i
THREEDIMENSIONAL MODELING OF ELASTOPLASTIC SINUSOIDAL
CONTACT UNDER TIME DEPENDENT DEFORMATION INCLUDING
BOTH STRESS RELAXATION AND CREEP ANALYSIS
by
Amir Rostami
A Thesis Submitted to the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 3, 2013
Keywords: 3D Sinusoidal Contact, Elastoplastic, Stress Relaxation, Creep, Finite Element Model
Copyright 2013 by Amir Rostami
Approved by
Robert L. Jackson, Chair, Associate Professor of Mechanical Engineering
Hareesh V. Tippur, McWane Professor of Mechanical Engineering
Jeffery C. Suhling, Quina Distinguished Professor and Department Chair
ii
ABSTRACT
Computational modeling of contact between rough surfaces has attracted a great deal of
attention due to the developing technological needs of industry. Most of the early models of rough
surface contacts assumed a cylindrical or spherical/ellipsoidal shape for the asperities on the
surfaces. Due to high memory space and computational time requirements, researchers use
simplified geometries to model the asperities or peaks on rough surfaces. Recent works tried to use
a sinusoidal shape for asperities to improve the previous models. The sinusoidal geometry gives a
better prediction of asperity interaction, especially for heavily loaded contacts. The effect of
adjacent asperities is considered in sinusoidal contacts by using a symmetric boundary condition.
Also, most of the multiscale contact models for rough surfaces use the Fourier series or Weierstrass
profile to transform a rough surface to combination of sine and cosine functions. Therefore, it
seems more reasonable to use a sinusoidal shape for asperities.
In the current work, the transient effect of creep and stress relaxation in contact between
sinusoidal surfaces is studied using FE simulations. A threedimensional sinusoidal asperity is
created, and is modeled in contact with a rigid flat surface. The material of the sinusoidal surface is
modeled as elastoplastic, bilinear isotropic hardening solid. The Garofalo formula is used in the
current work to model the transient behavior of creep and stress relaxation. Two load steps are
used in commercial software ANSYS (version 13.0) to model the effect of creep and stress
relaxation. The first load step is static deformation or the stress buildup stage that is used to
pressurize the asperity by the rigid flat surface. The second load step is the transient process during
which creep and stress relaxation occur. To verify the model, the results for the purely elastic and
elastoplastic cases (without the creep and stress relaxation effects) are compared to the previous
works in the literature.
Transient results under both constant displacement (stress relaxation) and constant force
(creep) boundary conditions are presented and discussed. A parametric study is done to analyze the
effect of the different material and geometrical properties and also the Garofalo constants on the
iii
transient results. In the end, empirical equations are developed for both contact area and contact
pressure based on the FEM results. The empirical equations are dependent on the surface
separation, aspect ratio, and the Garofalo formula constants. In the contact area and contact
pressure results for stress relaxation, a critical interference or surface separation was found that the
contact area and contact pressure showed different behaviors above and below this value. The
aspect ratio rate, ( )ldtd /D , is introduced as a parameter that is independent from the height of the
asperity during the stress relaxation process. This rate can be used in a multiscale contact model for
rough surfaces to predict the real contact area as a function of time.
iv
ACKNOWLEDGEMENTS
This thesis is based on the project that has been assigned and supported by the Siemens
Corporate Technology under the title of ?The Development and Application of Multiscale Friction
Prediction Methods to Dynamic Actuator Systems?. I started working on this project from January 2012
along my M.Sc. studies in Auburn University under the supervision of Prof. Robert L. Jackson. I
should thank Prof. Jackson who helped me a lot during this year and half, and I?ve benefited from
his suggestions and ideas countlessly during the course of the project. He has been completely
involved in every step of the project, and always supported me during the difficulties of the work. I
should thank Mr. Andreas Goedecke and Mr. Randolf Mock in Siemens Corporate Technology.
They have always supported and helped us with the data and info we needed during the project. I
should also thank Prof. Hareesh V. Tippur and Prof. Jeffrey C. Suhling for serving on my thesis
committee. In the same way, I should thank my lab mates in the Multiscale Tribology Laboratory
at Auburn University for their friendship and support and many shared laughs and hard work during
this year and half.
