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A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters

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A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters

Abstract and Figures

Since the early study by Grosch in 1963 it has been known that rubber friction shows generally two maxima with respect to speed-the first one attributed to adhesion, and another at higher velocities attributed to viscoelastic losses. The theory of Klüppel and Heinrich and that of Persson suggests that viscoelastic losses crucially depend on the "multiscale" aspect of roughness and in particular on truncation at fine scales. In this study, we comment a little on both theories, giving some examples using Persson's theory on the uncertainties involved in the truncation of the roughness spectrum. It is shown how different choices of Persson's model parameters, for example the high-frequency cutoff, equally fit experimental data on viscoelastic friction, hence it is unclear how to rigorously separate the adhesive and the viscoelastic contributions from experiments.
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lubricants
Article
A Discussion on Present Theories of Rubber Friction,
with Particular Reference to Dierent Possible
Choices of Arbitrary Roughness CutoParameters
Andrea Genovese 1, Flavio Farroni 1, Antonio Papangelo 2,3 and Michele Ciavarella 2, *
1
Department of Industrial Engineering, University of Naples Federico II, Via Claudio, 21-80125 Naples, Italy;
andrea.genovese2@unina.it (A.G.); flavio.farroni@unina.it (F.F.)
2Department of Mechanics, Mathematics and Management, Politecnico di Bari, Via Orabona 4,
70125 Bari, Italy; antonio.papangelo@poliba.it
3
Department of Mechanical Engineering, Hamburg University of Technology, Am Schwarzenberg-Campus 1,
21073 Hamburg, Germany
*Correspondence: michele.ciavarella@poliba.it
Received: 24 July 2019; Accepted: 24 September 2019; Published: 26 September 2019


Abstract:
Since the early study by Grosch in 1963 it has been known that rubber friction shows
generally two maxima with respect to speed—the first one attributed to adhesion, and another at
higher velocities attributed to viscoelastic losses. The theory of Klüppel and Heinrich and that of
Persson suggests that viscoelastic losses crucially depend on the “multiscale” aspect of roughness
and in particular on truncation at fine scales. In this study, we comment a little on both theories,
giving some examples using Persson’s theory on the uncertainties involved in the truncation of the
roughness spectrum. It is shown how dierent choices of Persson’s model parameters, for example
the high-frequency cuto, equally fit experimental data on viscoelastic friction, hence it is unclear
how to rigorously separate the adhesive and the viscoelastic contributions from experiments.
Keywords: rubber friction; viscoelasticity; roughness
1. Introduction
After the fundamental study by Grosch [
1
], who took advantage of the other fundamental study of
Williams Landel & Ferry [
2
] which relates temperature and rate dependence of viscoelastic properties,
it is known that rubber friction shows two maxima as a function of speed, one more easily attributed
to adhesion with the track, and another at high velocities due to viscoelastic losses. In the search
for more quantitative models, two main theories appeared, one by Klüppel and Heinrich [
3
], and
another by Persson [
4
], both concentrating heavily on the multiscale nature of surfaces and their fractal
roughness, which introduced more than one complication. For about 20 years, we have had to solve
several mathematical problems to deal with the fractal roughness issue, how to rigorously define the
measurement and the mathematical structure, and all theories have experienced several problems to
define cutos to the fractal scaling, particularly at short scales [
5
], where generally friction would tend
towards extremely high values and hence the choice of the truncation is critical.
Klüppel and Heinrich’s early theories [
3
] recognize this problem more explicitly than Persson’s
original paper [
4
] as an upper cutofrequency
ωmax
=2
π
v/
λmin
is introduced into the integral of their
Equation (35), where
λmin
is the lower cutolength of the excitation spectra and vis the velocity. They
showed that the results strongly depend on
λmin
as v
2
in their Equation (36) does depend on
λmin
. On
the other hand, dierently from Persson’s theory, they use the elastic contact model of the Greenwood
and Williamson (GW) asperity model for the contact mechanics, which has later found to have some
limitations and to be inaccurate, especially for broad spectra of roughness. A profound analysis of
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their theory is not simple: GW seems to lead to reasonable predictions for the mean penetration depth,
which appears to depend only on macroscopic wavelengths, as does Persson’s theory much later [
6
]
(see also [
7
]). For the cuto
λmin
, they introduce an energy balance [
3
] in the presence of adhesion,
which turns out to be relevant only in the normal force regime, i.e., when the external load is dominant
for the formation of elastic contacts, giving a cuto
λmin
very sensible to the surface fractal dimension,
especially in the limit of D
2, which in turns is the most common case for real surfaces. Contrary to
Persson [
4
], Klüppel and Heinrich [
3
] suggest that adhesion-induced hysteretic losses may play a role
only for extremely smooth surfaces, such as glasses with D=2, but they do not seem to be relevant
on rough surfaces such as road tracks with typical D=2.2. Klüppel and Heinrich [
3
] try to apply
the theory to the original cases of Grosch, namely friction on the silicon carbide paper and the glass
surface, by changing the broadness of the spectrum, although only very qualitatively, and not really
explaining the appearance of the two maxima in the case of Grosch data on silicon carbide: choices of
cutos are not immediately clear to the reader.
