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Theory of rubber friction and contact mechanics

10 Aug 2001-Journal of Chemical Physics (American Institute of Physics)-Vol. 115, Iss: 8, pp 3840-3861
TL;DR: In this article, the authors consider the case when the substrate surface has a self affine fractal structure and present a theory for the area of real contact, both for stationary and sliding bodies, with elastic or elastoplastic properties.
Abstract: When rubber slides on a hard, rough substrate, the surface asperities of the substrate exert oscillating forces on the rubber surface leading to energy “dissipation” via the internal friction of the rubber. I present a discussion of how the resulting friction force depends on the nature of the substrate surface roughness and on the sliding velocity. I consider in detail the case when the substrate surface has a self affine fractal structure. I also present a theory for the area of real contact, both for stationary and sliding bodies, with elastic or elastoplastic properties. The theoretical results are in good agreement with experimental observation.

Summary (3 min read)

Introduction

  • Theory of rubber friction and contact mechanics B. N. J. Persson Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany ~Received 12 March 2001; accepted 7 June 2001!.
  • The nature of the friction when rubber slides on a hard substrate is a topic of considerable practical importance, e.g., for the construction of tires,1 wiper blades,1 and in the cosmetic industry.
  • Surface roughness, but explicit results are presented for self affine fractal surface profiles.
  • It is clear that even if the rubber is able to make direct contact with the substrate in the large cavities, the pressure acting on the rubber at the bottom of a large cavity will be much smaller than the pressure at the top of a large asperity.
  • Note that the longest roughness wavelength possible are of order ;L .

II. SELF-AFFINE FRACTAL SURFACES AND CONTACT THEORIES

  • It has been found that many ‘‘natural’’ surfaces, e.g., surfaces of many materials generated by fracture, can be approximately described as self-affine surfaces over a rather Downloaded 21 Dec 2006 to 134.94.122.39.
  • A self-affine fractal surface has the property that if the authors make a scale change that is different for each direction, then the surface does not change its morphology,10,11 see Fig.
  • Since H,1 ~H51 correspond to D f52!, these theories predict that the area of real contact increases faster than linear with the load.
  • Roberts20 has shown that polar substances like soaps prevent direct contact between track and rubber ~see Sec. VII!; this explain why the friction is slightly lower for the wet 15% detergent case, compared to the wet, clean carborundum surface.

III. AREA OF REAL CONTACT: QUALITATIVE DISCUSSION

  • I have already emphasized the importance of knowing the nature of the area of real contact when discussing rubber friction.
  • This is at least one order of magnitude smaller than the ~static or lowfrequency!.
  • Thus, the deformations induced by the largest asperities are relatively well described by using the low frequency elastic modulus E(v)'E(0).
  • In Sec. V and Appendices B and C, I develop a new contact theory for surfaces with roughness on many different length scales.

IV. SLIDING FRICTION

  • Using the theory of elasticity ~assuming an isotropic elastic medium for simplicity!, one can calculate the displacement field ui on the surface z50 in response to the surface stress distributions s i5s3i .
  • Note that, in accordance with the discussion in Sec.
  • The friction coefficient m can be obtained by dividing the frictional shear stress ~21! with the pressure s0 , m5 1 2 E d2q q2 cos f C~q !.
  • This has a simple but important physical origin: Consider two cosine-surface corrugations, where the ‘‘wave vector’’ points ~a!.

V. CONTACT THEORY FOR RANDOMLY ROUGH SURFACES

  • If A0 denotes the nominal contact area, the load FN5s0A0 .
  • Now, assume that the macroscopic pressure s0 depends on the lateral position x in the nominal contact region, as would be the case if, e.g., a rubber ball is squeezed against a nominally flat substrate @where s0(x) is given by the Hertz expression#.
  • Elastic modulus E , the rubber friction coefficient is ~nearly!.
  • Numerical evaluation of Eq. ~31! shows that Eq. ~34! is an accurate representation of P(q) for all q ~or, equivalently, all G!.
  • This will, of course, not occur in real systems, where there always exist an upper cutoff zmax5q1 /q0 in the integral over z.

VI. NUMERICAL RESULTS

  • This model is, in fact, not a very good description of real rubbers, since the transition with increasing frequency from the rubbery region to the glassy region is much too abrupt, leading to a much too narrow ~and too high! m(v) peak.
  • Later the authors will use experimental data for E(v) for two different rubbers, illustrating how the results based on the present model ~Fig. 10! are quantitatively modified.
  • When s0 becomes of order E(0) the friction coefficient is no longer independent of s0 , but decreases with increasing s0 ~see 10 MPa curve in Fig. 16!.
  • This nonlinearity is associated with the breakdown of the filler network, which occurs in the range of a few % strain amplitude.
  • Is about half as large as when the calculation is based on the 8% stain amplitude data.

VII. DISCUSSION

  • The theory developed above can be used to estimate the kinetic friction coefficient for rubber sliding on a rough hard substrate.
  • In a typical case an equilibrium film of liquid some 200 Å thick becomes established between the surfaces.
  • The generation of repulsive forces between rubber and glass surfaces means that the pair will make a microconforming contact with a uniform thin film of liquid between them ~see Fig. 18!.
  • For this reason it is plausible that a liquidlike water layer may exist at the rubber–ice interface but not when ice is in contact with a hard, high energy solid surface, such as glass or metal oxides.
  • 38 Finally, for practical applications it is necessary to study the heating of the rubber during sliding.

