Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 2, pp. 277-288, Warsaw 2007

AXIALLY SYMMETRIC CONTACT INVOLVING FRICTION

AND BOUNDARY ROUGHNESS

Andrzej Brzoza

Volodymyr Pauk

Faculty of Civil and Environmental Engineering, Technical University of Kielce

e-mail: pauk@tu.kielce.pl

In the paper, the problem of axially symmetric contact involving partial
slip for a rigid flat-ended cylinder and elastic half-space is considered.
Friction forces are taken into account. In addition, the surface of the
half-space is assumed to be covered by some roughness. The model of
the roughness involving the friction forces is developed. The problem is
reduced to integral equations, which are solved numerically.

Key words: contact problem, boundary stick, slip

1. Introduction

Thewell-known Hertzian model of elastic contact (Hertz, 1981) provides that
contacting surfaces are perfectly smooth and frictionless. The corresponding
formulations involving friction forces in the contact area in the axially sym-
metric problem are presented inMossakovskij (1954), Spence (1968, 1975). It
was shown that for the limiting coefficients of friction, the contact area consi-
sts of the central region (0< r < c) with sticking conditions and an annular
zone (c< r<a), in which the contacting surfaces slip, see Fig.1. The size of
the stick region c depends on the friction coefficient andPoisson’s ratio. Even
so friction is taken into account, the formulations presented in Mossakovskij
(1954), Spence (1968, 1975) consider, from the geometrical point of view, ide-
ally smooth contacting surfaces. However, real engineering surfaces have some
geometrical microstructure, which consists of waviness and roughness. These
geometrical imperfections play important role in the real contact.

Theaimof this paper is to consider contact of a rigid flat punchandan ela-
stic half-space involving simultaneously friction forces and surface roughness,
Fig.1.



278 A. Brzoza, V. Pauk

Fig. 1.

There are many approaches which enable one to take into account the
boundary roughness in the contact area of elastic bodies. The well-known
Greenwood-Williamson model (Greenwood and Williamson, 1966) treats the
roughness as elastic asperities with a constant curvature and a Gaussian di-
stribution of heights, which are very densely distributed over the contacting
surface. TheGWmodel hasmanymodifications involving plastic deformation
and other random distributions. Numerical solution to the normal spherical
contact problam, involving the boundary roughness according to theGWmo-
del presented in Greenwood and Tripp (1967), shows a bigger contact radius
and lower normal pressure in comparison to the Hertz solution. However, the
GWmodel cannot be applied to the formulation of the contact problem invo-
lving friction forces, because it omits tangential forces and horizontal defor-
mation on the rough surfaces. In addition, theGWmodel has been developed
for parabolic bodies and cannot be used for flat-ended geometry.

In this paper, in order to take into account friction forces in the contact of
a flat-ended rigid cylinder and an elastic rough half space, we propose another
model of the boundary roughness. Thismodel is phenomenological and states
a modification of the Shtayerman model (Shtayerman, 1949) in the case of
tangential forces and deformation. It permits one to reduce the considered
contact problem to integral equations which can be solved numerically.

2. Problem formulation

Contact geometry of the considered problem is shown in Fig.1 in the cylindri-
cal co-ordinates Orϕz. We study an axially symmetric indentation of a rigid
flat-ended cylinder (punch) into an elastic half-space. The punch is centrally
pressed by the normal load P. As a result, the normal p(r)=−σzz(r,0) and
the radial q(r) = σrz(r,0) tractions are aroused in the common contact area



Axially symmetric contact involving friction... 279

0 ¬ r ¬ a. It is assumed that the contacting surfaces have some roughness.
The proposedmodel of deformation of the rough half-space is inspired by the
one given in Shtayerman (1949) for the normal frictionless contact.
According to this model, the vertical uz and radial ur displacements of

the rough surface of the half-space under the punch consist of two parts

uz(r,0)=u
(a)
z
(r)+u(b)

z
(r,0) 0¬ r¬ a

(2.1)

ur(r,0)=u
(a)
r
(r)+u(b)

r
(r,0) 0¬ r¬ a

where u
(a)
z (r) and u

(a)
r (r) describe displacements due to deformation of aspe-

rities, and u
(b)
z (r,0) and u

(b)
r (r,0) are bulk displacements. For the first ones,

a phenomenological model is used, according to which

u(a)
z
(r)=αp(r) u(a)

r
(r)=βq(r) 0¬ r¬ a (2.2)

where α, β are called roughness parameters.
The bulk displacements can be obtained as elastic solutions for the half-

space subjected to action of the traction p(r) and q(r) (Johnson, 1985)

u(b)
z
(r,0)=

4(1−ν2)
πE

a
∫

0

p(ξ)K1(ξ,r) dξ−
(1−2ν)(1+ν)

