Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  

Series: Mechanical Engineering Vol. 16, N
o
 1, 2018, pp. 93 - 98 

https://doi.org/10.22190/FUME180105009P 

© 2018 by University of Niš, Serbia | Creative Commons Licence: CC BY-NC-ND 

Short communication 

SOLUTION OF ADHESIVE CONTACT PROBLEM ON THE BASIS 

OF THE KNOWN SOLUTION FOR NON-ADHESIVE ONE 

UDC 539.6 

Valentin L. Popov  

Berlin University of Technology, Berlin, Germany 

Abstract. The well-known procedure of reducing an adhesive contact problem to the 

corresponding non-adhesive one is generalized in this short communication to contacts 

with an arbitrary contact shape and arbitrary material properties (e.g. non homogeneous 

or gradient media). The only additional assumption is that the sequence of contact 

configurations in an adhesive contact should be exactly the same as that of contact 

configurations in a non-adhesive one. This assumption restricts the applicability of the 

present method. Nonetheless, the method can be applied to many classes of contact 

problems exactly and also be used for approximate analyses. 

Key Words: Adhesion, Normal Contact, Heterogeneous Media, Effective Surface Energy 

1. INTRODUCTION 

The present short communication is publication of a short memo written on May, 14
th

, 

2015, and not published at that time as the field of applications of the obtained results 

seemed to be very narrow. It was communicated privately to colleagues and published for 

a restricted set of adhesive contact problems in [1]. Since then, it has been applied to a 

variety of problems including axially symmetric contact ones without a compact contact 

area [2] just as it has been systematically applied to a large variety of contact problems in 

the recent Handbook on Contact Mechanics [3]. However, the derivation and results are 

more general than the cases considered in [2] and [3]. They are based solely on existence of 

some force-indentation and area-indentation dependencies for non-adhesive contacts. In the 

present paper we provide a general derivation which is even applicable to the situations 

where the surface energy is a function of the coordinates. 

                                                           
Received January 05, 2018 / Accepted January 31, 2018 

Corresponding author: Valentin L. Popov  

Technische Universität Berlin, Sekr. C8-4, Straße des 17. Juni 135, D-10623 Berlin  

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



94 V. POPOV 

2. AXIALLY SYMMETRIC CONTACT PROBLEM 

 Consider indentation of a profile z = f(r) into an elastic medium with plane surface. 

The medium is assumed to be homogeneous in-plane, but may be heterogeneous in the z-

direction (e.g. a layer or a gradient medium and so on). We assume that indentation depth 

d is much smaller than the characteristic size of heterogeneity, while this may be not the 

case for contact radius a. We assume that the non-adhesive normal contact problem for 

this shape and this medium has been solved, so that the dependences of normal force FN 

and of the contact area on the indentation depth are known. Each of these three quantities 

determines uniquely two others, so that we can consider normal force FN,n.a.(a) and 

indentation depth dn.a.(a) as known functions of the contact radius, too. We can further 

define the potential energy of non-adhesive contact, Un.a.(a) and the contact stiffness  

 
, .

. .

d ( ( ))
( )

d

N n a

n a

F a d
k a

d
 . (1) 

which also can be considered as a known function of the contact radius. 

Now let us consider an adhesive contact under assumptions of the JKR-theory (range 

of interaction of adhesive forces much smaller than any characteristic size of the problem, 

so to say zero) and characterize adhesion with the work of detachment of surfaces per unit 

area, .  

The solution to this problem is given by the following set of equations: 

 
. .

( ) ( )
n a c

d d a l a  , (2) 

 
, . . . .

( ) ( ) ( ) ( )
N N n a n a c

F a F a k a l a    (3) 

with lc(a) given by the following equation: 

 
. .

4
( )

d ( ) / d
c

n a

a
l a

k a a

 
  . (4) 

Eqs. (2) and (3) give in implicit form the dependence of the normal force (in the 

adhesive contact) on the indentation depth thus solving the adhesion problem. 

