A DETERMINISTIC-STATISTICAL MODEL FOR TRIBO-CONTACTS …

The Pennsylvania State University

The Graduate School

College of Engineering

A DETERMINISTIC-STATISTICAL MODEL FOR TRIBO-CONTACTS

IN BOUNDARY LUBRICATION

WITH LUBRICANTSURFACE PHYSICOCHEMISTRY

A Thesis in

Mechanical Engineering

by

Huan Zhang

copy 2004 Huan Zhang

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

May 2005

The thesis of Huan Zhang was reviewed and approved by the following

Liming Chang Professor of Mechanical Engineering Thesis Advisor Chair of Committee

Marc Carpino Professor of Mechanical Engineering

Seong H Kim Assistant Professor of Chemical Engineering Richard C Benson Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering

Signatures are on file in the Graduate School

iii

ABSTRACT

The boundary-lubricated surface contact is truly an interdisciplinary process

involving deformation heat transfer physicochemical interaction and random-process

probability The objective of this thesis is to develop a surface contact model as a

theoretical platform upon which to carry out the boundary lubrication research with a

balanced consideration of all the four key aspects of the contact process The modeling

consists of three successive steps ndash (1) elastoplastic finite element analysis of frictional

asperity contacts (2) modeling of contact systems with friction and (3) modeling of a

boundary lubrication process

Finite element analysis of frictional asperity contacts ndash A finite element model is

developed and systematic numerical analyses carried out to study the effects of friction

on the deformation behavior of individual asperity contacts The study reveals some

insights into the modes of asperity deformation and asperity contact variables as

functions of friction in the contact The results provide guidance to analytical modeling of

frictional asperity contacts and lay a foundation for subsequent work on system contact

modeling

Modeling of contact systems with friction ndash Analytical equations are developed

relating asperity-contact variables to friction using contact-mechanics theories in

conjunction with the finite element results A system-level model is then derived from the

statistical integration of the asperity-level equations The model is a significant

advancement of the Greenwood-Williamson types of system models by incorporating

iv

contact friction It also serves as the platform in the final step of model development for

the boundary lubrication problem

Modeling of a boundary lubrication process ndash On the basis of the above

mechanical modeling an asperity-based model is developed for the boundary-lubricated

contact by incorporating other key aspects involved in the process Four variables are

used to describe an asperity contact under boundary lubrication conditions including

micro-contact area friction force load carrying capacity and flash temperature In

addition three probability variables are used to define the interfacial state of an asperity

junction that may be covered by various types of boundary films Governing equations

for the seven key asperity-level variables are derived based on first-principle

considerations of asperity deformation frictional heating and formationremoval of

boundary lubricating films These coupled asperity-level equations some of which are

nonlinear are solved iteratively and the solution is then statistically integrated to

formulate the contact model for boundary lubrication systems

The results obtained from the model suggest that it may provide a framework for

future investigation of the boundary lubrication process by integrating research advances

in contact mechanics tribochemistry and other related fields

v

TABLE OF CONTENTS

List of Figures vii

List of Tables ix

Nomenclaturex

Acknowledgementsxii

Chapter 1 Introduction 1

11 Boundary Lubrication and Boundary-Lubricated Contact 1 12 Important Aspects of Boundary-Lubricated Contact Literature Review 4

121 Mechanisms and Efficiency of Boundary Lubrication4 122 Contact Modeling Unlubricated Surfaces 11 123 Contact Modeling Boundary-Lubricated Surfaces14 124 Flash Temperature 16 125 Summary18

13 Research Objective Approach and Outline 18

Chapter 2 Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts22

21 Introduction 22 22 The Model Problem24 23 Results and Analysis27

231 Mode of Asperity Deformation 27 232 Shape of the Plastic Zone 30 233 Contact Size Pressure and Load Capacity 33

24 Summary37

Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction 48

31 Introduction 48 32 Modeling51

321 Model Structure 51 322 Asperity Contact Pressure 53 323 Asperity Area of Contact55 324 Critical Normal Approaches60 325 System Variables 65

33 Result Analysis68

vi

34 Summary76

Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86


421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100

43 Result Analysis104 44 Summary113

Chapter 5 Summary and Future Perspective121

51 The Deterministic-Statistical Model121 52 Perspective on Future Development123

Bibliography 126

vii

List of Figures

Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches

(a) linear scale (b) logarithmic scale 40

Figure 24 Plastic zones of the frictionless contact

(a) elastic-plastic transition (b) onset of full plasticity 41

Figure 25 Plastic zones of the contact with micro = 02






Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under

combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )

80

Figure 34 Dimensionless first critical normal approach 2D finite element

results against 3D theoretical analysis 81

Figure 35 Dimensionless second critical normal approach finite element results

and curve-fitting 81

Figure 36 Surface mean separation as a function of load and friction coefficient 82

viii

Figure 37 Asperity height distribution and mode of deformation of contacting

asperities 83

Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area

of contact 84

Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height

(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118

Figure 45 System-level contact and lubrication variables as functions of load

(a) degree of boundary protection (b) surface separation (c) real area of contact

119

Figure 46 State of boundary lubrication in the operating parameter space

(a) system-level friction coefficient (b) system boundary-lubrication protection

120

ix

List of Tables

Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85

x

Nomenclature

lA = area of asperity contact

nA = nominal contact area

tA = real area of contact

1E 2E = elastic modulus

lowastE = equivalent elastic modulus 1

2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν

tF = total friction force H = indentation hardness

aH∆ = lubricantsurface adsorption heat

rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct

AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact

mFP = asperity contact pressure at the onset of plastic flow

mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature

cR = molar gas constant (831451 ( )KmolJ sdot )

aS = probability of an asperity contact being covered by an adsorbed film

aS prime = survivability of the adsorbed layer in an asperity contact

atS prime = survivability of the adsorbed layer at the system level

nS = probability of an asperity contact with no boundary protection

ntS = probability of contact with no boundary protection at the system level

rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level

bT = bulk temperature

lT = contact temperature of an the asperity junction

1T∆ = asperity flash temperature V = sliding velocity

tW = total contact load a = radius of an asperity contact

0b = adsorption coefficient

123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk

c = substrate specific heat

xi

d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height

h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=

bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight

ct = duration of an asperity contact

ft = time to the break of the substratereacted film bonding z = asperity height

sz = distance between the mean of asperity heights and that of surface heights

α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact

1δ = first critical normal approach 2δ = second critical normal approach

η = area density of asperities κ = substrate thermal diffusivity

lmicro = local friction coefficient

tmicro = system friction coefficient

21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights

aσ = standard deviation of asperity heights

eσ = effective stress

aτ = shear strength of the adsorbed layer

mτ = average shear stress of an asperity contact

nτ = shear strength of the substrate material

rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )

xii

Acknowledgements

The completion of the thesis brings me to the end of my student life I would like

to take this opportunity to express my appreciation to all those who helped and supported

me during my journey of learning Without their guidance help and patience I would not

be able to go this far

First and foremost I am very grateful to my thesis advisor Prof Liming Chang

for introducing me to the exciting and challenging project for his continuous guidance

and encouragement from the day I met him more than five years ago Since then he has

inspired me in my research with his interest dedication and enthusiasm for this study At

each stage of the research I have benefited tremendously from his academic expertise

professional rigor and solid grasp of the big picture I especially appreciate the time and

effort he put into reading and commenting many drafts of the thesis as it was taking

shape I want to also thank him for his knowledgeable advice and constructive criticism

on every aspect of academic life which broadened my perspective improved my research

skills and prepared me for future challenges

I would like to thank other members of my thesis committee Professor Richard

Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable

suggestions during the course of my research and generously sharing with me their deep

understanding of this topic I want to express my sincere thanks to Dr Martin Webster

and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent

support and insightful comments

xiii

My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze

University for his encouragement advice and fruitful discussions during his stay here at

the Penn State University and when he is back in China Many thanks are also due to my

fellow students and research associates and all other friends at State College who have

offered immediate and continuous support throughout the past five years

I wish to acknowledge ExxonMobil Technology Company for the financial

support of the research project I also would like to thank Prof Stefan Thynell Professor-

in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in

my abilities and selecting me as a Graduate Teaching Fellow during the last semester of

my PhD This program has taught me many things which I cannot learn from any other

experience

I am indebted to my parents brother and sister for their enduring love and

support to my daughter for not spending as much time as I should and to my dear wife

Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally

I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two

years ago

Chapter 1

Introduction

11 Boundary Lubrication and Boundary-Lubricated Contact

Boundary lubrication provides the basic protection to the bearing surfaces of

machine components which operate at high load low speed or high temperature such as

o Geartooth camtappet and piston-ringliner contacts

o Rolling element bearing at the pure sliding sites

o Journal bearings during the periods of start-up and shutdown

The effectiveness of boundary lubrication is critical to the service life of these

components In addition boundary lubrication also plays an important role in the

following devices or operations

o MEMS [1] and headdisk interface [2]

o CMP and the metal cutting and formation operations [3]

o Natural and artificial joints such as those in the hip and in the knee after periods

of inactivity such as sleeping [4]

Therefore knowledge of the surface contact behavior in boundary lubrication is essential

to improve the performance of the above systems and procedures addressing the

efficiency safety environment and other concerns For example such knowledge is

invaluable in developing the strategies for controlling tribo-failure and minimizing wear

2

and in designing the environmentally benign lubricants and additives The objective of

the current research is to enhance the understanding in the area by developing a

theoretical model for the boundary-lubricated sliding contact of two rough surfaces

Figure 11 Boundary lubricated contacts of two rough surfaces

The nominally flat bearing surfaces usually deviate from their prescribed

geometry with microscopic irregularities Under boundary lubrication conditions two

rubbing surfaces make frequent and random micro-contacts at their high spots or the

asperities (as shown in Fig 11) The load applied to the system is then mainly carried by

the discrete asperity contacts and the total friction force is also the integration of local

tangential resistance During each asperity contact a series of micro-scale processes of

different nature proceed simultaneously and interact with each other in a number of ways

The direct mechanical response of two contacting asperities is their elastic or inelastic

deformation which results in the asperity load support This response is accompanied by a

group of physical and chemical reactions among the substrate additives lubricants and

environment leading to the formation of low shear-modulus films in the contact junction

These films protect asperities from direct contact and effective lubrication is thus

achieved The protective boundary films may be ruptured and then the asperity contact

takes place directly between the opposite metallic substrates The local friction resistance

may thus come from the shearing within the boundary films andor that occurring at the

3

metallic surfaces The shear stress along with the sliding velocity generates frictional

heating in micro contact regions As a result high local temperatures of short duration or

so-called flash temperatures may be aroused The frictional heating process may

facilitate the formation of the boundary lubricating films or deteriorate them by

dissociation desorption or oxidation The state of these films or their integrity also

depends on the levels of contact pressure and shear stress This state in turn largely

determines the shear stress and thus affects other micro-contact variables In summary

the system-level tribological behavior under boundary lubrication conditions is

collectively governed by multiple interactive asperity-level processes

On the other hand the micro-contact processes may also be affected by the

evolution of system features For example in the course of an asperity-to-asperity contact

the asperity temperature is composed of two components the flash temperature and the

bulk temperature The latter is largely system specific and governed by the overall heat

generation and transfer In addition the geometrical characteristics of the rubbing

surfaces may experience continuous progression resulting in dynamically changing

conditions at each asperity contact

The above discussion indicates that the boundary lubrication processes exhibits

diversity in their natures and scales The corresponding contact modeling is therefore a

truly interdisciplinary subject The model should be developed based on the knowledge

of the mechanisms of boundary films the contact of rough surfaces and the flash

temperatures of asperity contacts Significant advances have been made in these areas

and the current understanding of each is summarized below from the modeling viewpoint

to establish the theoretical framework and methodological focus for this thesis research

4

12 Important Aspects of Boundary-Lubricated Contact Literature

Review

121 Mechanisms and Efficiency of Boundary Lubrication

In boundary lubrication two different types of protective films may be formed in

an asperity junction to prevent the surface damage during sliding A layer of organic

compounds with polar end groups may be adsorbed on the surface Meanwhile an

inorganic film may be produced by the chemical reaction between the substrate and the

additives or lubricants These boundary films usually reduce friction and increase the

resistance of the system to surface failure such as seizure For example the formation of

Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel

system by a factor of 3-8 [5] The system performance is thus largely controlled by the

properties of the two types of boundary lubricating films including their composition

structure effectiveness and shearing behavior The generally accepted ideas about these

important issues and the recent developments are briefly reviewed below for the adsorbed

layer and the reacted film in sequence

A conceptual model has been proposed to explain the mechanism of boundary

lubrication by the adsorption [6] According to this model the polar ends of organic

lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon

chains projected vertically upward The molecular layers adsorbed on the opposite

surfaces are only weakly interacted The sliding of the two surfaces is then accomplished

between the adsorbed layers resulting in a low interfacial friction Therefore the

measured friction coefficient has often been used to characterize the relative lubrication

5

effectiveness of the adsorbed layers for various combinations of base lubricants polar

additives and surfaces It has been found that the effectiveness depends on the chain

length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type

of polar groups [12 13]

The adsorbed layer is generally effective up to a critical interfacial temperature

[14-16] It is because high temperature corresponds to strong thermal desorption leading

to a reduced fraction of surface that is covered by the adsorbed molecules The fractional

surfactant surface coverage θ or defect θminus1 has often been related to the interfacial

temperature and the free energy of adsorption of the additive or lubricant to the surface

The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]

which assumes that the surface is energetically homogeneous and there is very small or

zero net lateral interaction between adsorbate molecules The applicability of the

Langmuir isotherm in boundary lubrication studies has been verified experimentally for

different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm

may be more suitable in the case of heterogeneous surfaces and strong lateral interaction

within the adsorbed layer [11 13] Another model is proposed to determine the fractional

coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]

In addition to the interfacial temperature and adsorption energy this model also accounts

for the effect of sliding velocity

Assuming that the adsorbed layer is the only boundary lubricating film direct

metallic contact may occur as a result of the partial failure of this layer The interfacial

friction may then arise from both the shearing of the layer and the metallic contact The

6

overall friction force can thus be related to the fractional surfactant surface coverage and

the relation is given by [21]

