International Journal of Pavement Engineering

ISSN: 1029-8436 (Print) 1477-268X (Online) Journal homepage: http://www.tandfonline.com/loi/gpav20

Contribution to pavement friction modelling: an

introduction of the wetting effect

Malal Kane, Minh-Tan Do, Veronique Cerezo, Zoltan Rado & Chiraz Khelifi

To cite this article: Malal Kane, Minh-Tan Do, Veronique Cerezo, Zoltan Rado & Chiraz
Khelifi (2017): Contribution to pavement friction modelling: an introduction of the wetting effect,
International Journal of Pavement Engineering, DOI: 10.1080/10298436.2017.1369776

To link to this article: http://dx.doi.org/10.1080/10298436.2017.1369776

Published online: 06 Sep 2017.

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 International Journal of Pavement Engineering, 2017
https://doi.org/10.1080/10298436.2017.1369776

Contribution to pavement friction modelling: an introduction of the wetting effect

Malal Kanea, Minh-Tan Doa, Veronique Cerezoa, Zoltan Radob and Chiraz Khelifia
a
IFSTTAR, AME, EASE, F-44344, Bouguenais, France; bLarson Institute, Pennsylvania State University, PA, USA

ABSTRACT ARTICLE HISTORY

This paper presents a friction model describing the tyre rubber/road interaction that takes into account the Received 7 February 2017
viscoelasticity of the tyre rubber, the texture of the road surface and a water layer between the tyre/road Accepted 9 August 2017
interface by introducing explicitly a computation of the water layer effect in the calculation process of the
KEYWORDS
hysteretic friction. The geometry of the wetted portion of the interface model is simplified by transforming Friction; pavement; rubber;
it into an equivalent hydrodynamic bearing. Utilising the Reynolds equation, the bearing load capacity is texture; wet; hydrodynamic
calculated and the resulting forces are subtracted from the contact load when calculating the forces of the
Downloaded by [UC Santa Barbara Library] at 03:41 08 September 2017

effect
hysteretic friction. The mechanical behaviour of the rubber is represented in the model by Kelvin–Voigt
model. The frictional forces due to hysteresis are calculated at any given operating conditions (load, slip
speed, etc.) from the contact geometry of rough surfaces caused by the viscoelastic behaviour of rubber. To
validate the model, a set of surfaces including real pavements and artificially textured slabs were selected
covering a wide range of microtexture and macrotexture combinations and the computed and measured
friction compared. To describe the contact geometry of rough surfaces using macrotexture and to measure
actual friction, the Circular Track Meter and the Dynamic Friction Tester devices were used, respectively. The
friction coefficients computed using the model were compared to the measured friction coefficients. The
obtained results are presented in the paper and proved to provide high correlation between the measured
and modelled friction. The model is capable to predict wet friction at low as well as high speeds on wet
surfaces, thus proving to be capable to take adequately the wetting effect on the variation of friction with
increasing speed. Recommendations are provided to improve the model and extend it to a tyre friction
model.

