Friction Modeling
of
Elastomer

Metal
Contact
by
Katie Sherrick
Table of Contents
Table of Contents
................................
................................
................................
............................
2
Abstract
................................
................................
................................
................................
...........
3
Introduction
................................
................................
................................
................................
.....
3
Single

Asperity Contact Mechanics
................................
................................
................................
3
Elastic Contact Model
................................
................................
................................
.................
3
Effect of Viscoelasticity
................................
................................
................................
...............
4
Viscoelastic

Rigid Single Asperity Contact Model
................................
................................
......
5
Normal Loading
................................
................................
................................
.......................
6
Tangential Loading
................................
................................
................................
..................
9
Static Friction in Viscoelastic

Rigid Single Asperity Contact
................................
................
10
Summary
................................
................................
................................
................................
...
12
Multi

Asperity Contact Mechanics
................................
................................
................................
12
APPENDIX
................................
................................
................................
................................
....
14
Works Cited
................................
................................
................................
................................
...
16
Abstract
A common tribology problem is that of an elastomeric solid in contact with a metallic solid.
The
bulk of research prior to the 1960s was based on metal

metal contact and therefore in many
engineering
applications,
friction forces were calculated based on metal

metal contact rather
than metal

elastomer contact. While in certain applications this is an acceptable approximation
of friction behavior, in many instances such application of metal

metal friction laws to m
etal

elastomer contact has been incorrect.
1
A more accurate model of friction in elastomer

metal
contact may be obtained using a single

asperity contact model incorporating viscoelasticity,
then expanding to a multi

asperity/macroscopic model based on th
e stick

slip principle.
Introduction
Elastomers, including rubbers, are commonly

used engineering materials, finding application in
a wide variety of industries. However, the frictional behavior of elastomers can be significantly
different from fricti
onal behavior of other engineering materials.
It has been known for some time that friction between an elastomer and a rigid solid does not
obey the same laws of friction as two rigid solids (i.e. metals). For rubbers and synthetic
elastomers, it has b
een generally accepted that two distinct mechanisms of friction exist; the
friction arising from rubber sliding on a very smooth surface and that arising from rubber sliding
on a very rough surface. The friction arising from sliding on smooth surfaces is
related to
adhesion, while that arising from sliding on rough surfaces is c
aused by a loss of energy and is
referred to as the hysteresis component
.
2
Grosch demonstrated experimentally that both
components of rubber friction are related to the viscoela
stic properties of the material.
The static friction force of the elastomer material is of interest in cases where
positioning control
is important.
Single

Asperity Contact M
echanics
Elastic Contact Model
Single

asperity models are the first step in fr
iction modeling. When a single

asperity contact
pair is modeled as two elastic spheres in normal loading, the Hertz model is the most widel
y

used model for describing the contact stresses in the spheres.
The Hertz solution for elastic

1
(Smith)
2
(Savkoor)
rigid contact wit
h respect to contact radius, normal approach (penetration), and pressure
distribution is given as:
√
(
)
√
The elastic sphere was assumed to be incompressible (
ν
=0.5) with a shear modulus G.
3
Effect of Viscoelastici
ty
Viscoelasticity is the property of materials that exhibit both viscous and elastic properties when
undergoing deformation. There are several mechanical models that may be used to predict a
material’s response to loading conditions, including the Maxw
ell model, the Kelvin

Voigt model,
and the Standard Linear Solid model. The viscoelastic behavior of the solid is represented by
springs and dashpots in different combinations.
The Standard Linear Solid model
(also called the Zener model)
is genera
lly considered to be the
most accurate for rubber and similar elastomeric materials.
4
The Maxwell model (spring and
dashpot in series) does not describe creep phenomena that are observed in such materials, and
the Kelvin

Voigt model (spring and dashpot i
n parallel) does not describe stress relaxation.
The
use of the SLS model for a rubber compound is validated by experimental data obtained at a
reasonable strain rate.
The SLS model consists of a spring in parallel with a spring and dashpot that are in
series
, as
shown in the figure below
. Utilizing Hooke’s Law, the stress

strain relationship with respect to
time is obtained.
5
3
(Deladi, de Rooij and Schipper)
4
(Bui and P
onthot)
5
(Roylance)
Figure
1