I should also mention the support of the most important people in my life, my parents,
during my Master studies which of course they are far away from me, but, they have always been
close to me in my heart. I will try my best to be a person that will make them proud.
v
TABLE OF CONTENTS
ABSTRACT ?????????????????????????????.? ii
ACKNOWLEDGEMENTS ???????????????????????... iv
LIST OF FIGURES ???????????????????????????. vii
LIST OF TABLES ???????????????????????????.. xi
NOMENCLATURE ??????????????????????????... xii
1 INTRODUCTION ????????????????????????? 1
2
3
LITERATURE REVIEW ??????????????????????...
2.1 Introduction ?????????????????????????..
2.2 Elastic Sinusoidal Contact of Single Asperity ?????????????.
2.3 Elastoplastic Sinusoidal Contact of Single Asperity ??????????...
2.4 Average Surface Separation in Sinusoidal Contacts ???????????
2.5 Creep and Stress Relaxation Effects in Single Asperity Contacts ??????
METHODOLOGY ????????????????????????....
3.1 Introduction ?????????????????????????..
3.2 Modeling and Simulation of the Static Deformation Step (Load Step 1) ???
3.3 Verification of the Model Accuracy (Elastic Case) ???????????.
3.4 Verification of the Model Accuracy (Elastoplastic Case) ????????...
3.5 Surface Separation Results for the Elastic Case ????????????..
3.6 Surface Separation Results for the Elastoplastic Case ?????????...
4
4
8
9
11
13
21
21
21
26
28
29
30
vi
3.7 Modeling and Simulation of the Stress Relaxation and Creep Effects (Load
Step 2) ??????????????????????????????.
33
4 RESULTS ????????????????????????????...
4.1 Introduction ?????????????????????????..
4.2 Stress Relaxation Results ????????????????????....
4.3 Empirical Equations for the Stress Relaxation Case ??????????...
4.3.1 Empirical Fit for contact Area ??????????????????...
4.3.2 Empirical Fit for contact Pressure ?????????????????.
4.4 Base Heightdependency of the Stress Relaxation Results ????????..
4.5 Creep Results ?????????????????????????.
4.6 Comparison between the Stress Relaxation and Creep Results ??????...
36
36
36
52
52
57
63
70
74
5 CONCLUSIONS ?????????????????????????? 78
BIBLIOGRAPHY ???????????????????????????... 80
APPENDICES ????????????????????????????? 85
A ?APDL? CODE USED FOR MODLING THE EFFECT OF STRESS
RELAXATION ??????????????????????????..
85
vii
LIST OF FIGURES
1.1 Change in the stress and strain in a material due to (a) stress relaxation, and (b) creep ?.. 2
2.1 Contour plot of the threedimensional sinusoidal surface geometry ????????.. 7
2.2 Rigid flat and the sinusoidal asperity (a) before contact and (b) during contact where
surface separation, amplitude and the wavelength of the sinusoidal asperity are shown
schematically ?????????????????????????????
12
2.3 A hemispherical asperity with radius, R , before and after loading, showing the contact
radius, a , the displacement, ? , and the load, F ???????????????..
15
2.4 Relaxation of force (solid line) and evolution of contact area (dashed line) with respect to
normalized creep time in a (a) logarithmic and (b) conventional scale ???????...
17
3.1 Threedimensional plot of the sinusoidal surface ???????????????.. 22
3.2 The steps to create the sinusoidal asperity: (a) creating the keypoints, (b) creating the
lines, (c) creating the sinusoidal surface, and (d) adding volume to the sinusoidal surface..
24
3.3 The element plot of the sinusoidal asperity and the rigid flat including the boundary
conditions that are used for the geometry ??????????????????.
25
3.4 Comparison of the FEM elastic contact area results with JGH data and JacksonStreator
empirical equation ???????????????????????????
27
3.5 Comparison of the FEM elastoplastic contact area results with empirical equation
provided by Krithivasan and Jackson ???????????????????...
28
3.6 Comparison of the FEM elastic results for average surface separation (shown by circles)
with the JGH data (shown by crosses) and the new fit given by Eq. (38) (shown by solid
line) ????????????????????????????????..
29
3.7 The FEM elastoplastic results for average surface separation for various yield strengths.. 30
3.8 The FEM elastoplastic results for average surface separation for various aspect ratios ? 31
viii
3.9 The comparison of FEM elastoplastic results for average surface separation (shown by
small circles) with the new fit given by Eq. (39) (shown by solid line) ???????...
33
4.1 Plot of the threedimensional sinusoidal asperity under constant displacement boundary
condition ??????????????????????????????..