1. Introduction lubricant and can prevent these bonds to from. The hysteresis
component is due to viscoelastic energy losses when the rubber
Skid resistance describes the contribution that the road makes
is deformed cyclically by the pavement surface texture (Rohde
to tyre/road friction. Essentially, it is a measurement of friction
1976, Taneerananon and Yandell 1981, Kane et al. 2014, Rado
obtained under specified, standardised conditions, generally cho-
sen to fix the values of many of the potential variable factors so et al. 2014). These papers highlighted the role of microtexture
that the contribution that the road provides to tyre/road friction (small texture scales) on hysteretic friction generation. The hys-
can be isolated. It remains one of the main factors contributing teresis friction component develops even when the substrate
to road safety. It is mainly related to frictional characteristics of is wet (Greenwood and Tabor 1958, 1961). Depending on the
the tyre–road contact interface (road surface texture, tyre rubber operating conditions (mainly at high speeds) and the surface
type, etc.), but also depends on the tyre characteristics (geom- roughness (low macrotexture), hysteretic friction can drop too
etry, inflation pressure, tread depth, etc.), the operating condi- due to the water hydrodynamic effect that tends to lift up the
tions (speed, slip ratio, load, etc.) and environmental conditions tyre from the road (Moore 1975).
(water, dust, etc.). Under specific operating and environmental Recently, a contribution to friction modelling has been pro-
conditions, skid resistance can drop down to very low levels in posed by (Kane and Cerezo 2015). The proposed model is based
particular when the pavement surface is covered by water cou- on a simplified viscoelastic contact model of rough surfaces to
pled with a high speed. predict friction generated in a dynamic contact between a rubber
Thus, wetting of pavement surfaces is of paramount impor- pad and a rough surface. The rubber behaviour was modelled
tance in regard to skid resistance. Indeed, friction of polymeric through Kelvin–Voigt elements and the friction force was esti-
materials comes from two basic phenomena so called adhe- mated from the contact model describing the rough geometry
sion and hysteresis (Sabey 1958, Savkoor 1990). The adhesion of the moving rubber on the surface asperities. However, in this
component comes from inter-molecular bonds at the rubber/ model, the hydrodynamic effect was not explicitly included.
pavement contact interface and is highly dependent on the real To overcome this limitation, the present work proposes to
contact area. The presence of water at the interface acts as a redefine the hysteretic friction and introduce the hydrodynamic

CONTACT Malal Kane malal.kane@ifsttar.fr

© 2017 Informa UK Limited, trading as Taylor & Francis Group
 2  M. KANE ET AL.

effect in the calculation process. The geometry of the wet portion Figure 1 and is discretised in independent elements (Kane and
of the contact is approached by converting the contact geometry Cerezo 2015). For this study, the behaviour of each element is
into an equivalent hydrodynamic bearing model. Consequently, modelled by a Kelvin–Voigt model, which allows capturing the
using the Reynolds equation, the equivalent bearing load capac- viscoelastic behaviour of the rubber. The width of each element
ity can be calculated and subtracted from the overall load dis- is chosen to match the horizontal resolution of the measured
tributed over the contact. To clearly explain the model and the texture profiles. More complex rheological models could be used
performed work, the description of the proposed new model is but this will not change the approach.
organised in three parts: During the sliding of the rubber pad on the pavement surface,
the governing equations in the contact area are:
• The first part will focus on the redefinition of hysteretic
F ���⃗ + R
���⃗ij + T ���⃗ij + FR
�����⃗ij = 0⃗ (1)
friction. The basic governing equations of the model and ij
the calculation algorithm will be briefly discussed. where:
• The second part will detail how the hydrodynamic effect is
taken into account. The assumptions to transform the wet • F
���⃗ij is the force applied by the rubber element on the road
portion of the contact and to establish a ‘pseudo’ hydrody- surface. Its formulation depends on the rheological behav-
namic bearing equivalent will be detailed. iour considered for the rubber elements. In the present
• The third part will discuss the validation of the model. The work, it is calculated using a ‘Kelvin–Voigt’ model, where
experimental devices and test procedures will be described K is the spring’s elastic modulus per unit length and it
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first and a comparison between the experiments and the determines the elastic behaviour of the material. C is the
model predictions will be presented. dashpot’s viscosity per unit length and determines the
• The last section is devoted to a discussion on the limita- viscous effect of the material. F ���⃗ij is balanced by the load
tions and possible future improvements. through the contact pressure pij (see Equation (2)).
Although in this study the Kelvin–Voigt model was utilised
any other viscoelastic model would work. The impact of the
2. Redefining the hysteretic friction
chosen model appears in the calculation of the contact pres-
The governing equations are derived from the force balance at sure and displacement distributions (Equations (3), (4) and (8)).
any time instance of the forces acting in the contact between the The Kelving–Voigt model is very convenient for a first approach,
surface profile and the rubber pad moving on it. To take into because of the limited number of parameters (two) to be deter-
account the particularity of the DFT (see Section 5 and ASTM mined. Section 6 of this article is dedicated to the method used
2009), the geometry of its rubber pad is modelled as shown in to characterise them.