( )[ ]mbrAF τθθτ minus+= 1 (11)

where rA is the real area of contact bτ the shear strength of the boundary lubricating

film and mτ that of the substrate material By assuming that the surfaces are fully

covered by the adsorbate the shear strength bτ may be determined on the basis of the

measured frictional force and the knowledge of the real area of contact rA However this

is difficult in real engineering situations due to the uncertainty involved in the estimation

of rA and the possible desorption during the contact In order to overcome this difficulty

a feasible approach is to deposit monolayers or multilayers of organic films on very

smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere

against a plane For these types of contact configuration the area of contact could be

calculated using the well-known Hertzian solution and the calculation may be verified

experimentally for example by multiple-beam interferometry This approach was first

used to study the shearing behavior of calcium stearate monolayers deposited on

atomically smooth mica sheets [22] and then extended to a variety of other organic films

[23-26] The results of these studies show that the film shear strength is dependent on the

contact pressure and may be expressed in the following form [27]

sum+=j

njb

jPmicroττ 0 (12)

where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small

compared to other terms The coefficients and exponents of the series in this expression

7

characterize the mechanical or rheological properties of the boundary lubricating films In

addition to the experimental studies a theoretical model has been proposed relating the

friction of two adsorbed layers on the opposite surfaces to the energy barrier between two

adjacent equilibrium positions [28] Without considering the dislocations and energy

conservation the predictions from this theory are much higher than the experimental

results

Compared to the adsorbed layers the reacted films in boundary lubrication

systems are much more complex in terms of the formation composition structure

effectiveness and mechanical properties Typically the reacted films are generated from

the chemical reaction between the metal surface and the additive with one active element

such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation

process starts with the chemisorption of the additive on the metal surface This is

followed by the decomposition of the additive molecules leaving the active element

chemically bonded to the surface A thin film of metal salts is then formed and it may be

mixed with oxides in the presence of moisture or in air atmosphere Further growth of the

film involves the diffusion of the active elements and metallic ions Such a formation

process is similar to that of the oxide layer on the surface The growth of the film

thickness may follow a linear law initially and a parabolic law afterwards and may thus

be described by the following equation [31]

n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8

where An is the Arrhenius constant and Qn the activation energy of reaction These two

parameters are closely related to the type of metallic salt which strongly depends on the

availability of the active elements and the temperature at the interface On the other hand

the reacted films may also be formed by a multifunctional additive containing two or

more active elements The most widely used multifunctional additives are the alkyl and

aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating

film of a multilayer structure Starting from the substrate this type of film composes of

an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor

long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-

phosphate The transition between the two adjacent layers is gradual The portion of each

layer within the film depends not only on the properties of the lubricant additive and

substrate material but also the severity of the sliding contact More detailed information

can be found in [30] and [32-34] on the structure and composition of the ZDTP films and

the mechanism of action at the molecular level In addition the reacted films may include

a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick

layer of high-molecular weight organometallic compounds by the polymerization of

additive-free oil minerals [37 38]

The diversity of the reacted films formed in the boundary lubricated contact

suggests that they may work by different mechanisms depending on their form structure

and properties A very thin film of metal salts or oxides may act as a sacrificial layer of

low shear strength It is easily removed by the shear or cavitational forces along with the

friction heating but is able to be reformed immediately to sustain continuous sliding A

prime example is the boundary film formed from the extreme pressure additives [39] The

9

high-molecular polymeric film generated from base oil molecules may also work on the

basis of repeated removal and repair [40] In contrast the metal salt-films derived from

the antiwear additives are relatively thicker and usually much more tenacious They are

not easily removable during the sliding and the wear is thus controlled As for the

multilayer film resulting from ZDTP each layer has different properties and functions

[41] The metal salts such as FeS has sufficiently high shear strength and serves as an

adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has

high viscosity and its hardness is comparable to the mean contact pressure It can flow

plastically and may thus act as a protective layer against wear by eliminating the abrasive

contribution of oxides The outermost organic layer is mobile and has varying viscosity

similar to the base oil ensuring that the shear plane is located within the boundary

lubricating film This layer also serves as a reservoir for the regeneration of

polyphosphates

The reacted films described above may fail to provide effective protection to the

surfaces when the films are removed during the contact The failure process is strongly

affected by the level of interfacial shear stress frictional heating [29 42] and contact

pressure and plastic deformation [43 44] A number of models have been proposed to

explain the film-failure in terms of the friction-induced temperature rise andor the

mechanical stresses Accordingly a group of criteria has been defined The failure has

often been attributed to the imbalance between the formation and the removal of the

reacted films Based on this hypothesis a critical temperature condition has then been

determined In one of such studies [45] both the formation and removal rates have been

measured and modeled as a function of interfacial temperature using the Arrhenius-type

10

expression in the form of Eq (13) The failure occurs above a critical temperature when

the removal rate is greater than the formation rate For the system running at low speeds

the effects of frictional heating or interfacial temperature are negligible The reacted films

fail when the maximum interfacial stress exceeds the film or substrate shear strength and

a stress criterion has thus been defined [46 47] The film failure has also been viewed as

the result of the destruction of the chemical bonds between the active elements of

additive molecules and the metal surface [48 49] From the energy transfer point of view

these mechanically stressed bonds can be broken by the combined action of the thermal

energy from frictional heating and the distortion energy due to shearing According to the

thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents

their resistance to the destruction and may thus be used to characterize the film-failure

The three types of models described above are deterministic but the information about

many of their input parameters is incomplete and the failure process itself also involves a

certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate

to assess the likelihood of failure of the reacted films This likelihood may be expressed

as a probability similar to the fractional defect of the adsorbed layer The probability may

also be used to model the interfacial friction in combination with the knowledge of the

film shearing properties

In addition to the formation structure and effectiveness of the reacted films their

shearing behavior and other mechanical properties are also the key to understanding the

mechanism of boundary lubrication These aspects have thus been studied by many

researchers for the reacted films formed during tribological testing using conventional

tribometers and innovative scanning probe techniques With a ball-on-flat configuration

11

Tonck et al [51] measured the tangential stiffness by a microslip method for four types of

tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier

respectively The elastic shear moduli of these films were also determined and were

found similar to those of high molecular weight polymers such as polystyrene In

addition the results showed that the values of shear modulus would increase with the

load except in the case of the friction modifier More recently nanoindentation has been

widely used to measure the mechanical properties of the reacted films generated from a

variety of lubricant additives [52-55] It was observed that the film hardness and elastic

modulus would increase with depth up to a few nanometers beneath the surface

Correspondingly the resistive forces within the films might increase during the loading

stage of the indentation to accommodate the increasing applied pressure On the other

hand the lateral force microscopy has been used in combination with the atomic force

microscopy to examine the frictional properties of the tribo-films formed in reciprocating

Amsler tests [56 57] A linear relationship was revealed between the load and the friction

force measured for micro regions of the tribo-films This may be explained by the

distribution of the hardness and modulus in depth observed in the nanoindentation tests

Therefore the shearing behavior of the reacted films may also be described by Eq (12)

in its linear form Furthermore the friction coefficient of the micro regions was found in

good agreement with the macro results The overall friction coefficient is thus indeed

determined by the shearing of the reacted films covering the asperities

122 Contact Modeling Unlubricated Surfaces

For two nominally flat surfaces without lubrication their contact takes place at

distributed asperity junctions The contact models predict the mechanical responses of

12

surfaces to the applied loading These responses including the size and spatial

distribution of asperity contact spots and the surface and subsurface stress fields around

them are dependent on the topography of surfaces and their material properties

Two major approaches have been used to model the contact of rough surfaces

stochastic and deterministic The stochastic contact models can be further classified into

two groups statistical and fractal These approaches or models are distinguished by the

use of surface descriptions The basic features of different approaches are briefly

summarized below A more comprehensive review including the discussion on their

advantages and disadvantages can be found in ref [58]

The statistical approach was first proposed by Greenwood and Williamson [59]

In this approach the surface roughness is represented by asperities of simple geometrical

shape and with predefined radii of curvature The asperity heights are assumed to follow

a statistical distribution A rough surface is thus characterized by statistical parameters

such as the standard deviation of surface heights and correlation length A single asperity-

to-asperity contact is reduced to the deformation of two curved bodies in contact Its

solution may either be determined analytically using contact mechanics or expressed by

the empirical formula from the finite element simulation The surface contact is then

modeled by relating the load and the real area of contact to their asperity-level

counterparts by statistical integration

In many situations the statistical parameters of surfaces have been found strongly

dependent on the resolution of roughness-measuring instruments [60-62] This

phenomenon is due to the multiscale nature of the surface roughness which may be better

13

described by fractal geometry [63 64] The surface contact models are then developed

based on the use of power spectrum and scaling laws characterized by scale-invariant

quantities such as fractal dimension [65-69] These models also take the system variables

to be the integration of the asperity solution However each asperity is now represented

by the size of the contact spot based on which its amplitude of deformation and radius of

curvature are defined

The deterministic approach analyzes the computer generated surfaces or those

represented by the digitized output of roughness measurement The surface contact

behavior may then be predicted numerically by the method of influence coefficients [70-

77] and that based on the variational principle [78] Compared to the statistical and fractal

contact models the numerical simulation uses the digital maps of rough surfaces and

does not require any assumptions on asperity shape and distribution In addition this type

of analysis may be able to naturally account for the interaction of deformation of adjacent

contact spots

Significant advances have been made with the above approaches in the study of

both frictionless and frictional dry contacts of rough surfaces However the models

developed so far for the frictional contact appear to be largely oversimplified with some

major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in

modeling the frictional surface contact One is that contacting asperities may deform

elastically elastoplastically or plastically According to the results of frictionless

indentation of a sphere on a plane the normal load leading to initial yielding needs to

increase more than 400 times to cause fully plastic flow [79] The application of friction

reduces the first critical normal load [80-82] and thus the elastic deformation regime The

14

friction may also reduce the critical load related to plastic flow and the elastoplastic

deformation regime However this transition regime may still be significant compared to

the elastic regime Hence a high percentage of contacting asperities may be in the state

of elastoplastic deformation for the contact of rough surfaces with or without friction

Moreover a significant portion of asperities in contact may deform plastically in the

frictional situation For the frictionless contact all the three possible deformation modes

have been incorporated into several statistical models based on approximate analytical or

finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is

no similar model for the frictional contact due to the lack of a systematic study of the

elastoplastic behavior of contacting asperities with friction The other key phenomenon is

that the friction may significantly change the asperity pressure and contact area for those

asperities in elastoplastic and particularly fully plastic deformation Both experimental

and theoretical studies have shown that for a frictional plastic contact the interfacial

shear stress would lead to the growth of the asperity junction and reduction of the contact

pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for

asperity junctions under the combined normal and tangential loading The pressure and

contact area of the plastic junctions have also been solved using slip-line field theory [90-

95] and upper bound plasticity analysis [96] For the surface contact the effects of

friction on the subsurface stresses have been modeled but the contact pressure and area

are usually considered not to be altered by the friction In summary a mathematical

model accounting for these two important issues should be formulated for the frictional

contact of rough surfaces

123 Contact Modeling Boundary-Lubricated Surfaces

15

Under boundary lubrication conditions the contact of two rough surfaces is also

present in the form of distributed asperity contacts In addition to the asperities the

boundary films covering them may be involved in the contact process However these

films are very thin and thus it is reasonable to assume that the contact pressure and area

are mainly determined by the asperity deformation The contact response is mainly

affected by the boundary films through their effects on the interfacial friction Thus the

three approaches discussed in the last section may also be used to model the boundary-

lubricated surface contact if the shearing behavior of the boundary films is known

Many contact models have been developed for the boundary lubrication system

using the statistical approach [97-104] Besides the general contact response these

models predict the friction force as a function of load by summing up the local tangential

resistance The pressure and area of a single asperity contact are usually determined using

the Hertzian elastic solution In comparison the finite element method has been used to

analyze the mechanical responses of contacting asperities with nonlinear material

properties [104] For the determination of the friction force at the asperity junctions there

are several different formulations available For example Ogilvy [97] calculated the local

friction force by assuming constant film shear strength and using the energy of adhesion

Blencoe and Williams [101] related the interfacial shear strength to the contact pressure

according to empirical relations and Ford [103] took account of the contribution from

both interfacial adhesion and asperity deformation In addition to the statistical models

direct numerical simulation has also been performed for the contact of rough surfaces to

calculate the friction force resulting from adhesion and deformation [105] This

16

deterministic model extends the method of influence coefficients to account for the

effects of shear force on contact deformation

The study of the boundary-lubricated surface contact with the above models has

provided some insights into the effects of the rheology of boundary layers the substrate

material properties and the surface roughness on the system tribological behavior

However there are significant rooms for advancements in many aspects and

mathematical models with more insights may be developed First as mentioned in the

last section a large population of contacting asperities may be in either elastoplastic or

fully plastic deformation These two types of asperity contacts have not been properly

considered The important phenomena related to the two deformation modes such as the

pressure-shear stress coupling and the friction-induced junction growth also need to be

incorporated in to the model Second the adsorbed layer may be desorbed and the reacted

film may be ruptured during the asperity contacts Thus the effectiveness of boundary

lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of

theoretical and practical significance to capture this uncertainty by modeling the kinetic

behavior of the boundary lubricating films Third localized temperature rise or flash

temperature may be caused by the intensive shear stress at asperity junctions The

increasing contact temperature in turn may significantly affect the kinetics of the

boundary films and thus the interfacial shear stress As reviewed in the next section the

flash temperature has been calculated or measured by a number of researchers However

its interaction with the evolution of the boundary films has not been studied adequately in

contact modeling

124 Flash Temperature

17

The localized temperature rise due to frictional heating is an important

characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough

surfaces The rising temperature can be viewed as the thermal response of the contact and

it may strongly affect the behavior of lubricating films the properties of substrate

materials as well as most surface phenomena Thus the prediction of the interface

temperature plays an important role in modeling the sliding contact behavior

The maximum or average temperature rise of single asperity contacts has been

estimated based on the laws of energy conservation and heat conduction [106-115] Most

of these analyses focused on the flash temperature of an individual square or circular

contact Gecim and Winer considered the cooling-off effect between two consecutive

asperity contacts [112] Bhushan proposed an approach to include the effects of frictional

heating by neighboring asperity contacts [114] The analysis of asperity flash

temperatures has also been incorporated into different types of surface contact models to

predict the interfacial temperature distribution [67 68 and 116-118] For example the

fractal contact model developed by Wang and Komvopoulos [67 68] included the

analysis of the distribution of temperature rise at the interface Based on a statistical

contact model Yevtushenko and Ivanyk [116] determined the temperature rise of

contacting asperities and their thermal deformation for the sliding contact of rough

surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]

calculated the temperature rise at asperity contact spots which were the solution provided

by a deterministic surface contact model [71]

