Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 16, No 1, 2018, pp. 19 - 28 
https://doi.org/10.22190/FUME180102008P 

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

                                                          

Original scientific paper ∗

ADHESION BETWEEN A POWER-LAW INDENTER 
AND A THIN LAYER COATED ON A RIGID SUBSTRATE  

UDC 539.6 

Antonio Papangelo1,2  
1Politecnico di Bari, Department of Mechanics, Mathematics and Management, Italy 

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

Abstract. In the present paper we investigate indentation of a power-law axisymmetric 
rigid probe in adhesive contact with a "thin layer" laying on a rigid foundation for both 
frictionless unbounded and bounded compressible cases. The investigation relies on the 
"thin layer" assumption proposed by Johnson, i.e. the layer thickness being much smaller 
than the radius of the contact area, and it makes use of the previous solutions proposed 
by Jaffar and Barber for the adhesiveless case. We give analytical predictions of the 
loading curves and provide indentation, load and contact radius at the pull-off. It is 
shown that the adhesive behavior is strongly affected by the indenter shape; nevertheless 
below a critical thickness of the layer (typically below 1 µm) the theoretical strength of 
the material is reached. This is in contrast with the Hertzian case, which has been shown 
to be insensitive to the layer thickness. Two cases are investigated, namely, the case of a 
free layer and the case of a compressible confined layer, the latter being more "efficient", 
as, due to Poisson effects, the same detachment force is reached with a smaller contact 
area. It is suggested that high sensitive micro-/nanoindentation tests may be performed 
using probes with different power law profiles for characterization of adhesive and 
elastic properties of micro-/nanolayers. 

Key Words: Adhesion, Layer, JKR model, Adhesion Enhancement 

1. INTRODUCTION 

Adhesion is a much debated topic in contact mechanics covering different fields of 
application, from adhesion of rough surfaces [1-4] to bioinspired adhesive mechanisms [5, 6]. 
Nature has inspired different researchers to try to reproduce the same design strategy adopted 
by insects such as the "famous" gecko, or to develop an "optimal" profile to reach theoretical 
adhesive strength on a substrate [6-8]. The progress of technology allows us today to "design" 

 
Received January 02, 2018 / Accepted February 02, 2018 
Corresponding author: Antonio Papangelo  
Department of Mechanics, Mathematics and Management, Politecnico di Bari, Viale Japigia 182, 70126 Bari, Italy 
E-mail: antonio.papangelo@poliba.it 



20 A. PAPANGELO 

surface topography down to the nanoscale. Nanopatterned surfaces, with repeating pillars [9] or 
dimples [10], are nowadays inspiring many researchers to develop pressure sensitive adhesive 
mechanisms [10-12]. The majority of scientific literature has focused on the case of halfspace 
geometry; nevertheless, the development of microelectromechanical systems (MEMS), 
anti-wear coatings, microelectronics, pressure sensitive adhesive, multilayer coatings calls for a 
detailed understanding of the contact behavior of the layered surfaces in presence of adhesion. 
Some authors have dealt with axisymmetric contact of an elastic layer supported by a rigid 
foundation in adhesiveless and adhesive cases, both analytically [13, 14] and numerically [15]. 
It has been shown that for the Hertzian profile the pull-off force does not depend on the elastic 
properties of the material, similarly to the classical solution of Johnson-Kendall-Roberts (JKR) 
valid for halfspace geometry [16, 17]. Argatov et al. [18] also studied the indentation of an 
elliptic paraboloid profile in contact with a transversely isotropic layer supported by a rigid 
foundation in the compressible and incompressible case. To unveil the effect of the indenter 
profile, in this paper we study the adhesive indentation of an axisymmetric frictionless rigid 
punch with a power-law profile, which indents a compressible "thin layer" on a rigid 
foundation in both the bounded and unbounded case. The "thin layer" approximation was first 
proposed by Johnson [19], who assumed that layer thickness b is much smaller than radius of 
contact a, i.e. b<<a, so that the plane sections remain plane upon deformation. The same 
hypothesis was used by Jaffar [20], who expanded the analysis to the axisymmetric case and 
later by Barber [21] who generalized the formulation for an arbitrary three dimensional 
problem. Following Barber [21] we will derive the adhesive solution in the case of short range 
adhesion (the so-called JKR limit). To this end the energetic derivation of the original JKR 
model will not be repeated anew. It is in fact known that the adhesive JKR solution can be 
directly obtained from the adhesiveless solution [22] (see also [23-25]), i.e. indentation δ is  

