75 kN, and 85 kN. Before the test, a
large prepressure was applied to the rubber spring, and it
vibrated for 20 cycles under the maximum excitation, and
the test began 10 min later. In the formal test, 10 cycles were
tested in each working condition, and the force-displace-
ment data of the last 3 cycles were recorded; the interval
between each working condition was 3 min, so as to avoid
affecting the analysis due to obvious stress softening.
Formula (1) was used to process the force-displacement
data obtained from the test to obtain the equivalent stiffness
k. Periodic energy consumption Wis the area Senclosed by
the force-displacement hysteretic curve, which was obtained
by computer numerical integration with computer drawing
software origin2019b and was calculated by arithmetic mean
value for the data of three cycles tested in the same working
condition. In order to facilitate comparative analysis, con-
sidering that too small measurement amplitude will lead to a
large relative error of the test equipment and too large
measurement amplitude will lead to significant stress soft-
ening, the static stiffness analysis in this paper takes the
amplitude of 1 mm as the calculation standard when there is
no specified amplitude (Figure 4).
2.3. Preliminary Analyses of Test Results. When the dynamic
frequency is represented by iand the amplitude is represented
by j, the dynamic stiffness and equivalent damping coefficient
at different frequencies and amplitudes are represented in kij
and cij, respectively. According to formulas (1)–(3), the
stiffness, equivalent damping, and loss factors are calculated.
Wij is calculated by numerical integration of origin software.
Since there are many test conditions, the calculation results of
some typical conditions are selected as shown in Figure 5.
Figure 5(a) shows the curve of dynamic stiffness kij changing
with prepressure, Figure 5(b) shows the curve of periodic
energy consumption Wij changing with prepressure,
Figure 5(c) shows the curve of equivalent damping coefficient
cij changing with prepressure, and Figure 5(d) shows the
curve of loss factor ηchanging with prepressure.
According to the analyses in Figure 5, the stiffness, cycle
energy consumption, and equivalent damping coefficient of
the rubber pad all increase with an increase in prepressure.
For different combinations of frequency and amplitude, the
dynamic parameters have similar increasing proportions
with the change of prepressure. e loss factor decreases
with an increase in prepressure, but the change range is not
large; that is, the loss factor is less sensitive to the change of
prepressure. Based on the above analyses, the prepressure
has an important influence on the dynamic and static
characteristics of the rubber pad; therefore, it is necessary to
study these relationships quantitatively.
3. Research on the Parameter Calculation Model
Based on Prepressure Change
According to the preliminary analyses of 2.3, the stiffness,
periodic energy consumption, and equivalent damping co-
efficient of the rubber spring all increase with the increase of
prepressure, and the increase proportions of dynamic and
static parameters are also similar. Based on the above analyses,
70mm
150mm
260mm
Figure 2: Rubber pad of China standard EMU.
Figure 3: Test equipment.
5
6
7
8
9
10
11
Amplitude of static test
0.2mm
0.5mm
0.7mm
1mm
2mm
Stiness (kN/mm)
50 60 70 80 9040
Prepressure (kN)
Figure 4: e curve of quasi-static stiffness with prepressure.
Advances in Materials Science and Engineering 3
we make the following assumptions: (a) Assume that the
dynamic stiffness and static stiffness have the same proportion
of increase with the change of prepressure. (b) Assume that
the dynamic periodic energy consumption, equivalent
damping coefficient, and dynamic stiffness have the same
proportion of increase with the change of prepressure.
When the prepressure changes, according to the as-
sumptions from the static stiffness change law, dynamic
parameters change law can be obtained, so accurate cal-
culation of rubber spring static stiffness is very necessary.
3.1. Static Stiffness Calculation Based on Rectangular
Hypothesis. For rail vehicle bogies, annular rubber pad is a
common rubber spring. In the static compression process,
the actual size of the rubber pad is constantly changing, so
stiffness is a function of deformation rather than a fixed
value. rough a large number of tests, the relationship
between static stiffness and prepressure is nonlinear. When
calculating the static stiffness of rubber spring, it is generally
assumed that the change of ring rubber pad is regular; that is,
the vertical section of the rubber pad is rectangular, and
6
8
10
12
0.5Hz, 1mm
1Hz, 1mm
2Hz, 0.5mm
2Hz, 2mm
5Hz, 1mm
7Hz, 0.5mm
1Hz, 2mm
2Hz, 1mm
5Hz, 0.5mm
5Hz, 2mm
7Hz, 1mm0.5Hz, 2mm
7Hz, 2mm
10Hz, 0.5mm
10Hz, 1mm
10Hz, 2mm
Stiness (kN/mm)
50 60 70 80 9040
Prepressure (kN)
(a)
0
2
4
6
8
10
12
0.5Hz, 1mm
1Hz, 1mm
2Hz, 0.5mm
2Hz, 2mm
5Hz, 1mm
5Hz, 2mm
0.5Hz, 2mm
1Hz, 2mm
2Hz, 1mm
5Hz, 0.5mm
7Hz, 0.5mm
7Hz, 1mm
7Hz, 2mm
10Hz, 0.5mm
10Hz, 1mm
10Hz, 2mm
Energy consumption (kN·mm)
50 60 70 80 9040
Prepressure (kN)
(b)
0.00
0.04
0.08
0.12
0.16
0.20
Damping coefficient (kN·s/mm)
0.5Hz, 1mm
1Hz, 1mm
2Hz, 0.5mm
2Hz, 2mm
5Hz, 0.5mm
5Hz, 1mm
1Hz, 2mm
2Hz, 1mm
0.5Hz, 2mm
5Hz, 2mm
7Hz, 0.5mm
7Hz, 1mm
7Hz, 2mm
10Hz, 0.5mm
10Hz, 1mm
10Hz, 2mm
50 60 70 80 9040
Prepressure (kN)
(c)
0.10
0.12
0.14
0.16
0.18
Loss factor
0.5Hz, 1mm
0.5Hz, 2mm
1Hz, 2mm
2Hz, 0.5mm
2Hz, 2mm
5Hz, 2mm
1Hz, 1mm
5Hz, 0.5mm
7Hz, 0.5mm
7Hz, 2mm
10Hz, 1mm
2Hz, 1mm
5Hz, 1mm
7Hz, 1mm
10Hz, 0.5mm
10Hz, 2mm
50 60 70 80 9040
Prepressure (kN)
(d)
Figure 5: e change curve of dynamic parameters with prepressure.
