Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  

Series: Mechanical Engineering Vol. 12, N
o
 1, 2014, pp. 1 - 14 

METHOD OF DIMENSIONALITY REDUCTION IN CONTACT 

MECHANICS AND FRICTION: A USERS HANDBOOK. 

I. AXIALLY-SYMMETRIC CONTACTS 1 

UDC (539.3) 

Valentin L. Popov, Markus Hess 

Technical University Berlin 

Abstract. The Method of Dimensionality Reduction (MDR) is a method of calculation 

and simulation of contacts of elastic and viscoelastic bodies. It consists essentially of 

two simple steps: (a) substitution of the three-dimensional continuum by a uniquely 

defined one-dimensional linearly elastic or viscoelastic foundation (Winkler foundation) 

and (b) transformation of the three-dimensional profile of the contacting bodies by 

means of the MDR-transformation. As soon as these two steps are completed, the 

contact problem can be considered to be solved. For axial symmetric contacts, only a 

small calculation by hand is required which does not exceed elementary calculus and 

will not be a barrier for any practically-oriented engineer. Alternatively, the MDR can 

be implemented numerically, which is almost trivial due to the independence of the 

foundation elements. In spite of their simplicity, all the results are exact. The present 

paper is a short practical guide to the MDR. 

Key Words: Normal Contact, Tangential Contact, Adhesion, Friction, Partial Slip, Stress 

1. INTRODUCTION

In the recently published book [1], the so-called method of dimensionality reduction 

(MDR) is described for the first time in detail. MDR can be traced back to the solution of 

the normal contact problem by Galin (Russian Academy of Sciences) in the 1940s [2]. 

His results were later published by Sneddon and, in this way, made public to the western 

world [3]. The method of dimensionality reduction takes these results and puts them into 

such a form that even a layman in the field of contact mechanics can use them for a 

multitude of contact mechanical problems. In doing this, it merges the ideas and results 

from Cattaneo [4], Mindlin [5], Jaeger [6], and Ciavarella [7] about a close relationship 

between normal and tangential contacts, the solutions of Galanov and Borodich [8, 9, 10] 

Received February 20, 2014 / Accepted March 18, 2014
Corresponding author: Valentin Popov 

TU Berlin, Department of System Dynamics and Physics of Friction, Berlin, Germany 

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

Original scientific paper



2 V. POPOV, M. HESS 

for adhesive contacts of axially-symmetric profiles of power functions (later found 

independently by Yao and Gao [11]), as well as the theory of Lee and Radok about the 

relationship between elastic and viscoelastic contacts [12, 13]. 

The book [1] contains all of the necessary evidence and many examples of how to apply 

the MDR. However, it has proven to be too comprehensive for practical users. There is a 

need for the fundamental ideas and "recipes" of the MDR to be presented in a concise way 

without extensive reasoning or proof, a sort of "user's handbook." This work is dedicated to 

exactly such a practical instruction for the method of dimensionality reduction. 

2. TWO INTRODUCTORY STEPS OF THE MDR 

We consider a contact between two elastic bodies with moduli of elasticity of E1 and E2, 

Poison's numbers of 1 and 2, and shear moduli of G1 and G2, accordingly. In this work, we 
restrict ourselves to the axially-symmetric profiles, which is not necessarily required. A 

generalization to profiles that are not axially-symmetric is possible, but is not considered in 

this work. We denote the difference between the profiles of bodies as z = f(r). In the 

framework of the MDR, two independent steps are conducted: 

The first step: First, the three-dimensional elastic (or viscoelastic) bodies are replaced 

by a one-dimensional linearly elastic foundation. This is considered to be a linear array of 

elements having independent degrees of freedom and a sufficiently small separation 

distance x, Fig. 1. 

 
 a) b) c) 

Fig. 1 One-dimensional foundation of different materials: elastic foundation (a), purely 

viscous foundation (b), and viscoelastic foundation (c) with an example rheology 

according to Kelvin-Voigt 

In the simplest case of the elastic contact, the foundation consists of linearly elastic 

spring elements that have normal stiffness kz
 
and tangential stiffness kx

 
 (Fig. 1a): 

