Tribology International 33 (2000) 431–442
www.elsevier.com/locate/triboint
Contact modeling — forces
G.G. Adams *, M. Nosonovsky
Department of Mechanical, Industrial and Manufacturing Engineering, Northeastern University, Boston, MA 02115, USA
Abstract
This paper reviews contact modeling with an emphasis on the forces of contact and their relationship to the geometrical, material
and mechanical properties of the contacting bodies. Single asperity contact models are treated first. These models include simple
Hertz contacts for spheres, cylinders, and ellipsoids. Further generalizations include the effects of friction, plasticity, adhesion, and
higher-order terms which describe the local surface topography. Contact with a rough surface is generally represented by a multiasperity contact model. Included is the well-known Greenwood–Williamson contact model, as well as a myriad of other models,
many of which represent various modifications of the basic theory. Also presented in this review is a description of wavy surface
contact models, with and without the effects of friction. These models inherently account for the coupling between each of the
contacting areas. A brief review of experimental investigations is also included. Finally some recent work, which addresses the
dynamics and associated instabilities of sliding contact, is presented and the implications discussed.  2000 Elsevier Science Ltd.
All rights reserved.
Keywords: Contact; Contact mechanics; Contact pressure
1. Introduction
This paper provides a review of contact modeling with
an emphasis on contact forces, rather than on the detailed
state of stress in the contacting bodies. Related to contact
modeling is contact mechanics in which the two contacting bodies are topographically smooth and the
emphasis is on determining the relationship between the
applied load, contact area, and contact stress. Due to the
mathematical complexity involved, such problems are
typically restricted to linear elasticity, although the finite
element method and the boundary element method have
also been used in order to obtain solutions to problems
with complicated geometries and material behaviors.
The monographs by Johnson [1] and Hills et al. [2] provide comprehensive treatments of contact mechanics,
whereas those of Gladwell [3] and Galin [4] give more
mathematical descriptions of contact problems, and
Kikuchi and Oden [5] and Khludnev and Sokolowski [6]
provide variational and finite element treatments. Finally
the review articles by Bhushan [7] for single asperity
* Corresponding author. Tel.: +1-617-373-3826; fax: +1-617-3732921.
E-mail address: [email protected] (G.G. Adams).
contact and Bhushan [8] for multi-asperity contacts give
comprehensive reviews of contact mechanics of rough
surfaces.
In problems involving topograpically smooth surfaces
the real area of contact is the same as the apparent area
of contact. Real surfaces, however, always possess some
degree of roughness. Thus contact between two bodies
always occurs at or near the peaks of contacting
asperities and so the real area of contact will generally be
much less than the apparent contact area. Thus contact
modeling consists of two related steps. First the equations representing the contact of a single pair of
asperities are determined. In general this procedure
includes elastic, elastic–plastic, or completely plastic
deformation. Depending on the scale of the contact, plasticity effects may be penetration depth dependent. For
namometer scale contacts the effects of adhesion on the
normal force may also be important. The applied force
may be normal to the contacting area or it may include
a tangential component. The tangential component is
resisted by friction. Although the effect of surface layers
may be important in many applications, a review of that
area is outside the scope of this work. The monographs
[1,2] and the review article [8] are valuable sources of
information for study in that area. Because real surfaces
have roughness, it is necessary to combine the effects of
0301-679X/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 1 - 6 7 9 X ( 0 0 ) 0 0 0 6 3 - 3
432
G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
a large number of asperity contacts. In many instances
it may be possible to treat these contacts as uncoupled
from each other, whereas in other instances the effect of
coupling is very important. Finally dynamic instabilities
have recently been shown to occur under nominally steady sliding conditions. These instabilities can lead to differences between the measured friction force and the
resultant of interface shear stresses that would be
obtained with Coulomb’s sliding friction law.
2. Contact mechanics for a single asperity
2.1. Hertz contact
Consider two rough solid bodies brought into physical
contact through the action of applied forces. Contact
between the two bodies occurs over many small areas,
each of which constitutes a single asperity contact. It is
necessary to relate the force acting on a single asperity
to its deformation and contact area. The well-known solution of this problem was developed in the late nineteenth century by Hertz [9]. The assumptions for what
has become known as the Hertz contact problem are: (1)
the contact area is elliptical; (2) each body is approximated by an elastic half-space loaded over the plane
elliptical contact area; (3) the dimensions of the contact
area must be small compared to the dimensions of each
body and to the radii of curvature of the surfaces; (4)
the strains are sufficiently small for linear elasticity to
be valid; and (5) the contact is frictionless, so that only
a normal pressure is transmitted. Two contacting solids
are shown after deformation in Fig. 1. The point of first
contact is taken as the origin of a cartesian coordinate
system with the x–y plane as the common tangent plane
and the z–axis directed downwards. Using the notation
of Johnson [1], during compression by the normal load
P, distant points T1 and T2 displace distances d1 and d2
respectively parallel to the z–axis towards O. The quan-
Fig. 1. Hertz contact of two nonconforming elastic bodies.
tity d⬅d1+d2 is called the normal approach or the interference.
