12699


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 22, No 3, Special Issue, 2024, pp. 529 - 545  

https://doi.org/10.22190/FUME240322020W 

© 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

SLIDING FRICTION IN CONTACTS WITH  

ONE- AND TWO-DIMENSIONAL VISCOELASTIC 

FOUNDATIONS AND VISCOELASTIC HALF-SPACE 

Toshiki Watanabe1, Shintaro Hatanaka1,  

Ken Nakano2, Valentin L. Popov3 

1Graduate School of Environment and Information Sciences,  

Yokohama National University, Yokohama, Japan 
2Faculty of Environment and Information Sciences,  

Yokohama National University, Yokohama, Japan 
3Department of System Dynamics and Friction Physics,  

Technische Universität Berlin, Berlin, Germany 

ORCID iDs: Toshiki Watanabe   https://orcid.org/0009-0006-4442-3074  
 Shintaro Hatanaka   https://orcid.org/0009-0006-4989-6065    
 Ken Nakano   https://orcid.org/0000-0001-7437-7984     
 Valentin L. Popov   https://orcid.org/0000-0003-0506-3804    

Abstract. Solid viscoelasticity is one of the essential origins of sliding friction, as every solid 

exhibits energy dissipation due to it during deformation processes. In this paper, we first show 

theoretical solutions for one-dimensional (1D) problems of viscoelastic friction with a 1D 

viscoelastic foundation. Then, we extend the 1D model to a two-dimensional (2D) model to 

find theoretical solutions for 2D problems of viscoelastic friction. Finally, we apply the 

Method of Dimensionality Reduction (MDR) to the theoretical solutions for the 1D problems 

to discuss three-dimensional (3D) problems of viscoelastic friction. 

Key words: Sliding friction, Viscoelastic foundations, Viscoelastic half-space, MDR 

1. INTRODUCTION 

Solid viscoelasticity is one of the essential origins of sliding friction, as every solid 

exhibits energy dissipation due to it during deformation processes [1,2]. In theoretical 

investigation, a viscoelastic foundation consisting of independent rheological elements can 

be considered an asymptotic model of contacts of a rigid probe with a very thin, 

compressible viscoelastic layer [3-5]. Another limiting case is a very thick layer, namely 

the viscoelastic half-space [6-9]. Real systems are in between these two limits. 

 
Received: April 03, 2024 / Accepted May 09, 2024  

Corresponding author: Valentin L. Popov 
Technische Universität Berlin, Strasse des 17 Juni 135, 10623 Berlin, Germany 

E-mail: v.popov@tu-berlin.de 

https://orcid.org/0009-0006-4442-3074
https://orcid.org/0009-0006-4989-6065
https://orcid.org/0000-0001-7437-7984
https://orcid.org/0000-0003-0506-3804


530 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

Contact mechanics of viscoelastic foundations is simple [10]. By capitalizing on their 

simplicity, we can obtain a rough but clear picture of the basics of sliding friction due to 

solid viscoelasticity [11-14]. For example, we need only five system parameters to describe 

the sliding contacts of a rigid probe with a Kelvin-Voigt viscoelastic foundation [15]. This 

simplicity brings master curves for the velocity dependences of several quantities (e.g., the 

contact width, indentation depth or normal load, friction force, and friction coefficient) 

described by reasonable dimensionless numbers, showing the effects of the inlet slope and 

vertical lift on the friction coefficient [16]. This analytical method has been extended to the 

standard linear solid viscoelasticity, showing how mechanical and material origins for bell-

shaped velocity-dependent friction compete [17]. 

Contact mechanics of half-spaces is much more complicated compared with elastic 

foundations due to non-locality of elastic and viscoelastic interactions. However, for 

axisymmetric normal contacts, there exist a transformation reducing a three-dimensional 

(3D) contact into a contact with a one-dimensional (1D) viscoelastic foundation. This 

transformation is given by the so-called Method of Dimensionality Reduction (MDR) [18]. 

