Special Issue Article

Proc IMechE Part J:

J Engineering Tribology
Friction/water depth relationship—In situ 2014, Vol. 228(11) 1285–1297
! IMechE 2014

observations and its integration in tire/ Reprints and permissions:

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road friction models DOI: 10.1177/1350650114544533

pij.sagepub.com

V Cerezo1, MT Do1, D Prevost2 and M Bouteldja3

Abstract
The purpose of this paper is to provide new experimental evidences about the friction/water depth relationship and to
improve the formulation of friction models in terms of consideration of the effect of water depth. Tests are conducted on
test tracks. Friction forces are measured by means of a dedicated trailer providing a locked-wheel (full sliding) friction
coefficient. The test surfaces are wetted by an on-board wetting system providing water depths varying from 0.1 mm to
1.50 mm. Effect of the road surface texture on the friction/water depth variation is shown and commented. Inputs newly
provided by field tests, compared with laboratory tests, are highlighted. The obtained friction/water depths curves are
assimilated to Stribeck curves and analyses, assuming conditions of a starved lubricated contact, are conducted to
determine the lubrication regimes experienced by the tire/road contact when the road surface changes from dry to
wet. A new friction model is formulated using the three-zone description of the tire/road contact area. The formulation
is focused on the water drainage term reflected by a so-called hydrodynamic term FHL. It was assumed that FHL is a
product of elementary functions expressing the respective effects of water depth, speed, tire tread depth, and road
surface macrotexture on water drainage. Form of the elementary functions is derived from experimental evidences and
consideration of previous friction models. Fitting of the new model to experimental data is shown and comparison with
previous models is discussed.

Keywords
Friction, water depth, model, in situ measurements, pavement texture

Date received: 26 July 2013; accepted: 9 June 2014

likely to cause wet road accidents): without microtex-

Introduction
ture, the friction coefficient drops as soon as the sur-
It is well known that tire traction is reduced when the face is wetted, whereas the presence of microtexture
road surface is wet. Road authorities and car and tire helps to maintain the friction coefficient at a value
manufacturers are concerned by the issue of water close to that obtained on a dry surface, until a critical
drainage and in particular its prediction. Current WD is reached.
tire/road friction models express explicitly only the Do et al.’s findings represent a progress with regard
effect of the vehicle speed, and the other influencing to previous works.2,3 However, their tests were per-
factors such as the tire wear, the road surface texture, formed on surfaces composed of mosaics of coarse
and the water depth (WD) are considered in an impli- aggregates. This type of surface, which highlights
cit way (i.e., they are correlated to the parameters of mainly the effect of the microtexture, can be
friction-speed models). As a matter of fact, it is not used for an exploratory work, but results need to be
obvious to study the effect of each influencing factor confirmed by tests on real road surfaces which include
and their interaction on friction. another texture scale called macrotexture (see ‘‘Road
Research has been started at IFSTTAR for a texture scales’’ for definition). Moreover, the friction
couple of years to better understand the variation of
friction with WD. Do et al.1 developed a laboratory
1
test method (see more details in ‘‘Friction/water depth LUNAM Université, IFSTTAR, AME-EASE, Bouguenais, France
2
variation’’) and provided experimental evidence into Renault Technocentre, Guyancourt, France
3
Department Laboratoire de Lyon, CEREMA, Bron, France
the way friction varies with WD. These authors high-
lighted the effect of road surface microtexture (see Corresponding author:
‘‘Road texture scales’’ for definition) when the road V Cerezo, IFSTTAR, Route de Bouaye, CS4, 44344 Bouguenais, France.
surface evolves from dry to just wet state (which is Email: veronique.cerezo@ifsttar.fr
1286 Proc IMechE Part J: J Engineering Tribology 228(11)

