Embed Size (px)
Citation preview
Finite element estimation of hysteretic loss and rollingresistance of 3-D patterned tire
J. R. Cho • H. W. Lee • W. B. Jeong •
K. M. Jeong • K. W. Kim
Received: 27 March 2013 / Accepted: 13 September 2013 / Published online: 19 September 2013
� Springer Science+Business Media Dordrecht 2013
Abstract Some of energy supplied to the vehicle
driven on road is dissipated through rolling tires due to
the hysteretic loss of rubber compounds, so the
hysteretic loss is considered as a sort of pseudo-force
resisting the tire rolling. This paper is concerned with
the numerical prediction of the hysteretic loss and the
rolling resistance (RR) of 3-D periodic patterned tire.
A 3-D periodic patterned tire model is constructed by
copying 1-sector tire mesh in the circumferential
direction. Strain cycles during one revolution are
approximated by utilizing the 3-D static tire contact
analysis, for which the strain values at Gaussian points
in the elements which are sector-wise repeated in the
same circular ring of elements are taken. The strain
amplitude during one revolution of tire is determined
by taking the maximum principal value of the half
amplitudes of each strain components in the multi-
axial state of strain. The hysteretic loss during one
revolution is predicted in terms of the loss modulus of
rubber compound and the maximum principal value of
the half amplitudes of six strain components. Through
the numerical experiments, the validity of the pro-
posed prediction method is examined by comparing
with the experiment and the dependence of RR on the
tread pattern is also investigated.
Keywords Hysteretic loss � Patterned tire �Half strain amplitudes � Maximum principal
value � 1-Sector mesh � Tread pattern
List of symbols
FR Total rolling resistance
FZ Vertical reaction
RR Rolling resistance
qr Effective radius of tire
W Total hysteretic loss of tire during one
revolution
CR Coefficient of rolling resistance
DW Hysteretic loss density during one
revolution
d Phase lag
Tc;x Period and angular velocity
r0; e0 Stress and strain amplitudes_Q Heat generation per unit volume
G� Complex modulus of rubber compound
G0;G00 Storage and loss moduli of rubber
compound
f Frequency of external excitation
T Temperature
ca Peak-to-peak strain amplitude
J. R. Cho (&) � H. W. Lee � W. B. Jeong
School of Mechanical Engineering, Pusan National
University, Pusan 609-735, Korea
e-mail: [email protected]
J. R. Cho
Research and Development Institute of Midas IT,
Gyeonggi 463-400, Korea
K. M. Jeong � K. W. Kim
R&D Center, Kumho Tire Co., Ltd., Kwangju 506-711,
Korea
123
Int J Mech Mater Des (2013) 9:355–366
DOI 10.1007/s10999-013-9225-y
e1;K Maximum principal value of the half
amplitudes of six strain components in
the K-th element
Ns Number of tire sectors
Nog �ð Þ Global element number
C Circular path
=C Set of the sector-wise periodic elements
on the same circular path C
XCJ
J-th element in =C
@C‘
Set of all the ‘-th Gaussian points of all
the elements in =C
Deij
� �C
K;‘Half amplitudes of strain components at
the ‘-th Gaussian point of the K-th
element in sector 1
M‘ Total number of Gaussian points within
an element
Deij
� �C
KElement-wise averaged half amplitudes
of strain components
DwK Hysteretic loss per unit volume in XK
DWK Total hysteretic loss of the K-th element
XK
Vol XKð Þ Volume of the K-th element XK
=CI
Set of the elements in sector I which track
the path C
nC Number of distinct circular paths
Num =CI
� �Number of the elements belonging to =C
I
W Jið Þ Strain energy density functional
Ji Invariants of Green–Lagrangian strain
tensor
C10;C01 Moonley–Rivlin constants
D1 Penalty-like parameter
j; s Shear and bulk moduli of tire
g Loss factor of rubber compound
1 Introduction
Elastomeric components of automobile tire rolling on
the road exhibit the cyclic variations in deformation,
strain and stress fields, and accordingly the inherent
phase delay between strain and stress time histories
produces the hysteretic loss. This hysteretic loss
dissipates some of energy supplied to the vehicle
(Whicker et al. 1981; Clark 1999), making the gas
mileage of vehicle lower, so it is considered as a sort of
pseudo-force resisting the tire rolling, called the
rolling resistance (RR) (Pacejka 2002). The RR is
defined by the hysteretic loss occurred during one
revolution divided by the distance traveled by rolling
tire during the same period of time. According to the
worldwide trend for developing the energy-efficient
economic tire with higher gas mileage, the reduction
of RR is nowadays a great challenging subject for both
tire and car makers (Ebbott et al. 1999; Hublau and
Barillier 2008).
