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fact, is influenced by a great number of
variables and parameters, often hard to be
controlled and measured [3]: macro and
micro roughness of the bodies in contact,
pressure arising at their interface, materials
stiffness characteristics and their frequen-
cy and temperature dependence, relative
motion direction and speed are only a small
number of the factors that take part in a
phenomenon involving contact mechanics,
thermodynamics, polymers chemistry and,
from a wider point of view, vehicle dyna-
mics.
The first attempts to model friction pheno-
mena hails from the studies of C. A. Cou-
lomb, who, after the experimental activity
of G. Amontons [4], theorized that, for
metals, friction force was independent from
contact area and directly proportional to
the applied normal load by means of a
coefficient [5], expressing for the first time
the well known law:
Dependence on sliding velocity was not
taken into account, but a first distinction bet-
ween static and dynamic friction coefficient
was proposed and analyzed. Bowden and
Tabor [6][7] and Rabinowicz [8] introduced
the theme of adhesion in polymers contact,
investigating the frictional behaviour of rub-
ber and highlighting its strong dependence
from loads, temperature and relative speed.
A generalized friction model was proposed
by H. W. Kummer [9] during his activity in the
field of tyre/road interaction; his model (Fig.
1) considered for the first time the resistant
force as composed by three components:
adhesion, deforming hysteresis and wear.
In dry conditions, tyres friction forces take
the following form:
(2)
A Real-time physical analytical
Grip Model for Tyre Rubber
in sliding Contact
with Road Asperities
Vol. 67 - n. 7/9 luglio-settembre 2014 31
Flavio Farroni
Michele Russo
Francesco Timpone
Riccardo Russo
This paper deals with the frictional behaviour
of a tyre tread elementary volume in sliding
contact with road asperities. Friction is supposed
as composed by two main components:
adhesion and deforming hysteresis.
The target, fixed in collaboration with a motorsport
racing team and with a tyre manufacturing
company, is to provide an estimation of local grip
for on-line analyses and real time simulations and
to evaluate and predict adhesive and hysteretic
frictional contributions arising at the interface
between tyre tread and road.
A way to approximate asperities, based on
rugosimetric analyses on macro and micro scale,
has been introduced.
The adhesive component of friction has been
estimated by means of a new approach based
on two different models found in literature,
whose parameters have been identified thanks to
a wide experimental campaign previously
carried out.
The hysteretic component of friction has been
estimated by means of an energy balance taking
into account rubber viscoelastic behaviour and
internal stress / strain distribution, due to
indentation with road. The correct reproduction
of friction phenomenology and the model
prediction capabilities are highlighted making
particular reference to grip variability, due to
changes in working conditions.
flavio.farroni@unina.it
michele.russo@unina.it
francesco.timpone@unina.it
ricrusso@unina.it
Department of Industrial Engineering,
University Federico II, Naples, Italyi
Introduction
Rubber frictional behaviour in tyre/road inte-
raction is one of the main topics in a wide ran-
ge of research fields. Knowledge about phe-
nomena concerning with adherence is a key
factor in the development of braking/traction
and stability control systems adopted in auto-
motive industry [1], such as in the study of
innovative tyre structures and compounds,
able to minimize braking distances, to preser-
ve vehicle stability in panic situations and to
guarantee optimal roadholding on wet/icy sur-
faces [2].
Moreover, the continuous drivers’ seeking of
the optimal grip for each different driving
condition makes the development of a phy-
sical grip model an essential instrument for a
top-ranking racing team, in particular thanks
to the definitely lower resources needed by
simulations than by experimental tests car-
ried out in order to acquire information about
tyres behaviour. Rubber/asphalt friction, in
30 ATA - Ingegneria dell’Autoveicolo
(1)
Michele
Russo
Francesco
Timpone
Riccardo
Russo
Flavio
Farroni
ATA_Vol_67_Lug_Sett_2014_Layout 1 22/09/14 09.52 Pagina 30
lastic, soft and virtually incompressible
material (Poisson's ratio ν= 0.5). The rubbe-
ry state of a polymer is determined by the
so-called glassy transition temperature Tg.
When the working temperature is above Tg,
the polymer shows a rubbery behaviour.
Otherwise, it is a glassy one.