coefficient in Do et al.’s work was measured by fric-

tion sliders of some square centimeters in surface,
Tire/road contact area
which is small compared with the dimensions of tire/ The tire/road contact area is usually represented as a
road contact area. As a consequence, the water intru- three-zone area (Figure 2) in which:
sion and evacuation mechanisms are not the same as
those involved on real roads (for example, in labora- . zone 1 (wheel front part) is occupied by a thick
tory tests, the water film that builds up and tends to water film (>1 mm in depth);
lift the friction sliders cannot be evacuated by the sur- . zone 2 is occupied by a thin water film (<1 mm and
face macrotexture). down to 0.01 mm in depth);
The purpose of this paper is to confirm Do . in zone 3, the water film is broken but water drop-
et al.’s laboratory results by in situ observations. lets can still be present at the top of surface aspe-
By conducting field tests with full-scale tires, it is rities. Friction forces are generated in this zone
expected that some further information in terms of (wheel rear part).
combined effects of WD, speed, and macrotexture
can be provided. In the second part of the paper, Water drainage in zone 1 depends essentially on four
attempt is made to propose another formulation of factors: the WD, the speed, the tire tread depth, and
friction models taking into account more explicitly the road surface macrotexture; the first two factors
the effect of WD. aggravate the water intrusion and the other two
help to mitigate it. Actually, WD increases the
amount of fluid needed to be drained and water
Background speed decreases the time available to drain water; as
a consequence, the contact area is eroded by the water
Road texture scales
film. In the meantime, water can be evacuated, thanks
Road surface texture is separated into two scales4 to the voids provided by the tire tread depth and the
(Figure 1): road surface macrotexture.
Even if the road surface macrotexture and the tire
. the macrotexture, defined as surface asperities that tread depth provide adequate drainage, thin water
range from 0.1 mm to 20 mm in height and from films can still be formed continuously in zone 2 and
0.5 mm to 50 mm in width. locally at the asperity tips in zone 3.5 Due to their
. the microtexture, defined as surface asperities that viscosity (induced by their low thickness), these
range from 0.001 mm to 0.5 mm in height and less water films cannot be evacuated by drainage.
than 0.5 mm in width. Delubrication of thin water films can be done only

Figure 1. Road surface texture scales.

Figure 2. Tire/road contact area.

Cerezo et al. 1287

by the road surface microtexture. Savkoor5 defined A critical WD is defined as the moment at which
two delubrication criteria: delubrication in zone 2 the friction coefficient drop represents 0.75 of the dif-
(continuous film) is only possible if (original nota- ference (0–F). Based on their experiments, these
tions) hm < , where hm is the minimum film thickness authors found that the critical WD is between
calculated on the basis of smooth profiles and  is the 0.005 mm and 0.05 mm. Field tests confirmed the
root mean square value of peak heights of the micro- exponential variation of friction with WD and pro-
texture; delubrication in zone 3 (local films on asperity vided critical depths between 0.025 mm and 0.23 mm.3
tips) is only possible if the intensity of local pres- The results from field tests are judged more reliable
sure—exerted by microtexture asperities—is high than laboratory results because the field tests are con-
enough so that the film breaks down. Moore6 con- ducted at high speeds and with full-scale tires rather
firmed Savkoor’s criteria and provided values of  than with a rubber slider.
which must range from 10 mm to 100 mm. With respect Tests performed by Veith2 on real roads varying
to the local pressure, these two authors agree that the the WD, the vehicle speed, the tire tread depth, and
asperity slope and the curvature at the asperity tip are the pavement texture showed that friction coefficient
the important parameters. Yet, they recognized that varies linearly with the logarithm of WD, meaning
the detailed action and the geometrical requirements that the variation is exponential (Figure 5). The
of microtexture are still far from clear. road surface was wetted by sprinklers and the WD
was measured by means of depth gages3 and adsorb-
ent pads of known weight (the difference between dry
Friction/WD variation and wet weights gives the volume of adsorbed water
WD is usually defined by either of the following that can be then divided by the pad surface). Veith’s
methods:2 the depth above the asperity height results also pointed out an interaction between the
(Figure 3(a)) or the total depth (Figure 3(b)). From speed and the WD: friction does not depend on the
field measurements, Veith2 showed that there is no WD at low speed (32 km/h) whereas it decreases shar-
clear relationship between above-asperity and total ply with WD at high speed (96 km/h).
WDs as their values depend on the road surface More recently, research work was conducted at
macrotexture. Care must be taken when comparing IFSTTAR to tackle the question of thin water films
friction/WD relationships published in the literature and their effect on tire traction.1 Thin-film traction is
as authors do not always define the adopted definition a relevant issue as accidents are more likely on just
of WD. wet roads than on flooded ones; Sabey et al.7 actually
Kulakowski and Harwood3 conducted laboratory said ‘‘. . . about 60% of the wet road skidding acci-
and field tests to assess the relationship between fric- dents occur. . . when the road are wet but it is not
tion and WD. The wetting protocol is not described. raining.’’ Friction laboratory tests were performed
These authors developed a device using a motorized to plot the friction/WD curve for WDs varying from
depth gage to measure WDs in the range from 0 mm to 1 mm. A spray is used to wet the surface
0.025 mm to 0.5 mm. As the gage tip needs to touch (Figure 6) and then the amount of sprayed water is
an asperity summit to evaluate the WD, this method known by weighing. Dividing the volume of water by
refers to the above-asperity depth (Figure 3(a)). the wetted area, an average WD can be calculated.
Figure 4 shows a typical result obtained in the labora-
tory. The friction coefficient decreases exponentially
from an initial value 0 to a final value F, where it
does not evolve significantly with increasing WDs.