 
2 2

* 1 2

*

1 2

1 11
with

z
k E x

E EE

 
     , (1) 

 
* 1 2

*

1 2

2 21
with

4 4
x

k G x
G GG

 
     . (2) 

Starred values E
* 

and G
*
 denote the effective elastic moduli. Incompressible linearly 

viscous materials are presented by a linear damping element with damping coefficient  

(Fig. 1b), which is dependent on the viscosity  of the viscoelastic partner according to: 



 Method of Dimensionality Reduction in Contact Mechanics and Friction: A Users Handbook 3 

 4 x    . (3) 

Arbitrary combinations of these two base elements are also possible in order to satisfy 

the most complicated elastomers – Fig. 1c, for example, shows a viscoelastic foundation 

built out of elements of in parallel connected springs and dampers (Kelvin-Voigt model). 

In this paper, we will restrict ourselves to the case of "elastically similar" materials: 

 
2

2

1

1
2121

GG





, (4) 

which guarantees the independence of the normal and tangential contact problems [14]. 

This condition is always met in important cases of contacts between the bodies with the 

same elastic properties or those between a rigid body and an elastomer. 

The second step: In the second step, three-dimensional profile z = f(r) (Fig. 2, left) is 

transformed into a one-dimensional profile (Fig. 2, right) according to: 

 





x

r
rx

rf
xxg

0
22

d
)(

)( . (5) 

The reverse transformation is: 

 



r

x
xr

xg
rf

0
22

d
)(2

)( . (6) 

 

Fig. 2 The three-dimensional profile is transformed into a one-dimensional profile using the MDR 

For a less trivial example, we consider the contact of a parabolic profile with a worn 

tip (Fig. 3): 

 2 2

0 for 0

( )
for

2

r b

f r r b
b r a

R

 


  
 



. (7) 

The MDR transformed profile according to Eq. (5) is given by: 

 
2 2

0 for 0

( )
for

x b

g x x
x b b x a

R

  


 
  



. (8) 



4 V. POPOV, M. HESS 

 

Fig. 3 Parabolic indenter with a "worn" tip: original (solid line) and equivalent  

(dashed line) profiles for comparison 

Examples for the MDR transformation 

By inserting profiles into Eq. (5) that correspond to a cylinder, paraboloid, cone, or an 

arbitrary power function z ~ r
n
, we obtain the MDR transformed one-dimensional profiles 

which are summarized in Table 1, where: 

 
























2

1

2

2

2 n

n
n

k
n

   and   





0

1
d:)( tetn

tn
 is the gamma function. 

Table 1 The three-dimensional profile is transformed into a one-dimensional profile using 

the MDR  

 

 

3. CALCULATION STEP OF THE MDR USING THE EXAMPLE OF  

A NORMAL CONTACT WITHOUT ADHESION 

The one-dimensional profile according to Eq. (5) is now pressed into an elastic 

foundation corresponding to Eq. (1) with normal force FN (see Fig. 4). The normal 

surface displacement at point x within the contact area results from the difference between 

indentation depth d and profile form g: 



 Method of Dimensionality Reduction in Contact Mechanics and Friction: A Users Handbook 5 

 

Fig. 4 Equivalent model for a Hertzian contact 

 )()( xgdxu
z

 . (9) 

At the edge of non-adhesive contact x =  a, the surface displacement must be zero: 

 )(:0)( agdau
z

 . (10) 

This equation determines the relationship between the indentation depth and contact 

radius a. It should be noted that this relationship is independent of the rheology of the 

medium. The force of a spring at the point x is proportional to the displacement at this point:  

 xxuExukxF
zzzz

 )()()(
*

, (11) 

and the sum of all spring forces must correspond to the normal force in equilibrium. In the 

limiting case of very small spring separation distances x  dx, the summation becomes 

the integral: 

 
* *

0

: ( )d 2 ( ( ))d

a a

N z

a

F E u x x E d g x x


    . (12) 

Equation (12) provides the normal force in dependence on the contact radius and on the 

indentation depth, if Eq. (10) is taken into account. 

