FACTA UNIVERSITATIS  

Series: Mechanical Engineering Vol. 17, N
o
 1, 2019, pp. 95 - 102 

https://doi.org/10.22190/FUME190118011T 

© 2019 by University of Niš, Serbia | Creative Commons Licence: CC BY-NC-ND 

Original scientific paper

 

ON ADHESIVE THEORIES IN MULTILAYERED INTERFACES, 

WITH PARTICULAR REGARD TO "SURFACE FORCE 

APPARATUS" GEOMETRY 

 

Michele Tricarico
1
, Antonio Papangelo

2
,  

Andrei Constantinescu
3
, Michele Ciavarella

1
 

1
Politecnico di Bari, Department of Mechanics, Mathematics and Management, Italy 

2
Hamburg University of Technology, Department of Mechanical Engineering, Germany 

3
Laboratoire de Mécanique des Solides, Ecole Polytechnique, Palaiseau cedex, France  

Abstract. Adhesion is a key factor in many tribological processes, especially wear.  We 

generalize a recent formulation for the indentation of a multilayered material using an 

efficient integral transform method, to the case of adhesion, using a simple energetic 

transformation in the JKR regime. Then, we specialize the study for the geometry of the 

Surface Force Apparatus, which consists of two thin layers on a substrate, where the 

intermediate layer is softer than the other two. We find the pull-off force under "force 

control" (i.e. for "soft" loading systems), as well as under "displacement control" (i.e. for 

"rigid" systems), as a function of the geometrical thicknesses and material properties 

ratios, and the method is fully implemented in a fast Mathematica code, available to the 

public (see Appendix). 

Key Words: JKR Theory, Surface Force Apparatus, Adhesion 

1. INTRODUCTION 

Adhesion forces are more and more of interest in many areas of engineering, both as the 

basic building block to the theories of frictional interaction, or wear, particularly at micro 

and nano scales, i.e. at the level of asperities. Wear in particular does not occur when 

asperities deform plastically but when they adhere to the countersurface so strongly that they 

in fact detach a particle of material.  

                                                           
Received January 18, 2019 / Accepted March 08, 2019 

Corresponding author: Michele Ciavarella 

Department of Mechanics, Mathematics and Management, Politecnico di Bari, Viale Japigia, Italy  

E-mail: mciava@poliba.it 



96 M. TRICARICO, A. PAPANGELO, A. CONSTANTINESCU, M. CIAVARELLA 

In a recent paper [1], the role of adhesion in contact mechanics has been reviewed 

starting in particular from the fundamental contribution of Johnson, Kendall and Roberts 

(JKR) [2], who generalized Hertz' theory to include van der Waals forces described as 

infinitely short range forces, so that a "contact area" can still be defined, which includes both 

compressive and tensile stresses. JKR theory has been confirmed in a number of 

investigations: in principle it should hold only for soft materials and large sphere radius, but 

in practice for a single, smooth asperity, it holds approximately even for dimensions 

appropriate to the Atomic Force Microscope (AFM), or the Surface Force Apparatus (SFA). 

The latter is a scientific instrument which measures the interaction force of two surfaces as 

they are brought together and retracted using multiple beam interferometry to monitor 

surface separation, directly measure the contact area and observe any surface deformations 

occurring in the contact zone. Developed by Tabor and Winterton [3] and Israelachvili and 

Tabor [4], it comprises thin sheets of molecularly smooth mica, or similar material, glued to 

the cylindrical glass lenses of equal radii, which are then pressed into elastic contact with 

their axes at right angles. The SFA is frequently used in conjunction with the JKR theory to 

extract the surface energy of the contacting sheets. Errors may arise since the JKR theory 

accounts for the contact of homogeneous, isotropic and elastic cylinders (equivalent to the 

contact of a sphere with a flat surface). Sridhar et al. [5] extended the JKR theory to the 

layered structure of the SFA.  