For normal contact, the MDR provides the exact solution of the initial 3D contact problem 

[18]. There exists no exact procedure for applying MDR to sliding contacts. However, there 

are reasons to believe that a formal application of MDR to sliding systems gives 

qualitatively correct results. Since the famous work by Grosch [19], it is widely accepted 

that the energy dissipation in rubber friction is mostly due to internal dissipation in the 

viscoelastic material. Based on this idea and foregoing works [20], Persson suggested in 

1998 a simple idea of estimating energy dissipation during sliding [21]. He used the Hertzian 

solution and assumed that it is realized dynamically with the frequency corresponding to the 

time of contact of "asperities". This means that he suggested considering the sliding contact 

as a "repeated normal contact", which surely is not the exact picture of what happens in 

sliding but a very good qualitative presentation. This idea was used in a later paper by 

Persson and Tosatti [22] for treating more complicated topographies. This basic idea by 

Persson means approximate reduction of sliding contact problem to the normal contact 

problem. However, the normal contact problem is mapped exactly to a problem with a 1D 

foundation via MDR. This means, the sliding MDR solution will have the same accuracy 

as the initial concept by Persson. This hypothesis was checked by Kürschner [23] by 

comparing the formal sliding MDR-solution with the complete 3D Boundary Element 

Method (BEM) simulations. For contact with a simple fluid, he has shown that the MDR 

solution and the "numerically exact" solution differ only by a constant factor (depending 

on the probe shape). Kusche [24] could confirm a very good coincidence of MDR-solution 

with complete 3D BEM simulation for a number of more complicated rheology. These 

results show that the MDR procedure can also be used to sliding contacts with elastomers 

after correcting them by a constant factor of the order unity. 

In this paper, we first show theoretical solutions for 1D problems of viscoelastic friction 

with a 1D viscoelastic foundation (Section 2). Then, we extend the 1D model to a two-

dimensional (2D) model to find theoretical solutions for 2D problems of viscoelastic 

friction (Section 3). Finally, we apply the MDR to the theoretical solutions for the 1D 

problems to discuss 3D problems of viscoelastic friction (Section 4). 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 531 

2. SLIDING CONTACTS OF RIGID PROBES WITH A 1D VISCOELASTIC FOUNDATION 

2.1. Model for 1D Problems 

In Section 2, we consider simple 1D problems. Fig. 1 shows the analytical models for 

sliding contact between a rigid probe and a 1D viscoelastic foundation. We consider 

triangular and parabolic probes: 

 ( ) tanh x x= , (1) 

 
2

( )
2

x
h x

R
= , (2) 

respectively, where θ is the slope angle of the triangular probe, and R is the tip curvature 

radius of the parabolic probe. The viscoelastic foundation is a 1D array of independent 

Kelvin-Voigt elements along the x-axis, characterized by a stiffness (per unit area) K and 

a damping coefficient (per unit area) C. The indentation depth is δ, and the drive speed of 

the foundation is V. In this situation, the normal load (per unit length) W' and friction force 

(per unit length) F' are [16]: 

 
d

( ) d
d

b

a

h
W K h CV x

x


−

 
 = − − 

 
 , (3) 

 
d d

( ) d
d d

b

a

h h
F K h CV x

x x


−

 
 = − − − 

 
 , (4) 

respectively, where a and b represent the positions of the leading and trailing edges of the 

sliding contact, respectively. The leading edge is determined geometrically, while the 

trailing edge is determined dynamically (see Eq. (6)). The friction coefficient is obtained 

by μ = F'/W'. 

 

Fig. 1 Analytical models for sliding contact between a rigid probe and a 1D viscoelastic 

foundation; (a) triangular and (b) parabolic probes. 



532 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

2.2 Theoretical Solutions for 1D Problems 

2.2.1. Triangular Probe 

In this section, we show the theoretical solutions for the 1D sliding problem of the triangular 

probe. First, under constant-δ conditions (where the indentation depth is controlled to be 

constant), the dimensionless drive speed is [16]: 

 
tanCV

V
K






 . (5) 

Then, the normalized outlet contact width, dimensionless normal load, dimensionless 

friction force, and normalized friction coefficient under constant-δ conditions are given as 

the following explicit functions of the dimensionless drive speed [16]: 

 
1    for   0 1

0   for   1

V Vb

a V

 




 −  

 = 


, (6) 

 

2

2

1
1    for   0 1

tan 2

1
   for   1

2

V V
W

W
K

V V

 



 






