rsos.ro yalsocietypublishing .org
Resear ch
Cite this ar ticle: P opov VL, L yashenk o IA,
Filippo v AE. 2017 Influence of tangential
displacement on the adhesion str ength of a
contact between a parabolic pr ofile and an
elastic half-space. R. Soc . open sci. 4 : 161010.
http://dx.doi.or g/10.1098/rsos.161010
Receiv ed: 7 December 2016
Acc epted: 1 August 2017
Subjec t Categor y :
Physics
Subjec t Areas:
applied mathema tics/mathematical
modelling/mechanics
Key words:
adhesion, friction, tribology , shear forc e,
numerical simulation, method of
dimensionality reduction
Author for correspondence:
V alen tin L. P opov
e- ma i l: v [email protected]
Influenc e of tangential
displac ement on the
adhesion str ength of a
c ontac t bet w een a par abolic
pr ofile and an elastic
half-space
V alentin L. P opov 1,2,3 , Iak ov A. L yashenk o 1,4 and
Alexander E. F ilippov 1,5
1 Institut für Mechanik, FG Systemdynamik und Reibungsphysik, T echnische Univ ersität
Berlin, Sekr . C8-4, Raum M 122, Straße des 17. Juni 135, 10623 Berlin, Germany
2 National Resear ch T omsk State Univ ersity , 634050 T omsk, Russia
3 National Resear ch T omsk Polyt echnic University , 634050 T omsk, Russia
4 Depar tment of Applied Mathematics and C omplex S ystems Modelling , F aculty of
Electronics and Informational T echnologies , Sumy Stat e University , 40007 Sumy ,
Ukraine
5 National A cademy of Science, Donetsk Institute for Physics and Engineering ,
83114 Donetsk, Ukraine
VLP , 0000-0003-0506-3804 ;I A L , 0000-0001-7511-3163
The adhesion stre ngth of a contact between a rotationally
symmetric indenter and an elastic half-space is analysed
analytically and numerically using an extension of the
method of dimensionality reduction for superimposed
normal/tangential adhesive contacts. In particular , the
dependence of the critical adhesion force on the simultaneously
applied tangential for ce is obtained and the relevant
dimensionless parameters of the pr oblem are identified.
The fracture criterion used coincides with that suggested by
Johnson. In this paper , it is used to develop a method that is
applicable straightforwardly to adhesive contacts of arbitrary
bodies of r evolution with compact contact area.
1. Intr oduc tion
Johnson, Kendall and Roberts developed in 1971 their classical
theory of normal adhesive contact between two parabolic,
isotropic elastic bodies (JKR theory) [ 1 ] using the similarity
between the boundary of an adhesive contact and the tip of a
2017 The Authors . Published by the Ro yal Society under the terms of the Cr eative C ommons
Attribution License h ttp://creativ ecommons.or g/licenses/by/4.0/, which permits unr estric ted
use, pr ovided the original author and sourc e are cr edited.

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mode I crack (opening mode). They applied the same idea of the energy balance that Grif fith used in his
classical theory of cracks [ 2 ]. In the subsequent years, the theory of adhesive contacts developed rapidly
[ 3 ], mostly using various concepts developed in fracture mechanics.
In the JKR theory—just as in the theory of Griffith—the equilibrium configuration of an adhesive
contact is determined by minimizing the total energy of the system including the ener gy of elastic
deformation of contacting bodies, the interface energy and the work of external for ces [ 4 ]. As this
energy does not depend on the tangential displacement, the adhesive contact formally has no ‘tangential
strength’. This appar ently contradicts experimental observations. The contradiction is due to the
microscopically heter ogeneous structur e of any interface (at the atomic scale, if not earlier), which
provides finite contact str ength in the tangential dir ection.
From the micr oscopic point of view , one can interpret the fractur e criterion of Grif fith as the
requir ement that the str ess at a fixed distance (of atomic order) fr om the actual ‘crack tip’ reaches
some critical value (stress criterion). This condition leads to the macr oscopic dependence of the critical
stress on crack length, which is identical to r elations obtained fr om the macroscopic ener gy balance
[ 5 ]. Alternatively , one could r equir e that the relative displacement of the faces of the crack achieve
some critical value (deformation criterion). The stress and deformation criteria ar e equivalent for purely
elastic bodies, but can lead to differ ent fracture conditions in elastomers [ 5 ]. In this paper , we will only
deal with purely elastic bodies, so we can apply either the str ess or the deformation criterion without
loss of generality . Johnson studied the pr oblem of adhesive contact under superimposed normal and
tangential loading [ 6 ] and concluded that, ‘when tangential for ces are applied to an adhesive contact the
consequences are not at all well understood’. This situation has not changed much until now .