Diagram of Standard Linear Solid model of viscoelasticity
Viscoelastic

Rigid
Single
Asperity Contact Model
Due to
the correspondence principle,
the solution to the elastic problem
may be
used to obtain
results for the viscoelastic problem. Since the SLS model is most representative of the
viscoelastic behavior of elastomers in creep an
d relaxation effects, it is the material model
selected for this purpose.
Figure
2

Asperities in contact (left) and top view of contact area (right)
6
The solution to the elastic

rigid contact with respect to contact radius a,
normal approach
(deformation)
δ
n and pressure distribution p is given by Hertz as indicated in the previous
section. In this case, the elastic sphere is assumed to be incompressible (
ν
=0.5) with a shear
modulus G. The viscoelastic solution is provided
for two cases: when the variation of
6
(Deladi, de Rooij and Schipper)
deformation is prescribed (displacement

controlled case) and when load variation is prescribed
(load

controlled case).
For the SLS model, the creep and relaxation functions of the viscoelastic material are
respective
ly defined
as:
(
)
[
(
)
]
(
)
[
(
)
]
w
here g
1
and g
2
are the spring elasticities,
η
is the dashpot viscosity, and
⁄
is the
relaxation time.
Normal Loading
Applying these viscoelastic operators to the
load controlled case, the contact radius, pressure
distribution, and normal load are given by the following simplified equations
7
assuming that the
normal force varies in time according to
(
)
(
)
where H(t) is the Heaviside step
function:
Contact
Radius
(
)
√
(
)
Pressure Distribution
(
)
√
(
)
(
)
For the displacement controlled case, the normal displac
ement (indentation) is taken as
(
)
(
)
Contact Radius
√
Pressure Distribution
(
)
(
)
√
Normal Force
(
)
(
)
For an asperity couple with the viscolastic sphere having the following
properties:
7
(E. Deladi)
Appendix B, p. 155

156
The pressure gradient under the sphere at t=10 seconds is:
Tangential Loading
If, subsequently
to the application of the normal force P
, a tangential force Q is applied to the
asperity couple, it is assumed that the asperities stick together in a central region while an
annulus of slip develops at the outer region of contact. The distribution of s
hear stresses in the
contact is given by Mindlin
8
:
(
)
{
(
)
[
√
√
]
Where c is the radius of the “stick” zone:
√
8
(Deladi, de Rooij and Schipper)
This results in a piecewise function where the shear stress is dependent on whether or not the
point r is within the stick or slip region.
Static Friction in Viscoelastic

Rigid Single Asperity Contact
The mechanism assumed for static friction starts wit
h a very low tangential load, when the
asperities are sticking together within the contact area. As tangential load continues to
increase, an annulus of slip will develop along the outer edge of the contact and in this annulus
microslip is occurring betwe
en the two asperities. When the tangential load reaches a critical
value equal to the static friction force, the entire contact area between the asperities has
transitioned to slip and there will be macroslip between the two asperities
in the tangential
direction
. A plot of this mechanism is shown in the figure below:
Figure
3

Friction force versus tangential displacement
9
According to this mechanism, the static friction force is obtained from the equation
∫
(
)
when the stick area becomes zero (i.e. the contact radius of the stick zone
c
becomes zero).
A displacement

controlled case is used to validate the model by Deladi, de Rooij, and Schipper,
using a Maple worksheet to simulate a single pair of asperiti
es.
First, for a given displacement
the normal force (P) is calculated, with the tangential force (Q) assumed to be the maximum
allowed (
μ
*P). Applying the appropriate viscoelastic operator and calculating the shear stresses
9
(Deladi, de Rooij and Schipper)
over the maximum contact area
(c=0), the static friction force may be obtained.
The material
properties and loads are as given in the following table:
Figure
4

Material properties used for single

asperity model
The Maple worksheet was used to evaluate
the static friction force at several different normal
approach displacements. The worksheet can be found in Appendix A. The results of this
analysis are as shown below:
Figure
5