We now define linear force density qz(x): 

 )(
)(

:)(
*

xuE
x

xF
xq

z
z

z





 . (13) 

The stress distribution in the original three-dimensional system can be determined 

with the help of the one-dimensional distributed load using the integral transformation 

[15]: 

 
2 2

1 ( )
( ) d

r

q x
p r x

x r




 


 . (14) 

The normal surface displacement (both inside and outside of the contact area) is given by 

the transformation: 

 
2 2

0

( )2
( ) d

r

z
z

u x
u r x

r x



 . (15) 



6 V. POPOV, M. HESS 

We point out once more that one-dimensional values contain the independent variable 

x, while three-dimensional values contain radial variable r; function uz(x) inside the 

integrand of Eq. (15) means the surface displacement of the linearly elastic foundation. 

We will carry out the last two transformations using the example of a cone. The three-

dimensional profile in this case is f(r) = rtan; the MDR transformed profile is 

( ) tan
2

g x x


 . The displacement within the contact area is ( ) tan
2

z
u x d x


  . The 

linear force density is 
*

( ) tan
2

q x E d x



 

  
 

 and its derivative is 
*

( ) tan
2

q x E


    

(for positive x). Insertion into Eq. (14) and Eq. (15) results in: 

 

2* *

2 2

d
( ) tan tan ln 1

2 2

a

r

E x E a a
p r

r rx r
 

 
       

    
 , (16) 

 

2

2 2
0

2 ( / 2) tan 2
( ) d arcsin 1

a

z

d x d a r r
u r x

r a ar x

 

 

  
             

        
 . (17) 

Equation (17) gives the normal surface displacement outside of the contact area. A 

similar calculation for a parabolic profile f(r) = r
2
/(2R) initially provides uz(x) = d-x

2
/R 

and, after insertion into Eq.  (15), results in: 

 

2 2

( ) 2 arcsin 1
z

d r a r
u r

a r a

       
                    

. (18) 

For the contact of a flattened paraboloid, Eq. (7), we obtain the contact radius by 

using Eq. (10): 

 
22

)(:0)( ba
R

a
agdau

z
 . (19) 

and the normal force using Eq.(12): 

 
* *

* 2 2 2 2 2 2

0 0

2 2
2 d d (2 )

3

a a

N

E E
F E d x x x b x a b a b

R R
        . (20) 

These results correspond of course with those of Eijke [18] from the three-dimensional 

theory. 

Examples for normal contacts 

Insertion of the MDR transformed profiles into Eqs. (10) and (12) and an elementary 

integration provides the results for the "classical profiles" of a cylinder [17], sphere [16], 

and cone [3] as well as the general power profile [2], which are summarized in Table 2. 

The order of the rows corresponds to the order of the calculation steps. 



 Method of Dimensionality Reduction in Contact Mechanics and Friction: A Users Handbook 7 

Table 2 Solutions to the normal contact problem for simple profiles  

 

4. ADHESIVE NORMAL CONTACT 

The MDR rule for the mapping of adhesive contacts will be formulated in the following. 

As with the non-adhesive contact, the MDR transformed one-dimensional profile is brought 

into contact with the linearly elastic foundation defined in Section 2. Now, it will be assumed 

that all of the springs in the contact adhere to the indenter, then the contact radius remains 

the same after a successive decrease in the normal force. From the edge of the contact 

towards the middle, however, more and more springs will be loaded in tension. As soon as 

the change in length of the outer springs reaches the maximum allowable value:  

 
max *

2
( ) ( ) :

a
l a l a

E

 
     , (21) 

there will be a state of indifference between adhesion and separation (Fig. 5). Here,  is 

the separation energy of the contacting bodies per unit area. This state corresponds 

exactly to the equilibrium state of the three-dimensional adhesive contact [15]. 