37
4.2 The FEM results for the contact area and contact pressure versus time for reference
parameters ??????????????????????????????
38
4.3 The FEM results for contact pressure versus time for different surface separations
(penetrations) ????????????????????????????....
40
4.4 The FEM results for the contact area versus time for different surface separations
(penetrations) over a short duration of time ?????????????????..
41
4.5 The FEM results for the contact area versus time for different surface separations
(penetrations) over a longer duration of time ?????????????????
42
4.6 The arrows show how the deformed material displaces to the void volume between the
sinusoidal asperity and the rigid flat ????????????????????..
43
4.7 The FEM results for the contact area and pressure versus time for different yield
strength values ????????????????????????????..
44
4.8 The FEM results for the contact area and pressure versus time for different Garofalo
constant,
2
C
, values ( GPaS
y
1= )?????????????????...????
45
4.9 The FEM results for the contact area and pressure versus time for different Garofalo
constant,
2
C
, values ( GPaS
y
2= ) ?????????????????????.
46
4.10 The FEM results for the contact area and pressure versus time for different Garofalo
constant,
2
C
, values ( GPaS
y
4= ) ?????????????????????.
47
4.11 The FEM results for the contact area and pressure versus time for different Garofalo
constant,
1
C
, values ??????????????????????????...
48
4.12 The FEM results for the contact area and pressure versus time for different elastic
modulus, E , values ??????????????????????????...
49
4.13 The FEM results for the contact area and pressure versus time for different Poisson?s
ratio, n , values ????????????????????????????..
50
4.14 The FEM results for the contact area and pressure versus time for different aspect ratio,
l/D ,values ?????????????????????????????...
51
ix
4.15 The FEM results and corresponding curvefits for the normalized contact area for
different surface separations, D/
o
g ????????????????????..
53
4.16 The FEM results and corresponding curvefits for the normalized contact area for
different aspect ratios, l/D ???????????????????????..
54
4.17 The FEM results and corresponding curvefits for the normalized contact area for
different,
2
C , values ??????????????????????????..
56
4.18 The FEM results and corresponding curvefits for the normalized contact pressure for
different aspect ratios, l/D ???????????????????????..
58
4.19 The FEM results and corresponding curvefits for the normalized contact pressure for
different surface separations, D/
0
g ????????????????????..
59
4.20 The FEM results and corresponding curvefits for the normalized contact pressure for
different
2
C values ??????????????????????????...
61
4.21 The element plot of the sinusoidal asperity for a case with the (a) smaller height, and the
case with the (b) doubled height ?????????????????????...
63
4.22 The von Mises stress plot for case with the (a) smaller height, and the case with the (b)
doubled height ????????????????????????????..
64
4.23 Contact area results for the case with the (a) smaller height, and the case with the (b)
doubled height ????????????????????????????..
65
4.24 Contact pressure results for case with the (a) smaller height, and the case with the (b)
doubled height ????????????????????????????..
65
4.25 The aspect ratio rate results for the case with the (a) smaller height, and the case with the
(b) doubled height ???????????????????????????
66
4.26 The normalized amplitude rate results for the different values of Garofalo constant,
2
C ... 67
4.27 The normalized amplitude rate results for the different values of aspect ratio, l/D ??... 68
4.28 The normalized amplitude rate results for the different values of surface separation,
D/
o
g ???????????????????????????????....
69
4.29 Plot of the threedimensional sinusoidal asperity under constant force boundary
condition ??????????????????????????????..
70
4.30 The FEM results for the contact area versus time for reference parameters, and contact
pressure equal to GPap 8.0= ??????????????????????..
71
x
4.31 The FEM results for the contact area versus time for different constant contact loads ?. 72
4.32 The FEM results for the contact area versus time for different Garofalo constant,
2
C
values ???????????????????????????????...
73
4.33 The FEM results for the contact area versus time for different aspect ratio values, l/D ... 74
4.34 The von Mises stress (MPa) plot for the case under (a) constant force, and (b) constant
displacement boundary conditions ????????????????????...
75
4.35 Contact area results for the case under (a) constant force (creep), and the case under (b)
constant displacement (stress relaxation) boundary conditions ??????????.
76
4.36 Contact pressure results for the case under (a) constant force (creep), and the case under
(b) constant displacement (stress relaxation) boundary conditions ?????????