Figure 1. Up-Left: Picture of the real DFT measuring pad. Up-Right: Deformed geometry of the rubber. It is considered as composed by a set of independent viscoelastic
elements (the red line represents the pavement profile). Down: Forces acting in the contact between a rubber element and a pavement profile.
 INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING  3

Fij (t) = l × dx × pij (t) (2) The two-dimensional system is coupled the condition of
���⃗ij described by the following equation:
�����⃗ij = 𝜇loc R
FR
with ( ) ( )
duij (t) sin 𝛼j + 𝜇loc cos 𝛼j
pij (t) = Kuij (t) + C (3) Tij (t) = Fij (t) ( ) ( ) (6)
dt cos 𝛼j − 𝜇loc sin 𝛼j
And
when an element is not in contact with the pavement surface, its
uij (t) = 𝛿(t) − hi + zj (4) contact pressure is nil and the element is subjected to a relaxation
where t represents the time. uij (t) is the displacement of the rub- phase. Its position on the Z axis is determined by solving:
ber ith element contacting jth element on the pavement at time t.
δ (t) is the solid (or global) displacement of the DFT pad at time
Fij (t) = 0 (7)
t. hi represents the discretised h(x) at the ith point representing That is equivalent to:
then the DFT pad geometry. zj is the height of the jth point of
duij (t)
the pavement surface. Kuij (t) + C =0 (8)
dt
���⃗ is the traction force needed to move the element. This
• 
T ij At any time t and irrespective of the location of the pad on the
force must be just greater than the global friction force
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profile, the total load W applied on the DFT pad must be bal-
opposing that movement (Tij ≥ FFij where FF ij is the fric- anced by the normal contact pressure:
tion force).
• 
���⃗ij is the surface reaction.
R ∑
N

• 
�����⃗ij is a local friction force. FR
FR ���⃗ij when the ele-
�����⃗ij = 𝜇loc R W= Fij (t) (9)
ment is moving on that ‘pseudo smooth inclined plan’ i

where 𝜇loc represents a friction coefficient of two different where N is the number of discret elements composing the rubber
components: pad.
◦ adhesive component due to the molecular bonding of At the minimum needed traction to move an element of the
the two contacting surfaces. It may be close to zero or rubber, the traction and the friction forces are almost equal:
even nil when the contact is wet. ( ) ( )
◦ local hysteretic component of all texture scales smaller sin 𝛼j + 𝜇loc cos 𝛼j
than the resolution with which the profile is measured FFij (t) = Fij (t) ( ) ( ) (10)
and recorded (Example: If the topography is recorded at cos 𝛼j − 𝜇loc sin 𝛼j
1 mm of resolution, this will represent the contribution
of all scales smaller than 1 mm) (Rado 1996, Persson Thus, the global friction coefficient 𝜇j (t) then can be calculated
2001, Villani et al. 2011). using the following formula:
Figure 2 illustrates the multiscale representation of the local ∑N
scale from larger to smaller resolutions. FFij (t)
𝜇j (t) =
i
(11)
Projection of Equation (1) of local contact coordinates onto W
global contact axes x and z leads to the system of two equations
below: And the averaged global friction coefficient μav for each pavement
profile is calculated by averaging the friction coefficient 𝜇j (t) at
⎧ � � � � any time:
⎪ −Fij + Rij cos 𝛼j − FRij sin 𝛼j = 0
� � � � (5) 1 ∑
M
⎨
⎪ Tij − Rij sin 𝛼j − FRij cos 𝛼j = 0 𝜇av = 𝜇
⎩ M j j (12)

Figure 2. Fractal representation of the local hysteretic contribution of all texture scales smaller than the resolution with which the profile is recorded from left to right,
larger to finer scales (here, scales 0 to 2 are illustrated).
 4  M. KANE ET AL.

where M is the number of elements of the discretised pavement x( )