Figure 4. Relationship between friction and water depth, and

definition of a critical water depth.2
Source: Reprinted, with permission, from Frictional interaction of
Figure 3. Definitions of water depth: (a) above-asperity tire and pavement, copyright ASTM International, 100 Barr
depth; (b) total depth. Harbor Drive, West Conshohocken, PA 19428, USA.
1288 Proc IMechE Part J: J Engineering Tribology 228(11)

Figure 5. Friction versus water depth for full and half skid depth tires.3
Source: Reprinted, with permission, from Surface characteristics of roadways: international research and technologies, copyright ASTM
International, 100 Barr Harbor Drive, West Conshohocken, PA 19428, USA.

Existing friction models

Veith2 said that the types of rubber/pavement contact
in the footprint are either hydrodynamic or boundary
layer. Hydrodynamic lubrication dominates in zone 1,
boundary layer in zone 3, and zone 2 is a mixture of
both types. On this basis, this author postulated that2

 ¼ BL  ð1  FHL Þ ð1Þ

where BL is called the boundary-layer friction coef-

ficient; and FHL is the fraction of contact area in
hydrodynamic lubrication mode.
Examples of friction models appearing under
the form expressed by equation (1) are shown
Figure 6. Water spraying to create thin water films.3
in Table 1.
Modeling of the (BL) term is a complex task
since it involves the interaction between the road sur-
face microtexture and the tire rubber.8–10 In practice,
This WD can be assimilated to the total WD defined BL is represented by a low-speed friction coefficient
in Figure 3(b). Experimental evidence (Figure 7) which is measured2,11 or estimated from measure-
showed that the variation of friction with WD is not ments performed at higher speeds;12 this is also the
always exponential as published in previous studies. reason why low-speed friction is commonly used to
On a smooth surface (specimen E0), the friction coef- characterize in an indirect manner the road surface
ficient drops as soon as the surface is wetted and microtexture.
decreases continuously. On rough surfaces (specimens Works have been published more extensively on
E1, E2, and E3), the friction coefficient remains stable the hydrodynamic term FHL. However, as it can be
(mainly on specimen E3) and then decreases for seen from Table 1 (except for the Horne and
increasing WDs. The effect of the surface microtex- Buhlmann model13), speed is the most visible param-
ture can be seen too, as specimen E3 is the roughest eter in FHL expression. A possible explanation of the
surface and its friction-WD curve is the most stable in exclusive consideration of speed is that wet road skid
the beginning and remains thereafter higher than the resistance is often considered by road engineers under
other curves. the angle of vehicle speed dependence.
Cerezo et al. 1289