In his paper of 1997, Johnson approaches the problem of adhesive contact under superimposed
normal and tangential loading by considering the complete energy r elease rate at the boundary of an
adhesive contact [ 6 ]
Q = 1
2 E ∗  K 2
I + K 2
II + 1
1 − ν K 2
III  , (1.1)
where K I , K II and K III ar e the str ess concentration factors defined as
K I,II,III = F I,II,III
2 a √ π a , (1.2)
with a being the contact radius and F I , II , III the components of the applied force in the normal (I) and
tangential directions (II, radial dir ection; III, tangential to the boundary line). The problem of a cir cular
contact remains axially symmetric only if the Poisson number is zer o, ν = 0. In this case, the stress
concentration factors for the modes II and III are e qual along the entire boundary line. In the case
of arbitrary Poisson number , Johnson suggests to evaluate the average values of K II and K III ar ound
the periphery of the contact area, simplifying (1.1) to
Q = 1
2 E ∗  K 2
I + 2 − ν
2 − 2 ν K 2
II  . (1.3)
In terms of energy r elease rates, the condition of fracture can be formulated by equating the ener gy
release rate to some critical value r elated to the work of adhesion γ . W e would like to stress that this
approach is by no means self-evident. Physically , it means that elastic energy components due to normal
and tangential loading contribute in equal manner to the destruction of interfacial bonds. This may be
true in some cases. For example, if some polymer molecules that have to be br oken by sufficiently lar ge
elongation mediate the adhesion of surfaces, then both normal and tangential deformations have to be
considered when applying the ‘fractur e criterion’. The same may be valid in a contact of atomically
smooth surfaces with equal characteristic range of atomic interaction in normal and tangential dir ections.
In this case, the tangential displacement of atoms at the interface will bring them into a higher ener getic
position compared with ‘unstr essed’ atoms. Subsequently , a smaller amount of work will have to be
performed by normal for ces to complete the detachment. In this case, too, one can at least qualitatively
assume that both normal and tangential parts of elastic energy give appr oximately the same contribution
to the overall detachment energy . In other situations, however , this criterion can fail completely . Thus,
if the characteristic range of atomic interactions in the in-plane dir ection is much smaller than in the
normal direction, then the work of adhesion will be practically independent of the tangential loading
and the criterion (1.3) will not be valid. One can also imagine a physical model, in which there is some
‘microscopic friction’ between surfaces that ar e pr essed to each other by relatively long-range van der
W aals forces. In this case, the work of detachment will depend on the exact ‘dir ection of detachment’,
as was suggested in [ 7 ]. Thus, the correct condition for the equilibrium of an adhesive contact under

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tangential loading cannot be determined from pur ely theor etical considerations, as it may depend on the
specific physics of the interface.
Another important question in considering adhesion is what happens after the detachment takes
place at some position on the boundary of adhesive contact. If the detachment occurs due to combined
action of normal and tangential loading, then it may well be the case that the adhesion bonds will be
restor ed after the medium has r elaxed the tangential part of elastic energy . This rebinding can actually
take place, if it is not prevented by other factors. The simplest such factor may be a rapid change of
the surface (e.g. due to oxidation). Another reason may be irr eversible changes of surface topography
during detachment (so that the surfaces become incongruent and cannot r estore the initial configuration).
Finally , the actual work of d etachment may be much lar ger than the pure surface ener gy . In this case, the
main part of elastic energy will disappear irr eversibly and the relatively weak interfacial interactions will
not be able to restor e the integrity of the interface again. In all these cases, we would have an irreversible
adhesive contact .
In this paper , we consider exactly this case of irr eversible adhesion. One can interpret this case as a
fracture pr oblem of initially glued contact.
In the following, we do not consider the physics of the interface in detail, but just make assumptions
similar to those of Johnson, and use the method of dimensionality r eduction (MDR) for analysis of
critical detachment conditions in analogy to the MDR formulation for the normal adhesive contact [ 8 ].