H(x) = Hin − H − Hout (14)
profile. L in

𝛼6𝜂V 𝛽 lL2 [ a−1

]
3. Introducing the hydrodynamic effects of water Wh = log (a) − 2 (15)
2
Hout (a − 1)2 a+1
Depending on the operating conditions, contact geometry and
where, a = H in , with Hout and Hin, respectively, the outlet and inlet
H
the water volume, a hydrodynamic pressure will be generated in
the lubricant trapped between the rubber element and the pave- water thicknesses of the ‘pseudo’ bearing. Coordinate (x) deter-
out

ment surface. This hydrodynamic pressure exerts a force to lift up mines the position in the water flow direction. H(x) is the water
the rubber from the pavement and thus decreases the penetration thickness at a point localised at x in the bearing. η is the water vis-
depth of pavement asperities into the rubber elements reducing cosity. L and l are the length and the width of the bearing (these
therefore the hysteretic contribution to the tyre rubber/pavement two dimensions are equivalent to the DFT length and width).
skid resistance (Rohde 1976, Taneerananon and Yandell 1981). α et β are two emperical coefficients (105 and 1.75, respectivly)
An extreme case is when the rubber is fully separated from the added to modify the analytical expression of the hydrodynamic
surface or hydroplaning. load capacity of a classical bearing (equal to 1 in the case of a
Below hydroplaning speed, the tyre/pavement contact inter- classical bearing) to calculate the load of the ‘pseudo’ bearing.
face is in a mixed mode (wet and dry). Within this state the It can be noticed that when Hout is equal to Hin, the water will
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surface asperities are in direct contact with the tyre while the not carry any load, the ‘pseudo’ hydrodynamic load capacity Wh
valleys are filled by water. The generated hydrodynamic pressure is equal to zero (To verify that, the Equation[ (15)(can)be rewritten]
in these wet valleys can be computed by the Reynolds equation
𝛽 2
by replacing a with H in → Wh = 𝛼6𝜂V lL 2 log H in − 2 Hin +Hout
H H H −H

(Sabey 1958). out ( in out )

H +H out in out

in that case, when Hout = Hin → Wh = 0).

One of the requirements to use this equation is a continuous
Also, the Wh is calculated and taken into account only when
lubricant film. In the case of three-dimensional contact model, it
the amount of water present on the road surface (independent
is possible to directly carry out this calculation by imposing zero
of the range of the inlet water thickness) is enough to fill the
speed conditions of the lubricant particles at the dry/wet bor-
available spaces between the deformed measuring pad surface
ders. This will allow the lubricant to continue to flow bypassing
and the valleys of the road surface texture. Otherwise, Wh is equal
these dry/wet borders. However, in a two-dimensional contact
to zero and the calculation is done as a dry contact.
model, this condition of continuity cannot be ascertained, thus
it is necessary to find an equivalent two-dimensional model to
overcome this problem. 4. Algorithm of the model
The idea proposed in this work is to convert the wet por-
Figure 4 displays the new algorithm introducing the hydrody-
tion of the contact area into a ‘pseudo’ hydrodynamic bearing
namic effect. The added part ‘in comparison the one proposed
with an equivalent continuous lubricant film. To do so, first, the
in (Kane and Cerezo 2015)’ is identified by the red block (dashed
inlet water thickness is taken to be equivalent to the water depth.
line frame). The hydrodynamic load is subtracted from the initial
Second, the outlet lubricant thickness is set to keep the same water
load at each cycle of computational loop to calculate the friction
volume to the calculated volume that is trapped in the wet valleys
force. For the calculation of the friction coefficient, the friction
of the rough contact model, taking the viscoelastic deformation of
force is divided by the global load.
the rubber into account. With this equivalent configuration, the
It can be noticed that in the case of dry contact, the hydro-
hydrodynamic load can be computed with bearing load and sub-
dynamic load is not taken into account., the calculation repre-
tracted from the load applied on the moving DFT pad (Equation
sented by the red block in the algorithm scheme in the Figure
(15)). Figure 3 illustrates how the equivalency is set.
4, is skipped.
To calculate the hydrodynamic pressure ph (and there-
fore the ‘pseudo’ hydrodynamic load capacity Wh), one inte-
grates the two-dimensional Reynolds equation (Equation (13)) 5. Experiments
with the geometry of a hydrodynamic bearing film (Equation (14)).
( ) Experiments were conducted to validate the proposed model.
d dp (x) dH(x) Friction- and texture-measuring devices: The Dynamic
H(x)3 h = 6𝜂V
dx dx dx (13) Friction Tester (DFT) was used to measure friction and the