In situ observation of friction/WD

variation
Test surfaces
Friction tests were performed on the IFSTTAR test
track. Test sections are 3–3.5 m in width and 50–200 m
in length (Figure 8(a)). Test surfaces include wearing
courses representative of French road network
(asphalt concrete, surface dressing, etc.) and special
surfaces (epoxy, painted surfaces, etc.). The longitu-
dinal slope of the test tracks is null and the crossfall
value is around 1.5%. Surfaces can be wetted by
means of sprinklers (Figure 8(b)).
Asphalt concrete surfaces and painted surfaces
were tested. Their characteristics are given in
Table 2. As there is currently no consensus about
the direct measurement of microtexture (devices
Figure 7. Friction-water depth curves for different surface exist for measurements in the laboratory; however,
microtextures (E0: smooth; E1 to E3 are sandblasted with, their characteristics and their cost limit large-scale
respectively, increasing number of sandblasting cycles).3 deployments and on-road measurements), indirect
measurements using low-speed friction coefficients
are still used to characterize this texture scale.14
In this study, the microtexture is characterized by a
Table 1. Friction models appearing under the form expressed theoretical longitudinal friction coefficient at speed
by equation (1). zero symbolized by LFC0. The approach used by
Leu and Henry12 was adopted to determine LFC0.
Authors Model
These authors used the regression curve—exponential
12 V
Leu and Henry  ¼ 0 e V0 function—adjusted on friction coefficients measured at
where 0 is a theoretical friction different speeds and considered its intersection with the
coefficient at speed zero; V is the friction axis (speed equal to zero); the intersection
speed; and V0 is a constant related point gives LFC0. In the present study, as there are
to the road surface macrotexture. only two test speeds (60 km/h and 90 km/h, see section
Mancosu et al.11  ¼ ref  b0 þ b1
ðb3 þb4 V Þ ‘‘Friction measuring devices’’), the regression curve is
1þb2 e
simply replaced by a line linking the two experimental
where ref is a friction coefficient
points. Values of friction coefficients obtained at WD
measured under specific conditions;
V is the speed; and bi (i ¼ 1–4) are
of 0.1 mm were used to be as close as possible to a dry
constants to state; values of LFC0 are reported in Table 3.
h be adjusted. i
Horne and Buhlmann13  ¼ dry  1  Cmac pp1 þ Cmic pp2
The macrotexture is characterized by a normal-
ized parameter called mean profile depth (MPD)4
where dry is a friction coefficient (Figure 9). An elementary profile section defined
measured at very low speed on a by the profile baseline is divided into two halves
damp surface; p is the tire inflation of equal length. For each half, the maximum profile
pressure; p1 (resp. p2) is the fluid height is determined giving rise, respectively, to the
pressure generated under zone 1 first and second peak levels. The MPD, expressed in
(resp. zone 2) of the tire/road con- millimeters, is determined from road profiles by the
tact area; Cmac (resp. Cmic) expresses following4
the water drainage provided by road
surface macrotexture (resp. micro-
MPD ¼ mean ð1st peak level, 2nd peak levelÞ
texture), value 0 (resp. 1) meaning
perfect drainage (resp. no drainage).  average level
ð2Þ

It is obvious that a high MPD means large voids,

Despite the fact that water drainage depends on and so a high macrotexture.
four factors (see ‘‘Tire/road contact area’’), only
speed is explicit in existing tire/road friction models.
There is then a need to improve the model formula-
Friction measuring devices
tion to incorporate the three other factors as expli- The ADHERA friction monitoring device was used
citly—in the FHL term—as possible. for measurements on the test track. The machine
1290 Proc IMechE Part J: J Engineering Tribology 228(11)

Figure 8. IFSTTAR test track: (a) top view (test sections are on the straight part); (b) surface wetting by sprinklers.

