Prediction of the friction coefficient of filled rubber sliding on dry and wet surfaces with self-affine large roughness

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The friction of filled rubber on a rough surface is mainly determined by the rubber viscoelasticity and the surface property of multiple-scale asperities that can be represented by the power spectral density of the surface profile (i.e., power spectrum of surface roughness). This paper investigates a prediction model of rubber friction on dry and wet surfaces with large roughness under lightly squeezing, and finds a high stationary friction coefficient that depends on sliding speed. To this end, we demonstrated friction testing at low velocities with carbon-black-filled rubber and a hard substrate having self-affine surface roughness. From the experiment results, we estimated the hysteresis friction coefficient related to energy dissipation resulting from cyclic deformations of the viscoelastic rubber by applying the theory developed by Persson [(J. Chem. Phys. 115, 3840 (2001)]. We discussed the additional factor, an adhesion force, which also increases the friction coefficient. We concluded that the hysteresis loss of rubber viscoelastic deformation contributes most of the friction force, accounting for the nonlinear viscoelastic behavior of filled rubber, and that the operative surface wavelength extends to the order of micrometers.

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# _Mechanical Engineering Journal_

## Prediction of the friction coefficient of filled rubber sliding on dry and wet surfaces with self-affine large roughness

Hiro TANAKA* , **, Kimiyasu YOSHIMURA*, Ryo SEKOGUCHI*, Jumpei ARAMAKI*,

Asuka HATANO*, Satoshi IZUMI*, Shinsuke SAKAI* and Hiroshi KADOWAKI***

∗ Department of Mechanical Engineering, The University of Tokyo

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

∗∗ (present) Department of Mechanical Engineering, Osaka University

2-1 Yamadaoka, Suita, Osaka 565-0871, Japan

E-mail: htanaka@mech.eng.osaka-u.ac.jp

∗∗∗ Central Research Division, Bridgestone Corporation

3-1-1 Ogawahigashi-cho, Kodaira-shi, Tokyo 187-8531, Japan

Received 28 January 2015

Abstract
The friction of filled rubber on a rough surface is mainly determined by the rubber viscoelasticity and the surface
property of multiple-scale asperities that can be represented by the power spectral density of the surface profile (i.e.,
power spectrum of surface roughness). This paper investigates a prediction model of rubber friction on dry and wet
surfaces with large roughness under lightly squeezing, and finds a high stationary friction coefficient that depends
on sliding speed. To this end, we demonstrated friction testing at low velocities with carbon-black-filled rubber
and a hard substrate having self-affine surface roughness. From the experiment results, we estimated the hysteresis
friction coefficient related to energy dissipation resulting from cyclic deformations of the viscoelastic rubber by
applying the theory developed by Persson [(J. Chem. Phys. 115, 3840 (2001)]. We discussed the additional factor,
an adhesion force, which also increases the friction coefficient. We concluded that the hysteresis loss of rubber
viscoelastic deformation contributes most of the friction force, accounting for the nonlinear viscoelastic behavior
of filled rubber, and that the operative surface wavelength extends to the order of micrometers.

_Key words_ : Rubber friction, Viscoelasticity, Power spectrum of surface roughness, Self-affine fractal, Friction test

1. Introduction

Tire rubber is the sole mechanical component of a vehicle that is in contact with the road and sustains the weight of
the vehicle. From a mechanical viewpoint, the friction coefficient at the interface between the rubber and road is a prime
factor determining the driving performance of a vehicle in terms of braking, operability and environmental friendliness.
It is thus important that engineers understand the dependence of rubber-road friction on the viscoelasticity of the rubber
and surface characteristics of the road in the overall design of a tire from the material development of rubber compounds
to the patterning of tire treads.

Finite element analysis (FEA) can give an approximate solution to the continuum behaviors of the contact mechanics
and frictional system of a rubber material. However, on a microscale, there are remaining issues such as the real contact
area accounting for repulsive and attractive interactions. For example, the contact between an elastic material and a rough
substrate such as a self-affine surface have been extensively studied in FEAs (Hyun et al, 2004, Pei et al, 2005, Hyun
et al, 2007) and molecular dynamics simulations (Yang et al., 2006, Yang and Persson, 2008, Pastewka and Robbins,
2014). Furthermore, there remain considerable temporal and spatial challenges in elucidating the comprehensive friction
mechanism of rubber sliding over surface asperities of multiple sizes employing such numerical procedures (Mo et al,
2009,P´alfi et al., 2012).

Paper No.15-00084
J-STAGE Advance Publication date: 21 January, 2016

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

Meanwhile, two theoretical models of rubber friction on dry and rough surfaces have been developed by Kl¨uppel
and Heinrich (2000) and by Persson (2001), who proposed a hysteresis friction coefficient relating to the viscoelastic
deformation of rubber along self-affine asperities in different ways. In the past decade, the two models and their extensions
with adhesion effects have been experimentally validated for several combinations of rubber materials and substrates; e.g.,
unfilled or filled rubber materials and dry or wet surfaces (Westermann et al., 2004, Gal et al., 2005, Gal and Kluppel,
2008,Lorenz et al., 2011,Lorenz et al., 2013a). Furthermore, two new theories are being applied in developing a practical
model for rubber friction in tire traction (Heinrich and Kl¨uppel, 2008) and tire body dynamics (Persson, 2011). However,
most of these related studies have been still performed by confined research groups, e.g., (Busse et al., 2010,Lorenz et al,
2013b,Scaraggi and Persson, 2014), and exhaustive information is required to construct a common framework for rubber–
road friction. In this context, our study proposes such a concept of the friction mechanism to engineers widely, providing
measurements and predictions for rubber friction under conditions different from those of previous studies (Westermann
et al., 2004,Gal et al., 2005,Gal and Kluppel, 2008,Lorenz et al., 2011,Lorenz et al., 2013a).

In pioneering work by Grosch (1963), the author stated that rubber friction on a rough surface, which arises from
the interfacial adhesion and hysteresis deformation of the rubber, depends on velocity and temperature, and that the
obtained master curves of the friction coefficients are strongly correlated to the viscoelastic properties of the rubber. In
our experiment, as seen in Fig. 1, the friction of filled rubber on a rough surface is measured with and without lubrication
at low sliding velocities. While the friction coefficient for a dry surface increases with increasing velocity (solid line),
it remains constant for the lubricated surface (dashed line). One possible reason for this result is that the hard additive
particles in the lubricant oil, which were organomolybdenum compounds in this case, infilled the surficial cavities of
microscopic size, and prevented the rubber from penetrating the cavities, which consequently weakened the hysteretic
behavior.