4Advances in Materials Science and Engineering
further assume that the changes of inner circle radius and
outer circle radius are the same, which can be expressed by
the following formula:
R1−R�r−r1�d, (4)
where Ris the initial outer circle radius, ris the initial inner
circle radius, R1is the outer circle radius after compression,
r1is the inner circle radius after compression, and dis the
change in radius. According to the invariance of the volume,
the following relation can be obtained:
πR2
1−r2
1
H−hpre
�πR2−r2
H,
d�1
2
(R−r)hpre
H−hpre
,
R1�R+1
2
(R−r)hpre
H−hpre
,
r1�r−1
2
(R−r)hpre
H−hpre
,
(5)
where hpre represents the precompressed amplitude and His
the initial height of the rubber pad. e following calculation
is performed according to the empirical formula:
S1�Ac1
Af1
�πR2
1−r2
1
2πR1+r1
H−hpre
�R1−r1
2H−hpre
,
kst1 �Ac1μ1E
H−hpre
�Ac1 1+2S2
1
E
H−hpre
�πR2
1−r2
1
E
H−hpre
·1+R1−r1
2
2H−hpre
2
⎡
⎢
⎢
⎢
⎢
⎣⎤
⎥
⎥
⎥
⎥
⎦,
(6)
where S1is the area ratio, Af1is the sum of the internal and
external free areas, kst1 is static stiffness, Ac1 is the bearing
area, and μ1is the vertical shape coefficient. At this point, the
prepressure Fst1 can be calculated as follows:
Fst1 �hpre
0
Kst1dz�hpre
0
πR2
1−r2
1
E
H−z1+R1−r1
2
2(H−z)2
dz.
(7)
Due to the influence of the friction of the end surface, the
actual bearing area after the deformation is generally less
than the calculated value. In particular, the bearing area of
the rubber pad with cover is still the original value, and so
Ac1 �π(R2−r2), but the free area is variable. In this case,
formulas (6) and (7) can be modified as follows:
kst1 �Ac1μ1E
H−hpre
�πR2−r2
E
H−hpre
1+(R−r)2
2H−hpre
2
⎡
⎢
⎢
⎢
⎢
⎣⎤
⎥
⎥
⎥
⎥
⎦,
Fst1 �hpre
0
kst1dz�hpre
0
πR2−r2
E
H−z1+(R−r)2
2(H−z)2
dz.
(8)
3.2. Elliptic Hypothesis and Convexity Coefficient. In practice,
the compression deformation of the ring rubber pad is ir-
regular, and many rubber pads have rubber covers glued
together at both ends; therefore, it is unreasonable to assume
rectangular deformation. In this study, in order to better
characterize the deformation of the rubber pad, the elliptic
deformation hypothesis is adopted for calculation and
analysis.
Figure 6(a) shows the initial state of the annular rubber
pad, Figure 6(b) shows the state based on the rectangle
hypothesis, and Figure 6(c) shows the state based on the
ellipse hypothesis.
When the rubber spring is compressed, the bearing area
does not change because the area of the rubber cover is
constant, but the free area changes, and the middle of the
rubber spring expands outward. Under general pressure,
there is enough expansion space inside the annular rubber
pad. Two assumptions are made based on the vertical section
profile in Figure 7: (a) Assume that the vertical section of the
rubber pad extruded part is semiellipse. (b) Assume that the
sizes of the inner and outer semiellipses are the same. In
Figure 7, the center of the rubber pad is selected as the origin
to establish a rectangular coordinate system; the transverse
half axis of the semiellipse is represented by a, and the
vertical half axis is represented by b. e analytic geometric
expression of the contour curve in Figure 7 is given by
(x−R)2
a2+z2
b2�1, R ≤x≤R+a,
z�±b, −R<x<−r∪r<x<r,
(x−r)2
a2+z2
b2�1, r −a≤x≤r,
(x+r)2
a2+z2
b2�1,−r≤x≤−r+a,
(x+R)2
a2+z2
b2�1,−R−a≤x≤−R.
(9)
According to the volume invariance of rubber spring, it
can be calculated as follows:
V�πR2−r2
H�2b
0
πx2
1−x2
2
dz
�2πb R2−r2
+abπ(R+r).
(10)
.
