SURFACE SEPARATION AND CONTACT RESISTANCE CONSIDERING 
SINUSOIDAL ELASTIC-PLASTIC MULTISCALE ROUGH  
SURFACE CONTACT 
 
 
 
Except where reference is made to the work of others, the work described in this thesis is 
my own or was done in collaboration with my advisory committee. This thesis does not 
include proprietary or classified information. 
 
 
 
       
William Everett Wilson 
 
 
 
 
Certificate of Approval: 
 
 
             
George T. Flowers     Robert L. Jackson, Chair 
Professor      Assistant Professor 
Mechanical Engineering    Mechanical Engineering 
 
 
 
             
Dan B. Marghitu     George T. Flowers 
Professor      Dean 
Mechanical Engineering    Graduate School 
 
SURFACE SEPARATION AND CONTACT RESISTANCE CONSIDERING 
SINUSOIDAL ELASTIC-PLASTIC MULTISCALE ROUGH 
SURFACE CONTACT 
 
W. Everett Wilson 
 
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 
December 19, 2008 
 iii 
SURFACE SEPARATION AND CONTACT RESISTANCE CONSIDERING 
SINUSOIDAL ELASTIC-PLASTIC MULTISCALE ROUGH 
SURFACE CONTACT 
 
William Everett Wilson 
 
Permission is granted to Auburn University to make copies of this thesis at its 
discretion, upon the request of individuals or institutions and at 
their expense.  The author reserves all publication rights. 
 
 
 
      
Signature of Author    
 
 
 
      
       Date of Graduation    
 iv 
VITA 
 
 W. Everett Wilson, son of Jim and Betty Wilson was born on August 22, 1984, in 
Birmingham, Alabama.  He graduated high school from Gardendale High School, 
Gardendale, Alabama in May 2002.  He attended Troy University, Troy, Alabama and 
graduated in May 2006 with the degree of Bachelor of Science in Mathematics, minoring 
in Business Administration.  He joined the Masters program in the department of 
Mechanical Engineering at Auburn University in August 2006. 
 v 
THESIS ABSTRACT 
SURFACE SEPARATION AND CONTACT RESISTANCE CONSIDERING 
SINUSOIDAL ELASTIC-PLASTIC MULTISCALE ROUGH 
SURFACE CONTACT 
 
W. Everett Wilson 
Master of Science, December 19, 2008 
(B.S., Troy University, 2006) 
95 Typed Pages 
Directed by Robert L. Jackson 
 
 This thesis considers the multiscale nature of surface roughness in a new model 
that predicts the real area of contact and surface separation as functions of load.  This 
work is based upon a previous rough surface multiscale contact model which used 
stacked elastic-plastic spheres to model the multiple scales of roughness.  Instead, this 
work uses stacked 3-D sinusoids to represent the asperities in contact at each scale of the 
surface.  By summing the distance between the two surfaces at all scales, a model of 
surface separation as a function of dimensionless load is obtained.   Since the model 
makes predictions for the real area of contact, it is also able to make predictions for 
thermal and electrical contact resistance.  For the specific case of thermal contact 
resistance, scale-dependent surface characteristics are taken into account in this model.
 vi 
In the field of contact mechanics, concern has been voiced that the iterative calculation of 
the real contact area in multiscale methods does not converge.  This issue has been 
addressed with results not only confirming convergence but also giving the conditions 
necessary for the sinusoidal based multiscale method to converge. 
 To further verify the results of this new method, all results and calculations are 
compared to previous works that were based upon statistical mathematics to model 
contact area and load.  These comparisons have given qualitative support to the 
sinusoidal multiscale technique featured here as well as revealing some possible short-
comings of the statistical techniques, particularly in the area of surface separation 
calculations.  Upon further investigation, a correction is proposed in this work that 
alleviates this short-coming for statistical contact modeling.  The multiscale sinusoidal 
based elastic-plastic modeling technique is calculated and compared for a variety of 
surfaces, each with a differing roughness with appropriate results.  Finally, in an effort to 
experimentally validate the electrical contact resistance theoretical results, the initial 
setup and outline behind an experimental test rig is explained. 
 vii 
ACKNOWLEDGEMENTS 
 The time I spent at Auburn University studying with the goal of obtaining a 
Master?s Degree in Mechanical Engineering would have seen much less success without 
the excellent guidance I received along the way.  The credit for that goes primarily to Dr. 
Robert L. Jackson who provided me with the opportunity and means to study at this 
esteemed university but also served as a diligent mentor, director, professor, and friend 
along the way.  His assistance and encouragement to constantly expand my efforts and 
overcome my obstacles has been a deeply rewarding experience.  Special thanks also go 
to Dr. George T. Flowers and Dr. Dan B. Marghitu for serving on my graduate committee.  
The faculty and staff of the Mechanical Engineering Department at Auburn University 
also deserve recognition for their services provided throughout my graduate student 
years.  I would also like to thank Santosh Angadi, Jeremy Dawkins, Vijaykumar 
Krithivasan, and Saurabh Wadwalkar for their friendship and many shared laughs and 
frustrations in the Multiscale Tribology Laboratory at Auburn University. 
 Without a doubt, the most influential people in my life have been my loving 
parents, Jim and Betty Wilson.  They have constantly told me of their dreams of success 
for me and ensured I was aware of their pride in me and that support and love has been 
invincible in my life?s quests and journeys.  
 viii 
 Style manual or journal used Guide to Preparation and Submission of Theses and 
Dissertations and Bibliography as per ASME 
Computer software used Microsoft Office 2003, Matlab 7.0.4
 ix 
 
 
 
TABLE OF CONTENTS 
 
LIST OF FIGURES ix 
LIST OF TABLES xii 
NOMENCLATURE xiii 
1 INTRODUCTION 1 
2 BACKGROUND 3 
 2.1 Introduction 3 
 2.2 Statistical Models 3 
 2.3 Fractal Models 5 
 2.4 Multiscale Models 7 
3 OBJECTIVES 10 
4 METHODOLOGY 12 
 4.1 Introduction 12 
 4.2 Multiscale Sinusoidal Perfectly Elastic Contact 12 
 4.3 Multiscale Sinusoidal Elastic-Plastic Contact 16 
 4.4 Statistical Perfectly Elastic Contact 18 
 4.5 Statistical Elastic-Plastic Contact 21 
 4.6 Electrical Contact Resistance 22 
  4.6.1 Multiscale Electrical Contact Resistance 23 
  4.6.2 Statistical Electrical Contact Resistance 25 
 4.7 Thermal Contact Resistance 26 
 4.8 Scale Dependent Thermal Contact Resistance 27 
 4.9 Convergence of Real Area of Contact 28 
 4.10 Adjusted Statistical Surface Separation 29 
5 EXPERIMENTAL DESIGN 31 
6 RESULTS 34 
 6.1 Introduction 34 
 6.2 Convergence of Real Area of Contact 35 
 6.3 Contact Resistance Model Comparison 39 
 x 
  6.3.1 Calculated Real Area of Contact 41 
  6.3.2 Surface Separation 42 
  6.3.3 Electrical Contact Resistance 45 
  6.3.4 Thermal Contact Resistance 47 
 6.4 Comparison between Multiple Surfaces 48 
  6.4.1 Calculated Real Area of Contact 50 
  6.4.2 Surface Separation 55 
  6.4.3 Electrical Contact Resistance 64 
  6.4.4 Thermal Contact Resistance 69 
7 CONCLUSIONS 75 
BIBLIOGRAPHY 77 
 
 xi 
 
 
 
LIST OF FIGURES 
1.1 A schematic depicting the decomposition of a surface into superimposed 
sine waves. 
 
1 
2.1 Spherical contact model before contact (a), during mostly elastic 
deformation (b), and during mostly plastic deformation (c). 
 
4 
4.1 Graphical explanation of common terms used for the sinusoidal based 
multiscale contact model. 
 
13 
4.2 Graphic depicting comparison of JGH asymptotic solutions with Eq. 
(4.10). 
 
16 
4.3 Schematic of ?bottlenecked? current flow through asperities. 
 
23 
4.4 Graphical comparison of surface separation and adjusted surface 
separation. 
 
29 
5.1 Schematic of electrical contact resistance test apparatus. 
 
32 
6.1 Contact area ratio as a function of wavelength for perfectly elastic 
multiscale method where ? is varied in Eq. (31).  
 
35 
6.2 Contact area ratio as a function of wavelength for elastic-plastic 
multiscale method where ? is varied in Eq. (4.53). 
 
36 
6.3 Contact area ratio as a function of wavelength for perfectly elastic 
multiscale method where ? is varied in Eq. (4.53). 
 
37 
6.4 Contact area ratio as a function of wavelength for elastic-plastic 
multiscale method where ? is varied in Eq. (4.53). 
 
38 
6.5 Non-dimensional area vs. load. 
 
41 
6.6 Area compared to Surface Separation including the adjusted separation 
for Statistical Contact Methods. 
 
42 
6.7 Non-dimensional surface separation vs. load. 
 
44 
 xii 
6.8 Electrical contact resistance as a function of non-dimensional load. 
 
45 
6.9 Thermal contact resistance as a function of non-dimensional load 
including scale-dependent results. 
 
47 
6.10 Surface profile data with different roughness values. 
 
49 
6.11 Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method 
for elastic-plastic material deformation. 
 
51 
6.12 Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method 
for perfectly elastic material deformation. 
 
52 
6.13 Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the JG statistical method for elastic-
plastic material deformation. 
 
53 
6.14 Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the GW statistical method for perfectly 
elastic material deformation. 
 
54 
6.15 Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method 
for elastic-plastic material deformation. 
 
56 
6.16 Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method 
for perfectly elastic material deformation. 
 
57 
6.17 Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the JG statistical method for elastic-
plastic material deformation. 
 
58 
6.18 Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the GW statistical method for perfectly 
elastic material deformation. 
 
59 
6.19 Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the sinusoidal based multiscale method 
for elastic-plastic material deformation. 
 
60 
 xiii 
6.20 Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the sinusoidal based multiscale method 
for perfectly elastic material deformation. 
61 
6.21 Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the JG statistical method for elastic-
plastic material deformation. 
 
62 
6.22 Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the GW statistical method for perfectly 
elastic material deformation. 
 
63 
6.23 Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based 
multiscale method for elastic-plastic material deformation. 
 
65 
6.24 Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based 
multiscale method for perfectly elastic material deformation. 
 
66 
6.25 Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the JG statistical method 
for elastic-plastic material deformation. 
 
67 
6.26 Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the GW statistical method 
for elastic-plastic material deformation. 
 
68 
6.27 Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based 
multiscale method for elastic-plastic material deformation. 
 
70 
6.28 Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based 
multiscale method for perfectly elastic material deformation. 
 
71 
6.29 Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the JG statistical method 
for elastic-plastic material deformation. 
 