Meanwhile, the dynamic deformation of rolling tire
is complicated non-sinusoidal three-dimensional,
even at constant rolling speed, and furthermore the
loci of deformation, strains and stresses at a material
point in tire are not bisymmetrical with respect to the
tire vertical axis. Besides the non-sinusoidal non-
bisymmetrical three-dimensional dynamic deforma-
tion, the complicated tread blocks (Cho et al. 2004,
2007) increase the complexity of the problem for
predicting the hysteretic loss and RR. It has been
reported that the tire tread pattern which contacts
directly with the ground takes a significant portion
over 60 % of the total hysteretic loss (Ebbott et al.
1999), depending on the type of tire and the rolling
conditions. Therefore, the detailed tread blocks should
be considered to secure the reliability of the prediction
of the hysteretic loss and RR and the design quality of
the fuel-efficient tire. Because of the complexity of the
tire cyclic deformation, the most studies on the RR
estimation have been made numerically or experi-
mentally (Pillai 1995; Pacejka 2002). In the numerical
prediction of the RR, the main issue becomes the
calculation of the tire hysteretic loss because the
hysteretic loss is essential for predicting the RR.
The numerical methods which are widely used have
in common the fact that time histories of strains must
be approximated to calculate the hysteretic loss and
RR during one revolution, but those can be classified
according to how to approximate the strain cycles
along a circular path during a revolution. The main
stream was to approximate the strain cycles using
either tire sector-wise continuous polynomials or
Fourier series, by utilizing the static tire contact
analysis of simple tire models in which the effects of
tread pattern are neglected (McAllen et al. 1996; Park
et al. 1997; Ebbott et al. 1999; Shida et al. 1999). One
critical limitation of the conventional numerical
methods is that the detailed tread blocks are not fully
considered in 3-D tire modeling. As a result, the
prediction accuracy is still needed to be improved, and
furthermore the effects of the tread design parameters
356 J. R. Cho et al.
123
on the RR are not possible to investigate. The tread
pattern contributes to the major portion of the total
hysteretic loss of tire as mentioned earlier, so the
consideration of detailed tread blocks is essential not
only for securing the prediction accuracy but for
investigating the effect of tread pattern.
The purpose of the current study is to present a
numerical method for predicting the hysteretic loss
and the RR using 3-D full periodic patterned tire
model (Cho et al. 2004). The 3-D periodic patterned
tire model is constructed by copying 1-sector tire mesh
in the circumferential direction, and the strain cycles
for each circumferential ring of elements are approx-
imated making use of the static tire contact analysis.
The elaborate consideration is paid to extract the strain
values from the elements which are sector-wise
repeated in the same circular ring of elements. The
maximum principal value of the half amplitudes of six
strain components (Sonsino 1995; Song et al. 1998) is
used to calculate the hysteretic loss which a material
point dissipates during one revolution of tire. The
validity of the proposed method is examined by
comparing with the experiment, and the hysteretic
losses and RRs of two different periodic patterned tire
models are compared to investigate the influence of
tread pattern.
2 Hysteretic loss-induced rolling resistance of tire
2.1 Hysteretic loss and rolling resistance
The road reaction resultant exerted on the rolling tire is
composed of the vertical reaction FZ (that is, the wheel
load) and the total RR FR as depicted in Fig. 1a, and
the total RR FR is caused by the friction and the
hysteretic loss. In this paper, the RR is meant by the
resistance force due to the hysteretic loss. This RR is
considered as a pseudo-force resisting the tire rolling
motion from the fact that the hysteretic loss dissipates
the energy supplied to the tire axle. When W is denoted
as the hysteretic loss of tire during one revolution, the
RR is defined by
RR ¼ W
2pqr
ð1Þ
with qr being the effective radius of tire (Pacejka
2002; Cho et al. 2006).
Meanwhile, there exists another measure of the RR,
called the coefficient of RR, which is introduced for
practical use. Referring to Fig. 1b, the coefficient CR
of RR is defined by the ratio of the total RR with
respect to the wheel load FZ . It approximately
becomes the inclined angle h of the link connected
between the center of rolling tire and the horizontal bar
which is pulled by vehicle driven on the road. Once the
coefficient of RR is determined by the proving ground
test, the total hysteretic loss W can be roughly
predicted using Eq. (2) which is derived by letting
FR be RR. However, it should be noted that the total
hysteretic loss calculated in this manner becomes
higher than the actual hysteretic loss.