Considering a sinusoidal deformation acting
on a viscoelastic material with angular fre-
quency ω, the induced stress σcan be
expressed as the sum of two contributions,
one in phase with the imposed deformation
and a second in quadrature phase:
(3)
where;
• E’, said storage modulus, is the elastic
modulus part relative to the in phase
response of the material
• E”, called loss modulus, represents the
elastic modulus of the part in quadrature
phase.
A very common index used to describe the
dissipative attitudes of a compound, is the
loss angle tangent, defined as:
(4)
The stiffness parameter, adopted for visco-
elastic materials in place of Young’s modulus
is the complex dynamic modulus:
Vol. 67 - n. 7/9 luglio-settembre 2014
Tyre
33
where:
•F
Tis the total frictional resistance devel-
oped between a sliding tyre and road,
•F
ADH is the frictional contribution due to
Van der Waals’ adhesion bonds between
the two surfaces,
•F
Hb is the frictional contribution from bulk
deformation hysteresis in the rubber,
•F
Cis the cohesion contribution linked with
rubber wear, usually negligible.
Kummer postulated that FADH and FHb are
not independent, because adhesion is able
to increase the extension of the contact area
and consequently the zone in which the
hysteretic deformations occur.
Thanks to Kummer and Savkoor’s work [10],
Moore [11] hypothesized that the different
components were predominant on different
scales: the macro-roughness affects the
deformations related to hysteresis and the
micro-roughness affects the intermolecular
bonds characterizing adhesion. For this rea-
son, the two aspects may conceptually be
split and treated applying a sort of superpo-
sition principle (Fig. 2).
In this paper, a tyre/road friction physical
model, named Gr.E.T.A. (Grip Estimation for
Tyre Analysis) Model, will be presented. The
model, developed in collaboration with a
motorsport racing team and a tyre manufactu-
ring company, bases on the previous conside-
rations, providing an effective calculation of
the power dissipated by road asperities inden-
ted in tyre tread and taking into account the
phenomena involved with adhesive friction.
Model definition
and basic hypotheses
In order to model the complex interactions bet-
ween tyre and asphalt at a microscopic level, it
has been necessary to focus initially on the
behaviour of an elementary volume of rubber in
sliding contact with a limited portion of road.
Modelling asphalt, as commonly found in
literature [13], as the sum of sinusoidal
waves distributed in the space characteri-
zing the different roughness scales (Tab. 1),
tread elementary volume has been defined
as a square-based parallelepiped. Its height
is equal to tyre tread thickness and the base
side to road macro-roughness wavelength
λMACRO (Fig. 3).
Wavelengths λand roughness indices Ra [14]
characterizing soil profile, have been estima-
ted by means of proper algorithms employed
to analyze data acquired experimentally by
laser scan on different dry tracks and to repro-
duce the best-fitting sinusoidal waves corre-
sponding to macro and micro profiles (Fig. 4).
Material Characterizatio
Rubber employed in tyre treads is a hypere-
ATA - Ingegneria dell’Autoveicolo
Tyre
32
Fig. 1
Kummer's model for
rubber friction mechanisms.
Fig. 2
Influence of the
roughness scales
on the different friction
mechanisms
Range Size
Wavelength Ra
Mega-texture 50 – 500 mm 0.1 – 50 mm
Macro-texture 0.5 – 50 mm 0.1 – 20 mm
Micro-texture 0 – 0.5 mm 1 – 500 µm
Tab. 1
Road texture scales
dimensions [12]
Fig. 3
Elementary tread volume
and coordinates system
Fig. 4
Road acquired profile
and analysis of a 2D
section of it
ATA_Vol_67_Lug_Sett_2014_Layout 1 22/09/14 09.52 Pagina 32
molecules of the track.
For this reason, a satisfying modeling of
such friction mechanism cannot exclude the
knowledge of the complex phenomena con-
cerning chemistry of polymers and molecu-
lar physics. With the aim to reproduce the
functionalities between adhesive friction and
the main variables influencing it (i.e. sliding
velocity Vs, contact pressure p and tempera-
ture T), a model, which takes into account
both the approach of Le Gal and Klüppel [16]
and the one of Momozono and Nakamura
[17], has been adopted:
(8)
in which, from Le Gal’s model,
(9)
and, from Momozono’s model, the ratio of
the real actual contact area Acto the appa-
rent contact area A0is approximated by:
(10)
Thanks to a wide experimental campaign,
performed by the authors using a pin on disk
machine [18][19], it has been possible to
identify the parameters of Le Gal’s model
most difficult to define, i.e. the interfacial
shear strength τs,0, the critical velocity vcand
the viscoelastic dissipation parameter n.