For the case of solids of revolution, the contact area
is circular. The interference, contact radius (a), and
maximum contact pressure are given by [1]
冉 冊 冉 冊
冉 冊
d⫽
9P2
16RE∗2
p0⫽
6PE∗2
p3R2
1/3
, a⫽
1/3
,
1/3
3PR
4E∗
,
1 1−n21 1−n22 1 1 1
⬅
⫹
, ⬅ ⫹
E∗ E1
E2 R R1 R2
(1)
where p0 is the maximum contact pressure (which occurs
at r=0), E* is the composite Young’s modulus, E1,E2 and
n1,n2 are the Young’s modulii and Poisson’s ratios for
the lower and upper body respectively, R is the composite radius of curvature and R1,R2 are the radii of curvature of the lower and upper bodies respectively. Thus
the contact area and the interference each vary as the
2/3 power of the applied force. The contact pressure distribution is semi-elliptical with radius r and has a
maximum value at the origin equal to 3/2 of the average
contact pressure.
Analogous expressions may be written for the contact
of two cylindrical bodies whose long axii are parallel to
the y–axis. The results for the half-width of the contact
strip (a) and the maximum contact pressure are [1]
冉 冊
a⫽
4P⬘R
pE∗
1/2
,
p0⫽
冉 冊
P⬘E∗
pR
1/2
(2)
where P⬘ is the applied load per unit length of y–direction. The contact pressure distribution is again semielliptical, this time with a maximum value at the origin
equal to 4/p times the average contact pressure. The normal approach d is, however, indeterminate. This indeterminacy is a general consequence of two-dimensional
loading of an elastic half-space — the approach of distant points in the cylinders can take on any value
depending upon the choice of datum.
These Eqs. (1) and (2) are special cases of the more
general results for nonconformal contact of bodies of
general ellipsoidal profiles. The contact area is elliptical
and the contact pressure distribution is semi-ellipsoidal.
Detailed results for contact area and interference vs. normal force are fairly complicated and are given by Johnson [1] and Cooper [10]. For moderately elliptical contacts, Greenwood [11] showed that the contact pressure
and approach can be approximated by using the circular
contact formulas with an equivalent radius of curvature
equal to (AB)⫺1/2, where A and B are the principal relative curvatures. In [12] Greenwood compares different
approximate methods for calculating stresses in elliptical
Hertzian contacts and concludes that the method of [11]
gives less than a 3% error for 1ⱕB/Aⱕ5. Approximations which are more accurate for higher ellipticities
are also given.
G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
2.2. Elastic–plastic and fully plastic contacts
The solutions for Hertz contact remain valid until the
applied load is sufficiently large so as to initiate plastic
deformation [13]. The Tresca maximum shear stress
theory states that plastic deformation begins at a point
in the body at which the maximum shear stress reaches
a critical value, i.e.
max{|s1⫺s2|, |s2⫺s3|, |s3⫺s1|}⫽Y
(3)
where s1,s2,s3 are the principal stresses and Y is the
yield stress in the simple tension test. Another theory,
the von Mises criterion, states that yielding occurs when
the distortional strain energy reaches a critical value. The
result for the initiation of yielding is
(s1⫺s2)2⫹(s2⫺s3)2⫹(s3⫺s1)2⫽2Y2
(4)
which, for pure shear, predicts yielding at a stress 15.5%
higher than does the Tresca criterion.
For the Hertz contact of two spheres, the maximum
shear stress for n=0.3 occurs at a depth of 0.48 a and
has a value of 0.31 p0. Thus both the Tresca and von
Mises theories predict the onset of yielding when [7]
(p0)Y⫽1.60Y, PY⫽21.2
R2Y3
,
E∗2
冉冊
dY⫽6.32R
Y
E∗
2
(5)
Yielding will initiate in the material with the lower
yield strength. Equations analogous to Eqs. (3)–(5) are
given by Johnson [1] and Bhushan [7] for the plane
strain contact of two cylinders.
As the load continues to increase, the size of the plastic zone also increases. However, until the plastic zone
reaches the surface, it is constrained by the surrounding
elastic material. An analytical solution has been obtained
for full plasticity by Ishlinsky [14]; the contact pressure
in the middle is somewhat higher than the mean contact
pressure (pm). Thus while elastic–plastic behavior
initiates at pm=1.07 Y, fully plastic behavior corresponds to
pm⫽H⫽2.8Y
(6)
where H is called the hardness of the lower yield
strength material. Chang et al. [15] give a relationship
between pm and H which depends upon the Possion’s
ratio. Tabor [16] showed that the load increases by a
factor of about 300 and the contact radius increases by
a factor of about 10 from the onset of yielding until fully
plastic deformation. Work-hardening materials which
strain-harden according to a power law were considered
by Matthews [17].
Indentation testing has been known to give hardness
values which are depth dependent (e.g. Bhushan [18]);
the smaller the indentation depth the greater is the measured hardness. Conventional plasticity theories lack a
433
length scale and so are incapable of predicting this
effect. Recently several strain–gradient theories of plasticity have been developed which provide the needed
length scale. These papers are reviewed by Hutchinson
[19].