The results in Fig. 1 raise the question of how the hysteresis loss of the rubber sliding on the rough surface contributes
to the frictional force, or in other words, we wish to know in which operative range of the corrugation sizes does the
rubber undergo hysteretic deformation along the sliding surface. The two new theories described previously (Kl¨uppel and
Heinrich, 2000,Persson, 2001) provide good prediction models that address the question. Experimental validations of the
two models have shown fair agreement between theory and measurements of the friction coefficient of carbon-black-filled
rubber sliding on asphalt (Westermann et al., 2004,Gal et al., 2005). We here adopt the rubber friction model developed by
Persson (2001) because we can explicitly give the required range of frequencies for the calculation, which corresponds to
the operative range of the corrugation sizes causing the hysteresis loss. To compare with the Persson model, we carry out
a friction test using carbon-black-filled rubber and dry and wet substrates with large roughness under a low load condition
in the normal direction. The adhesion force for rubber friction is in general proportional to the true contact area, and we
thus select a weak normal force to weaken the adhesive effect on the total friction force.

The remainder of the paper is organized as follows. In Section 2, we review Persson’s friction theory (Persson,

2.4

2.0

1.0

: dry surface
: lubricant surface

-4 -3 -2

log _v_ [m/s]

Fig. 1 Correlations of rubber friction on a rough surface under dry and lubricated conditions at room temperature.
We greased the same rough surface with lubricant oil including organomolybdenum compound additives.
The dimensions of the rubber block were 20 × 20 × 4 mm 3 and the normal load stress was 2450 Pa.

[DOI 10 1299/ j 15 00084] © 2016 Th J S i t f M h i l E i

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

2001,Persson et al., 2005a) and describe the key equations required for the calculation of the hysteresis friction coefficient.
In Section 3, we explain the materials we employ in the friction test. We first provide surface information, measuring the
power spectrum of surface roughness. We next describe the carbon-black-filled rubber compound and identify its complex
elastic modulus with a generalized Maxwell model. In Section 4, we conduct a friction test and assess the numerical
hysteresis part of the friction coefficient for comparison with the measured friction coefficient. In Section 5, we discuss
the additional factor as an adhesive shear force, which increases the friction coefficient. Section 6 presents the summary
and conclusion.

2. Mathematical descriptions of Persson’s friction theory
2.1. Power spectrum of surface roughness with a self-affine fractal
Considering a rough surface with asperities at multiple scales, the power spectrum of surface roughness provides
good guidance for the wide-scale surface characteristic in wavenumber space. The power spectrum of surface roughness
_C_ ( _q_ ) was defined by Nayak (1971) as

1
_C_ ( _q_ ) =
(2π) 2

∫

d 2 _x_ ⟨ _h_ ( _x_ ) _h_ ( 0 )⟩ _e_ − _i_ _q_ · _x_, (1)

where _z_ = _h_ ( _x_ ) ∈ R is the substrate height at position _x_ ∈ R 2, measured from the average surface plane, and _q_ ∈ R 2 is
a wavevector. Furthermore, ⟨·⟩ indicates ensemble averaging. Assuming that the surface profile has inplane isotropy, we
only treat the amplitude of the wavevector as the wavenumber _q_ = | _q_ |. We also assume that the surface is a self-affine
fractal with fractal dimension _D_ f in some wavenumber region [ _q_ 0, _q_ 1]. If _q_ 0 ≪ _q_ 1, Equation (1) can be approximated with
the exponent _H_ (= 3 − _D_ f) in the following form (Persson, 2001):

_q_ 0

)−2( _H_ +1)

. (2)

_C_ ( _q_ ) ≃ _H_

2π

( _h_ 0

_q_ 0

)2 ( _q_

Here ⟨ _h_ 2 ⟩ = _h_ 2 0 /2. Equation (2) shows that a self-affine surface follows a power law described by _C_ ( _q_ ) ∼ _q_ −2( _H_ +1).

It is known that many surfaces tend to be a nearly self-affine fractal. In our experiment, the surface specimen
indeed exhibits a feature similar to a power law as we will mention in Section 3.1. Figure 2(a) illustrates the surface
roughness power spectrum for a typical rough surface. The log-log plot of _C_ ( _q_ ) has two types of wavenumber regions.
One is a macroscopic area, referred to as Region I and corresponding to the range [ _qL_, _q_ 0] where _qL_ ≈ 2π/ _L_ and _L_ is the
characteristic length of a specimen. The arithmetic average roughness (Ra) is mainly determined by the surface profile in
Region I. We refer to the next range of [ _q_ 0, _q_ 1] as Region II; these wavenumbers are referred to as the lower cut-off and
upper cut-off wavenumbers, respectively. The surface property in Region II is a self-affine fractal represented by Eq. (2).

2.2. Rubber friction model of Persson’s theory
When squeezing and sliding rubber as a hyperelastic material on a rigid and rough surface, the rubber cyclically
deforms along the surface corrugations with different wavelengths and amplitudes at each magnification ζ, as shown in

b ¾0
v

ζ =1

ζ =10

ζ = ζmax

ζ =100

_q_ = ζ _q_ 0

_qL_ _q_ 0 _q_ 1
log _q_

Fig. 2 (a) A schematic drawing of the power spectrum of surface roughness for a conventional rough surface;
(b) a conceptual sketch of the mechanism of rubber friction for a rubber block sliding on multiple-scale
asperities that form a self-affine fractal surface, following Persson’s idea (Persson, 2001,Persson et al.,
2005a).

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

Fig. 2(b). In this context, Persson developed rubber-friction theory to predict the friction coefficient due to pure hysteresis
of rubber deformation without adhesion (Persson, 2001). In the theory, let _A_ 0 and _A_ ( _q_ ) be the macroscopic (nominal)
contact area and the contact area at some wavenumber scale, respectively, the relative area of real contact is

_A_ ( _q_ )

_A_ 0

= 2

≃ (1 + [π _G_ ( _q_ )] 3/2 ) −1/3 . (3)

∫ ∞

exp
0 d _x_ sin _x_ _x_

[

- _x_ 2 _G_ ( _q_ )

]

In Eq. (3), _G_ ( _q_ ) is

�����, (4)

_G_ ( _q_ ) = 1

8

_E_ ( _q_ v cos ϕ)
�����

(1 − ν 2 )σ0

_E_ ( _q_ v cos ϕ)
�����

(1 − ν 2 )σ0

∫ _q_

_q_ 0 d _qq_ 3 _C_ ( _q_ )

∫ _q_

∫ 2π

dϕ
0

∫ 2π

where v, σ0 and ν are the sliding velocity, nominal stress and Poisson’s ratio of the rubber, respectively. Furthermore,
_E_ ( _q_ v cos ϕ) = _E_ (ω) ∈ C, which is the complex elastic modulus as a function of the angular frequency ω. With the three
elements mentioned above, namely _C_ ( _q_ ), _A_ ( _q_ ) and _E_ (ω), the hysteresis friction coefficient µh can be expressed by