The adhesion variations with sliding velocity
are explicit in the model, while the thermal
effect is modeled by means of the variation
induced by temperature in rubber dynamic
modulus E. Glassy region modulus E∞and
rubbery region modulus E0have been identi-
fied by means of the visco-elastic characte-
rization tests; m2is the second profile
moment of the probability distribution fun-
ction of the track (equivalent to the rms slo-
pe), has been computed elaborating road
data in accordance with the ASME B46.1.
Hysteresis model
The modelling of the hysteresis starts from
the expression of the power dissipated by a
rubber block that slides with speed Vsunder
a vertical load FZover a generic macro-rough
surface. Because of the complexity of the real
track surface, each elementary volume of the
deformed compound block is subjected to a
local stress/strain field (variable with the time)
resulting in a dissipated power, due to the
visco-elastic behaviour of the polymers.
Considering the volume VTOT of the elemen-
tary tread element and hypothesizing the Vs
constant in the sliding over a single asperity,
it is possible, thanks to [20], to formulate the
final hysteretic friction expression:
(11)
in which p represents the average contact
pressure in the nominal elementary area A0,
equal to (λMACRO)2.
In order to estimate this friction coefficient,
knowing polymer characteristics, road wave-
length and input variables, it is necessary to
provide the stress σ1in each point of the
discretized elementary tread volume, and, in
particular, of the stress components along x,
y and z directions. Although many authors
have proposed formulations able to estimate
stress distributions and contact area exten-
Vol. 67 - n. 7/9 luglio-settembre 2014
Tyre
35
(5)
E’ and E” values and tan(δ) are strongly
dependent on the temperature and on the
frequency at which the rubber is stressed, as
schematically represented in Fig. 5.
As regards the influence of temperature, the
typical behaviour of polymers is characteri-
zed by a dynamic modulus decrement with
increasing temperature, while the phase
angle increases until it reaches a maximum
before decreasing again.
For common passenger tyres, glass transition
temperature is often below 0°C, so that usual
working conditions are localized in the rubbe-
ry zone in order to provide optimal frictional
performances. Sport and high-performance
tyres, characterized by the employment of
different rubber compounds and fillers, exhi-
bit lower values of the dynamic modulus, that
make the tread manifesting softer and highly
wearable, a more hysteretic attitude and a
definitely higher Tg and, consequentially, a
higher thermal optimal working range.
When both the frequency and the temperatu-
re vary, it is possible to make use of the pro-
perty, whereby an appropriate shift operation
is capable of combining the effect of them:
the main element on which the temperature -
frequency equivalence principle is based is
that the values of the complex modulus com-
ponents, at any reference frequency and
temperature (f1, T1) are identical to the ones
observable at any other frequency f2at a pro-
perly shifted value of temperature α(T1):
(6)
The most widely relationship, used to descri-
be the equivalence principle, is the Williams-
Landel-Ferry (WLF) transform [15]. For pas-
senger tyre rubber, it can be employed in a
simplified way in order to determine the
unknown equivalent temperature T*= α(T1):
(7)
in which a common ΔT value, identifiable by
means of DMA tests at different frequencies,
is about 8°C.
The physical meaning of the law is that rubber
stressed at high frequency behaves like if the
stress is applied at lower frequency but at the
same time, at a colder working temperature.
High frequency acts reducing the time betwe-
en two consecutive stresses, not allowing the
complete relax of the rubber, in the same way
as a low working temperature would do.
Adhesion model
Adhesive friction, regarded as being the pri-
mary contributor when a rubber block slides
over a smooth unlubricated surface, is
usually pictured as being due to molecular
bonds between the rubber chains and the
ATA - Ingegneria dell’Autoveicolo
Tyre
34
Fig. 5
E’ and tan(δ)
relationship with
frequency and temperature
variations
ATA_Vol_67_Lug_Sett_2014_Layout 1 22/09/14 09.53 Pagina 34
and the traction state behind this last, due to
the presence of the adhesive contribution.
