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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 pAV

(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

sV

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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