2E

a
∫

r

q(ξ) dξ

(2.3)

u(b)
r
(r,0)=

4(1−ν2)
πE

a
∫

0

q(ξ)K2(ξ,r) dξ−
(1−2ν)(1+ν)

rE

r
∫

0

ξp(ξ) dξ

where E, ν areYoung’smodulusandPoisson’s ratio of thehalf-spacematerial.
The kernels K1,K2 are known

K1(ξ,r)=
ξ

ξ+r
K(k) k=

2
√
ξr

ξ+r
(2.4)

K2(ξ,r)=
ξ

ξ+r

[( 2

k2
−1
)

K(k)−
2

k2
E(k)

]

Here, K(x) and E(x) are the complete elliptic integrals of the first and second
kind, respectively.
As we see, the model proposed here states extension of Shtayerman’s one

on upon action of friction forces and involves radial deformation of the rough-
ness due to these forces. It depends on two parameters α, β which should be
determined experimentally. Notice that the first one describes stiffness of the
asperities in the normal direction and the second one – in the radial direction.
Another way to determine these parameters is presented in Pauk and Zastrau



280 A. Brzoza, V. Pauk

(2004). Comparing the solution to the normal contact problem obtained in
the framework of the GW model and Shtayerman’s one, it is possible to find
relations between the roughness parameter α and the parameters from the
GW model. Relations (2.2) describe also the well-known model of Winkler’s
foundation. This means that the developed model treats the roughness as a
set of springs or rods distributed over the surface.

3. Problem of normal contact

For further analysis, wemake the following assumption: the frictional traction
q(r) has no effect on the normal displacements. It is known (Johnson, 1985)
that this effect is generally small, whichmeans that the second term in formula
(2.3)1 can be omitted. This assumption permits one to consider separately the
problem of determination of the normal pressure p(r).

Satisfying with the help of formulae (2.1)1, (2.2)1, (2.3)1 and with the
above assumption the boundary condition of the normal problem

uz(r,0)= δ 0¬ r¬ a (3.1)

where δ is the rigid approach of the punch, we arrive at the integral equation
of the Fredholm type of the second kind

αp(r)+
4(1−ν2)
πE

a
∫

0

p(ξ)K1(ξ,r) dξ= δ 0¬ r¬ a (3.2)

This equation with the equilibrium condition

2π

a
∫

0

p(r)r dr=P (3.3)

states a closed system for the unknown function p(r).

Introducing dimensionless variables and functions

s=
r

a
η=
ξ

a
p∗(s)=

a2p(r)

P

α∗ =
αE

4(1−ν2)a
δ∗ =

aδE

4(1−ν2)P
k∗ =

2
√
ηs

η+s

K∗1(η,s) =
η

η+s
K(k∗)

(3.4)



Axially symmetric contact involving friction... 281

the system of integral equations (3.2), (3.3) can be rewritten in the form

α∗p∗(s)+
1

π

1
∫

0

p∗(η)K∗1(η−s) dη= δ
∗ 0¬ s¬ 1

(3.5)

2π

1
∫

0

p∗(s)s ds=1

Equations (3.5) were solved numerically for different values of the dimen-
sionless roughness parameter α∗. The effects of this parameter on the normal
pressure distribution are presented in Fig.2. For α∗ =0, thewell-known solu-
tion for the rigidflat-endedcylinder restingon the smoothhalf-space (Johnson,
1985)

p∗(s)=
1

2π
√
1−s2

0¬ s< 1 (3.6)

is obtained. This classical solution is unbounded for s → ±1. But in the
problem under consideration the contact pressure is bounded at the punch
edge. This result is due to the boundary roughness and was first discovered
by Shtayerman (1949).

Fig. 2.

4. Punch perfectly connected with half-space

Let us initially consider a situation in which friction forces are sufficiently big
to prevent slip in thewhole contact area. In this case, the boundary conditions
of the tangential problem are as follows (0¬ r¬ a)



282 A. Brzoza, V. Pauk

ur(r,0)= 0 q(r)<fp(r) (4.1)

where f is the Coulomb friction coefficient.
Satisfying condition (4.1), with the help of presentation (2.1)2, (2.2)2,

(2.3)2 we obtain the following integral equation for the unknown function q(r)
(0¬ r¬ a)

βq(r)+
4(1−ν2)
πE

a
∫

0

q(ξ)K2(ξ,r) dξ=
(1−2ν)(1+ν)

rE

r
∫

0

ξp(ξ) dξ (4.2)

Here, the contact pressure p(r) is assumed to be known from the previous
Section.
The total friction force Q is unknown and can be found as

Q=2π

a
∫

0

q(r)r dr (4.3)

Equation (4.2) was solved numerically similarly to equation (3.5)1 in nor-
malized variables (3.4) and

q∗(s)=
a2q(r)

P
β∗ =

βE

4(1−ν2)a
(4.4)

Figure 3 presents the distribution of frictional traction for some values
of the parameters β∗ and for α∗ = 0.5, ν = 0.3. In contrast to the case of
the perfectly smooth surface β∗ = 0 (Johnson, 1985), the friction forces in
the problem under consideration are bounded at the punch edge. Upon the
increasing parameter β∗, these forces decrease.