Persson’s theory, on the other hand, in the latest forms (Lorenz et al. [
8
]), seems to have a
surprisingly simple criterion for truncation wavenumber q
1
=2
π
/
λmin
: it is defined where the rms
slope reaches
H0rms (q1)=1.3 (1)
although other authors (Carbone & Putignano [
9
]) are more cautious about many possible choices to
the truncation cuto, e.g., small dirt particles or rubber wear particles.
However, more recently, both Klüppel and co-authors [
10
12
] and Persson and Volokitin [
13
]
seem to attribute a lot more importance to the adhesive term than the viscoelastic one. For example,
Lang & Klüppel [
12
], has an adhesion contribution attributed to peeling eects at the edges which
depends on velocity and viscoelastic properties in the simplified form
τs=τs0 1+E/E0
(1+vc/v)n!(2)
where
τs0
is a static shear stress for very low velocities and v
c
is a certain critical velocity to a plateau.
There are explicit expressions for
τs0
based on contact angle measurements; obviously E
/E
0
is the ratio
of dynamic modulus in a glassy and rubbery state, ncan be estimated from the power law behavior of
the relaxation time spectra H(
τ
) in the glass transition range. Notice that this term is then multiplied
by the real contact area, and hence the dependence on roughness spectrum and its truncation is very
important. Even the viscoelastic properties, which in Equation (2) are limited to two elastic moduli,
with no reference to relaxation spectra, could have more eect on the contact area. The agreement with
an extensive set of measurements is quite good over an extensive range of velocities, except perhaps at
very low ones of the order of 104m/s, where the discrepancy is significant.
Turning back to recent Persson’s theories [
8
,
13
], the dominant mechanics is also adhesion, and
for it, based on ideas such as stretching, detaching, relaxing, and reattaching of rubber molecules
of Schallamach/Cherniak (see Persson and Volokitin [
13
]), they suggest an adhesive contribution
proportional to the frictional shear stress, which is a Gaussian-like curve as a function of the logarithm
sliding velocity:
µadh =τf
p0
A
A0
(3)
where
τf=τf0 exp"clog10
v
v2#(4)
and where A/A
0
is the relative contact area, p
0
is the nominal contact pressure and vis the sliding velocity.
This adhesive contribution on one hand, is also very simple in form, and basically as semi-empirical as
is Klüppel’s one, but on the other, is symmetrical contrary to the Lang & Klüppel [
12
]. It maintains the
complications (or the eects) of the full-contact mechanics theory with roughness, since the contact
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area A/A
0
depends in principle on the full viscoelastic spectrum, and on the full roughness spectrum.
However, in the end, the choices of the factors
τf0
,c, and v* are not from independent experiments
such as contact angle measurements in Lang & Klüppel [12].
In this paper, short of making a full comparison of the Klüppel and Persson’s theories, which both
seem (at least in the presentation of their authors) to fit friction measurements quite well, we discuss
how the choice to be made about the truncation of the spectrum influences the determination of the
various coecients, with reference in particular to Persson’s theory. Here, we shall use in particular a
simplified version for the viscoelastic losses, since they are no longer considered so relevant. Persson,
indeed, in recent papers (Tolpekina & Persson, [
14
], see also [
15
]), seems to agree with Klüppel [
10
12
]
(just looking at the how the friction data are separated into viscoelastic and adhesive contributions)
that the contribution due to adhesion is now considered prevalent over the hysteretic one. Obviously,
this shift is also a sign of the uncertainty of the choice on how to interpret the two contributions
(viscoelastic and adhesive ones), which appear to have a similar bell-shape. In this paper, we shall
discuss exactly this point, where for brevity we consider only Persson’s theories.
2. Experimental Data
To analyze typical compounds of interest in asphalt vs. tire contact and to give a quantitative
example of the analysis made, all of the following is referred to the experimental data obtained by
Tolpekina and Persson [14] (TP in the following) for three dierent reference compounds A, B, and C
(see [
14
]), on which quite general information are provided: “B is a summer tread compound filled
with carbon black. A is a summer tread compound filled with silica and containing a traction resin.
C is a winter tread compound, filled with silica and containing the same traction resin (in the same
relative volume fraction) as for Compound A”. In the following subsection the power spectral density
(PSD) of a rough surface, the viscoelastic properties of the rubber, and experimental friction data used
as references for theoretical results are shown.