In a series of papers, Popov and co-authors have shown that contact pr oblems of axially symmetric
three-dimensional bodies (under the additional assumption of compact contact area) can be equivalently
repr esented by contacts with one-dimensional series of independent springs [ 5 ]. It is important to note
that the results for axially symmetric contacts obtained with MDR ar e exact , and not an appr oximation,
as is often believed. The MDR was first proposed 2007 for non-adhesive contacts [ 9 ], in which case it
simply coincides with the solution of Galin & Sneddon [ 10 , 11 ]. In his dissertation of 2011, Heß derived
the MDR formulation for adhesive contacts of arbitrary axis-symmetric bodies [ 12 ]. A short re view of the
MDR for contacts of bodies of revolution can be found in [ 8 ].
Let us briefly mention previous appr oaches to the pr oblem of adhesion under superimposed normal
and tangential load. In [ 13 ], the authors used the discrete element method for modelling contacts
between cohesive, frictional particles with normal and tangential loading taking into account adhesion
forces between the particles. In [ 14 ], a model of tangential adhesion contact was pr oposed, which,
however , r equires the assumption that the ef fects of normal and tangential for ce can be considered
independently . In this investigation, the authors showed that in the tangential contact problem the
influence of adhesion can be appr oximately described in terms of equivalent load. In a series of articles
by Guduru and co-authors [ 15 – 18 ] the work of adhesion was consider ed as a function of ‘mode-
mixing’, which means that the work of adhesion depends on the dir ection of detachment [ 7 , 19 ]. In [ 15 ],
the theory of Guduru et al. was verified experimentally . T angential adhesion effects wer e investigated
numerically within the framework of coupled Eulerian–Lagrangian method in [ 20 ]. In [ 21 ], adhesion-
induced plastic deformation due to tangential loading was considered. T angential adhesion ef fects
have been investigated in the context of biological systems [ 22 , 23 ] and physics of particle interactions
[ 13 , 24 , 25 ].
This paper is organized as follows. In §2, we r ecapitulate briefly the MDR appr oach for normal
adhesive contacts and extend it for the case of superimposed normal and tangential loading under
load-controlled and displacement-contr olled conditions. The model is then studied numerically and
analytically and the dependence of the adhesive for ce on the tangential force is established in pr oper
dimensionless variables. In §3, we describe the numerical procedur e in detail and compar e the numerical
and analytical results. Section 4 concludes the paper .
2. Method of dimensionalit y r educ tion formulation f or adhesive c ontact
and analytical solution
2.1. Method of dimensionality reduction for normal adhesive con tac ts
W e start our consideration with a short introduction to the MDR. Let us consider a contact between
an elastic continuum and a rigid, axially symmetric indenter having the shape f ( r ), where r is the polar
radius in the contact plane. If the penetration depth d is known as a function of the radius a of the contact:
d = g ( a ), (2.1)

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d
a D l max ( a )
g ( x )
0
x
z
F z
F x
f ( r )
r
0
r
z
F z
F x
( a )( b )
Figu re 1. MDR transformation of ( a ) the original thr ee - dimensional profile f ( r )i n t o( b ) a one - dimensional image g ( x ) and replac ement
of the elastic half-space by an elastic founda tion. In the presenc e of normal and tangential for ce and adhesion, the springs of the elastic
foundation will be displaced both in the normal and tangential dir ec tions. In this figur e, only v er tical displacements ar e shown.
then the normal force F N can be r epr esented as a function of penetration depth by the trivial equation
F N =  F N
0
d ˜
F N =  a
0
d ˜
F N
d ˜
d
d ˜
d
d ˜
a d ˜
a =  a
0
k ( ˜
a ) d g ( ˜
a )
d ˜
a d ˜
a
=  a
0
d k ( ˜
a )
d ˜
a ( d − g ( ˜
a ))d ˜
a ,
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(2.2)
where k ( ã ) is the stif fness of a cylindrical punch with radius ã . Equations (2.1) and (2.2) can be interpr eted
as the result of the indentation of a modified pr ofile g ( ã ) into an elastic foundation with independent
springs with spacing d ã and stiffness (1/2)(d k ( ˜
a )/d ˜
a )d ˜
a . This interpretation is the basis of the MDR for
both homogeneous [ 8 ] and non-homogeneous media [ 26 ]. In accordance with equations (2.1) and (2.2),
the use of MDR is possible under two conditions: (i) the contact stif fness k ( ã ) of a cylindrical stamp with
ar a d i u s ã must be known and (ii) the rule of determining the modified pr ofile g ( ã ) is known. Depending
on the circumstances, these two steps may be performed analytically , numerically or experimentally .