(Figure 10(a)) measures locked-wheel (full sliding) (0.1 and 0.25 mm) before the friction coefficient
longitudinal friction coefficient (LFC) on wet sur- collapses for WDs above 0.5 mm. Surfaces E2
faces.15 Full-scale standardized smooth tire is used and M2 show no abrupt transition in the variation.
for the measurement. LFC is defined as the ratio . in terms of friction level, surface C is well above
between the horizontal force (Fh) due to friction in surfaces E2 and M2 for WDs of 0.1 mm and
the tire/road contact and the vertical load (Fv) exerted 0.25 mm then, above 0.5 mm of WD, surface C is
on the measuring wheel (2500 N) (Figure 10(b)). below the two others.
Surfaces are wetted by a nozzle connected to a . in terms of friction decrease rate, one can observe
water tank located in the vehicle and the WD is regu- an increasing difference between the curves of sur-
lated by a water pump (to adapt the water flow to the faces M2, E2, and C as the WD increases. More
vehicle speed). A calibration procedure helps to deter- precisely, for WDs above 0.5 mm, the curve of sur-
mine the relationship between water flow and WD, face M2 remains stable whereas the curves of E2
assuming that the road surface is smooth. As the and C surfaces continue to decrease, the most dras-
WD is determined from a ratio between water tic friction reduction being observed for surface C
volume and wetted surface, it corresponds to a total (friction drops from 0.8 to 0.2 for WD increasing
depth (Figure 3(b)). Tested WDs are 0.1, 0.25, 0.5, from 0.1 mm to 1.5 mm).
0.75, 1, and 1.50 mm. The operator leaves at least
5 min between two successive measurements to avoid It can be seen that the behavior of surfaces C, E2, and
water accumulation on the test surfaces. M2 changes, mainly for surface C, at WDs around
LFC measurements were performed at 60 and 0.5 mm. If one looks at LFC0 and MPD values of
90 km/h and symbolized, respectively, LFC60 and the three surfaces, it can be said that micro- and
LFC90. Data are recorded on 20 m in length, where macrotexture affect these behaviors:
the test wheel is locked. The wetting process begins at
least 10 m before the vehicle reaches the test section . below 0.5 mm of WD, the ranking between the
and ends 10 m after the test section. This protocol three surfaces (C > M2 > E2) reflects that of
helps to provide a water film as homogeneous and LFC0 (respectively 0.82, 0.55, and 0.5).
constant as possible under the measuring tire during . above 0.5 mm of WD, the ranking is M2 > E2 > C
the measurements. which corresponds to that of MPD (respectively
1.12, 0.72, and 0.48 mm).
Results The effect of surface microtexture is further confirmed
Typical plots of LFC versus WD are presented in in Figure 12. Surfaces G1 and G2 are made from the
Figure 11. For the three surfaces, the friction coeffi- same asphalt formulation. They are painted after the
cient decreases with increasing WDs. Nevertheless, construction using two types of paint: G1 is covered
some differences can be noticed between the three by a paint used for pedestrian crosswalks without
surfaces: any inclusion (Figure 13(a)) and G2 is covered by
the same paint including micro-glass spheres
. in terms of shape, surface C exhibits high and (Figure 13(b)). As it can be seen in Table 3, G1 micro-
stable friction coefficients at the first two WDs texture is lower than G2 microtexture (LFC0 ¼ 0.31
Cerezo et al. 1291

Table 2. Characteristics of test surfaces.

Section Mix type Photo MPD (mm) LFC at 40 km/h

C Surface dressing 0.8/1.5 0.48 0.69

E2 Semi coarse asphalt 0.72 0.48

concrete 0/10 ‘‘new’’

M2 Semi coarse asphalt 1.12 0.48

concrete 0/10 ‘‘old’’

G1 Asphalt concrete covered by paint 0.49 0.25

G2 Asphalt concrete covered by paint 0.49 0.64

including glass spheres

Table 3. Theoretical longitudinal friction and 0.53 respectively). As a matter of fact, the friction
coefficient at speed 0. coefficient on G2 remains stable until 0.2 mm of WD
is reached, whereas the friction coefficient on G1
Surface LFC0
drops rapidly.
C 0.82 More data points would be needed to draw more
E1 0.59 conclusive observations about the effect of micro- and
M2 0.55 macrotexture, for example, the effect of micro-
E2 0.50 texture on the quick drop in friction coefficient at
A 0.45 low WDs as observed by Do et al.1 However, it was
not feasible to control the water flow of the measuring
1292 Proc IMechE Part J: J Engineering Tribology 228(11)

Figure 9. Definition of road surface mean profile depth.

Figure 10. ADHERA device: (a) vehicle; (b) trailer and measuring wheel.

Figure 12. LFC versus water depth for painted surfaces

(same macrotexture, different microtextures).

Figure 11. LFC versus water depth for asphalt surfaces. effect is negligible for surface M2, whereas it becomes
significant for surface C as the WD increases (the
device to obtain more WD levels (for example, every curve at 60 km/h is well above that at 90 km/h). The
0.05 mm). difference between the two surfaces can be again
The combined effects of WD, speed, and macrotex- attributed to their macrotexture: the MPD of surface
ture can be seen from graphs in Figure 14. The speed M2 is actually higher than that of surface C (Table 2).
Cerezo et al. 1293

Figure 13. Painted surfaces: (a) G1 pure paint; (b) G2 paint including glass spheres.

