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```International Journal of Engineering Science 55 (2012) 54–65
Contents lists available at SciVerse ScienceDirect
International Journal of Engineering Science
journal homepage: www.elsevier.com/locate/ijengsci
Adhesive full stick contact of a rigid cylinder with an elastic half-space
O.I. Zhupanska ⇑
Department of Mechanical and Industrial Engineering, University of Iowa, 2416A Seamans Center for the Engineering Arts and Sciences, Iowa City, Iowa,
IA 52242-1527, USA
a r t i c l e
i n f o
Article history:
Received in revised form 29 October 2011
Accepted 9 February 2012
Available online 14 March 2012
Keywords:
Contact mechanics
Friction
JKR model
Fourier integral transform
Exact solution
a b s t r a c t
The problem of adhesive contact of a rigid cylinder with an elastic half-space is considered.
The proposed adhesive contact model differs from the Johnson–Kendall–Roberts (JKR)
model by preserving the inﬂuence of the contact shear stresses in the problem formulation
and considering the so-called full stick contact due to the large values of the friction coefﬁcient between contacting surfaces, as opposed to the frictionless contact assumed in the
JKR model. An analytical treatment of the problem is presented, with the corresponding
boundary-value problem formulated in the bipolar coordinates. A general solution in the
form of Papkovich–Neuber functions and the Fourier integral transform is used to obtain
an exact solution to the formulated boundary-value problem. Comparison of the results
with the JKR model shows that accounting for the contact shear stresses leads to smaller
contact areas as compared to those predicted by the JKR model.
1. Introduction
Interaction between ideal surfaces in contact is characterized by an interplay of competing attractive and repulsive forces
which lead to the so-called equilibrium separation. At distances below equilibrium separation the surfaces will repel, and at
distances greater than equilibrium separation the surfaces will attract, resulting in the so-called adhesive contact. In many
traditional areas of application of contact mechanics (railways, gears, bearings), this surface attraction (i.e., adhesion) is
insigniﬁcant and therefore ignored; thus, only contact pressure is taken into consideration when the corresponding contact
problems are formulated. On the other hand, there has been an increasing interest to the problems of adhesive contact in the
last several decades due to technological advances in manufacturing of MEMS (Zhao, Wang, & Yu, 2003), surface polishing
(Rimai, DeMejo, & Bowen, 1990; Zhang, Busnaina, & Ahmadi, 1999), design of adhesive surfaces (Koberstein et al., 1998;
Tsibouklis et al., 1999), and instrumented depth-sensing indentation (Borodich & Galanov, 2008; Ebenstein & Wahl, 2006;
Wahl, Asif, Greenwood, & Johnson, 2006) among others.
It is well known that presence of the adhesive forces in normal frictionless contact results in a larger contact area as compared to that predicted by the classical Hertz theory (Hertz, 1881; Johnson, 1985). There have been a number of contact
adhesion models developed in the literature, among which are the prominent models of Johnson, Kendall, and Roberts
(1971), Derjaguin, Muller, and Toporov (1975) and Maugis (1992). For instance, in the Johnson–Kendall–Roberts (JKR) model
(Johnson et al., 1971), the total free energy balance is established by accounting for an elastic strain energy stored in the
deformed bodies due to mutual contact compression and a surface energy, which accounts for surface adhesion. Meanwhile,
the inﬂuence of contact shear stresses on the distribution of normal contact pressure and size of the contact area may be
quite signiﬁcant.
⇑ Tel.: +1 319 335 9350; fax: +1 319 335 5669.
doi:10.1016/j.ijengsci.2012.02.002
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
55
Contact shear stresses are usually a consequence of the action of the tangential force, but they also are present under normal contact, if elastic properties of the contacting bodies are different. Depending on the coefﬁcient of friction between the
contacting surfaces, the presence of the contact shear stresses will be manifested in the so-called stick and/or slip contact
conditions. Stick is characterized by the absence of the relative displacements of the points on the contact surface and occurs
at relatively large magnitudes of the coefﬁcient of friction that prevent sliding of the contacting surfaces (Johnson, 1985);
otherwise, partial slip is present in the contact area. Investigations of stick–slip contact problems started with the pioneering
work of Galin (1953), see also a recent revised translation by Galin and Gladwell (2008), and followed by many others (see,
e.g., a recent review in Zhupanska (2008)).