2.3. Friction and tangential loading
Consider now the application of a tangential load (F)
to a Hertzian contact. The first situation to be treated is
applicable for any of three cases — (1) a pair of identical
materials; (2) one rigid material and the other incompressible (n=1/2); or (3) both materials incompressible.
In these cases normal stresses do not cause relative tangential displacements and shear stress do not produce
relative normal displacements. This uncoupling greatly
simplifies the analysis. In the absence of a tangential
force, contacting points will not tend to undergo tangential displacements and therefore slip does not tend to
occur regardless of whether or not friction is present. For
the plane strain contact of two cylinders, it was shown
independently by Cattaneo [20] and Mindlin [21] that
there is a central stick region surrounded by two slip
zones. As the tangential force increases, the size of the
stick region decreases until overall sliding of the asperity
begins. This sliding occurs with Coulomb’s law of sliding friction satisfied (F=mP), where m is the coefficient
of sliding friction and with no distinction between static
and kinetic friction.
After sliding of an asperity is initiated, the effect of
friction is to superimpose a stress which is caused by
the tangential contact stress q. This tangential contact
stress alters the stresses in the half-spaces and hence
changes the load at which plastic deformation is
initiated. Furthermore for sufficiently high friction
(m⬎0.3) the maximum shear stress occurs at the interface [22], rather than sub-surface, and hence the transition from elastic–plastic to fully plastic behavior
occurs more rapidly than without friction. The details of
the contact stresses for sliding of dissimilar materials
was determined by Bufler [23] and are recorded by Johnson [1].
The effect of dry friction without tangential loading
has been incorporated into a study of contacting spheres
by Goodman [24]. For a pair of different elastic
materials the coupling between normal and shear stresses
is small and is sometimes neglected [24]. Complete solutions which include the effect of shear tractions on normal pressure have been obtained by Mossakovski [25]
and Spence [26,27]. They show that the effect of friction
is to increase the load required to produce a contact of
a given size by less than 5%.
2.4. Non-Hertzian elastic contacts
In the above description of Hertzian contacts it has
been tacitly assumed that the contacting bodies are non-
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G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
conforming. The local geometry of such non-conforming
contacts can be characterized by the radii of curvature
at the contacting points. Now consider an example of a
contact in which the profile(s) of the contacting bodies
cannot be adequately represented by a second-degree
polynomial, i.e. the gap between the undeformed axisymmetric bodies is given by
[29] and, using a different technique, Sneddon [30], who
determined the relation between the contact radius, the
applied force and the interference, i.e.
h(r)⫽Anr2n
Note that the contact area is proportional to the applied
force and the interference varies as the square-root of the
force. The results for the corresponding two-dimensional
contact of a blunt wedge with an elastic half-plane are
given by Johnson [1] as
(7)
where n is a positive integer. Such a surface has a curvature which increases from zero at its peak and may be
useful in modeling a highly burnished asperity. The solution of Stuermann [28], in which the bodies are modeled as elastic half-spaces, can be applied, i.e.
P⫽
4AnE∗ngna2n+1
, d⫽Angna2n
2n+1
(8)
where
1
1
P⫽ pa2E∗cot a, d⫽ pa cot a
2
2
P⬘⫽aE∗cot a
(11)
(12)
Although the contact stress is singular under the apex of
the cone/wedge, the maximum shear stress is bounded.
2·4…2n
gn⬅
1·3…(2n−1)
2.5. Contact at the nanometer scale — adhesion
For the two-dimensional contact problem with an
initial gap given by
At a scale of many nanometers, the solid bodies can
still be treated as a continuum, but the effects of surface
forces in the immediate vicinity of the contact region
can become important [31]. The adhesive stress s(z) is
typically represented by the Lennard–Jones potential
h(x)⫽Anx2n
(9)
the force per unit length is related to the half-width of
contact by
P⬘⫽npE∗Ana2n/gn
(10)
For n=1 the above reduces to Hertz contact, whereas
for large n, the stress distribution approaches that of a
flat-ended punch which has singular stresses at the corners.
The Hertz contact theory is restricted to cases in which
the surface profile has continuous displacement and
slope. Consider now the contact of a blunt cone (the
half-cone angle a is close to 90°) with an elastic halfspace (Fig. 2). This problem was considered by Love
冋冉 冊 冉 冊 册
8w z
s(z)⫽⫺
3z0 z0
−3
⫺
z
z0
−9
(13)
where z is the separation between atomic planes, z0 is
the equilibrium separation, and w is the work of
adhesion, i.e.
w⫽⌬g⬅g1⫹g2⫺g12
(14)
In Eq. (14) g1,g2,g12 are the corresponding surface
energies.
A model of the adhesion force was developed by
Bradley [32] for rigid spheres which gives
P⫽⫺
冋冉 冊 冉 冊 册
8pwR 1 z
3 4 z0
−8
⫺
z
z0
−2
(15)
The corresponding pull-off force PC occurs when z=z0
and is given by PC=2pwR.