77
xi
LIST OF TABLES
2.1 Overview of parameter ranges used for the FE simulation [1] ??????????... 16
4.1 Reference properties ??????????????????????????. 38
4.2 Overview of the parameter ranges used for the FE simulations (constant displacement
B.C.) ????????????????????????????????..
39
4.3 Overview of the parameter ranges used for the FE simulations (constant force B.C.) ?... 72
xii
NOMENCLATURE
A
area of contact
B
aspect ratio
B?
creep constant of the power law
B ??
creep constant of the exponential law
b creep constant of the exponential law
C critical yield stress coefficient
( )41
~
, =iCC
ii
creep constants of the Garofalo, strain hardening, and modified
time hardening model
d
material and geometry dependent exponent
E
elastic modulus
E?
reduced or effective elastic modulus
f
spatial frequency (reciprocal of wavelength)
F
contact force
g
average surface separation
G normalized surface separation
h
height of the sinusoidal surface
n creep constant of the Garofalo law
H
hardness
*
p
average pressure for complete contact (elastic)
xiii
*
ep
p
average pressure for complete contact (elastoplastic)
ave
porp
average pressure over entire asperity
P
normalized contact pressure
y
S
yield strength
t contact time
t dimensionless contact time
D
amplitude of the sinusoidal surface
l
asperity wavelength
n
Poisson?s Ratio
s stress
e strain
d
interference between sinusoidal asperity and rigid surface
ba, curvefitting parameters for contact area
dcba ???? ,,, curvefitting parameters for contact pressure
Subscripts
0 initial or at 0t =
c
critical value at onset of plastic deformation
cr creep dependent parameter
e elastic
ep
elastoplastic
JGH
from model by Johnson et al. [2]
1
CHAPTER 1
INTRODUCTION
The topic of contact between surfaces has been popular to researchers for many years.
There exists many works on modeling the contact of surfaces starting around 1888 by Hertz [3]
(originally developed to model optical contacts). All engineering surfaces are rough to some degree,
therefore it is important to develop a model for the contact between rough surfaces. The main goal
in modeling the contact between rough surfaces is to find a simple closed form solution for the real
contact area. Most tribological effects such as friction, wear, adhesion, and electrical and thermal
contact resistance are dependent on the real contact area between two contacting surfaces.
Creep and stress relaxation are time dependent phenomenon which cause changes in the
stress and strain in a material over time. For contacting surfaces, creep and stress relaxation cause
changes in the contact area and contact pressure between surfaces as time passes. Stress relaxation
refers to the stress relief of a material under constant strain condition (Fig. 1.1a), and creep describes
how strain in a material changes under constant stress condition (Fig. 1.1b). Any material can
experience creep if certain conditions are met. It could be metals at high temperatures, polymers at
room temperatures, and any material under the effect of nuclear radiation. Although there is no
recovered creep strain or reversible behavior under normal operating conditions, elastic
deformations are still recovered. The goal of this thesis is to model the effect of creep and stress
relaxation in contact between sinusoidal surfaces.
2
The change in contact area due to creep is considered as the reason why static friction
changes over time, and why the dynamic friction is dependent on velocity [46]. The previous works
on the creep effect in contact between surfaces used a cylindrical [4, 710] or spherical [1, 11, 12]
geometry for the asperities. In this work, a sinusoidal geometry is developed and analyzed which is
believed to be a more realistic geometry. The interaction between adjacent asperities, which is
ignored in most previous works, is considered by assuming a sinusoidal asperity. Two kinds of
boundary conditions are used in the current work to model the time dependent deformations: (i)
constant displacement boundary condition (stress relaxation) and (ii) constant force boundary
condition (creep). The hyperbolic sine function also known as the Garofalo formula is used to
model the effect of the creep and stress relaxation [1]. The results for the contact area and contact
pressure as they change over time are presented and discussed. Empirical equations are developed
by fitting to the FEM results. These empirical equations can be put in a multiscale model to obtain
the real contact area for a specific surface in contact as a function of time. This real contact area can
Fig. 1.1 Change in the stress and strain in a material due to (a) stress relaxation, and (b) creep
(a) (b)
3
be used to obtain a prediction for the time dependent static friction or the velocity dependent
dynamic friction between surfaces.
4
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Most of the previous models on the contact between rough surfaces assume a spherical
shape [1320] or ellipsoidal shape [2123] for the geometry of the asperities on the surfaces. Archard
[14] used a stacked model of spherical asperities and showed that although the relation between the
contact area and load for a single asperity is nonlinear, by using a multiscale model this relation
becomes linear.