+   

 = 
 + 


, (7) 

 

2

2

1
2    for   0 1

2

1
   for   1

2

V V V
F

F
K

V V

  



 




−   

 = 
 + 


, (8) 

 

2

2

4
   for   0 1

2
tan

1   for   1

V V
V

V

V

 











 −
 

+ = 




. (9) 

Then, under constant-W conditions (where the normal load is controlled to be constant), 

based on the relationship between δ and W' given by Eq. (7), we obtain the following 

dimensionless drive speed [16]: 

 

1/2

2

1/2

1/2

1
1    for   0 1

2tan

1
   for   1

2

W

V V V
CV K

V
K W

V V V

  

  



−

−

  
+    

   
 =      

+  
 

. (10) 

From Eq. (10), we obtain the dimensionless drive speed for constant-δ conditions as an 

explicit function of that for constant-W conditions [16]: 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 533 

 

2

2 2

2 2
   for   0

32

1 1 2
2    for   

2 2 3

W W

W

W W W W

V V
V

V

V V V V




 

−
= 


+ + 


. (11) 

Therefore, we obtain the following explicit forms for the other dimensionless quantities 

under constant-W conditions [16]: 

 

2

2 2
1    for   0

32

2
0   for   

3

W W

W

W

V V
Vb

a
V




−  

−
 = 






, (12) 

 

2
1/2

2

1 2
1    for   0

2 3

tan 2
2    for   

3

W W

W

W W W

V V
K

W
V V V

 



−  

 
 =    

+ − 

, (13) 

 

2 21 2
4 2    for   0

2 3

tan 2
1   for   

3

W W W W

W

W

V V V V
F

F
W

V



− −  

 
 = 

 


, (14) 

 

2 21 2
4 2    for   0

2 3

tan 2
1   for   

3

W W W W

W

W

V V V V

V







− −  


 = 




. (15) 

Fig. 2 shows the above theoretical solutions for the sliding problem between a triangular 

probe and a 1D viscoelastic foundation. The black solid line shows the solution (i.e., the 

dimensionless master curve). The blue and red broken lines show the asymptotic solutions 

for the low- and high-speed regions, respectively. When the drive speed is high, we see the 

outlet peeling of the viscoelastic foundation from the probe surface (i.e., decrease in b) in (a) 

and (b) and the vertical lift of the probe (i.e., decrease in δ) in (b). The friction coefficient 

linearly increases for low speed and shows constant for high speed in (a) and (b). 

2.2.2. Parabolic Probe 

In this section, we show the theoretical solutions for the 1D sliding problem of the 

parabolic probe. First, under constant-δ conditions, the dimensionless drive speed is [16]: 



534 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

 

Fig. 2 Theoretical solutions for the sliding problem between a triangular probe and a 1D 

viscoelastic foundation under (a) constant-δ and (b) constant-W conditions. 

 
2

CV
V

K R



 . (16) 

Then, the other dimensionless quantities under constant-δ conditions are given as the 

following explicit functions of the dimensionless drive speed [16]: 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 535 

 21
b

V V
a

   = − + + , (17) 

 ( )3 2 3/2

3

2
1 (1 )

32

W
W V V

K R
  




 = − + + , (18) 

 ( )2 4 2 3/2

2

2
2 3 2 2 (1 )

3

F
F V V V V V

K
     




 = − − + + , (19) 

 
2 4 2 3/2

3 2 3/2

2 3 2 2 (1 )2

1 (1 )

V V V V VR

V V

    


 

 


− − + +
 =

− + +
. (20) 

Then, under constant-W conditions, based on the relationship between δ and W' given 

by Equation (18), we obtain the following dimensionless drive speed [16]: 

 ( )
1/3

1/3
3 2 3/22

1 (1 )
3

WV V V V  

−
− 

= − + + 
 

. (21) 

However, it is not easy to obtain the dimensionless drive speed for constant-δ conditions 

as an explicit function of that for constant-W conditions. In this case, it is reasonable to 

take the following implicit form [16]: 

 

1/3

1/3

2
W

CV K
V V W

K RW
 

− 
 = 

 
. (22) 

This leads to the following implicit forms for the other dimensionless quantities [16]: 

 21
b

V V
a

   = − + + , (23) 