Let now prove the Eq. (4). If we indent the profile up to contact radius a, then the 

potential energy in this state will be Un.a.(a) and indentation depth dn.a.(a). The force in 

this moment will be FN,n.a.(a). Now let lift the indenter by l without changing the contact 

area. During this process the stiffness of the contact remains constant and equal to kn.a.(a). 

Therefore, the force will change according to  

 
. . . .

( ) ( )
N n a n a

F a F k a l    (5) 

and the potential energy will be equal to  

 
2

. . , . . . .
( ) ( ) ( ) ( )

2
n a N n a n a

l
U a U a F a l k a


    . (6) 



 Solution of adhesive contact problem on the basis of the non-adhesive solution 95 

The new indentation depth will be  

 
. .

( )
n a

d d a l  . (7) 

Let us now solve (7) with respect to l und insert it into Eq. (6): 

 
2

. .
. . , . . . . . .

( ( ) )
( ) ( ) ( )( ( ) ) ( )

2

n a
n a N n a n a n a

d a d
U a U a F a d a d k a


    . (8) 

The total energy (with consideration of the adhesion energy is equal to 

 
2

2. .
. . , . . . . . .

( ( ) )
( ) ( ) ( )( ( ) ) ( )

2

n a
tot n a N n a n a n a

d a d
U a U a F a d a d k a a


       . (9) 

Equilibrium value of a corresponds to the minimum of this energy with respect to a for a 

constant indentation depth d. To determine the minimum, we let the derivative be zero: 

2
, . . 2. . . . . . . .

, . . . .

, . .. . . . . . . .

, . . . .

d ( )

d ( )d ( ) d ( ) d ( ) d ( )
( ) ( )

d d d d 2 d

d ( )d ( ) d ( ) d ( ) d ( )
( ) ( )

d d d d d

tot

N n an a n a n a n a

N n a n a

N n an a n a n a n a

N n a n a

U a

a

F aU a d a k a d al
l F a k a l a

a a a a a

F aU a d a d a k a
F a l k a l

a a a a a




        

  
        
   

2

2
2

0

l
a


  



 (10) 

It is easy to see that the terms in brackets are identically zero, thus we get 

 
2

. .
d ( )

2
d 2

n a
k a l

a
a


   . (11) 

Solution with respect to l provides Eq. (4). Eqs. (5) and (7) coincide with Eqs. (3) 

and (2) and provide the solution to the problem. Note that this equation is applicable not 

only to homogeneous media but to all the media for which the dependence of the contact 

stiffness on radius is known, in particular of coated, multi-layer or gradient media [3]. In 

the case of a homogeneous medium dkn.a.(a)/da=2E
*
, where E

*
 is the effective elastic 

modulus responsible for a normal contact problem [4]. In this case we come to the known 

rule of Heß [5]. 

3. GENERAL CASE (NOT AXIS-SYMMETRIC OR NON-COMPACT CONTACT AREA)  

If the set of contact configurations of an adhesive contact would repeat that of contact 

configurations of the normal one for the same shape (which, regrettably, will generally not 

be the case!), then the adhesive contact could be solved in the following way. For 

simplicity, we consider here homogeneous media. We assume that the normal contact 

problem was solved so that the dependence of normal force FN,n.a. and contact area A on 

indentation depth d is known: 



96 V. POPOV 

 
, . . , .

( )
N n a N n a

F F d ,  (12) 

 
. . . .

( )
n a n a

A A d .  (13) 

Now we define the incremental stiffness 

 
, . .

. .

d

d

N n a

n a

F
k

d
   (14) 

and the formal "effective contact radius" (which in general case has of course nothing to 

do with any radius, but is just a formally defined quantity): 

 
. .

*

( )

2

n a
k d

a
E

 .  (15) 

The normal contact now can be described by the Method of Dimensionality Reduction 

with the equivalent profile z=g(x), where function g(x) is defined according to  

 ( )d g a ,  (16) 

(just by solving Eq. (15) with respect to d, for details see [5]) . 