In contrast to the algorithm for non-adhesive contact, Equation (10) must only be 

replaced by: 

 )()(:)()(
maxmax

alagdalau
z

  (22) 

in order to calculate the indentation depth. The normal force is given as before by Eq. (12): 


















 

aaa

N
alaxxgaagExxgadExxgdEF

0

max

*

0

*

0

*
)(d)()(2d)(2d))((2  (23) 



8 V. POPOV, M. HESS 

 

Fig. 5 Illustration for the MDR rule for an adhesive contact 

or 







 

a

N
alaxxgxEF

0

max

*
)(d)(2 . (24) 

If the force is controlled during the separation, then critical value ac of the contact radius 

at the moment of the loss of stability is determined by condition dFN /da = 0: 

 
*

d ( ) 9

d 2

g a

a aE

 
 . (25) 

Inserting the critical radius obtained from this equation into Eq. (23) results in the 

maximum negative force. We will call its magnitude the adhesion force FA: 

 
*

max

0

2 ( ) ( )d

a

A c c

c

F E a l a xg x x  
 
 
 

 . (26) 

The simplest example is calculating the adhesion force between a cylinder with the 

radius a and an elastic half-space. In this case, the integral in Eq. (26) is equal to zero and 

the adhesion force is only given by the first term: * 3 *
max

2 ( ) 8
A

F E a l a a E     , which 

corresponds to the result of Kendall [20]. Calculations for other profiles are just as simple 

and are summarized in Table 3. 

In order to stress the generality and simplicity of the calculation method, we will now 

conduct the calculation for a parabolic profile with a worn tip (Fig. 6). If we take the 

equivalent profile from Equation (8) into account, the resulting indentation depth is: 

 
2 2

max *

2
( ) : ( ) ( )

a a
d a g a l a a b

R E

 
     , (27) 

 

Fig. 6 Qualitative presentation of the adhesive contact of a parabolic profile with a flattened tip 



 Method of Dimensionality Reduction in Contact Mechanics and Friction: A Users Handbook 9 

and from Eq. (12), the normal force is: 

 
*

* 2 2 2 2 3 *2
( ) ( )d (2 ) 8

3

a

N z

a

E
F a E u x x a b a b a E

R
 



      . (28) 

This equation corresponds to the results from the three-dimensional theory [22]. 

Examples for adhesive normal contacts 

Inserting the MDR transformed profiles into Eqs. (25) and (26) and conducting an 

elementary integration provides the results summarized in Table 3 for the "classical profiles" 

of the cylinder [20], sphere [19], and cone [21], as well as the general power function profile 

[10, 11]. The order of the rows corresponds to the order of the calculation steps. 

Table 3 Solutions for adhesive contacts for simple profiles  

 

 

5. TANGENTIAL CONTACT 

We now consider an axially-symmetric indenter that is initially pressed into an elastic 

half-space with normal force FN and subsequently loaded with tangential force Fx in the x-

direction (Fig. 7). We assume that Coulomb's law of friction is valid in the simplest  

 

Fig. 7 Qualitative presentation of the tangential contact 



10 V. POPOV, M. HESS 

form in the contact: as long as tangential stress  is smaller than the coefficient of friction 

 times normal stress p, the surface is in a state of sticking. After slip sets in, the 

tangential stress remains constant and equal to p: 

 ( ) ( )r p r   for stick, (29) 

 ( ) ( )r p r  for slip. (30) 

It is known that for a small tangential force at the edge of the contact, a ring-shaped 

slip domain develops, which expands inwards for increasing force until the complete slip 

is exhibited. We denote the inner radius of the slip domain (or the radius of the stick 

domain) with c. 

The MDR is applied to the tangential contact as follows. The modified profile g(x) is 

pressed into the linearly elastic foundation with force FN and then tangentially displaced 

by ux
(0)

. The linearly elastic foundation is denoted by the stiffnesses according to (1) and 

(2). Every spring sticks to the indenter and is displaced along with it as long as the 

tangential force Fx = kx ux
(0)

 is smaller than Fz. After the adhesion force is reached, 

the spring begins to slip and the force remains constant and equal to Fz. This rule can 
also be incrementally formulated so that it can be applied for arbitrary loading histories: 

for a small displacement of indenter of ux
(0)

, we obtain: 

 

(0)
( ) , if ( )

( )
( ) , in a state of slip

x x x x z

z
x

x

u x u k u x f

F x
u x

k





   


 

. (31) 

The sign of the last equation is dependent on the direction of motion of the indenter. By 

following the incremental changes in the position of the indenter, we can explicitly determine 

the displacement of all the springs in the contact area; with this, all tangential forces: 