72 
6.30 Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the GW statistical method 
for perfectly elastic material deformation. 
73 
 xiv 
LIST OF TABLES 
6.1 Material Properties of Tin 39 
6.2 Rough Surface Characteristics and Convergence Variables. 49 
6.3 Rough Surface Characteristics and Adjusted Separation Values (Eq. 4.54). 63 
 
 xv 
NOMENCLATURE 
 
A  area of contact 
A  individual asperity area of contact 
nA  nominal contact area 
a  radius of the area of contact 
B  material dependant exponent 
C  critical yield stress coefficient 
Cp heat capacity per unit volume 
D  contact area factor 
d  separation of mean asperity height 
E  elastic modulus 
E?  )1/( 2vE ?  
Er  electrical contact resistance 
ye  yield strength to elastic modulus ratio, ES y ?/  
F Constant found from slope of Fourier Series 
f  spatial frequency (reciprocal of wavelength) 
G  asymptotic solution from JGH 
GH  geometrical hardness limit 
 K  hardness factor 
 Kn Knudsen number 
 k thermal conductivity 
L  scan length 
M  spectral moment of the surface 
N  total number of asperities 
P  contact force 
P  individual asperity contact force 
p  mean pressure 
?p  average pressure for complete contact 
 q elastic-plastic variable 
R  radius of hemispherical asperity 
yS  yield strength 
Tr thermal contact resistance 
sv  solid speed of sound 
sy  distance between the mean asperity height and the mean surface height 
z  height of asperity measured from the mean of asperity heights
 xvi 
Greek Symbols 
? coefficient of surface spectrum equation 
? exponent of surface spectrum equation 
? area density of asperities 
? separation of mean surface height 
? standard deviation of surface  heights 
?s standard deviation of asperity heights 
? asperity wavelength 
?MFP phonon mean free path 
? density of surface material 
?L electrical resistivity of surface material 
?T thermal resistivity of surface material 
? asperity amplitude 
? distribution function of asperity heights 
? interference between hemisphere and surface 
?  Poisson?s ratio 
 
Subscripts 
E elastic regime 
P plastic regime 
c critical value at onset of plastic deformation 
i frequency level 
JGH from Johnson, Greenwood, and Higginson [10} 
JG from Jackson and Green [15] 
asp asperity 
sur surface 
L electrical 
T thermal 
SD scale-dependent 
1 
CHAPTER 1 
INTRODUCTION 
 
Figure 1.1: A schematic depicting the decomposition of a surface into superimposed sine 
waves. 
 
 There are many different methods to model the contact of rough surfaces 
including statistical [1-4], fractal [5-8], and multiscale models [9-11].  Statistical 
modeling techniques use mathematical parameters of the surface to generalize the surface 
Each Line 
Represents a 
Different Scale of 
Roughness 
2 
into a statistical probability to determine the amount of contact and force.  The fractal 
mathematics based methods were derived to account for different scales of surface 
features not accounted for by the statistical models.  The multiscale models were 
developed to alleviate the assumptions imposed by fractal mathematics and to also 
improve how the material deformation mechanics are considered.  This work uses a 
Fourier transform to convert the data into a series of stacked sinusoids, as shown in Fig. 
1.1.  In a previous work [11] a method to calculate the surface separation from the 
multiscale model was not provided.  It is in the current work. In addition, this work 
differs from a previous multiscale model [11] in that it uses sine shaped surfaces instead 
of spherical shaped surfaces to model contact of the asperities.  The current work also 
provides a methodology for calculating the electrical and thermal contact resistance using 
the multiscale methodology.  This provides a method for including the effect of the scale 
dependent thermal properties [12-16].  Also, the surface characteristics necessary to 
obtain convergence of the iterative multiscale scheme is examined.   
3 
CHAPTER 2 
BACKGROUND 
2.1 Introduction 
 This chapter is devoted to the background material considered in the modeling of 
rough surface contact, surface separation and contact resistances seen in this thesis.  The 
first task will be to give an overview to a few of the many various contact mechanics 
techniques available.  The primary models discussed here will include the multiscale, 
statistical, and fractal methods.  Each of these methods is unique in its assumptions and 
mathematical techniques despite considerable qualitative agreement in their results.  The 
surface separation and electrical and thermal contact resistance will be discussed later in 
this thesis in the methodology section.   
 
2.2 Statistical Methods 
 One of the earliest works in the field of contact mechanics has been credited to 
Heinrich Hertz in his paper titled, On the contact of elastic solids, 1882.  Based upon 
finding of interference fringes between glass lenses, his work displayed elastic 
displacement in surfaces that were compatible with his proposed elliptical pressure 
distribution.  This distribution is, in fact, currently known as the Hertzian contact solution 
[17].  Since this finding, many models have been developed to expand the Hertzian 
contact solution from a single asperity or raised portion on a surface into a network of 
4 
related asperities that can more accurately describe the topography seen on engineering 
surfaces.  One of the very popular expansion techniques is the statistical contact model.   
 One such statistical effort is given by Greenwood and Williamson [1].  In their 
work, known throughout this thesis as the GW model, the interaction between two planes 
is considered.  One of these is a perfect flat while the other is covered in spherically 
shaped asperities. The primary assumptions of this model are that all the asperities must 
have the same radius of curvature, each asperity behaves independently of its neighbors, 
and the substrate material is not allowed to deform, only the asperities.  With these 
assumptions, the contact area is determined through statistical mathematics since the 
asperity heights are presumed to fit a Gaussian distribution.  Therefore, the Gaussian 
distribution gives the percentage of the surface in contact at each from the flat to the 
rough surface generally in terms of standard deviation.  This work gives results for elastic 
deformation because it uses a Hertzian contact solution at the spherical tips which 
assumes that the surface returns to its exact original profile and shape after a loading 
cycle.  The mathematical equations and results for this model will be given later in this 
thesis. 
 
Figure 2.1: Spherical contact model before contact (a), during mostly elastic deformation 
(b), and during mostly plastic deformation (c). 
 5 
 In order to further refine the statistical model, the effects of elastic-plastic 
deformation have been included by numerous researchers.  One such model is given by 
Jackson and Green (JG) [18], which establishes that the range for which the statistical 
model remains perfectly elastic is limited to lower loads.  There are many differing 
models that include the effects of plasticity in statistical modeling such as those offered 
by Chang, Etsion, and Bogy [19] and Kogut and Etsion [20] but these methods are not 
considered in detail for this work.  As the two surfaces increase contact pressure, the 
internal stresses of the asperities will eventually cause the material to yield and deform 
plastically.  The statistical models rely on the interference value between the two surfaces 
which describes the amount of material that must deform for two surfaces to maintain a 
given separation.  In other words, this is the material that would overlap if the two 
surfaces could pass into one another without deforming seen as the gray area in Fig. 2.1.  
To determine the onset of plasticity, a critical interference is calculated based upon 
common surface material parameters that determine when the equation formulated by 
GW must be altered from the Hertzian solution (perfectly elastic) to JG solution which 
gives elastic-plastic results.  This model is based upon the assumption of the GW 
statistical model and is limited to relatively small deformations; the contact radius can 
only be 41% of the radius of curvature. 
 
2.3 Fractal Models 
 The statistical models have shown to be a reliable and easily implemented 
technique but do have some short comings.  For example, the assumptions made are 
essentially averaging an entire rough surface into a single radius of curvature.  
 6 
Essentially, this means statistical models neglect the effects of different scales of features 
on a surface.  Close examination of any surface shows this to be quite inaccurate since 
the topography of a surface in fact appears quite random.  However, it is very difficult to 
calculate surface characteristics for a real engineering surface due to its random nature.  
This is the cause for the advent of the fractal modeling techniques. 
 The current research does not actually model a fractal technique but it is included 
here simply to compare with the models and assumptions made in this work.  One such 
fractal method is Majumdar and Bhushan (MB) [21].  Through the course of their work, 
they found that a surface is multiscale in nature in that as a surface is viewed with a 
higher magnification, each new ?scale? will show a topographical roughness.  To assist in 
modeling this phenomenon, the fractal methods assume that a true rough surface appears 
and behaves like a mathematical fractal equation, hence their name.  In the case of MB, 
the equation is the Weierstrass-Mandelbrot function.  The fractal equation assumes self-
affinity but not self-similarity.  This means that each scale of the surface is related by the 
fractal equations but the relation is different in the normal and lateral directions.  The 
parameters necessary for this equation are garnered from a comparison of the power 
spectrum fit of the rough surface data with the power spectrum of the Weierstrass-
Mandelbrot function. 
 From this point MB calculate elastic-plastic contact mechanics through 
mathematical relations to the Weierstrass-Mandelbrot function and the power spectrum of 
the surface.  This does alleviate some of the assumptions made in statistical models in 
that the surface parameters, specifically radius of curvature of asperities is now 
dependent upon the size of contact.  However, it is possible that a surface may not have 
 7 
an appropriate power spectrum and therefore cannot be related to the fractal equation and 
this model is continually self-affine with no bounds to how small a scale can be 
considered in this model.  In addition, the MB fractal models use a very primitive contact 
mechanics model that basically assumes that the real area of contact can be calculated by 
simply truncating a surface through the fractal described surface. 
 
2.4 Multiscale Models 
 Although the fractal technique is technically a multiscale modeling technique 
since it recognizes surface geometry at every scale available for contact, it has been 
singled out from the multiscale models for the reasons of its primary assumptions.  The 
fractal models carry the self-affinity principle too far.  The model has no stopping point 
although the physical world does.  At some scale, the surface is viewed so closely that the 
only remaining topography is the individual molecules.  Logically the scale modeling 
must stop around this point.  There is no smaller surface characteristic to view.  Also, all 
the scales of a surface will never be perfectly described by a single fractal equation.  The 
multiscale models developed in this work are ideal for this situation. 
 The multiscale modeling technique is initiated from Archard?s ?protuberance 
upon protuberance? modeling scheme [22].  In an early multiscale non-fractal technique, 
Archard expanded the Hertzian sphere against flat contact to feature a sphere of a certain 
radius coated with hemi-spheres of another radius which are all then coated with smaller 
hemi-spheres of a third radius.  This is the basis of a multiscale technique.  Each set of 
spheres with their own unique radius is a ?scale? and as load in increased the small scales 
are pressed into complete contact where the next layer begins to compress.  Archard also 
 8 
proved experimentally that rougher surfaces require greater loads to flatten the asperities 
and that the relationship between area and load approaches linearity.   
 One of the primary modeling techniques featured in this work is given by Jackson 
and Streator [11].  Their work refines the multiscale modeling technique further by 
developing a model more readily adaptable to real rough surfaces.  This model uses a 
series of stacked three dimensional sinusoidal waves to describe the multiple scales of 
contact.  The necessary assumptions for this type of model require (1) that the smaller 
asperities are stacked on top of larger asperities, (2) load is distributed equally over all 
asperities at that level, (3) all levels carry the same overall load, and (4) a smaller asperity 
level is not capable of extending the contact area beyond that capable of the larger scale 
below.  Other assumptions are required for each specific model based upon the desired 
deformation technique such as Jackson and Streator [11] use the Johnson, Greenwood, 
and Higginson [23] asymptotic solutions for perfectly elastic deformation derived from 
their work on 3-D wavy surfaces (JGH) [23].  The JGH asymptotic solutions are given 
for high and low loads so Jackson and Streator [11] fit a polynomial linking equation in 
between to model the complete range of contact.  For this modeling technique, the areal 
density of asperities and radius of curvature depend upon the frequency of each level of 
sine waves.  This is done by converting the data into a series of sine waves through a 
discrete Fourier transform which results in a series of frequencies and amplitudes used to 
calculate contact area for levels of load iteratively.  The JGH asymptotes and linking 
equation will be discussed in detail later in Chapter 4. 
 Finally, the above work by Jackson and Streator [11] was modified to include 
plasticity by Krithivasan and Jackson [24].  The framework of the perfectly elastic 
 9 
sinusoidal model given by JGH [23] can be further refined to more realistically model 
rough surfaces by including the contact solutions found by Krithivasan and Jackson [24] 
in place of the asymptotes derived from Hertzian contact.  The elastic-plastic solutions 
were found through analysis of the finite element modeling (FEM) of a three dimensional 
sinusoidal asperity.  Similar to the JG model for statistical elastic-plastic deformation, the 
model remains in a perfectly elastic deformation regime until critical values are reached.  
The current multiscale model doesn?t rely on the interference of the two surfaces for 
establishing contact area and load.  Instead it iteratively calculates area for each load 
level.  Therefore, the model by Krithivasan and Jackson [24] are adjusted to include the 
critical contact pressure or load at which the surfaces enter the elastic-plastic regime.  The 
equations used for this and the preceding modeling techniques are discussed in detail 
later in this thesis work. 
  