CR ¼FR
FZ
¼ tan h ffi h [W
2pqrFZ
; W ffi 2pqrFZh
ð2ÞFigure 2a represents the time histories of uni-axial
strain and stress of a viscoelastic body subject to 1-D
sinusoidal excitation. The phase difference d between
strain and stress causes the hysteretic loss depicted in
Fig. 2b, and the hysteretic loss density DW per unit
volume during a period Tc ¼ 2p=x is calculated by
Fig. 1 Representation:
a the total rolling resistance,
b its measurement by
experiment
Finite element estimation of hysteretic loss and rolling resistance 357
123
DW ¼ZTc
0
r sð Þ de sð Þds
ds
¼ZTc
0
r0e0 sin xsþ dð Þ cos xsð Þ ds ¼ pr0e0 sin d
ð3Þ
with r0 and e0 being the stress and strain amplitudes.
However, 3-D viscoelastic bodies used in various
engineering applications are subjected to more com-
plicated multi-axial cyclic excitations (Mars 2001), so
the variations of strains and stresses are neither one-
dimensional nor sinusoidal. Thus, in such cases, the
hysteretic loss DW per unit volume in Eq (3) is
expressed in a generalized form given by
DW ¼ZTc
0
rij sð Þ deij sð Þds
ds ð4Þ
The hysteretic loss is converted to the heat gener-
ation, and the heat generation rate _Q per unit volume
during a cycle is calculated by
_Q ¼ DW
Tc
¼ 1
Tc
ZTc
0
rij sð Þ deij sð Þds
ds ð5Þ
Differing from the previous 1-D sinusoidal excita-
tion, the rubber compounds of rolling tire exhibit the
complicated 3-D dynamic viscoelastic deformation.
And, their dynamic viscoelastic properties are usually
characterized by the responses to the sinusoidal strains
and stresses which are constituted in terms of the
complex modulus G� ¼ G0 þ iG00. Here, G0 and G00 are
called the storage (in-phase) modulus and the loss
(out-of-phase) modulus respectively, and the complex
modulus is a function of the strain amplitude,
frequency f, and temperature T (Kramer and Ferry
1994). The storage and loss moluli are correlated in
terms of the phase difference d as follows:
tan d ¼ G00
G0; G00 ¼ G�j j sin d ð6Þ
A typical elliptic hysteresis loop of linear visco-
elastic materials subject to a sinusoidal shear excita-
tion with the stain amplitude e0 is illustrated in Fig. 3.
It has been found, from the shear oscillating test with
the peak-to-peak stain amplitude ca ¼ 2e0, that a
moderately loaded elastomeric material produces the
hysteresis loop close to the elliptic-type hysteresis
loop (Ebbott et al. 1999). Thus, the stress–strain
relation for the elastomeric materials subject to non-
sinusoidal cyclic excitations can be approximated by
the linearized viscoelasticity as
Fig. 2 a 1-D sinusoidal
excitation, b hysteretic loss
of viscoelastic material per
unit cycle
Fig. 3 Strain amplitude-dependent elastic modulus of elasto-
meric compound
358 J. R. Cho et al.
123
r sð Þ ffi <e 2G� ca; f ;Tð Þ e0eixs� �
¼ 2G0e0 cosxs� 2G00e0 sinxs¼ <e r0ei xsþdð Þh i
ð7Þ
with the stress amplitude r0 ¼ 2G0e0= cosd¼ 2 G�j je0.