In formula (10), Vrepresents the volume of the rubber
pad, x1represents the coordinates of the points on the outer
contour of the rubber pad, and x2represents the coordinates
of the points on the inner contour of the rubber pad. bis
given by
b�H−hpre
2.(11)
By further calculation, we can get
Advances in Materials Science and Engineering 5
a�(R−r)(H−2b)
b�2(R−r)hpre
H−hpre
.(12)
e total free area Af2 of the rubber spring is the sum of
the outer and inner sides of the ellipse; it can be calculated by
Af2 �4πb
0
x1+x2
dz�4bπ(R+r) � 2π(R+r)H−hpre
.
(13)
By comparison, the results of the ellipse hypothesis and
rectangle hypothesis are the same for the calculation of the
free area:
Af2 �Af 1 �2π(R+r)H−hpre
.(14)
e conclusion formula (14) is based on the assumption
that the changes of inner and outer dimensions are equal. If
this assumption is not established, then the difference be-
tween Af2 and Af 1 is always small.
Based on the above analysis, the conclusion can be
drawn: under the assumption that the internal and external
expansions are equal, the static stiffness calculation formulas
of the two shape assumptions are the same; if only the ellipse
assumption is used, the accuracy of the static stiffness cal-
culation cannot be improved.
In practical applications, the vertical stiffness of rubber
spring increases with an increase in prepressure due to the
influence in geometric and material nonlinearity. In formula
(8), although shape variation and area ratio are considered,
there is still a large error between the calculated value and the
actual value, so maybe it is not enough to use area ratio to
correct the formula. When the ellipse hypothesis is con-
sidered, it is necessary to use the degree of ellipse defor-
mation to further modify the calculation formula (8).
In this study, the additional convexity coefficient is in-
troduced to modify the stiffness calculation formula, which
can also be understood as the modification of modulus in
case of large preload; this understanding does not affect the
calculation results. Convexity coefficient is a function, not a
specific value. e higher the compression degree of rubber
spring is, the more obvious the sides protrude inward and
outward and the greater the change of rubber performances
is. e effective coefficient of convexity should be able to
represent these changes, so the coefficient of convexity μcon is
defined as
μcon �1+a
b.(15)
In this case, the stiffness kst2 of the rubber pad can be
calculated as follows:
kst2 �Ac2μ2E
H−hpre
μcon �πR2−r2
E
H−hpre
1+R2−r2
2
8b2(R+r)2
1+a
b
�πR2−r2
E
H−hpre
1+(R−r)2
2H−hpre
2
⎡
⎢
⎢
⎢
⎢
⎣⎤
⎥
⎥
⎥
⎥
⎦1+a
b
,
(16)
where Ac2 represents the bearing area,
Ac2 �Ac1 �π(R2−r2),μ2is the vertical shape coefficient,
and the calculation method refers to formula (6). e re-
lationship between prepressure Fst2 and precompression
amplitude hpre can be given by
Fst2 �hpre
0
kst2dz�hpre
0
πR2−r2
E
H−z1+(R−r)2
2(H−z)2
1+a
b
dz.
(17)
In formula (17), aand bcorrespond to formula (12) and
formula (11), respectively, and hpre in aand bis replaced by
the variable z.
If prepressure is given instead of the precompression
amplitude value, the precompression amplitude value can be
calculated by using the inverse function hpre �f−1(Fst2)of
the function Fst2 �f(hpre), and then calculate the stiffness
according to formula (16). Due to difficulty in calculation for
the inverse function, hpre can be obtained by computer
numerical processing.
3.3. DFCC and Parameter Calculation. According to the
static stiffness calculation formula (16), the static stiffness
under different prepressures is different. In order to compare
Rubber
cover plate
Rubber
(a)
Prepressure
(b)
Prepressure
(c)
Figure 6: Compression deformation diagram of annular rubber pad.
0x1x2
rR
ba
Z
X
Figure 7: e vertical section profile of Figure 6(c).
6Advances in Materials Science and Engineering
the changes of static stiffness under different prepressures,
the DFCC λN•jz is defined as
λN•jz �kN
kjz
�
1/ H−hpre
1+ (R−r)2/2 H−hpre
2
(1+(a/b))
1/ H−hjz
1+ (R−r)2/2 H−hjz
2
1+ajz/bjz
,(18)
where kN,hpre,a, and b, respectively, represent the static
stiffness, precompression amplitude, ellipse horizontal half
axis, and vertical half axis when the prepressure is N, and
kjz,hjz ,ajz, and bjz are the corresponding parameters of the
fiducial prepressure.
e dynamic characteristics of the rubber pad have a
complex relationship with the amplitude and frequency. In
order to highlight the research focus and more clearly reveal
the change rule of performance parameters with the pre-
pressure, select a specific frequency and amplitude for
comparison; only consider the relative change of perfor-
mance parameters when the prepressure changes. According
to the assumptions that the parameters of dynamic-static
parameters increase in the same proportion with the change
of prepressure, and according to formula (18), the following
formula can be obtained:
kN�kjzλN•jz ,
kN•ij �kjz•ijλN•jz ,
cN•ij �cjz•ijλN•jz ,
WN•ij �Wjz•ijλN•jz .
(19)
In formula (19), the working conditions of performance
parameters are expressed by subscripts, iis frequency, jis
amplitude, Nis prepressure of the condition to be predicted,
and jz is fiducial prepressure.