Maple model of static friction force as a function of normal approach
Compare Figure 3 above to the plot from Deladi et al.:
0.00E+00
5.00E08
1.00E07
1.50E07
2.00E07
2.50E07
3.00E07
3.50E07
4.00E07
0.00E+00
5.00E09
1.00E08
1.50E08
2.00E08
2.50E08
3.00E08
3.50E08
Static friction force (N)
Normal Approach (m)
Static friction force as a function of normal approach
The outcome of the Maple worksheet evaluation is comparable to that obtained by Deladi, de
Rooij, and Schipper.
Summary
In
summary, when a viscoelastic

rigid single asperity couple is loaded normally and then
tangentially, the slip annulus increases with increasing tangential load until reaching a critical
state where the entire contact area between
asperities is in slip. Th
e force at this point is
considered the static friction force of the asperity couple.
Multi

Asperity Contact
Mechanics
The natural progression of the single

asperity contact model is to a multi

asperity model.
The
Greenwood

Williamson approach is used
to model the contact of multi

asperity surfaces.
The
assumptions inherent to the G

W approach are that 1) the rough surface is covered with a large
number of asperities which are approximately spherical, 2) asperity summits have a constant
radius, 3) the
ir heights vary randomly, and 4) the engineering surface has a Gaussian
distribution of peak heights.
10
The surface properties that are considered in this approach to
multi

asperity contact are the average peak radius (
β
), the standard deviation of asperity
heights
(
σ
), and the summit density (
η
s
).
11
It is assumed that the peak radius is constant, and that the
10
(Bhushan)
130

131
11
(Handzel

Powierza, Klimczak and Polijaniuk)
standard deviation of heights is Gaussian, and this is not always the case for materials in
contact. However, this simplified model of multi

asperit
y contact allows for an initial evaluation.
The contact mechanics of any given asperity couple are based on the compression (s

h) or
normal approach (
δ
) of the viscoelastic surface on the rigid surface. The critical asperity height
is calculated from m
aterial properties and surface roughness properties as in the following
equation
, also considering the surface separation
d
:
(
(
)
)
At this critical asperity height,
the individual asperity contacts can be classified as either partially
slip or total sliding.
The asperity contacts with a height larger than this critical height are in the
partial slip regime, while those with a lower height are in total sliding. Therefore the entire
friction force can be described as a summation of the
partial slip and total sliding contacts:
The friction force at which the partially

slip component becomes zero will be the maximum static
friction force, since all asperities are assumed to be in full slip. Depending on whether the load

controlled or displacement

controlled condition is used, th
e appropriate viscoelastic operator
(either creep compliance or stress relaxation) is substituted into the above equation for the
elastic modulus
G
.
The influence of surface roughness on multi

asperity friction modeling is studied by varying the
standar
d deviation of asperity heights and the mean asperity radius. This model indicates that
the static friction force is larger for a rougher surface (i.e. when standard deviation of asperity
heights is larger).
12
12
(Deladi, de Rooij and Schipper)
APPENDIX
Works Cited
Bhushan, Bharat.
Introduction to Tribology.
New York: John Wiley & Sons, 2002.
Bui, Q.V. and J.P. Ponthot. "Estimation of rubber sliding friction from asperity interaction
modeling."
Wear 252
(2002): 150

160.
Deladi, E.L., M.B. de Rooij and D.J. Schipp
er. "Modelling of static friction in rubber

metal
contact."
Tribology International 40
(2007): 588

594.
Deladi, Elena.
Static Friction in Rubber

Metal Contacts With Application to Rubber Pad Forming
Processes.
PhD Dissertation. Twente, The Netherlands: Uni
versity of Twente, 2006.
Handzel

Powierza, Z., T. Klimczak and A. Polijaniuk. "On the experimental verification of the
Greenwood

Williamson model for the contact of rough surfaces."
Wear 154
(1992): 115

124.
Roylance, David. "Engineering Viscoelasticity."
24 October 2001.
Department of Materials
Science and Engineering, MIT.
3 December 2009 <http://ocw.mit.edu/NR/rdonlyres/Materials

Science

and

Engineering/3

11Mechanics

of

MaterialsFall1999/038732E6

CF1E

4BD0

A22E

39123ADD3337/0/visco.pdf>.
Savkoor, A.R. "O
n the Friction of Rubber."
Wear 8
(1964): 222

237.
Smith, Robert Horigan.
Analyzing Friction in the Design of Rubber Products and Their Paired
Surfaces.
Boca Raton: Taylor and Francis Group, 2008.
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