Because of the self-orientation of the elemen-
tary volume, it is possible to state that in the
plane y-z sliding velocity components are
absent: it means that applying Kuznetsov’s
equations in this plane, imposing K equal to
zero, the perfectly symmetric tangential
stress so obtained can be considered as an
estimation of the stress component σ1y.
With the aim to extend the bi-dimensional
results to the whole three-dimensional ele-
mentary volume (Fig. 10), the planar compo-
nents have been scaled, reducing stress entity
at increasing distance from asperity apex, by
means of a quadratic function identified on the
basis of proper tests carried out with a FEM
solver [22]. The tests, characterized by high
computational loads, confirmed the goodness
of the stress fields calculated much more easi-
ly with Kuznetsov's equations, that, once
implemented, represent an optimal solution
for the needs of a real time physical model.
Results and validation
The connection of the presented grip model
with both an interaction model [23] - able to
provide for each steptime the contact pres-
sure and the sliding velocity at which each
tread element interacts with the correspon-
ding asperity - and with a thermal model [24]
- whose output is tread temperature in the
same steptime, supposed to be uniform in
the neighbourhood of the contact area with
a single road asperity - gives the possibility
to estimate friction arising at tyre/road inter-
face as the sum of the adhesive contribution
and of the hysteretic one.
As shown in Figg. 11 and 12, the model is able
to give a coherent response to input varia-
tions; in particular, in Fig. 11 is reported a 3D
plot obtained for a passenger GT tyre, sliding
at different Vs and P over a road profile,
described by macro and micro sinusoids
having λMACRO = 9.9 mm, Ra MACRO = 0.8 mm,
λMiCRO = λMACRO/100 and Ra MACRO =
Ra MACRO/100, chosen basing on a similarity
conception introduced by Persson [2].
Figg. 12a and 12b highlight friction depen-
dence on temperature and on rubber Stora-
ge Modulus. As expected for polymers, SBR
Vol. 67 - n. 7/9 luglio-settembre 2014
Tyre
37
sion, a multi-dimensional approach did not
result able to satisfy the real-time require-
ments and the low computational loads that
the applications for which the model has
been developed need. Thanks to the studies
of Y. A. Kuznetsov [21], it is possible
to calculate the stress state indu-
ced in a rubber elastic body by a
periodic sinusoidal perfectly rigid
indenter in sliding contact with it.
Once determined the radius of cur-
vature R of the road sinusoidal
indenter at the apex, it is possible to
estimate,by means of Kuznetsov
formula, the half-length N of the
contact area (Fig. 6) as a function of
the radius R, of the average contact
pressure p in the nominal area A0, of
the rubber dynamic modulus E cal-
culated taking into account of the
working conditions acting on the
examined elementary volume and of
the macro asperity wavelength:
(12)
Kuznetsov's method for planar stress calcu-
lation can be used to determine the three-
dimensional field starting from the vertical
planes x-z (with sliding velocity) and y-z
(without sliding velocity) shown in Figg. 7
and 8, localized under asperity apex.
The strong relationship between adhesive
and hysteretic friction is taken into account
by means of the Kuznetsov's parameter K,
that is supposed to be equal to the adhesive
friction coefficient. In Fig. 9 the effect of
adhesion on the stress profile is shown: the
increase of the adhesive component causes
a progressive asymmetrisation of the stress
field in the direction of the sliding.
Applying Kuznetsov equations in the plane
x-z, it is possible to calculate the stress com-
ponents σ1x and σ1z generated by the sliding
indenter, noticing in Fig. 7a the compressive
tangential stress localized before the indenter
ATA - Ingegneria dell’Autoveicolo
Tyre
36
Fig. 7
σ1x distribution in the
x-z plane, localized as
shown in Fig. 7b, under a
pressure of 125 KPa
Fig. 6
Detail of the contact
between sinusoidal asperity
apex and tyre tread
Fig. 10
left – σ1x stress in the
x-z plane for K=1 under a
pressure of 125 KPa and a
sliding velocity of 1 m/s
directed from left to right.