VIII. SUMMARY AND CONCLUSION

  • There is at present a strong drive by tire companies to design new rubber compounds with lower rolling resistance, higher sliding friction, and reduced wear.
  • At present these attempts are mainly based on a few empirical rules and on very costly trial-and-error procedures.
  • I have shown that for stationary surfaces ~or low sliding velocity!, and for typical pressures in the contact area between a tire and a road, the rubber will only make ~apparent!.
  • Contact with about 5% of the road surface.
  • I have developed a contact theory which describes how the ~apparent!.

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Theory of rubber friction and contact mechanics
B. N. J. Persson
Institut fu
¨
r Festko
¨
rperforschung, Forschungszentrum Ju
¨
lich, D-52425 Ju
¨
lich, Germany
Received 12 March 2001; accepted 7 June 2001
When rubber slides on a hard, rough substrate, the surface asperities of the substrate exert oscillating
forces on the rubber surface leading to energy ‘dissipation’ via the internal friction of the rubber.
I present a discussion of how the resulting friction force depends on the nature of the substrate
surface roughness and on the sliding velocity. I consider in detail the case when the substrate surface
has a self affine fractal structure. I also present a theory for the area of real contact, both for
stationary and sliding bodies, with elastic or elastoplastic properties. The theoretical results are in
good agreement with experimental observation. © 2001 American Institute of Physics.
DOI: 10.1063/1.1388626
I. INTRODUCTION
The nature of the friction when rubber slides on a hard
substrate is a topic of considerable practical importance, e.g.,
for the construction of tires,
1
wiper blades,
1
and in the cos-
metic industry. Rubber friction differs in many ways from
the frictional properties of most other solids. The reason for
this is the very low elastic modulus of rubber and the high
internal friction exhibited by rubber over a wide frequency
region.
The pioneering studies of Grosch
2
have shown that rub-
ber friction in many cases is directly related to the internal
friction of the rubber. Thus experiments with rubber surfaces
sliding on silicon carbide paper and glass surfaces give fric-
tion coefficients with the same temperature dependence as
that of the complex elastic modulus E(
) of the rubber. In
particular, there is a marked change in friction at high speeds
and low temperatures, where the rubbers response is driven
into the so-called glassy region. In this region, the friction
shows marked stick-slip and falls to a level of
0.4, which
is more characteristic of plastics. This proves that the friction
force under most normal circumstances is directly related to
the internal friction of the rubber, i.e., it is mainly a bulk
property of the rubber.
2
The friction force between rubber and a rough hard
surface has two contributions commonly described as the
adhesion and hysteretic components, respectively.
1
The hys-
teretic component results from the internal friction of the
rubber: during sliding the asperities of the rough substrate
exert oscillating forces on the rubber surface, leading to cy-
clic deformations of the rubber, and to energy ‘dissipation’
via the internal damping of the rubber. This contribution to
the friction force will therefore have the same temperature
dependence as that of the elastic modulus E(
) a bulk prop-
erty. The adhesion component is important only for clean
and relative smooth surfaces.
Because of its low elastic modulus, rubber often exhibit
elastic instabilities during sliding. The most well-known in-
volves the compressed rubber surface in front of the contact
area undergoing a buckling which produces detachment
waves which propagate from the front-end to the back-end of
the contact area. These so called Schallamach waves
3
occur
mainly at ‘high’ sliding velocity and for very smooth sur-
faces, but will not be considered further in this paper.
In three earlier papers we have studied both the adhesion
and hysteretic components of rubber friction.
4–6
Other stud-
ies of this topic are presented in Refs. 1, 79; reference 4
considered only the interaction between a flat rubber surface
and a single surface asperity or many identical asperities.In
Ref. 6 we studied the hysteretic contribution to the friction
for viscoelastic solids sliding on hard substrates with differ-
ent types of idealized surface roughness.
In this paper I develop a theory of rubber friction when a
rubber block is slid over a hard rough surface, with rough-
ness on many different length scales . The theory is valid
for arbitrary random surface roughness, but explicit results
are presented for self affine fractal surface profiles.
10,11
Such
surfaces ‘looks the same’ when magnified by a scaling fac-
tor
in the xy-plane of the surface and by a factor
H
where
0H 1 in the perpendicular z-direction. I note that many
materials of practical importance have approximately self-
affine fractal surfaces. Thus, for example, road surfaces and
the surfaces of many cleaved brittle materials tend to be self
affine fractal with the fractal dimension D
f
3 H2.2
2.5. In practice there is always a lower,
1
, and upper,
0
,
cutoff length, so that the surface is self-affine fractal only
when viewed in a finite length scale interval
1
0
. For
surfaces produced by brittle fracture, the upper cut off length
0
is usually identical to the lateral size L of the fracture
surface. This seems also to be the case for many surfaces of
engineering importance see, e.g., Ref. 14. However, for
road surfaces the upper cutoff
0
is of order a few mm,
which corresponds to the size of the largest sand particles in
the asphalt. Less is known about the short distance cutoff
1
,
but I will argue later that in the context of rubber friction it
may be taken to be of order a few
m, so that the length
scale region over which the road surface may be assumed to
be fractal may extend over 3 orders of magnitude.
When rubber slides on a hard rough surface with rough-
ness on the length scales , it will be exposed to fluctuating
forces with frequencies
v
/. Since we have a wide dis-
tribution of length scales
1
0
, we will have a corre-
JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 8 22 AUGUST 2001
38400021-9606/2001/115(8)/3840/22/$18.00 © 2001 American Institute of Physics
Downloaded 21 Dec 2006 to 134.94.122.39. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