Subsequently two different models were proposed for
the contact of elastic spheres. These models were due to
Johnson, Kendall and Roberts (JKR) [33] and Derjaguin,
Muller and Toporov (DMT) [34]. These theories
appeared at first to be contradictory until it was pointed
out by Tabor [35] that these models were valid for different ranges of the parameter m defined by
Fig. 2. A blunt elastic wedge (cone) pressed against an elastic
half-space.
冉 冊
m⫽
Rw2
E∗2z30
1/3
(16)
G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
The parameter m represents the magnitude of the elastic deformation compared with the range of surface
forces. The JKR theory assumes that the adhesive forces
are confined to inside the contact area and thus gives a
pull-off force of 1.5pwR. The DMT model assumes that
the adhesive forces act outside of the contact area and
yields PC=2pwR. The validity of the DMT model was
brought into question by Muller et al. [36] and Pashley
[37]. For small m the elastic deformation is negligible
and the Bradley model provides a reasonable approximation of adhesive forces, whereas for large values of
m the JKR model is appropriate. Experimental observations of contact area and load have been obtained
using the surface force apparatus [38]. Results agree well
with the JKR theory.
The investigation of Muller, Yushenko, and Derjaguin
(MYD) [39] uses a Lennard–Jones potential and allows
for a continuous variation of m between the limits of the
DMT and JKR models. Greenwood [40] conducted the
MYD analysis more accurately and in greater detail,
showing that the load-approach curve is S–shaped, leading to a pull-in as well as a pull-off force. It is noted
that the JKR theory does not allow for the existence of
a pull-on force whereas other theories do [39,41]. Analytical results for the transistion between DMT and JKR
were presented by Maugis [41] using a simplified model
of adhesion based upon the Dugdale [42] crack model.
The above discussion of adhesion assumes that loading and unloading occurs elastically. However inelastic
deformation leads to “adhesion hysteresis” [43]. For
inelastic unloading the energy released must overcome
dissipation as well as the work of adhesion and consequently additional work is needed to separate these
deformed surfaces. Ductile separation has been observed
with an atomic force microscope [44] at a scale of 2 nm.
Molecular dynamics simulations of a small number of
atoms also show this phenomenon [45].
Johnson [43] extended the models of adhesion to
include static and sliding friction. The approach is
through the concept of fracture mechanics, in which the
elastic strain energy release rate is equated to the work
done against surface forces (both adhesive and
frictional). There is some experimental evidence that
under tangential loading an adhesive contact will tend
to peel apart [46], suggesting an interaction between normal adhesive and tangential frictional forces. Recently
the use of the surface force apparatus (SFA) Homola et
al. [38] and the atomic force microscope [47] has made
it possible to measure friction and adhesion forces in a
sliding experiment. The theory of [43] is in good agreement with experimental findings.
435
coupled. Uncoupled contact models represent surface
roughness as a set of asperities, often with statistically
distributed parameters such as height or summit curvature (Fig. 3). The effect of each individual asperity is
local and considered separately from other asperities; the
cumulative effect is the summation of the actions of individual asperities. Coupled contact problems with rough
surfaces are more complicated mathematically because
the equations of elasticity must be solved for the entire
body simultaneously. This procedure leads to mixed
boundary value problems which can be solved analytically only for simple configurations. One of the most
tribologically important results of using these asperity
models is the calculation of the true contact area which
differs significantly from the nominal contact area. These
quantities differ because contact between rough surfaces
takes place only at and near the peaks of the asperities.
It is the real contact area which has a profound effect
on friction and wear. Recently fractal analyis methods
have also been used to model contacts.
3.1. Uncoupled multi-asperity models
3.1.1. Elastic contacts
Various statistical models of contact have been
developed which are related to the pioneering work of
Greenwood and Williamson [48] in 1966. These models
assume some distribution laws for asperity heights and
for asperity curvatures. The density of surface asperities
and the material and mechanical properties are also
important. In general, the result of this type of analysis
is that if the number of asperities (N) in contact is constant and the deformation is elastic, the true area of contact A is proportional to P2/3, where P is the applied load.
If the number of asperity contacts increase, but the average size of each asperity contact remains constant, then
A is proportional to P regardless of whether the deformation is elastic or plastic. This proportionally is
important because it allows an adhesion based friction
theory to be consistent with the observed Amontons–
Coulomb friction law.
The statistical models are based on the calculation of
probability of contact (P) at a given asperity of height
z, for two surfaces separated by a distance d, i.e.
冕
⬁
P(z⬎d)⫽ p(z)dz
(17)
d
3. Multi-asperity contact models
Conventional multi-asperity contact models may be
categorized as predominately uncoupled or completely
Fig. 3.
A rough half-space and a flat body soon to be in contact.
436
G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
where p(z) is the probability density function of
asperity heights.
The Greenwood and Williamson (GW) model [48]
assumes that, in the contact between one rough and one
smooth surface, (1) the rough surface is isotropic; (2)
asperities are spherical near their summits; (3) all
asperity summits have the same radius of curvature
while their heights vary randomly; and (4) there is no
bulk deformation and thus no interaction between
neighboring asperities. Thus the total area of true contact is
冕
A⫽pNR (z⫺d)p(z)dz
(18)
d
and the total load is
冕
⬁
P⫽(4/3)E∗NR1/2 (z⫺d)3/2p(z)dz
(19)
d
Two distributions of the asperity heights were considered — the exponential distribution
p(z)⫽e−z, z⬎0
(20)
and the Gaussian distribution
1
冑
e−z
2p
A⫽CP10/11
2/2
(21)
The exponential distribution leads to a linear dependence of the true contact area on the applied load,
whereas the Gaussian distribution yields an almost linear dependence.