More recent models consider a sinusoidal shape for the asperities because it seems to model
the geometry of real surfaces better, especially for heavily loaded contacts [2428]. It is shown for
twodimensional sinusoidal surfaces [26] and threedimensional sinusoidal surfaces [27] that the
average contact pressure increases past the conventional hardness, H , limit of
y
S3? obtained by
assuming spherical geometries [29]. Several works have shown experimentally measured contact
pressures much higher than three times the yield strength (
y
S3? ) [30, 31]. Furthermore, the
interaction between adjacent asperities is addressed by using a sinusoidal geometry with accurate
boundary conditions which is overlooked in works based on spherical asperities. Also, most of the
models consider the multiscale nature of surface roughness by using a Fourier series or Weierstrass
profile [25, 3234], and since these series use superimposed harmonic waves, it is logical to use a
sinusoidal shape instead of a spherical shape in modeling the asperities.
The first models on the contact between rough surfaces using sinusoidal shaped asperities
were mostly on the purely elastic contact. The elastic contact of twodimensional sinusoidal surfaces
was first solved by Westergaard [24]. Johnson et al. (JGH) [2] presented two asymptotic solutions
5
for the elastic contact of threedimensional sinusoidal surfaces, but no analytical solution is available
for the entire load range. Jackson and Streator [34] developed an empirical equation based on the
JGH data [2] for the whole range of loading from early contact to complete contact.
When two surfaces come into contact, it is primarily the peaks or asperities that carry the
load and make the contact. Therefore, the small contact area caused by the asperities carries very
high pressures, and plastic deformations are practically inevitable in most cases of contact between
metallic rough surfaces. Gao et al. [26] considered plastic deformations in their sinusoidal contact
model. They modeled a twodimensional elasticperfectly plastic sinusoidal contact using the finite
element method (FEM). Krithivasan and Jackson [27] and Jackson et al. [28] considered both elastic
and elastoplastic contacts in three dimensions in their work, and presented empirical equations for
the contact area as a function of contact pressure for the whole range of contact. Their equations
are used to verify the current model in the next sections.
One of the important stages in contact between surfaces is when ?complete contact?
happens. The definition of complete contact between rough surfaces depends on the specific case
and geometry of the surfaces in contact. In contact between sinusoidal surfaces, it is defined as the
state in which sinusoidal surfaces have completely flattened out and there is no gap in between the
surfaces. The average pressure (including both contacting and noncontacting regions) that causes
complete contact is an important parameter used to interpret the results, and is denoted by
*
p .
Recent works extended the asperity contact models by including both normal and tangential
loading [35], and loading and unloading [3638]. However, the transient behavior of asperities under
loading such as creep and stress relaxation has been neglected in these works. There have been a
few works on the computational modeling of the creep [11, 12] and stress relaxation [1] effect in the
contact between surfaces but their concentration has been mostly restricted to a single spherical
asperity contact. Most of the earlier models on the effect of the creep in contact between asperities
have assumed a rigid spherical punch indenting an elastoplastic flat surface [3943].
6
Different effects in contact between rough surfaces such as the dwelltime dependent rise in
static friction [5, 44], the velocity dependent dynamic friction [69] or friction lag and hysteresis [45]
can be explained by the creep theory. Many experimental studies concerning the increase of friction
with dwell time are published i.e. [4650]. Malamut et al. [44] studied the effect of dwell time on the
static friction coefficient due to creep for spherical contacts.
The creep behavior depends on the temperature and stress level to which the material is
exposed, and depends noticeably on the time duration of application of these conditions. The
change in the real contact area between two solids due to time of stationary contact is the main
motivation for modeling the creep and stress relaxation effects.
The current analysis uses the same geometry used in Johnson et al. [2] and Krithivasan and
Jackson [27] in order to compare the results in this paper to their works. The sinusoidal geometry is
described by:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?y
?
?x
?h
2
cos
2
cos1
(2.1)
where h
is the height of the sinusoidal surface, D
is the amplitude of the sinusoidal surface, and l
is the wavelength of the sinusoidal surface. The contour plot of the sinusoidal surface is shown in
Fig. 2.1.