4. Theoretical Analyses of Model Hypotheses
4.1. Influence Mechanism of Prepressure on the Characteristics
of Rubber Spring. Rubber material has entropy elasticity and
energy elasticity. e nature of high elasticity is entropy
elasticity [30, 31]; owing to the hot motion of the polymer
chain, the chain segment rotates around a polymer chain
axis in a small range, and the conformation changes. e
more convoluted the chain, the more possible conforma-
tions and the higher the entropy of the system. On the other
hand, the straighter the chain, the fewer the possible con-
formations and the smaller the entropy of the system. is
elasticity due to the thermal motion of the molecules, that is,
the increase in entropy of the system, is called entropy
elasticity. In the initial free state of the rubber spring,
considering the effect of thermal motion, the mesh chain
between the crosslinking points can be regarded as an ir-
regular line group, and its terminal distance conforms to the
Gauss distribution, with high conformational entropy.
When the rubber spring is compressed, all mesh chains
deform. As a result of mesh chains deformation, the order of
mesh chains is improved, and the conformational entropy of
rubber spring is decreased. According to the entropy in-
crease theory, the rubber spring produces a rebound force.
In the actual rubber deformation, the extension of the
molecular chain will also cause some changes in bond length,
bond angle, and intermolecular interaction, resulting in the
change of energy in the system. Changes in internal energy
also affect the elastic force, which is the elastic energy; the
strain energy density function is commonly used to describe
the elasticity of rubber [32, 33]. When the deformation is
small, the contribution of energy elasticity to elastic force is
obvious, while when the deformation is large, entropy
elasticity dominates.
e stiffness of the rubber spring increases with an
increase in the prepressure due to geometric nonlinearity
and material nonlinearity. When the rubber element is
compressed, the molecular chain tends to be more trans-
verse, and the increase of the ordered chain helps the ad-
jacent mesh chains to bind together, thus playing an
additional crosslinking role. e increase in the crosslinking
degree will lead to the increase of modulus and stiffness,
which will eventually lead to the increase in compression
stiffness of rubber. In addition, with the increase in the
compression amount, the molecular chains of the rubber
spring become less and less curly, and the lateral elongation
becomes more and more difficult, so more and more
pressure is needed to produce the same compression dis-
placement; that is, the stiffness of the rubber spring in the
precompression position becomes larger and larger.
4.2. eoretical Analysis of Stiffness Variation Rule Hypothesis.
Whether on dynamic or static conditions, when the pre-
pressure is the same, the shape variable of the rubber spring
in the balance position is the same. In other words, the
transverse elongation of rubber spring molecular chain is
the same, so it can be considered that the influences of
prepressure on static stiffness and dynamic stiffness are
similar. If the stress relaxation of rubber spring is simply
described by Maxwell model, the following formula can be
obtained:
E(t) � Ee(−t/τ),(20)
where E(t)is the stress relaxation modulus at tand τis the
time constant.
In fact, the static state is a quasi-static state with very low
frequency, so, at any displacement, it can be considered that
the rubber has a long time to relax, and the dynamic state has
less time to relax at any position.
Advances in Materials Science and Engineering 7
If very small displacement range Δxis considered as a
point for calculation, assuming that the relaxation time of
static condition at each point is t1and the relaxation time of
dynamic condition at each point is t2, then t1>t2is always
true if the amplitudes are the same:
E t1
�Ee −t1/τ
( ) <E t2
�Ee −t2/τ
( ),
E t2
E t1
�e−t2+t1
( )/τ
( ) >1.
(21)
According to formulas (16), (17), and (21), we can get
kd�e−t2+t1
( )/τ
( )kst .(22)
From formula (22), when the amplitude is the same, the
dynamic stiffness kdequals the static stiffness kst times the
frequency-dependent coefficient e((− t2+t1)/τ). When only the
prepressures change, it is considered that e((− t2+t1)/τ)is a fixed
value, the dynamic and static stiffness change laws are
similar, and the dynamic stiffness is greater than the static
stiffness.
Based on the above analysis, when only the prepressures
change, it is reasonable to assume that dynamic stiffness and
static stiffness have the same change proportion.
4.3. eoretical Analysis of Dynamic Parameter Variation
Rule Hypothesis. For dynamic conditions with the same
frequency and amplitude, the equivalent stiffness of rubber
springs is different under different prepressure. e greater
the prepressure, the greater the equivalent stiffness, as shown
in Figure 5(a). In Figure 8, the precompression position is set
to the displacement 0 point, FNmax is the spring force
corresponding to the maximum displacement when the
preload is N,Fjz max is the spring force corresponding to the
maximum displacement when the preload is fiducial pre-
pressure Njz, and kNand kjz are the corresponding
equivalent stiffnesses. e DFCC λN•jz can be calculated as
follows:
λN•jz �kN
kjz
�FNmax/x0
Fjz max/x0
�FNmax
Fjz max
.(23)
In an ideal case, considering the same displacement
point (distinguishing compression and rebound travel),
FN�λN•jzFjz holds. According to the calculation formula
W�Fdxof periodic energy consumption, we can get
WN
Wjz
�FNdx
Fjzdx
�λN•jzFjz dx
Fjzdx
�λN•jz.(24)
e following formula can be obtained by combining
formulas (2) and (24):
cN�λN•jzcjz .(25)
According to the above analysis, it can be seen that, in
the case of the same dynamic frequency and amplitude, with
only the change of prepressure, the increases in dynamic
equivalent stiffness and equivalent damping are similar, so,
theoretically, if only the prepressures change, it is reasonable
to assume that the damping coefficient and dynamic stiffness
have the same change proportion.