sponding wide distribution of frequency components in the
Fourier decomposition of the surface stresses acting on the
sliding rubber block. The contribution to the friction coeffi-
cient
from surface roughness on the length scale , will be
maximal when
v
/␭⬇1/
, where 1/
is the frequency where
Im E(
)/
E(
)
is maximal, which is located in the transition
region between the rubbery region low frequencies and the
glassy region high frequencies. We can interpret 1/
as a
characteristic rate of flips of molecular segments configura-
tional changes, which are responsible for the visco-elastic
properties of the rubber. Since the flipping is a thermally
activated process it follows that
depends exponentially or
faster on the temperature
exp(E/k
B
T), where E is the
barrier involved in the transition. In reality, there is a wide
distribution of barrier heights E and hence of relaxation
times
, and the transition from the rubbery region to the
glassy region is very wide, typically extending over 3 orders
of magnitude in frequency.
The following observation is of great importance for
rubber friction. Consider the contribution to the rubber fric-
tion from surface roughness of different wavelength and
amplitude h, see Fig. 1. If we assume that the applied pres-
sure is so high that the rubber is squeezed into complete
contact with the substrate, it follows from dimensional argu-
ments that the magnitude of the hysteretic contribution to the
friction coefficient only depends on h/, i.e., surface rough-
ness of different length scale contribute equally to the fric-
tion force if the ratio between the amplitude and wavelength
is constant. Thus, roughly speaking, we may state that sur-
face roughness of all length scales are equally important. Of
course, the different wavelength contributions to
(
v
) will
peak at different sliding velocities determined by
v
/␭⬇1/
,
i.e., the different wavelength contributions to
(
v
) are
shifted relative to each other along the
v
-axis, see Fig. 1c.
We may summarize these results by writing
f(
v
/,h/).
These profound results imply that it is very important
not to a priori exclude any roughness length scale from the
analysis. The distribution of different length scales will
broaden the
(
v
) curve, and also increase the peak maxi-
mum. However, let us note the following: Consider a surface
with surface roughness on two different length scales as in-
dicated in Fig. 2. Assume that a rubber block is squeezed
against the substrate and that the applied pressure is large
enough to squeeze the rubber into the large ‘cavities’ as
indicated in the figure. It is clear that even if the rubber is
able to make direct contact with the substrate in the large
cavities, the pressure acting on the rubber at the bottom of a
large cavity will be much smaller than the pressure at the top
of a large asperity. Thus while, because of the high local
pressure, the rubber may be squeezed into the ‘small’ cavi-
ties at the top of a large asperity, the pressure at the bottom
of a large cavity may be too small to squeeze the rubber into
the small-sized cavities at the bottom of a large cavity.
Hence, during sliding the small-scale roughness may give a
contribution to the pulsating deformations of the rubber and
hence to the friction force, only at the top of the big asperi-
ties. This important fact is taken into account in the analysis
presented in this paper. Thus, if A() is the apparent area
of contact on the length scale ␭关more accurately, I define
A() to be the area of real contact if the surface would be
smooth on all length scales shorter than , see Fig. 3, then I
will study the function P(
) A()/A(L) which is the rela-
tive fraction of the rubber surface area where contact occurs
on the length scale L/
where
1, with P(1) 1.
Here A(L) A
0
denotes the macroscopic contact area L is
the diameter of the macroscopic contact area so that A(L)
L
2
. I will show that for an ideal elastic body no plastic-
ity squeezed against a rigid self affine fractal surface with-
out a short-distance cut off, P(
) 0as
. This result is
FIG. 1. Rubber dotted area sliding on a hard corrugated substrate. The
magnitude of the contribution to the friction from the internal damping in
the rubber is the same in a and b because the ratio between the amplitude
and the wavelength of the corrugation is the same. c shows the
(
v
)
curves for the roughness profiles in a and b兲共schematic.
FIG. 2. Rubber sliding on a substrate with roughness on two different length
scales. The rubber is able to fill-out the long-wavelength roughness profile,
but it is not able to get squeezed into the small-sized ‘cavities’ at the
bottom of a big cavity.
3841J. Chem. Phys., Vol. 115, No. 8, 22 August 2001 Theory of rubber friction and contact mechanics
Downloaded 21 Dec 2006 to 134.94.122.39. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