 )()(
*

xuxGxukF
xxxx

 . (32) 

and the linear force density (distributed load): 

 )()(
*

xuG
x

F
xq

x
x

x





  (33) 

are also known. The distribution of tangential stress (r) as well as displacements ux(r) in 
the original three-dimensional contact are determined as follows [1]: 

 



r

x
x

x
xr

xu
ru

0
22

d
)(2

)( . (34) 

 
*

2 2 2 2

( ) ( )1
( ) d dx x

r r

q x u xG
r x x

x r x r


 

 
 

   
 

  . (35) 

If the indenter is displaced in one direction from the equilibrium position, then radius c of 

the stick domain is determined from the condition that tangential force kxux
(0)

 is equal to  
times normal force kzuz(c) (Fig. 8): 



 Method of Dimensionality Reduction in Contact Mechanics and Friction: A Users Handbook 11 

 
* (0) *

( ( ))
x

G u E d g c  . (36) 

The tangential displacement is equal to: 

 

(0)

*

*

, for

( )
( ( )), for

x

x

u x c

u x E
d g x c x a

G


 


  
   

 

, (37) 

the distributed load is: 

 
* (0)

*

, for
( )

( ( )), for

x
G u x c

q x
E d g x c x a

 
 

  
, (38) 

and the resulting tangential force is
2
: 

 
*

0

2 ( )d 2 ( ( )) ( ( ))d .

a a

x x

c

F q x x E c d g c d g x x
 

     
 

   (39) 

The normal force is still given by Equation (12) and ratio Fx /(FN) is given by:  

 

0

( )d

( ) ( )d

a

x c

a

N

xg x x
F

F
ag a g x x













. (40) 

The relative displacement ux = ux
(0)

 – ux(x) of the surfaces in contact is obtained by 

subtracting ux
(0)

 from Eq. (37): 

 

Fig. 8 Equivalent model for the classical tangential contact according to Cattaneo and Mindlin 

 
 

*

*

0, for

( ) ( ) , for
x

x c

u E
g c g x c x a

G





   
   

 

. (41) 

                                                           
2
 We stress once more that all macroscopic values obtained using the procedure described above correspond 

exactly to the three-dimensional solutions of Cattaneo [4], Mindlin [5], Jaeger [6] and Ciavarella [7].  



12 V. POPOV, M. HESS 

The relative displacement in the original three-dimensional system is calculated using Eq. 

(34) as: 

 
*

* 2 2
0

2 ( ) ( )
( ) d , for 

r

x

E g c g x
u r x c r a

G r x




  
   

 
 . (42) 

For example, for a conical, we obtain: 

 

2*

*
( ) tan 1 arcsin ,

2
x

E r c
u r c for c r a

c rG


 

      
            
       

. (43) 

Examples for tangential contacts 

Inserting the MDR transformed profiles into Eqs. (36) and (40), results in the 

relationships between radius c of the stick domain, ratio Fx /(FN), and tangential 
displacement ux

(0)
. The results for the "classical profiles" of the sphere [4, 5], cone [23], 

as well as the power function profile are summarized in Table 4. The order of the rows 

corresponds to the order of the calculation steps. 

Table 4 Solutions for the tangential contacts of simple profiles  

 

6. CONCLUSIONS 

In the present paper, we have limited ourselves to the essential rules and procedures of 

the method of dimensionality reduction. Evidence for the statements herein can be found 

in the works [1] and [15]. 

The possibilities of the MDR are much more expansive as presented in this 

composition. Further successful applications have been found for the rolling contact [24, 

25], contacts with elastomers [26], contacts of rough surfaces [27, 28], elastomer friction 

[29], thermal effects in contacts [1], acoustic emission in rough contacts [31], and wear 

[32]. Interested readers are referred to the cited literature as well as the book [1]. In many 

cases, the MDR allows an analytical solution to the problem, as shown in this work. 



 Method of Dimensionality Reduction in Contact Mechanics and Friction: A Users Handbook 13 

However, it can also be easily implemented numerically and used for the investigation of 

systems with complex dynamic loadings [33].  