7
Quarter Used in FE Model
Fig. 2.1 Contour plot of the threedimensional sinusoidal surface geometry
8
2.2 Elastic Sinusoidal Contact of Single Asperity
As mentioned before, Johnson et al. [2] developed asymptotic solutions for contact area of a
perfectly elastic contact of threedimensional sinusoidal shaped surfaces. In their work, p
is defined
as the average pressure on the surface (considering both contacting and noncontacting regions), and
*
p
is defined as the average pressure that when applied to the surface causes complete contact.
*
p is
given as:
?fE?2p
*
?= (2.2)
where D is the amplitude of the sinusoidal surface, f is the frequency or reciprocal of the asperity
wavelength, l , and E
?
is the equivalent elastic modulus which is given by:
2
2
2
1
2
1
111
EEE
nn 
+

=
?
(2.3)
11
,nE and
22
,nE are the elastic modulus and Poisson?s ratio of the contacting surfaces.
The flat surface is considered to be rigid in this work: ( )??
2
E , and Eq. (2.3) reduces to:
2
1
1 n
=?
E
E (2.4)
The Johnson et al. [2] solutions are applicable when *<< pp i.e. at the early stages of contact, and
when p approaches
*
p *? pp i.e. near the complete contact. The equations are given as shown:
*<< pp :
3/2
*2
1
8
3
)(
?
?
?
?
?
?
?
?
=
p
p
f
A
JGH
p
p
(2.5)
9
*
pp ? :
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
*2
2
1
2
3
1
1
)(
p
p
f
A
JGH
p
(2.6)
Jackson and Streator [34] developed an empirical equation based on the experimental and numerical
data provided by Johnson et al. [2], linking Eqs. (2.5) and (2.6):
for 8.0
*
<
p
p
:
04.1
*
2
51.1
*
1
)(1)(
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
p
p
A
p
p
AA
JGHJGH
(2.7)
for 8.0
*
?
p
p
:
2
)(
JGH
AA = (2.8)
2.3 ElastoPlastic Sinusoidal Contact of Single Asperity
Threedimensional elastoplastic contact between sinusoidal surfaces has been investigated
by Krithivasan and Jackson [27] and Jackson et al. [28]. They showed that for the elastoplastic
contact, complete contact occurs much earlier (at the lower pressures) than elastic contact. Jackson
et al. [28] presented an empirical equation for calculating the average pressure,
*
ep
p , that causes
complete contact for the elastoplastic case. The equation is given below:
( )
5/3
*
*
7/4
11
?
?
?
?
?
?
?
?
+DD
=
c
ep
p
p
(2.9)
where
c
D is the analytically derived critical interference, and is given by:
f
e
E
S
y
c
n
p
3/2
3
2
?
=D (2.10)
10
Jackson and Krithivasan?s [27] empirical equation for the contact area versus contact
pressure for the elastoplastic case is given by:
( ) ( )
04.1
*
2
51.1
*
1
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
ep
JGH
ep
p
p
p
A
p
p
AA (2.11)
where
p
A is given by
d1
d
2
y
d1
1
c
p
?
4CS
p3.
2
A
2A
+
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= (2.12)
In the above equation,
c
A is the critical contact area for the sinusoidal contact based on spherical
contacts [19], and is given by:
2
2
8
2
?
?
?
?
?
?
?
?
?
=
E?f
CS
?
A
y
c
(2.13)
where the constant C
is related to the Poisson?s ratio by:
( )?..C 7360exp2951=
(2.14)
The value of the constant d in Eq. (2.12) is given by:
2
1
d
y
S
E
dd
?
?
?
?
?
?
?
?
D?
=
l
(2.15)
where, 8.3
1
=d and 11.0
2
=d are constants which are obtained empirically by curvefitting to FEM
results.
11
In the current work, Eq. (2.11) is used to confirm the first load step or stress buildup stage
of the finite element model (before creep and stress relaxation initiate at 0=t ).
2.4 Average Surface Separation in Sinusoidal Contacts
In many applications that require tight tolerances such as sealing and lubricated bearings, it is
important to be able to predict the surface separation between contacting surfaces. The average
separation also determines the volume of the space trapped between two surfaces which is a very
important parameter in lubricated surfaces. Little work has been done to characterize the surface
separation or gap between sinusoidal surfaces as a function of load. In Fig. 2.2, surface separation
between a sinusoidal asperity and a rigid flat surface in twodimensions is shown. The average
surface separation before the contact is equal to the amplitude of the sinusoidal asperity, D=g , (Fig.
2.2a), and during the contact the average surface separation is less than the amplitude, D