the friction coefficient remains stable and close to that

obtained on a dry surface, and then, once the WD
reaches a ‘‘critical’’ value, the friction coefficient
decreases. The occurrence of either variation depends
on the microtexture of the road surface: a high-micro-
texture surface tends to have a two-phase variation,
whereas a low-microtexture surface tends to have an
exponential variation.
In situ observations also bring out other aspects
that could not be observed in the laboratory. They
clearly show the effects of road surface macro- and
microtexture on the friction-WD variation: the micro-
texture controls the beginning of the friction-WD
curve (where the road surface evolves from dry to
just wet), whereas the macrotexture controls the rate
at which friction decreases with WD, mainly at high
WDs. As explained in ‘‘Tire/road contact area,’’ the
macrotexture is associated to thick WDs (evacuation
Figure 14. LFC versus water depth at different speeds. by drainage) and the microtexture is associated to thin
WDs (evacuation by squeezing, when drainage is no
more possible); the observations made in this study
The speed effect on lubrication at the tire/road inter- reflect completely these associations. It can also be
face is well known (speed decreases the time available noticed that the degree of slipperiness of a surface
to drain water) (see section ‘‘Tire/road contact area’’ can change drastically depending on its micro- and
and Figure 2). This effect, unfavorable for driver macrotexture: actually, surface C appears to be safe
safety, can be either accentuated on low-macrotexture at low WD (high friction) but exhibits a drastic fric-
surface (C) or compensated by a high-macrotexture tion reduction at high WD; on the contrary, the fric-
surface (M2). tion coefficient of surface M2 is low but remains
It could be noticed that no analysis is done regard- stable.
ing the effect of tire tread depth. It is due to the Field tests also provide further information in
friction-measuring device used for the study which terms of combined effects of WD, speed, and macro-
operates with blank tires. Even if the use of this texture. These effects could not be assessed in the
type of tire is guided historically by the need to per- laboratory as, even if some laboratory devices can
form skid resistance measurements in worse driving perform friction tests at different speeds, the contact
conditions, patterned tires would be more relevant area between the friction sliders and the test surface is
as they correspond to daily used tires. It is expected small (few square centimeters) compared with the tire/
that the effects of WD and speed discussed above will road contact patch. As a consequence, the water film
be mitigated by the presence of tire patterns; future that builds up and tends to lift the friction sliders are
tests will help to confirm this assumption. not evacuated by the surface macrotexture as would
be the case on real roads. In situ observations con-
tribute in this way to better identify the influencing
Discussions
factors.
Graphs in Figures 11, 12, and 14 confirm results Compared with previous studies dealing also with
obtained in the laboratory by Do et al.1 Actually, field tests,2 the tests conducted in this study help to
the friction/WD variation is not always exponential; better characterize the friction/WD relationship.
it comprises—at least—two phases: a first stage where Indeed, instead of observing the classical exponential
1294 Proc IMechE Part J: J Engineering Tribology 228(11)

Figure 15. Influence of starvation on the Stribeck curve.15

decrease of friction with WD, the present study profile root mean square lies between 6 mm and 18 mm.
showed another behavior which can be assimilated, Using the minimum estimated water thickness and the
in the spirit, to the well-known Stribeck curve. maximum root mean square, the ratio of lubricant
Can the curves obtained in this study be considered thickness/surface roughness is about 1:1.
as Stribeck curves? Stribeck curves are usually plotted As test conditions in Beautru’s work are similar to
with the V/p term on the X-axis, where  is the lubri- those in the present study (locked-wheel friction meas-
cant viscosity, V is the relative sliding speed, and p is urements on road surfaces), the calculation above
the average contact pressure. Nonetheless, Schipper16 indicates that the quantity of poured water would be
said that there is not much difference between the use high enough to activate hydrodynamic actions in the
of the lubricant thickness (i.e., the WD in the present tire/road contact area. Back to the curves in Figures
study) and the V/p term, as these two parameters are 11, 12, and 14, one can then say that the first stage of
strongly related. The curves presented in Figures 11, the friction/WD variation corresponds to a boundary
12, and 14 can be then treated as Stribeck curves. lubrication regime and that the critical WD lies at the
With respect to the nature of the lubrication, as transition between the boundary and mixed lubrica-
only a limited quantity of water is poured in front of tion regimes, where the friction coefficient starts to
the measuring tire, it can be said that the friction tests decrease with increasing WDs.
conducted in the present study are starved lubricated The concept of critical WD developed in this paper
tests; Savkoor5 also said that, once the road surface is different from that defined by Kulakowski and
macrotexture and the tire grooves have eased the flow Harwood.3 These authors defined a WD as critical
of water, the remaining water part is considered as when the friction lost represents 75% of the total
‘‘starved.’’ For starved contacts, as shown in available (see ‘‘Friction/water depth variation’’ for
Figure 15,17 the shape of the Stribeck curve depends detailed definition). In this paper, the critical WD is
on the ratio between the lubricant thickness and the defined as the transition between boundary and mixed
surface roughness. If the amount of supplied lubricant lubrication regimes; in terms of friction variation, it
is high enough (ratios of lubricant thickness/surface represents the transition from a stable stage to a
roughness above 1.4 in Figure 15), hydrodynamic sudden drop. With respect to driver safety, the sur-
action can take place. If the amount of supplied lubri- prise caused by a sudden drop of available friction can
cant is low (ratio of lubricant thickness/surface rough- be fatal; the critical depth defined in this study seems
ness of 0.6 in Figure 15), only boundary lubrication then to be more relevant. It is the hope of this study to
takes place, i.e., the Stribeck curve transforms into a identify new parameters that could help to better
horizontal friction line. determine the moment at which tire traction is no
In a recent work,18 attempt was made to estimate longer enough to maintain the trajectory of the
the thickness of the water film trapped between the vehicle.
tire and the road surface asperities. Using a model
developed by Moore,19 Beautru18 found that the
trapped-water thickness lies between 20 mm and
Towards a comprehensive friction model
60 mm. This author also determined the roughness of As it was stated in section ‘‘Existing friction models,’’
the road surfaces tested in his work and found that the most of current friction models display only the effect
Cerezo et al. 1295