Right – Extension of σ1x to
the whole 3D elementary
volume
Fig. 8
σ1y distribution in the
y-z plane, localized as
shown in Fig. 8b, under a
pressure of 125 KPa
Fig. 9
σ1zin the x-z plane for
adhesion coefficient
(Kuznetsov’s parameter K)
equal to 0, 0.5 and 1
under a pressure of
125 KPa and a sliding
velocity of 1 m/s directed
from left to right
Fig. 11
3D plot reporting friction
coefficient for a passenger
GT tyre as a function of
sliding velocity Vs and
contact pressure p, at a
tread average temperature
of 25°C
l
π
ATA_Vol_67_Lug_Sett_2014_Layout 1 22/09/14 09.53 Pagina 36
Bibliography
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[7] F.P. Bowden, D. Tabor, “The Adhesion of
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[17] S. Momozono, K. Nakamura, K. Kyogoku,
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[18] F. Farroni, M. Russo, R. Russo, “Tyre-Road
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Temperature”, In: Proceedings of the ASME
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[19] F. Farroni, E. Rocca, R. Russo, “Experimental
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Vol. 67 - n. 7/9 luglio-settembre 2014
Tyre
39
is highly sensitive to working temperature,
confirming that the selection of the proper
thermal range, in which the compound will
work, is the main key factor for the grip per-
formances maximization. For what concerns
the effect of a variation of the storage modu-
lus, the response of the model reproduces
the expected physical behaviour, describing
the decrease of the frictional attitude of the
tyre consequently to an increase in E' value,
that generates a lower indentation level and
a less adhesive interaction.
Conclusions
A model, called GrETA, aimed to evaluate the
local grip in the interaction between tyre tread
and road, has been described in this paper.
The main task, that the developed model has
had to deal with, has concerned the need to
describe the adhesion and hysteresis frictio-
nal mechanisms adopting a physical-analyti-
cal formulation as efficient as possible, in
order to make the model employable in dri-
ving simulations and real time analyses.
Road profiles, after a rugosimetric analyses
stage, have been modelled by means of a
double sinusoidal wave, able to take into
account of the macro and micro roughness
scales, that, applying a sort of superposition
principle, can be attributed respectively to
hysteretic and adhesive submodels. SBR
polymers employed in tread production have
been characterized providing storage modu-
lus E' and tan(δ) measurements.
As concerns the adhesive component of fric-
tion, the work has based on models available
in the literature, resulting in an original formu-
lation, which summarizes peculiar aspects of
the various studies. Adhesion model, being
its parameters identified in order to fit experi-
mental data provided thanks to previous
experimental tests, results self-validated.
Hysteretic friction model, applied in innovati-
ve conditions thanks to the estimation of 3D
stress/strain states by means of properly
adapted Kuznetsov equations, has been
validated and the test procedure and results
have been provided and discussed.
Final results, consistent with the theory and
in good agreement with the expected physi-
cal behaviour, have been reported. Further
developments will concern with the investi-
gation of the wet contact phenomena and
with the implementation and validation of
submodels, which are able to describe them.
ATA - Ingegneria dell’Autoveicolo
Tyre
38
Fig. 12
a – Passenger GT tyre
friction coefficient as a
function of sliding velocity
Vs for different tread
Temperatures, under a
pressure of 300 KPa.
b – Influence of Storage
Modulus at 25°C
and 300 KPa
b
a
Symbols
µ = friction coefficient [-]
λ= roughness wavelength [m]
Ra= roughness index [m]
ɛ= strain [-]
σ= stress [Pa]
E' = rubber storage modulus [Pa]
E'' = rubber loss modulus [Pa]
E = rubber complex dynamic modulus [Pa]
δ= phase angle between storage and loss
modulus [rad]
f = frequency [Hz]
T = temperature [K]
t = time [s]
FT= tyre friction force [N]
FADH = adhesion force [N]
FHb = hysteretic force [N]
FC= cohesion force linked with rubber wear [N]
FN= normal load [N]
Vs= sliding velocity [m/s]
p = average contact pressure [Pa]
Ac= real actual contact area [m2]
A0= apparent contact area [m2]
τs,0 = interfacial shear strength [Pa]
vc= the critical velocity [m/s]
n = viscoelastic dissipation parameter [-]
E∞= glassy region dynamic modulus [Pa]
E0= rubbery region modulus [Pa]
m2= track rms slope [-]
WDISS = dissipated power [W]
VTOT = total tread volume in contact with an
asperity [m3]
N = contact half length [m]
R = curvature radius of asperity apex [m]
T0= period of stress/strain cycle [s]
ω= stress angular frequency [rad/s]
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