For homogeneous media, the rule for finding the modified pr ofile g ( ã ) is known explicitly . In the
following, we will denote the ar gument of this function by x as is usually done in the MDR. However ,
in this context x does not denote any spacial coordinate but the internal variable of the MDR. The initial
three-dimensional pr ofile f ( r ) (shown in figur e 1 a ) is first replaced by the one-dimensional pr ofile g ( x )b y
means of the transformation [ 8 ]:
g ( x ) =| x |  | x |
0
f  ( r )
 x 2 − r 2 d r . (2.3)
If needed, the original surface z = f ( r ) can be always restor ed fr om its MDR-transformed one-dimensional
profile by
f ( r ) = 2
π  r
0
g ( x )
 r 2 − x 2 d x . (2.4)
In this paper , we will limit ourselves to parabolic pr ofiles of the form f ( r ) = r 2 /(2 R ). However ,
generalization to arbitrary rotationally symmetric profiles is straightforward. In the case of the parabolic
profile, transformation (2.3) leads again to a parabolic pr ofile g ( x ) with a changed coefficient:
f ( r ) = r 2
2 R ⇒ g ( x ) = x 2
R . (2.5)
In the second step [ 8 ], the elastic half-space must be replaced by an elastic foundation, as shown in
figure 1 , consisting of independent springs having normal and tangential stiffness
k z = E ∗  x and k x = G ∗  x , (2.6)
where  x is the spacing of the springs and the ef fective moduli E *a n d G * are defined as
E ∗ = E
1 − ν 2 = 2 G
1 − ν , G ∗ = 4 G
2 − ν , (2.7)

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so that
G ∗ = E ∗ 2 − 2 ν
2 − ν . (2.8)
Note that in the case of ν = 0 both effective moduli coincide: E * = G *. For definiteness and simplicity , all
numerical simulations and analytical calculations below are performed under this assumption.
Before pr oceeding to tangential contacts, we will first r ecapitulate the application of the MDR to
normal adhesive contacts. If the MDR-transformed profile g ( x ) is indented into the elastic foundation by
an indentation depth d , then the displacement of individual springs inside the contact will be determined
by the equation
u z ( x ) = d − g ( x ) = d − x 2
R . (2.9)
The size of the adhesive contact at a given indentation depth can be easily found from the principle
of virtual work: the springs at the boundary of contact are str etched by  l =− u z ( a ). The ener gy released
through detachment of two boundary springs is equal to E ∗  l 2  x . Through detachment, the fr ee surface
energy 2 π a  x γ is cr eated (this ener gy can only be defined in the original, three-dimensional system).
According to the principle of virtual work, the system will be in equilibrium if these two energies
are equal:
E ∗  l 2  x = 2 π a  x γ . (2.10)
It follows that the condition of equilibrium of boundary springs can be written as
 l =  l max ( a ) =  2 π a γ
E ∗ . (2.11)
This condition is known as the rule of Heß [ 12 ]. Combining (2.9) and (2.11), we get
u z ( a ) = d − a 2
R =−  l max ( a ) =−  2 π a γ
E ∗ (2.12)
or
d = a 2
R −  2 π a γ
E ∗ . (2.13)
The normal for ce can be calculated as the sum of all spring forces:
F z ( a ) = E ∗  a
− a
u z ( x )d x = 2 E ∗  a
0  d − x 2
R  d x = 4 E ∗ a 3
3 R −  8 π a 3 E ∗ γ . (2.14)
Later we will consider a more general situation, wher e the indenter is also displaced in the tangential
direction by u (0)
x . It is convenient to present both analytical and numerical r esults in terms of
dimensionless variables:
˜
a = a
a 0
, ˜
F z = F z
F 0
, ˜
d = d
d 0
, ˜
u (0)
x = u (0)
x
d 0
, ˜
u z = u z
d 0
, (2.15)
where F 0 , a 0 and d 0 ar e the critical values of the for ce, the contact radius and the absolute indentation
depth at the moment of detachment of the parabolic profile fr om the elastic half-space under for ce-
controlled conditions [ 4 ]:
F 0 = 3
2 π R γ , a 0 =  9 π R 2 γ
8 E ∗  1 / 3
, d 0 =  3 π 2 R γ 2
64 E ∗ 2  1 / 3
. (2.16)
In dimensionless variables, equations (2.13) and (2.14) take the form
˜
d = 3 ˜
a 2 − 4 ˜
a 1 / 2 (2.17)
and
˜
F = ˜
a 3 − 2 ˜
a 3 / 2 . (2.18)
These results of course coincide with the classical solution of Johnson et al. [ 1 ]. The dependence of
the dimensionless normal force on the dimensionless appr oach (indentation depth) implicitly defined
by equations (2.17) and (2.18) is shown in figure 2 and will be used for testing numerical pr ocedures
described in §3.