of vehicle speed, the other influencing factors being

considered implicitly. Based on the new experimental
evidences presented in this study, it is now tempting to
develop a friction model where the effect of factors
such as WD and road surface macrotexture is more
explicit, so users can clearly see how each factor
affects friction and how the influencing factors
interact.
In the following sections, we try first to develop a
model relating the friction coefficient to the WD.
Then, we try to see how this new model is compar-
able—in terms of formulation—to friction-speed
models. Finally, a new formulation is proposed to
include the effects of WD, speed, and macrotexture.
As mentioned in ‘‘Results’’ section, the conducted test
program does not allow study the effect of tire tread
depth; it is the reason why this effect is not included in
the proposed model for the moment. Figure 16. Comparison between model 1 (equation (5)) and
measured friction (same data as Figure 11).

Model describing the dependence of

Looking at Leu and Henry model12 in Table 1, it
friction on WD
can be said that the FHL term of their model is
Based on their laboratory results, Do et al.1 proposed h i
the following model VV
FHL ðVÞ ¼ 1  e 0
ð5Þ
h   i
WD
 WD0
 ¼ ð 0   F Þ  e þ F ð3Þ where FHL(V) is the hydrodynamic term expressing
the speed effect.
where F is the final friction coefficient, 0 is the fric- Based on a very exhaustive friction test campaign
tion coefficient at WD ¼ 0; and WD0,  are constants. covering a large range of road surfaces, tires, speeds,
These authors observed that their model is a gen- and WDs,20 Do et al.21 found that the Leu and Henry
eralization of the model developed by Kulakowski model can be generalized to better fit the experimental
and Harwood3 data expressed in terms of friction-speed curve (some
curves actually presented a shape similar to Stribeck
 ¼ ð0  F Þ  e½WD þ F ð4Þ curve and cannot be fitted by an exponential func-
tion). These authors proposed to add simply an expo-
where  is a constant. nent to the term V/V0, meaning that
Actually, equation (4) is a special case of equa-    
tion (3) (for  ¼ 1 and WD0 ¼ 1/). Fitting of equa- V

tion (5) to experimental data of the present study FHL ðVÞ ¼ 1  e
V0
ð6Þ
(Figure 16) shows that this model still works on in
situ observations. However, the model presents the where  is a constant.
same drawback as previous models (Table 1): it The following assumptions are then made to derive
expresses explicitly only the effect of a single factor. the FHL terms for the WD and the macrotexture:
There is then a need to formulate a more general
model. . if two factors act in the same way, in terms of water
intrusion or drainage, the respective FHL terms
should have the same form;
Formulation taking into account the combined . if two factors act in an opposite way, the FHL term
of one factor has the form of (1  FHL) of the other
effect of speed, macrotexture, and WD factor.
As stated in ‘‘Existing friction models,’’ most friction
models can be presented under the form2:  ¼ BL.(1– These assumptions mean that (as the WD acts as the
FHL) (equation (1)). We will assume, as it was also speed)
said in ‘‘Existing friction models,’’ that BL is repre-
sented by a low-speed friction coefficient. The discus- h   i
sion which follows is then focused on the formulation  WD
WD0
of FHL term. FHL ðWDÞ ¼ 1  e ð7Þ
1296 Proc IMechE Part J: J Engineering Tribology 228(11)