 11 2 P/Aw ′′′−= δδ  (1) 

where δ1 is the adhesiveless indentation, w is the work of adhesion, A′ is the first derivative 
of contact area and P1″

 the second derivative of the adhesiveless load with respect to δ1. 
Then, adhesive load P is  
 111 2 P/AwPPP ′′′′−=  (2) 

where P1′ is the first derivative of the adhesiveless load with respect to indentation δ1. 
Notice that (1, 2) are approximated for a general three-dimensional contact, but they are 
exact for an axisymmetric or line contact; hence we are not adding any other approximation 
except the "thin layer" assumption. Recently a further generalization of the approach suggested  
by Argatov [22] has been proposed [26], which, in boundary element simulations,  introduces a 
mesh dependent criterion (based on the balance between elastic energy release and adhesion 
work) to numerically solve adhesive contacts of complex shapes, which in principle can be 
easily generalized to the case of layered systems. 

In this paper we will extend the asymptotic solutions for adhesiveless thin layer 
problems found by Johnson [19], Jaffar [20], Barber [21] to the JKR adhesive case. We 
shall then discuss implications, particularly regarding the dependence of pull-off and 
loading curves on the geometry of the profile suggesting the strategies to obtain "optimal" 
adhesion. The analytical results obtained may be of interest for elastic and adhesive property 
characterization of thin films, through nano/microindentation test, particularly when a 
polymeric coating is adopted over metals. 



 Adhesion between a Power-Law Indenter and a Thin Layer 21 

2. METHODS 

We study indentation by a rigid frictionless punch on a thin layer on a rigid substrate in 
presence of short range adhesion (JKR type [16]). To this end the adhesiveless relations 
load-indentation and area-indentation are needed. We follow the line of argument of 
Barber [21], who provided the solution of the adhesiveless problem, for a general three 
dimensional profile of the punch, for the case of both a frictionless unbounded layer and a 
bounded compressible layer. The Cartesian coordinate system is set as follows: the contact 
surface of the layer coincides with plane z=0 while two-dimensional spatial coordinates x1, 
x2 lie in the plane of the layer. We restrict our analysis to the case where layer thickness b is 
much smaller than contact radius a, i.e., b<<a, so that the deformation through the layer is 
homogeneous; thus, the plane section remains plane upon compression. This corresponds 
to the original Johnson's approximation, which implies that the in-plane displacements of 
the layer with components u1, u2 are independent of z. For the case of frictionless 
foundation Barber [21] showed that the contact pressure is 

 ( ) ( 2121 x,xb
E

x,xp ξ
∗

= )  (3) 

where indentation ξ is the local interpenetration between the indenter and the layer if it did 
not deform, E* is the effective elastic modulus E*=E1/(1-ν

2), ν the Poisson's ratio, b  the 
layer thickness. For the case of bounded compressible layer Barber [21] obtained 

 ( ) ( 2121 x,xb
ZE

x,xp ξ
∗

= )  (4) 

where Z=(1-ν)2/(1-2ν). The general three dimensional adhesiveless solution could be 
transformed into the corresponding JKR adhesive solution with the approximation that the 
"contact shape" is the same in the adhesive and adhesiveless solutions [22-25]. However, 
here we are dealing with axisymmetric contact; therefore, no approximations are introduced 
[22-25]; thus, the solution will be exact, provided that the original Johnson's approximation 
of a "thin layer" is fulfilled. 

3. AXISYMMETRIC CONTACT WITH POWER LAW PROFILE 

3.1. Frictionless unbounded layer 

Consider the indentation by a rigid axisymmetric frictionless indenter with power law 
profile rk/(kRk-1) on a thin layer supported by a rigid foundation (r is the radial coordinate 
and R a reference length, see Fig. 1). 