The condition for the equilibrium of an adhesive contact can be obtained from the 

standard balance of energy at small variation of the "contact radius". We assume that the 

boundary springs (in the MDR picture) detach when they achieve critical length lc, 

which is determined by equating the relaxed elastic energy  

 
2. .

d
2

2d

n a

el c

k
U x l

a
       (17) 

to the change of adhesive energy: 

 
d d d d d ( )

d d d d d
adh

A A d A g a
U A x x x

a d a d a
            . (18) 

Equating (17) and (18) we get 

 
2. .

d d
2

2d d

n a

c

k A
x l x

a a
      , (19) 

hence 

 
 

 
. .

. .

2 d / d

d / d

n a

c

n a

A a
l

k a


  .  (20) 

In the case of axis-symmetric profiles with a compact contact area we have of course 

trivially A = a
2
, dA/da = 2a and 

*

c
E/al Δγπ2Δ  , which coincides with the "rule of 

Heß" [5].  



 Solution of adhesive contact problem on the basis of the non-adhesive solution 97 

4. GENERALIZATION FOR ARBITRARY MEDIA 

Note that in the systems with a complicated “microstructure”, the surface energy also 

can depend on the size of the contact, as e.g. illustrated for different shapes with internal 

damages as well as for “brushes” in [7]. In this case Eq. (22)is modified to   

 
 

na

2 ( ) d ( ) / d
( )

d ( ) / d
c

a A a a
l a

k a a

 
  , (21) 

where (a) is the size-dependent effective surface energy which can be determined by 

means of the concept of the “filling factor” the use of which has been validated in many 

examples in [7]. Together with Eqs. (2) and (3) this solves the problem. 

5. APPLICABILITY OF THE MACROSCOPIC APPROACH TO CONTACT OF ROUGH SURFACES 

Recently, Ciavarella has come independently to a very similar approach [7]. He has 

also provided an extensive and instructive discussion of applicability of the macroscopic 

approach to adhesive contacts of rough surfaces. The main assumption of the approach is 

that the sequence of contact configurations of an adhesive contact is the same as in the 

case of a non-adhesive contact. This condition is clearly not fulfilled even in the case of 

“asperity models” like Greenwood and Williamson. There are no reasons to assume that 

in the case of a more general roughness the contact configurations of an adhesive contact 

will repeat those of a non-adhesive contact, so that in the general case, this assumption is 

not fulfilled, either. That the present approach cannot be generally applied to contacts of 

rough surfaces is already clear from the fact that in the present approach there is no 

“hysteresis of the force of adhesion” (thus, the force of adhesion does not depend on the 

loading history which is not the case in real rough contacts as discussed in [9, 10]. 

However, there can be some situations where the above assumption is fulfilled or 

approximately fulfilled. For example, if the Johnson parameter [11] is overcritical then a 

complete contact can be realized in spite of roughness [12]. Furthermore, the concept can 

be applied to rough surfaces by using the concept of the filling parameter as discussed in 

[7]. However, this approach uses the notion of a “real contact area” which is a poorly 

defined quantity (an excellent discussion of this property and the ways of its proper 

physical definition can be found in [13]. E.g. one of the “regularizing factors” may be the 

final range of adhesive forces which substantially modifies the contact situation at a small 

scale [14, 15]. Further investigation of this problem, especially using the now available 

numerical technique of the Boundary Element method for adhesive contacts [7] is needed. 

6. CONCLUSIONS 

Equations (2), (3) and (22) provide a simple solution for all the problems that the 

normal contact problem has been solved for – either analytically or numerically. This 

includes all the contacts with a homogeneous continuum, coated medium, gradient 



98 V. POPOV 

material, plates, thin layers, membranes or living cells, and so on. The only restriction of 

the method is that the sequence of the contact configuration is the same as in the non-

adhesive problem. This is valid for compact axisymmetric contacts and for some other 

cases of axisymmetric contacts analyzed in [2].  

Acknowledgments: The author acknowledges financial support by the Deutsche Forschungs-

gemeinschaft (DFG PO 810-55-1). The author acknowledges very useful discussions with M. 

Ciavarella. 

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axialsymmetrischer Kontaktprobleme, Springer, Berlin, 341 p. 

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niedrigerer räumlicher Dimension, Cuvillier, Berlin, Germany 

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