The main condition for applying the JKR is perhaps that surfaces should be very smooth, 

and this is why the surface layer of the SFA is an extremely flat, optically transparent, mica 

layer (in turn backed with an ultra-thin silver layer to reflect light), and further material or 

molecules of interest are then coated or adsorbed onto the mica layer. The mica layer is 

mounted on a glass cylinder and two such cylinders are put in contact in cross-perpendicular 

configuration, which is equivalent to the contact of a sphere with a plane (see Fig.1). SFA is 

extremely sensitive as it uses piezoelectrics to position with a force accuracy in forces at the 

10
-8

 N level, and optical interferometry to measure distances to within 0.1 nanometer, and is 

similar in some respects to the Atomic Force Microscope (AFM), except that it is a surface-

surface apparatus rather than a tip-surface one, and it can measure much longer-range forces, 

although as a technique it is probably more laborious. 

  

Fig. 1 The geometry of the SFA apparatus contact. In each of the crossing cylinders, a 

mica layer is covering a glass cylinder (a bilayer is in this configuration adsorbed) 



 On Adhesive Theories in Multilayered Interfaces... 97 

In the SFA, many authors probably use the JKR original formulation for homogeneous 

halfspaces to extract the surface work of adhesion of the contacting sheets. Sridhar et al. [5] 

already proposed a hybrid Finite Element Method-analytical technique to obtain an 

extension of the JKR solution. Their work plots the force as a function of the contact area, 

and hence reveals only the maximum force under force control (when the loading system can 

be considered "soft"), which they find only weakly dependent on moduli or thicknesses 

ratios. 

As an example, we can consider the case of Functionally Graded Materials with power 

law elastic modulus, where E0 is the characteristic modulus at length z0  

 
0

0

( )

k

z
E z E

z

 
  

 

 with 1 1k    (1) 

This case can be solved analytically [6], to obtain that the pull-off force remains 

independent on the elastic modulus 

 
3

2
c

k
P R 


   (2) 

[for example under force control] where ω is the work of adhesion of the interface. Hence, 

Pc is very weakly dependent on power exponent k and for example for a case k = 0.5 the 

pull-off changes from the JKR value 
3

2
c

P R    by only +14%. 

However, a combination of elastic moduli which is not monotonically varying leads to 

more interesting results. For example, Stan and Adams [7] use a mathematical solution 

similar to what we shall use here, and show the results for three samples having layer 

structure (from top to bottom) indicated in Tab. 1. The samples differ from each other 

through the elastic modulus of the second layer where sample #3 is very soft. Each layer was 

2 nm thick, the tip was considered rigid and of radius 20 nm, and w = 0.1J/m
2
 for all the 

samples. Sample #1, #2 give almost identical results, while sample #3 gives a very different 

force-indentation curve, with less pull-off under force control, but much more pull-off under 

displacement control, perhaps by +30-40%, than the other two. 

Table 1 Examples of 3 layers structure on a substrate of much larger modulus (100 GPa, 

Poisson’s ratio 0.25), in Stan and Adams [7]  

E[GPa] #1 #2 #3 
Top 15 15 15 

Mid 50 25 5 

Bott 10 10 10 

 

McGuiggan et al.[8] used the FEM technique of [5] to find more extensive results for the 

SFA including some experiments, and found that, realistically, the pull-off force can vary 

between 

 ... 2
c

P R R       (3) 



98 M. TRICARICO, A. PAPANGELO, A. CONSTANTINESCU, M. CIAVARELLA 

i.e. a variation with respect to the JKR value of −33%... + 33%. Material properties and layer 

thicknesses in [8] are reported in Tab. 2, and can be considered typical for SFA. Between the 

glass substrate (silica) and the mica surface, which are almost of the same material 

properties, there is an intermediate layer with more than one order of magnitude softer 

modulus, which is typically an epoxy glue. 

Table 2 Examples of 3 layers typical of SFA with their properties according to [8] 

layer Thickness [µm] E [GPa] ν 

mica 5.5 62 0.21 

epoxy 25 3.4 0.5 

silica substrate 72 0.25 

However, notice once again that all the results are typically plotted in terms of contact 

area vs. load, which does not permit us to distinguish between force or displacement control. 