18

125 Summary

The above literature review shows that significant progress has been made in the

understanding of different boundary lubrication mechanisms the modeling of rough

surfaces and the calculation of flash temperature Research has also been initiated to

address the integral effects of these important aspects For example a failure criterion of

boundary lubrication has been incorporated into a thermal contact model of rough

surfaces [117] However only the elastic deformation and thermal desorption are

considered More recently an asperity-contact model has been designed to calculate the

tribological variables by simultaneously simulating the key processes involved but the

solution obtained is not suitable to be integrated into a system model [119] In summary

a comprehensive contact model needs to be developed to include the effects of multiple

deformation modes of contacting asperities the uncertainty of the boundary lubricating

films the flash temperature due to friction and their interaction

13 Research Objective Approach and Outline

This thesis aims to develop a surface contact model for the boundary lubrication

system to gain more insights into its tribological behavior For a given load the model

should be able to predict the asperity contact variables and their distribution and the

system friction coefficient and area of contact The model should also factor in surface

topography material and lubricant properties and other operating conditions in addition

to the system load

In this research the statistical approach is selected to relate the system contact

variables to their asperity-level counterparts The reason is that the statistical models are

19

able to identify the important trends in the effects of surface properties on the system

contact behavior with relatively simple calculation The key component of the research is

thus the development of a deterministic model for a single asperity contact under

boundary lubrication conditions

At the asperity level the model needs to capture the characteristics of

fundamental mechanical physiochemical and thermal processes involved in the

boundary-lubricated contact From the mechanical point of view the model to be

developed should cover the three possible deformation modes of contacting asperities

under combined normal and tangential loading For this purpose the effects of friction on

the pressure area and deformation mode of a single asperity contact are first explored

using the finite element method since it is impossible to obtain the analytical solution

directly The finite element results are then combined with the contact mechanics theories

to derive model equations for a frictional asperity contact involving the three possible

deformation modes These pure mechanical equations are used to describe the boundary-

lubricated asperity contact in conjunction with the expressions developed to calculate the

flash temperature and to characterize the behavior of boundary films The solution of all

the asperity-level modeling equations is finally used to formulate the contact model for

the boundary lubrication system by means of statistical integration

In summary the thesis comprises three layers of modeling and analysis ndash (1)

elastoplastic finite element analysis of frictional asperity contacts (2) modeling of

contact systems with friction and (3) modeling of a boundary lubrication process Each

layer of analysis is presented as a chapter in the main text and briefly described below

20

Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite

element model is developed and systematic numerical analyses carried out to study the

effects of friction on the contact and deformation behavior of individual asperity contacts

The study reveals some insights into the modes of asperity deformation and asperity

contact variables as function of friction in the contact The results provide guidance to

analytical modeling of frictional asperity contacts and lay a foundation for subsequent

work on system modeling

Chapter 3 Modeling of contact systems with friction ndash Analytical equations are

developed relating asperity-contact variables to friction using the theory of contact-

mechanics in conjunction with the finite element results in chapter 2 By statistically

integrating the asperity-level equations a system-level model is developed and used to

study the effects of the friction on the system contact behavior It serves as the platform

in the final step of model development for the boundary lubrication problem

Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous

two layers of modeling a deterministic-statistical model for the boundary-lubricated

contact is developed by incorporating the essential aspects of boundary lubrication Four

variables are used to describe a single asperity contact including micro-contact area

pressure shear stress and flash temperature In addition three probability variables are

introduced to define the interfacial state of an asperity junction that may be covered by

various boundary films Governing equations for the seven key asperity-level variables

are derived based on first-principle considerations of asperity deformation frictional

heating and kinetics of boundary lubrication films These asperity-scale equations are

coupled and some of them are nonlinear Their solution is thus obtained by an iterative

21

method and is statistically integrated to formulate the contact model for boundary

lubrication systems The model is then used to study the effects of surface roughness and

operation parameters on the system tribological behavior

Each of the above three chapters is relatively self-contained though they are also

well-connected Finally Chapter 5 concludes the thesis with a summary of the main

contributions and some suggestions for future work

22

Chapter 2

Effects of Friction on the Contact and Deformation Behavior

in Sliding Asperity Contacts

21 Introduction

It is quite well recognized that the solid-to-solid contact between the surfaces of

machine components is made at their surface asperities These asperity contacts often

play a significant role in the tribological performance of mechanical systems especially

under dry and boundary lubricated conditions Greenwood and Williamson [56]

established a framework for the statistical asperity-contact based models of two

contacting surfaces The concept was used in many areas of micro-tribology modeling

such as machine components in mixed lubrication [122] head-disk interface of computer

disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name

just a few

The model of reference [56] does not include friction which can significantly

affect the behavior of the asperity contacts A number of researchers have studied the

effects of friction For elastic contacts the theory of elasticity is used to obtain closed-

form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the

subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton

and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-

dimensional problem The results show that the friction brings the point of the maximum

shear stress closer to the surface and increases the compressive stress at the leading edge

23

and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied

the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and

Sackfield and Hills [80] analyzed the problem for point contacts The results show that

the yielding would start at lower normal loads and the points of the initial yielding would

move to the surface when the friction coefficient exceeds 03

For fully plastic contacts the theory of plasticity may be used to obtain

approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in

plastic contacts using the octahedral shear stress theory The results show that for a given

normal load the friction reduces the contact pressure and increases the contact area

Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]

derived a flow equation for asperity junctions under the combined normal and tangential

loading With this equation he explained the phenomenon of the junction growth and the

high friction between clean metal surfaces that were observed in experiments Johnson

[92] and Collins [93] also solved the plastic frictional contact problems using the theory

of slip-line field In addition to the pressure reduction and junction growth they

concluded that the friction coefficient would reach a high value of about unity in the

extreme

A large number of asperity contacts in a dry or boundary-lubricated system may

be in elastic-plastic deformation In this mode of deformation analytical solutions are not

readily available The methods of finite elements are often used to study the effects of

friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the

contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]

simulated the collision process of a pair of asperities Nagaraj [133] and many others

24

analyzed contact problems with stick and slip These numerical studies however largely

focused on special problems Fundamental issues have not been adequately addressed

such as the effects of friction on the mode of the asperity deformation shape and size of

the plastic zone in the micro-contact and the asperity pressure contact area and load

capacity

In this chapter a systematic finite element analysis is carried out to study sliding

asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis

focuses on the above fundamental issues of the effects of friction to reveal some insights

into the behavior of sliding asperity contacts The modeling and results are presented in

the next two sections

22 The Model Problem

The model of a deformable half-cylinder in sliding contact with a rigid flat is used

in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should

capture the essential effects of the friction on the contact and deformation behavior of an

asperity contact while significantly simplifying the computational complexity The

material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a

ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high

value of YE would result in a plastically deformed region in the contact that is much

smaller than the cross-section area of the half-cylinder so that the results will be fairly

independent of the latter and of the boundary conditions away from the contact

Furthermore the results in the dimensionless form presented later in the chapter are

essentially independent of the YE ratio so long as the region of plastic deformation is a

25

very small proportion of the bulk material which is the case in actual asperity contacts

The normal loading to the contact is prescribed in terms of the approach of the rigid flat

to the cylinder δ which is more meaningful than specifying a normal load for asperity

contacts between two surfaces The tangential loading F is given in terms of a shear

stress distribution in the contact proportional to the pressure distribution

( ) ( )xpx microτ = (21)

where micro is a prescribed coefficient of friction and the pressure distribution is to be

determined in the solution process It should be pointed out that the contact between two

bodies in gross sliding is of interest in this thesis study In such a contact the assumption

of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The

ratio of the local shear stress to the local pressure in a sliding contact can be extremely

complex and often exhibits significant random behavior A uniform micro as a parameter

would represent a stochastic average that can be sensibly used to study the effects of

friction on the contact

The solid modeling software I-DEAS is used to generate the finite element mesh

of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane

strain elements with a total number of 2713 nodes A substantial number of elements are

allocated in the region around the contact The commercial finite element code ABAQUS

is used to simulate the sliding contact problem and small deformation is assumed in the

finite element calculations Zero-displacement boundary conditions are prescribed for the

nodes at the bottom of the finite element model The rigid-surface option is employed to

mimic the rigid flat which is constrained to move vertically The normal loading to the

26

model asperity by means of a normal approach is realized by enforcing a vertical

displacement to the flat The adaptive automatic stepping scheme is implemented for

loading More detail descriptions of algorithms used to determine the contact nodes and

contact conditions are given in the ABAQUS manual [134] For a given combination of

the normal approach and friction coefficient the finite element calculations yield the

pressure distribution and the width of the contact and the nodal von Mises stresses Mσ

Then the average pressure and load capacity of the contact can be calculated

Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the

initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to

determine the shape and size of the plastic zone

The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]

pointed out that the maximum relative error may be expressed as one-half of the ratio of

the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and

under frictionless normal loading about 12 surface nodes come into contact with the rigid

flat when the initial yielding occurs in the model asperity The error under this condition

would then be under 10 Indeed the finite element results for an elastic frictionless

contact compare favorably with the results from the Hertz theory including the pressure

distribution contact width and location of the material point of initial yielding

Considering that a large portion of the analyses will be carried out for a greater number of

surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate

The adequacy of the finite element mesh is studied with additional evaluations First the

results are essentially independent of the direction of sliding from either left or right

Second the results are also essentially independent of the history of normaltangential

27

loading (ie changes of δ and micro ) which is sensible for small deformation of a non-

work-hardening asperity Finally the plastic zones for fully plastic contacts compare

reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]

23 Results and Analysis

The contact pressure and sub-surface stresses are calculated for a range of the

normal approach δ and friction coefficient micro The results are presented and analyzed

to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of

micro-contact plastic zone and (3) the pressure size and load capacity of the asperity

contact

231 Mode of Asperity Deformation

The state of the asperity deformation may be categorized into three regimes ndash

elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all

material points are less than the uni-axial yield strength of the material In an elastic-

plastic contact plastic yielding occurs at some material points marking a transition from

the elastic to fully plastic deformation In a fully plastic contact all material points

around the contact enter plastic deformation and the ability of the asperity to take

additional load is largely lost For a frictionless contact the transition from elastic-plastic

to full plastic contact is often defined to be the point when all the nodal pressures in the

contact largely reach the value of the material hardness which is considered to be about

equal to 28Y [79] For a frictional contact this definition may not be used as the

tangential loading can substantially bring down the pressure that can be developed In this

chapter the elastic-plastic to full plastic transition is defined to be the condition under

28

which the von Mises stresses of all surface nodes in the contact region have reached the

uni-axial yield stress of the material It is noted from numerical results that under the

above condition the contact pressure distribution is fairly uniform corresponding to full

plasticity

Two critical values of the normal approach are defined to describe the modes of

the asperity deformation The first critical normal approach 1δ corresponds to the

condition under which the initial yielding occurs in the contact and the second one 2δ

the condition under which the contact becomes fully plastic The effects of the friction on

the state of the asperity deformation may be studied by examining the values of the two

critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of

the friction coefficient up to micro = 10 this micro value may be considered to be an upper

bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ

which is the first critical normal approach for the frictionless contact For micro = 0 the

normal approach causing the onset of fully plastic deformation of the contact is about

forty times of 10δ This large value of 2δ which is of the same order of magnitude as

those obtained for 3D circular contacts [84 137] suggests a rather long transition from

the elastic contact to the fully plastic contact However the elastic-plastic transition is

rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is

further reduced to one half of 10δ at micro = 10 The normal approach or the contact force

causing the initial yielding of the contact is also reduced significantly by the friction At

micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This

reduction accelerates at high friction values At micro = 10 1δ is reduced to only about

29

014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in

Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical

approximations dividing the regimes of asperity deformation Numerical errors arise from

the sizes of the finite element meshing and the stepping size of the normal approach δ∆

in the solution process The results of Fig 23 are obtained with a maximum stepping size

of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given

the assumptions and idealizations of the model problem This is further supported by the

fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact

derived analytically using the equations in references [79 80]

The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity

deformation at a given friction coefficient and normal approach of the contact The rapid

reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the

friction on the deformation in an asperity contact With high friction the contact may

change from the state of elastic deformation to the state of fully plastic deformation with

little elastic-plastic transition as the normal approach or the contact force increases The

large reductions of the two critical approaches with friction also signify significant

reductions of the contact pressures at the points of transition of the mode of the asperity

deformation In a frictionless contact the average contact pressure at the elastic-to-

elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the

elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123

and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in

30

the pressure are evidently due to the large shear stresses that are developed in the asperity

contact

The finite element results may also be used to study the equation of the full plastic

flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the

contact This equation may be expressed as

222 Hp =+ατ (22)

where α is a constant s the interfacial shear stress and H the indentation hardness of the

material or the maximum pressure that can be developed in the contact Taking

YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)

can be determined for a given friction coefficient using the calculated pressure and

surface shear stress at the normal approach of 2δδ = For the model problem with a

friction coefficient up to micro = 10 the calculations of the nine data points along the

microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These

fairly uniform values of α lie in the range of values discussed in [89]

