A Friction Physical Model For The Estimation Of Hysteretic Dissipations Arising At The Contact Between Rigid Indenters And Visco-Elastic Materials

Abstract

Knowledge about phenomena concerning with adherence is a key factor in the automotive field and in particular in the braking/traction and stability control systems design. Moreover, the continuous drivers’ seeking of the optimal grip conditions, makes the development of a physical friction model an essential instrument for the investigation of the factors acting on indentation and adhesion mechanisms on which tyre/road interaction is based. Rubber/asphalt friction, in fact, is influenced by a great number of variables and parameters, often hard to be controlled and measured: macro and micro roughness of the bodies in contact, pressure arising at their interface, materials stiffness characteristics and their frequency and temperature dependence, relative motion direction and speed.
The possibility offered by a physical model to provide a better comprehension of the cited factors allows to act on them with a wide range of aims: studying soil textures structured in order to increase drivers' safety both in dry and in wet conditions,
producing more performing rubber compounds, able to optimize frictional behaviour under certain temperatures or frequencies and, in particular in race applications - for which the presented studies have been originally carried out - in order to configure optimal vehicle setup and driving strategies.
A deep knowledge of the mechanisms involved with tyre/road friction is a key factor in the design of the suspension system: an optimal setting of tyre working angles, operated in order to optimize temperature, contact pressure and sliding velocity distributions, can be efficiently provided by a physical grip model able to indicate the best wheel configuration at the boundary conditions changes.

Full text

World Tribology Congress 2013
Torino, Italy, September 8 – 13, 2013
A Friction Physical Model For The Estimation Of Hysteretic Dissipations Arising
At The Contact Between Rigid Indenters And Visco-Elastic Materials
Flavio Farroni*, Michele Russo, Riccardo Russo and Francesco Timpone
Department of Industrial Engineering, University of Naples Federico II,
Via Claudio 21, 80125 Naples, Italy
*Corresponding author: flavio.farroni@unina.it
1. Introduction
Knowledge about phenomena concerning with
adherence is a key factor in the automotive field and in
particular in the braking/traction and stability control
systems design [1, 2]. Moreover, the continuous drivers’
seeking of the optimal grip conditions, makes the
development of a physical friction model an essential
instrument for the investigation of the factors acting on
indentation and adhesion mechanisms on which
tyre/road interaction is based. Rubber/asphalt friction, in
fact, is influenced by a great number of variables and
parameters, often hard to be controlled and measured:
macro and micro roughness of the bodies in contact [3,
4], pressure arising at their interface [5], materials
stiffness characteristics [6] and their frequency and
temperature dependence [7], relative motion direction
and speed [8].
The possibility offered by a physical model to
provide a better comprehension of the cited factors
allows to act on them with a wide range of aims:
studying soil textures structured in order to increase
drivers' safety both in dry and in wet conditions,
producing more performing rubber compounds, able to
optimize frictional behaviour under certain temperatures
or frequencies and, in particular in race applications -
for which the presented studies have been originally
carried out - in order to configure optimal vehicle setup
and driving strategies [9] [10].
A deep knowledge of the mechanisms involved with
tyre/road friction is a key factor in the design of the
suspension system: an optimal setting of tyre working
angles, operated in order to optimize temperature,
contact pressure and sliding velocity distributions, can
be efficiently provided by a physical grip model able to
indicate the best wheel configuration at the boundary
conditions changes.
2. Basic Hypotheses
Asphalt and common road profiles are widely
considered as well described by the sum of two
sinusoidal waves characterizing the macro and the
micro roughness scales; for a sort of superposition
principle it can be stated that macro roughness
(modelled as a sinusoid characterized by wavelength
value in the range 0.5 – 50 mm) is responsible of the
indentation phenomena concerning with hysteretic
component of rubber friction, while micro roughness
(modelled as a sinusoid characterized by wavelength
value in the range 0 – 0.5 mm) is linked with the
attitude that bodies in sliding contact have to link each
other by means of intermolecular adhesive bonds [11].
As concerns the modelling of global tyre system
hysteretic dissipation due to interaction forces with road
[5], an analytical model has already been developed [12,
13]. In the presented study the analyses will be focused
on the only hysteretic friction component due to asphalt
macro roughness and consequently a single tread
element will be considered.
In order to model the interactions between a tyre
tread element and an asphalt asperity, it has been
necessary to focus on the behaviour of an elementary
parallelepiped of rubber in sliding contact with a 3D
sinusoidal wave. 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 (Figure 1).
Figure 1 - Elementary tread volume
and coordinates system.
Wavelength λ and roughness index Ra [14]
characterizing soil profile have been estimated by
means of proper algorithms employed to analyze data
acquired experimentally by laser scan on different dry
tracks and to reproduce the best-fitting sinusoidal wave
corresponding to the road profile (Figure 2).
Figure 2 - Acquired road profile
and plot of a 2D section.
The chosen Cartesian reference system, as showed
in Figure 1, has its origin in the centre of the upper
parallelepiped face; x-axis is in the tread surface plane,
oriented in the sliding direction of the indenter, z-axis is