A realistic setup, of course, is somewhere in between load control and displacement control, 

the latter being the limit when the stiffness of the system tends to very high values. 

In this paper, we shall extend the method published by Constantinescu et al. [9] to 

adhesive configurations and study some implications for SFA apparatuses JKR adhesive 

curves. 

2. FROM NON-ADHESIVE TO ADHESIVE SOLUTION 

The original adhesionless method [9] writes the harmonic Papkovich-Neuber 

displacement potentials as the Hankel transform of four unknown arbitrary functions 

1 2 3 4
( ), ( ), ( ), ( )

i i i i
A A A A     for each layer and two other functions 5 ( )A   and 6 ( )A   for 
the substrate. In order to get the solution of the problem, the determination of 4n+2 

unknown functions is needed. Then we write the boundary conditions of continuity of 

displacements and traction among the layers, and finally the contact problem at the 

surface. The algebraic computation of displacements and stresses is done symbolically in 

Mathematica. The system has a solution in terms of 
1

1
( )A  , which depends on an unknown 

pressure distribution (function H()). This function is the solution of the Fredholm equation 

of the second kind 

 
1

0

1
( ) ( , ) ( ) ( )H M y H y dy F  


 

, for 0 1   (4) 

where F depends only on the imposed indentation depth and on the indenter shape, and τ 

is the normalized radial coordinate in the contact area. For kernel M(y,τ), infinite integrals 

are given in Constantinescu et al. [9] which unfortunately contain highly oscillatory 

integrand if a/h1 (the contact area over thickness of the first layer) is high. To improve the 

range of a/h1 < 100 with sufficient accuracy, we modified the original code in the 

supplementary material of Constantinescu et al. [9], by splitting the integration intervals 

of the infinite integrals (which are obviously already truncated in practice to where the 

integrand is significantly nonzero) in 10 parts, where on each of them Gauss-Legendre 

quadrature with 25 points is done. 



 On Adhesive Theories in Multilayered Interfaces... 99 

2.1. Transformation into adhesive solution 

The original JKR theory is derived and it determined the elastic strain energy U for the 

sphere problem by following a two-step scenario as follows. We first load the contact in 

compression to load P1 up to a contact area A1. We then hold the contact area constant as 

to make the system linear, and then reduce the load to P2. Hence we can write 

 

1

2 1 1 2
( )

P
P P



 


 
    

 

 (5) 

Assuming the final value of displacement δ2 [displacement control], contact areas A1 

and δ1 are obtained by minimizing the total potential energy 

 
1

U A    (6) 

Therefore, we obtain 

 

1

0
A





 and hence 1

1 1 1

U U

A A






 
 

  
 (7) 

and hence 

 
2

1 1
2 1 2

1 1

2
A P

  
 

 
 

 
 (8) 

as obtained in [10], which permits us to derive a general relation between the adhesive 

solution and that without adhesion, which is exact in axisymmetric problems as is the 

present one. A similar derivation was later also suggested by Popov [11], and more 

restricted cases also treated in [12,13]. 

We use non-dimensional parameters according to [1]: 

 2ˆˆ ˆ; ;
P a

P a
R R R


  

 
    (9) 

where 

 
1/3

*

1
E R




 
  
 

 (10) 

with * 1
1 2

1
1

E
E





 being the reduced elastic modulus of the first layer. 

We obtain that pull-off under displacement control is ˆ 5 / 6AP    and under force control 

it is ˆ 1.5AP    for the original JKR spherical case. 



100 M. TRICARICO, A. PAPANGELO, A. CONSTANTINESCU, M. CIAVARELLA 

3. SOME EXAMPLES 

Let us gives a few examples of interest for SFA apparatuses. In particular we consider 

a rigid sphere of radius R = 4h1 indenting a layered flat surface. We take, therefore, as in 

Fig.1, 2 layers on a substrate having the same Young's modulus of to the top layer, with E1 

= E3 = 70 GPa, ν1 = 0.21, ν2 = 0.5, ν3 = 0.25 and we fix E1/E2 = 20 and h = 1 nm, varying 

the thickness of intermediate layer h2/h1 = {1, 5, 10, 20, 50}.  