232 Shape of the Plastic Zone

The behavior of the two critical normal approaches shown in Fig 23 is closely

related to the effects of the friction on the shape and size of the plastic zone in the

asperity contact The problem of a frictionless contact is first studied The location of the

initial yielding is in the central region of the contact about 067 times the contact-half-

width beneath the surface Figure 24 shows the plastic zones for two values of the

normal approach One is at the halfway between 1δ and 2δ and the other at 2δ

31

corresponding to the mode of elastic-plastic deformation and the onset of full plastic

flow respectively Under both loading conditions the plastic zones are similar and are

nearly of a circular shape In the former the subsurface initiated plastic deformation has

grown substantially and has largely propagated to the contact surface except a thin layer

that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has

also become plastic while the plastic zone expands further with a diameter nearly three

times as that of the former

The problems with friction are studied next Figure 25 shows the results obtained

with a friction coefficient of micro = 02 the direction of the friction force is from the left to

the right The location of the initial yielding is shifted towards the leading edge of the

contact at 053 times the contact-half-width beneath the surface and 065 to the right

With a normal approach corresponding to halfway into the elastic-plastic transition the

surface material at the trailing one half of the contact has become plastic while a surface

layer at the leading one half is still elastic This is in contrast to its frictionless counterpart

of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the

contact As the normal approach further increases the plastic zone rapidly propagates

towards the surface on the leading side When full plasticity is reached in the contact the

plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of

a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant

tangential loading in the contact the value of the normal approach to bring about full

plasticity is reduced to about 025 of that of the frictionless contact and the width of the

contact to about 027

32

Figure 26 shows the results with a higher friction coefficient of micro = 05 With

this high friction the plastic yielding is initiated at the surface one site at the leading

edge and another immediately occurring thereafter at the trailing edge The result of the

two-site plastic yielding is consistent with an analytical approximation [79] The two

plastic sub-zones propagate and eventually unite as the normal approach increases

Halfway into the elastic-plastic transition the plastic deformation is largely confined to

near surface and a small segment at the leading edge of the contact remains elastic

When full plasticity is reached the plastic zone has not significantly propagated into the

depth aside from a protruding-wing region that is developed towards the leading edge of

the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser

magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-

perfectly plastic contact with high friction The width of the contact in this case is only

about 005 of that of its frictionless counterpart at the condition of full plasticity Figure

27 shows the results with an even higher friction coefficient of micro = 10 Similar to the

problem of micro = 05 the yielding initiates at the surface at both the leading and trailing

edges of the contact The two plastic sub-zones have not yet connected halfway into the

elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic

zone of a significant magnitude is developed at the leading edge The width of the contact

is about 004 of the size for the frictionless problem when full plasticity is reached and

the plastic deformation is largely confined to a very thin surface layer in the contact

region

33

233 Contact Size Pressure and Load Capacity

It is of interest to study the effects of the friction on the contact variables

including the junction size pressure and load capacity of the asperity For a meaningful

study and results comparison the normal approach is held constant while the friction

coefficient is varied Figure 28 shows the results obtained at a relatively low level of

loading the normal approach is set equal to the normal approach causing plastic yielding

in a frictionless contact 10δ The results are plotted in the scale of their corresponding

values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the

effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the

contact pressure starts to decrease while the contact junction grows At micro = 047 for

example the pressure is reduced to 084 of its frictionless value and the junction is

increased to 119 However the load carried by the asperity is essentially unaffected due

to the compensating effects of the pressure reduction and junction growth At the higher

level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth

in the contact size becomes more intensified to about one half and two times their

frictionless values at the extreme The change in the load capacity is only modest with a

maximum reduction of about 11 at micro = 10

The reduction of the pressure with friction in Fig 28 may be studied with Eq

(22) For a normal approach of 10δδ = the contact is largely elastic when the friction

coefficient is small Therefore it can accommodate some tangential traction without

bringing about significant plastic deformation (ie 22 ατ+p is significantly less than

2H ) Consequently the pressure is not affected by the friction As the level of friction

34

increases the amount of plastic deformation increases At micro = 05 for example

101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly

plastic with the current normal approach of 10δδ = As a result the coupling between the

normal and tangential loading in the asperity contact is more pronounced and the increase

in the surface shear stress would be at the expense of the contact pressure The contact

eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the

tangentialnormal coupling is even stronger and follows Eq (22)

The growth of the contact junction with friction may be studied by examining the

shift of the junction in the direction of the friction force Figure 29 shows the sizes of the

contact junction at different levels of the friction coefficient along with the center

locations of the junction Up to a friction coefficient of micro = 038 the junction

experiences little growth and its center location is virtually unchanged This result may be

attributed to the fact that the junction is largely elastic up to this level of the friction The

results however show a significant trend of the junction growth with the friction

coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not

visible An examination of the critical normal approaches shown in Fig 23 suggests that

with 10δδ = the degree of plastic deformation in the contact increases significantly in

this range of the friction coefficient Thus the increase in the junction size is attributed to

the contact becoming more plastic as for a given normal approach (in a frictionless

contact) the junction size is about twice as large for a plastic contact than for an elastic

contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29

show that the junction growth becomes more pronounced accompanied by a significant

35

shift of the center of the junction which is an indication of tangential plastic flow In this

range of the friction coefficient the contact eventually reaches the state of full plasticity

The accelerated junction growth is attributed to two factors One is the growth associated

with the further increase of plastic deformation in the contact and the other the tangential

plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the

trend of the junction growth and the shift of the center of the junction become somewhat

moderated In this range of the friction coefficient the contact is now in the mode of full

plasticity and the junction growth is primarily due to the friction-induced tangential

plastic flow

Figure 210 shows the effects of the friction on the contact variables at a relatively

high level of loading The normal approach in this case is three times as large as that with

which the results of Fig 28 are obtained At this loading level the pressure reduction

and junction growth take place in the low range of the friction coefficient but the load

capacity is virtually unchanged In the median range of the friction the pressure and the

contact size become significantly more sensitive to the friction coefficient At micro = 05

the pressure is reduced to 058 of its frictionless value while the junction size increased to

154 The load capacity of the junction is still maintained at its frictionless level up to micro

= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher

friction coefficients the pressure reduces further and so grows the junction However the

results suggest that the junction growth in this case is not as pronounced as the pressure

reduction in comparison with the results from the previous case of low loading The

results further show a limited junction growth at the high-end of the friction coefficient

As a result the compensation of the junction growth to the pressure reduction becomes

36

less effective at this level of loading and the load capacity of the junction is significantly

reduced by the effect of friction At micro = 10 for example the load capacity is reduced to

061 of its value for the frictionless contact

The limit in the junction growth shown in Fig 210 for relatively high contact

loading is possibly due to the geometric effect of the asperity A higher loading produces

a larger contact size and a larger surface slope at the edges of the contact junction

particularly the leading edge because of the friction-induced tangential plastic flow The

tangential plastic flow and the surface slope are the two competing factors that determine

the size and the growth of the contact junction When the contact size is small the slope

is small and the junction growth is largely governed by the plastic flow leading to a large

increase of the junction with friction When the contact size is large the surface slope at

the leading edge is large and would ultimately limit further growth of the junction

It should be pointed out that a majority of the contacting asperities in the contact

of rough surfaces might experience a level of loading that is significantly above that with

which the contact-variable results in Fig 210 are obtained For machine components

such as bearings and engine cylinders the radius of surface asperities may be taken as of

the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the

normal approach causing plastic yielding of the contact in the absence of friction is of the

order of magnitude of 01010 =δ microm [79] For relatively highly finished machine

components the surface RMS roughness is often significantly larger than 01 microm and

thus the normal approaches of many contacting asperities can be significantly above 001

microm In this situation the loss of load capacity to the friction by these contacting asperities

37

could be more severe than that predicted in Fig 210 As a result the average gap

between the two surfaces would reduce so as to bring additional asperities into contact to

support the applied load in the system

24 Summary

This chapter conducts a finite element analysis of the effects of friction on the

contact and deformation behavior in sliding asperity contacts The analysis is carried out

using two input variables One is the normal approach of a rigid surface towards the

asperity and the other the coefficient of friction in the contact Results are presented and

analyzed to reveal the effects of friction on the mode of asperity deformation the shape

of micro-contact plastic zone the contact pressure and size and the asperity load

capacity The results lead to the following conclusions

1) The friction in the contact can significantly reduce the normal approach that

initiates the plastic yielding in the asperity and the normal approach that causes

the asperity to become fully plastic The reduction is more pronounced for the

second critical normal approach so that with a relatively high friction coefficient

the contact may change from the state of elastic deformation to the state of fully

plastic deformation with little elastic-plastic transition as the normal approach or

the contact force increases

2) The friction can significantly change the shape and reduce the size of the

plastically deformed region in the asperity when the contact becomes fully plastic

The reduction is most pronounced at high friction coefficients and the plastic

deformation is largely confined to a thin surface layer in the contact

38

3) The friction can have a large effect on the contact size pressure and load capacity

of the asperity At low friction and a relatively small normal approach these

contact variables are not affected With medium friction the pressure is reduced

and the contact size is increased however the influence on the asperity load

capacity is small due to a compensating effect between the pressure reduction and

junction growth With high friction the pressure reduction continues but the

junction growth is limited particularly for a large normal approach the limit in the

junction growth appears to be due to a geometric effect of the asperity

Consequently the effect of the pressure-junction compensation becomes less

effective and the asperity load capacity can be lost significantly

It should be emphasized that the finite element results presented in the

dimensionless form given in this chapter are sufficiently general Essentially the same

results are obtained with different radii or material parameters of the model asperity as

long as the region of plastic deformation in the contact is small so that the half-space

assumption is fairly valid Although the analyses are conducted using a line-contact

model the effects of friction in sliding asperity contacts of three-dimensional geometry

should be basically the same and the same conclusions would have been reached

Therefore the finite element results are used in the next chapter to guide the development

of analytical modeling equations for frictional asperity contacts that lay a foundation for

subsequent work on system contact modeling

39

Rigid flat

δ

Figure 21 Half-cylinder contact model

Sliding direction of the rigid flat

Figure 22 Finite element mesh of the model problem

40

Figure 23 Effects of friction on the critical normal approaches

(a) linear scale (b) logarithmic scale

35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes

Friction coefficient

41

Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity

(the top figure shows the zoom-in of the region in the dashed rectangle in (a))

(a)

(b)

Contact width

Elastic deformation Plastic deformation

Rigid flat

Asperity

42

Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity

(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)

(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44


(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)

(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2

25 PressureContact size Load capacity


Con

tact

var

iabl

es

Figure 28 Contact variables with 10δδ =

46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038

Contact center Friction force

Contact size

Fric

tion

coef

ficie

nt

Figure 29 Shift and growth of the contact junction with 10δδ =

47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3

A Mathematical Model of the Contact of Rough Surfaces with

Friction

31 Introduction

The contact between two nominally flat but rough surfaces is of great importance

in the study of the tribological behavior of mechanical systems Since the true contacts

are made at randomly distributed surface peaks or asperities asperity-based models have

often been used to study surface contact phenomena

A typical asperity contact-based model incorporates individual asperity contact

solutions into statistical descriptions of surfaces Greenwood and Williamson initiated

this approach in 1966 [59] In the GW model the rough surface was taken to consist of

hemispherically tipped asperities with an identical radius The asperity heights were

assumed to follow an isotropic Gaussian distribution The contact between two rough

surfaces was further converted to a contact between an equivalent rough surface and a

rigid flat plane By applying the Hertzian elastic contact solution to the distributed

asperities the GW model related the real area of contact and system contact load to the

mean separation of the surfaces Handzel-Powierza et al [139] verified this model

experimentally within the range of elastic deformation and for quasi-isotropic surfaces

However they also found that the theoretical prediction by the GW model would become

invalid when a significant portion of contacting asperities no longer deform elastically

The GW model has been extended mainly in two ways One is to treat other asperity

49

contact geometries including random radii of asperity curvatures [140] elliptic

paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider

asperity inelastic deformation such as an elastic-plastic model based on the volume

conservation of plastically deformed asperities [144] and a model incorporating the

transition from elastic deformation to fully plastic flow [84]

The aforementioned models assume frictionless contacts However any sliding

contact of surfaces involves friction which can be significant For a surface contact with

friction an asperity-based model may also be developed from the variables of frictional

asperity contacts A number of researchers have studied frictional contact of surfaces

using such a scheme For elastic contacts the asperity pressure and area are slightly

affected by the friction [79] and the two variables may be determined using the Hertz

theory Using this relation in combination with the expressions for adhesive forces

Francis [99] and Ogilvy [97] modeled the system contact variables and the friction

coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled

a plastic contact system by assuming that all contacting asperities deform plastically and

that the asperity pressure and contact area are not affected by the friction Chang et al

[145] devised an elastic-plastic frictional surface model in which some asperities deform

elastically and others in full plastic flow It is assumed that the area of asperity contact is

determined from the Hertz solution and that only elastically deformed asperities

contribute to the friction force

The above researchers have made some fundamental contributions to the study of

frictional effects in the contact of rough surfaces However they have not considered two

key phenomena in frictional contacts One is that a contacting asperity may deform

50

elastically elastoplastically or plastically and the friction can largely change the mode of

the asperity deformation Johnson [79] showed that in a frictionless asperity contact the

contact force causing fully plastic flow could be 400 as large as the contact force leading

to the initial yielding According to the finite element study in the last chapter the

difference between the two contact forces is reduced by friction but is still significant

Thus a high percentage of the asperity contacts of rough surfaces may be in the state of

elastoplastic deformation The other key phenomenon is that the friction may

significantly change the asperity pressure and contact area for those asperities in

elastoplastic and particularly fully plastic deformation Both experimental and

theoretical studies have shown that for a frictional plastic contact the interfacial shear

stress can cause large growth of the asperity junction and large reduction of the contact




95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical

model including these two key phenomena has not been formulated for the frictional


In Chapter 2 a finite element model has been used to study the effects of friction

on the asperity contact in all the three modes of deformation This chapter uses the finite

element results in conjunction with the theory of contact mechanics to model frictional

asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation

including the junction growth and the coupling between contact pressure and shear stress