Greenwood and Williamson [48] also use the plasticity index f given by
f⫽(E∗/H)(s/R)1/2
(22)
where s is the standard deviation of asperity heights.
The plasticity index is responsible for the transition from
elastic to plastic deformation — low values of f correspond to elastic deformations whereas high values are
associated with plastic deformation. During the initial
contact of most metal surfaces prepared with an engineering finish the deformation will be predominantly plastic [49]. However repeated loading introduces permanent
deformations and residual stresses which can cause steady state stresses to become elastic [1].
Even before the well-known classic work of Greenwood and Williamson, a linear distribution of heights of
aligned spherical asperities
2
p(z)⫽ 2(L⫺z), 0⬍z⬍L
L
(23)
(24)
where C is a constant. Greenwood and Tripp [51]
extended Zhuravlev’s model for non-aligned asperities
and demonstrated that misalignment leads to a more
nearly proportional relation between contact area and
force, i.e. A=CP12/13. Ling [52] used a simple rectangular
height distribution
p(z)⫽
⬁
p(z)⫽
was considered in 1940 by Zhuravlev [50] and yielded
an almost linear result
1
, ⫺L⬍z⬍L
2L
(25)
and obtained similar results.
The contact of a rough sphere with a smooth sphere
was studied by Greenwood and Tripp [53]. They found
that the Hertzian results are valid at sufficiently high
loads, while at lower loads the effective pressure distribution is much lower and extends much further than for
smooth surfaces. Greenwood [54] studied the true area
of contact between a rough surface and a flat. He applied
a constriction resistance method to measure the area of
true contact and developed a method of finding the
resistance of a cluster of microcontacts. Greenwood [55]
also showed that for elastic solids with randomly distributed asperity heights, the average size of an asperity contact is almost independent of load. Therefore, like the
case of a pure plastic material, the dependence of the
true contact area on the load is almost linear. Greenwood
and Tripp [51] showed that the contact of two rough
surfaces can be modeled by the contact of one flat and
one rough surface. The equivalent rough surface is
characterized by an asperity curvature which is a sum
of the asperity curvatures of the two rough surfaces, i.e.
1 1 1
⫽ ⫹
R R1 R2
(26)
and the peak-height distribution of the equivalent surface
has a standard deviation given by
冑
sp⫽ s2p1+s2p2.
(27)
Whitehouse and Archard [56] considered the random
surface profile as a random signal characterized by a
height distribution and an autocorrelation function. This
is shown to be equivalent to asperities having a statistical
distribution of both heights and radii. Onions and Archard [57] studied a model with a Gaussian distribution of
surface heights (rather then asperity heights) and of
asperity peak curvatures. Gupta and Cook [58] permitted
the tip heights to be Gaussian distributed whereas the
asperity radii were log-normally distributed. Nayak [59]
considered a more sophisticated statistical model which
characterizes a random surface by three spectral
moments of the profile: m0, m2, and m4, which are equiv-
G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
alent to the variances of the distribution of profile
heights, slopes, and curvature respectively. This leads to
a distribution of peak heights which is different from
Gaussian. The summits are regarded as elliptical paraboloids with principal curvatures ␬1 and ␬2 in two orthogonal directions. The mean curvature is not constant but
varies with summit heights, higher summits have larger
mean curvature. The bandwidth parameter
m0 m 4
a⫽ 2
m2
(28)
is introduced, and it is shown that the real contact area
at a given separation depends only on a, while the load
depends on both a and m2. Bush et al. [60] used the
Nayak microgeometery assumptions to develop an elastic contact model which treated asperities as elliptical
paraboloids with random principal axis orientations and
aspect ratio. O’Callaghan and Cameron [61] and Francis
[62] extended the Bush et al. model for the case in which
both surfaces are rough and asperities need not contact at
their summits. They obtained corresponding equivalent
values for m0, m2, and m4 which reduce this case to that
of one smooth and one rough surface. They concluded
that this type of contact is negligibly different from the
GW model.
Tallian [63] developed a model for strongly anisotropic surfaces in which the surface is modeled as a random process with Gaussian distributed heights, and
found that surface frequency and not just roughness
determine the contact behavior. Hisakado [64] pointed
out that a Gaussian distribution of asperity heights and
curvatures for a given asperity shape may lead to a nonGaussian distribution of the surface height, which is
unrealistic for most engineering surfaces. He considered
a parabolic and a conical asperity shape. Bush et al. [65]
considered a rough surface with a random anisotropic
distribution of asperity radii. Such a distribution is
characterized by nine values of mij known as bispectral
moments. Compared with the asymptotic solution for the
isotropic case, their model gives a contact area which is
2% lower. Sayles and Thomas [66] investigated a deviation from isotropy which they called “elliptic anisotropy”. This term implies that contact spots have the
form of randomly oriented ellipses; their results for the
contact area are somewhat lower than that obtained with
the Bush et al. model.