A signiﬁcant contribution to the stick–slip contact problems was made by Spence (1968a, 1975), who originated a selfsimilarity approach, which yields geometrically similar stress and displacement ﬁelds at each step of application of progressive load for any indenter with a polynomial proﬁle. In this setting, Spence considered a full stick contact problem and the
problem with ﬁnite friction. The full stick contact problem was reduced to an integral equation, which was then solved using
the Wiener–Hopf technique (Spence, 1968b). Following Spence’s ideas of self-similarity, Nowell, Hills, and Sackﬁeld (1988)
constructed a solution of the contact problem for dissimilar cylinders. A completely independent analytical solution procedure for 2D stick–slip contact problems was suggested by Zhupanska and Ulitko (2005), who gave an exact solution to the
normal contact with friction of a rigid cylinder with an elastic half-space. It is also worth to mention contributions to the selfsimilarity approach by Borodich (1993), Borodich and Galanov (2002) and Borodich and Keer (2004). A self-similarity approach changed the way of treatment of contact problems that involve moving contact boundaries, and allowed one to avoid
an incremental step-by-step computation of the contact stresses performed simultaneously with the increase of the contact
area (e.g. Goodman, 1962; Mossakovski, 1954, 1963).
It is well known from the numerous studies of the stick–slip contact problems that when friction between contacting surfaces is taken into consideration in the normal non-adhesive contact of bodies with different elastic properties, the normal
contact pressure exerted at the interface is greater than that predicted by the Hertz theory. In other words, under a ﬁxed
normal compressive load, the actual contact area will be smaller than that predicted by the Hertz theory (see, e.g., Nowell
et al., 1988; Zhupanska, 2009; Zhupanska & Ulitko, 2005). Thus, it is reasonable to expect that the contact stress distributions
and the size of the contact area in a problem of adhesive contact will also depend on whether or not the assumption of
smooth (frictionless) contact holds. This observation served as a primary motivation for the present paper, where the problem of full stick with adhesion of an elastic half-space and a rigid cylinder is considered.
The adhesive contact model proposed in this paper incorporates Grifﬁth energy balance between elastic energy and surface energy, as it is done in the JKR model, to account for adhesive effects. Full stick contact conditions are formulated within
Spence’s self-similarity approach (Spence, 1968a) and the solution procedure follows the author’s earlier work (Zhupanska &
Ulitko, 2005), where a general solution in the form of Papkovich–Neuber functions and a special coordinate system, namely
the planar bipolar coordinates, were used along with the Wiener–Hopf technique. This allowed us to construct an analytical
solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space and obtain a simple exact
solution in the limiting case of full stick between the cylinder and half-space.
The present paper was also stimulated by the works of Chen and Gao (2006a, 2006b, 2007), who considered non-slipping
adhesive contact problems for an elastic cylinder on stretched substrates, mismatched elastic cylinders, and mismatched
elastic spheres. In the case of contact with a stretched substrate, the elastic deformation of the substrate was included in
the formulation of the boundary-value problem, and the effects of the resulting shear stresses on the adhesive contact were
examined. In the case of two mismatched cylinders, a mismatched constant strain was stipulated in the contact zone (as
opposed to the self-similarity argument used in the present study). The solution procedure for the considered contact problems consisted in the reduction of the formulated boundary-value problems to the system of coupled singular integral equations with respect to the unknown normal and shear stresses in the contact area that have been solved analytically. The
solution procedure employed in the present paper is much simpler, does not require solution of integral equations, reducing
merely to inversion of the Fourier integrals, and yields an exact solution.
The rest of the paper is organized as follows. Section 2 introduces the boundary-value problem under consideration. Section 3 describes the solution procedure that includes reformulation of the boundary-value problem in the bipolar coordinates, application of the Papkovich–Neuber general solution and Fourier integral transform to obtain the distributions of
normal and shear contact stresses. Non-adhesive full stick contact problem is discussed in Section 4. Finally, Section 5 provides a simple analytical solution to the adhesive full stick contact problem.
2. Problem formulation
Consider an elastic half-space indented by a rigid cylinder of radius R (Fig. 1). The force P applied to the center of the
cylinder is normal to the surface of the half-space, and the half-space deforms in the state of plane strain.
Assume that the coefﬁcient of friction between two contacting surfaces is large enough that there are no relative displacements of the points on the surfaces in the contact area a 6 x 6 a (i.e., the so-called full stick contact takes place). For ﬁnite
values of the friction coefﬁcient, two slip zones adjacent to the stick zone will arise, if the elastic properties of the contacting
materials are different. If the contacting bodies have the same elastic properties, the contact is frictionless. The focus of this
paper is on adhesive contact of a rigid cylinder with an elastic half-space with full stick contact conditions according to
56
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
y
-a
a
O
P
R
x
Fig. 1. Contact of an elastic half-space with a rigid cylinder.
Spense’s self-similarity approach (Spence, 1968a). The self-similarity approach postulates that the contact stress and strain
ﬁelds are geometrically similar under progressively increasing load, and the strain in the stick zone takes the form:
@ux ¼ C 0 jxj;
@x y¼0
a 6 x 6 a;
ð1Þ
where ux is the lateral displacement, C0 is a non-positive ‘‘frozen-in’’ strain constant. The linear dependence in (1) ensures a
constant strain at each point of the stick contact area for a progressively increasing load. In other words, the lateral strain at
any given point of the stick contact area does not change when the boundary of the contact area changes, which explains the
term ‘‘frozen-in’’. The value of the frozen-in strain constant C0 is unknown and has to be determined as a part of the solution.