Fig. 3.



Axially symmetric contact involving friction... 283

For the investigation of possible slip between the contacting surfaces, the
ratio of tangential to normal tractions should be investigated. In the classical
case (Mossakovskij, 1954; Spence, 1968, 1975), it is a function of Poisson’s
ratioo ν and the coordinate r, and it is unbounded at the punch edge r= a.
Thismeans that condition (4.1)2 cannotbesatisfied in thispoint forphysically
accepted values of the friction coefficient f < ∞. As a consequence, slip is
expected on the annulus of the contact area edge. The corresponding contact
problem involving partial slip was considered in Spence (1975).

Quite different situation is observed in the contact problem under con-
sideration. The ratio q(r)/p(r) is now a function of ν, r as well as of the
roughness parameters α∗ and β∗. This ratio at the punch edge is shown in
Fig.4 versus Poisson’s ratio for some sets of α∗ and β∗. We can see that it
is bounded, and there are values of parameters ν, α∗, β∗ for which stick con-
dition (4.1)2 is satisfied for the given friction coefficient f. For example, if
f = 0.1 and the roughness parameters α∗ = 0.2, β∗ = 0.5, there is a range
of Poisson’s ratio 0.255 < ν < 0.5 (see Fig.4), for which condition (4.1)2 is
satisfied and the contacting surfaces remain stuck within the whole contact
region. If 0< ν < 0.255, this condition is disturbed at r = a, and slip must
occur near the punch edge.

Fig. 4.

Such formulation of the contact will be considered in the next Section.The
possibility of expected slip for other sets of parameters f,α∗, β∗ can be read
fromFig.4. Generally, the possibility of slip occurence is bigger for smaller β∗

when α∗ is fixed, compare the curves α∗ =0.5, β∗ =0.2; α∗ =0.5, β∗ =0.5;
α∗ =0.5, β∗ =1.0.



284 A. Brzoza, V. Pauk

5. Partial slip contact problem

Let us now pass to the problem in which the contacting surfaces are stuck in
the central part of the contact area, while on the annulus of the punch edge
slip takes place. The boundary conditions of the considered problem are as
follows:

— for 0<r<c

ur(r,0)= 0 q(r)<fp(r) (5.1)

— for c< r<a

q(r)= fp(r) ur(r,0)> 0 (5.2)

Notice that the stick area size c is unknown.

Satisfying boundary condition (5.1)1 with the help of formulae (2.1)2,
(2.2)2, (2.3)2, we arrive at an integral equation for the frictional traction
(0<r<c)

βq(r)+
4(1−ν2)
πE

a
∫

0

q(ξ)K2(ξ,r) dξ=
(1−2ν)(1+ν)

rE

r
∫

0

ξp(ξ) dξ (5.3)

where the contact pressure p(s) as assumed to be known from the solution to
the normal problem.

Integral equation (5.3) can not be solved in the present form because it is
written within the stick area, while the unknown function q(s) is defined on
the whole contact region. To avoid this difficulty, the distribution of friction
forces is expressed as follows

q(r)=

{

fp(r)+q0(r) for 0<r<c

fp(r) for c< r<a
(5.4)

where q0(r) is a newunknown called the corrective traction. Notice that func-
tion (5.4) satisfies boundary conditions (5.1)2, (5.2)1 if q0(r)< 0.

Substituting (5.4) into (5.3), the integral equation for the corrective trac-
tion can be derived (0<r<c)

βq0(r)+
4(1−ν2)
πE

c
∫

0

q0(ξ)K2(ξ,r) dξ=

(5.5)

=
(1−2ν)(1+ν)

rE

r
∫

0

ξp(ξ) dξ−fβp(r)−
4(1−ν2)
πE

f

a
∫

0

p(ξ)K2(ξ,r) dξ



Axially symmetric contact involving friction... 285

The unknown total friction force Q can be now calculated from (4.3) and
(5.4) as

Q= fP +2π

c
∫

0

q0(r)r dr (5.6)

In the normalized variables and functions (3.4), (4.4) and

q∗0(s)=
a2q0(r)