µh = 1

2

_A_ 0

_qq_ 0 1 d _qq_ 3 _C_ ( _q_ ) _A_ _A_ ( _q_ 0 )

∫ 2ϕ _E_ (ω)

dϕ cos ϕIm
0 (1 − ν 2 )σ0

∫ _q_ 1

. (5)

In the case of σ0 ≪ _E_ (0), _G_ ( _q_ ) ≫ 1 for most _q_ -values of interest, and Eq. (3) can be written in the simpler form

_A_ ( _q_ )

_A_ 0

≃ 2

∫ ∞ [

0 d _x_ exp

∞ [

0 d _x_ exp

]

= [π _G_ ( _q_ )] −1/2, (6)

- _x_ 2 _G_ ( _q_ )

where sin _x_ ≃ _x_ . Substituting Eq. (6) into Eq. (5), σ0 vanishes, and µ thus does not depend on σ0. Furthermore, Eq. (5)
reveals that µ depends on _q_ and not on the magnitude of the complex elastic modulus, because _A_ / _A_ 0 ∝ α −1 as _E_ → α _E_,
where α is a scaling parameter.

Taking into account the self-affine fractal surface, Eq. (5) can be approximated using Eqs. (2)–(4) as

µh ≃ 1

4

_q_ 1/ _q_ 0 dζζ −2 _H_ +1 (

1 + [π _G_ (ζ)] 3/2)−1/3 ∫ 2π

2π dϕ cos ϕIm _E_ (ζ _q_ 0v cos ϕ)

0 (1 − ν 2 )σ0

(1 − ν 2 )σ0

1 ∫ _q_ 1/ _q_ 0

4π ( _q_ 0 _h_ 0)2 _H_ 1

, (7)

where _G_ (ζ) is

����� . (8)

1 ∫ _q_ 1/ _q_ 0
_G_ (ζ) = dζζ −2 _H_ +1
16π ( _q_ 0 _h_ 0)2 _H_ 1

∫ 2π

dϕ
0

∫ 2π

_E_ (ζ _q_ 0v cos ϕ)
�����

(1 − ν 2 )σ0

Note that _q_ is replaced with ζ _q_ 0, recalling ζ is the spatial magnification as shown in Figs. 2(a) and (b).

The fundamental idea in Persson’s theory is that the macroscopic work resulting from the shear friction force is
equivalent to the total dissipative energy built up from the hysteresis loss due to cyclic deformations of the interfacial
rubber across the surface at multiple scales. However, it is hard to accept that such a purely hysteretic behavior has a
nanometer feature because it becomes dominated by other phenomena such as an adhesion interaction, contamination
interference and inhomogeneous deformation of filled rubber at the microscale. Therefore, an upper cut-off wavenumber
should be considered as a free parameter that cannot be determined only by surface information. Persson clearly states
that an upper cut-off wavenumber may be related not only to surface properties but also rubber properties (Persson et
al., 2005a), and his group indeed determined _q_ 1 for the sizes of compact rubber-wear particles in a friction experiment
(Lorenz et al., 2011). In their study, it was estimated that _q_ 1 = 5 × 10 6 m −1, which was consistent with the diameter of 20
µm of the smallest wear particle. Our prime interest as regards Eq. (7) is how to adequately determine the free parameter
_q_ 1. In this regard, we will give a qualitative assessment of _q_ 1 with respect to the rate of increase in the friction coefficient
obtained from the measurements and simulations in Section 4.2.

3. Materials
3.1. Measurement of the power spectrum of surface roughness
We employed commercial anti-slip tape (Safety Walk type A TM, 3M Japan Ltd.) as a hard and rough surface. With
two types of laser microscopes, we measured the surface profiles at the three magnifications listed in Table 1. Figures
3(a)–(c) show the typical surface profiles and the probability distribution of the surface height, measured by microscopes
at 40- and 200-power magnification. Note that the raw data for 200-power magnification are contaminated by much
optical noise, and the measured height data were thus smoothed with a median filter. It is observed that the surface
profile for 200-power magnification has a shorter wavelength than that for 40-power magnification. Figure 4 shows the

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

Table 1 Conditions of the surface measurements.

instrument microscope VR-3100 laser microscope VK-9500
(manufacturer) (Keyence Co., Ltd.) (Keyence Co., Ltd.)
height resolution 0.1 µm 0.01 µm
XY calibration 7.403 µm 1.391 µm 0.137 µm
magnification ×40 ×200 ×2000
observed area 15.16 × 15.16 mm ~~2~~ 2.849 × 2.849 mm ~~2~~ 0.281 × 0.281 mm ~~2~~

measuring points 2048 × 2048 2048 × 2048 2048 × 2048
the num. of samples 2 3 6

Table 2 Measured surface properties of a road specimen.

name Safety-Walk type A ~~TM~~

(3M Japan Ltd.)
_h_ 0 [µm] 82.89
_q_ 0 [1/m] 5070
_b_ [m ~~4~~ ] −16.57 ± 0.19
_H_ 0.631 ± 0.014

#### a

0.3

0.2

0.1

–0.1

–0.2

–0.3

0 2 4 6 8 10 12 14 16 18 20

_x_ [mm]

0.4

0.3

0.2

0.1

–0.1

–0.2

–0.3

–0.4

5 ×10 -3

0 0.2 0.4 0.6 0.8 1.0

_x_ [mm]

×40 (2 samples)

×200 (3 samples)

0
-0.6 -0.3 0 0.3 0.6

_h_ [mm]

Fig. 3 Typical surface profiles measured by the two microscopes with (a) 40- and (b) 200-power magnifications.
Note that the scales on the two axes in (a) are unequal and that the measured height data of (b) are smoothed
using a median filter. (c) Probability distributions of the surface height.

calculated power spectrum of surface roughness for a magnifying power of 40, 200 or 2000. Note that we conducted
an averaging procedure for each _C_ ( _q_ ) and that we ignored the high-frequency components resulting from optical noise
at each magnification. The calculated _C_ ( _q_ ) well represents the self-affine fractal property. The scale of the measured
minimum wavelength is on the order of a few micron, corresponding to the maximum wavenumber, ∼ 10 6 1/m, which is
large enough to predict the experimental friction coefficients at the measured range of sliding velocities and temperatures
(see Section 4).

As shown in Fig. 2(a), _C_ ( _q_ ) is divided into Regions I and II. In each of the two regions, bilinear equations are
approximated for the measured _C_ ( _q_ ) through linear least squares fitting:



log _C_ ( _q_ ) = _b_, _qL_ ≤ _q_ ≤ _q_ 0,
(9)
log _C_ ( _q_ ) − _b_ = −2( _H_ + 1)(log _q_ - log _q_ 0), _q_ 0 ≤ _q_,

where _b_ is a constant as illustrated in the inset of Fig. 4. Table 2 gives the obtained surface property of the specimen.