The assumption on the relatively small size of the contact zone (a R) allows us to write the boundary condition for normal displacement uy in the entire contact area as
uy jy¼0 ¼
a2 x2
;
2R
a 6 x 6 a:
ð2Þ
Note that in this 2D formulation of the contact problem does not allow for determining the axial displacement of the halfspace (as opposed to the 3D axisymmetric formulation). This deﬁciency of the 2D model was resolved in the work of Argatov
(2001), where the approach between two contacting elastic bodies that deform in the state of plane strain was determined
by the so-called local compliances.
Since the full stick contact problem is considered, we also assume that in the contact area the magnitude of the shear
stress is insufﬁcient for causing slip, thus
jsxy j 6 ljry j;
a 6 x 6 a;
ð3Þ
where l is the coefﬁcient of interfacial friction between the half-space and the cylinder.
Outside the contact zone, the surface of the elastic half-space is free of stresses
ry jy¼0 ¼ sxy jy¼0 ¼ 0; a < jxj < 1:
ð4Þ
The equilibrium condition leads to the following equality to be satisﬁed by the normal stress and applied force
P¼
Z
a
a
ry jy¼0 dx:
ð5Þ
Eqs. (1)–(4) constitute a mixed boundary-value problem of planar elasticity for the case of plane strain. Our objective is to
solve the boundary-value problem (1)–(4) and obtain exact solutions for the corresponding non-adhesive and adhesive contact full stick contact problems.
3. Solution procedure
The solution procedure used in this work is based on that proposed by Zhupanska and Ulitko (2005) and consists of the
general solution in the form of Papkovich–Neuber functions, bipolar coordinate system, and the Fourier integral transform.
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
57
3.1. General solution in the bipolar coordinates
The Papkovich–Neuber solution (Gladwell, 1980; Sokolnikoff, 1956; Uﬂyand, 1965) is a general solution of elastostatics
equations that represents the components of displacement and stress ﬁelds by a combination of functions that are solutions
of the Laplace equation. It is known to provide advantages in solving boundary-value problems of elasticity for bodies
bounded by inﬁnite planar surfaces (half-space, layer, wedge, etc.). Generally, the Papkovich–Neuber solution has the form
u¼B
1
rðr B þ B0 Þ; r2 B ¼ r2 B0 ¼ 0:
4ð1 mÞ
ð6Þ
Here u is the displacement vector, B and B0 are vector and scalar harmonic functions, r is the position coordinates vector, and
m is the Poisson ratio. This solution simpliﬁes signiﬁcantly in the case of the half-space, and for an elastic half-space y P 0 in
the state of plane strain, which is considered in this work, the components of elastic displacement and stress on the surface
of the half-space become (Uﬂyand, 1965)
@ux @ U3 @uy @
;
½ð3
4
¼
e
¼
;
¼
m
Þ
U
U
x
2
3 @x y¼0
@y y¼0
@x y¼0 @x
y¼0
ry @
s
@
xy
;
½2ð1
½ð1
2
¼
m
Þ
U
U
;
¼
m
Þ
U
U
2
3 2
3 @x
2G y¼0 @y
2G
y¼0
y¼0
y¼0
ð7Þ
where U2 and U3 are two harmonic Papkovich–Neuber functions and G is the shear modulus of the elastic medium.
At the next step, planar bipolar coordinates (a, b) are introduced that relate to the Cartesian coordinates (x, y) as (see, e.g.,
Morse & Feshbach, 1953)
x¼a
sinh a
;
cosh a þ cos b
y¼a
sin b
;
cosh a þ cos b
1 < a < 1;
p 6 b 6 p;
ð8Þ
where the metric parameter a equals the half-size of the contact area. The relationship between the differentials in Cartesian
and bipolar coordinates is
@
1
@
@
;
þ sin b sinh a
¼
ð1 þ cosh a cos bÞ
@x a
@a
@b
@
1
@
@
¼
sin b sinh a
:
þ ð1 þ cosh a cos bÞ
@y a
@a
@b
ð9Þ
The bipolar coordinate system is shown in Fig. 2.
Introduction of bipolar coordinates has important mathematical consequences with regard to constructing an exact solution of the boundary-value problem (1)–(4). It enables one to reformulate the boundary-value problem (1)–(4) in three inﬁnite regions, namely b = p, 1 < a < 0, b = 0, 1 < a < 1, and use the Fourier integral representations to obtain the solution.
Note that the coordinate line b = 0 corresponds to the portion a 6 x 6 a of the x axis on the surface of the half-space (y = 0);
line b = p corresponds to the interval a < jxj < 1 outside of the contact area on the surface of the half-space (y = 0), a = 0 corresponds to the line x = 0, and the ‘‘inﬁnite’’ points a = ±1 correspond to x = ±a, respectively.