10 
CHAPTER 3 
OBJECTIVES 
The thesis work presented here is focused on further development of the 
sinusoidal based multiscale contact modeling technique originally presented by Jackson 
and Streator [11]. In their work, Jackson and Streator developed the necessary conditions 
and equations to determine the theoretical real area of contact for the sinusoidal 
multiscale modeling method.  The sinusoidal contact work was further built upon by 
Krithivasan and Jackson [24] to include the effects of plasticity in the individual asperity 
contact model.  For the thesis presented here, both the models mentioned above will be 
employed to calculate the real area of contact, contact pressure or load, surface 
separation, both electrical and thermal contact resistances, and finally the effects of scale 
dependent material properties will be evaluated for thermal contact resistance.   
In the field of contact mechanics, there exists some concern as to the validity of 
the multiscale modeling techniques due to the fact that, in some instances, they fail to 
converge.  This means that certain conditions prevent the contact area from reaching a 
final non-zero contact area equal to the nominal or apparent area of contact.  The 
convergence of both multiscale modeling techniques (perfectly elastic and elastic-plastic) 
will be examined and the necessary conditions for convergence will be compared for a 
variety of surface roughness.  The models themselves will also be compared for four 
separate sets of data gathered from a stylus profilometer, each with a varying roughness.   
11 
In addition to calculating the previously mentioned surface interactions for the 
sinusoidal based multiscale model, this work also compares all of the features mentioned 
to the pre-existing and well known statistical contact models.  In the case of perfectly 
elastic contact, the results of the sinusoidal multiscale method will be compared to that of 
the Greenwood and Williamson (GW) model [1].  However, the elastic-plastic 
deformation will be compared to the Jackson and Green model (JG) [18].  Surface 
separation, electrical contact resistance, thermal contact resistance, and scale dependent 
thermal contact resistance will be calculated from a presented statistical technique as 
well. 
During the course of this work, a possible error is exposed for the statistical 
models with respect to the prediction of surface separation.  The surface separation of 
both perfectly elastic (GW) and elastic-plastic (JG) statistical techniques does not reach 
zero when the calculated real area of contact is at its maximum value at complete contact.  
One would assume that at the maximum contact area the entire surface area available is 
in contact so there cannot be any separating gap between the two surfaces being forced 
together.  To alleviate the discrepancy, an adjusted separation model will be. 
Finally, an attempt will be made to further validate the theoretical models above 
by designing a test apparatus.  The test apparatus will be used for a comparison of 
electrical contact resistance as a function of load.  This will be accomplished by 
incrementally increasing the load while taking a measurement of the voltage drop across 
the interacting faces of two metallic surfaces at each load step.  The details of these 
techniques are now illuminated in the following sections. 
12 
CHAPTER 4 
METHODOLOGY 
4.1 Introduction 
 This chapter describes in detail the numerical models used to calculate rough 
surface contact.  The numerical techniques necessary for real area of contact, contact 
pressure or load, and surface separation are described for the unique cases of perfectly 
elastic and elastic-plastic deformation.  Furthermore, the fundamental theory and 
techniques of contact resistance are discusses for both electrical and thermal contact 
resistance. 
 
4.2 Multiscale Perfectly Elastic Contact 
 The employed multiscale model [11] uses the same direction of thought as 
Archard [22], but provides a method that can be easily applied to real surfaces.  First a 
fast Fourier transform is performed on the surface profile data to predict the terms for the 
Fourier series describing the surface.  This series describes the surface as a summation of 
a series of sine and cosine waves.  The complex forms of the sine and cosine terms at 
each frequency are combined using a complex conjugate to provide the amplitude of the 
waveform at each scale for further calculations.  Each frequency is considered a scale or 
layer of asperities which are stacked iteratively upon each other.  In equation form these 
relationships are given by: 
13 
n
i
i
ii AAA ???
?
???
?= ?
=
max
1
?         (4.1) 
1?= iii APP ?          (4.2) 
where A is the real area of contact, ? is the areal asperity density, P is the contact load, An 
is the nominal contact area, and the subscript i denotes a frequency level, with imax 
denoting the highest frequency level considered. 
0 2 4 6 8 10 12
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
 
Figure 4.1:  Graphical explanation of common terms used for the sinusoidal based 
multiscale contact model. 
 
   
  Wavelength, ? 
 
 
       Amplitude 
              ? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 Frequency is the inverse of wavelength, f=1/? 
 14 
 Each frequency level is modeled using a sinusoidal contact model.  Previously 
derived [11] equations fit to the data and asymptotic solutions given by Johnson, 
Greenwood, and Higginson (JGH) [23] are used.  The first equation is derived from Hertz 
contact and is used for low loads where ?<< pp : 
( ) 3
2
21 8
32 ?
?
?
??
?=
?p
p
fAJGH pi
pi        (4.3) 
However, at higher loads where the contact is nearly complete, p approaches ?p , and the 
following equation must be implemented: 
( ) ??
?
?
???
?
??
?
??
? ??=
?p
p
fAJGH 12
311
22 pi       (4.4) 
 Fortunately, JGH provide experimental and numerical data to support their 
asymptotic solutions which Jackson and Streator [11] used to fit a linking equation for the 
asymptotes Eq. (4.3) and Eq. (4.4) as follows: 
For   
( ) ( ) 04.1251.11 1 ??
?
?
???
?+
?
?
?
?
?
?
?
?
??
?
??
??=
?? p
pA
p
pAA
JGHJGH      (4.5) 
For  
( )2JGHAA =              (4.6) 
where p* is the average pressure to cause complete contact between the surfaces and is 
given by [23] as: 
fEp ??=? pi2         (4.7) 
8.0* <pp
8.0* ?pp
 15 
The current work also fits a new equation to the surface separation results given by JGH 
[23].  In previous works, the multiscale model was used to relate area to load.  However, 
for many applications such as those requiring tight tolerances like sealing and lubricated 
bearings, it is also important to be able to predict surface separation.  JGH gave 
asymptotic solutions for the surface separation.   As ?pp  approaches zero, the solution is: 
( )[ ] ??
?
?
???
??++
???
?
???
??=
?? p
p
p
pG 12ln43
2
11 3/22
1 pi      (4.8) 
While as ?pp  approaches 1 the solution given by [23] is: 
2/52/3
22 12
3
15
16 ?
?
?
??
? ??
?
??
?
?=
?p
pG
pi       (4.9) 
In the current work an equation is then fit to join these two solutions: 
( )847.0158.0*696.0*1 ????
?
?
???
? +?
???
?
???
? ??=
p
p
p
p?      (4.10) 
As seen in Fig. 4.2, ?  appears to be a good fit to the asymptotic functions given by Eqs. 
(4.8-4.9). 
 
 16 
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p?/p*
?/?
curve fit to JGH asymptotes, ?
JGH low load asymptote, G1
JGH high load asymptote, G2
 
Figure 4.2: Graphic depicting comparison of JGH asymptotic solutions with Eq. (4.10). 
 
 The separation height, H, between the two surfaces is calculated by subtracting 
the ? value from the amplitude, ?, at each scale level and then summing them together 
over all frequency scales as follows: 
( )?
=
??=
max
1
i
i
iH ?         (4.11) 
 
4.3 Multiscale Elastic-Plastic Contact 
 As noted previously, many of the asperities at the different frequency levels 
undergo plastic deformation.  Therefore an elastic-plastic sinusoidal contact model is 
 17 
needed to consider this effect.  The equations used in the current work to calculate the 
elastic-plastic contact are derived from FEM results by Krithivasan and Jackson [24] and 
Jackson, Krithivasan and Wilson [25].  The methodology is very similar to that of the 
perfectly elastic case with the exception that a different set of formulas is used once a 
calculated critical pressure is reached.  The critical average contact pressure (Pc), critical 
average pressure over the nominal area ( cp ), the critical load and critical area (Ac) are 
given by:     
32
2 2
1
6
1 ?
?
??
?
? ?
???
?
???
?
??= yc S
C
EfP pi        (4.12) 
2
28
2
???
?
???
?
??= Ef
CSA y
c pi         (4.13) 
( )
( )2
32
2
22
24
1
3
22
822 Ef
CSCS
Ef
CSffpAp yyy
ccc ??=???
?
???
?
???
?
???
?
??== pipi    (4.14) 
where ( )vC 736.0exp295.1 ?= . 
 At low loads, P<Pc, and consequently small areas of contact, it is acceptable to 
assume that any deformation of the asperities in contact will behave perfectly elastically.  
However, as load increases to the critical value, plastic deformation will begin to occur 
within the asperities.  To evaluate the plastic deformation we replace Eq. (4.3) with: 
( ) q
q
y
qcp
CS
pAA ++
???
?
???
?= 12
1
1
4
32 ?        (4.15) 
11.0
8.3 ??
?
?
???
? ????=
?yS
Eq         (4.16) 
 
 18 
This replacement results in the following equation for contact area: 
( ) ( ) 04.1251.11 ??
?
?
???
?+
?
?
?
?
?
?
?
?
??
?
??
??=
??
PP p
pA
p
pAA
JGHP     (4.17) 
The pressure to cause complete contact during elastic-plastic deformation is then given 
by [25] as: 
53
* 74
11
??
?
?
?
??
?
?
?
+???
=
?
cp
pP         (4.18) 
Where fE
vS
c
y
'3
3
2exp2
pi
???????
=?  and is the critical amplitude, below which the sinusoid will 
always deform in the elastic regime.  Plastic deformation is caused by stress initially 
below the surface and building as more pressure is added.  To determine the critical 
amplitude where plasticity begins, the maximum von Mises stress is equated to yield 
strength and the resulting formula solved for critical amplitude, c? .  Surface separation 
is calculated exactly as before by using Eq. (4.10), except the separation for pressures 
greater than cp  must have *p  replaced by ?Pp . 
 