 

1/3 2/32
2/3

2

32 4

39
W

RK
W

W
  −   

 =   
   

, (24) 

 

1/3
2

4/3

4

4
W

R K
F F F W

W
 

− 
 = 

 
, (25) 

 

1/3
2

4/34
W

R K
F W

W
   − 

 = 
 

. (26) 

Fig. 3 shows the above theoretical solutions for the sliding problem between a parabolic 

probe and a 1D viscoelastic foundation. When the drive speed is high, we see the outlet 

peeling of the viscoelastic foundation in (a) and (b) and the vertical lift (i.e., decrease in δ) in 

(b). The friction coefficient linearly increases for low speed in (a) and (b). For high speed, it 

shows constant in (a) but decreases (b). The latter property, termed the velocity-weakening 

friction, is known for the source of stick-slip instabilities in sliding systems [25-32]. 



536 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

 

Fig. 3 Theoretical solutions for the sliding problem between a parabolic probe and a 1D 

viscoelastic foundation under (a) constant-δ and (b) constant-W conditions. 

 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 537 

3. SLIDING CONTACTS OF RIGID PROBES WITH A 2D VISCOELASTIC FOUNDATION 

3.1. Model for 2D Problems 

In Section 3, we extend the 1D problems to consider the 2D problems for conical and 

paraboloidal probes: 

 
2 2( , ) tanh x y x y= + , (27) 

 
2 2

( , )
2

x y
h x y

R

+
= . (28) 

The viscoelastic foundation is a 2D array of independent Kelvin-Voigt elements in the xy-

plane, characterized by a stiffness (per unit area) K and a damping coefficient (per unit 

area) C. The normal load W and friction force F are: 

 ( )
(0) ( )

(0) ( )

d d

a b y

a a y

h
W K h CV x y

x


− −

 
= − − 

 
  , (29) 

 ( )
(0) ( )

(0) ( )

d d

a b y

a a y

h h
F K h CV x y

x x


− −

  
= − − − 

  
  . (30) 

The x-positions of leading and trailing edges are dependent on their y-position:  a = a(y) 

and b = b(y), respectively. The former is determined geometrically, while the latter is 

determined dynamically (see Eq. (31)). The friction coefficient is obtained by μ = F/W. 

3.2. Theoretical Solutions for 2D Problems 

3.2.1. Conical probe 

In this section, we consider the 2D sliding problem of the conical probe. To determine 

the integral range of Eqs. (29) and (30), we must solve the following nonlinear equation: 

 
2 2 2 4

4 3 2 2

2 2 2 4
2 2 1 2 0

y y y y
V V V

a a a a
     

 
− + + − − − + = 

 
, (31) 

where a = a(0) and β = β(y) = b(y)/a(0). This difficulty comes from the fact that the vertical 

section of a cone is (not triangular but) hyperbolic, except for the vertical section by the 

plane that intersects the cone tip. However, considering the limiting cases (i.e., low- and 

high-speed cases leading to β = 1 and 0, respectively) enables us to obtain asymptotic 

solutions. The asymptotic solutions for constant-δ conditions are: 

 
2

3

   for low speedtan
3

   for high speed

W
W

K
V













 = 



, (32) 



538 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

 
3

   for low speed
tan 2

   for high speed
4

V
F

F
K

V
















 = 



, (33) 

 

3
   for low speed

2

tan
   for high speed

4

V











 = 



. (34) 

The asymptotic solutions for constant-W conditions are: 

 

1/3
2

1/3

1/2

1   for low speed
3 tan

   for high speed
3

W W

W

W

K V


  



−

−


  

 =    
   

 

, (35) 

 

1/3
9

   for low speed
8

tan
   for high speed

4

W

W

V
F

F
W



 

 
   = 



, (36) 

 

1/3
9

   for low speed
8

tan
   for high speed

4

W

W

V





 

 
   = 



. (37) 

Fig. 4 shows the theoretical solutions for the sliding problem between a conical probe 

and a 2D viscoelastic foundation, showing similar velocity dependences to those for the 

1D problem of the triangular probe (see Fig. 2). A distinct difference can be seen in the 

velocity dependence of the indentation depth for high speed (i.e., δ is proportional to V–1 

for the 1D problem and V–1/2 for the 2D problem). 