Upon substituting representations (8) into the boundary conditions (1)–(4) and using relationships (9), the original
boundary-value problem (1)–(4) in terms of harmonic functions U2 and U3 can be rewritten in the bipolar coordinates in
the form
tanh a
@ U3 2
2
¼
C
a
; 1 < a < 1;
0
@b b¼0
cosh a þ 1
@
a2 tanh a2
½ð3 4mÞU2 U3 ¼
; 1 < a < 1;
@a
R cosh a þ 1
b¼0
@
½2ð1 mÞU2 U3 ¼ 0; 1 < a < 1;
@b
b¼p
@
½ð1 2mÞU2 U3 ¼ 0; 1 < a < 1:
@a
b¼p
ð10Þ
The following integral representation for the harmonic Papkovich–Neuber functions in the bipolar coordinates is used (Uﬂyand, 1965)
58
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
Fig. 2. Planar bipolar coordinates.
cosh a2 þ sin 2b
U2 ða; bÞ ¼ C 2 ln
!
Z
1
þ pﬃﬃﬃﬃﬃﬃﬃ
2p
2
!
a
cosh 2 þ sin 2b
cosh a sin b
2
U3 ða; bÞ ¼ ð1 2mÞC 2 ln
cosh a2 sin 2b
1
½A2 ðkÞ cosh kb þ B2 ðkÞ sinh kbeika dk;
1
1
þ pﬃﬃﬃﬃﬃﬃﬃ
2p
Z
1
ð11Þ
½A3 ðkÞ cosh kb þ B3 ðkÞsinh kbeika dk:
1
Expressions (11) contain the regular Fourier expansions and logarithmic terms that replicate the singularities of the elastic
ﬁeld at the inﬁnity. The constant C2 in (11) is unknown at the moment and is determined as a part of the solution of the
boundary-value problem (10).
Next, the unknown densities A2,3(k) and B2,3 (k) are found by replacing functions U2 and U3 with expressions (11) in the
equalities (10) and inverting the obtained Fourier integrals:
"
pﬃﬃﬃﬃﬃﬃﬃ C 0 a2 Z
kB3 ðkÞ ¼ 2p
p
sinh a2
1
0
3a
2
2cosh
kA3 ðkÞ ¼ ð3 4mÞkA2 ðkÞ #
C2
;
cosh pk
pﬃﬃﬃﬃﬃﬃﬃ a2
k2
2p
;
R sinh pk
pﬃﬃﬃﬃﬃﬃﬃ (
i
2p a2 2 h
2
2
k 2ð1 mÞcosh pk ð1 2mÞsinh pk þ ð1 2mÞC 2 sinh pk
kA2 ðkÞ ¼
DðkÞ R
)
Z 1
sinh a2
C 0 a2
cosh pksinh pk
3a
p
0 2cosh
2
ð12Þ
pﬃﬃﬃﬃﬃﬃﬃ (
2
2
2ð1 mÞcosh pk ð1 2mÞsinh pk C 0 a2
2p a2 2
2
½2ð1 mÞcosh pk
k cosh pk þ ð1 2mÞC 2
cosh pk
DðkÞ R
p
)
Z 1
sinh a2
2
;
cos
k
a
d
a
ð1 2mÞsinh pk
3
0
2cosh a2
kB2 ðkÞ ¼ where
2
DðkÞ ¼ 4ð1 mÞ2 cosh
pk ð1 2mÞ2 sinh2 pk:
ð13Þ
The following known integrals (Erdelyi, 1954) were used to obtain integral densities (12)
Z
1
0
Z
0
1
cos ka
p
da ¼
;
cosh a2
cosh pk
sinh a2
2cosh
3a
2
pk2
:
sinh pk
ð14Þ
Substituting Papkovich–Neuber functions (11) into the general representations (7), we can write the general solution for
stresses and displacements at the surface of the half-space in the bipolar coordinates in the form of Fourier integrals
Z 1
@ux cosh a þ 1 ð1 2mÞC 2
1
ika
p
ﬃﬃﬃﬃﬃﬃ
ﬃ
¼
þ
kB
ðkÞe
dk
;
3
a
@x b¼0
cosh a2
2p 1
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
1
uy jb¼0 ¼ pﬃﬃﬃﬃﬃﬃﬃ
2p
ry 2G
¼
b¼0
syx 2G
1
½ð3 4mÞA2 ðkÞ A3 ðkÞeika dk;
1
Z 1
cosh a þ 1 C 2
1
ika
p
ﬃﬃﬃﬃﬃﬃ
ﬃ
þ
k½2ð1
m
ÞB
ðkÞ
B
ðkÞe
dk
;
2
3
a
cosh a2
2p 1
¼
Z
59
b¼0
Z
cosh a þ 1 i
pﬃﬃﬃﬃﬃﬃﬃ
a
2p
1
k½ð1 2mÞA2 ðkÞ A3 ðkÞeika dk;
1
Z 1
@ux cosh a 1 1
p
ﬃﬃﬃﬃﬃﬃ
ﬃ
¼
k½A3 ðkÞsinhpk þ B3 ðkÞ cosh pkeika dk;
a
@x b¼p
2p 1
uy jb¼p ¼ 2ð1 mÞC 2 ln
ry 2G
b¼p
syx 2G
b¼p
¼
¼
cosh a2 þ 1
1
þ pﬃﬃﬃﬃﬃﬃﬃ
cosh a2 1
2p
cosh a 1 1
pﬃﬃﬃﬃﬃﬃﬃ
a
2p
cosh a 1 i
pﬃﬃﬃﬃﬃﬃﬃ
a
2p
Z
Z
1
Z
1
½ðð3 4mÞA2 ðkÞ A3 ðkÞÞ cosh pk þ ðð3 4mÞB2 ðkÞ B3 ðkÞÞ sinh pkeika dk;