very important, because it shows that even without any short
distance cut off it is possible for the sliding friction force to
remain finite, since the rubber will not make contact with or
experience the very short-wavelength surface roughness.
Note that the longest roughness wavelength possible are of
order L. This correspond to the wave vector q
L
2
/L.If
we define q q
L
, we can consider P(
) P(q/q
L
)asa
function of q; I denote this function by P(q) for simplicity.
This paper focuses mainly on rubber friction, but, as
indicated above, I also present a new theory of contact me-
chanics see Appendices B and C, valid for randomly rough
e.g., self-affine fractal surfaces. In the context of rubber
friction, mainly elastic deformation will occur in the
substraterubber contact areas. However, the contact theory
developed in this paper can also be applied when both elastic
and plastic deformation occur in the contact areas. This case
is, of course, relevant to almost all materials other than rub-
ber.
This paper is organized as follows: In Secs. II and III, I
present some basic results related to self-affine fractal sur-
faces and contact theories, which form a necessary back-
ground for the theory developed in Secs. IV and V. In Sec.
IV, I derive a general formula for the hysteretic contribution
to rubber friction. This formula contains the function P(
)
introduced above, which is derived in Sec. V and Appendix
B for randomly rough e.g., self-affine fractal surfaces. Sec-
tion VI contains numerical results for the velocity dependent
friction coefficient. Section VII presents some general com-
ments about rubber friction, and Sec. VIII is the summary
and conclusion. In Appendix C, I present a new contact me-
chanics theory for randomly rough surfaces, when both elas-
tic and plastic deformation occurs in the contact areas. In
Appendix E, I study the contribution to rubber friction from
the emission of elastic waves from the sliding interface.
II. SELF-AFFINE FRACTAL SURFACES AND
CONTACT THEORIES
It has been found that many ‘natural’ surfaces, e.g.,
surfaces of many materials generated by fracture, can be ap-
proximately described as self-affine surfaces over a rather
wide roughness size region. A self-affine fractal surface has
the property that if we make a scale change that is different
for each direction, then the surface does not change its
morphology,
10,11
see Fig. 4. Thus, the statistical properties of
the surface are invariant under the scaling transformation,
x
x, y
y, z
H
z, 1
where the exponent H can be related to the fractal dimension
via D
f
3 H. Since we expect 2 D
f
3 it follows that
0 H 1. Recent studies have shown that asphalt road tracks
are approximately self-affine in a finite surface roughness
interval, with an upper cut-off of order a few mm.
12
In order to study rubber friction on a hard self-affine
fractal surface, it is first necessary to be able to describe the
contact mechanics. A simple model of contact mechanics for
fractal-like surfaces was studied as early as 1957 by
Archard.
13,14
He showed that the area of real contact A is
nearly proportional to the load or normal force, AF
N
.
In a recent series of papers by Roux et al.
15
and Bhushan and
co-workers,
16
it is claimed that for self-affine surfaces the
area of real contact depends nonlinearly on the load. Assum-
ing only elastic deformation they found
AF
N
2/(1 H)
,orF
N
A
(1 H)/2
. 2
Since H 1 H 1 correspond to D
f
2, these theories pre-
dict that the area of real contact increases faster than linear
with the load. This is usually not observed experimentally. In
my opinion, the theories of Roux et al. and of Bhushan and
co-workers are based on questionable assumptions see Ap-
pendix D. The contact theory developed in this paper see
Appendices B and C predict AF
N
, unless the load F
N
is
so large that the contact area A is close to the nominal con-
tact area A
0
.
In the theory developed in this paper, the friction coeffi-
cient is given by a sum over different length scales. Now in
most cases the upper limit in the sum is quite obvious. For
example, for an asphalt road track the upper cutoff is of the
order of a few mm the typical grain sizes as observed in
surface profile measurements. In a recent measurement, an
asphalt road surface was observed to be a self-affine fractal
down to the shortest length-scale studied approximately
0.03 mm.
8
The short distance cutoff in the sum over length
scales may, however, not be determined by the intrinsic cut-
FIG. 3. A rubber ball squeezed against a hard, rough, substrate. Left: the
system at two different magnifications. Right: the area of contact A()on
the length scale is defined as the area of real contact when the surface
roughness on shorter length scales than has been removed i.e., the surface
has been ‘smoothened’ on length scales shorter than ␭兲.
FIG. 4. Elastic contact between a flat rubber surface and a hard solid sub-
strate. The surface is assumed to be self-affine fractal with an upper cutoff
0
L. The system is shown on the length scale
0
. Increasing the magni-
fication shows that within an apparent contact area, the rubber will only
make partial contact with the substrate see text.
3842 J. Chem. Phys., Vol. 115, No. 8, 22 August 2001 B. N. J. Persson
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off of the fractal nature of the surface which could be an
atomic distance, but by surface contamination,
17,18
or com-
pressed air pockets in small sized cavities, or by a thin
‘skin’ on the rubber surface with strongly modified proper-
ties. For example, if the rubber surface is covered by small
uniformly sized dust particles e.g., talc or carbon or silica
particles from the fillers, or pulverized stone from a road, or
carbon particles from the automobile exhaust, then the low
distance cutoff is obviously determined by the particle size
see Fig. 5a兲兴, since the particle covered rubber surface can-
not penetrate into surface cavities smaller than the typical
particle diameter. In fact, it is known that the tire-road fric-
tion increases when a road surface has dried up after a strong
rain fall. Presumably, the rain washes away contamination
particles from the road and tire surface. On the other hand,
if the surface is covered by water or some other ‘lubrica-
tion’ fluid e.g., oil or grease, which fills out the small sur-
face cavities, then the low distance cutoff will be determined
by the smallest asperities which can penetrate above the con-
tamination layer Fig. 5b兲兴. Thus, the contamination layer
will remove the contribution to the energy dissipation from
the small surface asperities and cavities, and reduce the fric-
tion force. This effect is illustrated in Fig. 6 with experimen-
tal results for a rubber block sliding on dry clean dashed
line, dusted dasheddotted, and wet solid line carborun-
dum stone surfaces.
19
The figure also show results for wet
surfaces with an added 5% detergent. Roberts
20
has shown
that polar substances like soaps prevent direct contact be-
tween track and rubber see Sec. VII; this explain why the
friction is slightly lower for the wet 5% detergent case,
compared to the wet, clean carborundum surface.
III. AREA OF REAL CONTACT: QUALITATIVE
DISCUSSION
I have already emphasized the importance of knowing
the nature of the area of real contact when discussing rubber
friction. In this section, I discuss some basic results of con-
tact theory, which form a necessary background for the
theory presented in Secs. IV, V, and Appendices B and C.
Consider first a flat rubber surface squeezed against a
hard surface with a periodic corrugation with wavelength
and amplitude or height h; see Fig. 7. If A
0
is the nominal
contact area i.e., the area of the bottom surface of the
rubber block, and F
N
the load, then we define the average
perpendicular stress or pressure