W e now analyse the condition of instability of the contact, i.e. the conditions under which the
possibility of the adhesive contact to sustain equilibrium is lost. W e will consider both displacement-
controlled and load-contr olled contacts. Under displacement-controlled conditions, the macr oscopic

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–2 0 2 4 6
–1
0
1
2
d
~
d
~
F
~
z
F
~
z
–2 0 2 4 6
–1
0
1
2
( a )( b )
Figu re 2. Dependence of the normal for ce on the indentation depth for the normal contact with adhesion. Solid lines show the analytical
solution defined by equations (2.17) and (2.18). Cir cles r epresent r esults of numerical experiments for displac ement- contr olled ( a )a n d
load- contr olled ( b ) conditions as described in §3.
displacement of the indenter is imposed by a very stiff external system. Physically , this means that during
the movement of the system towards the equilibrium state, the displacement is kept exactly constant.
Load-controlled c onditions are physically r ealized by applying the given force thr ough a very soft spring.
Thus, in conditions of load control, the force during the r elaxation to the equilibrium r emains fixed. The
method described in this section has been used successfully for modelling the influence of adhesion on
impact between elastic particles under displacement-controlled conditions [ 25 ].
2.2. Superimposed normal and tangential loading
Let us now assume that the loading of the profile consists of superimposed normal for ce and tangential
displacement u (0)
x . The energy r eleased by detachment of two boundary springs will now be equal to
E ∗ u z ( a ) 2  x + G ∗ u (0)2
x  x . Equating this to the work of adhesion 2 π a  x γ , we arrive at the following
equilibrium condition:
E ∗ u z ( a ) 2 + G ∗ u (0)2
x = 2 π a γ . (2.19)
This rule is exactly equivalent to the rule obtained by Johnson on the basis of the ener gy r elease rate (1.3).
From (2.19), for the elongation of the boundary springs we get
| u z ( a ) |=  2 π a γ
E ∗ − G ∗
E ∗ u (0)2
x . (2.20)
Using (2.9), we can write the relationship between the indentation depth and contact radius in the form
d = a 2
R −  2 π a γ
E ∗ − G ∗
E ∗ u (0)2
x . (2.21)
The normal and tangential forces ar e functions of contact radius a :
F z = 2 E ∗  ad − a 3
3 R  (2.22)
and
F x = 2 G ∗ a · u (0)
x . (2.23)
In dimensionless variables (2.15), equations (2.21)–(2.23) can be written as
˜
d = 3 ˜
a 2 −  16 ˜
a − G ∗
E ∗ ˜
u (0)2
x , (2.24)
˜
F z = ˜
a
2 ( ˜
d − ˜
a 2 ) (2.25)
and ˜
F x = G ∗
2 E ∗ ˜
a · ˜
u (0)
x . (2.26)
These equations determine the normal for ce–indentation relation in the pr esence of tangential
displacement. Note that substitution of (2.24) into (2.25) at ˜
u (0)
x = 0 reduces to the result (2.18) for the
normal contact.

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F
~
z
D F
~
z
F
~
x F
~
x
01 2345
–1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5
–0.5
0
0.5
02
46 81 0
0.1
0.2
0.3
0.4
0.5
( a )( b )
Figu re 3. ( a ) The dependence of normaliz ed critical normal forc e ˜
F z on normalized critical tangen tial force ˜
F x , for the case E * = G *.
Analytical results ar e repr esented b y solid lines and the results of numerical simulation b y open circles , diamonds, stars and triangles .