The asperity-scale equations are then used to build a mathematical model for the

51

frictional contact between two nominally flat surfaces The modeling is described next

and results presented

32 Modeling

321 Model Structure

In this chapter the framework established by Greenwood and Williamson [59] is

used to model the sliding contact between two rough surfaces As illustrated in Fig 31

the concept of equivalent rough surface is used The material properties of the equivalent

surface are taken to be a combination of those of the two surfaces in contact

Consider a single contact point of the surface shown in Fig 31 The normal

loading to the contact is prescribed in terms of the approach of the rigid flat to the

asperity

dz minus=δ (31)

where z is the height of the asperity and d the distance from the mean plane of asperity

heights to the rigid flat The friction force F is measured in terms of the average

interfacial shear stress in the asperity contact that is assumed to be proportional to the

average contact pressure

mm Pmicroτ = (32)

where micro is the coefficient of friction taken to be an input parameter in this chapter It

should be pointed out that the frictional sliding contact between two surfaces is studied

52

In such a contact the assumption of a uniform friction coefficient for all asperities is

theoretically feasible to study the effects of the frictional loading

The asperity pressure and area of contact depend on both the normal approach and

the friction coefficient Or

( )microδ mm PP = (33)

( )microδ ll AA = (34)

For a given surface separation d and friction coefficient micro the real area of contact and

the contact load of the system are calculated by statistically integrating the above two

asperity contact variables

( ) ( ) ( )dzzfdzAAdAd lnt intinfin

minus= microηmicro (35)

( ) ( ) ( )dzzfdzWAdWd lnt intinfin


where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the

asperity contact force which is equal to the product of asperity contact pressure and area

A key component of the modeling is to develop expressions for the asperity

contact variables in terms of normal approach and friction coefficient With a given

friction coefficient a contacting asperity experiences three deformation stages as the

normal approach increases elastic elastic-plastic and fully plastic The transition of the

deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2

The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely

53

decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are

first formulated as functions of δ and micro in each of the three deformation regimes Then

the dependence of the two critical normal approaches on the friction coefficient is

modeled Finally the equations used to determine the system variables from the asperity

contact solutions are presented

322 Asperity Contact Pressure

Consider a contacting asperity in elastic deformation It is defined by the normal

approach δ below ( )microδ1 Under such a condition the tangential loading generally has

small effects on the contact pressure and area [79] Therefore the two variables are

assumed to be only dependent on the normal approach The asperity contact pressure is

then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ

microδ δ le ( )microδ1 (37)

When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless

contact the asperity contact pressure at 02 )(

==

micromicroδδ or 20δ reaches its maximum

possible value or the indentation hardness of the material H Thus the frictionless

asperity contact pressure for 20δδ ge can be written as

( ) HP m ==0

micro

microδ 20δδ ge (38)

54

For a frictional contact the asperity pressure in fully plastic deformation depends on how

much interfacial shear stress is developed in the contact The pressure and shear stress

may be related by the Tabor equation [89]

222 HP mm =+ατ ( )microδδ 2ge (39)

Combining this equation with mm Pmicroτ = yields a general expression for the asperity

pressure in a fully plastic contact

( )( ) 2121

αmicro

microδ+

=HPm ( )microδδ 2ge (310)

With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a

pressure expression can be obtained for a contact in elastoplastic deformation For a

frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic

function of the normal approach Based on that Zhao et al [84] derived an expression of

pressure in terms of the first and second critical approaches 10δ and 20δ

( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus

minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)

where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and

mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic

relation also holds when friction is present Equation (311) may then be generalized to

calculate the contact pressure of a frictional asperity contact in the elastoplastic regime

For a given normal approach and friction coefficient the pressure expression is given by

55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ

microδδmicromicromicromicroδ

12

1

lnlnlnlnminus

minusminus+= mYmFmYm PPPP

( ) ( )microδδmicroδ 21 ltlt (312)

In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and

( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)

323 Asperity Area of Contact

The asperity contact area is determined first for a frictionless contact When the

normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]

( ) δπmicroδmicro

RAl ==0

10δδ le (313)

With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow

Its area of contact may be determined by the Abbott and Firestone model [147] and is

given by


RAl 20=

= 20δδ ge (314)

For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng

and Wang [148] modeled the area of contact using a polynomial function which smoothly

joins Eqs (313) and (314) The resulting area expression is given by

( ) δπδδmicroδmicro

RAl )231( 320

primeprimeminusprimeprime+==

2010 δδδ lele (315)

where ( ) ( )102010 δδδδδ minusminus=primeprime

56

Next the area of a frictional asperity contact is modeled According to previous

experimental and theoretical studies [87-89] the tangential loading would cause the

growth of the asperity junction The amount of junction growth depends on the interfacial

shear stress and the mode of deformation Thus the asperity contact area may be

expressed as the frictionless area ( )0

=micro

microδlA multiplied by a junction growth factor that

is a function of both the normal approach and the friction coefficient ( )microδ Ak

( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)

A model for )( microδAk is developed below to calculate the asperity contact area from the

above equation For elastic deformation the area of contact is assumed to be unaffected

by the tangential force Furthermore there is no growth at 0=micro Therefore

( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)

Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure

and shear stress remains constant for a given friction coefficient Therefore it is

reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=

Or

( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)

Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the

normal approach and is approximated by a logarithmic function of δ according to Eq

(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same

range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ

microδδmicromicroδ

12

1

lnlnlnln11minus

minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)

The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to

be modeled to complete the modeling of the asperity contact area This function may be

determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime

For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)

yields a relation between the shear stress and the friction coefficient

( )( ) 2121

αmicro

micromicroδτ+

=H

m ( )microδδ 2ge (320)

Now consider an asperity subjected to both normal and tangential loading and is in fully

plastic flow Under such a condition the characteristics of the junction growth may be

captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by

Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic

zone to be shifted in the direction of the force and a volume of material to be

agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating

process is also revealed by the finite element results in the last chapter This process is

intensified as the shear stress increases and is likely to be the cause of the friction-

induced junction growth Both the slip-line field solution and the finite element results

show that the shift of the plastic-zone and the agglomeration of the material level off as

the interfacial shear stress approaches to the shear strength of the substrate oτ At this

point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk

which is estimated next

58

Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-

plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic

deformation is largely confined to the thin surface layer [92] The finite element results in

Chapter 2 also exhibit similar features Consequently volume conservation requires that

the material agglomerated at the leading edge occupies a volume equal to that of the apex

segment of the wedge that would have penetrated into the flat surface The slip-line

solution further suggests that the shape of the agglomerated material is similar to that of

the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be

approximated by

( )w

ibl

αsin=∆ (321)

where ib is the semi-width of the frictionless contact at the given normal approach of the

wedge The size of contact with friction is then given by

( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)

The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so

( )wAlk

αsin2110 += (323)

A cylindrical asperity may be approximated as a wedge with a semi-angle Wα

approaching o90 Equation (323) then yields 510 =Alk for this case A value of

410 =Alk is chosen in this study to model the junction growth of spherical asperities

59

The choice is based on the above order-of-magnitude analysis in conjunction with the

consideration that the asperity load-capacity decreases with friction

For an asperity contact in fully plastic deformation the upper-bound junction

growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear

stress mτ increases from zero to oτ This increase may be divided into two stages based

on the analysis of the junction growth by Kayaba and Kato [149] and the finite element

results in the last chapter In the first stage the junction growth is very mild before the

shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it

largely accelerates to reach the maximum value of 0Alk Therefore the following

piecewise linear function is used to model ( )mAlk τprime

( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+

ltlesdotminus+=prime

cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)

In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in

the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-

bound junction growth function )(microAlk using Eq (320) yields

( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)

where cmicro from Eq (320) is related to cτ by

60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)

The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth

can take place at a modest friction coefficient Equations (316) (319) and (325) form a

complete set to model the junction growth of the asperity contact area

The frictional asperity contact pressure and area have been expressed above in

terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and

( )microδ 2 The two critical normal approaches are determined in the next section using

contact-mechanics theories in conjunction with finite element results

324 Critical Normal Approaches

The first and second critical normal approaches divide the asperity deformation

into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of

them decrease as the friction coefficient increases Their dependence on the friction

coefficient is modeled below Consider the first critical normal approach ( )microδ1 It

corresponds to the initial yielding of a contacting asperity The yield of material is

assumed to be governed by von Misesrsquo shear strain-energy criterion [135]

3

2

2YJ = (327)

where 2J is the second stress tensor invariant and Y the yield strength of the material

This invariant is defined in terms of the stress components by

61

( ) ( ) ( )[ ] 222222

2 6 zxyzxyxxzzzzyyyyxxJ τττ

σσσσσσ+++

minus+minus+minus= (328)

For a frictionless contact the von Mises criterion may be simplified to a linear relation

between the contact pressure and the yield strength [144]

YkP YmY = (329)

A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for

( ) 1001 δmicroδmicro

==

is obtained and is given by

REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)

When friction exists the von Mises yielding criterion should be applied to the

resultant stresses caused by both normal and tangential loading In the case of elastic

deformation Hamilton [128] assumed that the actions of these two types of loading are

largely independent of each other Under this assumption the principle of superposition

is applicable and the resultant stress filed is given by

Tij

Nijij σσσ += (331)

where Nijσ and T

ijσ are the stress fields induced in the asperity by the normal and the

tangential loading respectively For a spherical asperity Hamilton [128] derived the

expressions of Nijσ and T

ijσ which may be written in the following functional form

( ) mijLij PZYX microσσ primeprimeprime= (332)

62

where ijLσ is a dimensionless function of the friction coefficient and the position within

the asperity The position is defined by the coordinates normalized by the radius of the

asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress

tensor invariant can also be expressed in a similar functional form

( ) 222 mL PZYXJJ microprimeprimeprime= (333)

where LJ 2 is also a dimensionless function of position and friction coefficient With the

pressure mP given by Eq (37) 2J is shown to be a linear function of the normal

approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝

⎛primeprimeprime= (334)

For a given friction coefficient the initial yielding takes place at the position

( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs

(327) and (334) yields the condition of initial yielding of a frictional asperity contact

( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)

From this equation the first critical normal approach is determined and is given by

( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)

The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by

63

( ) ( )( )micromicroδ

max2

max21

0

L

L

JJ

=prime (337)

Due to the complexity of the original stress expressions only numerical results are

available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the

normalized first critical normal approach ( )microδ1prime for a range of friction coefficient

Similar results are obtained for a cylindrical asperity by the finite element method in

Chapter 2 as illustrated in Figure 34

The second critical normal approach ( )microδ 2 defines the onset of fully plastic

deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a

criterion for the onset based on a group of experimental and numerical results The

criterion is given by

402 asymplowast

YRaE (338)

where 2a is the radius of the contact area This radius is related to the frictionless second

critical normal approach 20δ by Eq (314) to give

( ) 21202 2 δRa = (339)

Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given

by

REY 2

20 800 ⎟⎠⎞

⎜⎝⎛asympδ (340)

64

With the availability of 20δ the second critical approach ( )microδ 2 can now be

determined The determination is based on the results that the theoretically determined

)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is

sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also

similar to that obtained from the finite element results An approximate expression can

then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model

in the last chapter to give

( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)

Equation (341) is obtained by a least-square regression of the data points using a

quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned

that Eq (341) is derived for the friction coefficient up to 10 as the finite element

calculation has only been performed in this range For the friction coefficient larger than

10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or

( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ

microδmicroδ 01gemicro (342)

Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this

approximation should not cause any significant error Using Eqs (340) to (342) along

with Eq (336) ( )microδ 2 is determined for any given friction coefficient

In summary the asperity contact pressure is expressed in terms of the normal

approach and the friction coefficient by Eqs (37) (310) and (312) depending on the

value of δ It is presented below for convenience

65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro

microδδmicroδmicroδmicroδ

microδδmicromicromicro

microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)

The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA

and the junction growth function )( microδAk The expressions of the two functions are also

repeated below

( ) ( )⎪⎩

⎪⎨

⎧

geltltprimeminusprime+

le=

=

20

201032

10

0

2231

δδδπδδδδπδδ

δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)

where )(microAlk is given by Eq (325)

325 System Variables

The asperity contact equations developed in previous sections are now used to

model the frictional sliding-contact between two nominally flat rough surfaces The real

area of contact and contact load of the system are related to the corresponding asperity-

level variables by Eqs (35) and (36) The two system variables are functions of the

66

surface separation and friction coefficient They are also dependent on both material and

topographical properties of the surfaces The material characteristics are described by

Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity

contact is expressed in terms of its height the probability distribution of asperity heights

is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the

parameters based on the asperity heights are used to describe the surface However the

surface is usually characterized by the parameters related to the surface heights

Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the

second set of surface parameters such as the standard deviation of surface heights σ The

relation between these two sets of surface parameters was provided by Nayak [150]

The two surface contact variables may be normalized by the system parameters

The real area of contact is normalized by the nominal contact area nA and the contact

load by the product of nA and lowastE The following steps are taken to complete the

normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus

lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other

variables of length scale in Eqs (35) and (36) are normalized by σ The resulting

dimensionless system contact variables are given by

( ) ( ) ( )

dzzfdzAdAd lt intinfin

minus= microβmicro (346)

( ) ( ) ( ) ( )

dzzfdzPdzAdWd mlt intinfin

minusminus= micromicroβmicro (347)

67

where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and

σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus

and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough

surface and the rigid flat and sz the difference between the mean plane of surface heights

and that of asperity heights If the asperity heights follow a Gaussian distribution their

probability distribution function is given by

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)

And the dimensionless distribution function )( zf is given by

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛= lowastlowastlowast 2

2

50exp21 zzf

aa σσ

σσ

π (349)

Four surface parameters including β aσσ sz and Rσ are needed to determine the

system contact solution from Eqs (346) and (347) However three of them β aσσ

and sz are all dependent on another parameter sα which measures the spectrum

bandwidth of the surface roughness [150] Their expressions in terms of sα are given by

[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)

The surface roughness is therefore characterized by two independent parameters sα and

Rσ

33 Result Analysis

The model developed above is uedd to investigate the frictional contact behavior

of two nominally flat surfaces Using numerical integration the surface separation and

real area of contact are obtained and presented over a range of loading conditions and a

set of surfaces characterized by plasticity indices The statistical features of individual

asperity contacts are also examined to provide insights into the effects of friction on the

system contact behavior

The contact of steel-on-steel surfaces is considered with Youngs modulus

1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio

3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by

considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr

a very high friction coefficient of the order of 10 or higher is observed for clean metal

surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ

and the shear flow is observed As a result the real area of contact increases substantially

and the pressure much reduced In the extreme the Taborrsquos equation yields

( )20τα H= (353)