2.1. Surface PSD
For rough surface for the rubber friction analysis, a concrete block was considered. This type of
substrate is very stable (negligible wear) and is easily available in a large number of nominally identical
blocks. The most important information about the substrate is the surface roughness power spectrum.
Notice that it is possible to find dierent normalization of the PSD and therefore it is important to pay
attention to the formulations used for the calculation of the quantities such as the root-mean-square
(RMS) height h
rms
, the RMS slope h
0rms
, and the RMS curvature h
rms
[
16
]. Referring to the Persson
formulation, Figure 1shows the 2D surface roughness power spectrum of the concrete surface. The
significant range of interest of this self-ane fractal power spectrum can be assumed to be a power law:
C(q)=C0·q2(1+H)(5)
where the Hurst exponent H=0.86 and C
0
=0.001152 mˆ
(22H)
. The wavenumber q
0
considered in this
study is q
0
=10
2.7
[1/m] but fortunately the choice of this truncation is not very relevant for friction
estimation, while the choice of the large wavenumber cutois extremely more sensible and arbitrary,
as debated in the next section.
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Figure 1. The surface roughness power spectrum of the concrete substrate.
2.2. Material Viscoelastic Properties
A full viscoelastic rubber characterization is fundamental for the friction calculations. The
knowledge of the complex elastic modulus of the rubber over a rather large frequency range is
necessary, and more recently it has been suggested also its behavior at different strain values when
the non-linear effects related to the viscoelastic modulus of the rubber should be taken into account.
The standard way of measuring the viscoelastic modulus is to deform the rubber sample in an
oscillatory manner with a constant strain or stress amplitude. This is done at different frequencies
and then repeated at different temperatures. The results measured at different temperatures can be
shifted according to the timetemperature superposition principle to form a master curve covering a
wide range of frequencies at the chosen reference temperature. TP have performed measurements of
the viscoelastic modulus in both shear and in tensile (elongation) modes, providing both the shear
modulus G(ω) and the Young’s modulus E(ω). These moduli are related, for the lack of better
approximations, via E = 2G (1 + ν) where ν = 0.5 is the Poisson ratio.
In their paper, TP [14] tested three different compounds: A is a summer tread containing silica
and resin, B is a summer tread filled with carbon black and C is a winter tread filled with silica and
traction resin. The viscoelastic moduli of the three different rubber compounds A, B, C, are detailed
in Figure 2, as storage (Re E) and loss (Im E) moduli.
Let us consider for the compound under investigation a power law approximation for the
moduli, at low frequency. This approximation, which appears quite good in the practical range of
interest for rubber compounds used in tires (as depicted in Figure 2), also permits a simple estimate
of the relevant equations in closed form at low speeds. The power law behavior considered is:
𝑅e 𝐸(𝜔)=10αr𝜔βr and 𝐼m 𝐸(𝜔)=10αiωβi
(6)
In Table 1 the parameters of the power law approximation for the three compounds are
summarized.
Table 1. Parameters for power law approximation.
Compound
αr
αi
βi
Compound A
1.4193
0.5375
0.0939
Compound B
1.3262
0.2312
0.0507
Compound C
1.4140
0.4713
0.0737
Figure 1. The surface roughness power spectrum of the concrete substrate.
2.2. Material Viscoelastic Properties
A full viscoelastic rubber characterization is fundamental for the friction calculations. The
knowledge of the complex elastic modulus of the rubber over a rather large frequency range is
necessary, and more recently it has been suggested also its behavior at dierent strain values when the
non-linear eects related to the viscoelastic modulus of the rubber should be taken into account. The
standard way of measuring the viscoelastic modulus is to deform the rubber sample in an oscillatory
manner with a constant strain or stress amplitude. This is done at dierent frequencies and then
repeated at dierent temperatures. The results measured at dierent temperatures can be shifted
according to the time–temperature superposition principle to form a master curve covering a wide
range of frequencies at the chosen reference temperature. TP have performed measurements of the
viscoelastic modulus in both shear and in tensile (elongation) modes, providing both the shear modulus
G(
ω
) and the Young’s modulus E(
ω
). These moduli are related, for the lack of better approximations,
via E=2G(1 +ν) where ν=0.5 is the Poisson ratio.
In their paper, TP [
14
] tested three dierent compounds: A is a summer tread containing silica
and resin, B is a summer tread filled with carbon black and C is a winter tread filled with silica and
traction resin. The viscoelastic moduli of the three dierent rubber compounds A, B, C, are detailed in
Figure 2, as storage (ReE) and loss (ImE) moduli.