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

–14

–16

–18

–20

–22

–24

–26

3.0 4.0 5.0 6.0

log _q_ [1/m]

_b_

: surface (×40)
: surface (×200)
: surface (×2000)

–2( _H_ +1)

_q_ L

_q_ 0

Fig. 4 Power spectra of surface roughness for 40-, 200- and 2000-power magnifications.

Here _h_ 0 is calculated using Eqs. (2) and (9) as

~~√~~

2π
_H_ _q_ 0 210 _b_ ≈ 82.886 µm. (10)

_h_ 0 =

From the raw surface-height data, we can confirm that _h_ 0 = 2⟨ _h_ 2 ⟩≈ 94.877 µm, which is of the same order as the result

obtained with Eq. (10), where we remove the top and bottom 1% of the height distribution data to filter noise. The surface
specimen has a wavelength corresponding the lower cut-off _q_ 0 as λ0 = 2π/ _q_ 0 ≈ 0.813 mm, and the exponent _H_ ≈ 0.63
corresponds to the fractal dimension _D_ f ≈ 2.37, which is within the allowable region of 2.2 to 2.8 that is common for
self-affine fractal surfaces (Persson et al., 2005a). In particular, our specimen is categorized as a rough surface because
_H_ - 0.7 in the two earlier studies (Westermann et al., 2004,Lorenz et al., 2011).

From the raw surface-height data, we can confirm that _h_ 0 =

√

3.2. Complex elastic modulus of the rubber sample
We present the relaxation stress and modulus of the rubber sample employing a generalized Maxwell model with _n_
terms as illustrated in Fig. 5, each of which is

)

(

)

, _E_ ( _t_ ) = _E_ 0 +

∑ _n_ _Ei_ exp

_i_ =1

- _t_

τ _i_

- _t_

τ _i_

- _t_

σ( _t_ ) = σ0 +

∑ _n_

_i_ =1

σ _i_ exp

, (11)

where σ _i_, _Ei_ and τ _i_ are the stress, Young’s modulus and relaxation time of the _i_ -th term, respectively. The relaxation time
has the relationship τ _i_ = η _i_ / _Ei_, where η _i_ is the _i_ -th coefficient of viscosity. Applying dynamic deformation to the rubber,
the complex elastic modulus can be described as

1

1 1
_~~E~~_ _i_ + _~~i~~_ ωη _i_

_E_ (ω) = _E_ ′ + _iE_ ′′ = _E_ 0 +

∑ _n_

_i_ =1

, (12)

where _E_ ′ and _E_ ′′ are storage and loss elastic moduli, respectively.

For a sliding substance, we employed carbon-black-filled rubber with the compound formulation listed in Table 3.
To obtain the viscoelasticity of this rubber sample, we conducted a torsional shear test using a commercial rheometer
and measured the dynamic elastic modulus at several frequencies _f_ (= ω/2π) and temperatures _T_ . The dynamic strain
amplitude was 0.1%. The shift factor _aT_ as a function of _T_ can be expressed by the Williams–Landel–Ferry equation
(Willimas, 1955):

log _aT_ = − _C_ 1( _T_ − _T_ s)

_C_ 2 + _T_ - _T_ s

, (13)

where _T_ s is a reference temperature. The employed rubber sample has a shift factor with _C_ 1 = 10.36 and _C_ 2 = 144.5
at _T_ s = 25 ◦ C. Using _aT_, we obtained the master curve of viscoelastic data for _E_ ′, _E_ ′′ and tan δ = | _E_ ′′ / _E_ ′ |, as shown in
Fig. 6. Note that we used the isotropic relationship _G_ = _E_ /2(1 + ν) for ν = 0.5 to convert the measured shear modulus _G_
to _E_ . On the basis of the measurements, we construct the fitting curves with the 20–term generalized Maxwell model of

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

_E_ 0 _E_ 1 _E_ 2

_Ei_ _En_

η1 η2 η _i_ η _n_

Fig. 5 Schematic of the _n_ –term generalized Maxwell model.

Table 3 Compound formulation of a rubber sample.

SBR [phr] 100
carbon black [phr] 70
softener [phr] 40
sulfur [phr] 1.5
others: age inhibitor, accelerator, etc.

1.0E+9

1.0E+8

1.0E+7

1.0E+6

1.0E+5

1.0E+4

: Re _E_ [Pa]
: fitting curve
: Im _E_ [Pa]
: fitting curve
: tan δ
: fitting curve

1.0E+3-4 -2 0 2 4 6 8

log _f_ [Hz]

0.8

0.6

0.4

0.2

Fig. 6 Complex elastic properties of a rubber specimen: the measured data at a strain amplitude of 0.1% and a
reference temperature of 25 °C, and fitting curves obtained with a 20–term generalized Maxwell model.

Eq. (12), where _E_ (0) = _E_ 0 ≈ 26.72 MPa. Note that the fitted _E_ ′′ is linearly interpolated in the lower frequency region to
prevent undulation due to the higher-order terms of the generalized Maxwell model.

It is well known that the storage elastic modulus _E_ ′ of filled rubber depends on the strain amplitude, which is so-called
nonlinear viscoelastic behavior. The storage elastic modulus in general is lower under large stain (Payne, 1964, Payne,
1965,Westermann et al., 2004,Lorenz et al., 2013a). As we mentioned in Section 2.2, the friction coefficient of Persson’s
theory does not depend on | _E_ | in the case of σ0 ≪ _E_ (0). However, it is unclear how µ depends on the relative change in
the amplitudes between _E_ ′ and _E_ ′′ . We here modify the storage elastic modulus as β _E_ ′ to concisely represent nonlinear
viscoelastic behavior with a scaling parameter, 0 ≤ β ≤ 1. The corrective complex elastic modulus is then written as
_E_ = β _E_ ′ (1 + _i_ tan δ/β). When σ0 ≪ _E_ (0), Eqs. (4) and (6) with β _E_ ′ are rewritten as

. (14)

)]−1/2

[ π

8

∫ _q_

_q_ 0 d _qq_ 3 _C_ ( _q_ )

_A_ ( _q_ )

_A_ 0

= (1 − ν2)σ0

∫ 2π

dϕ _E_ ′2
0

(

1 + tan2 δ
β 2

For a common type of rubber, tan δ is small at very low frequency. Hence, _A_ / _A_ 0 ∝ 1/β and furthermore µh ∝ 1/β for
_E_ ′ → β _E_ ′ when (tan δ)/β ≪ 1 at very low frequency corresponding to lower sliding speed. It is natural that the real
contact area increases as rubber becomes softer, thus increasing the friction coefficient.