1
k½ð2ð1 mÞA2 ðkÞ A3 ðkÞÞ sinh pkþð2ð1 mÞB2 ðkÞ B3 ðkÞÞ cosh pkeika dk;
1
1
k½ðð1 2mÞA2 ðkÞ A3 ðkÞÞ cosh pk þ ðð1 2mÞB2 ðkÞ B3 ðkÞÞ sinh pkeika dk:
ð15Þ
1
3.2. Contact stress distributions
The distributions of normal and shear contact stresses are obtained by substitution of the densities A2,3(k) and B2,3 (k) into
the corresponding equations for contact stresses (15) and subsequent evaluation of the resulting complex integrals using
calculus of residuals. In this section we outline the main mathematical steps of this procedure and derive analytical expressions for the contact stress distributions.
First, we rewrite normal and shear contact stresses in the complex form:
(
"
iha #)
sinh a2
cosh a þ 1 a @ 2
e
C 2 eiha
cosh ph eiha
@J 1 ða;hÞ
;
þi
þ
¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R @ a2 cosh a þ 2ð1 mÞ a cosh a C 0 a sinh ph
3
2p
@a
sinh a2
3 4m
2cosh a2
b¼0
2
2
ry þ isyx 2G
0 6 a 6 1:
ð16Þ
Here J1(a, h) denotes
J 1 ða; hÞ ¼
Z
1
0
1
2
2cosh 2x
eihðaþxÞ
eihðaxÞ
a
þx a
x dx
sinh 2
sinh 2
ð17Þ
and
1 2m
sinh ph ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;
3 4m
2ð1 mÞ
cosh ph ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
3 4m
ð18Þ
Integral (17) is evaluated in the following way. First, we use the representation
eihx
¼i
sinh 2x
Z
1
tanh pðh þ kÞeikx dk:
ð19Þ
1
This representation is obtained using direct and inverse Fourier transforms, calculus of residuals, and the integral (Erdelyi,
1954)
Z
0
1
ept dt
2
cosh t
¼
p
½wðp=4 þ 1=2Þ wðp=4Þ 1;
2
Rep > 2:
ð20Þ
Here w(p) is the logarithmic derivative of the Gamma function, for which the following series representation (Bateman,
1953) is used
wðpÞ ¼ C þ ðp 1Þ
1
X
n¼0
1
;
ðn þ 1Þðp þ nÞ
ð21Þ
where C is the Euler constant. Substituting of (19)–(21) into the integral (17) and using calculus of the residuals, we arrive at
the following expression for J1(a, h)
60
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
J 1 ða; hÞ ¼ 2
1
X
eiha
2p
@ eiha
d
1
1
1
þ4
þ
ð
Þ
4
p
i
tanh
p
h
ðk þ ihÞqk eðkþ2ihÞa ;
a
a
a
@
d
1
þ
e
2
sinh 2 cosh ph a cosh 2
a
k¼0
a > 0;
ð22Þ
where
qk ¼ 2
1
X
n¼0
ð1Þn
:
n þ k þ 12 ih
ð23Þ
Eventually, substituting (22) into (16), the distributions of the normal and shear contact stresses are obtained in the form
ry þ isxy 2G
b¼0
( eiha
a
a 1 2
1
a
C2
a
þ 4ð1 mÞ
2 cosh
h ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ih
sinh
cosh
cosh a2
2 4
2
a
2
3 4m R
"
2a
3a
cosh 2
cosh ph cosh 2
p
a 1 2
1
a
h
iC 0 a
ð1
2ihÞ
þ
þ
2ih
sinh
2
cosh
2
cosh a2
p
sinh a2
2 4
2
sinh a2 cosh ph
#)
2
1
X
1
1
k þ ih qk eðkþ2Þa ; 0 < a < 1:
2ðcosh a þ 1Þ
2
k¼0
ð24Þ
The statics condition (5) now becomes
P0
¼ 2pC 2 ;
2G
ð25Þ
Finally, we look at the asymptotic behavior of the contact stresses as a ? 1 to examine their behavior near the edges of the
contact area
( eihaþa=2 a 1
C2
cosh ph
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
:
h2 ih þ 2ð1 mÞ C 0 a
a
p
3 4m R 4
ry þ isxy 2G
b¼0
a ! 1ðx ! a 0Þ
!