Let us consider for the compound under investigation a power law approximation for the moduli,
at low frequency. This approximation, which appears quite good in the practical range of interest for
rubber compounds used in tires (as depicted in Figure 2), also permits a simple estimate of the relevant
equations in closed form at low speeds. The power law behavior considered is:
ReE(ω)=10αrωβrand ImE(ω)=10αiωβi(6)
In Table 1the parameters of the power law approximation for the three compounds are summarized.
Table 1. Parameters for power law approximation.
Compound αrβrαiβi
Compound A 1.4193 0.0820 0.5375 0.0939
Compound B 1.3262 0.0501 0.2312 0.0507
Compound C 1.4140 0.0639 0.4713 0.0737
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Figure 2. Real (solid blue) and imaginary (solid red) parts of the viscoelastic modulus of rubber
Compound A (a), B (b), C (c) in TP [14]. The dashed lines indicate the power law approximations at
low frequencies.
Notice that TP affirms that when a rubber tread block is sliding on a road surface, the strain in
asperity contact regions is typically of order one, ε 1. To take into account the effect of these large
strain, they perform strain sweep measurements up to a strain of order one, resulting in a reduction
of the small strain modulus E(ω) of a factor Sf 0.1, where Sf is the strain softening factor [15].
Figure 2.
Real (solid blue) and imaginary (solid red) parts of the viscoelastic modulus of rubber
Compound A (
a
), B (
b
), C (
c
) in TP [
14
]. The dashed lines indicate the power law approximations at
low frequencies.
Notice that TP arms that when a rubber tread block is sliding on a road surface, the strain in
asperity contact regions is typically of order one,
ε
1. To take into account the eect of these large
strain, they perform strain sweep measurements up to a strain of order one, resulting in a reduction of
the small strain modulus E(ω) of a factor Sf0.1, where Sfis the strain softening factor [15].
Lubricants 2019,7, 85 6 of 12
2.3. Friction
The available measured data for low velocities (LV) and higher velocity (HV) was obtained by
TP [
14
] using two dierent experimental setups. Friction for LV was obtained by means the Leonardo
da Vinci experiment, which allows only measurement of the friction coecient on the branch of the
µ
(v)-curve where the friction coecient increases with increasing sliding speed. The friction coecients
for the sliding speeds of 0.1, 0.3, 1, and 1.8 m/s were obtained by means of a linear friction tester (LFT).
Figure 3shows the measured data and calculated results, by using Persson formulation, for rubber
sliding on concrete. Notice that the total friction curve is calculated as a sum of the viscoelastic and
adhesive contribution plus a constant term
µconst
=0.2 (an empirical choice, for the lack of better data),
which the authors refer to the “scratching of the concrete surface by the hard filler particles”.
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2.3. Friction
The available measured data for low velocities (LV) and higher velocity (HV) was obtained by
TP [14] using two different experimental setups. Friction for LV was obtained by means the Leonardo
da Vinci experiment, which allows only measurement of the friction coefficient on the branch of the
µ(v)-curve where the friction coefficient increases with increasing sliding speed. The friction
coefficients for the sliding speeds of 0.1, 0.3, 1, and 1.8 m/s were obtained by means of a linear friction
tester (LFT).
Figure 3 shows the measured data and calculated results, by using Persson formulation, for
rubber sliding on concrete. Notice that the total friction curve is calculated as a sum of the viscoelastic
and adhesive contribution plus a constant term µ const = 0.2 (an empirical choice, for the lack of better
data), which the authors refer to the “scratching of the concrete surface by the hard filler particles”.
Figure 3.
The measured (symbols, from [
14
]) and calculated (lines) friction coecient using the Persson
formulation for Compound A (a), B (b), C (c).
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The parameters adopted for the friction calculation are summarized in Table 2.
Table 2. Summary of the reference parameters adopted for the friction calculation.
Compound q1h0rms µconst log10 v* (m/s) τf0 (MPa) Sf
Compound A 2·1061.3 0.2 2.47 3.6 0.1
Compound B 2·1061.3 0.2 1.97 4.0 0.1
Compound C 2·1061.3 0.2 2.53 4.1 0.1
3. Discussion on the Viscoelastic and Adhesive Contributions
In this section, we want to point out that the recognition of the importance of the adhesive
and viscoelastic contribution is not at all simple and obvious. Indeed, starting from the viscoelastic
properties of the rubber and the PSD of a rough surface, we propose dierent set of arbitrarily
determined parameters, which return a dierent combination of Gaussian-like curves of the two
contributions, allowing an equally good fit of the experimental data.