2
oriented in the direction of tread width, and y-axis is
oriented in order to obtain a right handed coordinates
system.
Rubber and road are considered as isotropic and
homogeneous materials; moreover, road is modelled as
perfectly rigid.
3. Visco-Elastic Phenomena Characterization
Rubbery state of a polymer is determined by the
so-called glass transition temperature Tg. If the working
temperature is above Tg the polymer shows a rubbery
behaviour, below Tg a glassy one.
Polymers, in particular in the neighbourhood of the
thermal transition zone, do not follow reversible
stress-strain behaviour, but the strain lags behind the
stress with a delay of δ/ω ,where ω is the pulsation
(rad/sec). This visco-elastic effect cannot be described
by the classical perfectly elastic dynamic modulus E: in
order to model this hysteresis it is necessary to
introduce a dynamic storage modulus E' (Figure 3) and
a dynamic loss modulus E''. An index frequently used to
estimate hysteretic attitude of polymers is the loss angle,
defined as tan(δ) = E''/E' (Figure 4).
Figure 3 - Passenger tyre E’
thermal characterization data.
Figure 4 - Passenger tyre tan(δ)
thermal characterization data.
When both the frequency and the temperature vary,
it is possible to make use of the property 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 components
at any reference frequency and temperature (f1, T1) are
identical to the ones observable at any other frequency
f2 at a properly shifted value of temperature α(T1):




1 1 2 1
, ,
E f T E f T

 

 
(1)
The most widely used relationship able to describe
the equivalence principle is the Williams-Landel-Ferry
(WLF) transform [15]. For passenger tyre rubbers it can
be employed in a simplified way in order to determine
the unknown equivalent temperature T*= α(T1):
*
2 1
1
f T T
log
f T
 


  
 
(2)
in which a common ΔT value, identifiable, once known
rubber thermodynamics characteristics [16], by means
of DMA and DSC tests [17] 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 between two consecutive stresses, not
allowing the complete relax of the rubber, in the same
way as a low working temperature would do.
Rubber dynamic modulus is modelled by means of
temperature and frequency DMA characterizations,
whose outputs have been showed in Figure 3 an 4.
Defining stress frequency f of sliding tyre tread over
rough surface as the ratio between sliding velocity Vs
and macro-scale wavelength λ, visco-elastic material
model - and consequently the whole hysteretic friction
model - is able to take into account of the effects that
temperature, sliding velocity and road roughness have
on indentation phenomena.
4. Hysteretic Power Dissipation Model
The modelling of the hysteresis starts from the
expression of the power dissipated by a rubber block
that slides with speed Vs under a vertical load FZ over a
generically rough surface.
In general, considering the volume VTOT of the
elementary tread element, it is possible to express the
dissipated power Wdiss at time t as [18]:




   
, , ,
, , , , , ,
TOT
TOT
diss
V
V
W t w x y z t dV
d
x y z t x y z t dV
dt


 
 
 
 

 (3)
in which W (x, y, z, t) represents the dissipated power in
each point of the deformed volume at time t.
Hypothesizing the Vs constant in the sliding over a
single asperity, each stress/strain cycle can be
considered as performed in a period T0 equal to the
inverse ratio of the frequency f at which rubber is
stressed if sliding on macro roughness.
The average value of dissipated power must be thus
evaluated over such this time period:
   
0
00
1
, , , , ,
T
w x y z w x y z t dt
T

(4)
In the same time period, considering tangential force
FT as constant, the global dissipated power can be
expressed as:
0
diss T s N s s
W F V F V pA V
 
   (5)

Associazione Italiana di Tribologia (http://www.aitrib.it/) 3
in which p represents the average contact pressure in the
nominal area A0, equal to λ2. In this way, expressing the
stress as σ and the strain as ε, the balance between
global and local dissipated powers is:
   
0
000
1
, , , , , ,
TOT
T
s
V
d
pAV x y z t x y z t dt dV
T dt

 
 

 
 
 
  (6)
allowing to formulate the final hysteretic friction
expression:
   
0
00
0
1
, , , , , ,
TOT
T
V
s
d
x y z t x y z t dt dV
T dt
pAV



 
 
 
 
 
(7)
In order to estimate this friction coefficient, knowing
polymer characteristics, road wavelength and input
variables, it is necessary to provide the stress and the
strain derivative in each point of the discretized
elementary tread volume, and, in particular, of their
components along x, y and z directions. Thanks to the
studies of Y. A. Kuznetsov [19, 20, 21], it is possible to
calculate the stress state induced in a rubber elastic body
by a periodic sinusoidal perfectly rigid indenter in sliding
contact with it.
Once determined the radius of curvature R of the road
sinusoidal indenter at the apex:
 
2
1
2
λa
R
R


 
 
 
 
 
 