Figure 3. Left: Representation of the real DFT measuring pad. Right: Representation of the ‘pseudo’ hydrodynamic bearing with an equivalent continuous lubricant film.
 INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING  5

Circular Track Meter (CTM) to measure texture profiles Test surfaces: The test surfaces were chosen to cover a large
of the test surfaces (Figure 5). Texture is measured with a combination of microtexture and macrotexture with different
horizontal resolution of 0.87 mm. For details about these levels in order to cover a wider range of friction coefficient val-
devices, please refer to (Hanson and Prowel 2004, ASTM ues. The selected surface is composed of six different surface
2004, 2009). morphologies:

Mechanical characteristic, Initialization of

Geometry of the DFT rubber and Hydrodynamic Load to
DFT Slip speed and Normal load CTM pavement profile Zero
The solid (global)
penetration of the rubber
in the pavement profile

The displacement of each element “i” of the

rubber, positioned at the “jth” element on the
pavement profile
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Any negative
displacement is Set of a new solid
set to zero (global) penetration
of the rubber in the
pavement

The local contact pressure of each element “i”

of the rubber, positioned at the “jth” element
on the pavement profile

- Any negative pressure is set Move all DFT elements to

to zero their next positions on the
- The corresponding profile
displacement is recalculated

Calculation of the Hydrodynamic load

DFT normal load = DFT load – Hydrodynamic load

Comparison between the integrated pressure

in the whole contact length and the DFT
normal load

Equality?

Calculation of the Instantaneous friction

by dividing the Hysteretic Force by the
DFT Initial Normal Load

Calculation of
the averaged
friction

Figure 4. Improved algorithm. The added part is identified by the red block in the organigram. The hydrodynamic load is subtracted to initial load at each loop to calculate
the hysteretic friction force.
 6  M. KANE ET AL.

• Three artificially textured surfaces extracted from the same W =𝛾 ×u (16)

large granite slab (polished noted GSP in Figure 6(a), sawn
noted GSS in Figure 6(b) and sandblasted noted GSSB in With γ representing the slope of the load/displacement line that
Figure 6(c)), is equal to the following equation:
• Two real road surfaces from two different formulations (a
‘very thin asphalt surfacing’ noted VTAC in Figure 6(d) 𝛾 =S×K (17)
and a ‘medium coarse asphalt’ noted MCA in Figure 6(e)),
From Equation (17), K is deducted: K = 1.4 × 108 N/m3
• One slab made by sticking rounded and sandblasted
(Representing the elastic modulus per width of the DFT pad,
aggregates on a support noted RSBA (See Figure 6(f)).
hence the in N/m3 instead of N/m2 that is more common)
Figure 6 displays a picture of the six surfaces and the corre- (Figure 8).
sponding profile (measured with the CTM) underneath. To determine C, an estimation of its value was performed
On each surface the texture is measured using the CTM, then using the proposed model. Using the Sandblasted Granite Slab,
the friction coefficient is measured at the exact same physical the C value was select such that model results fit the measured
same track location using the DFT. Extreme care is exercised to experimental value of the friction coefficient at 20 km/h (thus,
place DFT device on the exact location where the CTM measured Sandblasted Granite Slab designates as ‘GSSB’ was chosen as the
the surface texture. reference surface). The computed value of C = 102 Ns/m3 was
then used in all model calculations as a constant. (Representing
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6. Determining the model parameters the viscosity per width of the DFT pad, hence the in Ns/m3
instead of Ns/m2 that is more common)
Characterising rheological model parameters of the rubber: The Characterising the surface ‘local friction coefficients’: It is
rubber material is modelled using a ‘Kelvin–Voigt’ viscoelastic admitted that the contribution to friction of the microtexture
material model. Thus, only two model parameters have to be scales of the surface is pronounced when measuring friction at
determined the spring’s elastic modulus and the dashpot’s vis- very low speeds (17 km/h). Thus, to characterise the local fic-
cosity (K and C, respectively). tion coefficient μloc of the surfaces, friction measurements were
To determine the spring’s elastic modulus, loads ranging from performed using the British Pendulum Tester. The measure-
10 to 50 N were applied to the rubber against a smooth surface ment principle is based on a pendulum rotating about a spindle
and the corresponding displacements measured. This operation attached to a vertical pillar. At the end of the tubular arm a head
is performed using a UMT tribometer (Figure 7). of known mass is fitted with a rubber slider. The pendulum is
The following assumptions were made: released from a horizontal position so that it strikes the sample
surface with a low velocity (17 km/h). The distance travelled by
(1) The contact pressure distribution is uniform,
the head after striking the sample is determined by the friction
(2) The displacement of the rubber due to the applied load
of the sample surface. A reading of Skid Resistance (named SRT)
is measured at the permanent regime of the elements.
was obtained for each test surface (Figure 9).
Due to the second assumption, the second term of Equation Summary of model parameters: Input parameters for the
(3) is eliminated. So, the integration of the remaining term on simulations for this paper are displayed in the following table
the whole contact surface S transforms to: (Table 1).