For all the three compounds, the theoretical results obtained calculating the viscoelastic
contribution using the simplified formulation were first investigated
µvisc =h0
rms(q1)·K·ImE(q1v)
E(q1v)
(7)
where h
0rms
is the rms slope of the surface that depends on the cutowavenumber q
1
and E(q
1
v) is the
complex viscoelastic modulus of the rubber (for a given temperature), at the circular frequency 2
π
f
=q
1
v. Adopting the power law approximation of the moduli at low frequencies, the coecient Kis
calculated as:
K=1H
1H+βiβr
Γ(1+βi/2)
Γ(3/2+βi/2)
rΓ(1/2+βr)
Γ(1+βr)
ππ
2(βr+1H)
(8)
Again, the total friction curve is calculated as a sum of the viscoelastic and adhesive contributions
plus the constant term
µconst
=0.2. The parameters are the same as those used for the calculation with
the Persson theory and reported in Table 2.
For each compound, Figure 4shows the measured and calculated friction coecient on the
concrete surface as a function of the logarithm of sliding speed, using the simplified formulation. The
lower red curve represents the (calculated) viscoelastic contribution to the friction coecient, while the
green is the adhesive contribution and the upper blue curve is the total calculated friction.
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Figure 4. The measured (symbols, from [14]) and calculated (lines) friction coefficient on concrete as
a function of the logarithm of the sliding speed, using the simplified formulation for the viscoelastic
contribution, for Compound A (a), B (b), C (c).
The results obtained with Persson theory and with our simplified assumption for the loss and
storage modulus do not show significant difference for low-velocity friction data being these results
considered to be principally influenced by the adhesion. For high sliding velocity some difference
can be appreciated but these data are often not predicted appropriately.
Figure 4.
The measured (symbols, from [
14
]) and calculated (lines) friction coecient on concrete as a
function of the logarithm of the sliding speed, using the simplified formulation for the viscoelastic
contribution, for Compound A (a), B (b), C (c).
The results obtained with Persson theory and with our simplified assumption for the loss and
storage modulus do not show significant dierence for low-velocity friction data being these results
Lubricants 2019,7, 85 9 of 12
considered to be principally influenced by the adhesion. For high sliding velocity some dierence can
be appreciated but these data are often not predicted appropriately.
Here it should be emphasized that these results are strongly influenced by the value of some
parameters that are fixed in a non-rigorous way. In particular:
-
the choice of the cutowavenumber q
1
influences strongly both viscoelastic and
adhesive contributions;
-
the term
µconst
=0.2 which is attributed to scratching of the concrete surface by the hard filler
particles is quite arbitrary;
- The reference velocity v* and the τf0 significantly influence the adhesive curve;
-
The assumption that for sliding friction on the rough surfaces the deformation is
ε
1 implies a
reduction of the strain modulus E(
ω
) of a strain factor S
f
0.1. Due to the non-linear eects related
to the viscoelastic modulus of the rubber, lower strain values would cause sensible variation of
the Sfand therefore of the adhesive friction.
Adopting the simplified formulation (7) for the viscoelastic friction and Equations (3) and (4)
for the adhesion, dierent combinations in the choice of these parameters are proposed, which are
able to give an equally good fitting of the available experimental data. In Tables 35, five dierent
parameters set used for simulation for Compound A, B, and C, respectively, are reported. Figure 5
reports the corresponding friction results. All the proposed fits predict quite accurately the low-velocity
experimental data, while at high velocity the fit is generally poor. Notice that we have varied the
high-wavenumber cutoq
1
of several orders of magnitude obtaining almost equally good fits for the
friction coecient. This poses fundamental questions on the physical meaning of this quantity and on
the role that it plays in modern theories of viscoelastic friction.
Table 3.
Summary of the dierent parameter sets adopted for the friction calculation of the Compound A.
SET q1h0rms µconst log10 v* (m/s) τf0 (MPa) Sf
A2·1093.6 0 1.5 8 0.1
B2·1061.3 0.3 2.2 4.1 0.1
C2·1071.85 0.3 1.9 7 0.1
D2·1050.9 0.35 2.1 7 0.3
E3·1040.63 0.35 2.1 7 0.5
Table 4.
Summary of the dierent parameter sets adopted for the friction calculation of the Compound B.
SET q1h0rms µconst log10 v* (m/s) τf0 (MPa) Sf
A2·1010 4.5 0 1 4.0 0.1
B2·1061.3 0.3 1.80 8 0.19
C2·1050.9 0.3 1.8 8 0.3
D3·1040.63 0.35 1.8 8 0.5
E1071.66 0.25 2 5.5 0.1
Table 5.
Summary of the dierent parameter sets adopted for the friction calculation of the Compound C.