4. Experimental results and numerical analysis
4.1. Friction testing
Using a tribological testing instrument (TL201Ts, Trinity-Lab Inc.), we conducted a friction test for the surface and

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

disc weights

equilibrator with
a horizontal load cell

rubber block

movable stage
rough surface

Fig. 7 Schematic of a rectilinear friction test with a tribological testing instrument (TL201Ts, Trinity-Lab Inc.).

Table 4 Conditions of friction tests for three types of rubber blocks.

rubber edge block dimension normal force sliding-velocity sliding distance trials
sample treatment (mm 3 ) (g) (mm/s) (mm)
R100 no 20.0 × 20.0 × 4.0 100 0.1 ∼ 1.0 20.00 3
R100’ yes 20.0 × 20.0 × 4.0 100 0.1 ∼ 1.0 20.00 3
R200 no 28.3 × 28.3 × 4.0 200 0.1 ∼ 1.0 20.00 3
R200’ yes 28.3 × 28.3 × 4.0 200 0.1 ∼ 1.0 20.00 3

rubber specimens described in Section 3. Figure 7 is a schematic of the friction test; the stage with the rough surface
can linearly move in contact with a fixed rubber block. While sliding the rubber block, a normal force was applied to the
rubber by weighting equipment; i.e., the friction test was performed under load control. Table 4 gives the experimental
conditions. We prepared two rubber blocks (R100 and R200) having different dimensions of contact area, and provided
a load force, 100 or 200 g, to each rubber block so that the nominal stress was σ0 = 2450 Pa in each case. Note that the
friction coefficient of rubber is substantially independent of such low weight (Grosch, 1963). We also chamfered the front
edge on the contact side of additional rubber blocks to compare the friction factor of the rubber with and without edge
treatment. We refer to the two additional rubber blocks with edge treatment as R100’ and R200’. To eliminate the effects
of other phenomena such as an increase in the interfacial temperature, which is the so-called _flash temperature_ (Persson,
2006), and stick-slip vibration over a sliding surface, which reduce µ, we first selected a range of low sliding velocities
from 0.1 to 1.0 mm/s. In this friction test, µ was obtained as the ratio of shear to normal forces, where the shear force was
the average of the steady-state forces measured by a horizontal load cell as illustrated in Fig. 7.

Figures 8(a) and (b) show the measured friction coefficient as a function of the logarithm (base 10) of the sliding
velocity for rubber blocks (a) R100 and R100’ and (b) R200 and R200’ with dry surfaces. Note that each of the friction
coefficients in these plots is an average taken over three trials and that the range of these sliding velocities are shifted using
Eq. (13) at a reference temperature of 25 °C. Comparing the results obtained with and without edge treatment, it is found
that the friction coefficient of edge-treated rubber was higher at all sliding velocities because the treatment prevented the
front edge of the rubber block from sticking to the surface excitations. Conversely, µ was similar for R100 and R200 and
for R100’ and R200’, and the nominal contact area was thus large enough to present the reproducible contact of surface
asperities. Hence, the nominal stress σ0 can be applied to the framework of Persson’s theory. In Figs. 8(a) and (b), each
error bar shows the average standard deviation of µ taken over the three trials. Figures 8(c)–(f) show the measured µ of
R100, R200, R100’ and R200’ for each trial, respectively. Note that the three trials correspond to three different surface
specimens under dry conditions. The magnitude of µ thus depends on a geometrical factor such as the macroscopic surface
variation and/or the chamfered edge profile. However, the rate of the monotone increase in µ was similar among the three
curves. In the next section, we compare µh derived from Persson’s theory with the average friction coefficient of R200’
because the standard deviations were smallest for this rubber block; the standard deviation was less than 5% in each trial.
The increment of µ(v) for R200’ is described as ∆µ = µ(vmax) − µ(vmin) ≈ 0.1507, where vmin ≈ 1.496 × 10 −4 m/s and
vmax ≈ 1.496 × 10 −3 m/s.

We next performed the additional friction tests for R200’ under both dry and wet conditions at the wider range of
sliding velocities from 0.1 to 10 mm/s, where we prepared the wet surface to mist with pure water. Figure 9 shows the
average of measured friction coefficients on dry and wet surfaces for three trials as a function of the logarithm (base 10)
of the sliding velocity. The wet surface lubricated with water results in the drop in the friction coefficient throughout the
measuring velocities. Thus, the adhesion effects were removed at the wet interfaces and we assume that the difference

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

1.8

1.6

1.4

1.2

a 1.8

1.6

1.4

1.2

c 1.8

1.6

1.4

1.2

e 1.8

1.6

1.4

1.2

: R100 (average of trials)
: R100’ (average of trials)

-4 -3

log _a T_ _v_ [m/s]

_T_ = 23.8 ˚C

: R100, 1st trial
: R100, 2nd trial
: R100, 3rd trial

-4 -3
log _v_ [m/s]

_T_ = 22.1 ˚C

: R100’, 1st trial
: R100’, 2nd trial
: R100’, 3rd trial

-4 -3
log _v_ [m/s]

: R200 (average of trials)
: R200’ (average of trials)

-4 -3
log _a T_ _v_ [m/s]

_T_ = 24.4 ˚C

: R200, 1st trial
: R200, 2nd trial
: R200, 3rd trial

-4 -3
log _v_ [m/s]

_T_ = 22.6 ˚C

: R200’, 1st trial
: R200’, 2nd trial
: R200’, 3rd trial

-4 -3
log _v_ [m/s]

_T_ s = 25.0 ˚C

d 1.8

1.6

1.4

1.2

f 1.8

1.6

1.4

1.2

Fig. 8 (a) Average friction coefficients of R100 and R100’ at a reference temperature of 25 °C; (b) average friction
coefficients of R200 and R200’ at a reference temperature of 25 °C; friction coefficients of (c) R100, (d)
R200, (e) R100’ and (f) R200’ for each trial.

1.8
: R200’, dry
: R200’, wet

1.6

1.4

1.2

-4 -3
log _a T_ _v_ [m/s]

_T_ s = 25.0 ˚C

-2

Fig. 9 Average friction coefficients of R200’ under dry and wet conditions where the reference temperature is 25
°C, and the sliding velocity ranges from 0.1 to 10 mm/s.

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

between the friction forces on the dry and wet surfaces is equal to the adhesive shear force. As mentioned previously,
flash temperature reduces the friction coefficient at higher sliding speed. It is true that the rate of increase in µ reduced
with increasing v and then stick-slips occurred when log v - −2.5, which indicates a bifurcation from steady sliding to
stick-slip instability (Baumberger et al., 1994,Heslot et al., 1994,Ozaki and Hashiguchi, 2010,Ozaki et al., 2013).