5. Test Analysis of the Calculation Model
5.1. Error Analysis of the CCCF. For the analysis of the
validity of the modified formula, the CCCF, unmodified
formula, and experimental data were compared to analyze
the calculation errors. Figure 9 shows the results.
From Figure 9, it can be seen that CCCF is very close to
the test results, and the maximum error of CCCF in the
commonly used prepressure range (according to the train
load) is within 6%; the unmodified rectangular approximate
calculation method is almost about 50%; only in the case of
very small prepressure, the two calculation methods are
relatively close. Based on the comparative analysis, CCCF is
more accurate to calculate the static stiffness of rubber pad,
F
X
0x0
W = S
FN
kjz
Fjz
kN
–x0
–Fjz
–FN
Figure 8: Dynamic hysteresis curve under different prepressures.
0 20406080
e test results
CCCF
Uncorrected formula
Stiffness (kN/mm)
Prepressure (kN)
2
4
6
8
10
Figure 9: Comparison curves of different models.
8Advances in Materials Science and Engineering
and the error is greatly reduced compared with the original
calculation method.
In order to further verify the accuracy of CCCF calcu-
lation, annular rubber pads A and B of a metro train are
selected for static stiffness analysis. e results show that the
formula is still very accurate. As shown in Table 1, the
calculation errors of CCCF are only 5.17% and 0.62%.
erefore, CCCF may be universal to the annular rubber
pad, and its better calculation accuracy is not an accidental
phenomenon, which can reflect some basic physical laws of
the rubber pad after compression.
5.2. Test Verification of DFCC. According to the test data, kij,
Wij, and cij of 45 kN prepressure were selected as the
calculation bases, and these parameters are expressed as
k45ij,W45ij , and c45ij. According to formula (2), for specific
frequency and amplitude, the difference between W45ij and
c45ij is a fixed scale factor (1/(πωx2
0)), and the result after
conversion is exactly the same, so the conversion of W45ij
will not be discussed in detail. ηij is less sensitive to the
change of prepressure and will not be discussed here. Owing
to the problems such as equipment error in the test, too
small dynamic amplitude, too high frequency, and pre-
pressure force may cause irregular jump of measurement
data. e fiducial prepressure was selected as 45 kN to
calculate the DFCC under different working conditions,
with the frequency of 0.5 Hz, 1 Hz, 2 Hz, 5 Hz, 7 Hz, and
10 Hz, the amplitude of 0.5 mm, 1 mm, 1.5 mm, and 2 mm,
and the prepressure of 45 kN, 57 kN, 65 kN, and 75 kN.
Table 1: eoretical and experimental comparison of A and B stiffness.
Basic parameters Prepressure (kN) Static stiffness (kN/mm)
Test results CCCF Unmodified formula
AR�113 mm; r�40.5 mm;
H�20.5 mm; E�2.28 MPa 34 49.68 52.25 32.88
BR�102 mm; r�40.5 mm;
H�22 mm; E�2.28 MPa 34 32.40 32.12 18.28
0 5 10 15 20 25
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Stiness ratio
Sampling sequence
45kN
65kN
57kN
75kN
(a)
1.10 1.15 1.20
0
2
4
6
Count
Value of ratio
1.20 1.25 1.30 1.35
0
2
4
6
Count
1.40 1.45 1.50 1.55
0
2
4
6
Count
57kN
Value of ratio
65kN
Value of ratio
75kN
(b)
Figure 10: e curve of the stiffness ratio and interval statistics. (a) e curve of the stiffness ratio. (b) Chart of frequency distribution.
Advances in Materials Science and Engineering 9
Taking the different combinations of frequency and am-
plitude as sample points, there are 24 samples in total, and
each sample has 4 state DFCC values. Figures 10 and 11 are
given based on the DFCC calculation results: Figure 10
shows the DFCC of dynamic stiffness and Figure 11
shows the DFCC of damping coefficients.
It can be seen from Figures 10 and 11 that, regardless of
the combination of frequency and dynamic amplitude, the
conversion coefficient under the same prepressure is obvi-
ously stable in a small fixed range. erefore, it can be
considered that the prepressure has a significant effect on the
DFCC of stiffness and damping coefficient, and the degrees
of influence for different combination sequences of fre-
quency and amplitude are similar. In the frequency and
amplitude range of the test, the effect of frequency and
amplitude on DFCC is not significant. To further analyze the
influence of prepressure on dynamic performance, the
arithmetic mean values of DFCC at different sampling
points are calculated under the same preload, as shown in
Table 2.