69

Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]

the value of α is selected to be 27 according to the above equation The surface

asperities are assumed to have a Gaussian distribution As mentioned in the modeling

section the surface geometry is thus described by two parameters Rσ and sα Based

on experimental data given in [152] the value of Rσ is chosen to be in the range of

41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of

surface contacts [84 138] show that the other parameter sα takes a value ranging from

15 to 10 It is also known that this parameter would tend to be a constant for a given type

of finishing operation [138] Without loss of generality sα = 5 is used in the calculation

According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are

00455 1104 and 1009 respectively

The combined effect of surface roughness and material properties may be

measured by the plasticity index defined by [59]

( ) 2110δσψ a= (354)

According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index

measures the relative degree of surface roughness to material strength For a frictionless

contact it is also directly related to the likelihood that plastic deformation takes place

The contact is purely elastic if ψ is substantially less than one and a significant number

of asperity contacts are plastic when ψ is around unity The results of the system contact

variables are presented next for surfaces with a number of ψ values

70

Figure 36 examines the effects of friction on the relation between the separation

and load The results are obtained for the contact at three different values of the plasticity

index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three

values of the plasticity index correspond to low medium and high degrees of surface

roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is

not affected by friction when the friction coefficient is sufficiently small particularly for

a low plasticity index With a high plasticity index however the effects of friction on the

surface separation become significant Relatively large reductions of the surface

separation are predicted particularly under high contact load The results of Fig 36 may

be analyzed by examining the asperity-scale contact behavior and its statistical

characteristics

Referring to Fig 31 the asperities with heights larger than the separation d are

in contact Among them those with heights ranging from d to 10δ+d deform elastically

when there is no friction Figure 37 shows the distribution curve of the asperity heights

normalized by aσ The area below the curve to the right of ad σ gives the percentage of

the asperities that are in contact With 00=micro the elastically deformed asperities fall in

the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve

within this interval corresponds to the population of the asperities in frictionless elastic

contact Thus the percentage of all the contacting asperities in elastic deformation eφ is

given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)

Table 32 presents the values of eφ for different plasticity indices and a number of

loading conditions defined by the surface separations

In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that

without friction the majority of contacting asperities would deform elastically When

friction is present an effective plasticity index may be similarly defined following Eq

(354)

( ) ( )[ ] 211 microδσmicroψ ae = (356)

In addition to surface roughness and material properties this effective plasticity index is a

function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an

increase of the effective plasticity index As a result some of the asperities originally in

the elastic regime now deform at least partially plastically For a friction coefficient

smaller than 30=micro the asperities experiencing the deformation transition are in the

early stage of elastic-plastic regime Their contact pressure might decrease slightly but

compensated by the friction-induced junction growth so that the load capacities of these

asperities are not reduced For a higher friction coefficient a certain percentage of

asperities go deep into the elastoplastic regime or even fully plastic The increase in the

contact area can no longer compensate the reduction of the contact pressure As a result

these asperities lose a significant part of their load capacity To support the given load

72

the separation of the surfaces is reduced to bring more asperities into contact and to have

the asperities of smaller heights carry a larger portion of the load

For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is

about 11 Referring to Table 32 a substantial population of contacting asperities

undergoes inelastic deformation at 00=micro although the majority still deform elastically

With friction the deformation becomes more severe and more asperities become

elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According

to Eq (356) the effective plasticity index only increases about 5 This implies that

there is only a small portion of asperities in severe elastoplastic deformation for the

friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant

reduction of the surface separation develops and the reduction becomes more pronounced

with a higher friction coefficient In the case of 70=micro for example the reduction

reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an

even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in

Table 32 suggest that the elastically deformed asperities only make a small contribution

to the overall load capacity in the case of 00=micro Therefore the percentage of asperities

with a decreased load capacity is significant even at a relatively low friction level Fig

36 (c) shows that a large reduction of the surface separation is generated with a modest

friction coefficient of 30=micro

The friction-induced reduction of the surface separation can be examined by

considering the load-redistribution among asperities of different heights Let the load

taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of

73

heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density

function may be defined to characterize the load distribution among asperities of different

heights and is given by

( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)

where tW is the system load Figure 38 shows the distribution function )( microzfW along

the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction

coefficients As the friction coefficient is increased the distribution curve shifts towards

the asperities of smaller heights and its peak value decreases This shift is accompanied

by the reduction of the surface separation that brings additional asperities into contact A

close examination of the distribution curves however reveals that the load carried by

these additional asperities is a small portion of the total load This portion of the load is

geometrically equal to the area below the curve to the left of point od It is 03 with

30=micro and 45 with 70=micro Thus the friction largely causes the applied load to

redistribute among the asperities that have already been in contact The shift of the

distribution curves in the manner shown in Fig 38 implies that the asperities of larger

heights give up some load which is redistributed among asperities of smaller heights

The load-redistribution is closely associated with the change of the modes of deformation

of the asperities which provides a measure of the contact severity In the case of 00=micro

about 30 of the total load is carried by the asperities in elastic contact and the

remaining by the asperities in elastoplastic deformation At 50=micro the contacting

asperities deforming elastically carry only 03 of the system load the asperities in

74

elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic

asperities As the friction coefficient is further increased to 70=micro these three

percentages change to 01 100 and 899 respectively and the contact severity is

much increased

In addition to reducing the surface separation and changing the asperity load

distribution the friction increases the total real area of contact This increase consists of

two parts One part is due to the reduction of surface separation As a result a larger

population of asperities is brought into contact and the asperities originally in contact are

subjected to higher normal approaches The other part is due to the friction-induced

junction growth of the asperities in elastoplastic and fully plastic contacts This part is

more critical as the contribution from the junction growth to the total real area of contact

reflects the degree of tangential flow and thus provides a measure of the friction-induced

contact instability The friction-induced junction growth may be characterized at the

system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)

where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart

Figure 39 shows Ajφ as a function of the contact load at different friction levels

and for the three plasticity indices The results indicate that the junction growth mainly

depends on the friction and the plasticity index and is not very sensitive to the applied

load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth

due to friction contributes very little to the total contact area for the friction coefficient up

75

to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real

area of contact tA to the nominal contact area nA is about 466 in the frictionless case

At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This

can be explained by the fact that the frictionless second critical normal approach 20δ is

very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is

larger than 200 according to Eqs (330) and (340) If there is no friction most of the

contacting asperities are in elastic deformation as shown in Table 32 The additional

tangential loading reduces both the first and second critical normal approaches and a

certain population of asperities deform inelastically Then the junction growth occurs at

these asperities The higher the friction coefficient the larger the population of asperities

in inelastic deformation and so is the contribution made by the junction growth

However even with 50=micro most of the elastically-deformed asperities are still in the

early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized

density function given by Eq (349) has a value below 4102 minustimes at an asperity height of

az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =

As a result the friction only

causes very small junction growth suggesting that the contact system with a low plasticity

index remains fairly stable up to a relatively large friction coefficient With an even

larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the

junction growth may eventually become significant At a friction coefficient of 70=micro

for example the value of nt AA becomes 57 and that of Ajφ is increased to about

10 Since this amount of junction growth is concentrated on asperities of large heights

the local instability developed at these asperities may induce some adverse tribological

76

behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much

reduced Table 32 shows that the frictionless contact already involves a significant

population of asperities in elastoplastic or fully plastic deformation The number of these

asperities is further increased by friction Thus a larger portion of the real area of contact

comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the

contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly

more severe than the case of =ψ 066 With an even higher plasticity index the friction-

induced junction growth could be much more pronounced At ψ = 186 as shown in Fig

39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a

friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of

friction-induced junction growth and tangential plastic flow would likely be a source of

tribo-instability that can lead to scuffing failure of the system

34 Summary

This paper develops an asperity-based model for the frictional sliding-contact of

rough surfaces Model equations for asperity contact variables are first derived using

theories of contact mechanics in conjunction with finite element results The equations

include the effects of friction on the modes of deformation of the asperity and asperity

pressure and area of contact The asperity-scale equations are then used to formulate a

contact model of the surfaces by means of statistical integration The model is used to

study the effects of the friction on the system contact behavior The results lead to the

following conclusions

77

1) For a contact system with a friction coefficient lower than 10=micro the friction

has little impact on the contact behavior even for a relatively rough and soft

surface with a plasticity index around =ψ 20

2) For a contact system of a given plasticity index the friction beyond a certain level

can significantly reduce the surface separation and increase the real contact of

area The reduction of the surface separation is closely associated with the load-

redistribution among asperities of different heights which increases system

contact severity

3) The percentage contribution to the real area of contact of the surfaces by the

friction-induced junction growth increases with the friction coefficient and the

plasticity index Since this increase is closely associated with the degree of

tangential flow of the surface materials it may provide a measure of friction-

induced contact instability of the tribo-system

The contact model presented in this chapter assumes a uniform friction

coefficient In reality the friction coefficient in an asperity junction may vary

significantly depending on the local contact conditions particularly in boundary

lubrication It can reach a very high value in severe situations such as metal-to-metal

contact due to the damage of boundary lubrication films The junction growth or local

instability may lead to system-level instability even though the overall friction

coefficient is not too high Therefore the surface contact model for boundary lubrication

systems should be able to take account of the variation and distribution of friction

78

coefficients among all contacting asperities A model of this ability is developed in the

next chapter based on the above modeling of contact systems with friction

79

Figure 31 Schematic of the equivalent contact system

Figure 32 Critical normal approaches and modes of asperity deformation

0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation

Elastic-plastic ( ) 102 δmicroδ

( ) 101 δmicroδ

micro

10δδ

δ

Mean plane of surface heights Mean plane of asperity heights

h sz

dz

Equivalent rough surface Rigid flat

80

Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )

(redrawn from ref [92])

αw αw

P

F

Plastically deformed region

(b) 2bi

αw αw

P

Q


(a)

∆l

81

Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis

Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting

0 02 04 06 08 101

05

1

Finite element resultsTheoretical rsults

micro

0 02 04 06 08 110-2

10-1

100Finite element resultsCurve-fitting results

micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2

Figure 36 Surface mean separation as a function of load and friction coefficient

micro = 00 ~ 03 micro = 07 nt AEW lowast

(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83

Figure 37 Asperity height distribution and mode of deformation of contacting asperities

Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )

-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025

Figure 39 Contribution of the friction-induced junction growth to the real area of contact

Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ

micro = 04 micro = 05

micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85

Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062

Table 32 Percentage of elastically-deformed asperities in frictionless contact

lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4

A Deterministic-Statistical Model of Boundary Lubrication

41 Introduction

Mathematical modeling is an important element to study the tribological behavior

of boundary-lubricated systems In boundary lubrication the surface asperities carry a

large portion of the applied load and the friction force is the sum of individual asperity-

level tangential resistance Therefore a sensible approach to model a boundary

lubrication system is to incorporate individual asperity contact solutions into statistical

descriptions of surfaces Such an approach was first proposed by Greenwood and

Williamson [59] for the frictionless contact of surfaces

Following the framework of the GW model [59] many asperity contact-based

models have been developed for the boundary lubrication system [97 101 104 105 120

and 121] In these models the system-level load and tangential force and the real area of

contact are solved by integrating the corresponding asperity-level variables For each

contacting asperity the contact pressure and area are usually determined using the

Hertzian elastic solution In comparison there are several different formulations for the

determination of the friction force at the asperity junctions For example Ogilvy [97]

calculated the local friction force by assuming constant shear strength of the interfacial

film and using the energy of adhesion Blencoe and Williams [101] related the interfacial

shear strength to the contact pressure according to empirical relations and Komvopoulos

87

[120] took account of the local resistance from both the asperity deformation and the

interfacial adhesive shearing

For the boundary lubrication systems the asperity contact-based models

developed so far have provided some insights into the effects of the rheology of boundary

layers the substrate material properties and the surface roughness on the system

tribological behavior However significant room exists for advancement in many aspects

and mathematical models with more insight can be developed First a large population of

the contacting asperities may be in either elastoplastic or fully plastic deformation

Important phenomena related to the two deformation modes such as the pressure-shear

stress coupling and the friction-induced junction growth have not been adequately

studied Second the contacting asperities under boundary lubrication are protected by

physically adsorbed or chemically reacted interfacial films The shear strength of these

films is dependent on the contact pressure and the dependence has been incorporated into

some surface contact models [101] On the other hand the adsorbed layer may be

desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts

Thus the effectiveness of boundary lubrication at an asperity junction is characterized by

intrinsic uncertainty It would be of theoretical and practical significance to capture this

uncertainty by modeling the kinetic behavior of the boundary lubricating films in

conjunction with probability theory Third the intensive shear stresses at the asperity

junctions can generate high flash temperatures which in turn affect the integrity of the

boundary films and thus the interfacial shear stresses and asperity pressure Although the

flash temperature has been calculated or measured by a number of researchers [106-115]

its interdependence with the state of the boundary films has not been studied In

88

summary the mode of micro-contact deformation the kinetics of the adsorbed layers and

the reacted films and the temperature rising due to friction are all important aspects in

boundary lubrication Although extensive work has been conducted on each of these

aspects respectively research addressing their integral effects is limited Recently a

micro-contact model [119] has been designed to fill this gap It calculates the tribological

variables during a collision of two asperities by simultaneously simulating the key

processes involved However the approach is not suitable for an asperity-based contact

model of surfaces

A mathematical model is presented in this chapter for the contact of rough

surfaces in boundary lubrication The surface contact is viewed as distributed asperity

contacts in a random process Seven asperity event-average variables are defined to

characterize an individual asperity contact in boundary lubrication The governing

equations for the seven variables are derived from first-principle considerations of the

asperity deformation frictional heating and the state of boundary films These equations

are solved simultaneously and the asperity-level solution is further integrated to calculate

the tribological variables at the system level The modeling process is described next

followed by results and discussion

42 Modeling

421 Modeling Strategy

This chapter develops an asperity-contact based model for the boundary-

lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to

the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a

89

single equivalent rough surface is used The contact between two rough surfaces is

converted to a contact between an equivalent rough surface and a rigid flat plane Each

contact point of the equivalent surface corresponds to a sliding contact between two

asperities on the original surfaces

The modeling starts by considering an individual boundary-lubricated asperity

contact illustrated in Fig 41 During the course of the contact several processes proceed

simultaneously and interact with each other in a number of ways The asperity deforms

under the combined action of tangential and normal loading The temperature in the

micro-contact rises as a result of the frictional heating The stresses and temperature

affect the state of the boundary film in the asperity junction which in turn affects the

mechanical and thermal behavior of the micro-contact Four micro contact variables are

used to characterize the asperity-level event involving these processes They are the

asperity contact pressure and area mP and 1A shear stress mτ and flash temperature

