A Generalised Asperity-Based Friction Model

Abstract

Despite considerable research effort, the use of physics-based modelling to predict frictional behaviour is still a debatable
question in modern tribological research. This article presents a dry-friction model, based on physical phenomena such as
adhesion, elastic–plastic contact and deformation. This contribution offers a means to simulate all kinds of frictional behaviour
that is observed in experimental research. The contact of two bodies through their surfaces is transformed into the contact
of a body that is provided with asperities and containing material and geometrical information of both of the mating surfaces,
and a counter profile, holding solely geometrical information. The local adhesion between the asperity tips and the counter
profile, together with the elastic–plastic behaviour of the asperities themselves, form the basis for this model. The simulation
results show qualitatively good agreement with experimental study. Friction and contact phenomena such as normal creep, increasing
static coefficient of friction with increasing dwell time, pre-sliding hysteresis with nonlocal memory, Stribeck and viscous
effect, frictional lag, stick–slip and dynamical oscillations are revealed by this model. Furthermore, future improvement
and refinement of the model is possible (and ongoing) so as to incorporate lubrication and asperity wear.

KeywordsFriction mechanisms-Dynamic modelling

Full text

Page 1

ORIGINAL PAPER
A Generalised Asperity-Based Friction Model
Kris De Moerlooze•Farid Al-Bender•
Hendrik Van Brussel
Received: 19 April 2010/Accepted: 11 June 2010/Published online: 29 June 2010
? Springer Science+Business Media, LLC 2010
Abstract
physics-based modelling to predict frictional behaviour is
still a debatable question in modern tribological research.
This article presents a dry-friction model, based on physical
phenomena such as adhesion, elastic–plastic contact and
deformation.Thiscontributionoffersameanstosimulateall
kindsoffrictionalbehaviourthatisobservedinexperimental
research. The contact of two bodies through their surfaces is
transformed into the contact of a body that is provided with
asperities and containing material and geometrical infor-
mation of both of the mating surfaces, and a counter profile,
holding solely geometrical information. The local adhesion
between the asperity tips and the counter profile, together
with the elastic–plastic behaviour of the asperities them-
selves, form the basis for this model. The simulation results
showqualitativelygoodagreementwithexperimentalstudy.
Friction and contact phenomena such as normal creep,
increasing static coefficient of friction with increasing dwell
time, pre-sliding hysteresis with nonlocal memory, Stribeck
and viscous effect, frictional lag, stick–slip and dynamical
oscillations are revealed by this model. Furthermore, future
improvement and refinement of the model is possible (and
ongoing) so as to incorporate lubrication and asperity wear.
Despite considerable research effort, the use of
Keywords
Friction mechanisms ? Dynamic modelling
List of Symbols
dj
dn
Spring deformation of the j-th element
Normal interference depth
Dj
Maximum spring deformation of the j-th
Maxwell-Slip element
Main damping ratio
Tangential damping ratio of the i-th asperity
Normal damping ratio of the i-th asperity
Mean counter profile wavelength
Coefficient of friction
Static coefficient of friction
Yielding force
Non-dimensional time
Main eigenfrequency of the system
Tangential eigenfrequency of the i-th asperity
Normal eigenfrequency of the i-th asperity
Eigenfrequency of counter profile of the i-th
asperity
Amplitude of the applied excitation
Dimensional tangential and normal damping of
the i-th asperity
Dimensional main tangential damping
Damping matrix
Horizontal asperity distribution
Frequency of the applied excitation
Non-dimensional global friction force
Non-dimensional global normal contact force
Local normal contact force
Adhesion force of the i-th asperity
Force vector
Elastic spring stiffness
Plastic spring stiffness
Profile stiffness of the i-th asperity
Dimensional tangential and normal stiffness of
the i-th asperity
Dimensional main tangential stiffness
Stiffness matrix
Dimensional asperity length
f
fTi
fNi
k
l
ls
rNi, rTi
s
xN
xnTi
xnNi
xnPi
A
cT,i, cN,i
CT
C
dxT,i
ft
Ffric
Fnorm
Fn
Fadh,i
F
kel
kpl
kp,i
kT,i, kN,i
KT
K
li
K. De Moerlooze (&) ? F. Al-Bender ? H. Van Brussel
Mechanical Engineering Department, Division PMA, Katholieke
Universiteit Leuven, Celestijnenlaan 300B, 3001 Heverlee,
Belgium
e-mail: Kris.DeMoerlooze@mech.kuleuven.be
123
Tribol Lett (2010) 40:113–130
DOI 10.1007/s11249-010-9645-x