Figure 5. Up: dynamic friction tester, Up-Left: side view, Up-Right: bottom view, Up-Middle: measuring rubber pad. Down: Circular Track Meter, Down-Left: side view,
Down-Right: bottom view.
 INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING  7

(a) Sandblasted Granite Slabnoted “GSSB” (b) Sawn Granite Slabnoted “GSS”

GSSB GSS
2 2

height mm
height mm

0 0
-2 -2
-4 -4
100 150 200 250 300 100 150 200 250 300
distance mm distance mm
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(c) Polished Granite Slab noted “GSP” (d) Very Thin Asphalt Surfacing noted “VTAC”

GSP VTAC
2 2
height mm

height mm

0 0
-2 -2
-4 -4
100 150 200 250 300 100 150 200 250 300
distance mm distance mm

(e) Medium Coarse Asphalt noted “MCA” (f) Rounded Sandblasted Aggregates noted “RSBA”

MCA RSBA
2 2
height mm

height mm

0 0
-2 -2
-4 -4
100 150 200 250 300 100 150 200 250 300
distance mm distance mm

Figure 6. The test surfaces and their profiles at the bottom.

 8  M. KANE ET AL.

Figure 7. Tribometer UMT used to determine the spring’s elastic modulus.

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60
Displ (mm) Load (N) y = 44.45x - 7.1429
0.39 10.5 R² = 0.9971
50
0.4 11.5
0.44 12.5
0.64 20 40
Load (N)

0.85 30
1.27 50 30

20

10

0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Displacement (mm)

Figure 8. Load/displacement curve and data table.