0
F
N
/A
0
. Let us now
study under which conditions the load F
N
, and the rubber
substrate adhesion forces, are able to deform the rubber so
that it comes in direct contact with the substrate over the
whole surface area A
0
Fig. 7b兲兴, i.e., under which condi-
tions the rubber is able to deform and fill out all the surface
‘cavities’ of the substrate.
Assume first that a uniform stress
acts within a circular
area radius R centered at a point P on the surface of a
FIG. 5. Influence of contamination on the rubbersubstrate interaction.
Contamination particles a, or trapped liquid b, will inhibit the rubber to
get squeezed into the small sized surface cavities.
FIG. 6. The kinetic friction coefficient for rubber sliding on a carborundum
surface under different conditions from Ref. 19.
FIG. 7. A rubber block squeezed against a substrate with a cosines corru-
gation. In a the applied pressure is too small to squeeze the rubber into
complete contact with the substrate, while in b it is high enough to do so.
3843J. Chem. Phys., Vol. 115, No. 8, 22 August 2001 Theory of rubber friction and contact mechanics
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semi-infinite elastic body with the elastic modulus E. This
will give rise to a perpendicular displacement u of P by a
distance which is easy to calculate using continuum mechan-
ics, u/R
/E. This result can also be derived from simple
dimensional arguments: First, note that u must be propor-
tional to
since the displacement field is linearly related to
the stress field we assume here, and in what follows, that
linear elasticity theory is valid. However, the only other
quantity in the problem with the same dimension as the
stress
is the elastic modulus E so u must be proportional to
/E. Since R is the only quantity with the dimension of
length we get at once u(
/E)R. Thus, with reference to
Fig. 7, if h/␭⬇
0
/E, the perpendicular pressure
0
will be
just large enough to deform the rubber to make contact with
the substrate everywhere.
In the case of passenger tires one typically has
0
0.2 MPa, and in the case of truck tires 0.8 MPa. This is at
least one order of magnitude smaller than the static or low-
frequency elastic modulus E10 MPa of filled rubbers but
only a little smaller than that of unfilled rubber where E
1 MPa. We conclude that the pressure
0
is in general not
able to deform the rubber to fill out the large surface cavities
on a road, since in this case one typically has h/␭⬇1, which
according to the discussion above would require a local pres-
sure of order
E. However, according to the contact
theory of Greenwood,
14,17
the average pressure which acts in
the rubbersubstrate contact area at the largest asperities is
of order (/R)
1/2
E, where is the rms surface roughness
amplitude and R the average radius of curvature of the
largest surface asperities see Fig. 4. Since for a road surface
we expect ⌬⬇R it is clear that the local pressure in the
contact area of the large surface asperities will be of order of
E, i.e., just large enough in order for the rubber to deform
and fill out at least some of the smaller sized surface cavities.
The way the apparent contact area varies with the observa-
tion length scale L/
is described by the function P(
).
Next, let us consider the role of the rubbersubstrate
adhesion interaction.
21
When the rubber deforms and fills out
a surface cavity of the substrate, an elastic energy E
el
Eh
2
will be stored in the rubber. Now, if this elastic
energy is smaller than the gain in adhesion energy E
ad
⬇⌬
2
as a result of the rubbersubstrate interaction
which usually is mainly of the van der Waals-type, then
even in the absence of the load F
N
the rubber will deform
spontaneously to fill out the substrate cavities. The condition
E
el
E
ad
gives
4,5
h/␭⬇(
/E)
1/2
. For the rough surfaces
of interest here we typically have h/␭⬇1, and with E
1 MPa and the surface free energy change
3 meV/Å
2
the adhesion interaction will be able to deform
the rubber and completely fill out the cavities if
0.1
m. However, I believe that because of surface con-
tamination there will be a low distance cut-off in the sum
over length scales which is larger than 0.1
m, and for this
reason, in the context of the tire-road friction, I do not be-
lieve that the adhesion rubbersubstrate interaction is impor-
tant. The same conclusion has been reached by Fuller and
Tabor in an experimental study of the dependence of rubber-
substrate adhesion on the surface roughness.
22
The discussion above is for stationary surfaces. During
sliding we must take into account that the elastic modulus E
depends on the perturbing frequency
, and that E(
)isa
complex quantity with an imaginary part related to the inter-
nal friction of the rubber. In a first approximation we may
still use the estimates presented above for the deformations
induced by the largest asperities if we replace E E(0) with
E(
)
, where the frequency
v
/
0
. Now, for a typical
rubber at room temperature E(
)E(0) for
c
10
5
s
1
. When the frequency increases towards the glassy
region (
10
9
s
1
),
E(
)
increases by a factor of
1000. In a typical case, for a tires sliding on a road with
v
10 m/s one gets
v
/
c
0.1 mm. Thus, the deformations
induced by the largest asperities are relatively well described
by using the low frequency elastic modulus E(
)E(0).
However, the rubber will be much harder to deform by the
small sized asperities since the effective elasticity
E(
)
may depending on the size of the asperities be up to 1000
times higher than the low-frequency modulus. On the other
hand, in the antilock braking system ABS of automobile
tires on dry or wet road
v
1 cm/s in the incipient part of the
footprint area, and in this case
v
/
c
0.1
m so that surface
cavities with linear size larger than 0.1
m will experience
relative ‘soft’ rubber. These aspects of the frequency-
dependent deformation of the rubber by the substrate asperi-
ties is taken fully into account in the theory developed below.
In Sec. V and Appendices B and C, I develop a new
contact theory for surfaces with roughness on many different
length scales. The contact theory of Greenwood was origi-
nally developed for surfaces with roughness on a single
length scale. Thus, in this theory the surface asperities are
‘approximated’ by spherical caps of identical radius of
curvature but with a Gaussian height distribution. The
Greenwood theory has been applied to real surfaces with
roughness on many different length scales, by defining an
average radius of curvature R see, e.g., Ref. 23. However,
it turns out that R depends strongly on the resolution of the
roughness-measuring instrument, or any other form of filter-
ing, and hence is not unique. The contact theory developed in
this paper is based on a completely different physical ap-
proach, and gives well defined results for surfaces with arbi-
trary surface roughness.
IV. SLIDING FRICTION
Using the theory of elasticity assuming an isotropic
elastic medium for simplicity, one can calculate the dis-
placement field u
i
on the surface z 0 in response to the
surface stress distributions
i
3i
. Let us define the Fou-
rier transform,
u
i
q,
1
2
3
d
2
xdtu
i
x,t
e
i(q"x
t)
,
and similar for
i
(q,
). Here x (x,y) and q(q
x
,q
y
) are
two-dimensional vectors. In Appendix A, I have shown that
u
i
q,
M
ij
q,
j
q,
,
or, in matrix form,
u
q,
M
q,
q,
,
where the matrix see Appendix A,
3844 J. Chem. Phys., Vol. 115, No. 8, 22 August 2001 B. N. J. Persson
Downloaded 21 Dec 2006 to 134.94.122.39. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Citations
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TL;DR: In this article, the authors presented calculations on the global fuel energy consumption used to overcome friction in passenger cars in terms of friction in the engine, transmission, tires, and brakes.