Page 2

Li
mi
M
M
N
q
t
x, y
xi, yi
Non-dimensional asperity length
Dimensional mass of the i-th asperity
Dimensional main mass
Mass matrix
Number of asperities
State vector
Dimensional time
Dimensional position
Dimensional asperity mass position of the i-th
asperity
Dimensional position of the free end of the main
spring
Dimensional main mass position
Non-dimensional position
Dimensional time derivation
Non-dimensional time derivation
xm, ym
xt, yt
X, Y
.
0
1 Introduction
The innovative study of Leonardo da Vinci [1], Guillaume
Amontons [2], Leonard Euler [3] and Charles Augustin de
Coulomb [4], concentrated on investigating the complex
mechanics of the relative motion between contacting
bodies. However, over the last 500 years, the main research
effort was focused on the mechanisms giving rise to the
frictional resistance, in particular adhesion and deformation
[5], while friction as a dynamical process evolving in the
contact was often overlooked. The profound understanding
of this dynamical behaviour is of crucial importance in
diverse applications, ranging from high precision machine
elements over biomedical and automotive applications to
earthquake engineering. The exigency for a model capable
of simulating the dynamical behaviour is evident for
studying stick–slip motion, where the unstable motion
originates from the interaction of the frictional dynamics
and the rest of the mechanical system. In order to study this
phenomenon on a theoretical basis, a realistic model is
needed for the frictional behaviour in the contact of the
mating surfaces.
Despite numerous attempts, the modelling of dry fric-
tion between solids remains a pertinent issue in modern
tribological research. The reason may be found in the
complexity of the processes taking place in any tribological
contact, on different space and time scales, including
elastic and plastic deformation of bodies, material proper-
ties, lubrication, thermal effects, humidity, fracture and
wear, reintegration of particles into the surface, chemical
reactions and mechanical interactions [6, 7, 8].
Another difficult issue which Popov and Psakhie [6]
mention, is the multi-scale nature of friction. Microscopic
scales affect the friction properties, such as a thin layer of
impurities on a metal surface drastically change the friction
properties [9]. Conversely, the time and space scales of
contact mechanics can differ by many orders of magnitude
[10].
The asperity-based modelling has a long history. One of
the earliest studies on the modelling of systems at the
asperity level, is probably the study of Abbott and Fire-
stone [11], where the contact was simulated with a network
of spheres, which are truncated by the indentation with a
rigid flat surface. More than three decades later, Kragelsky
[12], considers a rough surface as a composition of elastic
rods with different lengths, fixed from one end to a rigid
surface. Based on a statistical approach, the global coeffi-
cient of friction is function of the deformation of the
material, the geometrical properties and the local friction
force, of which the latter is considered to be pressure
dependent. This model shows an increasing coefficient of
friction with decreasing surface roughness, in accord with
experimental results.
Greenwood and Williamson [13] also base their study
on multi-asperity behaviour, considering the asperity tips
as hemispheres with a constant radius of curvature. The
contact between two rough surfaces is transformed into the
contact of an equivalent rough surface and a rigid flat
surface. Whitehouse and Archard [14] extend the model of
Greenwood and Williamson by studying a random varia-
tion of radius of curvature of the asperity tips. Ogilvy [15]
develops a numerical strategy to simulate the frictional
behaviour of rough surfaces, using Hertzian contact anal-
ysis [16] and simple plasticity theory. The coefficient of
friction is calculated on both the micro scale and the macro
scale. On the micro scale, the adhesive forces required to
break all junctions are summed up, while in the macro-
scopic approach, the friction force is related to the total
area of true contact. Agreement of the numerical results
with experimental study is found in the contact of sub-
strates with a MoS2coating.
Ford [17] bases his multi-asperity model on a combi-
nation of the models of Tabor [18] and Greenwood and
Williamson [13], to study the contact angle and its distri-
bution for pure elastic and pure plastic indentation. The
numerical code utilised, therefore, is based on the Ogilvy
model [15], where the asperities are no longer considered
as hemispheres of equal radius.
The CEB model of Chang, Etsion and Bogy [19]
incorporates the elastic–plastic behaviour of the contacting
asperities.
Tworzydlo et al. [21] present a constitutive model of
frictional interfaces, with the aim of providing the link
between micro scale effects at the asperity contacts and
macroscopic phenomena of the contacting materials. They
make use of a special finite element method, combined
114Tribol Lett (2010) 40:113–130
123