According to the analysis of Table 2, under the same
prepressure, the average DFCC of dynamic stiffness, peri-
odic energy consumption, damping coefficient, and static
stiffness is very close, and the maximum relative difference is
only 2.61%. e mean DFCC under each preload contains 24
combinations of frequency and amplitude, which may not be
accidental. e high approximation of DFCC may reflect
some deep relationship; that is to say, the influence trend and
degree of prepressure on dynamic stiffness, periodic energy
consumption, damping coefficient, and static stiffness are
similar, with similar quantitative relationship.
rough the above test analysis, it can be concluded
that the assumption that the dynamic and static param-
eters change in the same proportion is in line with the
actual test, and it is feasible to use DFCC with static
stiffness to calculate the dynamic characteristics under
different prepressures. It is complex and difficult to cal-
culate the dynamic characteristics directly by using the
foundation parameters; the calculation of dynamic
characteristics is affected by many factors, such as
0 5 10 15 20 25
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Damping ratio
Sampling sequence
45kN
65kN
57kN
75kN
(a)
1.05 1.10 1.15 1.20
2
4
6
8
Count
Value of ratio
1.16 1.18 1.20 1.22 1.24 1.26
0
0
2
4
6
Count
1.35 1.40 1.45 1.50
0
2
4
6
Count
57kN
Value of ratio
65kN
Value of ratio
75kN
(b)
Figure 11: e curve of the damping ratio and interval statistics. (a) e curve of the damping ratio. (b) Chart of frequency distribution.
Table 2: Arithmetic mean of DFCC under different prepressures.
45 kN 57 kN 65 kN 75 kN
Average DFCC of kij 1 1.139 1.259 1.493
Average DFCC of Wij 1 1.109 1.213 1.424
Average DFCC of cij 1 1.109 1.213 1.424
Average DFCC of kst 1 1.123 1.227 1.460
10 Advances in Materials Science and Engineering
geometric size, modulus, frequency, amplitude, pre-
pressure, temperature, and humidity. erefore, in this
study, based on the dynamic parameters of a certain
prepressure, the dynamic parameters under different
prepressures are calculated by DFCC, which is very
meaningful for the prediction of dynamic performances of
trains under different loads.
5.3. Error Analysis of the DFCC Method. Select 45 kN as the
conversion standard, and calculate λN•45 when the pre-
pressure is 57 kN, 65 kN, 75 kN, and 85 kN according to the
theoretical formula (18); use formula (19) to calculate the
dynamic parameters and compare with the test value, as
shown in Figure 12. is method is named the theoretical
DFCC prediction method (Method 1).
Select 45 kN as the conversion standard; according to the
test data and formula (1), the static stiffness kst under dif-
ferent prepressures is calculated, and the ratio of test stiffness
is further calculated to obtain λN•45; refer to Table 2; use
formula (19) to calculate the dynamic parameters and
compare with the test value, as shown in Figure 13. is
method is named the test DFCC prediction method (Method
2).
As shown in Figure 12(a), the dynamic stiffness obtained
by Method 1 is compared with the test value, the maximum
error is 10.53%, the average error is 5.85%, and the root
mean square error is 6.54%. As shown in Figure 12(a), the
dynamic damping coefficient obtained by Method 1 is
compared with the test value, the maximum error is 12.26%,
the average error is 9.36%, and the root mean square error is
9.56%.
As shown in Figure 13(a), the dynamic stiffness obtained
by Method 2 is compared with the test value, the maximum
error is 4.85%, the average error is 0.05%, and the root mean
square error is 2.25%. As shown in Figure 13(a), the dynamic
damping coefficient obtained by Method 2 is compared with
the test value, the maximum error is 5.82%, the average error
is 3.37%, and the root mean square error is 3.61%.
In summary, Method 1 and Method 2 have high ac-
curacy in fitting dynamic stiffness and damping coefficient
under different prepressures. In particular, the predicted
value of Method 2 is almost consistent with the actual test
results. erefore, when the prepressure is the same, the
DFCC of static stiffness is similar to the DFCC of dynamic
parameters. Some assumptions about DFCC are reason-
able, and it is effective to use the conversion coefficient of
static stiffness to describe the dynamic parameters. Both
Method 1 and Method 2 can be used successfully to de-
scribe rubber pad dynamic characteristics. Method 2 has a
better approximation degree due to the use of static test
values, but more data of static test is needed. Although
Method 1 is not as accurate as Method 2, it does not need
static test data and can be directly calculated according to
basic state and geometric parameters. Method 1 is more
convenient and fully meets the requirements of engi-
neering calculation. e method of DFCC is particularly
important for the calculation of the dynamic perfor-
mances of the rubber pad. By using this method in dif-
ferent prepressure conditions, only the parameters of the
fiducial prepressure conditions need to be tested. e
parameters of other prepressure conditions can be cal-
culated, and this could significantly reduce the test
workload.
45 50 55 60 65 70 75
6
7
8
9
10
Stiffness (kN/mm)
Prepressure (kN)
1Hz, 1mm Method 1
2Hz, 2mm Method 1
5Hz, 1mm Method 1
10Hz, 1mm Method 1
1Hz, 1mm test
2Hz, 2mm test
5Hz, 1mm test
10Hz, 1mm test
(a)
1Hz, 1mm Method 1
2Hz, 2mm Method 1
5Hz, 1mm Method 1
10Hz, 1mm Method 1
1Hz, 1mm test
2Hz, 2mm test
5Hz, 1mm test
10Hz, 1mm test
45 50 55 60 65 70 75
0.02
0.04
0.06
0.08
0.10
Damping coefficient (kN·s/mm)
Prepressure (kN)
(b)
Figure 12: Comparison between Method 1 and test value.