3.2.2. Paraboloidal Probe 

In this section, we consider the 2D sliding problem of the paraboloidal probe. First, 

under constant-δ conditions, we can take the same dimensionless drive speed as that for 

the 1D problem of the parabolic probe (see Eq. (16)): 

 
2

CV
V

K R



 . (38) 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 539 

 

Fig. 4 Theoretical solutions for the sliding problem between a conical probe and a 2D 

viscoelastic foundation under (a) constant-δ and (b) constant-W conditions. 

This simplicity comes from the fact that the vertical section of a paraboloid is always a 

parabola with an identical tip curvature radius. Then, we obtain the following explicit forms 

for the other dimensionless quantities under constant-δ conditions: 

 2 2 2 1

2

1 1
(1 ) (1 ) arctan( )

4 2 22

W
W V V V V

KR
    





− = + − + + , (39) 

 
2 4 2 2 1

5

5
(1 ) arctan( )

2 32

F
F V V V V V V

K R
      





− = − − + + , (40) 

 
2 2 2 1

2 4 2 2 1

3 6 (1 ) 6(1 ) arctan( )2

6 20 12 12 (1 ) arctan( )

V V V VR

V V V V V V

   


     


 

 

−

−

+ − + +
 =

− − + +
. (41) 



540 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

Then, under constant-W conditions, based on the relationship between δ and W given 

by Eq. (39), we obtain the following dimensionless drive speed: 

 

1/4

2 2 2 11 1
(1 ) (1 ) arctan( )

4 2 2
WV V V V V V    


−

− 
= + − + + 

 
. (42) 

However, it is not easy to obtain the dimensionless drive speed for constant-δ conditions 

as an explicit function of that for constant-W conditions, as in the case for the 1D problem 

of the parabolic probe (see Section 2.2.2). In this case, it is reasonable to take the following 

implicit form: 

 

1/4

1/4

2
W

CV K
V V W

K RW
 

−  
= =  

 
. (43) 

This leads to the following implicit forms for the other dimensionless quantities: 

 

1/21/2

0

W

WRK

W W






 

−
  

 =   
   

, (44) 

 

1/4
3

5/4

5

8
W

R K
F F F W

W
 

− 
 = 

 
, (45) 

 

1/4
3

5/48
W

R K
F W

W
   − 

 = 
 

. (46) 

Fig. 5 shows the theoretical solutions for the sliding problem between a paraboloidal 

probe and a 2D viscoelastic foundation, showing similar velocity dependences to those for 

the 1D problem of the parabolic probe (see Fig. 3). Some distinct differences can be seen 

in the velocity dependences of the indentation depth, friction force, and friction coefficient 

for high speed (i.e., δ is proportional to V–1 for the 1D problem and V–2/3 for the 2D problem; 

F and μ are proportional to V–1/2 for the 1D problem and V–1/3 for the 2D problem). 

4. SLIDING CONTACTS OF RIGID PROBES WITH A 3D VISCOELASTIC HALF-SPACE 

4.1. Basics of the Method of Dimensionality Reduction (MDR) 

In Section 4, we consider 3D problems between an axisymmetric rigid probe having 

the shape z = f(r) and a 3D viscoelastic half-space with the plane surface characterized by 

Kelvin-Voigt model with shear modulus G and dynamical viscosity . As most elastomers 

have Poisson ratio close to 1/2, we assume that the medium is incompressible. Under these 

assumptions, the application of the MDR consists of the following two steps [18]: 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 541 

 

Fig. 5 Theoretical solutions for the sliding problem between a paraboloidal probe and a 

2D viscoelastic foundation under (a) constant-δ and (b) constant-W conditions. 

 

Step I: The 3D viscoelastic body is replaced by a 1D viscoelastic foundation consisting of 

an array of non-interacting Kelvin-Voigt elements which stiffness k and damping 

coefficient c are defined as follows: 

 4k G x=  , 4c x=  , (47) 

where Δx is spacing between adjacent elements. 