 
(8)
it is possible to estimate by means of Kuznetsov formula,
the half-length N of the contact area (Figure 5) as a
function of the radius R, of the average contact pressure p
in the nominal area A0, of the rubber dynamic modulus E
calculated taking into account of the working conditions
acting on the examined elementary volume:
 
2
14 1
sin pR
l
Nl E


 

 

 

 
 
(9)
Figure 5 - Detail of the contact between
sinusoidal asperity apex and tyre tread.
Kuznetsov's method for planar stress calculation
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) showed in
Figure 6 and 7, localized under asperity apex. The model
does not need the direction of the sliding velocity on each
asperity as an input because the orientation of the
elementary volume adapts automatically to it,
considering x axis parallel to Vs. In this way the only
cinematic input is the modulus of the sliding velocity and
it is taken into account by means of the frequency effect
that it has on rubber for WLF law.
Figure 6 - σx distribution in the x-z plane
under a pressure of 125 KPa.
Figure 7 - σy distribution in the y-z plane
under a pressure of 125 KPa.
The essential relationship between adhesive and
hysteretic friction is taken into account by means of
Kuznetsov's parameter K, that is supposed to be equal to
the adhesive friction coefficient.
Thanks to a wide experimental campaign, performed
by the authors of the present paper with the aim to
investigate tyre/road adhesive component of friction
arising in the sliding contact with micro-rough profiles
[22, 23], it has been possible to identify the optimal
value of K for each kind of analysis.
Applying Kuznetsov equations in the x-z plane, it is
possible to calculate the stress components σx and σz
generated by the sliding indenter in the direction of
sliding velocity. Because of the self-orientation of the
elementary volume, it is possible to state that in the plane
y-z sliding velocity components are absent: it means that
Kuznetsov’s equations must be applied in this plane
imposing K equal to zero, obtaining perfectly symmetric
tangential stress that can be employed in order to provide
an estimation of the stress component σy.
With the aim to extend the bi-dimensional results,
reported in Figure 8, to the whole three-dimensional
elementary volume, the planar components have been
scaled basing on the information obtained by the stress
estimation in the perpendicular planes. As an example,
the extension of the σx component has been carried out as
follows: a scaling function SF, equal to :
( )
( )
max (abs ( ))
y
y
y
SF y


 (10)
has been defined with the aim to scale stress components
along y direction. Thanks to SF, it is possible to provide
an estimation of σx in the whole elementary volume:
(x, ,z) (x,z) SF(y)
x x
y
 
  (11)

4
showed in Figure 9.
Figure 8 - σx in the xz plane (left) and σy in the yz plane
(right), depth = 2mm, temperature = 40°C,
pressure = 150KPa.
Figure 9 - σx scaling procedure (left) and final σx
extension in the 3D space (right).
The same results can be applied extending σz along y
direction and σx along x direction. Stress components τxy,
τxz and τyz are usually neglected in power dissipation
analyses because their contribution is about one order of
magnitude lower than the one relative to components σ.
The knowledge of the described stress components
allows to calculate the numerical derivative of the strain
dε/dt of Equation 7 extended to the elementary tread
volume. At this aim, the volume has been discretized in
200 nodes along x and y and 50 along z; this number
represents an optimal trade off between the stability of
the results and the computation performances.
Figure 10 - Employment of a single volume strain
for the calculation of strain derivative
over multiple asperities.
In the hypothesis of Vs constant for the sliding over
two consecutive identical asperities, rubber deformations,
calculated as σ/E, can be supposed to be equal under each
different sinusoidal indenter. For this reason, instead of
considering ε values at different times in the numerical
derivative of strain:
0 0




t t t
ε ε
εt (12)
it will be enough to localize in the same elementary
volume the value assumed by ε after a time Δt (Figure
10) substituting Δx = VsΔt in the derivative:
0 0




x x x
ε ε
εx (13)
with Δt < 0.1 ms in order to avoid aliasing problems.
5. Results
The proposed model has been validated by means of
multiple solutions: FEM analyses performed thanks to a
commercial software have highlighted a good
correspondence with stress and strain distributions
obtained with the showed methodologies (Figure 11); an
experimental activity is actually providing comparisons
between model simulations results and friction tests
carried out with a modified skid pendulum and a pin on
disk tribometer. Moreover, the hysteretic friction trend
exhibited as a function of the main variables involved in
the phenomenon (Vs, T, p) is in good accordance with
the expected one, deductible from the wide
experimental results available in literature [24]; in
Figure 12 a map of hysteretic friction at a temperature
of 40°C is reported for a GT sport tyre; it is possible to
observe that the shape of the map along pressure axis is
strongly influenced by the indentation mechanisms
linked with road roughness and rubber deformability,
and that the decreasing trend along Vs axis is mainly
due to WLF law and polymer characterization.
Figure 11 - Comparison between proposed estimation
and FEM calculation of σx in xz plane
under a pressure of 125 KPa.
Figure 12 - Map of hysteretic friction
at a temperature of 40°C.

Associazione Italiana di Tribologia (http://www.aitrib.it/) 5
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