4.4 Statistical Perfectly Elastic Contact 
 To compare and contrast the results of the multiscale sinusoidal models, statistical 
contact models are also calculated using the same surface parameters and profilometer 
results.  For the perfectly elastic case, this work employs the Greenwood and Williamson 
[1] approach for asperity contact.  The GW method requires that a few crucial 
assumptions be made:  (1) each asperity is assumed to behave independently of 
 19 
neighboring asperities, (2) all asperities have the same radius of curvature, (3) the 
asperity heights from the surface follow a Gaussian height distribution, and (4) only the 
actual asperity may deform, all substrate material is rigid as well as the contacting 
surface.   
 Using the Greenwood and Williamson [1] type statistical method hinges upon 
obtaining statistical parameters that describe the surface.  The radius of curvature, R, and 
the areal asperity density, ?, are calculated by McCool [26] using the spectral moments of 
the surfaces: 
?
=
??????=
N
n ndx
dz
NM 1
2
2
1         (4.19) 
2
1
2
2
4
1
n
N
n dx
zd
NM ?= ???
?
???
?=         (4.20) 
Where N is the total number of asperities on the surface and z is the distance from the 
mean height of the surface to the asperity peak.  Then R and ? are found from: 
???
?
???
??
???
?
???
?=
36
1
2
4
pi? M
M         (4.21) 
5.0
4
375.0 ??
?
?
???
??=
MR
pi         (4.22) 
The Gaussian distribution for the asperity heights is given as follows: 
( )
?
?
?
?
?
?
?
?
???
?
???
??=? ? 22/1 5.0exp2
ss
z
??
pi .      (4.23) 
McCool [26] defines s?  to be the standard deviation of the asperity heights.  This is 
calculated from the standard deviation of the entire surface (RMS roughness): 
 20 
22
4
22 10717.3
Rs ???
??
+=         (4.24) 
 For the GW case, the area and load are calculated using an integral of ? and a 
function relating the z value to a value d.  d is defined as the value above which the 
asperities will be in contact with the rigid flat.  The compression distance, z-d, is the 
interference of the rigid flat with the asperity peaks and is known as ? for the remainder 
of this work.  The integrals used to find the contact area, A, and the contact pressure or 
load, P, for each d value are given below: 
( ) ( ) ( ) dzzAAdA
d
n ???= ?
?
??        (4.25) 
( ) ( ) ( ) dzzPAdP
d
n ???= ?
?
??        (4.26) 
For the perfectly elastic case, A  and P  are acquired from the Hertz solutions given as: 
?piRAE =          (4.27) 
( ) 2/334 ?REP E ?=         (4.28) 
Furthermore, surface separation can be obtained by relating the distance from the mean 
surface height to the rigid flat, ? , to d.  
syd +=?          (4.29) 
The value sy  is defined by Front [27] as follows: 
Rys ?
045944.0=         (4.30) 
 
 21 
4.5 Statistical Elastic-Plastic Contact 
 Similar to the multiscale model, some of the asperities will undergo plastic 
deformation as loads increase past the critical values.  This work uses the methodology of 
Jackson and Green [28] and [3], referred to as JG for the remainder, which replaces the 
Hertzian contact solution in the GW model with equations suited for elastic-plastic 
deformation after critical values have been reached.  The statistical method calculates 
load and area as a function of separation instead of area as a function of load as seen in 
the multiscale methods.  Therefore, instead of using the critical force to define the elastic-
plastic regime of contact, the critical interference is used.  The critical interference value 
is given by [28] as follows: 
RE SC yc ???
?
?
???
?
?
??= 2
2
pi?         (4.31) 
For interference ?<1.9?c, spherical contact is considered to effectively agree with the 
perfectly elastic Hertzian contact model.  However, when ??1.9?c the following 
equations from JG are used in place of Eqs. (4.27-4.28).  This substitution will provide 
the necessary values to calculate the elastic-plastic behavior of the asperities in contact. 
B
c
JG RA ??
?
?
???
?=
?
??pi         (4.32) 
??
???
??
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
???
???+
???
?
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
???
??=
ccy
G
cc
cJG CS
HPP
?
?
?
?
?
?
?
? 9/52/312/5
25
1exp14
4
1exp (4.33) 
where 
32
23
4 ?
?
??
?
??
?
??
?
?
?= yc S
C
E
RP pi        (4.34) 
 22 
( )yeB 23exp14.0=         (4.35) 
E
Se y
y ?=          (4.36) 
???
?
???
? ?
?
??
?
???=
R
a
S
H
y
G picos192.084.2        (4.37) 
???
?
???
?
?= cRR
a
?
??
9.1         (4.38) 
These equations are then used in Eqs. (4.25-4.26) for the single asperity area and load. 
 
4.6 Contact Resistance 
 One of the concerns of this work is calculating the effect of surface roughness on 
electrical resistance.  Therefore, the goal of this section is to determine how the flow of 
the current between surfaces is affected by the true area of contact for each load level.  
Since only a few, scattered asperities are actually in contact for any given load level, the 
current is restricted to very small contact patches when compared to the area of the entire 
surface.  As the current flows through these asperity peaks, it will be effectively 
?bottlenecked? resulting in some resistance to the conduction as seen in Fig. 4.3 on page 
23. 
 
 
 23 
 
Figure 4.3:  Schematic of ?bottlenecked? current flow through asperities. 
 
 Holm [29] gives a simple formula to calculate the electrical resistance due to 
asperity contact. 
aEr
LL
asp 4
21 ?? +=         (4.39) 
Where Er refers to the contact resistance value, a is the radius of contact, and ? is the 
specific electrical resistance, or resistivity, of the respective surfaces.  However, this 
equation is only good for a single asperity.  In the case of both multiscale and statistical 
techniques, additional equations are required to calculate resistance for the entire surface, 
(see the following sections). 
 
4.6.1 Multiscale Electrical Contact Resistance 
 The multiscale sinusoidal method presented here is an iterative method that 
calculates area and resistance for each particular frequency level.  To predict electrical 
contact resistance, the first step is to calculate the average radius of contact per frequency 
level i: 
 24 
2
12 fA
Aa
i
i
i ???=
?pi
        (4.40) 
 Once the average contact radius is established, Eq. (4.39) is implemented to 
calculate resistance per asperity per level.  For the sinusoidal case, it is assumed that the 
tip of the asperity is similar to a hemisphere so the radius of curvature at the tip is used.    
Oftentimes, an alleviation factor is used in thermal contact resistance calculations to 
account for the affect of a large contact radius, a, in relation to the asperity tip radius, R.  
Since electrical and thermal contact resistances are very mathematically similar, it stands 
to reason that the alleviation factor, ?, should also be included for electrical contact 
resistance.  Though there are various ways to calculate this factor [30], the simplified 
version offered by Cooper et al. [31] is chosen for this work: 
5.1
1
1 ??
?
?
???
? ?=?
?i
ii AA        (4.41) 
The alleviation factor, ?i, is combined with the resistance value and the result is summed 
over all possible iteration levels to find the total resistance for the entire surface in 
contact.  In equation form this is given as: 
?
=
??=
max
1
1
i
i
iiiitotal AErEr ?         (4.42) 
 It is important to note that this technique calculates the resistance for each 
frequency level and then sums them over all frequency levels to calculate the total.  
Another technique exists that only evaluates the resistance for the highest frequency level 
that still reduces contact area [32].  This alternative technique is not considered in this 
work.  Also, this methodology does not change depending on the inclusion of plasticity 
since all resistance calculations are done after obtaining the contacting area. 
 25 
 
4.6.2 Statistical Electrical Contact Resistance 
 To continue comparing the multiscale results with that of the earlier statistical 
method (see section 4.4 and 4.5), the electrical contact resistance is also obtained for 
statistical perfectly elastic and elastic-plastic methods.  Greenwood and Williamson [1] 
include a solution for contact conductance in their work.  Resistance is simply the inverse 
of conductance so the technique for calculating perfectly-elastic contact resistance is as 
follows: 
( ) ( )?
?
? ????=
d
Ln
e
dzzRAdEr 2121121 ???       (4.43) 
Note the inclusion of the alleviation factor, ?.  The statistical method is not a multiscale 
procedure so the alleviation factor is calculated as follows: 
5.1
1 ??
?
?
???
? ?=?
n
r AA         (4.44) 
 Contact resistance from elastic-plastic statistical contact is calculated in the same 
manner except using a different elastic-plastic asperity contact model.  One such 
technique is given by Kogut and Etsion [33], which is used as an outline for this work.  
However, there are dissimilarities since this work relies on the methodology of Jackson 
and Green [28] for single asperity contact.  Eq. (4.43) is still used but the contact radius, 
a, has changed in accordance with the work of Jackson and Green: 
RDaep ?=           (4.45) 
where if  9.1/0 ?? c?? , then 1=D , but if c?? 9.1? , then 
B
c
D ??
?
?
???
?=
?
?
9.1 .   
 26 
By applying the contact radius in Eq. (4.45) to the resistance calculation in Eq. (4.43), the 
elastic-plastic electrical contact resistance is obtained. 
( ) ( )?
?
????=
d L
ep
n
ep
dzzaAdEr ?? 21       (4.46) 
Similar to the multiscale model, the alleviation factor, ?, is included here as well.   
 