For the case of a frictionless unbounded layer the contact pressure is given by Eq. (3), 
and hence 

 ( ) ⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
−= −

∗

11 k

k

kR
r

b
E

rp δ  (5) 

where δ1 is the adhesiveless indentation. 



22 A. PAPANGELO 

 
Fig. 1 Schematic representation of an axisymmetric power law profile in contact with a 

thin elastic layer of thickness b supported by a rigid foundation 

The total load reads 

 1
2

1 2 −
+∗

+
= k

k

R
a

b
E

k
P

π
 (6) 

where a is the contact radius and the subscript "1" stands for the adhesiveless solution. In 
fact from p(a)=0, the adhesiveless indentation δ1 is  

 11
1

−
= k

k

R
a

k
δ  (7) 

Hence, we have for the repulsive solution  

 
( )

( )

k/

k

kk/k

Rb
E

k
k

P
1

12

2
1

2

1 2 ⎟
⎟
⎠

⎞
⎜⎜
⎝

⎛
+

=
−

+∗+ δ
π  (8) 

 
k/

k
k/

R
kA

2

1
12 ⎟
⎠
⎞

⎜
⎝
⎛

=
−

δ
π  (9) 

Computing derivatives A, P1′, P1″ and using Eqs. (1) and (2) after some algebra the 
adhesive solution is obtained 

 
( )

⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛ +
−⎟

⎠
⎞

⎜
⎝
⎛

+
=

∗−

+∗

E
w

b
k

k
Rk

k
b

E
P

k/

k

k/k

2
2

2 1
2

1
1

2

δ
δ

π  (10) 

 ∗−= E
w

b21δδ  (11) 

Under force control the pull-off is found imposing P′=0 which gives  



 Adhesion between a Power-Law Indenter and a Thin Layer 23 

 
k/

k
POPO, E

w
bRa;

E
w

b
k

1

1
1 222

2
⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛
==

∗
−

∗
δ  (12) 

Substituting the first equation from Eqs. (12) into Eq. (10), the pull-off force is obtained  

 ( )
( ) ( )k/k

k/kk/
PO E

w
bR

b
E

k
k

P
22

122 2
2

2
+

∗
−

∗

⎟
⎠
⎞

⎜
⎝
⎛

+
−= π  (13) 

For a Hertzian profile, i.e. imposing k=2, the following results are obtained 

 
41

1
2

222
/

POPOPO, E
bw

Ra;RwP;
E
w

b ⎟
⎠
⎞

⎜
⎝
⎛

=−==
∗∗

πδ  (14) 

Notice that for the Hertzian case, the pull-off does not depend on the elastic properties 
of the layer, but only on geometry "R" and surface energy "w " similarly to the classical 
JKR results valid for a sphere pressed against an halfspace, but with an enhanced adhesive 
strength (in the classical JKR theory the pull-off is equal to -3/2πRw). We define the 
dimensionless quantities 

 
Rw
P

P;
b

E
w π

δ
δ

22
==

∗

 (15) 

Dimensionless forms of the adhesive load, Eq. (10), and pull-off reads 

 
( )

( )
( ) ( )

( ) ⎟
⎠
⎞

⎜
⎝
⎛

−+⎟
⎠
⎞

⎜
⎝
⎛

+
=

−

∗
−

+

kE
w

bR
k

k
P

k/
k/k

k/k
k/k 2

12
2

2
22

2
2

δδ  (16) 

 ( )
( ) ( )k/k

k/kk/
PO

E
w

bR
k

k
P

22
22 22

2

−

∗
− ⎟

⎠
⎞

⎜
⎝
⎛

+
−=  (17) 

where Eq. (11) is used to express Eq. (10) in terms of δ . For a given geometry of the 
indenter and for given material properties one may think to optimize the thickness of layer 
b. Notice that: for k<2 increasing the layer thickness leads to higher pull-off, for the 
Hertzian case k=2 the pull-off is insensitive to the layer thickness, while for k>2 the layer 
thickness has to be reduced for increasing the pull-off force. The layer thickness in fact is 
raised at the power (2-k)/(2k), which is plotted in Fig. 2. For high values of exponent k we 
obtain  21 /PO bP −∝  , which suggests that in order to obtain high adhesive strength k>>2 
should be adopted together with a very thin layer. Bearing in mind that our argumentation 
is based on the thin layer approximation, in the rest of the paper we will focus on the case 
k>2.  