McCool [67] investigated the limit of applicability of
elastic contact models of rough surfaces, using a plane
strain solution from the literature for a sinusoidally corrugated half-space. The range of validity of the assumptions that the asperities are micro-Hertzian (i.e. that they
can be approximated by a second order polynomial in
the vicinity of the contact point) and that the asperities
deform elastically was shown to be related to the mean
square surface slope and to the macro-contact pressure.
McCool [68] also considered a general anisotropic
437
model and his results demonstrated very good agreement
with those of the simpler GW model. He showed also
that the separation d in the GW asperity model can be
related to the separation h of the Bush et al. surface
microgeometry model by
h⫽d⫹4
冉 冊
m0
pa
1/2
(29)
Ju and Farris [69] applied spectral analysis methods
and the Fast Fourier Transform (FFT) to characterize a
surface in two-dimensional contact problems. Bjorklund
[70] developed a contact model of one rough and one
perfectly flat elastic surface with random asperity height
distribution, which assumed that some asperities are in
stick contact while others are in slip contact, depending
on asperity height. Hagman and Olofsson [71] considered a model for micro-slip between contacting surfaces based on deformation of elliptical elastic asperities.
3.1.2. Plastic and elastic–plastic contacts
The basic plastic contact model is an outgrowth of the
“profilometric model” by Abbot and Firestone [72]. The
deformation of a rough surface against a flat is treated
as the truncation of the rough surface at its intersection
with the flat. The true area of contact is the geometric
intersection of the flat surface with the original profile
of the rough one. The pressure in the contact area is
just the indentation hardness, and thus the total load is
proportional to the true contact area. Nayak [73] applied
his “random process profile model” [59] to this plasticity model.
Pullen and Williamson [74] assumed that the area of
contact is the geometrical intersection of the two surfaces and that volume conservation during plastic deformation is obtained by a uniform rise of the noncontacting
surface. These assumptions may be correct for very
heavily loaded contacts. An elastic–plastic model based
on volume conservation of an asperity control volume
during plastic deformation, was introduced by Chang,
Etsion, and Bogy (CEB) [15]. The contact area, force,
and interference for a single asperity are related by
冉 冊
A⫽pRd 2⫺
dC
, P⫽AKH, d⬎dC
d
(30)
where dC is the critical interference at the inception of
plastic deformation and K relates the mean contact pressure to the hardness [15]. For an interference less then
the critical value, the contact is elastic, while for d⬎dC
the contact is plastic. They used the single asperity
results to develop a multi-asperity model for elastic–
plastic deformation using assumptions similar to those of
the GW model. An elastic–plastic contact model which
generalizes the CEB model by taking into account the
directional nature of surface roughness and by consider-
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G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
ing contact spots of elliptic form was proposed by Horng
[75]. Zhao et al. [76] developed a multi-asperity model
which incorporates the transition from elastic deformation to fully plastic flow.
3.1.3. Fractal analysis
In the past years models with fractal surface geometries have been developed which are based on the presumption that surface geometry replicates itself at different length scales. However, long before the discovery of
fractal objects by mathematicians in the 1970’s, Archard
[77] investigated a model of equidistant spherical
asperities of the same radius R1, which have asperities
of a smaller radius R2 on their surface, which in turn
have even smaller asperities of radius R3. Based on the
Hertzian elastic model, he calculated the dependence of
the true contact zone on the load for one, two, and three
sets of the asperities. The results for the contact of a
plane and a sphere are
A⫽CP2/3, A⫽CP8/9, A⫽CP26/27
(31)
and for the contact of one smooth and one rough plane
A⫽CP4/5, A⫽CP14/15, A⫽CP44/45
(32)
These dependencies tend to converge to a linear
dependence as the “order” of the asperities is increased.
It is known that surface roughness measurements
depend on the resolution of the measuring instrument
and hence traditional roughness data is scale-dependent.
Unlike statistical models, modern fractal models account
for the multi-scale nature of surfaces. Fractal analysis
characterizes surface roughness by two scale-independent parameters D and G, where D relates to distributions of different frequencies in the surface profile and
G to the magnitude of variations at all frequencies. The
fractal dimension D is in the range 1⬍D⬍2. Borodich
and Mosolov [78] studied a model for flat perfectly plastic asperities based on a Cantor set of repeatedly magnified scales. Majumdar and Bhushan [79] considered a
three-dimensional surface model based on the Weierstrass–Mandelbrot wave function. In order to handle this
function, a first approximation was considered. This
model also introduces a critical area for plastic deformation which is a function of D, G, the hardness, and
the moduli of elasticity of the bodies.