#!#)
"
1
1
X
X
ð1Þk k þ 14 þ h2
ð1Þk k þ 12 þ h2
p
1
p
2
þi 1þ
:
h
2
2
h 1þ
2
2
4
cosh ph
cosh ph
k þ 1 þ h2
k þ 1 þ h2
k¼0
k¼0
"
2
ð26Þ
2
As one can see, both normal and shear stresses have inﬁnite oscillations when approaching the edges of the contact area.
In the case of full stick without adhesion, the continuity of stresses across the boundary of the contact area is required,
and the complex-valued condition of ﬁniteness (vanishing) of the contact stresses (24) at the edges of the contact zone has
the form (Zhupanska & Ulitko, 2005)
1
ν=0.2, normal stress
ν=0.4, normal stress
ν=0.2, Hertzian normal stress
ν=0.4, Hertzian normal stress
0.8
0.6
0.4
ν=0.2, shear stress
ν=0.4, shear stress
0.2
0
0
0.2
0.4
0.6
0.8
1
x/a
Fig. 3. Distributions of the contact stresses for the non-adhesive full stick contact.
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
"
!
1
X
ð1Þk k þ 14 þ h2
a 1
C2
cosh ph
p
h2 ih þ 2ð1 mÞ C 0 a
h 1þ
2
2
R 4
a
p
cosh ph
k þ 12 þ h2
k¼0
!
!#
1
1
X
X
ð1Þk k þ 12
1
p
ð1Þk
2
2
2h
¼ 0:
h
2
þi 1 þ
1 2
1 2
4
cosh ph
þ h2
þ h2
k¼0 k þ
k¼0 k þ
2
61
ð27Þ
2
This complex-valued condition, together with the statics condition (25) are used to determine the values of the unknown
constants C2 and C0, and the half-size a of the contact area:
1 h
;
R MðhÞ
1
hKðhÞ a2
C2 ¼ 1 4h2 þ 4
;
8ð1 mÞ
MðhÞ R
2
pG
hKðhÞ a
1 4h2 þ 4
;
P0 ¼
2ð1 mÞ
MðhÞ R
C0 ¼ ð28Þ
ð29Þ
ð30Þ
where the following designations are introduced
!
1
X
ð1Þk k þ 14 þ h2
KðhÞ ¼
h 1þ
2
;
2
p
cosh ph
k þ 12 þ h2
k¼0
!
1
X
ð1Þk 14 h2 k þ 12 þ h2
cosh ph
1
p
:
h2
1þ
2
MðhÞ ¼
2
4
p
cosh ph
k þ 1 þ h2
k¼0
cosh ph
p
ð31Þ
2
The distributions of the normal and shear stresses in the contact area (0 < a < +1) in the case of non-adhesive full stick contact are determined by (24) with constants C2, C0, and a given by (28)–(30). Note that in this case of non-adhesive full stick
contact the contact stresses have ﬁnite-amplitude oscillationswhenapproaching the edges
of the contact zone.
r
sxy Fig. 3 shows the distributions of the contact normal stress Ra 2Gy and shear stress Ra 2G
along with the Hertzian dis
H
R ry
tribution of normal stress a 2G for Poisson’s ratios m = 0.2 and m = 0.4.
rH
The Hertzian distribution of normal stress Ra 2Gy is given by (Johnson, 1985)
R rHy
1
¼
a 2G
2ð1 mÞ
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x2
;
1
a
jxj 6 a:
ð32Þ
The results presented in Fig. 3 illustrate a well-known fact that a considerably higher compressive load is required in the
contact with friction (including the limiting case of full stick, when friction is large enough to prevent slip) comparing to
the compressive load in the Hertzian contact, to produce the contact area of the same size (e.g. Zhupanska & Ulitko,
2005). It is also worth mentioning that normal contact of a rigid cylinder with an incompressible (Poisson’s ratio m = 0.5) elastic half-space is frictionless. In this case, the shear stresses are absent in the contact area and the normal stress distribution
follows the Hertzian stress distribution (32). Obviously, the frozen-in constant C0 = 0 in this special case, as it follows from
(28).
5.1. Exact solution
To account for adhesion in contact, the approach of Chen and Gao (2006a), whose treatment of adhesive contact employed the similarities between contact and crack problems, is used in the present work. The use of the analogy between
an opening crack and contact of two bodies goes back to the works of Muskhelishvili (1953) and Erdogan (1963) and in
the context of the adhesive contact to Maugis, Barquins, and Courtel (1976, 1978), Greenwood and Johnson (1981).