where FHL(WD) is the hydrodynamic term expressing

the water effect; and WD0,  are constants.
And (as the macrotexture and the speed act in an
opposite way)
h   i
MPD
 MPD0
FHL ðMPDÞ ¼ e ð8Þ

where FHL(MPD) is the hydrodynamic term express-

ing the macrotexture effect; and MPD0,  are
constants.
It is quite interesting to notice that if F ¼ 0 in
equation (3), the FHL term that could be deduced
from this equation would have the same form as equa-
tion (7).
Back to equation (1), we assume further that FHL is
a product of elementary FHL terms expressing,
respectively, the effects of, on the one hand, WD Figure 17. Comparison between model 2 (equation (11)) and
and speed, and, on the other hand, tire tread depth measured friction (same data as Figure 11).
and macrotexture, meaning that

FHL ¼ FHL ðWDÞ  FHL ðVÞ  FHL ðtire tread depthÞ adjustment is done once for all surfaces and all
 FHL ðMPDÞ ð9Þ speeds, whereas the old model (equation (2)) must
be adjusted for each surface and each speed. Even
A product is used as it expresses the fact that the though the new model requires more parameters (six
four influencing parameters act simultaneously. in total) compared with the old model (four in total),
Using equations (6) to (8), equation (9) can now be the number of data points (72 ¼ 6 surfaces  2
written as speeds  6 WDs) used for adjustment is high enough
h   i !  h   i !
to avoid over-fitting. Correlation between measured
WD   MPD and calculated friction coefficients is fairly good
 V  
1  e ð Þ
WD0 MPD0
FHL ¼ 1  e V0 e (r2 ¼ 0.79), proving the relevance of the new model.
The results for surfaces C, E2, and M2 are shown
ð10Þ in Figure 17. It can be seen that the model fits remark-
ably well surfaces E2 and M2, and the beginning of
The FHL (tire tread depth) is not missing in equa- surface C. The model overestimates friction values
tion (11). As blank tires were used in the present obtained on surface C for WDs above 0.5 mm.
study (the tire tread depth is equal to zero), and that Compared with the model expressed by equation
FHL(tire tread depth) can be expressed as FHL(MPD), (4), which was developed expressly for Do et al.’s1
FHL(0) ¼ 1 laboratory results, the new model provides similar
Equation (10) is represented under a widely recog- results on surfaces E2 and M2, and fitting is less suc-
nizable form and it should help to better understand cessful on surface C at high WDs; yet, both models
and quantify the interaction between the influencing overestimate the friction value at 1.5 mm of WD.
factors. Despite this weakness, the new model is promising
because its formulation helps to see clearly the effect
of different influencing factors while previous models
Fitting of the model to experimental data
are dedicated to a singe factor—speed or WD—at the
The new friction model can be written as same time. Further efforts are obviously needed to
better consider low-macrotexture surfaces (like C) at
 ¼ BL  ð1  FHL Þ ð11Þ high WDs.

where FHL is expressed by equation (10).

As it was mentioned in ‘‘Existing friction models’’,
Conclusions
BL is not easy to obtain and is represented in practice The purpose of this paper is to provide new experi-
by a low-speed friction coefficient; in this study, it is mental evidences about the friction/WD relationship
assumed12 that BL ¼ LFC0 (see ‘‘Test surfaces’’ for and to improve the formulation of friction models in
definition). terms of consideration of the effect of WD.
Fitting of the new friction model to experimental Tests were conducted on test tracks including sur-
require adjustment of six parameters (, , , WD0, faces representative of French wearing courses and
V0, and MPD0). It should be noted that the special surfaces such as epoxy or painted surfaces.
Cerezo et al. 1297

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None declared. University of Twente, the Netherlands, 2005.
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