7. Results and discussion studies (Do et al. 2004). Despite the simplicity of the approach,
these results reinforce the results presented in the previous paper
Effect of lubricant: Figure 10 displays the variation of friction
that the model reproduces, with some limitations, the basic phys-
coefficient with speed in dry (dashed line) and wet (continuous
ics governing the skid resistance phenomena.
line) conditions for three test surfaces.
Comparison model/experiment: The two first plots of Figure 11
In ‘pseudo’ dry conditions (meaning very low water thickness,
(from Table 2 data) display the correlations of the friction
just enough to prevent adhesion to operate, but insufficient to create
­coefficients predicted by the model and the measurements from
hydrodynamic load), the effect of speed on the friction coefficient is
the experiments at 20 and 60 km/h. The last plot grouped corre-
very low, independent of the texture depth. The higher the micro-
lations on all the three measurement speeds 20, 40 and 60 km/h
texture and macrotexture are, the better the friction is. The rank is:
in the same plot.
(1) VTAC: high macrotexture and high microtexture, The results show very good correlations, the points follow
(2) GSSB: medium macrotexture and high microtexture, generally the bisector and the coefficient of determination R²
(3) GSP: low macrotexture and high microtexture. which measures the regression quality is very good (above 0.95)
for all speeds.
In wet condition and low speeds, the behaviour of the friction The continuous lines and the points of Figure 12 represent the
for all surfaces is similar to those on dry conditions. But when model predictions and experimental measurements, respectively,
the speed increases, the lower the macrotexture is, the higher is of the friction coefficient of the six surfaces at different speeds
the friction drop. For the high macrotextured surface (VTAC), (ranging from 0 to 60 km/h for the model and 20, 40 and 60 km/h
the friction coefficient remains almost the same at high speeds, for the experiments).
whereas for lower macrotextured surface (GSP), the friction is The predictions are good for the high macrotextured surfaces
almost nil at high speeds. at all speeds. For low macrotextured surfaces, the model overes-
The above described behaviours displayed by the model are timates the hydrodynamic effect at higher speeds and therefore
in agreement with what is usually considered in experimental underestimating the friction coefficients.
 INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING  9

Name SRT
GSP 0,3
GSS 0,55
GSSB 0,69
RSBA 0,73
MCA 0,75
VTAC 0,6

Figure 9. British Pendulum Tester and STR data obtained for each test surface.
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Table 1. Other inputs parameters of the simulations.

K C dx W h l L Hin 𝛿
Dashpot’s Measuring Applied load Thickness of Water
Spring’s elastic viscosity scale of the on the rubber the rubber Width of the Length of the ­thickness Dynamic
modulus (N/m3) (N s/m3) profiles (mm) (N) (mm) rubber (mm) rubber (mm) (mm) ­viscosity (Pa s)
1 × 4 × 108 102 0.87 11.8 6 16 20 10−3 10−3

CF vs Speed (Wet vs Dry) leads to a decrease in the frictional force due to the less deep
1.2
penetration of the road asperities into the rubber.
1 Discussion: Despite the demonstrated limitations of the
GSP_Wet
model, it can be concluded that it demonstrated its ability to
GSSB_Wet
0.8
VTAC_Wet
reproduce the physics governing wetted surface friction caused
GSP_Dry by hysteretic viscoelastic rubber behaviour well and thus open a
FC

0.6
GSSB_Dry
promising and attractive way to model tyre–road friction. This
VTAC_Dry
0.4 model is capable to take into account different parameters that
have an effect on the friction level:
0.2
• parameters related to the tyre: its geometry and the rubber
0 characteristics,
1 11 21 31 41 51 61
• parameters related to the road surface: its texture and the
measuring resolution,
Speed (km/h)
• parameters related to the contaminant: its viscosity and
thickness,
• and parameters related to the operating conditions: the
Legende surfaces speed and load.

1-GSP To overcome the limitations of the described model additional

2-GSS work for improvement should be considered. One such improve-
3GSSB ment would consider a model extension to a mixed calculation
4-RSBA of hydrodynamic forces instead of considering a hydrodynamic
5-VTAC bearing with a continuous lubricant film, especially when extend-
ing the model in three dimensions. The boundary condition at
the border between dry and wet portions of the contact area
would be to impose a zero flow velocity to the water particles
Figure 10. Up: The continuous lines and the points represent, respectively, the when they hit the asperities. This would allow the particles to
predictions from the model and experimental measures of friction coefficient of bypass the asperities and thus preserve the continuity of the
the three surfaces at different speeds. Down: 3D representation of the wet friction
coefficients of five of the test surfaces vs. speed.
lubricant film. Of course, the calculation would take more time,
but the algorithm would not change fundamentally.
Nevertheless, the model reproduces the drop of the friction Also, the coefficients α et β introduced to modify the analyt-
when the speed increases. The physical explanation is that when ical expression of the hydrodynamic load capacity of a classical
the speed increases, the hydrodynamic load increases too. This bearing needs depeer research to find physical justifications.
 10  M. KANE ET AL.