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Cites background from "Theory of rubber friction and conta..."

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Journal ArticleDOI
TL;DR: This work presents surface roughness power spectra of many surfaces of practical importance, obtained from the surface height profile measured using optical methods and the atomic force microscope, and shows how the power spectrum determines the contact area between two solids.
Abstract: Surface roughness has a huge impact on many important phenomena. The most important property of rough surfaces is the surface roughness power spectrum C(q). We present surface roughness power spectra of many surfaces of practical importance, obtained from the surface height profile measured using optical methods and the atomic force microscope. We show how the power spectrum determines the contact area between two solids. We also present applications to sealing, rubber friction and adhesion for rough surfaces, where the power spectrum enters as an important input.

866 citations

Journal ArticleDOI
26 Feb 2009-Nature
TL;DR: It is demonstrated that the breakdown of continuum mechanics can be understood as a result of the rough (multi-asperity) nature of the contact, and that roughness theories of friction can be applied at the nanoscale.
Abstract: Macroscopic laws of friction do not generally apply to nanoscale contacts. Although continuum mechanics models have been predicted to break down at the nanoscale, they continue to be applied for lack of a better theory. An understanding of how friction force depends on applied load and contact area at these scales is essential for the design of miniaturized devices with optimal mechanical performance. Here we use large-scale molecular dynamics simulations with realistic force fields to establish friction laws in dry nanoscale contacts. We show that friction force depends linearly on the number of atoms that chemically interact across the contact. By defining the contact area as being proportional to this number of interacting atoms, we show that the macroscopically observed linear relationship between friction force and contact area can be extended to the nanoscale. Our model predicts that as the adhesion between the contacting surfaces is reduced, a transition takes place from nonlinear to linear dependence of friction force on load. This transition is consistent with the results of several nanoscale friction experiments. We demonstrate that the breakdown of continuum mechanics can be understood as a result of the rough (multi-asperity) nature of the contact, and show that roughness theories of friction can be applied at the nanoscale.

802 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider contact problems for very smooth polymer (PMMA) and Pyrex glass surfaces prepared by cooling liquids of glassy materials from above the glass transition temperature.