Advances in Materials Science and Engineering 11
6. Conclusions
In this study, through theoretical analysis, CCCF for static
stiffness calculation is proposed, and DFCC method based
on the change of prepressure is further proposed. rough
the theoretical analysis and experimental verification of
CCCF and DFCC, the following conclusions can be drawn:
(1) e change of the prepressure has a significant effect
on the dynamic-static stiffness and damping coef-
ficient of the rubber pad but not on the loss factor.
e different combinations of dynamic frequency
and amplitude do not change the influence of pre-
pressure on dynamic characteristics.
(2) CCCF is reasonable and effective in the calculation of
static stiffness, which greatly improves the calcula-
tion accuracy compared with the rectangular ap-
proximation in common use. e error within the
range of common prepressure meets the engineering
requirements. CCCF does not depend on test data
and can be calculated directly from the basic data of
rubber pad, which has a certain engineering
significance.
(3) When the prepressure changes, the DFCC of static
stiffness, dynamic stiffness, periodic energy con-
sumption, and damping coefficient have very similar
change laws, which is established in theory and
practice. Test DFCC method is more effective than
theoretical DFCC method, but theoretical DFCC
method is more convenient in calculation and
application.
DFCC method is effective for the calculation and pre-
diction of rubber pad, but the applicability of other rubber
components needs further study. In addition, the
quantitative relationship among dynamic characteristic
parameters, frequency and amplitude, still needs to be
further studied in order to avoid considering different fre-
quencies and amplitudes in the dynamic fiducial prepressure
test in the current model.
Data Availability
e data used to support the findings of this study are
available from the corresponding author upon request.
Conflicts of Interest
e authors declare that they have no conflicts of interest.
Acknowledgments
is research was supported by the National Key R&D
Program of China (Grant nos. 2016YFB120404 and
2018YFB1201700) and by the Independent Subject of State
Key Laboratory of Traction Power (Grant no.
2018TPL_T04).
References
[1] G. M. Giovanni, E. Mele, G. Marrazzo, G. Brandonisio, and
A. d. Luca, “Stability issues and pressure–shear interaction in
elastomeric bearings: the primary role of the secondary shape
factor,” Bulletin of Earthquake Engineering, vol. 14, no. 2,
pp. 569–597, 2016.
[2] L. Nasdala, A. Kempe, and R. Rolfes, “An elastic molecular
model for rubber inelasticity,” Computational Materials Sci-
ence, vol. 106, pp. 83–99, 2015.
[3] M. Gajewski, R. Szczerba, and S. Jemioło, “Modelling of
elastomeric bearings with application of yeoh hyperelastic
45 50 55 60 65 70 75
6
7
8
9
10
Stiness (kN/mm)
Prepressure (kN)
1Hz, 1mm Method 2
2Hz, 2mm Method 2
5Hz, 1mm Method 2
10Hz, 1mm Method 2
1Hz, 1mm test
2Hz, 2mm test
5Hz, 1mm test
10Hz, 1mm test
(a)
45 50 55 60 65 70 75
0.02
0.04
0.06
0.08
0.10
0.12
Damping coefficient (kN·s/mm)
Prepressure (kN)
1Hz, 1mm Method 2
2Hz, 2mm Method 2
5Hz, 1mm Method 2
10Hz, 1mm Method 2
1Hz, 1mm test
2Hz, 2mm test
5Hz, 1mm test
10Hz, 1mm test
(b)
Figure 13: Comparison between Method 2 and test value.
12 Advances in Materials Science and Engineering
material model,” Procedia Engineering, vol. 111, pp. 220–227,
2015.
[4] P. Wagner, P. Wriggers, C. Klapproth, C. Prange, and B. Wies,
“Multiscale FEM approach for hysteresis friction of rubber on
rough surfaces,” Computer Methods in Applied Mechanics and
Engineering, vol. 296, pp. 150–168, 2015.
[5] O. P. Singh, T. Sreenivasulu, and M. Kannan, “e effect of
rubber dampers on engine’s NVH and thermal performance,”
Applied Acoustics, vol. 75, pp. 17–26, 2014.
[6] C. Li, G. T. Yang, and Z. Z. Huang, “Constitutive equations of
incompressible nonlinear super-elastic material,” Transaction
of Beijing Institute of Technology, vol. 31, no. 1, pp. 30–34,
2011, Chinese.
[7] Y. Q. Li and X.-L. Gao, “Constitutive equations for hypere-
lastic materials based on the upper triangular decomposition
of the deformation gradient,” Mathematics and Mechanics of
Solids, vol. 24, no. 6, pp. 1785–1799, 2019.
[8] R. Luo, H. L. Shi, J. Y. Guo, L. Huang, and J. Wang, “A
nonlinear rubber spring model for the dynamics simulation of
a high speed train,” Vehicle System Dynamics, pp. 1–8, 2019.
[9] D. W. Zhang and S. Y. Zhu, “A fractional derivative model for
rubber spring of primary suspension in railway vehicle dy-
namics,” ASCE-ASME Journal of Risk and Uncertainty in
Engineering Systems, Part B: Mechanical Engineering, vol. 3,
no. 3, 2017.
[10] H. Shi and P. Wu, “A nonlinear rubber spring model con-
taining fractional derivatives for use in railroad vehicle dy-
namic analysis,” Proceedings of the Institution of Mechanical
Engineers, Part F: Journal of Rail and Rapid Transit, vol. 230,
no. 7, pp. 1745–1759, 2016.