SET q1h0rms µconst log10 v* (m/s) τf0 (MPa) Sf
A2·1061.3 0.25 2.0 5.25 0.1
B1071.66 0.25 2 6.9 0.1
C3·1071.96 0.25 2 8 0.1
D1050.8 0.3 2.1 7.1 0.3
E1040.5 0.3 2.1 6.4 0.5
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Figure 5. The measured (symbols, form [14]) and calculated (lines) friction coefficient on concrete as
a function of the logarithm of the sliding speed considered five different sets of arbitrary parameters
for each Compound A (a), B (b), C (c).
4. Conclusions
In conclusion, the theories of rubber friction seem to have evolved over the years, but now the
emphasis on the viscoelastic losses for tireroad contact due to fractal roughness has been much
reduced, because the adhesive term is considered dominant. Perssons theories [4,1315] seem to have
evolved more rigorously in the contact mechanics treatment (they are approximate, but the
Figure 5.
The measured (symbols, form [
14
]) and calculated (lines) friction coecient on concrete as a
function of the logarithm of the sliding speed considered five dierent sets of arbitrary parameters for
each Compound A (a), B (b), C (c).
4. Conclusions
In conclusion, the theories of rubber friction seem to have evolved over the years, but now the
emphasis on the viscoelastic losses for tire–road contact due to fractal roughness has been much
reduced, because the adhesive term is considered dominant. Persson’s theories [
4
,
13
15
] seem to
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have evolved more rigorously in the contact mechanics treatment (they are approximate, but the
approximations have been corrected over the years), but the adhesion part is less fundamentally
rigorously introduced, having been less studied and compared with experiments, and perhaps more
eort in the coming years should concentrate on this. Klüppel’s theories [
3
,
10
12
], which seem instead
to remain attached to the GW asperity description of roughness, which today is considered inaccurate.
While this may not be very important for viscoelastic losses, since anyway there is an arbitrary
truncation to be defined, it is less clear how this has an impact in the real contact area estimation, and
hence the adhesive friction. While Klüppel’s theories have attempted to discuss at length the problem
of truncation of roughness—and the resulting conclusion is not so clear—Persson’s theory has recently
made a very abrupt choice of truncation at rms slope of 1.3, which is not well explained. We have
indeed oered some examples on how dierent choices of the parameters (for example varying by
orders of magnitude the high-frequency cuto) equally fit the data, and hence it is unclear in Persson’s
theory, for example, how to rigorously separate the two contributions from experiments.
Author Contributions:
Conceptualization, F.F., A.G., A.P. and M.C.; Data curation, A.G.; Investigation, A.G. and
F.F.; Methodology, A.P. and M.C.; Supervision, F.F. and M.C.; Writing–original draft, F.F. and M.C.; Writing–review
& editing, A.G., A.P. and M.C.
Funding:
A.P. is thankful to the DFG (German Research Foundation) for funding the project PA 3303/1-1. A.P.
acknowledge support from PON Ricerca e Innovazione 2014/2020-Azione I.2-D.D. n. 407 del 27/02/2018, bando
“AIM”. M.C. is supported by the Italian Ministry of Education, University and Research (MIUR) under the
Departments of Excellence. Grant No. L.232/2016.
Conflicts of Interest: The authors declare no conflict of interest.
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2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
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... Several research groups are paying their attention to understand how frictional resistance and adhesion interact [15,16,17,18,19,20]. Frictional sliding tests with polymeric slab (similarly to what happens in tire-road contact) have reveled that rubber friction has two peaks [21,22], one usually attributed to adhesion (at low sliding speed) an the other due to viscoelastic losses (at high speed) [23,24,25,26,21]. Despite the great effort spent to model [17,15,16,19,20], simulate [20,27] and test soft materials [15,16,18,28,29], predictive models are still out of reach, particularly because of the multiscale nature of the contact interface, hence numerical techniques [30,31,32] or direct measurements via costly experimental campaigns have to be performed [33]. ...
... Several research groups are paying their attention to understand how frictional resistance and adhesion interact [15,16,17,18,19,20]. Frictional sliding tests with polymeric slab (similarly to what happens in tire-road contact) have reveled that rubber friction has two peaks [21,22], one usually attributed to adhesion (at low sliding speed) an the other due to viscoelastic losses (at high speed) [23,24,25,26,21]. Despite the great effort spent to model [17,15,16,19,20], simulate [20,27] and test soft materials [15,16,18,28,29], predictive models are still out of reach, particularly because of the multiscale nature of the contact interface, hence numerical techniques [30,31,32] or direct measurements via costly experimental campaigns have to be performed [33]. ...