613 citations

Journal ArticleDOI
TL;DR: In this paper, a general formalism for the calculation of the power spectral density for the fluctuating electromagnetic field is presented and applied to the radiative heat transfer and the van der Waals friction using both the semiclassical theory of the fluctuated electromagnetic field and quantum field theory.
Abstract: All material bodies are surrounded by a fluctuating electromagnetic field because of the thermal and quantum fluctuations of the current density inside them. Close to the surface of planar sources (when the distance $d⪡{\ensuremath{\lambda}}_{T}=c\ensuremath{\hbar}∕{k}_{B}T$), thermal radiation can be spatially and temporally coherent if the surface can support surface modes like surface plasmon polaritons, surface phonon polaritons, or adsorbate vibrational modes. The fluctuating field is responsible for important phenomena such as radiative heat transfer, the van der Waals interaction, and the van der Waals friction between bodies. A general formalism for the calculation of the power spectral density for the fluctuating electromagnetic field is presented and applied to the radiative heat transfer and the van der Waals friction using both the semiclassical theory of the fluctuating electromagnetic field and quantum field theory. The radiative heat transfer and the van der Waals friction are greatly enhanced at short separations $(d⪡{\ensuremath{\lambda}}_{T})$ between the bodies due to the evanescent electromagnetic waves. Particularly strong enhancement occurs if the surface of the body can support localized surface modes like surface plasmons, surface polaritons, or adsorbate vibrational modes. An electromagnetic field outside a moving body can also be created by static charges which are always present on the surface of the body due to inhomogeneities, or due to a bias voltage. This electromagnetic field produces electrostatic friction which can be greatly enhanced if on the surface of the body there is a two-dimensional electron or hole system, or an incommensurate adsorbed layer of ions exhibiting acoustic vibrations. Applications of radiative heat transfer and noncontact friction to scanning probe spectroscopy are discussed. The theory gives a tentative explanation for the experimental noncontact friction data.

574 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, the authors proposed a new theory of elastic contact, which is more closely related to real surfaces than earlier theories, and showed how the contact deformation depends on the topography of the surface, and established the criterion for distinguishing surfaces which touch elastically from those which touch plastically.
Abstract: It is usually assumed that the real area of contact between two nominally flat metal surfaces is determined by the plastic deformation of their highest asperities. This leads at once to the result that the real area of contact is directlyproportional to the load and independent of the apparent area-a result with many applications in the theories of electric contacts and friction. Archard pointed out that plastic deformation could not be the universal rule, and introduced a model which showed that, contrary to earlier ideas, the area of contact could be proportional to the load even with purely elastic contact. This paper describes a new theory of elastic contact, which is more closely related to real surfaces than earlier theories. We show how the contact deformation depends on the topography of the surface, and establish the criterion for distinguishing surfaces which touch elastically from those which touch plastically. The theory also indicates the existence of an 'elastic contact hardness', a composite quantity depending on the elastic properties and the topography, which plays the same role in elastic contact as the conventional hardness does in plastic contact. A new instrument for measuring surface topography has been built; with it the various parameters shown by the theory to govern surface contact can be measured experimentally. The typical radii of surface asperities have been measured. They were found, surprisingly, to be orders of magnitude larger than the heights of the asperities. More generally we have been able to study the distributions of asperity heights and of other surface features for a variety of surfaces prepared by standard techniques. Using these data we find that contact between surfaces is frequently plastic, as usually assumed, but that surfaces which touch elastically are by no means uncommon in engineering practice.

5,371 citations

Journal ArticleDOI
TL;DR: In this paper, the authors compare and discuss recent experimental and theoretical results in the field of H2O-solid interactions, and emphasize studies of low (submonolayer) coverages of water on well-characterized, single-crystal surfaces of metals, semiconductors and oxides.

2,096 citations

Book
01 Dec 1997
TL;DR: In this article, a sliding system on clean (Dry) and lubed surfaces is presented. But it does not consider the effect of surface topography and surface contaminants.
Abstract: 1. Introduction.- 2. Historical Note.- 3. Modern Experimental Methods and Results.- 4. Surface Topography and Surface Contaminants.- 5. Area of Real Contact: Elastic and Plastic Deformations.- 6. Sliding on Clean (Dry) Surfaces.- 7. Sliding on Lubricated Surfaces.- 8. Sliding of Adsorbate Layers.- 9. Boundary Lubrication.- 10. Elastic Interactions and Instability Transitions.- 11. Stress Domains, Relaxation, and Creep.- 12. Lubricated Friction Dynamics.- 13. Dry Friction Dynamics.- 14. Novel Sliding Systems.- 15. Outlook.- References.

1,103 citations

Journal ArticleDOI
TL;DR: In this paper, the authors examined whether the hypothesis of elastic deformation of surface protuberances is consistent with Amontons's law, that the friction is proportional to the applied load.
Abstract: This paper examines whether the hypothesis of elastic deformation of surface protuberances is consistent with Amontons’s law, that the friction is proportional to the applied load. For a single elastic contact, the area of contact A is known to be proportional to the ⅔ power of the load W . Since the frictional force is generally assumed to be proportional to A , it has been thought that in elastic deformation Amontons’s law would not be obeyed. However, conforming surfaces usually touch at many points, and it is shown that in these circumstances A and W become nearly proportional. Experiments are described which show that the general law is that the friction is proportional to the true area of contact; whether or not Amontons’s law is obeyed depends upon the surface topography. For highly elastic materials such as Perspex, Amontons’s law is obeyed when contact is made at many points, and other relations between A and W are observed when the contacts are few. Experiments with lubricated brass specimens show that the same conclusions apply to carefully prepared or well run-in metal surfaces running in conditions where the damage is small.