The upper line ( diamonds and stars) corresponds to displacemen t control in both dir ections. The lo wer line ( open circles and triangles)
corr esponds to load contr ol in the vertical direction and displacement contr ol in the tangential dir ec tion. Diamonds and open circles (r ed
lines) corr espond to detachment at a negative indenta tion depth d . Stars and triangles correspond to detachment at a positive indentation
depth; ( b ) the differenc e between the normal forc es shown in ( a ) as a function of normalized tangential for ce ˜
F x .
Let us str ess that equation (2.19) assumes that the work of adhesion does not depend on the direction
in which the surfaces are detached fr om each other . This is a physical assumption that may be incorr ect
in some systems [ 7 , 19 ]. At this point, further investigation of the pr ocess of detachment would have to
be carried out. In the following, we remain in the framework of the ‘Johnson paradigm’, and use the
detachment condition (2.19) and equations based on it. W e will treat both the displacement-contr olled
and load-controlled cases in the normal dir ection, but will confine ourselves to displacement control in
the horizontal direction.
2.2.1. Adhesion for ce under load- contr olled conditions
Under load-controlled conditions, the instability occurs at the contact radius at which the normal force is
minimized. Thus, the condition of instability can be written as d F z /d a = 0. Dif ferentiating equation (2.22)
with respect to a and using equation (2.21) we arrive at the condition
2 a 2
c , fl
R  2 π γ a c , fl
E ∗ − G ∗
E ∗ u (0)2
x − 3 π γ a c , fl
E ∗ + G ∗
E ∗ u (0)2
x = 0, (2.27)
or , in dimensionless variables (2.15)
˜
a 2
c , fl  ˜
a c , fl − G ∗
16 E ∗ ˜
u (0)2
x − ˜
a c , fl + G ∗
24 E ∗ ˜
u (0)2
x = 0. (2.28)
This equation determines the dependence of the critical radius ˜
a c , fl on the tangential displacement ˜
u (0)
x .
The dependence of the adhesion for ce on the tangential force can be determined by substituting ˜
a c , fl into
equations (2.24)–(2.26). This dependence is shown in figur e 3 a (lower solid line). For ˜
F x = 0( o r ˜
u (0)
x = 0),
equations (2.24), (2.25) and (2.28) pr ovide the critical force ˜
F z (0) =− 1.
Let us note that the dependence shown in figur e 3 a determines the critical tangential force for both
positive and negative normal forces. In the case of the negative normal for ce, this critical value re ally
corresponds to loss of ‘adhesive’ contact, as destr uction of the contact will lead to movement of the
indenter away from the substrate. In the case of positive normal for ces, contact will not be lost at the
critical value, only the continuity of the contact will be lost, which can be understood as propagation of
a mode II (tangential) crack. The critical condition (2.19) does not differ entiate between these two cases,
so the resulting dependences of F z on F x ar e equally valid for positive and negative F z , although the
physical interpretation is dif fer ent.

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2.2.2. Adhesion under displacemen t- contr olled conditions
For the pure normal contact, this mode is shown in figur e 2 a ; the detachment occurs at F z / F 0 = ˜
F z =− 0.5.
The condition of instability is in this case formally given by d( d )/d a = 0. Thus, in the general case of
superimposed normal and tangential loading, differ entiating (2.24) gives
a 3
c , fg − G ∗ u (0)2
x
2 π γ a 2
c , fg − 2 π γ R 2
16 E ∗ = 0, (2.29)
or , in dimensionless variables (2.15)
˜
a 3
c , fg − G ∗
16 E ∗ ( ˜
u (0)
x ˜
a c , fg ) 2 − 1
9 = 0. (2.30)
This equation determines the dependence of the critical radius ˜
a c , fg on the tangential displacement ˜
u (0)
x .
For critical forces at the moment of detachment, we obtain
˜
F 2
x = 4 G ∗
E ∗  ˜
a 3 − 1
9  (2.31)
and
˜
F z = ˜
a 3 − 2
3 . (2.32)
From the two last equations, it follows
˜
F z = E ∗
4 G ∗ ˜
F 2
x − 5
9 . (2.33)
For the case E * = G *, the relationship between ˜
F z and ˜
F x under displacement control simplifies to
˜
F z = 1
4 ˜
F 2
x − 5
9 . (2.34)
This dependence is shown in figure 3 a (upper solid line). Additionally , in figure 3 b the differ ences
between the normal forces shown in figur e 3 a ,  ˜
F z , are shown as a function of normalized tangential for ce
˜
F x . At zero tangential for ce ˜
F x = 0, this differ ence is equal to  ˜
F z (0) = ˜
F fg
z (0) − ˜
F fl
z (0) =− 5/9 − ( − 1) = 4/9.