24 A. PAPANGELO 

 

Fig. 2 The pull-off force is proportional to  .bP k
k

PO
2

2−

∝  Here we plot the exponent (2-k)/(2k) 
versus k to show that for k<2(>2) thicker (thinner) layer increase the pull-off force 

Similarly to Gao and Yao [7] we assume w=10 mJ/m2, E*=1 GPa and for the layer 
thickness and reference length b=1 µm, R=1 mm. The loading curves are reported in Fig. 3 
for k=[1.9, 2, 2.1]. Two main points arise, i.e., firstly, there is no instability in displacement 
control (contrary to the classical JKR problem with halfspace), and secondly, the pull-off 
force is greatly affected by k. This is further confirmed in Fig. 4 where the pull-off force is 
plotted as a function of k: it appears that moving from k=2 to k=3 an enhancement of one 
order of magnitude is obtained. 

 
Fig. 3 Force-indentation curves for different power law profiles k=[1.9, 2, 2.1].  

Solid (dashed) curves are stable (unstable) under force control 



 Adhesion between a Power-Law Indenter and a Thin Layer 25 

 
Fig. 4 Pull-off as a function of the indenter shape, R/b=[102, 103, 104] 

The strong enhancement obtained increasing k calls for further investigations. First we 
compute the average tension acting within the contact area at pull-off |σPO| 

 
b

wE
k

k
a
P

PO

PO
PO

∗

+
==

2
22π

σ  (18) 

where we clearly recognize the toughness of material KIc=(E
*w)1/2 and the dependences on 

layer thickness b-1/2 and the shape of indenter "k", but independent of the other lengthscale 
"R" involved in the problem. For k>2, the "optimal" layer thickness, critical thickness bcr 
below which theoretical strength σth is reached, is easily obtained as 

 PO
th

Ic
cr a

K
k

k
b <<⎟⎟

⎠

⎞
⎜⎜
⎝

⎛
⎟
⎠
⎞

⎜
⎝
⎛

+
=

22

2
2

σ
 (19) 

We shall assume reasonable values for σth =10 MPa, w=10 mJ/m
2, E*=1 GPa as in Gao 

and Yao [7] which for k=2 gives bcr=1 µm and is further reduced for k>2. We recall here 
that the present analysis is valid within the Johnson's approximation b<<a. With the same 
set of parameters we estimate ratio b/aPO in the "critical condition", i.e. for b=bcr=1 µm, 

 
( )

1
2

2
112

21 <<⎟
⎠
⎞

⎜
⎝
⎛

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
+

=
−∗−

−
k/k

th

th

k/
k/

cr,PO

cr

R
/wE

k
k

a
b σ

σ
 (20) 

and plot Eq. (20) in Fig. 5 as a function of k and for  R/b=[102, 103, 104]. The curves show 
that for k>2 the "optimal design" is feasible within the thin layer approximation where the 
pull-off tension is close to the theoretical strength of the material. Also notice that we have 
used  b=bcr= 1 µm, but for any  b<bcr → |σPO |= σth; ; thus, the condition in Eq. (20) can be 
easily fulfilled for our reasonable set of parameter values. The "efficiency" of the adhesive 
mechanism is easily shown comparing the contact radius at pull-off with that obtained in 
the classical JKR theory, i.e. aJKR,PO=(9/8πR

2w/E*)1/3. In fact, ratio aPO/ aJKR,PO ∝ b
1/(2k) and 

hence, for a sufficiently thin layer, the same detachment force is reached with a much 
smaller contact area.  