Larsson et al. [80] investigated the inelastic flattening
of rough surfaces and compared the results of stochastic
and fractal models. For the fractal model they obtained
a non-linear relation between the impression depth h and
the area of true contact
冉
4(2−D)
h⫽GD−1 A 2
pc D
冊
1−D/2
(33)
which includes two scale-independent fractal parameters
D and G, where c2 is a function of the strain hardening
and creep exponents. In the case of perfectly plastic
behavior they find that the contact pressure reduces to
the hardness value for both the stochastic and fractal
models. However for hardening materials, the slope of
the contact area versus loading curve increases with the
load for the fractal model and is in contrast to the stochastic model. The nominal pressure based on this fractal
model does not converge unless the fractal dimension is
less then a certain number, which may indicate that the
fractal model is not sufficiently developed at this point.
Warren and Krajcinovic [81] introduced a model for
elastic–perfectly plastic contact of rough surfaces based
on the random Cantor set. Polonsky and Keer [82] studied scale effects in elastic–plastic asperity contacts.
Othmani and Kaminsky [83] as well as Podsiadlo and
Stachowiak [84] developed experimental techniques to
measure fractal parameters of a surface.
3.2. Coupled contact models
3.2.1. Analytical models
The coupled contact problems with asperities are more
complicated mathematically since the equations of elasticity must be solved in the whole body and thus the
boundary conditions, both unmixed and mixed, must be
applied to the entire surface. Therefore, in addition to
the non-mixed boundary conditions (e.g. continuity of
the normal and shearing stresses and Coloumb’s friction
law), two mixed boundary conditions must be applied.
Namely, normal displacements are continuous in the
contact zone(s) and the normal stress vanishes in the separation zone(s). Instead of a random distribution of
asperity heights only periodic interface profiles, e.g.
sinusoidal, are considered with this approach. Usually
these problems are solved by using the Green’s function
method, by applying series techniques, or by a complex
potential method, and lead to singular integral equations
for the contact pressure distribution. It is possible to
show that the problem with two elastic bodies in contact
can be reduced to an equivalent problem with one rigid
body and one elastic body with effective material parameters.
The frictionless two-dimensional elastic contact problem for a surface loaded by a periodic system of rigid
flat punches was solved for the contact pressure by
Sadowsky [85]. Westergaard [86] used the complex
stress function technique to obtain a closed-form solution for the two-dimensional frictionless contact problem of an elastic half-space with a wavy sinusoidal interface (Fig. 4). He showed that the normal stresses at the
contact zone of the surface are given by
syy⫽C cos x(sin2a⫺ sin2x)1/2
(34)
where C is a constant depending on the load and geometric parameters of the profile and a is the half-contact
G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
439
dimensional waviness. At light loads the contact area is
approximately circular and the Hertz theory can be
applied. When contact is almost complete, the separation
zones are almost circular and behave like pressurized
penny-shaped cracks. As the load is increased, the
numerical analysis demonstrates a change of the contact
area from almost circular to almost square, then to separation areas which are nearly circular, and finally to the
complete contact
Fig. 4.
space.
Contact of a wavy elastic half-space with a flat elastic half-
width. Eq. (34) yields a relation between the load and
the half-width of the contact area, i.e.
P⫽C
p
冑2
(1⫺ cos a),
3.2.2. Experimental results
The effect of interaction between neighboring asperity
contacts was studied by Leibensperger and Brittain [98]
using photoelasticity. Handzel–Powierza et al. [99] verified experimentally the GW model and obtained good
agreement with the theory within the range of elastic
deformations and for quasi-isotropic surfaces. A number
of experimental tests (see Woo and Thomas, [100]) of
statistical and fractal models have been made as well as
measurements of the topography of surfaces by stylus,
optical methods, or electrical contact, and by scanning
tunnelling microscopy methods.
(35)
As for the Hertz contact problem, two-dimensional elasticity does not allow for the solution of the interference.
Another approach to the mixed boundary conditions
is to use a series technique. Thus the Westergaard problem was solved by Dundurs et al. [87] using Legendre
polynomials. The complex potential approach and integral equation method were used by Soviet researchers
Muskhelishvili [88,89], Shtaerman [90], Lurie [91], and
Galin [4]. Shtaerman [90] showed that the frictionless
periodic contact problem can be reduced to a singular
integral equation which can be solved analytically for
sinusoidal waviness. Kuznetzov [92] obtained a solution
to the Westergaard problem using an alternative method.
The limits of applicability of uncoupled models for a
sinusoidal profile was investigated by Berthe and Vergne
[93] utilizing the results of Westergaard [86].
With steady sliding and Amontons–Coulomb friction
included in the analysis, a solution is also possible. Kuznetzov [94] considered the frictional (low velocity) sliding problem by using a complex potential which reduced
to the Westergaard’s solution in the case of zero friction.
Results were obtained only for contact pressures.
Nosonovsky and Adams [95] solved the frictional contact problem with a sinusoidal contact profile for arbitrary sliding velocities.
Manners [96] obtained a solution of the problem without friction for periodic profiles with higher harmonics
of waviness. Johnson et al. [97] obtained a numerical
solution, as well as asymptotic solutions for small and
large zones of contact for the frictionless case of two-
4. Frictional sliding contacts
The relative sliding motion of two surfaces is resisted
by a tangential force which is called the friction force.