4.7 Thermal Contact Resistance 
 Thermal contact resistance refers to the build-up of heat at the boundary between 
the two surfaces due to the same ?bottleneck? effect referred to in electrical resistance 
seen in Fig. 4.3 on page 23.  Technically, heat can flow across the gaps in the material as 
well as through the contacting asperities.  However, the heat transfer across the gaps is 
neglected since like electrical current, usually the majority of the heat flow will follow 
the path of least resistance or in this case the asperities in contact. Indeed, thermal and 
electrical contact resistances are very similar effects and are computed using very similar 
methods as well.   
 The thermal resistance values for the multiscale contact methods are found by 
modifying Eqs. (4.41-4.42) as follows: 
aTr
TT
asp 4
21 ?? +=         (4.47) 
??= maxi
i
iiiitotal ATrTr ?         (4.48). 
In this case, T?  refers to the thermal resistivity or inverse of the conductivity of the 
material. 
 27 
 Similarly, thermal contact resistance for statistical methods is obtained by 
substituting thermal resistivity for electrical resistivity in Eqs. (4.43-4.46). 
( ) ( )?
?
? ????=
d
Tn
e
dzzRAdTr 2121121 ???      (4.49) 
( ) ( )?
?
????=
d T
ep
n
ep
dzzaAdTr ?? 21       (4.50) 
 
4.8 Scale Dependent Thermal Contact Resistance 
 Scale dependency is an emerging topic in the field of contact resistance.  The 
concept behind scale dependency is that as a sample of a material is viewed at 
increasingly higher magnification the material properties actually change according to 
how small of a sample is viewed.  The reason for this is that at some point one is viewing 
actual atoms pressed against each other instead of the continuous material.  Therefore, at 
this point the scale is below that where most imperfections and features are seen, such as 
grain boundaries.  This impacts the multiscale contact model because it takes into account 
many different scales of asperities down to where this phenomenon is seen.  To include 
these effects, the thermal contact resistance, Eqs. (4.47-4.50), is replaced with a scale 
dependent value found in the work of Prasher and Phelan [34]: 
?????? +?= KnkaTrSD pi3812        (4.51) 
Kn refers to the Knudsen number ?
?
??
?
?
a
MFP? where 
MFP?  is the phonon mean free path. 
ps
MFP Cv
k
??
3=         (4.52) 
 28 
Here, ? is the density of the material, sv  refers to the solid speed of sound, Cp is the heat 
capacity, and k is the thermal conductivity.  Note also that the alleviation factor, ?, has 
been included in the scale-dependent calculations although it is not part of the Prasher 
and Phelan [34] solution. 
 
4.9 Convergence of Real Area of Contact 
 As mentioned previously, the data set used for this model is converted into a 
series of stacked sine waves using the discrete Fourier Transform.  All calculations for the 
model are then made based off the amplitude and wavelength of these sine waves.  The 
multiscale model considered here assumes that the predicted area converges as all scales 
are included in the model.  This is important because if the predictions do not converge 
then the area will approach zero as smaller scales are included, as was predicted by [35].  
In order to test convergence, a power fit is found for the nominal amplitude as a function 
of wavelength.   
???
ii =?          (4.53) 
where i?  is the wavelength (inverse of frequency) and both ? and ? are constants derived 
by fitting Eq. (4.17) to the Fourier series of the surface data.  For a particular surface the 
best fit was found with ?=.085 and ?=1.5.  Starting with this ?benchmark case?, the 
values ? and  ? are then varied individually to find any critical values at which point 
convergence is not possible.  This process is carried out for both perfectly elastic and 
elastic plastic cases.  The results are provided in Chapter 6.2 
4.10 Adjusted Statistical Surface Separation 
 29 
Through comparison to the sinusoidal multiscale technique, a potential error is 
brought to light in the statistical methods.  When calculating surface separation, the 
statistical models show a significant surface separation despite the calculated real area of 
contact reaching its maximum value.  This result is highly unexpected since there should 
be no separating gap if the two surfaces are entirely in contact as seen in the multiscale 
modeling results.  The reasons for difference amongst the statistical and multiscale 
methods are three-fold.  First, in the multiscale method, all asperities are loaded equally 
so they may actually be over-compressed.  Second, the statistical model does not consider 
the interactions between adjacent asperities.  At some loads, the valleys neighboring each 
peak may be rising as they fill in with the plastically deformed material. Third, believed 
to be the most important, the statistical method does not adjust the mean height based on 
asperity deformation. 
  
 
Figure 4.4: Graphical comparison of surface separation and adjusted surface separation. 
 
To improve agreement between the models, an adjusted separation is now 
implemented.  As shown schematically in Fig. 4.4, the adjusted separation, ?ADJ, is found 
by assuming that the mean height when considering deformation will locate at the 
 30 
midpoint between the deformed tips of the asperities and the deepest valleys of the 
asperities.  Therefore, as the surface deforms due to contact, the mean surface height 
actually changes from ??/2 to a smaller value. Using this concept, the surface separation 
predicted by the statistical model can be adjusted using the following simple equation: 
2
5.0 ???? +=
ADJ         (4.54) 
 As seen in Fig. 4.4, Eq. (4.54) takes the average of the mean surface height at zero 
deformation, ??/2, and the distance from the mean surface height to the peak of the 
deformed surface, ?.  The ? value is found not to be constant and varies not only for 
different surfaces but also depends on whether plasticity is considered.  As shown in Fig. 
4, ?? represents the peak to valley height of the surface.  Since statistically 68% of the 
asperities are accounted for between -2? and 2? and 99.7% are accounted for between -6? 
and 6?, one could make an approximation of ? between 4 and 12. 
31 
CHAPTER 5 
EXPERIMENTAL DESIGN 
 One of the many issues in the field of contact mechanics is the validation of the 
many different contact models.  Problems arise in experimental validation because the 
surface features and behaviors mentioned in works of this type are often at extremely 
small scales.  An additional dilemma is that typically metals are the material of choice for 
studying contact mechanics.  These and other issues result in no efficient and reliable way 
to ?view? the real area of contact to compare to theoretical results.  Therefore, the 
experimental test apparatus for this work is designed to measure electrical contact 
resistance.  This surface characteristic is much easier to accurately calculate and the 
choice of metals actually aids the process since most metals are very good conductors.   
 To compare with the theoretical calculations for electrical contact resistance, the 
testing procedure must be able to alter the load placed upon the test samples and measure 
the change in resistance at the interface of the test surfaces.  In this case, the test is also 
designed to garner measurements for a variety of roughness.  The samples used are all as 
close to identical as possible.  The roughness is changed by sanding each surface with a 
different grit of sandpaper to altering the roughness left from machining and selecting one 
surface to be the common base to use against the remainder.  After selecting and altering 
the samples, a stylus profilometer is used to measure the exact roughness and profile for 
32 
each sample.  Then a copper wire is attached to each sample which will be the connection 
for a multi-meter set up to measure the voltage drop across the contacting surfaces. 
 
Figure 5.1:  Schematic of electrical contact resistance test apparatus. 
 
 Finally, a modified drill press is used to press the two surfaces together for the 
desired loads as seen in Fig. 5.1.  A load cell is used to ensure accurate readings through 
Labview? software.  In this test, weights are applied to the drill press to increase the 
pressure upon the two surfaces in increments.  At each load increment, the multi-meter is 
used to calculate the electrical resistance of the samples in contact.  This value is a 
 33 
measure of the voltage drop across the surface due to incomplete contact and is expected 
to correspond to the theoretical results for the load applied.   
 Unfortunately, only a few initial runs were attempted with this test apparatus 
which proved to be unsuccessful.  The results were very inconsistent in that for the first 
few increments of load, the resistance would indeed drop but as greater and greater loads 
were added the resistance would increase, sometimes to its original value.  Also, the 
resistance values would not reach a consistent value.  Instead the resistance would rise 
and fall within a range that furthered the inaccuracy of the results.  One issue in setting up 
this experiment, which most likely led to the resistance issues, was the sample material 
available at the time of testing was a steel tab being pressed against a polished stainless 
steel block.  Getting a consistent connection from the copper wires of the multimeter to 
these steel samples was very difficult since tin solder does not hold on steel. Therefore, 
any connection made turned out to be extremely fragile.  To remedy these issues, this 
experiment will be attempted later using samples of different materials which should be 
easier to manipulate and measure. 
 
34 
CHAPTER 6 
RESULTS 
6.1 Introduction 
 This chapter is devoted to the calculated results from the theoretical models.  The 
first section reveals the results of a convergence analysis for the sinusoidal multiscale 
model.  For the second section, a single surface is used so the roughness remains 
constant.  This is done to compare the results of the sinusoidal based multiscale model 
with that of the statistical technique for both perfectly elastic and elastic-plastic 
deformations cases with the intention of validating the new multiscale technique with the 
familiar statistical model.  Third and final is a section that gives a direct view of the 
effects roughness has upon the calculated theoretical values.   
 
35 
 
 
 
 
6.2 Convergence of Real Contact Area 
10?4 10?3 10?2 10?1 100
10?6
10?5
10?4
10?3
10?2
10?1
100
Contact Area Ratio A
i/A
r
Wavelength, ?i/L  
Figure 6.1:  Contact area ratio as a function of wavelength for perfectly elastic multiscale 
method where ? is varied in Eq. (4.53). 
 
 
?=1000 
 
?=100 
    Iteration Direction 
    
?=10 
 
 
?=1.0 
 
 
?=0.1 
 
 
?=0.01 
 
 
?=0.001 
 36 
10?4 10?3 10?2 10?1 100
10?3
10?2
10?1
100
Contact Area Ratio A
i/A
r
Wavelength, ?i/L  
Figure 6.2:  Contact area ratio as a function of wavelength for elastic-plastic multiscale 
method where ? is varied in Eq. (4.53). 
 
 The first test of convergence is to characterize the role that ? plays in Eq. (4.53).  
For this test case, ? is varied from 10-3 to 103 by an order of magnitude at each step.  The 
fit value for ? was found to be 0.085, which falls within the range of values tested.  As 
seen in Fig. 6.1-6.2, ? does not appear to play a significant role in convergence since all 
the lines show a flattening trend at the smaller wavelengths.  This result is independent of 
plasticity since nearly the same trend is seen in both Figs. 6.1-6.2. 
 
 
 
?=1000 
 
 
 
?=100 
 
    Iteration Direction 
 
?=10 
 
 
?=1.0 
 
 
?=0.1 
 
 
?=0.01 
 
 
?=0.001 
 37 
 
 
 
 
 
10?4 10?3 10?2 10?1 100
10?12
10?10
10?8
10?6
10?4
10?2
100
Contact Area Ratio A
i/A
r
Wavelength, ?i/L  
Figure 6.3:  Contact area ratio as a function of wavelength for perfectly elastic multiscale 
method where ? is varied in Eq. (4.53). 
 
 
?=2.0 
 
?=1.75 
 
?=1.5 
 
?=1.25 
 
?=1.0 
 
 
?=0.75 
 
 
?=0.5 
 
 
?=0.25 
 
 
?=0.0 
 
 
 Convergence 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 No Convergence 
 38 
10?4 10?3 10?2 10?1 100
10?5
10?4
10?3
10?2
10?1
100
Contact Area Ratio A
i/A
r
Wavelength, ?i/L  
Figure 6.4:  Contact area ratio as a function of wavelength for elastic-plastic multiscale 
method where ? is varied in Eq. (4.53). 
 
 The second test of convergence is conducted by varying the value of ? in Eq. 
(4.53).  The results from this test, shown in Figs. 6.3-6.4, show that this exponent is the 
deciding factor for convergence in both perfectly elastic and elastic-plastic cases.  For 
this test ? is varied from 0 to 2 at intervals of 0.25.  For a good fit to the FFT data, ? was 
found to be 1.5, which is within the range of values tested.  ? values lower than 1 show a 
continual slope despite the wavelength size suggesting that convergence is not possible 
for these cases.  This result is shown to be similar for both perfectly elastic and elastic-
plastic cases. 
 