4.2. Numerical assessment
In Section 4.1, the experiment revealed that the increments of µ(v) did not vary greatly among all measurements in
the range of 10 −4 ≤ v ≤ 10 −3 mm/s, and the existence of adhesion friction was shown by the friction test held under a
wet condition. We here conduct sensitivity analyses of pure rubber hysteresis based on Persson’s theory and compare the
results with the experimental results in terms of the magnitude and increment of µ(v).

With the surface description given in Table 2 and the viscoelastic model shown in Fig. 6, we estimate the friction
coefficient from Eqs. (7) and (8) without nonlinear viscoelastic behavior; i.e., β = 1. In Fig. 10(a), the solid lines show the
calculated µh as a function of the logarithm (base 10) of v for the different magnifications ζmax, which corresponds to the
upper cut-off wavenumber according to _q_ 1 = ζmax _q_ 0, and the dot-dashed curve shows the measured friction coefficient for
R200’, which we here indicate as µexp. When ζmax = 3000, the amplitude of µh agrees well with that of µexp at the lower
side of sliding speed. However, the difference increases as v increases. Conversely, the rate of increase in µh comes close
to that of µexp with decreasing ζmax, and they best agree at ζmax = 250, although the value of µh becomes smaller across
the whole speed range. This means that the scale around ζmax = 250 is the valid magnification for the prediction of the
hysteresis friction. The obtained ζmax corresponds to _q_ 1 ≈ 1.27 × 10 6 m −1 (λ ≈ 4.96 µm), the order of which agrees well
with that of other experiment estimations (Westermann et al., 2004,Lorenz et al., 2011). In the case that v = 10 −4 ∼ 10 −2.7

m/s and ζmax = 250, the operative frequency for the calculation of µh, which corresponds to [ _q_ 0, _q_ 1], ranges from 0 to
403.4 Hz.
Figure 10(b) shows _A_ / _A_ 0, which is calculated using Eq. (3), as a function of ζ with varying sliding velocity. It
is clear that the ratio of the real contact area is of the order of 10 −4 ∼ 10 −5 for magnifications from 250 to 3000 and
that there are only a few differences in _A_ / _A_ 0 among the measuring velocities for the fixed magnification. Considering
ε0 = σ0/ _E_ ′ (0) = 2450/(26.722 × 10 6 ) ≈ 9.17 × 10 −5, the macroscopic normal strain of sliding rubber is infinitesimal.
However, the normal stress for the large magnification remarkably increases along with a sudden decrease in _A_ / _A_ 0.
Therefore, the normal strain should not be considered small at such a microscopic scale.

We next conduct sensitivity analysis of µh for a change in β, accounting for the effect of nonlinear viscoelastic behavior. Figures 11(a)–(c) show µh(v)–curves for the three representative parameters β = 0.8, 0.6 and 0.4, respectively. The
results in Figs. 11(a)–(c) reveal that all the hysteresis friction coefficients increase at each magnification with decreasing
β, which is attributed to the increasing real contact area described by Eq. (14). For example, the rate of increase can be
simply calculated as _A_ → 0.25 _A_ when ζmax = 250 and β = 0.8, because tan δ is at most 0.2 in the operative frequency
domain ranging from 0 to 403.4 Hz (see Fig. 6) and tan 2 δ/β 2 can be ignored in the integral range. Figure 11(d) focuses on
the increment of the friction coefficient at ζmax = 250; ∆µh changes from 0.085 to 0.182 as β decreases from 1 to 0.4. The

10-5

2000 3000

sliding velocity, _v_

10-3

10-4

10-5

#### a 3.0

2.0

1.0

ζmax = 3000

ζmax = 2000

ζmax = 1000

ζmax = 500
ζmax = 250

#### b

: Numerical Sol.,   = 1β
:     for R200’µexp

: 0.1 mm/s
: 0.2 mm/s
: 0.3 mm/s

: 0.4 mm/s
: 0.5 mm/s
: 1.0 mm/s

-4 -3
log _v_ [m/s]

-6

1000 2000
ζ

10
0 1000 2000 3000

Fig. 10 (a) Calculated µh curves as a function of v for five different magnifications (solid lines) and the
experimental curve of R200’ (dot-dashed curve); (b) the calculated relative area of real contact _A_ / _A_ 0
as a function of ζ at the different sliding velocities.

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

#### a 3.0

2.0

1.0

ζmax = 3000

ζmax = 2000

ζmax = 1000

ζmax = 500

ζmax = 3000

ζmax = 2000

ζmax = 1000

ζmax = 500

ζmax = 250

#### b 3.0

2.0

1.0

: Numerical Sol.,   = 0.8β
:     for R200’µexp

ζmax = 250 : Numerical Sol.,   = 0.8β

µexp

: Numerical Sol.,   = 0.6β
:     for R200’µexp

-4 -3
log _v_ [m/s]

#### c 3.0

2.0

1.0

ζmax = 2000

ζmax = 1000

ζmax = 500

ζmax = 250

#### d 0.2

0.1

(0.5199, 0.1507)

∆µ exp

: Numerical Sol.

ζmax = 250

: Numerical Sol.,   = 0.4β
:     for R200’µexp

-4 -3
log _v_ [m/s]

0.4 0.6 0.8 1

Fig. 11 Calculated µh(v) curves accounting for nonlinear viscoelastic behavior of the filled rubber: (a) β = 0.8; (b)
β = 0.6; (c) β = 0.4, and (d) the increment of the hysteresis friction coefficient, ∆µh = µh(vmax)             - µh(vmin),
simulated for a change in β to compare with measurements.

plot shows that ∆µh at β = 0.5199 is consistent with the measurement ∆µexp = 0.1507, accounting only for the hysteresis
contribution of friction.

5. Discussion

The previous section verified that the nonlinear viscoelastic behavior of filled rubber is an important factor determining µh because a reduction in β increases _A_ / _A_ 0 and µh. It is necessary for further detailed prediction to measure the
effective viscoelastic modulus for large strain and to estimate the effective strain the rubber undergoes during sliding. An
effective procedure was studied by Lorenz et al. (2013a). In our experiment, the nominal strain is ε0 ≈ 9.17 × 10 −5 .
Therefore, the complex elastic modulus measured at strain amplitude of 0.1 % is adequate on a macroscale. However, the
ratio of the real contact area is less than of order 10 −4 and effective strain at the microscale, ε = _A_ 0ε0/ _A_, is estimated to
be of order unity. Strain sweep testing in general shows that the dynamic storage modulus of carbon-black-filled rubber
decreases less than half under such large strain (Payne, 1964,Payne, 1965,Barr´es et al, 2003), which is acceptable for our
numerical assessment that the simulated rubber coefficient β < 1/2 is in good agreement with µexp. The complete prediction of µh requires the true complex elastic modulus as a function of the strain amplitude changing across the multiple
sizes of asperities.