Let us stress that the presented form of the MDR is only applicable to contacts with 

homogeneous elastic or viscoelastic half-spaces and it does not take into account size effects 

[34]. However, extensions to contacts with final bodies or heterogeneous media are also 

possible [35]. The application of the MDR to rough contacts requires a separate paper.  

REFERENCES  

1. Popov, V.L., Hess, M., 2013, Methode der Dimensionsreduktion in Kontaktmechanik und Reibung, 

Berlin Heidelberg: Springer-Verlag 

2. Galin, L.A., 1961, Contact Problems in the Theory of Elasticity, North Carolina State College, USA 

3. Sneddon, I.N., 1965, The relation between load and penetration in the axisymmetric Boussinesq 

problem for a punch of arbitrary profile, Int. J. Eng. Sci., 3(1), pp. 47-57. 

4. Cattaneo, C., 1938, Sul contatto di due corpi elastici: distribuzione locale degli sforzi, Rendiconti 

dell'Accademia nazionale dei Lincei, 27, pp. 342-348, 434-436, 474-478. 

5. Mindlin, R.D., , 1949, Compliance of elastic bodies in contact, Journal of Applied Mechanics., 16, pp. 

259–268. 

6. Jaeger, J., 1995, Axi-symmetric bodies of equal material in contact under torsion or shift, Archive of 

Applied Mechanics,  65, pp. 478-487. 

7. Ciavarella M., 1998, Tangential Loading of General Three-Dimensional Contacts. Journal of Applied 

Mechanics, 65, pp. 998-1003. 

8. Galanov, B.A., 1993, Development of analytical and numerical methods for study of models of 

materials, Report for the project 7.06.00/001-92, 7.06.00/015-92. Kiev, Institute for Problems in 

Materials Science, (in Ukrainian) 

9. Borodich, F.M., Galanov, B.A., 2004, Molecular adhesive contact for indenters of nonideal shapes, in: 

ICTAM04, Abstracts book and CD-Rom Proceedings, Warsaw, IPPT PAN 

10. Borodich, F.M., 2008, Hertz type contact problems for power-law shaped bodies, in:  Galin, L.A., 

Gladwell, G.M.L. (Eds.) "Contact Problems. The Legacy of L.A. Galin", Springer, 2008, pp. 261-292. 

11. Yao, H., Gao, H., 2006, Optimal shapes for adhesive binding between two elastic bodies, Journal of 

colloid and interface science, 298(2), pp. 564-572. 

12. Lee, E.H., 1955, Stress analysis in viscoelastic bodies, Quart. Appl. Math, 13, pp. 183-190. 

13. Radok, J.R.M., 1957, Viscoelastic stress analysis, Quart. Appl. Math, 15, pp. 198-202. 

14. Johnson, K.L., 1987, Contact Mechanics,  Cambridge University Press  

15. Hess, M., 2011, Über die exakte Abbildung ausgewählter dreidimensionaler Kontakte auf Systeme mit 

niedrigerer räumlicher Dimension, Cuvillier, Berlin, Germany 

16. Hertz, H., 1882, Über die Berührung fester elastischer Körper, Journal für die reine und angewandte 

Mathematik, 92, pp. 156-171. 

17. Boussinesq, V.J., 1885, Application des Potentiels a l'etude de l'equilibre et du Mouvement des Solids 

Elastiques, Gautier-Villar, Paris, France 

18. Ejike, U.B.C.O., 1981, The stress on an elastic half-space due to sectionally smooth-ended punch, 

Journal of elasticity, 11(4), pp. 395-402. 

19. Johnson, K.L., Kendall, K., Roberts A.D., 1971, Surface energy and the contact of elastic solids. 

Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 324(1558), 

pp. 301-313. 

20. Kendall, K., The adhesion and surface energy of elastic solids, Journal of Physics D: Applied Physics. 

1971, 4, pp. 1186-1195. 

21. Maugis, D., Barquins, M., 1981, Adhesive contact of a conical punch on an elastic half-space, Le 

Journal de Physique Lettres, 42(5), pp. 95-97. 

22. Maugis, D., Barquins, M., 1983, Adhesive contact of sectionally smooth-ended punches on elastic half-

spaces: theory and experiment, Journal of Physics D: Applied Physics, 16, pp. 1843-1874. 