The stress ﬁeld near a tip of interfacial crack possesses an oscillatory character with the energy release rate
G¼
1
cosh
2
pj
jKj2
:
2E
ð33Þ
Here K is a complex-valued stress intensity factor
K¼
pﬃﬃﬃﬃﬃﬃﬃ
2p limða xÞr ðry ðxÞ þ isxy ðxÞÞ;
x!a
ð34Þ
where a is the half-size of the contact area and r is the stress singularity near the edge of the contact area (both to be determined from the solution of the adhesive contact problem), and ry(x) and sxy(x) are, respectively, the normal and shear contact stresses corresponding to the case of adhesive contact.
The oscillation index j is deﬁned by the stress singularity as
62
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
1
þ ij:
2
r¼
ð35Þ
The effective Young’s modulus, E⁄, in the case of the contact of an elastic half-space with a rigid cylinder is deﬁned as
1 1 m2 1 m
¼
¼
;
E
E
2G
ð36Þ
where m, E and G are Poisson’s ratio, Young’s modulus, and shear modulus of the elastic half-space, respectively.
The equilibrium state of the contact is determined using Grifﬁth’s energy balance criterion
G ¼ Dc ¼ c1 þ c2 2c12 ;
ð37Þ
where Dc is the work of adhesion, and c1, c2 and c12 are the surface energies of the half-space, cylinder, and contact interface,
respectively.
Substitution (33) into (37) results in the equilibrium relationship for the adhesive contact problem
1
2
cosh
pj
jKj2
¼ Dc
2E
ð38Þ
from which the size of the contact area is determined.
Now this approach is applied to the adhesive contact problem for an elastic half-space indented by a rigid cylinder considered in this paper. The stress intensity factor (34) in the bipolar coordinates can be expressed as
K¼
pﬃﬃﬃﬃﬃﬃﬃ
2p lim ð2aÞr ear ðry ðaÞ þ isxy ðaÞÞ;
a!1
ð39Þ
where the normal and shear contact stresses are represented by (24), and their asymptotic behavior at a ? 1 is given by
(26). Analyzing (39) and (26), one can see that the stress singularity is
r¼
1
þ ih;
2
ð40Þ
where the oscillation factor h is determined by (18).
After substituting (24) into (39), the stress intensity factor becomes
pﬃﬃﬃﬃﬃﬃ K
2 pa a 1
C2
p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
¼
h2 ih þ 2ð1 mÞ C 0 aðKðhÞ þ iMðhÞÞ ;
2G
a
3 4m R 4
ð41Þ
where K(h) and M(h) are given by (31). Note that this expression contains three unknown constants, C2, C0, and a, which have
to be found as a part of the solution of the adhesive full stick contact problem.
The equilibrium relationship is obtained by combining (41) and (38) and accounting for (36):
3
2 3
2
a 1
a
1m 3
h2 C 0 a3 KðhÞ þ 2ð1 mÞC 2 a þ
h þ C 0 a3 MðhÞ ¼
a Dc :
R 4
R
pG
ð42Þ
The unknown constant C0 is found from the non-adhesive full stick contact problem and is given by (28); the static condition
(25) is used to determine the constant C2 in terms of the applied load P0 and the half-size of the contact area a.
Substituting (28) and (25) into (42) yields the ﬁnal form of the equilibrium relationship
2
2
a 1
a2 hKðhÞ 1 m P0
1m
h2 þ
¼
R 4
R MðhÞ
2G p
pG
ð43Þ
from which the relationship between the applied load P0 and the half-size a of the contact area is determined as
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
hKðhÞ a2
pG
2
P0 ¼
Dca:
1 4h þ 4
2
2ð1 mÞ
MðhÞ R
1m
pG
ð44Þ
When the load is reduced to zero (P0 = 0), the contacting surfaces still stay in contact due to adhesive forces, and the contact
area has a ﬁnite size determined by
a3
R2
¼
16ð1 mÞ
Dc
:
pG 1 4h2 þ 4 hKðhÞ2
MðhÞ
ð45Þ
Subsequent application of the tensile load leads to further decrease in the contact area until the limit of stability is reached
and spontaneous separation of the contacting surfaces occurs. The tensile force Pc that corresponds to the limit of stability is
called the ‘‘force of adhesion’’ (Johnson, 1985) or ‘‘elastic adherence force’’ (Barquins, 1988). The force of adhesion in the considered case of the adhesive full stick contact is
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
"
Pc ¼ 3
#1=3
RðDcÞ2
pG
63
ð46Þ
8ð1 mÞ 1 4h2 þ 4 hKðhÞ
MðhÞ
and the corresponding half-size of the contact area is
2
31=3
R 2 Dc
61 m
7
ac ¼ 4
5
pG 1 4h2 þ 4 hKðhÞ2
MðhÞ
ð47Þ
:
Note that the size of the contact area is such that dP0/da = 0.
To complete the analytical solution of the adhesive full stick contact problem, the normal and shear contact stresses are
found by substitution expressions (25) and (28) for the constants C2 and C0 and expression (44) for the half-size of the contact area a into the general complex-valued representation for stresses (24).