And, the hysteretic loss DW done by e0eixs and r sð Þin Eq. (5) during a non-sinusoidal cycle with a half
period Tc=2 becomes
DW ¼ <e
ZTc=2
0
r sð Þ de sð Þds
ds
¼<e
ZTc=2
0
i2x G�j je20eixsei xsþdð Þds ¼ p G00e2
0
ð8Þ
In order to predict the hysteretic loss of a rolling tire
exhibiting the multi-axial state (Sonsino 1995) of
strain and stress caused by a non-sinusoidal periodic
excitation, one may consider the use of the maximum
and minimum equivalent strains to determine the
amplitude e0 of strain cycle during one revolution. But,
the difference of the maximum and minimum equiv-
alent strains can not be used because the equivalent
strain is always positive (Song et al. 1998). For this
reason, the maximum principal value of the half
amplitudes of six strain components is used for the
current study. Letting e1 be the maximum principal
value of the half amplitudes of six strain components,
then one can calculate the hysteretic loss of a rolling
tire per unit volume during one revolution by replacing
e0 in Eq. (8) with e1:
DW ¼ p G00e21 ð9Þ
By introducing the shear modulus G and the loss
factor g of rubber compound, together with the
relation of G� ¼ G 1þ igð Þ ¼ G0 þ iG00, the previous
Eq. (9) ends up with
W ¼ p G00e21 ¼ pg Ge2
1 ð10Þ
2.2 Strain amplitude and hysteretic loss of 3-D
periodic patterned tire
A 3-D periodic patterned tire composed of a number of
uniform sectors is shown in Fig. 4, where points P and
P0 are located on the same circular path. Another point
Q is located on the same 2-D tire section–section as
point P but it is not on the circular path C. With these
three points, we compare the strain cycles along the
circular path between the smooth and patterned tires
during the steady-state rolling.
As represented in Fig. 5a, two points P and P0
within the smooth tire model produce the same
bisymmetrical strain distributions with only the angle
difference during the steady-state tire rolling. But,
differing from point P, point Q produces a bisymmet-
rical strain distribution with the different amplitude
even though no angle difference exists. Thus, in the
smooth tire model, the material points located on the
same circular path C produce the same strain cycle
during the steady-state rolling.
On the other hand, two points P and P0 within
the patterned tire model produce the same strain
distribution along the circular path C during the
steady-state rolling only when the tread pattern is
periodic and two points are periodically repeated in the
circumferential direction. It is because tread blocks of
3-D non-periodic patterned tire are not uniformly
distributed. Figure 5b shows the strain distributions
produced by three points P, P0 and Q when two points
P and P0 are not sector-wise periodic in the circum-
ferential direction. It implies that the material points
Fig. 4 A 3-D periodic
patterned tire model
Finite element estimation of hysteretic loss and rolling resistance 359
123
located on the same circular path C may not produce
the same strain cycle unless those points are sector-
wise periodic. In such a case, the strain cycles should
be taken from all the elements within the tire in order
to calculate the hysteretic loss. Fortunately, if the tread
pattern is sector-wise periodic then only the strain
cycles at all the elements within 1-sector tire mesh are
needed.
Figure 6 shows a 3-D periodic patterned tire model
composed of ns uniform sectors, where the total
element numbers of each sector are equally Ns. The
global element number Nog is assigned sector by
sector such that
Nog XK jSector I
� �¼ Nog XK jSector 1
� �þ I � 1ð Þ � Ns;
I ¼ 1; 2; . . .; ns ð11Þ
with XK jSector I being the K-th element within the I-th
sector. Then, along a circular path C one can extract ns
elements with the global element numbers which are
different exactly by I � 1ð Þ � Ns. The set =C of such
ns periodic elements XCJ can be defined by
=C ¼Yns
J¼1
XCJ ; XC
J \ C 6¼ ; and
Nog XCJ
� �� Nog XC
L
� ��� �� ¼ I � Ns
ð12Þ
with I being 1� I� ns � 1ð Þ. Note that XCJ is the J-th
element in =C while Nog XCJ
� �is the global element
number of XCJ in the whole 3-D tire mesh. Further-
more, let us denote x‘J be the ‘-th Gaussian point within
the J-th element in =C, then the set @C‘ of all the ‘-th
Gaussian points of all the elements in =C are denoted
by
@C‘ ¼
Yns
J¼1
x‘J ; x‘J 2 XCJ 2 =C; ‘ ¼ 1; 2; . . . ð13Þ
In the current study, the time histories of strain
components that a material point experiences along a
circular path C are approximated making use of the
3-D static tire contact analysis. Then, each circumfer-
ential ring=C of elements produces the bisymmetrical
strain distributions with respect to the tire vertical axis.
Once the static contact analysis is performed, ‘
numbers of strain cycles are approximated for each
strain component using the strain values at x‘J 2 @C‘
along a circumferential ring =C of elements. In other
words, a total of 6� ‘ strain cycles are approximated
for each circumferential ring =C of elements. The
approximation can be made either by sector-wise
polynomials (Park et al. 1997) or by cyclic functions
shown in Fig. 5 which are characterized by the
minimum and maximum stain values. In the current
study, the latter approach is adopted in connection
with the hysteretic loss prediction method described in
Sect. 2.1.