Page 3

with empirical findings for the local friction force and a
statistical homogenisation technique.
Karpenko and Akay [22] describe a computational
method for the simulation of the friction force in steady
sliding of three-dimensional rough surfaces. They assume
the friction force to originate from the elastic asperity
deformation and the shearing due to the adhesion occurring
the junctions. Based on the local friction dependency on
the normal pressure and the arbitrary orientation and con-
tact of asperities, they develop an iterative procedure uti-
lising a successive grid refinement.
Several research groups use a FEM-based approach to
model the contact between a rigid flat surface and a rough
surface, where the asperities are modelled as hemispheres.
Jackson and Green [23] develop empirical formulas for the
elastic–plastic contact in this manner. Quicksall et al. [24]
extend this study for a set of materials and compare the
results obtained with the Jackson and Green model.
None of the previously presented modelling strategies
are, however, capable of simulating dynamic frictional
phenomena such as the Stribeck effect, viscous sliding and
friction lag. In order to manage this, a completely different
modelling approach is necessary.
During the last two decades, another category of friction
models were being developed, namely heuristic models for
friction force dynamics, intended to be used for control
purposes [25]. Such models are not concerned with actual
contact mechanics, but are parametric mathematical for-
mulations that can be used for fitting experimentally
obtained friction behaviour. A special case of these is the
Generic model proposed in [26], which uses phenomeno-
logical mechanisms to formulate the dynamics of asperity
interaction.
The objective of this study is to formulate a model that
combines the approaches and features of both categories of
models reviewed above, so as to result in a most general
and physically realistic representation of the contact and
friction phenomenon.
This article presents a generic, wear-less dry friction
model, which incorporates both static (deformation, adhe-
sion) aspects and dynamic behaviour. With this model, one
is able to simulate the dynamics of the interaction of a body
comprising a mass and a set of asperities under normal
loading and tangential excitation. Despite relatively simple
assumptions, the model is able to simulate all experimen-
tally observed frictional behaviours. The article is struc-
tured as follows. In Sect. 2, the ingredients of the model
are discussed, leading to the renewed formulation of the
generic model. This is followed in Sect. 3 by an enuncia-
tion of the numerical strategy and the utilised normalisa-
tion. Section 4 overviews the obtained results to show the
relevance of this contribution. Finally, some appropriate
conclusions are drawn.
2 Model Description
Geike and Popov [27] show that three-dimensional contacts
have the same force—displacement law as a reduced one-
dimensional model. This grants one to reduce a three-
dimensional asperity based rough surface into a more
manageable and computable system, in which each asperity
is converted to a mass, connected via a normal and tan-
gential spring, to the main mass. This reduction allows us
to describe the dynamical frictional behaviour of rubbing
surfaces, which is mandatory for an accurate modelling of
the frictional behaviour of these contacts.
2.1 The Contact Scenario
The contact of two surfaces sliding over each other, can be
represented by a mass with a flexible surface, containing all
asperities, and a counter profile, determined by a certain
geometrical form and a certain stiffness, which is much
higher than the stiffness properties of the asperities.
The two first bodies are consequently transformed to an
upper body, containing all geometric and material infor-
mation, and a counter profile, containing only geometrical
information.
The top surface is represented by asperities, each of
which has a certain mass, stiffness, damping and length
which, combined with the shape of the counter surface,
forms the transformation of the original local asperity
contact. For the sake of simplicity, the asperity mass is
considered as a point mass, i.e. without geometrical
dimensions. This is illustrated in Fig. 1, where the varying
radius of the asperity is a measure for the stochastic dis-
tribution of the asperity masses.
The main mass of the sliding body M can be excited in
different ways. One can either control its motion by a direct
application of an external force or displacement, or realise
the same through a parallel connection of a spring KTand a
damper CTto the mass, while controlling the position of
the other (free) end of the spring/damper combination.
An asperity is represented by its mass mi, the normal and
tangential stiffness kNi, kTiand damping cNi, cTi, where i
stands for the i-th asperity of a set of size N. In that way,
normal and tangential behaviour of the asperity are
decoupled. The undeformed length of the asperity is
characterised by its relaxation length li. This is presented in
Fig. 1, where the rest length of asperity 1 is indicated by l1.
In contact modelling, two contacting rough surfaces are
often reduced to the contact of one surface, having the
equivalent roughness of both original surfaces, and a per-
fectly flat surface (e.g. [13, 28]). This reduction is plausible
for stationary contact simulation, but fails when consider-
ing frictional dynamics in sliding contacts, since it misses
the ability of asperity interlocking and an incorrect
Tribol Lett (2010) 40:113–130115
123