?=2.0 
 
?=1.75 
 
?=1.5 
 
?=1.25 
 
?=1.0 
 
 
?=0.75 
 
 
?=0.5 
 
 
?=0.25 
 
 
?=0.0 
 
 Convergence 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 No Convergence 
 39 
 The results of these tests have shown that convergence is dependent upon the 
exponent, ?, in the power function fit to the FFT data in Eq. (4.53).  As the scales are 
iteratively included, the contact area reduces.  Therefore, the average pressure continues 
to increase and may eventually become larger then the pressure to cause complete contact 
(p*).  If this pressure stays above the pressure to cause complete contact (p*) the area will 
no longer reduce and convergence is obtained.  Therefore, for convergence to occur, p* 
must stay constant or decrease as ? decreases. This suggests that the requirement for 
convergence depends upon the following relationship: 
1* ?=?? ???
?p         (4.55) 
As long as ?/? stays constant or decreases as ? decreases then the multiscale sinusoidal 
method will converge.  This condition is met as long as the ? value in Eqs. (4.53 & 4.55) 
are greater than or equal to 1, as shown by Eq. (4.55). 
 
PaS y 61014 ?=   
mL ???= ?8105.11?  
PaE 910369.41 ?=  
36.0=v  
 
Table 6.1:  Material Properties of Tin 
 
6.3 Contact Resistance Model Comparison 
 This part of the work assumes realistic material properties for all results gathered 
(see Table 1).  Since this work includes an examination of electrical resistance, the 
material of choice is Tin due to its common use in electrical connectors and circuits.  The 
surface profile is measured from machined metal samples using a stylus profilometer.  To 
ensure an accurate comparison of the contact models, the material properties and surface 
 40 
geometry are kept constant for all calculations.  For actual calculations, Matlab? is used 
for evaluating mathematical results. 
 Once the data is gathered, a Fast Fourier Transform is performed as mentioned 
before.  The result is then converted to amplitude via the complex conjugate.  Then 
multiscale models can be calculated as described above.  For the GW and JG models, the 
statistical parameters are acquired using McCool?s [26] methods by finding the spectral 
moments about the surface (see Appendix II).  From here, the GW model is fairly 
straightforward except for the integrals.  To solve these, numerical integration techniques 
are employed.  Simpson?s Method is used by first breaking the integral into 1000 sub-
intervals and performing the Simpson?s interpolation on each subinterval. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 41 
6.3.1 Calculated Real Area of Contact  
10?8 10?7 10?6 10?5 10?4 10?3 10?2
10?8
10?7
10?6
10?5
10?4
10?3
10?2
10?1
100
F / (A * E?)
A r
 / A
multi?scale perfectly elastic
multi?scale elastic?plastic
statistical perfectly elastic
statistical elastic?plastic
 
Figure 6.5: Non-dimensional area vs. load. 
 
 As seen in Fig. 6.5, higher loads result in a greater area of contact for the two 
surfaces.  The comparison of the two modeling techniques resulted in good qualitative 
agreement but poor quantitative agreement.  Greater contact area also results from the 
inclusion of plastic deformation.  This is caused by the behavior of the solid asperities to 
flow and flatten when plastic deformation is included.  The asperity material tends to 
?flow? resulting in greater deformation as well as ?filling in? the low spots around each 
asperity.  This combines with the higher loads to produce larger amounts of contact.  
These two techniques are calculated in very different manners using almost no common 
 42 
equations.  However, all the methods displayed here show the linear relation between real 
contact area and load.  Therefore, this qualitative agreement serves to confirm the 
accuracy of the trends of both methods for modeling the contact of rough surfaces.  
 
6.3.2 Surface Separation 
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
Ar/A
H/
?
multi?scale perfectly elastic
multi?scale elastic?plastic
statistical perfectly elastic
statistical elastic?plastic
stat. elastic adj.
stat elastic?plastic adj.
 
Figure 6.6: Area compared to Surface Separation including the adjusted separation for 
Statistical Contact Methods. 
 
 The surface separation as a function of the calculated real area of contact is shown 
in Fig. 6.4.  Although the overall trends of the two models are similar, the calculations for 
the statistical methods both show the surfaces in full contact before the surface separation 
 43 
has reached zero.  Logically, when contact is complete, the real area of contact should be 
at its maximum and surface separation should be zero.  To alleviate this possible error, 
Eq. (4.54) is used in the adjusted results seen above (bold lines in Figs. 6.6 and 6.7).  For 
the surface data considered in the current analysis a value of ?=5.2 was found to work 
well for the perfectly elastic case.  However, for elastic-plastic deformation, ?=10.02 
produced the appropriate results. These adjustments will be different for each case 
because the amount of compression and deformation will be different for the perfectly 
elastic and elastic-plastic cases. The resulting adjusted statistical model results are shown 
in Fig. 6.6.  
 As expected, greater loads as well as plastic deformation serve to decrease the gap 
between the surfaces as shown in Fig. 6.7.  This is logically correct since greater loads 
increase the area of contact through deforming the asperities.  One would expect the 
flattening of the asperities to allow the surfaces to come closer to each other and 
eventually contact completely when the calculated area is equal to the nominal area.  The 
adjustment given by Eq. (4.54) once again improves the agreement between the statistical 
and multiscale methods.   
 44 
10?6 10?4 10?2 100 102
0
1
2
3
4
5
6
7
F / (A * E?)
H / 
?
m?s perfectly elastic
m?s elastic?plastic
stat. perfectly elastic
stat. elastic?plastic
stat. elastic adj.
stat elastic?plastic adj.
 
Figure 6.7: Non-dimensional surface separation vs. load. 
 
 Perhaps surprisingly, the models predict similar results on the same order of 
magnitude.  Once again the overall trends of both the statistical and multiscale methods 
are confirmed by one another.  In this case though, the estimated behavior of the 
separation is fairly different.  Both methods show the decreasing gap with greater loads 
but they follow significantly different curves.   
 Without strong experimental confirmation, it is difficult to ascertain the accuracy 
of either model.  However, the current multiscale model will result in larger contact 
stiffness than predicted by the statistical models.  Stiffness is defined as the change in 
 45 
contact force per change in surface separation.  This appears to agree with experimental 
findings by Drinkwater et. al. [36] using acoustic methods. 
 
6.3.3 Electrical Contact Resistance 
10?8 10?6 10?4 10?2 100
10?8
10?7
10?6
10?5
10?4
10?3
10?2
10?1
100
Electrical Contact Resistance (
?)
F / (A * E?)
multi?scale perfectly elastic
multi?scale elastic?plastic
statistical perfectly elastic
statistical elastic?plastic
 
Figure 6.8: Electrical contact resistance as a function of non-dimensional load. 
 
 For the particular case of modeling the contact of an electrical connector, one 
concern was the amount of electrical resistance due to the ?bottleneck? effect of 
contacting asperities.  Fig. 6.8 shows the calculated results for this effect and compares 
the results for the multiscale and statistical models described in Chapter 4.  Since the 
electrical resistance is due to the gaps between the surfaces, it follows naturally that the 
 46 
electrical resistance decreases with load which as expected is inverse to the behavior of 
contact area.  Yet again, when compared to the perfectly elastic cases, the elastic plastic 
cases require a lower load for all resistance values which implies a greater contact area at 
each load.  The multiscale and statistical models also agree surprisingly well even though 
they make different predictions for contact area.  This suggests that it can be difficult to 
validate these models based on contact resistance measurements alone.  Interestingly, the 
ECR appears to decrease rapidly as complete contact is approached (as seen by the 
elbows at the end of each curve). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 47 
6.3.4 Thermal Contact Resistance 
10?8 10?6 10?4 10?2 100
10?4
10?2
100
102
104
106
Thermal Contact Resistance (K/W)
F / (A * E?)
multi?scale perfectly elastic
multi?scale elastic?plastic
statistical perfectly elastic
statistical elastic?plastic
bold represents scale?dependency
 
Figure 6.9: Thermal contact resistance as a function of non-dimensional load including 
scale-dependent results. 
 
 Thermal contact resistance is very similar to ECR because it is a measure of 
resistance to heat flow due to a reduced area of contact.  Figure 6.9 displays the model 
results concerning thermal contact resistance.  As can be seen, higher loads result in 
lower resistance values.  This is due to the increased amount of material in contact thus 
reducing the impedance to heat flow.  The values depicted are extremely similar to that of 
electrical resistance and this is to be expected given that they are calculated in similar 
manners and the flow of heat and flow of electricity are mathematically equivalent.   
 48 
 The bold lines shown represent the scale-dependent thermal contact resistance 
(see Eq. (4.51)).  Scale dependency results in only slightly higher resistance values and in 
some cases does not appear to have any affect at all.  Note that in the cases where scale 
dependency has no effect, the bold line in Fig. 6.9 is overlapping the non-scale dependent 
line preventing it from being seen.  The current work finds that the inclusion of scale 
dependant single asperity thermal contact resistance does not affect the predicted overall 
thermal contact resistance significantly for the sinusoidal based multiscale rough surface 
contact model.  This is also confirmed for the different spherical based multiscale contact 
model by Jackson, Bhavnani, and Ferguson [32]. 
 
6.4 Comparison between Multiple Surfaces 
 The next concern is what effect real multiscale roughness plays in the contact 
area, separation, and resistance for different surfaces.  To measure this effect, multiple 
surface profiles were obtained using a stylus profilometer on four surfaces with 
roughness varying from 0.24 ?m up to 5.82 ?m.  Fig. 6.10 shows that the smoother 
profiles are relatively flat where the roughest shows rather large changes in surface 
heights. 
 49 
0 50 100 150 200 250 300 350 400?1
?0.5
0
0.5
1
1.5 x 10
?5
Length of Scan (?m)
Profilometer Data (Angstroms)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.10: Surface profile data with different roughness values. 
 
Surface Roughness ? in Eq. 4.53 ? in Eq. 4.53 RMS Error of FFT to Fit 
1 0.24 ?m 0.13 1.60 0.3322 
2 0.34 ?m 0.085 1.5 0.4134 
3 1.05 ?m .037 1.4 0.4053 
4 5.82 ?m .006 1.0 0.0678 
 
Table 6.2:  Rough Surface Characteristics and Convergence Variables. 
 
 Table 6.2 provides a comparison between roughness and the coefficients of the 
power equation, Eq. (4.53), fit to the Fourier transform for each of the four surfaces 
 50 
examined in this work.  The root-mean-square error between the fit and FFT data is given 
as well.  Some of the error values are quite high due to the scattering of the FFT data 
which does not allow for a precise fit.  In application, this could result in large differences 
in the model predictions when a power equation is used versus the actual data.  In the 
current work the actual data is used in the multiscale model.  It is important to note that 
the values for ? vary seemingly independent of roughness whereas ? behaves inversely of 
roughness.  Therefore, the roughest surface has ?=1.0 which is at the limit of convergence 
as seen in Figs. 6.3-6.4.  Therefore there may be some very rough surfaces for which the 
multiscale technique will not converge.  This is not a concern for relatively smooth 
surfaces however. 
 