[11] D.-w. Sun, Z.-g. Chen, G.-y. Zhang, and P. Eberhard,
“Modeling and parameter identification of amplitude- and
frequency-dependent rubber isolator,” Journal of Central
South University, vol. 18, no. 3, pp. 672–678, 2011.
[12] J. Wu and W. B. Shangguan, “Modeling and applications of
dynamic characteristics for rubber isolators using viscoelastic
fractional derivative model,” Engineering Mechanics, vol. 25,
no. 1, pp. 161–166, 2008, Chinese.
[13] P. Spanos, “Cure system effect on low temperature dynamic
shear modulus of natural rubber,” Rubber World, vol. 229,
no. 2, pp. 22–27, 2003.
[14] M. Sj¨
oberg and L. Kari, “Nonlinear isolator dynamics at finite
deformations: an effective hyperelastic, fractional derivative,
generalized friction model,” Nonlinear Dynamics, vol. 33,
no. 3, pp. 323–336, 2003.
[15] M. Sj¨
oberg and L. Kari, “Non-linear behavior of a rubber
isolator system using fractional derivatives,” Vehicle System
Dynamics, vol. 37, no. 3, pp. 17–236, 2002.
[16] M. Sj¨
oberg, “Rubber isolators-measurements and modeling
using fractional derivatives and friction,” SAE Technical Pa-
pers, vol. 3518, pp. 2000-2001, 2000.
[17] Z.-L. Li, Z.-q. Liu, D.-G. Sun, B.-J. Yan, and J. Meng,
“Fractional Maxwell model of viscoelastic oscillator and its
frequency response,” Journal of Vibration Engineering &
Technologies, vol. 6, no. 1, pp. 1–6, 2018.
[18] T. L. Yu, D. Y. Yuan, and X. L. Sun, “e finished product of
rubber bearing mechanical performance analysis in the low
temperature,” Applied Mechanics and Materials, vol. 178–181,
pp. 2254–2259, 2012.
[19] W. Teng, F. Tan, and H. Shi, “Experimental study of high/low
temperature effects on the dynamic performance of rubber
spring for railway vehicles,” Journal of Physics: Conference
Series, vol. 1213, no. 5, Article ID 052042, 2019.
[20] R. ¨
Osterl¨
of, L. Kari, and H. Wentzel, “Temperature depen-
dency of a viscoplastic constitutive model for rubber with
reinforcing fillers,” in Proceedings of the 9th European Con-
ference on Constitutive Models for Rubbers, pp. 149–156,
ECCMR, Nantes, France, July 2015.
[21] A. Stevenson, “e influence of low-temperature crystalli-
zation on the tensile elastic modulus of natural rubber,”
Journal of Polymer Science: Polymer Physics Edition, vol. 21,
no. 4, pp. 553–572, 1983.
[22] A. Stevenson, “Effect of elastic strain on the low temperature
crystallization of rubber,” Plastics & Rubber Institute, vol. 6,
pp. 1–6, 1987.
[23] L. Kari, “e non-linear temperature dependent stiffness of
precompressed rubber cylinders,” KGK-kautschuk Gummi
Kunststoffe, vol. 55, no. 3, pp. 76–81, 2002.
[24] L. Kari, “An analytical temperature-dependent collocation
model for preloaded rubber cylinders,” e Journal of Strain
Analysis for Engineering Design, vol. 37, no. 4, pp. 289–299,
2002.
[25] Z. Y. Cheng, Z. D. Hu, and L. Li, “Study on the performance of
lead rubber bearing considering vertical force correlation,”
International Journal of Multiphysics, vol. 12, no. 3,
pp. 267–282, 2018.
[26] C. G. Koh and J. M. Kelly, “A simple mechanical model for
elastomeric bearings used in base isolation,” International
Journal of Mechanical Sciences, vol. 30, no. 12, pp. 933–943,
1988.
[27] K. L. Ryan, J. M. Kelly, and A. K. Chopra, “Nonlinear model
for lead-rubber bearings including axial-load effects,” Journal
of Engineering Mechanics, vol. 131, no. 12, pp. 1270–1278,
2005.
[28] D. Foti, A. Catalan Goni, and S. Vacca, “On the dynamic
response of rolling base isolation systems,” Structural Control
and Health Monitoring, vol. 20, no. 4, pp. 639–648, 2013.
[29] N. Menga, D. Foti, and G. Carbone, “Viscoelastic frictional
properties of rubber-layer roller bearings (RLRB) seismic
isolators,” Meccanica, vol. 52, no. 11-12, pp. 2807–2817, 2017.
[30] P. S. He, P. P. Zhu, and H. Yang, “Characteristics of rubber
elasticity,” Polymer Bulletin, vol. 12, pp. 68–71, 2009, in
Chinese.
[31] E. M. Arruda and M. C. Boyce, “A three-dimensional con-
stitutive model for the large stretch behavior of rubber elastic
materials,” Journal of the Mechanics and Physics of Solids,
vol. 41, no. 2, pp. 389–412, 1993.
[32] R. W. Ogden, “Recent advances in the phenomenological
theory of rubber elasticity,” Rubber Chemistry and Technology,
vol. 59, no. 3, pp. 361–383, 1986.
[33] M. Mooney, “A theory of large elastic deformation,” Journal of
Applied Physics, vol. 11, no. 9, pp. 582–592, 1940.
Advances in Materials Science and Engineering 13
Available via license: CC BY
Content may be subject to copyright.