877 citations

Journal ArticleDOI
TL;DR: In this paper, the authors describe a study of the adhesion between elastic solids and the effect of roughness on their adhesion, showing that roughness which is small compared with the overall deformation occurring at the region of the rubber-Perspex contact can produce an extremely large reduction in adhesion.
Abstract: This paper describes a study of the adhesion between elastic solids and the effect of roughness in reducing the adhesion. The experiments were carried out between optically smooth rubber spheres and a hard smooth flat surface of Perspex which could be roughened to various degrees. The radius of the rubber spheres was varied by a factor of 8, their elastic modulus by a factor of 10, while the centre line average (c.l.a.)of the roughened Perspex surface was varied from 0.12 to 1.5 μm. The results show that c.l.a. roughnesses which are small compared with the overall deformation occurring at the region of the rubber-Perspex contact can produce an extremely large reduction in adhesion. The effect is more marked for rubbers of higher modulus. On the other hand the curvature of the sphere (over the range examined) has little influence. For this reason and because the analytical problem of a sphere on a rough flat is extremely complicated a theoretical analysis has been developed for the simpler case of a smooth flat in contact with a rough flat surface. As in Greenwood & Williamson (1966) the rough surface is modelled by asperities all of the same radius of curvature and with heights following a Gaussian distribution of standard deviation σ. The overall contact force is obtained by applying the contact theory of Johnson, Kendall & Roberts (1971) to each individual asperity. The theory predicts that the adhesion expressed as a fraction of the maximum value depends upon a single parameter, 1/Δ e ,which is the ratio between a and the elastic displacement δ C that the tip of an asperity can sustain before it pulls off from the other surface. The analysis shows that the adhesion parameter may also be regarded as representing the statistical average of a competition between the compressive forces exerted by the higher asperities trying to prize the surfaces apart and the adhesive forces between the lower asperities trying to hold the surfaces together. Although the theory is derived for two nominally plane surfaces it is found to fit the experimental results for a sphere on a flat reasonably well.

838 citations

Frequently Asked Questions (17)
Q1. What is the friction coefficient for a rubber?

Since the stains involved in rubber friction when sliding on a road surface is of order unity ~or ;100%!, when calculating the tire-road friction coefficient the effective elastic modulus E(v) obtained from large amplitude stress-strain measurements should be used. 

In this paper, a general theory of the hysteretic contribution to rubber friction has been presented for rubber sliding on selfaffine fractal surfaces, e.g., a tire on a road surface. 

Using the theory of elasticity ~assuming an isotropic elastic medium for simplicity!, one can calculate the displacement field ui on the surface z50 in response to the surface stress distributions s i5s3i . 

the rubber surface is likely to be microscopically rough and the rubber molecules undergoes large thermal movement which may tend to break up any icelike structure at the interface. 

The temperature field T(x,t) must be determined by solving the heat diffusion equation with the heat source Q(x,t), and with the appropriate boundary conditions which depend on the external conditions ~e.g., road temperature!. 

however, a surface active agent ~e.g., sodium dodecyl sulphate!, is included in the electrolyte solution, monolayer protection prevents surfaces from coming into intimate contact at points where the separating liquid film is locally punctured. 

One problem with applying the present theory to filled rubbers is the strongly nonlinear relation between the shear stress and the shear strainDownloaded 21 Dec 2006 to 134.94.122.39. 

The complex elastic modulus E(v) used in the calculation was measured at 8% strain amplitude, which is so large that a complete break down of the filler network has occurred. 

There is at present a strong drive by tire companies to design new rubber compounds with lower rolling resistance, higher sliding friction, and reduced wear. 

Since for a road surface the authors expect D'R it is clear that the local pressure in the contact area of the large surface asperities will be of order of E , i.e., just large enough in order for the rubber to deform and fill out at least some of the smaller sized surface cavities. 

The authorbelieve that because of surface contamination there will be a low distance cut-off in the sum over length scales which is larger than 0.1 mm, and for this reason, in the context of the tire-road friction, The authordo not believe that the adhesion rubber–substrate interaction is important. 

In order to study rubber friction on a hard self-affine fractal surface, it is first necessary to be able to describe the contact mechanics. 

27By using a transparent substrate, it should be possible to study the asperity contact areas during squeezing and shearing of thin fluid films. 

according to the contact theory of Greenwood,14,17 the average pressure which acts in the rubber–substrate contact area at the largest asperities is of order '(D/R)1/2E , where D is the rms surface roughness amplitude and R the ~average! 

In Fig. 17 the authors only show m(v) up to the velocity where the friction coefficient is maximal; higher velocities are probably of no direct interest for tires since the region where m(v) decreases with increasing v should be avoided, as interfacial stick-slip may occur when m8(v) ,0, which may result in an enhanced wear rate, and a loud noise. 

The steady state kinetic friction coefficient for a flat rubber surface sliding on a nominally flat substrate is in the most general case is given byAIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspm5 1 2 EqL q1 dq q3 C~q ! 

It has been found that many ‘‘natural’’ surfaces, e.g., surfaces of many materials generated by fracture, can be approximately described as self-affine surfaces over a ratherDownloaded 21 Dec 2006 to 134.94.122.39.