Page 4

representation is given of the continuously changing rela-
tive topology [26]. Therefore, the asperity population,
representing the top surface, is given a certain length dis-
tribution, reflecting the equivalent roughness of the original
surfaces. On the other hand, the lower surface is proposed
as a profile with constant amplitude, representing a mea-
sure for the overlap between asperities during contact,
while the wavelength is chosen arbitrarily, from a given
distribution. Various distributions for the asperity length
are applicable, such as exponential and Gaussian distribu-
tions for the asperity heights, as Greenwood and Wil-
liamson [13] propose. Persson et al. [10] indicate the
Gaussian nature of the height distribution of natural sur-
faces, such as surfaces prepared by fracture or sputtering,
while polished rough surfaces show a non-symmetric
height distribution. Andrews and Sehitoglu [29] propose a
log-normal distribution for the asperity height, tip radius
and horizontal position. However, Lampaert [25] shows
that the type of distribution has only a minor influence on
the hysteresis friction behaviour.
In conformity with the Greenwood–Williamson assump-
tion [13], no interaction forces between asperities are
modelled. Contact between upper surface and lower surface
occurs when an asperity of the top surface penetrates (i.e.
interferes with) the lower surface. A proper choice of the
counter surface accelerates the calculation of the interfer-
ence depth d. An extra stiffness kpi, the profile stiffness,
affects the motion of the asperity mass miin the normal
direction of the counter profile with a force Fn= kpidn,
where dnis the normal interference. Besides the contact
interaction of the asperity and the counter profile, an addi-
tional adhesion mechanism is modelled. The tangential
adhesion law used by Al-Bender et al. [26] must be under-
stood as a tangential surface force, resisting tangential
motion. This adhesion law is depicted in the left panel of
Fig. 2. The case of the more classical adhesion laws [20, 30,
31, 32], where the asperity is subjected to an attractive–
repulsive force in the normal direction towards the contact-
ingsurface,isdepictedintherightpanelofFig. 2.Inorderto
describe the force–separation relationship, the Dugdale
approximation [33] is used, where the integral of the force–
separation curve is a measure for the work of adhesion.
Further below, the results yielded by these two approaches
are compared and discussed.
Fig. 1 The contact scenario
Fig. 2 Left panel the adhesion
law in the tangential direction,
opposite to the velocity of the
asperity, right panel the normal
adhesion law, an attractive force
is exerted to the asperity when
the adhesion zone is entered
116Tribol Lett (2010) 40:113–130
123