The solution is calculated over a list of adhesiveless indentation depth values. We use two 

discretizations: in the interval δ1 = [0.001, 2] nm we use a step of 0.02 nm, while for δ1 = [2, 50] 

we use a step of 0.5 nm. The finer discretization for small values of δ1 aims at obtaining a good 

estimation of the pull-off in displacement control. It should be noted that it is not possible to 

compute the solution at point δ1 = 0, because of numerical limitation in the code. 

As shown in Fig.2, the curves of load-indentation displacement or contact radius vs. 

displacement vary considerably their shape, and in particular the pull-off load vary both if 

under displacement control (the load at the lowest displacement, ˆAP ) or under force-

control ˆ
B

P  (the absolute minimum of the load). 

 

Fig. 2 Curves of load vs. displacement (a) and contact radius vs. load (b), for a case  

"SFA-like" (2 layers on a substrate having properties identical to the top layer), with 

E1 = 70 GPa; E1/E2 = 20; but varying the thickness of the intermediate layer h2/h1 = 

{1, 5, 10, 20, 50} 

 

Fig. 3 Pull-off load under displacement-control ˆAP  (a) or load control ˆBP  (b) for a case 

"SFA-like" (2 layers on a substrate having properties identical to the top layer), with E1 

= 70 GPa; E1/E2 = 20; but varying the thickness of the intermediate layer h2/h1  

a) b) 

a) b) 



 On Adhesive Theories in Multilayered Interfaces... 101 

In particular, Fig. 3 shows only the pull-off loads and it is clear that the variation is 

particularly relevant under displacement control, which is a case not so documented in the 

literature, whereas under force control the variation is relatively small, as already well-

known. 

We now move to consider a fixed ratio of thicknesses, namely h2/h1 = 5, which is 

typical for SFA, and vary modulus ratio E1/E2 = {1, 5, 10, 20, 50}, see Figs. 4 and 5. The 

variation of pull-off forces is shown to depend strongly on the modulus ratio initially and 

then asymptotically reach some limit values. 

 

Fig. 4 Curves of load vs. displacement (a) and contact radius vs. load (b), for a case 

"SFA-like" with h2/h1 = 5, and vary the modulus ratio E1/E2 = {1, 5, 10, 20, 50} 

 

Fig. 5 Pull-off load under displacement-control ˆAP  (a) or load control 
ˆ
B

P  (b) for a case 

"SFA-like" with h2/h1 = 5 and vary the modulus ratio E1/E2 = {1, 5, 10, 20, 50} 

4. CONCLUSIONS 

We have obtained a general method for solving multilayered problems indentation 

with JKR adhesion and we have made some considerations for the SFA type of geometry. 

It is shown that the thickness ratio and modulus ratio of the second to first layers 

(assuming the substrate is almost identical to the top layer) can vary in a relatively modest 

way the pull-off under force control, and in a more pronounced way the pull-off under 

displacement control. The curves vary their shape considerably. The model provides a fast 

evaluation of the entire curve. 

a) b) 

a) b) 



102 M. TRICARICO, A. PAPANGELO, A. CONSTANTINESCU, M. CIAVARELLA 

APPENDIX: MATHEMATICA CODE 

The code used for this work, implemented in the symbolic software Mathematica, is 

available as supplementary resource with the paper [13] at the following link: 

http://dx.doi.org/10.1016/j.ijsolstr.2013.04.017 

Acknowledgements: Antonio Papangelo is thankful to the DFG (German Research Foundation) for 

funding the projects HO 3852/11-1 and PA 3303/1-1. 

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http://dx.doi.org/10.1016/j.ijsolstr.2013.04.017