W ith increasing ˜
F x the value  ˜
F z monotonically decreases. Note that the lower dependence in figur e 3 a
(under load-controlled c onditions in the normal direction and displacement contr ol in the tangential
direction) is well approximated by equation (2.34) when value  ˜
F z shown in figure 3 b is close to zer o. At
˜
F x  1 both dependences shown in figure 3 a coincide and ar e described by equation (2.34).
3. Numerical pr ocedur e and c omparison with analytical r esults
In the following, we r eproduce the above r esults numerically and use the developed numerical procedur e
to extend them to a more complicated detachment condition. Let us first briefly describe the numerical
procedur e for the case of normal adhesive contact. In the first step, the modified parabolic pr ofile g ( x )
(2.5) is indented by d into the elastic foundation shown in figure 1 b . After this, the position of the
critical boundary springs (and thus the contact radius) is calculated using the condition (2.11). Given the
indentation depth and the contact radius, the normal and tangential forces can be obtained by summing
the forces of all springs in contact:
F z = E ∗  x 
cont
u z ( x i ) (3.1)
and
F x = G ∗  x 
cont
u x ( x i ), (3.2)
where u z ( x i )a n d u x ( x i ) are, r espectively , the normal and tangential displacements of individual springs
at the coordinate x i . In a pur ely normal contact u x ( x i ) = 0, and F x = 0.
When analysing the adhesive contact under displacement-controlled conditions, we displace the rigid
indenter in the vertical direction step by step with a chosen discr etization  z . The new configuration of
contact after each step is calculated using the rule (2.11) for the springs at the boundary of the contact. If
it happens that more than two springs are detached, we r eturn to the pr evious step and proceed further
with a discretization step  z /2 and, if necessary , decr ease it further until only one spring is lost on each
side. This procedur e is continued up to the point of the instability .
For load-controlled conditions, the contr olling parameter is the normal for ce F up .I ne a c hs t e p , F up is
increased by an incr ement  F , and the new equilibrium configuration of the contact is found. Similar

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................................................
to the pr ocedure for displacement contr ol, the incr ement of the force is decr eased if mor e than two
springs are detached in one step. The r esults of numerical calculations with displacement-contr olled and
load-controlled conditions for normal motion ar e presented together with analytical r esults in figure 2 .
Since the numerical r esults coincide with the analytical ones, this provides some validation for the above
numerical procedur e.
The procedur e in the case of combined normal and tangential loading is basically the same as
described above. The only differ ence is the use of a modified detachment condition
 l =  u 2
x + u 2
z =  l max . (3.3)
In figure 3 a , the results of numerical calculations ar e shown along with analytical results pr esented in
the previous section.
4. C onclusion
W e performed analytical and numerical analysis of the adhesive contact between two elastic bodies with
an axially symmetric gap profile under superimposed normal and tangential loading. The study was
performed under the simplest assumption that the surface energy does not depend on the dir ection of
detachment. However , the developed analytical method can be generalized straightforwar dly for more
complicated adhesive interactions, as for example suggested in [ 23 ]. Under the above assumptions, the
application of tangential force leads to a decrease of the normal adhesive for ce. W e considered dif ferent
combinations of contr olled load and controlled displacement in both normal and tangential dir ections
and derived for each case solutions in the appropriate dimensionless variables. In the case of mor e
general adhesive interaction than assumed in this paper , it would further be inter esting to take into
account the partial slip and frictional forces in the contact plane (as has been done for the special case of
the Dugdale adhesive potential in [ 27 ]).
Data acc essibilit y . No supporting data needed. All necessary algorithms ar e fully described in article.
Authors’ c ontributions. V .L.P . proposed the idea of the article, performed the analytical investigation and wrote the text of
the article; I.A.L. and A.E.F . carried out numerical analysis of the equations and performed numerical modelling. All
authors gave final approval for publication.
Competing in terests . W e declar e we have no competing interests.
Fun d in g. This work was supported in parts by the German Academic Exchange Service (DAAD), by the Ministry of
Education of the Russian Federation and by T omsk State University Academic D.I. Mendeleev Fund Pr ogram. I.A.L.
is grateful to MESU for financial support under the project 0116U006818.
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