Fig. 5 Circumferential
distributions of strain:
a smooth tire model,
b patterned tire model
Fig. 6 A 3-D periodic patterned tire model consisted of ns
periodic sectors
360 J. R. Cho et al.
123
The half amplitudes Deij
� �C
K;‘of each strain com-
ponent along a circular path C which the ‘-th Gaussian
point of the K-th element in sector 1 tracks during one
revolution are determined by
Deij
� �C
K;‘¼ max
xCm2@C
‘
eij � minxC
m2@C‘
eij
�����
�����
,
2; i; j ¼ 1; 2; 3;
‘ ¼ 1; 2; . . .; K ¼ 1; . . .; ns ð14Þ
By denoting M‘ be the total number of Gaussian
points within an element, the element-wise averaged
half amplitudes Deij
� �C
Kof each strain cycles that the
K-th element in sector 1 produces are calculated by
Deij
� �C
K¼XM‘
‘¼1
Deij
� �C
K;‘=M‘; K ¼ 1; 2; . . .; ns ð15Þ
And the maximum principal value e1;K of the
element-wise averaged half amplitudes Deij
� �C
Kof six
strain components which the K-th element within
sector 1 produces during one revolution is calculated
by solving the characteristic polynomial of principal
strains using the well-known closed-form formulas
(Malvern 1969; Simo and Hughes 1998). Then, the
hysteretic loss D wK per unit volume that the K-th
element XK in sector 1 dissipates during one revolu-
tion is calculated by
D wK ¼ pG00e21;K ð16Þ
Then, the total hysteretic loss DWK which the K-th
element in sector 1 dissipates during one revolution is
calculated by
DWK ¼ D wK � Vol XKð Þ ð17Þ
with Vol XKð Þ being the total volume of the K-th
element in sector 1. In this manner, the element-wise
averaged hysteretic losses for all the remaining
elements in sector 1 which track different circular
paths can be calculated.
Referring to Fig. 5, let us denote =C1 be the set of
the elements in sector 1 which track the same circular
path C:
=C1 ¼ XK : XK jSector 1\C 6¼ ;
� �ð18Þ
Because all the elements XK 2 =C1 track along the
same circular path C during the steady-state rolling,
their hysteretic losses DWK should be averaged. Then,
the total hysteretic loss W that a 3-D periodic patterned
tire dissipates during one revolution is calculated by
W ¼XnC
C¼1
XNum =C1ð Þ
K¼1
DWK
Num =C1
� �; XK 2 =C1 ð19Þ
In which nC and Num =C1
� �being the number of
distinct circular paths and the number of the elements
belonging to =C1 .
3 Numerical experiments
According to the numerical formulae described in the
previous Sect. 2, an in-house program for importing
the ABAQUS output file computing the hysteretic loss
and RR was coded in Fortran. Referring to Fig. 7a, an
automobile tire model P205/60R15 composed of a
single carcass layer and double tread belt layers is
taken for the numerical experiments, and its material
properties of base rubbers and reinforcement parts
may be referred to our previous paper (Cho et al.
2002). For the comparison purpose, another periodic
patterned tire models are also simulated.
A 3-D periodic patterned tire model is generated by
sequentially copying 1-sector mesh shown in Fig. 7b
in the circumferential direction, such that the nodes on
the common interfaces between two adjacent 1-sector
meshes should be exactly coincident. The material
properties and the boundary conditions specified to
1-sector mesh are also copied during the copying
process of 1-sector mesh by a commercial solid
modeler I-DEAS. The number of copying is deter-
mined by 2p=a with a being the angle of 1-sector
mesh. Referring to our previous paper (Cho et al.
2004), 1-sector patterned tire mesh is generated by
combining 1-sector tire body mesh and 1-sector tread
pattern mesh which is generated from a wire frame of
2-D 1-pitch tread pattern, according to a series of basic
meshing operations, geometry transformations and
other manipulations. The body and tread pattern
meshes are incompatible at the common interface, so
both meshes are combined by the surface-to-surface
tying algorithm provided by ABAQUS/Standard
(Hibbitt, Karlsson & Sorensen 2007).