P
c∗ =

c

a
Q∗ =

Q

P
(5.7)

equations (5.5), (5.6) read

β∗q∗0(s)+
1

π

c
∗

∫

0

q∗0(η)K
∗

2(η,s) dη=
1−2ν
4(1−ν)

s
∫

0

η

s
p∗(η) dη+

−fβ∗p∗(s)−
f

π

1
∫

0

p∗(η)K∗2(η,s) dη 0<s<c
∗

(5.8)

Q∗ = f+2π

c
∗

∫

0

q∗0(s)s ds

Integral equation (5.8)1 was solved numerically. The unknown normali-
zed size of the stick zone c∗ was calculated iteratively satisfying the physical
condition

q∗0(c
∗)= 0 (5.9)

which, according to formula (5.4), provides the continuity of frictional traction
within the whole contact area.
After calculation of the corrective traction, the distribution of friction for-

ces can be found from function (5.4).
The input data for calculations are: Poisson’s ratio ν, roughness parame-

ters α∗ and β∗, and friction coefficient f.
Figures 5a and 5b showdistributions of tangential traction for some values

of the friction coefficient and for two values of Poisson’s ratio.We observe this
traction decreasing upon a decrease in the friction coefficient. Simultaneously,
the stick zone shrinks. These phenomena aremore evident for lower Poisson’s
ratios. If the friction coefficient increases, the distributions of the shear trac-
tion tend to those obtained in the previous Section and are shown by dotted
curves.
The dimensionless stick region size c∗ and total shear force Q∗ are respec-

tively presented inFig.6a andFig.6b, as functions of the friction coefficient f
for some values of Poisson’s ratio. An increase of both these parameters with



286 A. Brzoza, V. Pauk

Fig. 5.

the friction coefficient is observed. At some values of the friction coefficient,
we obtain c∗ =1which means that the slip zone vanishes and the contacting
surfaces are in the stick conditions.

Fig. 6.

6. Conclusions

The axially symmetric contact of a rigid flat-ended punch and an elastic half-
space is studied. Simultaneous effects due to friction under the punch and
roughnessof thehalf-space surfaceare investigated.Anewmodel of the surface



Axially symmetric contact involving friction... 287

roughness is developed.Thismodel enables one to solve the considered contact
problems by the boundary integral equation method. The derived integral
equations are solved numerically.

New phenomena brought about to the surface roughness are observed. In
contrast to the corresponding classical contact, in the problem under conside-
ration the contact with perfectly connected surfaces is physically possible for
the limiting coefficients of friction.

References

1. Greenwood J.A., Tripp J.H., 1967, The elastic contact of rough spheres,
ASME J. Appl. Mech., 89, 153-159

2. Greenwood J.A.,Williamson J.B.P., 1966, Contact of nominally flat sur-
faces,Proc. Roy. Soc., London, Ser. A, 295, 300-319

3. Hertz H., 1981, On the contact of elastic bodies, Journal für die reine und
angewadte Mathematik, 92, 156-171 [in German]

4. Johnson K.L., 1985,Contact Mechanics, Cambridge University Press, Cam-
bridge, UK

5. Mossakovskij V.I., 1954, The fundamental mixed problem of the theory of
elasticity for a half-space with a circular line separating the boundary condi-
tions,Prikladnaya Matematika i Mekhanika, 18, 187-196 [in Russian]

6. Pauk V., Zastrau B., 2004, Plane contact problems with partial slip for
rough elastic half-space, J. Theor. Appl. Mech., 42, 107-124

7. Spence D.A., 1968, Self-similar solutions to adhesive contact problems with
incremental loading,Proc. Roy. Soc., London, Ser. A, 305, 55-80

8. Spence D.A., 1975, The Hertz contact problemwith finite friction, J. Elasti-
city, 2, 297-319

9. Shtayerman I.Y., 1949,Contact Problem in the Theory of Elasticity, Goste-
khizdat, Moscow [in Russian]

Osiowo-symetryczne zagadnienie kontaktu z uwzględnieniem tarcia

i chropowatości brzegu

Streszczenie

W pracy zajęto się zagadnieniem osiowo-symetrycznego kontaktu płasko zakoń-
czonego walca z podatną półprzestrzenią przy uwzględnieniu częściowego poślizgu



288 A. Brzoza, V. Pauk

oraz sił tarcia. Dodatkowo założono, że powierzchnia rozważanej półprzestrzeni jest
chropowata. Zaproponowano odpowiedni model chropowatości z siłami tarcia. Całe
zagadnienie sprowadzono do równań całkowych, które rozwiązano numerycznie.

Manuscript received June 29, 2006; accepted for print October 21, 2006