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Viscoelastic materials are receiving increasing attention in soft robots and pressure sensitive adhesives design, but also in passive damping techniques in automotive and aerospace industry. Here, by using the correspondence principle originally developed by Lee and Radok and further extended by Ting and Greenwood, we transform the elastic solutions of Persson for contact of nominally flat but randomly rough surfaces to viscoelastic indentation. As an example, the cases of step loading and of the response to a single cycle of harmonic loading are studied. For the latter, the effect of the loading frequency, of the ratio between the rubbery and the glassy moduli of the material, and of the mean normal load on the dissipated energy per cycle is studied in detail for a standard viscoelastic material. The results shown are significant for the engineering applications involving cyclic indentation of soft materials, such as in tire-road contact, seals, pick-and-place manipulators and grippers
... Indeed, the rubber molecular chains follow a cycle of stretching and braking, which generates friction phenomena between internal particles in a specific material volume (viscoelastic behaviour). The stress cycle provides three phases: the first phase consists in creating bonds between the tread and the road; in the second one, the molecular chains are stretched and a friction force is generated due to viscoelastic behaviour that opposes the skidding phenomenon; in the last phase, the bond breaks and forms again farther on (Genovese et al., 2019(Genovese et al., , 2020a. It is clear that the essential condition for adhesion to be operative is a direct contact between the tread rubber and the road surface (i.e. when the road is clean and dry). ...
... As said, there are two principal components of rubber friction: adhesion and hysteresis (Genovese et al., 2019). Depending on the surface, kinematic-dynamic working and temperature conditions, diverse adhesion and hysteresis friction terms contribute to the total grip amount. ...
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This chapter deals with tyre mechanics and it has a particular focus on thermal effects on its dynamical behaviour. In the first part the typical tyre structure is introduced together with the tyre mechanical/dynamical behaviour according to a classical approach, so recalling the main kinematic and dynamic quantities involved in tyre pure and combined interactions. The core of this chapter is the description of a physical-analytical tyre thermal model able to determine the thermal status in each part of the tyre useful for vehicle dynamics modelling and driving simulations in order to take into account thermal effects on tyre interactions and consequently on vehicle dynamical behaviour. Successively also the tyre wear modelling is faced, after a brief introduction to the different models available in literature some considerations are reported concerning the thermal effects on wear.
... Moreover, polymers are widespread in the transport sector since tires are made of polymers composite, assuming a crucial role in terms of safety, reliability and performance. Indeed, tire compounds are responsible for the tire/road interaction phenomena that significantly affect the vehicle dynamics; friction performance [Lorenz et al., 2015;Arricale et al., 2020;Genovese et al., 2019], rolling resistance and wear [Moore, 1980;Braghin et al., 2006;Farroni et al., 2017;Genovese et al., 2020b] are some examples of events in which the viscoelasticity of the tread is a key factor. It is evident how their great diffusion poses the study of the mechanics of polymers as an active and important area of research. ...
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A procedure of non-destructive experimental tests aimed at determining the viscoelastic characteristics of the rubbers is described. After recalling the basic principles of viscoelastic theory, the experimental setup and the measurements necessary for the test are described. The procedure consists in hitting the surface of the object under test with a specially instrumented indenter. The techniques for processing the acquired signals to identify the characteristic starting and ending points of indentation in the time-histories are then illustrated. By analyzing separately the phases of the free drop of the indenter and the phase of contact between indenter and rubber, the main mechanical characteristics, such as stiffness and damping of both the instrumented indenter and the bulk of rubber under test are estimated. The methods of calculating the viscoelastic parameters of the rubber starting from the knowledge of the above mechanical parameters are then illustrated. The determined results, in terms of storage and loss moduli, are compared with those provided by classic DMA type tests. The qualitative agreement is excellent. The quantitative agreement falls in the scattering range typical for this kind of measurement. The paper is completed with a discussion of the main causes of measurement uncertainty.
... Specific performance attributes of the models that are addressed include: (a) versatility, (b) number of parameters required, (c) computational efficiency and (d) integration with Persson's friction model [31]. Persson's model was selected as the "carrier" friction model, as it has been heavily relied upon in the literature for the prediction of rubber friction [32][33][34][35][36][37][38]. ...
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Up-to-date predictive rubber friction models require viscoelastic modulus information; thus, the accurate representation of storage and loss modulus components is fundamental. This study presents two separate empirical formulations for the complex moduli of viscoelastic materials such as rubber. The majority of complex modulus models found in the literature are based on tabulated dynamic testing data. A wide range of experimentally obtained rubber moduli are used in this study, such as SBR (styrene-butadiene rubber), reinforced SBR with filler particles and typical passenger car tyre rubber. The proposed formulations offer significantly faster computation times compared to tabulated/interpolated data and an accurate reconstruction of the viscoelastic frequency response. They also link the model coefficients with critical sections of the data, such as the gradient of the slope in the storage modulus, or the peak values in loss tangent and loss modulus. One of the models is based on piecewise polynomial fitting and offers versatility by increasing the number of polynomial functions used to achieve better fitting, but with additional pre-processing time. The other model uses a pair of logistic-bell functions and provides a robust fitting capability and the fastest identification, as it requires a reduced number of parameters. Both models offer good correlations with measured data, and their computational efficiency was demonstrated via implementation in Persson’s friction model.