The ratio of this tangential force to the normal force is
called the coefficient of kinetic friction (m). Although
this coefficient can easily be determined experimentally,
the mechanics of contact and friction is quite complex
as friction is a consequence of many interacting phenomena. Basically the friction force is attributed to tangential
adhesion forces. Thus the friction force should be proportional to the real area of contact. As has been previously mentioned, this proportionality is nearly true for
the static contact of elastic and plastically deforming
asperities. A complete review of friction is well beyond
the scope of this work. The reader is referred to the
review article by Tabor [49] and the monograph of Rabinowicz [101]. A recent paper by Bengisu and Akay
[102] develops a model for dry friction based upon
asperity interactions and adhesion forces. For reviews of
dynamic friction see Oden and Martins [103] and Martins et al. [104]. The following summarizes one interesting aspect of dynamic contact.
4.1. Dynamic instabilities in sliding contacts
Recent analysis as well as simulations have discovered dynamic instabilities in frictional sliding contacts.
These instabilities raise issues about the nature of
dynamic sliding and, perhaps more importantly from a
practical point of view, influence the predicted tangential
contact forces.
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G.G. Adams, M. Nosonovsky / Tribology International 33 (2000) 431–442
Martins et al. [105] investigated the sliding of elastic
and viscoelastic half-spaces against a rigid surface.
Dynamic instabilities were found for cases in which the
friction coefficient and the Poisson’s ratio were large.
These instabilities were thought to play a role in Schallamach waves [106]. In another investigation, Adams
[107] showed that the steady sliding of two elastic halfspaces is also dynamically unstable, even at low sliding
speeds. The instability mechanism is essentially one of
slip-wave destabilization. Steady-state sliding is shown
to give rise to a dynamic instability in the form of selfexcited motion. These self-excited oscillations are generally confined to a region near the sliding interface and
can eventually lead to either partial loss-of-contact or to
propagating regions of stick–slip motion. The existence
of these instabilities does not depend upon a friction
coefficient which decreases with increasing speed, nor
does it require a nonlinear contact model as with Martins
and Oden [104]. These analytical results are consistent
with the numerical simulations of Andrews and Ben–
Zion [108]. In a different investigation, Adams [109]
uses a simple beam-on-elastic-foundation model in order
to investigate instabilities caused by sliding of a rough
surface on a smooth surface. The mechanism of instability in that investigation is due to the interaction of a
complex mode of vibration with the sliding friction
force.
Adams [110] then investigated the sliding of two dissimiliar elastic bodies due to periodic regions of slip and
stick propagating along the interface. It was found that
such motion, which results from a self-excited instability, allows for the interface sliding conditions to differ
from the observed sliding conditions. In particular an
interface coefficient of friction (the ratio of interface
shear stress to normal stress) and an apparent coefficient
of friction (ratio of remote shear to normal stress) were
defined. The interface friction coefficient can be constant
or an increasing/decreasing function of slip velocity.
However the apparent coefficient of friction is less than
the interface friction coefficient. Furthermore the apparent coefficient of friction can decrease with sliding speed
even though the interface friction coefficient is constant.
Thus the measured coefficient of friction does not necessarily represent the behavior of the sliding interface.
Also the presence of slip waves may make it possible
for two frictional bodies to slide without a resisting shear
stress and without any interface separation. In the limit
as the slip region becomes very small compared to the
stick region, the results of Adams [110] become that of
a slip pulse travelling through a region which otherwise
sticks [111].
The possibility of two elastic bodies sliding relative
to each other, without slipping, due to a separation pulse
has been investigated by Comninou and Dundurs [112]
for identical materials and by Adams [113] for different
materials. The two semi-infinite isotropic elastic bodies,
of different material properties, satisfy Coulomb’s friction inequality at their common interface, and are subjected to applied normal and shear stresses which are
insufficient to produce global slipping. No distinction is
made between static and kinetic friction, and the friction
coefficient is speed-independent. Although Coulomb’s
inequality is satisfied at the interface, the force necessary
to produce relative motion is less than would be predicted by Coulomb’s law.
Adams [114] also investigated the role of elastic body
waves in the sliding of an elastic half-space against a
rigid surface and in the sliding of two elastic half-spaces
[115]. It was shown that steady sliding is compatible
with the formation of a pair of body waves (a plane dilatational wave and a plane shear wave) radiated from the
sliding interface. These waves radiate energy allowing
for sliding to occur with less energy dissipated due to
frictional heating than is supplied through the work done
by the external forces.
5. Conclusions
Contact modeling, with an emphasis on forces rather
than stresses, has been reviewed. Contact modeling has
been divided into two distinct phases — the contact of a
single asperity and the combined effects of a great many
contacts. Included are effects of elastic and plastic deformations, depth–dependent plasticity models, tangential
loading, non-Hertzian geometries, and adhesion. Multiasperity contact models include uncoupled contact models, fractal contact models, and convention coupled contact models. Experimental verifications are also briefly
discussed. Future work in contact modeling should
address issues at the nanometer scale (including nanometer scale plasticity and effects of contaminant films)
and contact dynamics.
Acknowledgements
The authors are grateful to the National Science Foundation for its support under Grant No. CMS-9622196 of
the Surface Engineering and Tribology Program.
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