1T∆ In addition the interfacial condition of an asperity junction may be in one of three

states or their combination The asperity may be covered by the lubricantadditive

molecules adsorbed on the surface protected by surface oxides or other reacted films or

in direct contact without boundary protections Because of the intrinsic uncertainty

involved in a boundary-lubricated asperity contact it may not be possible to determine

the state of micro-boundary lubrication in absolute terms Accordingly three probability

variables introduced in [119] are used to describe this state The first variable aS is the

probability of the asperity junction covered by an adsorbed film the second variable rS

the probability of the junction protected by a reacted film and the third nS the

90

probability of contact with no boundary protection These probability variables take

values of less or equal to one and they sum to unity

1=++ nra SSS (41)

The three probability variables may be interpreted using the fuzzy set theory [154]

Taking each of the three possible contact states as a fuzzy set the corresponding

probability variable may then represent the membership degree of the interfacial film as a

whole into this set

At a given moment the random asperity contacts developed in the contact of two

surfaces are in general at different stages of asperity collision A typical asperity contact

event may be meaningfully described using the time-averages of the four micro contact

variables and the three probability variables over the duration of the contact For

simplicity the same symbols are used to represent the corresponding asperity event-

average variables The next section derives the governing equations for the seven event-

average variables based on first-principle considerations of asperity deformation

frictional heating and asperity interfacial condition Since these processes are interrelated

the governing equations are coupled and an iterative procedure is then used to solve them

for the seven event variables of an individual asperity contact Finally the system-level

tribological and probability variables are determined by statistically integrating the

asperity-level results in the random process

422 Asperity Contact and Probability Variables

Consider the junction formed during an asperity-to-asperity contact which is

represented by a single asperity contact of the equivalent surface shown in Fig 31 The

91

area of the junction and the contact pressure may be expressed in terms of the asperity

normal approach δ and the local friction coefficient lmicro Such expressions have been

derived in the last chapter for the contacting asperity in any of the three modes of

deformation elastic elastoplastic or fully plastic The pressure expression is given by

[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

ndeformatioplasticFullyH

ndeformatioticElastoplasPPP

ndeformatioElasticRE

P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)

where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal

approaches categorizing the asperity deformation into the three deformation modes The

expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in

Eq (42) are defined in the nomenclature The area of the asperity contact is given by

( ) )0()( δmicroδmicroδ llAll AkA = (43)

where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth

function due to friction Of the two functions )0(δlA is derived in ref [84] and is given

by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92

where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is

formulated in the last chapter and is given by

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)

where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in

detail in Chapter 3

At a given δ the asperity contact pressure and area may be calculated from the

above three equations if the local friction coefficient lmicro is known For the current

problem mml Pτmicro = is a variable to be determined instead of an input parameter as in

the last chapter The asperity shear stress mτ which is needed to determine lmicro may be

considered as the interfacial shear strength in the sliding junction This shear strength

generally varies with the state of micro-boundary lubrication which is characterized by

the three interfacial probability variables defined earlier It may be estimated as the

weighted average of the shear strengths of the three possible interfacial states with aS

rS and nS being the weighting factors

nnrraam SSS ττττ ++= (46)

where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted

film and with no boundary protection respectively Among them nτ may be taken as

the shear strength of the substrate material The shear strengths of the boundary layers

93

aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-

pressure relations have been obtained for different lubricantsurface pairs by experimental

studies These relations can be written as a polynomial of the form [27]

)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣

⎡+= summicroττ i = a or r (47)

where 0τ is the shear strength at zero pressure In many cases of interest its value is

small compared to other terms The coefficients and exponents of the series in this

equation are parameters characterizing the rheological properties of the boundary

lubricant layers Various specific forms of Eq (47) have been used to study the effects of

boundary-film properties on the system tribological behavior [100 101] In this study the

linear form is used as a first-order approximation

The three probability variables in Eq (46) need to be modeled to determine the

interfacial shear stress mτ The modeling makes use of two additional probability

variables One is the survivability of the adsorbed film in the course of an asperity contact

aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of

unity if the integrity of the corresponding film is intact On the other hand aS prime goes to

zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is

mostly damaged The values of aS prime and rS prime are determined by modeling the thermal

desorption of the adsorbed layer and the damage of the reacted film

The survivability of the adsorbed layer aS prime is modeled first In an asperity

junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]

94

and substrate plastic deformation [26] It is sensible to equal the survivability of the

adsorbed layer to its fractional surface coverage which has been used to characterize the

effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an

appropriate adsorption model may be selected to determine aS prime based on the fundamental

aspects of the structure of adsorbed molecules and the interactions among them Of the

adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is

energetically uniform and no lateral interactions are involved between adsorbed

molecules It has the advantage of giving a simple equation for the adsorption process

and being used to directly analyze the experimental results [18] Therefore the

Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)

For a given contact pressure and temperature aS prime is solved from the above equation by a

numerical method

Next consider the survivability of the reacted film rS prime during an asperity contact

The film may be ruptured resulting from the destruction of the chemical bond between

the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm

bonding ft The bonding can be broken up by adsorbing the thermal energy from

frictional heating andor the distortion energy due to shearing According to the thermal

fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation

[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)

where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13

sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its

activation or fluctuation volume in which active failure occurs and eσ the effective

stress and lT the junction temperature representing the mechanical and thermal loading

on the film Since the rupture of the reacted film is more likely developed along the

interface the effective stress eσ in Eq (49) may be directly related to the interfacial

shear stress mτ In addition the film rupture usually starts from a micro defect in the

asperity junction and the micro defect may be viewed as a micro crack The development

of the micro crack is then controlled by the shear stress within a small element at the edge

of the crack Due to the existence of the micro crack eσ or the maximum shear stress at

the interface may be expressed as

mse C τσ = (410)

where sC is a factor reflecting the intensification of the shear stress within a small

element at the edge of a micro crack This factor is of the order of ddl λ where dλ is

the size of the small element at the crack edge and of the order of interatomic spacing or

100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value

of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may

now be estimated by comparing ft with the duration of the contact which is given by

96

Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability

that the reacted film will fail within t∆ is given by

fr ttS ∆=primeminus1 (411)

and the corresponding survivability of the film is equal to

fr ttS ∆minus=prime 1 (412)

Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through

the asperity contact is then given by

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)

The survivability in this form may also be deduced from the exponential failure-time

distribution model [156]

The two survivability variables aS prime and rS prime are now used to determine the three

contact probability variables According to the analysis by surface enhanced Raman

spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant

molecules usually occurs on the top of the reacted film Thus there is no effective

protection for the substrate surface if the reacted film is damaged and the probability of

contact without boundary protection is given by

rn SS primeminus= 1 (414)

97

By Eq (41) rS prime can then be expressed as the sum of aS and rS

rra SSS prime=+ (415)

The probability of contact covered by an adsorbed layer may then be written as

ara SSS primeprime= (416)

Combining Eq (415) and (416) the probability of contact protected by the reacted film

is given by

( )arr SSS primeminusprime= 1 (417)

Six of the seven asperity event-average variables have been modeled above The

last one the contact temperature lT in the asperity junction needs to be determined In

general lT comprises two components

lbl TTT ∆+= (418)

where bT is the bulk temperature and lT∆ is the flash temperature caused by the

frictional heating in the asperity contact In this study the bulk temperature is taken to be

an operating parameter while the flash temperature is determined based on a model

developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the

elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le

the pressure distribution at the asperity junction is parabolic and so is that of the shear

stress The flash temperature is thus calculated with a parabolic circular heat source and

is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆

τ ( )lmicroδδ 1le (419)

where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a

plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly

distributed over the asperity junction The expression for lT∆ is then derived with a

uniform circular heat source and is given by

2211 658065806880

ecec

ml PKPK

VaT

+++=∆

τ ( )lmicroδδ 2ge (420)

Additional derivation is needed for the elastoplastic contact with a normal approach of

( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as

the combination of a parabolic heat source and a uniform one It is sensible to assume the

corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore

a generalized expression of the flash temperature for the whole range of normal approach

is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)

In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as

TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge

and are called TpD and TpG respectively For an elastoplastic contact TD and TG may

be approximated by linear interpolation and are given by

99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12

1 ( ) ( )ll microδδmicroδ 21 ltlt (422)

and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12


The above modeling process provides a complete set of equations for the contact

and probability variables that characterize a single asperity contact under boundary

lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area

lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact

probability variables Equation (421) provides an expression for the flash temperature

lT∆ Supplementary equations are also developed to determine other variables involved

in the seven key equations such as the two survivability variables aS prime and rS prime Each one

of the modeling equations is coupled with some others and some of them are highly

nonlinear Thus these equations can only be solved iteratively for given material and

lubricant properties asperity geometry asperity normal approach and sliding velocity

Starting from initial estimates of the three interfacial probability variables an iteration

procedure is outlined below

1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear

stress for given normal approach and contact probability variables

2) Calculate the flash temperature lT∆ from the frictional asperity contact solution

using Eq (421)

100

3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)

4) Estimate the survivability of the reacted film rS prime using Eq (413)

5) Determine the three contact probability variables using Eqs (414) (416) and

(417)

6) Calculate the shear stress mτ using Eq (46)

7) Check the convergence by comparing the current shear stress result with its

previous value If the accuracy requirement is satisfied stop the iteration

Otherwise go back to step 1)

This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration

the seven asperity event-average variables and other supplementary variables are

determined They are the solution of an individual asperity contact


The tribological variables of the boundary lubrication system are determined next

Given a surface separation Fig 31 shows that there are many numbers of asperity

contacts of different normal approaches The variables in each of these contacts may be

determined using the procedure described in the preceding section The following

statistical integrals are then used to model the asperity-contact random process to

determine the load friction force and the real area of contact at the system level

( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin

η (424)

101

( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin

minusminus= τη (425)


minus=η (426)

where z is the height of the asperity ( )zf its probability distribution d the distance

from the mean plane of asperity heights to the rigid flat and dz minus the approach of the

rigid flat to the asperity or δ With the system load tW and friction force tF determined

the system-level friction coefficient may be calculated by

ttt WF=micro (427)

In addition the asperity-level probability variables may be integrated to generate a group

of system-level probability variables to measure the overall effectiveness of boundary

lubrication For example the system-level probability of contact with no boundary

protection and the system-level survivability of the reacted film and that of the adsorbed

layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102

Similarly the mean flash temperature among the contacting asperities may be calculated

by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)

The three system-level contact variables tW tF and tA may be normalized by

system parameters Their dimensionless expressions are given by

( ) ( ) ( ) ( )

dzzfdzAdzPdWd lmt intinfin

minusminus= β (432)

( ) ( ) ( ) ( )

dzzfdzAdzdFd lmt intinfin

minusminus= τβ (433)

( ) ( ) ( )

dzzfdzAdAd tt intinfin


where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =

ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31

of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ

The system-level probability variables and the mean flash temperature may also be

expressed in a similar dimensionless manner as follows

( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)

Finally assume that the asperity heights have a Gaussian distribution of standard

deviation aσ Their probability distribution function is given by

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)


system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of

them β aσσ and sz are related to the parameter measuring the spectrum bandwidth

of the surface roughness or sα Their expressions in terms of sα are given by [138]

πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)

It should also be noticed that the asperity flash temperature is related to the

absolute value of the contact size according to Eq (421) Thus the asperity radius R

needs to be given Based on the surface descriptions in refs [122 138] the area density

of the asperities η is specified and then R determined from Eq (441) in conjunction

with the Rσ parameter Therefore the surface roughness is characterized by three

independent parameters sα Rσ and η

43 Result Analysis

The model is used to study the sliding contact behavior between two rough

surfaces in boundary lubrication The results are obtained and presented for a set of

surfaces characterized by their plasticity indices and a range of system load and sliding

velocity


1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ

and tensile strength 3HY = The constant α in Eq (42) was estimated to be around

27 in the last chapter The substrate thermal properties are defined by the thermal

conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK

Two parameters are used to describe the surface adsorption of the lubricant molecules

They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate

The value of aH∆ is taken to be 40kJmol corresponding to relatively strong

105

physisorption of the lubricantadditive to the surface [159] The value of m is assumed to

be 600amu representative of the combination of general lubricants and additives [160]

Two other parameters the bond destruction energy rH∆ and the activation volume γ

are used to characterize the reacted film on the surface The value of rH∆ is chosen to be

120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the

experimental results of polymers [155] considering that the reacted film can be viewed

as high-molecular-weight organo-metallic polymers [161 162] The proportional

constant relating the interfacial shear strength to the asperity pressure in Eq (47) is

chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which

are reasonable values [163] The surface asperities are assumed to have a Gaussian

distribution As mentioned in the modeling section the surface geometry of this

distribution is described by three parameters Rσ sα and η Based on experimental

data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to

31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as

discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β

aσσ and sz are 00455 1104 and 1009 respectively The area density of surface

asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study

-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally

operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C

The effect of contact force on the system friction is studied first A higher load

dependence of the friction would suggest a higher degree of tribo-instability of the

boundary lubrication system Figure 43 shows the results for surfaces of different

106

degrees of roughness represented by a series of plasticity indices ψ = 066 093 186

and 255 The plasticity index is defined by [59]