Figure 8a shows a cross-section of 3-D patterned
tire which is composed of pure rubber parts and the
fiber-reinforced rubbers (FRRs), where the tread mesh
Finite element estimation of hysteretic loss and rolling resistance 361
123
and the tire body mesh with different mesh densities
are combined by the incompatible tying algorithm
(Cho et al. 2004; Hibbitt et al. 2007). Rubbers except
for the fiber-reinforced rubbers are modeled by the
Moonley–Rivlin material model defined by
W J1; J2; J3ð Þ ¼ C10 J1 � 3ð Þ þ C01 J2 � 3ð Þ
þ 1
D1
J3 � 1ð Þ2 ð20Þ
In which C10 and C01 are material constants
determined by experiments and D1 the parameter for
enforcing the material incompressibility, and Ji the
invariants of Green–Lagrangian strain tensor. The
penalty-like parameter D1 ¼ 2jð Þ is determined from
4j C10 þ C01ð Þ=s with j and s being the shear and bulk
moduli of rubber. For the given values of C10 and C01,
the material incompressibility increases in proportion
to j=s but the choice of j=s ¼ 100 is usually
recommended. The belt layers in underlying rubber
matrix are modeled as a single orthotropic shell layer,
while steel beads and underlying rubber in the bead
region are modeled as homogenized solid elements
based upon the linear rule of mixtures.
Two periodic patterned tire models consist of 71
periodic sectors and the total element and node
numbers of the entire FEM meshes are as follows:
77,859 and 102,675 for the patterned tire model 1 and
82,748 and 109,199 for the patterned tire model 2,
respectively. The other mesh and material properties
are given in Table 1, where all the element types are
supported in ABAQUS/Standard. Both the shear
modulus and loss factor are assumed to be independent
of strain amplitude, temperature and excitation fre-
quency, and those of each tire component are given in
‘‘Appendix’’. The viscoelastic properties were mea-
sured at the Kumho America Technical Center
(KATC) using the MTS 793.32 ADC Testware system.
The 3-D static contact analyses by ABAQUS/Standard
Fig. 7 a Periodic patterned
tire models 1 and 2,
b 1-sector mesh
Fig. 8 a 2-D section mesh
of the patterned tire,
b loading conditions
362 J. R. Cho et al.
123
are commonly composed of two steps in sequence.
Fist, the tire model is inflated up to the preset internal
pressure pi ¼ 30 psi with all the nodes being in contact
with rim fixed, and next, the rigid body is forced to
contact with the tire model by the vertical force
Fy ¼ 495 kgf . The friction coefficient l between the
tire model and the rigid body is set by 1.0.
Patterned tire model 1 is firstly simulated to validate
the in-house test program and to verify the reliability of
the numerical results. The overall distribution of the
maximum principle strain obtained by ABAQUS/
Standard is represented in Fig. 9a. It is observed that
the maximum principle strain shows the symmetry with
respect to the tire vertical line and it becomes highest in
the contact patch with the peak value of emax1 ¼ 0:388.
Figure 9b shows the 2-D sectional distribution of the
maximum principle strain at h ¼ 180�. Relatively high
strains are observed in the vicinity of bead and belt
edge and in the upper sidewall region, and the
distribution is shown to be almost bisymmetrical.
Using the in-house test program, the amplitudes of six
strain components are calculated at all the Gaussian
points of all the elements in sector 1. And the
maximum principal values e1;K of the half amplitudes
of six strain components for all the elements within
sector 1 during one revolution are calculated by
solving the characteristic polynomial of principal
strains using the well-known closed-form formulas.
With the computed maximum principal values e1;K
of the half amplitudes of six strain components for all
the elements within sector 1, the element-wise
hysteretic losses are computed and the total hysteretic
loss of 78:19J is obtained. The comparison of the RRs
obtained by the present method and experiment is
given in Table 2, where the experiment was carried
out at the R&D Center of Kumho Tire Co. in Korea
according to the experimental method given in a paper
by Pillai (1995). The numerically predicted value is
less than the experimental data with the relative
difference of 11.3 %, and this discrepancy is caused
by several factors. The dynamic rolling of tire
produces the asymmetric circumferential distribution
of strain, but the current numerical prediction using
static strains assumes the ideal symmetric sinusoidal
excitation. And, the real dynamic contact between tire
and the scratched ground surface is simplified as a
static contact with the smooth rigid surface by
introducing the frictional coefficient. Therefore, addi-
tional deformation and heat generation produced by
such a complex dynamic friction within the tire
contact patch are ignored. In addition, the storage
modulus G0 does also decrease with the temperature
increase, so the ignorance of the coupling between tire
deformation and temperature is this study does not
account for the variation of strain to the temperature
change. Furthermore, the reliability of experimental
data is also influenced by several factors such as road
condition, wind resistance and test equipment.