5.2. Comparisons with the JKR model
According to the JKR model, the equilibrium relationship in the case of adhesive contact of an elastic half-space and a rigid
cylinder is (Barquins, 1988)
PJKR
0
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
a2
pG
2
¼
Dca;
2ð1 mÞ R
1m
pG
ð48Þ
"
PJKR
c
pGRðDcÞ2
¼ 3
8ð1 mÞ
#1=3
ð49Þ
and the corresponding half-size of the contact area is
aJKR
¼
c
1=3
1m 2
:
R Dc
pG
ð50Þ
It is worth noting that in the case of an incompressible half-space (the value of Poisson’s ratio is m = 0.5), the solution for the
adhesive full stick contact reduces to the JKR model. In this case, the contact becomes frictionless (i.e., C0 = 0 as it follows
from (28)), and, therefore, the normal contact stress is distributed according to the classical Hertz theory and the shear stress
is absent. Moreover, h = 0 in (43), (45), (46), and these relations coincide with (48), (49), and (50), respectively.
Next, the results obtained in the case of adhesive full stick contact are compared with those due to the JKR model. Fig. 4
shows the ratio of the size of the contact area at the limit of stability in the adhesive full stick contact (Eq. (47)) to that preJKR
dicted by the JKR model (Eq. (50)), ac =aJKR
ratio is plotted against Poisson’s ratio.
c . The ac =ac
As one can see, the sizes of the contact areas in the adhesive full stick contact are at most 13.6% smaller than those predicted by the JKR model.
1
0.99
0.98
0.97
0.96
0.95
0.94
ac
JKR 0.93
ac
0.92
0.91
0.9
0.89
0.88
0.87
0.86
0
0.1
0.2
0.3
0.4
0.5
ν
Fig. 4. Ratio of the size of the contact area at the limit of stability in the adhesive full stick contact to that predicted by the JKR model.
64
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
3.5
ν=0.2
ν=0.4
JKR model
3
2.5
2
a0
1.5
1
0.5
0
-4
-3
-2
-1
0
1
2
3
4
P0
Fig. 5. Variations of the normalized contact area size with the applied load for the adhesive full stick and JKR models.
To compare equilibrium relationships for the adhesive full stick and JKR models, the following normalizations for the load
and half-size of the contact area are introduced
e0 ¼ P
P0
pGRðDcÞ2
8ð1mÞ
~0 ¼ a
1=3 ;
a
R2 Dcð1mÞ
pG
ð51Þ
1=3 :
These normalizations allow us to rewrite the equilibrium relationships (43) and (48) as
pﬃﬃﬃ
e 0 ¼ 1 4h2 þ 4 hKðhÞ a
~2 4 a
~
P
MðhÞ
ð52Þ
pﬃﬃﬃ
e JKR ¼ a
~2 4 a
~;
P
0
ð53Þ
and
which clearly highlight the difference between these two contact models. As before, when Poisson’s ratio m = 0.5 (i.e. h = 0),
the adhesive full stick contact model is reduced to the JKR model.
Fig. 5 shows variations of the normalized contact area size with the applied load for the adhesive full stick contact model
(Eq. (52)) for Poisson’s ratios m = 0.2 and m = 0.4 and for the JKR contact model (Eq. (53)).
Similarly to the non-adhesive full stick contact, for which under a ﬁxed normal load, the actual contact area is smaller
than that predicted by the Hertz theory (see, e.g., Nowell et al. (1988), Zhupanska & Ulitko (2005), Zhupanska (2009)), in
the case of adhesive full stick contact the contact area is smaller than that predicted by the JKR model. As it was pointed
out in the Introduction, the JKR model is based on the Hertz theory, where contact shear stresses are neglected and only
transmission of normal stress is taken into account, whereas the presented adhesive full stick contact model accounts for
the inﬂuence of the shear stress on the normal stress.
6. Conclusions
In this work, the problem of adhesive contact of a rigid cylinder with an elastic half-space has been considered. The inﬂuence of the shear stress on the normal stress in the contact area was preserved in the problem formulation by considering full
stick contact condition within Spense’s self-similarity approach (Spence, 1968a). The employed solution procedure consists
of the general solution in the form of Papkovich–Neuber functions, bipolar coordinate system, and the Fourier integral transform. This solution procedure yields an exact solution to the formulated boundary-value problems by means of the Fourier
integral transform. The mathematical simplicity of the solution procedure is contrasted with other more laborious methods
that commonly reduce contact problems to singular integral equations and rarely produce exact solutions. A detailed analysis of the solution is conducted. The results show that accounting for the contact shear stresses leads to smaller contact
areas as compared to those predicted by the JKR model, which neglects the inﬂuence of the shear stresses in contact. The
difference between the presented solution and the JKR model increases as Poisson’s ratio decreases, and is not negligible.
Given the simple analytical form of the obtained solution, we believe that it can be useful for practical applications, including
interpretation of experimental data obtained using various indentation techniques.
O.I. Zhupanska / International Journal of Engineering Science 55 (2012) 54–65
65
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