... 5 Furthermore, in vehicle dynamics, tires tread is made of different vulcanized polymers and fillers in order to ensure the vehicle performance and safe operation. [6][7][8] Actually, the friction phenomenon implicitly depends on the viscoelastic behavior of the tire tread compound, [9][10][11][12] where the hysteretic and adhesive phenomena respectively occur in the local contact patch due to tire block deformation and the chemical bound with road micro asperities. 13,14 In addition, the wear phenomenon of tire compound is mostly a result of strain energy loss due to viscoelastic friction mechanism as amount of an adhesive 15 and hysteretic ratio. ...
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Background The ultrasound technique, usually based on the transmission mode, is capable of providing the viscoelastic properties of polymers. Further techniques involving pulse-echo methods were also described in literature, but they still exhibit inaccuracies in the evaluation of the acoustic properties. Objective The manuscript focuses on an innovative approach for the characterization of the viscoelastic behavior of polymers employing the ultrasound methodology. The proposed procedure is based on the pulse-echo method in order to overcome possible inaccuracies in acoustic properties evaluation and in issues related to transmitter mode applications. Methods Starting from the pulse-echo method adopted for the acquisition, a novel formulation for data processing has been developed and described, allowing to determine the wave attenuation coefficient, in comparison to the commonly employed procedures involving ultrasound in polymers characterization, based on transmitter mode inspections. To carry out the study, a specifically designed ultrasound bench has been set up and three different polymers have been tested in the temperature range of interest. Results According to the proposed methodology, the loss factor towards the temperature is determined starting from the data acquired considering the identified attenuation coefficient and the measured sound velocities. The trustworthiness of the novel procedure has been proved comparing the obtained viscoelastic loss factor quantities to the reference master curves obtained by the standard Dynamic Mechanical Analysis characterizations carried out on the same polymer specimens. Conclusions A novel methodology involving ultrasound technology aiming to evaluate the viscoelasticity of the polymers using non-destructive approach has been developed. The results obtained are agreement with the standard viscoelastic master curves determined through the DMA.
... On one hand, empirical models rely on experimental measures to make the simulation more accurate, and on the other hand, physical models rely on physics to give more insight about tire behaviour. In regards to the calculation of the forces at the tire/road interface, complex multiscale friction models (Genovese et al., 2019) are based on the knowledge of the road surface and of the viscoelastic properties of tire tread (Genovese et al., 2020). Here, the Dugoff physical model to describe the friction forces between the tire/road interface is adopted. ...
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The temperature and load dependence of the sliding friction behaviour of a racing tire tread compound on coarse and fine rough granite substrates is analysed by experimental and theoretical techniques. Based on dry friction measurements at different temperatures, friction master curves are constructed by shifting the data horizontally on the velocity axis using the same shifting factors as found from viscoelastic master curves. The obtained isothermal friction curves increase rapidly with increasing sliding velocities and show a more or less pronounced plateau over a broad velocity range, which decreases with increasing load. For analysing this behaviour, the Klüppel and Heinrich theory of rubber friction and contact mechanics is applied, which considers the multi-scale contacts and excitations of the rubber sliding on rough surfaces in the frame of a linear viscoelastic approach. The extension of this theory to more realistic surfaces with two or more scaling ranges is described in some detail. It takes adhesion and hysteresis contributions into account referring to the viscoelastic response of the rubber on different frequency scales. The theory predicts that under isothermal conditions the coefficient of friction decreases with load, which is more pronounced for the adhesion than for the hysteresis contribution. This result is found to be in fair agreement with the measure friction curves confirming the contact mechanical approach of the theory.
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A basic theoretical concept of rubber friction on rough surfaces is presented that relates the frictional force to the dissipated energy of the rubber during sliding stochastic excitations on a broad frequency scale. It is shown that this is of high relevance for tire traction and allows for a prediction of the likely level of friction of tread compounds on the basis of viscoelastic data.. The impact of both, the frequency dependent loss- and storage modulus on the frictional force during sliding of tires on rough tracks, is demonstrated quantitatively for different sliding velocities. The effect of the surface roughness of road hacks is described by three characteristic surface descriptors, i.e., the fractal dimension and the correlation lengths parallel and normal to the surface. These descriptors can be obtained from a fractal analysis of the road texture via stylus- or laser measurements. In particular, it is shown that the applied model of rubber friction is in agreement with the classical friction data of Grosch, who found a broad maximum for the friction coefficient with increasing sliding speed. The broadness of the friction maximum is shown to be directly related to the broadness of the roughness scale of the surface.