Page 5

2.2 Elastic–Plastic Asperity Behaviour
Since the pioneering study of Greenwood and Williamson
[13], numerous attempts to model the contact of rough
surfaces were carried out. Several static models include the
elastic–plastic behaviour of the contacting asperities [19,
34, 24, 23]. On the other hand, viscoelastic materials can be
modelled by combining basic mechanical models, such as
the Voigt and Maxwell models. These elements consist of a
spring and a dashpot in parallel and in series respectively to
simulate the deformation behaviour of viscoelastic mate-
rials [35, 36]. In order to include the hysteretic elastic
plastic spring behaviour in our model, a Maxwell-Slip
element [37, 38] is used, which allows fast and accurate
simulation of hysteretic behaviour of a weakening spring.
The basic scheme of the Maxwell-Slip model is depicted in
Fig. 3. The model consists of a parallel connection of
elements, characterised by their stiffness kjand the maxi-
mum deformation of the spring Dj(with j = 1…M). This
latter parameter relates the statics of the element to its own
frictional behaviour, as the maximal force, which the ele-
ment can resist, is given by Fmax= Dj? kj. As state vari-
able, the spring deformation dj is chosen. When the
element sticks, its position is fixed and the spring defor-
mation djincreases, until its maximum value Djis reached,
as a result of which the element slips. Following Rizos and
Fassois [39], the state equation for each element can be
written in a dense way for both regimes as in Eq. 1:
djðtÞ ¼ sgnfxðtÞ ? xðt ? 1Þ þ djðt ? 1Þg
? minfjxðtÞ ? xðt ? 1Þ þ djðt ? 1Þj;Djg;
while the output is the sum of all spring forces:
ð1Þ
FðtÞ ¼
X
N
j¼1
kj? djðtÞð2Þ
This model structure is used to simulate the elastic–plastic
behaviour of both the normal and tangential asperity
deformation, one Maxwell-Slip model for each degree of
freedom. The deformation of the spring is imposed by the
motion of the asperity mass on the one hand, and the main
mass position on the other hand. Every Maxwell-Slip
model consists of two elements, of which one element is
purely elastic (which can be considered as a Maxwell-Slip
element with infinite maximum deformation), while the
second is able to yield and thus to dissipate energy. This is
depicted in Fig. 4, where the normal stiffness of an asperity
is represented as the parallel connection of a purely elastic
spring keland a Maxwell-Slip element with stiffness kpl.
When applying a periodic deformation to the asperity in the
normal direction, a force–deformation plot reveals the
hysteresis behaviour, where the area enclosed by the hys-
teresis loop is a measure for the dissipated energy during
the motion. Consequently, the asperity behaviour is
dependent on the loading trajectory followed. Although the
hysteresis model only comprises two elements, when
considering a large amount of asperities with stochasti-
cally distributed parameters, no significant difference is
observed when using more elements to model an asperity.
Let us note, however, that during simulation, the motion of
the asperity is not necessarily periodic, but depends on
several model parameters, such as the asperity mass and
stiffness, the motion of the main mass, and the shape of the
counter profile. However, the mechanism of energy dissi-
pation and hysteretic behaviour remains unaltered.
3 Numerical Strategy and Calculation Method
3.1 Contact Detection
The shape of the counter profile has to fulfil some condi-
tions to yield a computationally efficient algorithm, with-
outmortgagingthenumerical
necessary. The shape of the profile is of fixed pattern, with
randomly varying wavelength, and constant amplitude. The
stability morethan
F1
1
x
1
F,x
F2
F1
F3
FN
kN
k3
k2
k1
1
x1
Fig. 3 The model structure of
the Maxwell-Slip model
Tribol Lett (2010) 40:113–130117
123