23. Truman, C.E., Sackfield, A., Hills, D.A., 1995, Contact mechanics of wedge and cone indenters, Int. J. 

Mech. Sci., 37, pp. 261-275. 



14 V. POPOV, M. HESS 

24. Wetter, R., 2012, Shakedown and induced microslip of an oscillating frictional contact , Physical 

Mesomechanics, 2012, 15(5-6), pp. 293-299. 

25. Wetter, R., Popov, V.L., 2014, Shakedown limits for an oscillating, elastic rolling contact with 

Coulomb friction. International Journal of Solids and Structures, 2014, 51(5), pp. 930-935. 

26. Kürschner, S., Popov, V.L., 2013, Penetration of self-affine fractal rough rigid bodies into a model 

elastomer having a linear viscous rheology, Phys. Rev. E, 87, 042802. 

27. Pohrt, R., Popov, V.L., Filippov, A.E., 2012, Normal contact stiffness of elastic solids with fractal 

rough surfaces for one- and three-dimensional systems, Phys. Rev. E, 86, 026710. 

28. Pohrt, R., Popov, V.L., 2013, Contact stiffness of randomly rough surfaces. Sci. Rep. 3, 3293. 

29. Li, Q., Popov, M., Dimaki, A., Filippov, A.E., Kürschner, S., Popov, V.L., 2013, Friction Between a 

Viscoelastic Body and a Rigid Surface with Random Self-Affine Roughness, Physical Review Letters, 

111, 034301. 

30. Popov, V.L., Voll, L., Li,Q., Chai, Y.S., Popov, M., 2014, Generalized law of friction between elastomers and 

differently shaped rough bodies, Sci. Rep. 2014, 4, 3750. 

31. Popov, M., 2012, Contact force resulting from rolling on self-affine fractal rough surface, Physical 

Mesomechanics, 15(5-6), pp. 342-344. 

32. Popov, V.L., 2014, Analytic solution for the limiting shape of profiles due to fretting wear, Sci. Rep., 4, 

3749. 

33. Teidelt, E., Willert, E., Filippov, A.E., Popov V.L., 2012, Modeling of the dynamic contact in stick-slip 

micro-drives using the method of reduction of dimensionality, Physical Mesomechanics, 15(5-6), pp. 

287-292. 

34. Argatov, I., 2010, Frictionless and adhesive nanoindentation: Asymptotic modeling of size effects, Mechanics of 

Materials, 42 (8), pp. 807–815. 

35. Popov, V.L., 2014, Method of dimensionality reduction in contact mechanics and tribology. Heterogeneous media, 

Physical Mesomechanics, 17(1), pp. 50-57. 

METODA DIMENZIONALNE REDUKCIJE U KONTAKTNOJ 

MEHANICI I TRIBOLOGIJI: UPUTSTVO ZA KORISNIKE             

I. AKSIJALNO-SIMETRIČNI KONTAKTI 

Metoda dimenzionalne redukcije (MDR) predstavlja metodu za proračun i simulaciju kontakta 

elastičnih i viskoelastičnih tela. Suštinski se sastoji od dva jednostavna koraka: (a) zamene 

trodimenzionalnog kontinuuma jedinstveno definisanom jednodimenzionalnom linearno elastičnom ili 

viskoelastičom podlogom (Winklerova podloga) i (b) transformacije trodimenzionalnog profila 

kontaktnih tela pomoću MDR-transformacije. Nakon izvršenja ova dva koraka, kontaktni problem 

se može smatrati rešenim. Za aksijalno-simetrične kontakte samo mali deo proračuna prevazilazi 

granice elementarnog računa i taj deo neće predstavljati prepreku za praktično orijentisanog 

inženjera. Kao alternativno rešenje, MDR se može primeniti i numerički, što je gotovo trivijalno 

zbog nezavisnosti elemenata podloge. I pored jednostavnosti, svi rezultati su egzaktni. Rad 

predstavlja kratak praktični vodič kroz MDR.  

Ključne reči: normalni kontakt, tangencijalni kontakt, adhezija, trenje, delimično klizanje, napon