1.20 1.20
60 km/h
20 km/h
1.00 1.00

0.80 0.80

Model
Model
0.60 0.60
0.40 0.40
y = 0.9834x y = 0.9746x
0.20 R² = 0.9875 0.20 R² = 0.9301
0.00 0.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20
Experiment Experiment

1.20
20, 40 and 60 km/h
1.00
Model 0.80

0.60
y = 0.9818x
Downloaded by [UC Santa Barbara Library] at 03:41 08 September 2017

0.40
R² = 0.9653
0.20

0.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Experiment

Figure 11. The two first plots display the correlations of the friction coefficients predicted by the model and the measures obtained from the experiments (20 and
60 km/h). The last plot grouped all of them (all the three speeds 20, 40 and 60 km/h) in the same plot.

Table 2. Frictions coefficients – experimental and model data.

Experimental Model
Name SRT 20 km/H 40 km/H 60 km/H 20 km/H 40 km/H 60 km/H
GSP 0.3 0.12 0.08 0.07 0.11 0.17
GSS 0.55 0.28 0.17 0.16 0.20 0.14
GSSB 0.69 0.59 0.55 0.53 0.62 0.50 0.35
RSBA 0.73 0.85 0.81 0.78 0.82 0.81 0.79
MCA 0.75 0.74 0.70 0.64 0.74 0.71 0.69
VTAC 0.6 1.03 1.02 0.99 1.02 1.01 0.98

CF vs Speed (Exp vs Mod) The determination of μlocal is an open question and needs
GSP_Mod deeper investigations too:
GSS_Mod

GSSB_Mod
• It is necessary to find a way to define the limit wavelength
RSBA_Mod from which a section of profile between two points (cor-
MCA_Mod responding to the measuring resolution) can be consid-
VTAC_Mod ered as smooth. In others words, the limit scale from
FC

GSP_Exp which any smaller asperities will no longer participate to

GSS_Exp
the generation of hysteretic friction. This would eliminate
GSSB_Exp
the need to experimentally determine the μlocal model
coefficient.
RSBA_Exp

MCA_Exp

VTAC_Exp
• If research show that the required scale at which experimen-
tal determination of μlocal is smaller than what is feasible with
Speed (km/h)
today’s technology, it is necessary to determine which exper-
imental procedure would be best for determining the μlocal?
Figure 12. The continuous lines and the points represent, respectively, the model
predictions and experimental measures of friction coefficient of the six surfaces at The incorporation of the temperature effects would also
different speeds. be very beneficial. In fact, as the water will maintain constant
 INTERNATIONAL JOURNAL OF PAVEMENT ENGINEERING  11

the temperature, the WLF (Williams–Landel–Ferry) equation References

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 12  M. KANE ET AL.

Appendix A

Experimental
Name SRT 20 km/H 40 km/H 60 km/H
GSP 0.3 0.12 0.08 0.07
GSS 0.55 0.28 0.17 0.16
GSSB 0.69 0.59 0.55 0.53
RSBA 0.73 0.85 0.81 0.78
MCA 0.6 0.74 0.70 0.64
VTAC 0.75 1.03 1.02 0.99

Model V12- WET all speeds Model V13- DRY all speeds
Name 1 10 20 40 60 80 1 10 20 40 60 80
GSP 0.4913 0.3305 0.108 x x 0 0.431 0.431 0.431 0.4306 0.4305 0.431
GSS 0.7297 0.5553 0.196 x x 0 0.726 0.726 0.726 0.7257 0.7257 0.726
GSSB 0.8612 0.7919 0.621 0.475 0.375 x 0.862 0.862 0.862 0.8616 0.8615 0.861
RSBA 0.8273 0.8257 0.822 0.811 0.794 0.778 0.843 0.843 0.838 0.8387 0.8385 0.838
MCA 0.7562 0.7521 0.743 0.717 0.693 0.681 0.805 0.805 0.791 0.7913 0.7912 0.791
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VTAC 1.0293 1.0277 1.024 1.009 0.984 0.955 1.036 1.036 1.032 1.0324 1.0324 1.032