Table 1 Material and simulation data taken for the numerical
experiments
Item Parameters Patterned
tire 1
Patterned
tire 2
Mesh Element numbers per
sector
1,447 1,538
Element types C3D8H, C3D6H,
SFM3D4R
Material
properties
Shear modulus of
rubber G MPað Þ1.30–6.83
Loss factor g� 0.079–0.213
Effective radius qr mð Þ 0.29750 0.29749
* The values at T = 70 �C
Fig. 9 Distributions of the
maximum principle strain:
a overall, b section of
h = 180 �C
Finite element estimation of hysteretic loss and rolling resistance 363
123
Figure 10a represents the RR contributions of tire
components, where GUM and BC indicate gum chafer
and belt cushion, respectively. From the fact that
pattern and tread account for most of RR over 60 %, it
has been confirmed that the design of pattern, material
and geometry of tire tread is significantly important for
reducing the RR. Figure 10b shows the variation of
RR to the total number of elements, where the detailed
values of three points are 36:20N at 57.979, 41:42Nat
66,499 and 42:14N at 77,859, respectively. It is
confirmed that the mesh density taken for the numer-
ical experiment shows the convergence in predicting
the RR.
In order to investigate the dependence of RR on the
tread pattern, patterned tire model 2 with more coarse
tread blocks is simulated with the same loading
conditions. The sectional distribution of the maximum
principle strain of patterned tire model 2 is represented
in Fig. 11a. It is observed that the peak maximum
principle strain value of 0.421 occurs in the bead
region, and the strain distribution show the almost
bisymmetry. When compared with the previous pat-
terned tire model 1, the patterned tire model 2
produces the slightly different strain distribution,
implying that the sectional strain distribution is
influenced by the tread pattern. Figure 11b represents
the RR contributions of tire components for the
Table 2 Comparison with the experiment
Rolling
resistance (N)
Relative
difference (%)
Present 42.14 -11.30
Experiment 47.51 –
Fig. 10 Rolling resistance: a contributions of tire components, b convergence to the total number of finite elements
Fig. 11 Patterned tire 2: a distributions of the maximum principle strain, b rolling resistance contributions of tire components
364 J. R. Cho et al.
123
patterned tire model 2, where SKIM refers to belt skim
rubber. Compared with Fig. 10a of the patterned tire
model 1, it is observed that the contribution of tread
becomes smaller, justifying the above-mentioned
dependence of the hysteretic loss on the material
volume of tread pattern.
The total hysteretic losses and RRs of two different
tire models are compared in Table 3. When compared
with the patterned tire model 1, the total hysteretic loss
and RR of the patterned tire model 2 are smaller by
8.50 % respectively. It is because the patterned tire
model 2 has smaller rubber material volume and the
total hysteretic loss is proportional to the total material
volume of rubber compound as given in Eq. (17). The
comparative differences of the total hysteretic losses
and RRs given in Table 3 are consistent with such a
physical characteristic of the hysteretic loss of elas-
tomeric material. Consequently, from the comparison
of numerical results of two different tread patterns,
one can confirm the significant effect of tire tread
pattern on the hysteretic loss and RR.
4 Conclusion
In this paper, the hysteretic loss and RR of 3-D
periodic patterned tire was numerically predicted by
utilizing the static tire contact analysis. The periodic
patterned tire models were constructed by copying
1-sector tire mesh in the circumferential direction, and
the half amplitudes of strain cycles during one
revolution of tire were approximated with the strains
at Gaussian points of the elements which are sector-
wise repeated in the same circumferential ring of
elements. And the hysteretic loss during one revolu-
tion was computed with the maximum principal value
of the half amplitudes of six strain components which
represents the combined effect of individual strain
components on the hysteretic loss. Through the
numerical experiments, it has been verified that the
proposed numerical method predicts the RR with the
relative error less than 11.3 %, and it has been found
that pattern and tread account for most of RR over
60 %. From the parametric numerical experiment to
the tread pattern, it has been also found that the tread
pattern gives rise to the significant effect on the peak
maximum principle strain, RR and the RR contribu-
tions of tire components. It is convinced that the
proposed hysteretic loss estimate can be usefully
applied to the design of fuel-economic patterned tires.
Acknowledgments The financial and technical support for
this work by the R&D Center of Kumho Industrial Co., Ltd. is
gratefully acknowledged.
Appendix: Viscoelastic material properties
See Table 4.