6.4.1 Calculated Real Area of Contact 
 The following section describes the effects of surface roughness on each of the 
four modeling techniques, sinusoidal based multiscale elastic-plastic and perfectly elastic 
as well as the statistical elastic-plastic (JG) and perfectly elastic (GW).  Each model is 
compared for four surfaces of greatly varying roughness.  The values of the roughness 
can be seen above in Table 6.2.  In this case, all results are non-dimensional.  The y-axis 
features the calculated real area of contact divided by the apparent or nominal area of 
contact.  The nominal area is the maximum possible area that can come into contact.  The 
x-axis is the range of normalized loads over which the calculations are made. The load is 
normalized by dividing by nominal area of contact and the elastic modulus (E`). 
 
 
 51 
 
 
10?7 10?6 10?5 10?4 10?3 10?2 10?1
10?5
10?4
10?3
10?2
10?1
100
F / (A * E?)
A r
 / A
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.11: Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method for elastic-
plastic material deformation. 
 
 
 
 
 
 
 52 
 
 
10?8 10?6 10?4 10?2 100
10?8
10?7
10?6
10?5
10?4
10?3
10?2
10?1
100
F / (A * E?)
A r
 / A
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.12: Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method for perfectly 
elastic material deformation. 
 
 
 
 
 
 
 53 
 
 
10?8 10?7 10?6 10?5 10?4 10?3
10?5
10?4
10?3
10?2
10?1
100
F / (A * E?)
A r
 / A
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.13: Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the JG statistical method for elastic-plastic material 
deformation. 
 54 
10?10 10?8 10?6 10?4 10?2
10?8
10?6
10?4
10?2
100
F / (A * E?)
A r
 / A
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.14: Real area of contact as a function of dimensionless load for surfaces of 
different roughness modeled using the GW statistical method for perfectly elastic 
material deformation. 
 
 As seen in Fig. 6.11-6.14, the real area of contact changes, but remains in the 
same orders of magnitude despite the different roughness values of the four profiles.  This 
results in the predicted real area of contact being extremely similar despite the change in 
roughness between surfaces.  However, as is expected, the graph does show that the 
results are ranked in order of decreasing roughness with smoother surfaces showing a 
greater contact area for each load level.  It is interesting to note that the inclusion of 
plasticity (Figs. 6.11 & 6.13) do not show as great a separation between each surface as 
 55 
the perfectly elastic models (Figs. 6.12 & 6.14).  Also, the models vary when viewing 
what force is required to reach full contact (Ar/A=1) although, in general, the elastic-
plastic models require less force to reach full contact than the perfectly elastic models for 
either sinusoidal based multiscale or statistical techniques.  
 
6.4.2 Surface Separation 
 The results seen in this section show the effects of surface roughness on surface 
separation.  As described previously, this is the difference between the mean heights of 
the two surfaces as they are pressed together.  The first four graphs, Figs. 6.15-6.18, give 
results for surface separation as compared to non-dimensional load.  For these four the 
units of the y-axis are for dimensionless surface separation by dividing the calculated 
separation with the standard deviation, ?.  The remaining graphs of this section, Figs. 
6.19-6.22, display the same dimensionless surface separation but as a function of 
dimensionless area (Ar/A).  It is important to note that the statistical modeling results 
(Figs. 6.17, 6.18, 6.21, & 6.22) are all calculated using the adjusted method suggested by 
Eq. (4.54).   
 
 
 
 
 
 
 
 56 
 
 
 
 
10?8 10?6 10?4 10?2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F / (A * E?)
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.15: Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method for elastic-
plastic material deformation. 
 
 
 
 
 57 
 
 
 
 
10?8 10?6 10?4 10?2 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F / (A * E?)
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.16: Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the sinusoidal based multiscale method for perfectly 
elastic material deformation. 
 
 
 
 
 58 
 
 
 
 
10?10 10?8 10?6 10?4 10?2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F / (A * E?)
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.17: Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the JG statistical method for elastic-plastic material 
deformation. 
 59 
10?8 10?6 10?4 10?2 100
0
0.5
1
1.5
2
2.5
3
F / (A * E?)
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.18: Surface Separation as a function of dimensionless load for surfaces of 
different roughness modeled using the GW statistical method for perfectly elastic 
material deformation. 
 
 Figs. 6.15-6.18 show some interesting results when considering different surface 
roughness values.  All of the models show the ranking behavior seen in the area 
calculations above but only for very high loads when the surfaces reach zero separation.  
Aside from this phenomenon, there appears to be very little definitive effect of roughness 
on separation since the separation models appear to be fairly random.  The qualitative 
trends are the same for all the modeling techniques and roughness but there seems to be 
little order due to roughness.  This is especially true for the multiscale model results 
 60 
shown in Figs. 6.15-6.16.  However, the statistical models, whose results are shown in 
Figs. 6.17-6.18, do display the ranking behavior with the smoother surfaces having a 
decreased surface separation except for the smoothest surfaces which behaves 
independently of the others including a unique slope.  When recalling the similarities 
among the area calculations this seems to be quite strange behavior.  Also note that each 
modeling technique requires a unique load range to reach full contact or zero separation. 
 
10?8 10?6 10?4 10?2 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ar / A
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.19: Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the sinusoidal based multiscale method for elastic-
plastic material deformation. 
 
 61 
 
 
 
 
10?8 10?6 10?4 10?2 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ar / A
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.20: Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the sinusoidal based multiscale method for perfectly 
elastic material deformation. 
 
 
 
 
 62 
 
 
 
 
10?8 10?6 10?4 10?2 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ar / A
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.21: Surface separation as a function of real area of contact for surfaces of 
different roughness modeled using the JG statistical method for elastic-plastic material 
deformation. 
 
 63 
10?8 10?6 10?4 10?2 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ar / A
H / 
?
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.22: Surface separation as a function of real area of contact for surfaces of different 
roughness modeled using the GW statistical method for perfectly elastic material 
deformation. 
 
Surface 1 2 3 4 
Roughness 0.24 ?m 0.34 ?m 1.05 ?m 5.82 ?m 
? for GW model 153.7 5.2 8.8 6.8 
? for JG model 159.9 10.02 14.2 12.8 
 
Table 6.3:  Rough Surface Characteristics and Adjusted Separation Values (Eq. 4.54). 
 
 64 
 Figures 6.19-6.22 also have no ranking behavior.  Recall that the statistical results 
(Figs. 6.17, 6.18, 6.21, & 6.22) all feature the adjusted surface separation technique with 
great success since all the cases show zero surface separation when the contact area has 
reached one which is equivalent to full contact for the non-dimensional techniques 
displayed in Figs. 6.21-6.22.  The values of ? in Eq. 4.54 are vastly different for each 
roughness with no apparent correlation between these values and the roughness itself as 
seen in Table 6.3.  The ?-values are all reasonable with the exception of the smoothest 
surface.  As mentioned previously, a presumably acceptable ?-value lies between 4 and 
12. If ? equals 12, then the adjustment seen in Eq. 4.54 relates to +6?.  Statistical analysis 
will show that this value will include 99.7% of the asperities.  Therefore, the values for 
the smoothest surface seem quite extreme with no obvious explanation.  Future work in 
this area will be necessary to explain this discrepancy.  
 
6.4.3 Electrical Contact Resistance 
 In this section, the results for the electrical contact resistance (ECR) are compared 
for the four different rough surfaces used in this work.  As before, the results are given 
with the units of Ohms (?) for the ECR along the y-axis and non-dimensional load along 
the x-axis. 
 
 
 
 
 
 65 
 
 
 
 
10?8 10?6 10?4 10?2
10?8
10?7
10?6
10?5
10?4
10?3
10?2
10?1
100
Electrical Contact Resistance (
?)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.23: Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based multiscale method for 
elastic-plastic material deformation. 
 
 
 
 
 66 
 
 
 
 
10?8 10?6 10?4 10?2 100
10?12
10?10
10?8
10?6
10?4
10?2
100
Electrical Contact Resistance (
?)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.24: Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based multiscale method for 
perfectly elastic material deformation. 
 
 
 
 
 67 
 
 
 
 
10?10 10?8 10?6 10?4
10?6
10?4
10?2
100
102
Electrical Contact Resistance (
?)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.25: Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the JG statistical method for elastic-plastic 
material deformation. 
 
 
 68 
10?8 10?6 10?4 10?2 100
10?8
10?6
10?4
10?2
100
Electrical Contact Resistance (
?)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.26: Electrical contact resistance (ECR) as a function of dimensionless load for 
surfaces of different roughness modeled using the GW statistical method for elastic-
plastic material deformation. 
 
 In accordance with the real area of contact, electrical contact resistance is very 
similar for the four surfaces and is also ranked according to roughness with the smoothest 
surface having the least resistance values, seen in Figs. 6.23-6.26.  The different 
modeling techniques have already been established to show qualitatively similar trends 
for a common surface.  However, the behavior of the four different models for the various 
roughnesses is strikingly dissimilar.  The overall decreasing trend with a sudden drop is 
common, but the multiscale models (Figs. 6.23-6.24) react to each roughness with a 
 69 
nearly equidistant gap.  In comparison, the two statistical models (Figs. 6.25-6.26) show 
no gap for all but high loads for the smoothest surfaces but rather large gaps for the 
higher roughness values.  There is also a strange shoulder in the multiscale perfectly 
elastic solution (Fig. 6.24) which is not present in the other models.  However, the 
shoulder seems to be at nearly the same location when the multiscale elastic-plastic 
model (Fig. 6.23) shows the sudden drop in resistance for the very high loads.  This may 
be due to the discrete scales flattening out at higher loads.  Since the perfectly elastic 
contact cannot behave in this manner the next iteration requires the surface to return to 
nearly linear behavior till much higher loads. 
 
6.4.4 Thermal Contact Resistance 
 This section will display the results for the thermal contact resistance calculations 
when considering varied surface roughness.  This comparison is done for each model in 
order to show further distinction among their unique mathematical methods.  It is 
important to note that the following results are for thermal contact resistance without the 
scale-dependent surface characteristics included.  It is seen in the earlier thermal 
conductivity comparison that scale-dependency has negligible effects on most models, 
loads, and deformation characteristics and has therefore been omitted from these 
comparisons.  The units used for these comparisons are Kelvin meters squared per Watt 
along the y-axis (TCR) and non-dimensional load along the x-axis. 
 
 
 
 70 
 
 
 
 
10?8 10?6 10?4 10?2
10?3
10?2
10?1
100
101
102
103
104
105
Thermal Contact Resistance (m
2  * K/W)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
 
Figure 6.27: Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based multiscale method for 
elastic-plastic material deformation. 
 
 
 
 
 71 
 
 
 
 
10?8 10?6 10?4 10?2 100
10?6
10?4
10?2
100
102
104
106
Thermal Contact Resistance (m
2  * K/W)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.28: Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the sinusoidal based multiscale method for 
perfectly elastic material deformation. 
 