We next consider the additional factor pertaining to the adhesive effect, which is well known to be important to rubber
friction on a dry surface (Gal et al., 2005,Gal and Kluppel, 2008,Lorenz et al., 2011,Lorenz et al., 2013a). The adhesive
force at the interface directly contributes to the shear force of friction; i.e., the total friction coefficient is given by

µ = µh + µa, µa = τs

σ0

_A_
_A_ 0

, (15)

where µh is the hysteresis friction coefficient derived using Eq. (7) and µa is the adhesive friction coefficient due to the
shear stress τs on the real contact area _A_ . The true interfacial shear stress is attributed to microscopic bond breaking (i.e.,

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

### a -4 2.0

-4

ξmax= 250

-5

ξmax= 500

: log _A A_ / 0

: log _G G_ / 0

ξmax= 250

ξmax= 500

1.8

1.6

1.4

-6

-6 -5 -4 -3 -2 -1 0 1

log _v_ [m/s]

### b

0.3

0.2

0.1

ξmax= 250

: µa

: τ s 15

ξmax= 500

ξmax= 250

ξmax= 500

-6 -5 -4 -3 -2 -1 0 1

log _v_ [m/s]

Fig. 12 Elements required to predict the adhesive friction coefficient µa: (a) _A_ / _A_ 0 and _G_ / _G_ 0; (b) τs and µa. Each
function is calculated in the range of crack propagating velocity from 10 −6 to 10 m/s when ζmax = 250
with β = 0.4 or ζmax = 500 with β = 0.5. Other details of the analysis are described in Appendix A.

the origin of wear), and the predictive modeling of τs has been developed in terms of surface free energy (Gal et al.,
2005,Gal and Kluppel, 2008) or crack propagating energy (Lorenz et al., 2011,Persson and Brener, 2005b,Persson et al.,
2005c). We here adopt the latter adhesion model, and the concrete calculation of τs is summarized in Appendix A.
Figure 12(a) shows the ratio of crack propagating energy _G_ / _G_ 0 in the sliding velocity range from 10 −6 to 10 m/s,
estimated using Eqs. (A.1)–(A.4), when ζmax = 250 with β = 0.4 or ζmax = 500 with β = 0.5. These conditions of ζmax and
β will be later adopted to compare the calculated µ in Eq. (15) with the measured µ under dry and wet conditions. Note
that, in the calculation of µa, we assume that the crack propagating velocity is equal to the sliding velocity of the rubber
friction. In Fig. 12(a), we include _A_ / _A_ 0 calculated in Eq. (3) under the same conditions of ζmax and β. Subsequently, we
analyze τs and µa using Eqs. (A.5) and (15) as shown in Fig. 12(b).

The results in Figs. 12(a) and (b) reveal the following. The order of the calculated τs is similar to that obtained in an
earlier experiment (Lorenz et al., 2011). Nevertheless, µa predicted in this study is 6 ∼ 10 times smaller than the measured
friction coefficient (see Fig. 8) since the ratio of real contact area _A_ / _A_ 0 is small owing to the low load stress. Alternatively,
_A_ / _A_ 0 rapidly drops with increasing v in contrast to the increase in _G_ / _G_ 0, which explains why µa asymptotically approaches
zero as v increases. The obtained frictional property is the conventional behavior that the viscoelastic peak of the friction
coefficient due to adhesion occurs at a low sliding speed (Grosch, 1963, Moore, 1980). By comparing the cases for
ζmax = 250 with β = 0.4 and ζmax = 500 with β = 0.5, it is found that all the four elements, _A_ / _A_ 0, _G_ / _G_ 0, τ _s_ and µa, are
reduced with increasing the magnification ζmax throughout the range of the sliding velocity.

Figure 13 shows the calculated friction coefficient including the hysteresis and adhesion terms as µ = µh + µa for
ζmax = 250 and β = 0.4, and the result is that the magnitude of µ agrees well with that of µexp for R200’ as shown in
Fig. 8(b). To investigate whether the predicted adhesion friction is well estimated, Figures 14(a) and (b) compare the
calculated hysteresis and adhesion friction with the experiment value under the dry and wet conditions in Fig. 9. In Fig.
14(a), the predicted µ (bold solid line) for ζmax = 250 and β = 0.4 is also in good agreement with µexp on the dry substrate
at the lower sliding speed. However, the calculated µh (bold dashed line) is lower than µexp on the wet substrate at the
lower sliding speed, thus, µa is overestimated. On the other hand, in the case of ζmax = 500 and β = 0.5, the difference
of µexp between dry and wet conditions is well represented by µa (see Fig. 14(b)). However, the rate of increase in the
coefficient with respect to sliding velocity enlarges due to the double scale of ζmax, thus, µh becomes overestimated with
increasing v.

There are two possible explanations for this inconsistency. One explanation is that a temperature-induced vibration

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

1.8

1.6

1.4

1.2

: R200, dry
: R200’, dry

:
:

µh + µa

µh

1.0

-4 -3
log _a T_ _v_ [m/s]

_T_

Fig. 13 Correlations of the numerical hysteresis and adhesion friction and the measurements as shown in Fig.
8(b): the bold dashed line indicates the friction coefficient calculated with only the hysteresis term under
the conditions of ζmax = 250 and β = 0.4, and the bold solid line indicates both the hysteresis and
adhesion terms under the same conditions.

#### a 1.8

: R200’, dry
: R200’, wet

#### b 1.8

: R200’, dry
: R200’, wet

:
:

µh + µa

:
:

µh + µa

1.6

1.4

1.6

1.4

µh

_T_ s = 25.0 ˚C
ξmax= 250, β = 0.4

1.2

-4 -3
log _v_ [m/s]

-2

_T_ s = 25.0 ˚C
ξmax= 500, β = 0.5

1.2

-4 -3
log _v_ [m/s]

-2

Fig. 14 Correlations of the numerical hysteresis and adhesion friction and the measurements under dry and wet
conditions as shown in Fig. 9: the bold dashed and solid lines indicate the friction coefficients calculated
by µh and µ = µh + µa, respectively, under the conditions of (a) ζmax = 250 with β = 0.4 or (b) ζmax = 500
with β = 0.5.

arises at the local interface in the range of high sliding velocities. Since the rubber is softened by flash temperature, partial
local slips, which are referred to as _precursors_, easily occur (Ozaki et al., 2014). The propagation of these precursor
events along the sliding direction generates macroscopic stick-slip motion and reduces the friction coefficient (Rubinstein
et al., 2004, Ozaki et al., 2014). During sticking-slipping, the rubber friction is no longer dominated by pure hysteresis
deformation, and the predictions thus disagree with the measurements at a higher sliding velocity. The other explanation is
that the concise form of Eq. (14) with a scaling parameter β is not adequate to precisely represent the nonlinear viscoelastic
behavior of the filled rubber. Different from our model, which uniformly scales the storage elastic modulus, the complex
elastic modulus of filled elastomers indeed exhibits complicated amplitude- and frequency-dependent properties (H¨ofer
and Lion, 2009), where the hysteresis loss is strongly related to properties of the molecular chain microstructure of filled
rubber such as the volume fraction and the distribution morphology of carbon black (Tomita et al., 2008). The constitutive
approach of nonlinear viscoelasticity has been extensively developed to represent such an observed material behavior
(H¨ofer and Lion, 2009, Osterl¨of ¨ et al., 2014) and the frictional formulation of Persson’s theory should be enhanced to
extend the viscoelastic modulus with these appropriate models.