 

Step II: The 3D profile z = f(r) is transformed into a 1D profile g(x) according to: 

 
2 2

0

( )
( ) d

x
f r

g x x r
x r


=

−
 .  (48) 



542 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

In particular, triangular and parabolic profiles are transformed as follows [18]: 

 3 3( ) tan ( ) tan
2

D Df r r g x x


 =  = , (49) 

 
2 2

3 3

( ) ( )
2 D D

r x
f r g x

R R
=  = . (50) 

Now the transformed shape g(x) is pressed into the viscoelastic foundation by the depth δ 

and moved tangentially with a given velocity v. To obtain the results which apply to the 

initial 3D problem, the coefficients of friction obtained by the above MDR procedure 

should be multiplied with 0.67 for triangular probes and 0.55 for parabolic probes [23]. 

This means that the solutions obtained in Section 2 can be applied – after corresponding 

coordinate transformation – to contacts with the 3D half-space. 

4.2. Theoretical Solutions for 3D Problems 

4.2.1. Conical Probe 

Comparison of shape and material parameter definitions given by Eqs. (1) and (3) with 

those of the MDR, Eqs. (49) and (47), shows that the following variable change must be 

undertaken: 

 3tan tan
2

D


  , 4K G , 4C  , W W  , F F  . (51) 

For the main governing parameter, the dimensionless speed (5), we obtain: 

 3tan

2

DV
V

G


 


 .  (52) 

Applying variable transformation (51) to (9) and using the scaling coefficient 0.67 for 

conical probes [23] yields: 

 

2

2

3

4
   for   0 1

21.05 tan

1   for   1

D

V V
V

V

V

 






 

 −
 

+ 




. (53) 

This equation provides the coefficient of friction of a 3D cone in contact with a Kelvin-

Voigt half-space under conditions of fixed indentation depth. 

The governing parameter for the case of a fixed normal force, Eq. (10), becomes 

 

1/2

32 tan D
W

GV
V

G W

   
  

 
, (54) 

and the coefficient of friction (15) becomes: 

 

2 2

3

1 2
4 2    for   0

2 3
1.05 tan

2
1   for   

3

W W W W

D

W

V V V V

V

 


− −  


= 




. (55) 



 Sliding Friction in Contact with Viscoelastic Foundations and Viscoelastic Half-Space 543 

4.2.2. Paraboloidal Probe 

Comparison of shape and material parameter definitions given by Eqs. (2) and (3) with 

those of the MDR, Eqs. (50) and (47), shows that the following variable change must be 

undertaken: 

 3

2

DR
R  , 4K G , 4C  , NW F , xF F . (56) 

The dimensionless drive speed (16) becomes: 

 

3D

V
V

G R





 . (57) 

Applying transformations (57) and scaling with factor 0.55 for the parabolic profile, we receive: 

 
2 4 2 3/2

3 2 3/2

3

2 3 2 2 (1 )
0.55

1 (1 )D

V V V V V

R V V

    

 




− − + +
=

− + +
.  (58) 

For the case of fixed force, the governing dimensionless drive speed (22) becomes: 

 

1/3

3

4
W

D

V G
V

G R W

  
  

 
.  (59) 

The scaling (26) of the coefficient of friction becomes: 

 

1/3
2

34
0.55 D

W

R G

W
 

 
  

 
. (60) 

With definitions (59) and (60), the results reported above in Fig. 3 are applicable to the 3D case. 

5. CONCLUSION 

In this paper, we discussed 1D, 2D, and 3D sliding problems of typical rigid probes 

(triangular and parabolic probes in the 1D and conical and paraboloidal probes in the 2D 

and 3D) in contact with 1D and 2D viscoelastic foundations and a viscoelastic half-space, 

respectively. We found that extending the 1D to the 2D models does not change the master 

curves qualitatively, confirming that the simplest 1D model is a remarkable tool for 

understanding sliding friction due to solid viscoelasticity. Applying the MDR to the 1D 

model is expected to estimate the sliding friction more quantitatively. 

Acknowledgement: This study was supported by CREST (Grant Number JPMJCR2193) of the 

Japan Science and Technology Agency (JST), KAKENHI (Grant Number 21H01236) of the Japan 

Society for the Promotion of Science (JSPS), and by Deutsche Forschungsgemeinschaft (Grant 

Number 423918021). 



544 T. WATANABE, S. HATANAKA, K. NAKANO, V. L. POPOV 

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