 
 
 
 72 
 
 
 
 
10?9 10?8 10?7 10?6 10?5 10?4 10?3
10?2
10?1
100
101
102
103
104
105
106
Thermal Contact Resistance (m
2  * K/W)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.29: Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the JG statistical method for elastic-plastic 
material deformation. 
 
 
 73 
10?8 10?6 10?4 10?2 100
10?4
10?2
100
102
104
106
Thermal Contact Resistance (m
2  * K/W)
F / (A * E?)
Surface 1 Rq=0.24 ?m
Surface 2 Rq=0.34 ?m
Surface 3 Rq=1.05 ?m
Surface 4 Rq=5.82 ?m
Fi
gure 6.30: Thermal contact resistance (TCR) as a function of dimensionless load for 
surfaces of different roughness modeled using the GW statistical method for perfectly 
elastic material deformation. 
 
 Figures 6.27-6.30 display the results of thermal contact resistance (TCR) for each 
different model with the intent of comparing how each model reacts to a change of 
surface roughness.  At first glance, all the models appear to give the same or at least 
extremely similar results.  Indeed, all the models show a steady, nearly linear, decreasing 
trend for the majority of loads with a sudden drop for the higher loads.  However, upon 
closer inspection, TCR for the elastic-plastic models, Figs. 6.27 & 6.29, terminates at 
significantly lower loads than the perfectly elastic models, Figs. 6.28 & 6.30.  Also, the 
 74 
TCR models show shoulders for only the multiscale perfectly elastic case very near the 
values at which the multiscale elastic-plastic model drops suddenly for the higher loads.  
This correlates well with the electrical contact resistance since the calculations for both 
effects are very similar.  The same ranking behavior is seen here as in the previous 
calculations but the multiscale models show an almost uniform change from one 
roughness to another.  Specifically, the multiscale elastic-plastic model, Fig. 6.27, shows, 
for the most part, equal but small gaps between the four surfaces.  The statistical models 
seem to be much more sensitive to roughness changes since these models show almost no 
gap between the two very smooth surfaces (surfaces 1 & 2, 0.24 and 0.34 ?m roughness 
respectively) with a considerably larger jump in the between the remaining surfaces.  
Note that the results for ECR and TCR are all quantitatively similar for both models and 
all surface roughnesses. 
 
 
 
75 
CHAPTER 7 
CONCLUSIONS 
 The results from a multiscale model based on stacked sinusoidal surfaces have 
shown to be qualitatively similar in comparison with existing statistical contact models.  
When viewing surface separation as a function of dimensionless load, it seems that the 
multiscale models offer a differing description of how the surface behaves.  At high 
loads, the multiscale methods predict no separation between the surfaces which correlates 
exactly with the area of contact equaling the apparent area of contact (complete contact).  
However, even though the statistical methods show a similar trend as the maximum area 
is reached, there appears to still be some separation between the two surfaces.  This is 
most likely a result of the statistical methods being designed more for lightly loaded 
contacts and ignoring the change in overall peak to valley height between asperities at 
higher loads.  The adjusted statistical model separation calculation offered in this work 
takes this effect into account and does show zero separation at the maximum contact area.  
Electrical contact resistance predictions seem reasonable based on the similarity between 
statistical and multiscale methods.  Actually, the statistical and multiscale models predict 
very similar values, while the predicted contact areas are not as similar.  This suggests 
that using contact resistance measurements may not be an effective way of evaluating 
rough surface contact models.  
76 
 In response to concerns about the convergence of the multiscale techniques, this 
work relates a power fit to the FFT data which reveals that the sinusoidal multiscale 
technique will converge as long as the average pressure (proportional to amplitude) stays 
constant or decreases as the wavelength decreases.  This situation requires that the 
exponent in the power fit remain 1 or greater for the multiscale sinusoidal technique to 
converge.    
Finally, the multiscale sinusoidal method is used to generate results for a variety 
of real surfaces shows the overall expected trends for area, electrical contact resistance 
and surface separation.  As is expected, the results for the surfaces are ranked according 
to roughness yet produce extremely similar results.  Upon first inspection, it appears that 
surface separation does not match the ranking behavior for the various surfaces.  
However, at greater loads, lower roughness values do decrease surface separation. 
 
77 
 
 
 
BIBLIOGRAPHY 
 
1. Greenwood, J.A. and J.B.P. Williamson, Contact of Nominally Flat Surfaces. 
Proc. R. Soc. Lond. A, 1966(295): p. 300-319. 
 
2. Chang, W.R., I. Etsion, and D.B. Bogy, An Elastic-Plastic Model for the Contact 
of Rough Surfaces. ASME J. Tribol., 1987. 109(2): p. 257-263. 
 
3. Jackson, R.L. and I. Green, A Statistical Model of Elasto-plastic Asperity Contact 
between Rough Surfaces. Trib. Int., 2006. 39(9): p. 906-914. 
 
4. Kogut, L. and I. Etsion, A Finite Element Based Elastic-Plastic Model for the 
Contact of Rough Surfaces. Tribology Transactions, 2003. 46(3): p. 383-390. 
 
5. Kogut, L., and Jackson, R. L., A Comparison of Contact Modeling Utilizing 
Statistical and Fractal Approaches. ASME J. Tribol., 2005. 128(1): p. 213-217. 
 
6. Majumdar, A. and B. Bhushan, Fractal Model of Elastic-plastic Contact Between 
Rough Surfaces. ASME J. of Tribol., 1991. 113(1): p. 1-11. 
 
7. Wang, S., J. Shen, and W.K. Chen. Determination of the Fractal Scaling 
Parameter from Simulated Fractal Regular Surface Profiles Based on the 
Weierstrass-Mandelbrot Function (IJTC2006-12068). in STLE/ASME Int. Joint 
Trib. Conf. 2006. San Antonio, TX. 
 
8. Yan, W. and K. Komvopoulos, Contact Analysis of Elastic-Plastic Fractal 
Surfaces. Journal of Applied Physics, 1998. 84(7): p. 3617-3624. 
 
9. Bora, C.K., et al., Multiscale Roughness and Modeling of MEMS Interfaces. 
Tribology Letters, 2005. 19(1): p. 37-48. 
 
10. Ciavarella, M., et al., Linear Elastic Contact of the Weierstrass Profile. Proc. R. 
Soc. Lond. A, 2000. 1994(456): p. 387-405. 
 
11. Jackson, R.L. and J.L. Streator, A Multiscale Model for Contact between Rough 
Surfaces. Wear, 2006. 261(11-12): p. 1337-1347. 
 
12. Fleck, N.A. and J.W. Hutchinson, A Phenomenological Theory for Strain 
Gradient Effects in Plasticity. J. Mech. Phys. Solids, 1993(41): p. 1825-1857. 
 78 
13. Wei, Y. and J.W. Hutchinson, Hardness Trends in Micron Scale Indentation. J. 
Mech. Phys. Solids, 2003(51): p. 2037-2056. 
 
14. Fleck, N.A., et al., Strain Gradient Plasticity: Theory and Experiment. Acta 
Metall. Mater., 1994. 42(2): p. 475-487. 
 
15. Gao, H., et al., Mechanism-Based Strain Gradient Plasticity-I. Theory. J. Mech. 
Phys. Solids, 1999(47): p. 1239-1263. 
 
16. Gao, H., et al., Mechanism-Based Strain Gradient Plasticity- II. Analysis. J. 
Mech. Phys. Solids, 2000(48): p. 99-128. 
 
17. Johnson, K.L., Contact Mechanics. 1985, Cambridge, U.K.: Cambridge 
University Press. 452. 
 
18. Jackson, R.L. and I. Green, A Statistical Model of Elasto-Plastic Asperity Contact 
Between Rough Surfaces. Tribology Int., 2006(39): p. 906-914. 
 
19. Chang, W.R., I. Etsion, and D.B. Bogy, An elastic-plastic model for the contact of 
rough surfaces. ASME J. Tribol., 1987. 109: p. 257-63. 
 
20. Kogut, L. and I. Etsion, Elastic-plastic contact anaysis of a sphere and a rigid 
flat. J. App. Mech. Trans. ASME, 2002. 69(5): p. 657-62. 
 
21. Majumdar, A. and B. Bushnan, Fractal Model of Elastic-Plastic Contact Between 
Rough Surfaces. ASME J. Tribol., 1991. 113(1): p. 1-11. 
 
22. Archard, J.F., Elastic Deformation and the Laws of Friction. Proc. R. Soc. Lond. 
A, 1957(243): p. 190-205. 
 
23. Johnson, K.L., J.A. Greenwood, and J.G. Higginson, The Contact of Elastic 
Regular Wavy Surfaces. Int. J. Mech. Sci., 1985. 27(6): p. 383-396. 
 
24. Krithivasan, V. and R.L. Jackson, An Analysis of Three-Dimensional Elasto-
Plastic Sinusoidal Contact. Tribology Letters, 2007. 27(1): p. 31-43. 
 
25. Jackson, R.L., V. Krithivasan, and W.E. Wilson, The Pressure to Cause Complete 
Contact between Elastic Plastic Sinusoidal Surfaces. IMechE J. of Eng. Trib. - 
Part J. In Print. 
 
26. McCool, J.I., Comparison of Models for the Contact of Rough Surfaces. Wear, 
1986(107): p. 37-60. 
 
27. Front, I., The Effects of Closing Force and Surface Roughness on Leakage in 
Radial Face Seals, in Israel Institute of Technology. 1990, Technion. 
 79 
28. Jackson, R.L. and I. Green, A Finite Element Study of Elasto-Plastic 
Hemispherical Contact. ASME J. Tribol., 2005. 127(2): p. 343-354. 
 
29. Holm, R., Electric Contacts. 4th ed. 1967, New York: Springer Verlag. 21. 
 
30. Madhusudana, C.V., Thermal Contact Conductance. 1996, New York: Springer-
Verlag. 
 
31. Cooper, M.G., B.B. Mikic, and M.M. Yovanovich, Thermal Contact Conductance. 
Int. J. Heat Mass Transfer, 1969(12): p. 279-300. 
 
32. Jackson, R.L., S.H. Bhavnani, and T.P. Ferguson, A Multi-scale Model of Thermal 
Contact Resistance between Rough Surfaces. ASME J. Heat Transfer, 2007. 
 
33. Kogut, L. and I. Etsion, Electrical Conductivity and Friction Fore Estimation in 
Compliant Electrical Connectors. STLE Tribol. Trans., 2000. 43(4): p. 816-822. 
 
34. Prasher, R.S. and P.E. Phelan, Microscopic and Macroscopic Thermal Contact 
Resistances of Pressed Mechanical Contacts. J. App. Phys., 2006(100). 
 
35. Ciavarella, M., et al., Linear elastic contact of the Weierstrass Profile. Proc. R. 
Soc. Lond. A, 2000(456): p. 387-405. 
 
36. Drinkwater, B.W., R.S. Dwyer-Joyce, and P. Cawley. A Study of the Interaction 
between Ultrasound and a Partially Contacting Solid--Solid Interface. in 
Mathematical, Physical and Engineering Sciences. 1996: The Royal Society.