26 A. PAPANGELO 

 
Fig. 5 Ratio  bcr/ aPO,cr  plotted for different geometries of the indenter, with  

R/b=[102, 103, 104]. The thin layer approximation is fulfilled for bcr/aPO,cr <<1 

3.2. Bonded compressible layer 

Barber [21] has shown that if the layer is compressible ν≠0.5 and bounded to the rigid 
substrate, the contact pressure is given by Eq. (4), which differs from the frictionless 
behavior for a factor Z=(1−ν)2/(1−2ν). Thus the previous analysis can be repeated to obtain 
the total adhesiveless load 

 
( )

( )

k/

k

kk/k

Rb
E

Z
k

k
P

1

12

2
1

2

1 2 ⎟
⎟
⎠

⎞
⎜⎜
⎝

⎛
+

=
−

+∗+ δ
π  (21) 

Computing derivatives A, P1′, P1″ and using Eqs. (1) and (2) after some algebra the 
adhesive solution is obtained 

 
( )

⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛ +
−⎟

⎠
⎞

⎜
⎝
⎛

+
= ∗−

+∗

E
w

Z
b

k
k

Rk
k

Z
b

E
P

k/

k

k/k 22
2 1

2

1
1

2

δ
δ

π  (22) 

 ∗−= E
w

Z
b2

1δδ  (23) 

which corresponds to the Hertzian case for k=2. To find the minimum we impose P′=0, 
which gives 

 
k/

k
POPO, E

w
Z
b

Ra;
E
w

Z
b

k

1

1
1

2
2

22
⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛
==

∗
−

∗
δ  (24) 

and hence 

 ( ) ( ) ( )
( ) ( )k/k

k/kk/kk/
PO E

w
bZR

b
E

k
k

P
22

22122 2
2

2
+

∗
−−

∗

⎟
⎠
⎞

⎜
⎝
⎛

+
−= π  (25) 



 Adhesion between a Power-Law Indenter and a Thin Layer 27 

Notice that for k>2 pull-off PPO is increased and contact area πa
2
PO decreased with 

respect to the frictionless case, thus, due to Poisson's effects, the contact is more "efficient" 
in the bounded configuration. 

4. CONCLUSIONS 

In this paper we have studied the adhesive indentation by an axisymmetric frictionless 
rigid indenter with power-law profile on a thin layer supported by a rigid foundation. It has 
been shown that the detachment force at pull-off is strongly affected by the geometry of the 
tip. Nevertheless we noted that reducing the layer thickness, which also fulfills the "thin 
layer approximation", leads to reaching the theoretical strength at pull-off. Using 
reasonable data for the elastic and adhesive parameters of the layer, as in Gao and Yao [7], 
we have shown that the critical thickness of the layer below which the theoretical strength 
is reached is of the order of 1 µm. While for the case of Hertzian (parabolic) profile the 
detachment force is independent of the thickness of the layer and on its elastic properties 
(similarly to the classical results of Johnson-Kendal-Roberts valid for halfspace geometry) 
when a general power-law is used, this dependence arises. It has been shown that the 
adhesive mechanism is more efficient when compared to the JKR halfspace solution, 
particularly for bounded compressible layers, as the same detachment force is obtained 
with a much smaller contact area. This occurs because the dominant length scale for the 
stress intensity factor at the contact edge is the layer thickness. The presented analysis is 
particularly suited for polymeric coating of metallic samples with micro or nanometer 
thickness. Exploiting different probe profiles high sensitivity micro-/nanoindentation test 
may be performed to determine adhesive and elastic properties (w, E*) of the thin layers 
coated on "rigid" substrates. Possible extension of the present work may consider the 
adhesive indentation of multilayered systems, which are of interest in tribology, as for the 
"Surface Force Apparatus", often represented as a three-layer halfspace [27]. 

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	ADHESION BETWEEN A POWER-LAW INDENTER AND A THIN LAYER COATED ON A RIGID SUBSTRATE 
	Antonio Papangelo1,2 
	1. Introduction
	2. Methods
	3. Axisymmetric Contact with Power Law Profile
	3.1. Frictionless unbounded layer
	3.2. Bonded compressible layer

	4. Conclusions
	References