References
Ebbott, T.G., Hohman, R.L., Jeusette, J.P., Kerchman, V.: Tire
temperature and rolling resistance prediction with finite
element analysis. Tire Sci. Technol. TSTCA 27(1), 2–21
(1999)
Cho, J.R., Jeong, H.S., Yoo, W.S.: Multi-objective optimization
of tire carcass contours using a systematic aspiration-level
adjustment procedure. Comput. Mech. 29, 498–509 (2002)
Cho, J.R., Kim, K.W., Yoo, W.S., Hong, S.I.: Mesh generation
considering detailed tread blocks for reliable 3D tire ana-
lysis. Adv. Eng. Softw. 35, 105–113 (2004)
Cho, J.R., Choi, J.H., Yoo, W.S., Kim, J.G., Woo, J.S.: Esti-
mation of dry road braking distance considering frictional
energy of patterned tires. Finite Elem. Anal. Des. 42,
1248–1257 (2006)
Table 3 Comparison of the total hysteretic losses and rolling
resistances
Total hysteretic
loss (J)
Rolling resistance
(N)
Patterned tire 1 78.75 42.14
Patterned tire 2 72.06 (-8.50 %) 38.56 (-8.50 %)
Table 4 Shear moduli and loss factors of individual rubber
components
Rubber components Shear modulus
G MPað ÞLoss factor
g
Tread, pattern 1.480 0.167
Body ply, belt, FCAP
(cap ply)
1.430 0.079
Sidewall 1.227 0.092
Inner liner 1.300 0.129
Apex 6.830 0.115
BC (belt cushion), SKIM
(skim rubber)
1.300 0.129
GUM (gum chafer) 2.400 0.213
Finite element estimation of hysteretic loss and rolling resistance 365
123
Cho, J.R., Kim, K.W., Jeong, H.S.: Numerical investigation of
tire standing wave using 3-D patterned tire model. J. Sound
Vib. 305, 795–807 (2007)
Clark, S.K.: Rolling resistance of pneumatic tire. Tire Sci.
Technol. TSTCA 6(3), 163–175 (1999)
Hibbitt, Karlsson, Sorensen: ABAQUS/Standard User’s Man-
ual, Ver. 6.7. Pawtucket (2007)
Hublau, V., Barillier, A.: The equations of the rolling resistance
of a tire rolling on a drum. Tire Sci. Technol. TSTCA 36(2),
146–155 (2008)
Kramer, O., Ferry, J.S.: Dynamic mechanical properties. In:
Mark, J.E., et al. (eds.) Science and Technology of Rubber.
Academic Press, London (1994)
Malvern, L.W.: Introduction to the Mechanics of a Continuous
Medium. Prentice-Hall, New Jersey (1969)
Mars, W.S.: Multiaxial fatigue crack initiation in rubber. Tire
Sci. Technol. TSTCA 29, 171–185 (2001)
McAllen, J., Cuitino, A.M., Sernas, V.: Numerical investigation
of the deformation characteristics and heat generation in
pneumatic aircraft tires, part II. Thermal loading. Finite
Elem. Anal. Des. 23, 265–290 (1996)
Pacejka, H.B.: Tire and Vehicle Dynamics. Society of Auto-
motive Engineers, Oxford (2002)
Park, H.C., Youn, S.K., Song, T.S., Kim, N.J.: Analysis of
temperature distribution in a rolling tire due to strain
energy dissipation. Tire Sci. Technol. TSTCA 25(3),
214–228 (1997)
Pillai, P.S.: Total tire energy loss comparison by the whole tire
hysteresis and the rolling resistance methods. Tire Sci.
Technol. TSTCA 23(4), 256–265 (1995)
Shida, Z., Koishi, M., Kogure, T., Kabe, K.: A rolling resistance
simulation of tires using static finite element analysis. Tire
Sci. Technol. TSTCA 27(2), 84–105 (1999)
Simo, J.C., Hughes, T.J.R.: Computational Inelasticity.
Springer, Berlin (1998)
Song, T.S., Lee, J.W., Yu, H.J.: Rolling resistance of tires—an
analysis of heat generation. Technical Paper 980255, SAE,
Warrendale, PA 15096-0001 (1998)
Sonsino, C.M.: Multiaxial fatigue of welded joints under in-
phase and out-of-phase local strains and stresses. Int.
J. Fatigue 17(1), 55–70 (1995)
Whicker, D., Browne, A.L., Segalman, D.J., Wickliffe, L.E.: A
thermomechanical approach to tire power loss modeling.
Tire Sci. Technol. TSTCA 9(1–4), 3–18 (1981)
366 J. R. Cho et al.
123