6. Summary and conclusion

We studied a prediction model for the friction coefficient of a rubber block sliding over a rough and rigid surface,
employing Persson’s theory, and compared the results obtained with experimental measurements. First, we briefly ex

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

plained hysteresis rubber friction theory developed by Persson for rubber sliding over a nonadhesive surface with rigid
asperities of multiple scales, which was formulated with rubber viscoelasticity and the power spectrum of surface roughness. Subsequently, we measured the self-affine fractal property of the surface with laser microscopes and identified the
complex elastic modulus of the rubber sample employing a 20-term generalized Maxwell model, fitting the master curves
obtained in a vibration test. With surface and rubber specimens, we conducted friction tests at low sliding velocities with
light squeezing, and qualitatively assessed the experimental results using the prediction model described as the addition
of the hysteresis and adhesive friction coefficients: µ = µh + µa.

The main insights obtained can be summarized as follows. Employing Persson’s theory, we estimated the wavenumber range of the surface height as [ _q_ 0, _q_ 1]≈[5.07 × 10 3, 1.27 × 10 6 ] (1/m), where the hysteresis of rubber viscoelastic
deformation was purely integrated. The obtained upper cut-off wavenumber corresponded to the smallest wavelength of
about 5 µm, the order of which was similar to that of estimations in earlier studies (Westermann et al., 2004,Lorenz et al.,
2011). Accounting for nonlinear viscoelastic behavior, we modeled the corrective complex elastic modulus with a scaling
parameter β, and found that the proposed model modestly agreed with both the magnitude of the measured friction coefficient and the rate of increase in the coefficient with respect to sliding velocity when ζmax = _q_ 1/ _q_ 0 = 250 and β = 0.4 ∼ 0.5.
In addition, we simulated the adhesion force derived from the crack propagating energy (Lorenz et al., 2011). We then
showed that the adhesion friction for ζmax = 250 was overestimated to compare with the experiment measurements under
dry and wet conditions and that our model described using a scaling parameter β needs to be improved for the precise
prediction of rubber friction including the hysteresis and adhesion effects.

In this study, we adjusted the two parameters _q_ 1 and β to coincide with the measured friction coefficient. We thus
revealed the qualitative nature of friction for filled rubber on a rough surface, such as the change in both the friction
coefficient and the real contact area along with sliding velocity and nonlinear viscoelastic behavior of rubber. As it now
stands, it is hard to quantitatively distinguish these parameters because a single parameter of a friction coefficient is
consolidated from multiple physical elements containing the two parameters. To that end, we need to conduct further
experiments at several temperatures to investigate the rate of the friction coefficient in a wider range of sliding velocities.
We also need to extend the experiments using several combinations of types of compound rubber and surfaces, including
unfilled rubber and silicon-filled rubber, and lubricated surfaces, and to establish an assessment framework from the
resulting database.

Acknowledgements

The authors thank Dr. K. Yamagiwa (National Institute of Occupational Safety and Health) for the help with the
measurements of surface profiles. This work was partly supported by Bridgestone Corporation and by the Japan Society
for the Promotion of Science under Grants-in-Aid for Scientific Research (B) and Young Scientists (A) (JSPS KAKENHI
Grant Nos. 30322069 & 25709001).

Appendix A. Estimation of shear stress due to crack propagation by Persson’s thoery

Employing theory on energy dissipation at an opening crack between rubber and a substrate (Lorenz et al., 2011,
Persson and Brener, 2005b, Persson et al., 2005c), we here estimate the adhesive shear stress and the related friction
coefficient due to crack propagation of the interfacial rubber. In Ref. (Lorenz et al., 2011), the relationship between the
crack propagation energy _G_ (v) and the crack-tip radius _a_ (v) with crack propagating velocity v satisfies the similarity rule

_a_ (v)

_a_ 0

= _G_ (v), (A.1)

_G_ 0

where _G_ 0 and _a_ 0 relate to a very slowly propagating crack. With numerical consideration, the left-hand side of Eq. (A.1)
is described as (Lorenz et al., 2011)

1 − _E_ (ω0)
_E_ (ω1)

∫ ξ∗

0 dξ _F_ (ω0 _e_ ξ)Im[1/ _E_ (ω0 _e_ ξ)]

)

ξ∗, (A.2)

0 dξIm[1/ _E_ (ω0 _e_ ξ)]

_a_ 0

_a_ = 1 −

(

~~∫~~ ξ∗

where

_F_ (ω0 _e_ ξ ) =

[1 − _e_ 2(ξ−ξ∗)]1/2, (A.3)

and

Tanaka, Yoshimura, Sekoguchi, Aramaki, Hatano, Izumi, Sakai and Kadowaki,

Mechanical Engineering Journal, Vol.3, No.1 (2016)

)

, (A.4)

ξ ∗ = ln

( 2πv _aT_

_a_ ω0

where _aT_ is the shift factor in Eq. (13). Furthermore, _E_ (ω0) ≈ _E_ (0) and _E_ (ω1) ≈ _E_ (∞). In our calculation, we set
sufficiently small and large values as ω0 = 1.0 × 10 −6 rad/s in the rubbery region and ω1 = 1.0 × 10 9 rad/s in the glassy
region, respectively.

The shear stress τf, resulting from the crack propagation of the interfacial rubber, can be described as

τs = _G_ (v)

_G_ 0

_G_ 0

ℓ , (A.5)

where ℓ is the size of the real contact region. In the calculation of τs, _G_ (v)/ _G_ 0 was obtained from Eqs. (A.1)–(A.4). We
here set _a_ 0 = 1 nm and _G_ 0/ℓ = 0.25 MPa in reference to (Lorenz et al., 2011).

Equations (A.1)–(A.5) provide the numerical predictions of log _G_ / _G_ 0 and τs corresponding to our experiment data
(see Figs. 12(a) and (b)).

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