( ) 2110δσψ a= (444)

where 10δ is the first critical normal approach of a frictionless asperity contact with

which plastic yielding takes place In this study the values of the plasticity index chosen

above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and

31051 minustimes respectively For the relatively smooth surface with a low plasticity index the

results show that the friction coefficient at the system level is low and is almost

independent of the load At ψ = 066 for example the value of tmicro varies very slightly

around 0055 This value is close to the assumed ratio of the shear strength of the

adsorbed layer to the contact pressure It suggests that the surface is well protected by an

adsorbed layer of lubricantadditive molecules and the corresponding system-level

survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further

examination shows that most of the contacting asperities deform elastically The

correlation between the system tribological behavior and its asperity level origin will be

discussed in detail later In the case of ψ = 093 the mode of deformation of the

contacting asperities are basically elastic or early elastoplastic and similar results of the

system friction coefficient are obtained On the other hand the system friction coefficient

increases with the load for systems of plasticity index significantly higher than unity At

ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from

5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of

107

overall surface protection rtS prime decreases from nearly unity to 967 The probability of

unprotected contact at the system level ntS emerges and it is about 33 at the high end

of the load This probability is small but mainly contributed by the few asperities of large

heights which are in fully plastic deformation This group of asperities would carry a

significant portion of load if they are well protected by the boundary films However the

protection becomes damaged in these junctions and the shear stress approaches the shear

strength of the substrate As a result these asperities lose their load carrying capacity

causing the significant increase in the system friction coefficient With an even higher

plasticity index of ψ = 255 the friction coefficient at the system level increases

dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for

the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system

friction coefficient is above rmicro = 015 which is the assumed shear strength-contact

pressure ratio of the reacted film At this load a close examination reveals that the

boundary lubrication fails in a significant number of asperity junctions The

corresponding value of the probability of surface protection is about 994=primertS The

probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more

asperities lose their load capacity as the boundary films in these junctions are deteriorated

leading to the drastic increase of the friction which suggests a possibility of tribo-

instability

It should be pointed out that each of the above four groups of results is obtained

for a constant plasticity index In reality the continuous operation may change the

roughness of the bearing surfaces and the properties of the near-surface material leading

108

to an increasing or decreasing plasticity index A reduction of the plasticity index

corresponds to a healthy run-in process while an increase indicates some tribo-instability

For a given system the current model may be used to determine whether a run-in process

is needed by studying the friction behavior around the intended operating point If the

friction coefficient is sensitive to the operating parameters such as load or sliding velocity

the system should go through a run-in period at mild conditions to reduce its plasticity

index On the other hand the run-in may not be needed if the friction coefficient is

insensitive to the operating conditions as a result of the combined effects of boundary

lubricant material and surface finish

The behavior of the system friction with the load is rooted in the scattering

tribological behavior of distributed asperity contacts Figure 44 presents the shear stress

in an asperity junction as a function of asperity height the probability distribution

function of the asperity heights is also shown in the figure for reference The analysis is

performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For

each system the results are presented at three values of the surface separation =σh 05

10 and 20 which are used to represent different levels of loading In the system with ψ

= 066 almost all the contacting asperities deform elastically for the three given values of

σh The asperity pressures are not very high and the areas of contact are relatively

small In these asperity junctions both the adsorbed layer and the reacted film are largely

intact The interfacial shear stress increases continuously with the asperity height and the

asperity-level friction coefficients are slightly higher than amicro = 005 At the given

nominal sliding velocity of =V 10ms only low flash temperatures are generated The

low pressure friction and flash temperature of the asperity contacts suggest that there is

109

no significant coupling among the deformation the frictional heating and the condition

of the boundary films The contacting asperities can thus be viewed as very stable At the

system level the resulting friction coefficient also has a value close to amicro = 005 and it is

almost independent of the load as shown in Fig 43 Next the tribological behavior of the

asperity contacts is examined for the relatively rough system of ψ = 186 When the

asperity height is below some critical value Figure 44 (b) shows that the shear stress in

the asperity junction also increases continuously with the height similar to the case of ψ =

066 The asperities in this group may be considered as stable For the asperities with a

height above a critical value the shear stress jumps to a value close to the shear strength

of the substrate A close examination of the results reveals that these asperities are in

fully plastic deformation as a result of the strong coupling among the physical and

chemical processes involved The frictional heating accelerates the thermal desorption of

the adsorbed layer and the rupture of the reacted film The damage of these films in turn

increases the interfacial shear stress as well as the frictional heating Consequently the

boundary films in these asperity junctions fail to provide effective protection The shear

stress then approaches the substrate shear strength and the asperity contact pressure is

largely reduced leading to a high asperity-level friction coefficient This group of

asperities may thus be considered as unstable The size of the group is measured by the

area ua shown in Fig 44 (c) which increases as the surface separation decreases The

above two groups of results show that the emergence of unstable contacting asperities

and their population are related to the value of the plasticity index and the load The

system tribological behavior is thus also affected by these two parameters In practice the

possible variation of the plasticity index during the operation may significantly change

110

the number of the unstable asperities For example a successful run-in process reduces

the plasticity index and pushes to the right the critical position of the shear stress-asperity

height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a

low level so that they do not induce a tribo-instability to the system

It is interesting to examine how the condition of boundary lubrication may affect

the surface separation and the real area of contact of the system from the results of a

frictionless contact For illustration purposes the sliding velocity between the two

contacting surfaces is used to alter the condition of the boundary lubrication which may

be defined by the probability variable rtS prime of the overall boundary-film protection

Figure 45 present the rtS prime results as a function of the applied load for two sliding

velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area

of contact are also presented under these conditions as well as for frictionless contacts At

a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible

effect on the boundary film and the value of rtS prime decreases only slightly from 999 to

987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently

the calculated surface gap and the real area of contact are essentially the same as those

calculated assuming frictionless contact For heavier loads the sliding velocity may

increasingly deteriorate the boundary-film protection by thermal desorption of the

lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted

surface film As a result the asperity load capacity may be reduced leading to a

significant decrease of the surface separation and significant increase of the real area of

contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film

111

protection is 198=primertS with =V 10ms and decreases to 387=primertS when the

sliding velocity increases to =V 40ms For =V 10ms the gap between the two

surfaces is about the same as that for frictionless contact but it is reduced by about 27

when the system slides at =V 40ms Similar results are shown for the calculated real

area of contact With =V 40ms the area increases more than 50 from that for the

frictionless contact It should be pointed out that this increase is largely due to tangential

plastic flow of the asperity contacts that lose the boundary-film protection and it may

play a key role in the system tribo-instability An analysis of the contributions of the

tangential plastic flow to the real area of contact is presented in Chapter 3

The model may also be used to study the tribological behavior of the boundary

lubrication system in key parameter spaces The load and the sliding velocity are chosen

to define a key space since it is of particular interest to determine the limits of the two

operating parameters as guidelines for the design of tribological components [164 165]

Figure 46 presents the contours of the system friction coefficient tmicro and surface

protection probability rtS prime in this operating space The results show that the value of tmicro

increases with the two operating parameters and that of rtS prime decreases In addition a

given level of friction coefficient usually corresponds to a specific level of boundary

protection and is also related to a certain degree of plastic deformation

Considering 20=tmicro for example the corresponding value of the surface protection

probability is around 90=primertS and about 30 of the real area of contact is due to the

asperities in fully plastic deformation Based on experimental observations the surface

and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]

112

The scuffing may be more attributed to the tangential flow of the plastically deformed

asperities which may be measured by the contribution of the junction growth to the real

area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two

contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the

boundary lubrication system Accordingly the load-velocity plane may be divided into

two different regions In the high load-high velocity region the contours crowd together

and exhibit high gradients between adjacent levels The system may have a high

possibility of instability Left to this region this possibility decreases as the friction

coefficient and surface protection probability become insensitive to the two operating

parameters The transition regime between the above two regions may define the limits of

safe operation This transition regime has been related to the critical temperature for a

system in which the tendency to failure is controlled by the competitive formation and

removal of oxides [45] For a more general system considered in the current study the

transition regime may correspond to a critical level of plastic deformation or junction

growth which needs to be determined experimentally

It should also be mentioned that the above results are obtained for given bulk

temperature and surface plasticity index In reality the bulk temperature may be elevated

under high load andor high velocity since the system cooling in these severe situations is

not as effective as in the mild operations As a result the operating conditions may have

more dramatic effects on the system behavior in the high load-high velocity regime For

example the system friction coefficient may become even higher and its contours may be

more crowded compared to the results presented in Fig 47 (a) Separately the plasticity

index of the bearing surfaces may either increase or decrease during the operation The

113

pattern of the two types of contours and the region of high tribo-severity may thus change

accordingly Although limited by the lack of reliable data about the above two factors

more insight may be gained into their effects on the lubrication performance and the

effects of other factors through a systematic parametric study with the current model

Insights may also be gained by further developing the model considering the thermal

balance and the progression of surface topography

44 Summary

An asperity-based model is developed for the sliding contact of two rough

surfaces in boundary lubrication Four variables are used to describe an individual

asperity contact including micro-contact area pressure interfacial shear stress and flash

temperature Furthermore three probability variables are used to define the interfacial

state of the asperity junction The asperity-level modeling equations are derived from the

theories of contact mechanics flash temperature kinetics of boundary films and random-

process probability These equations are then used to formulate a contact model of the

surfaces by means of statistical integration Results from the model may be summarized

in the following

1) For relatively smooth and hard surfaces the boundary lubrication is effective at

both the asperity and system levels over a relatively wide range of load and

sliding velocity The resulting system friction coefficient is low and insensitive to

load and speed

2) For relatively rough and soft surfaces a significant group of contacting asperities

may lose boundary-film protection and experience a high level of local friction

114

At a given sliding velocity the number of these unstable asperities increases with

the load leading to a significant increase in the system friction coefficient

3) For a given system a friction coefficient sensitive to the operating parameters

suggests that the system should go through a run-in period to reduce the surface

plasticity index and thus the number of unstable asperity contacts On the other

hand the run-in may not be needed if this sensitivity is absent

4) The condition of boundary lubrication may strongly affect the system contact

behavior Under a given load an increase in the sliding velocity may deteriorate

the boundary-film protection leading to a significant decrease of the surface

separation and a significant increase of the real area of contact

5) The space of operating parameters may be divided into two regions according to

the tribo-severity evaluated from the contour pattern of the system friction

coefficient or the surface protection probability in this space The transition

between these two regions may be related to a critical degree of asperity plastic

deformation or junction growth

A more systematic parametric study can be conducted with the current model to

gain more insights into the effects of material and lubricant properties in boundary

lubrication The structure of the model is flexible enough for further development and

improvement by incorporating research advances in contact mechanics tribochemistry

and other related fields

115

Figure 41 An individual boundary-lubricated asperity contact

116

|error| lt ε

End

Initial guess of local contact probabilities

Start

Solve Pm Al and microl from Eqs (42) ndash (45)

Calculate ∆Tl with Eq (421)

Calculate Sa with Eq (48)

Calculate Sr with Eq (413)

Calculate Sa Sr and Sn with Eqs (414) (416) and (417)

Calculate τm with Eq (46)

error = τm ndash τm


τm = τm

Figure 42 Flowchart for the determination of the solution of an asperity collision

117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08

Figure 43 System-level friction coefficient as a function of load

( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05

Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution

( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012

Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact

(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07


(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)

(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5

Summary and Future Perspective

This thesis research develops an interdisciplinary surface contact model for

boundary lubrication systems based on a balanced consideration of key processes of

different natures involved in the contact The major efforts and conclusions of the

research are summarized below along with visions of future trends

51 The Deterministic-Statistical Model

The modeling process consists of three successive phases which are outlined as

follows

1) Finite Element Analysis of a Single Frictional Asperity Contact

A systematic finite element analysis is first carried out to study the effects of

friction on the deformation behavior of a single asperity contact The results show that

the friction in contact can significantly affect the mode of asperity deformation With a

relatively high friction coefficient the contact may change from the state of elastic

deformation to the state of fully plastic deformation with little elastic-plastic transition as

the contact force increases The friction can also significantly change the shape and size

of plastically deformed zone At high friction coefficients the plastic deformation is

largely confined to a thin surface layer in the contact In addition the friction causes the

reduction of pressure and the growth of asperity junction in the case of elastoplastic or

fully-plastic contact These results are presented in the dimensionless form and the

conclusions drawn from them are sufficiently general The insights gained in the analysis

122

are used in the second part as a foundation for the analytical modeling of frictional

asperity and surface contacts

2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction

A statistical asperity-based model is developed for the frictional contact between

two nominally flat surfaces using the finite element results in the first part and the theory

of contact mechanics This model significantly advances the Greenwood-Williamson

types of system contact models by adding the dimension of friction as well as

incorporating the three possible modes of asperity deformation The model is able to

capture the essential effects of friction on the surface contact behavior These effects are

reflected by the reduction of surface separation and the increasing real area of contact

The model is also able to determine the contribution from the friction-induced junction

growth to the real area of contact The level of this contribution may be a measure of the

system tribo-instability Moreover the model provides a basis for further refinement and

development Although assuming a uniform friction coefficient at the interface it lays a

foundation for the study of boundary lubrication in which the friction may vary

dramatically among contacting asperities

3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact

The third part of the modeling process is the core of this thesis It models the

boundary-lubricated surface contact by incorporating the physicochemical and thermal

aspects of the problem into the mechanical contact model developed in the second part

In this interdisciplinary model an individual asperity contact under boundary lubrication

conditions is viewed as an event A group of deterministic and probabilistic variables are

123

defined or selected to characterize such a contact process or event The governing

equations for these variables are derived based on a balanced consideration of asperity

deformation frictional heating and the kinetics of boundary films These asperity-level

equations are solved iteratively and the solution is then integrated to formulate the

contact model for the boundary lubrication system This model is capable of relating the

system tribological behavior defined by the friction coefficient the real area of contact

and the effectiveness of boundary films to surface roughness operation conditions and

material and lubricant properties It is thus able to evaluate the safety of operation and the

tribo-stability through parametric study or sensitivity analysis regarding the range of

different factors Furthermore the modeling equations of asperity variables and their

solution as well as the statistical integration can be viewed as interrelated modules The

model is thus an open-ended framework allowing each module to be updated by

incorporating research advances in related fields Some possible directions of future

development are discussed in the next section

52 Perspective on Future Development

The final model developed in this thesis provides a tool to study the tribological

behavior of the boundary lubrication system in a greater depth of understanding than any

previous model One of the immediate applications of the model is a systematic

parametric study or sensitivity analysis on the effects of various important factors

involved in the boundary-lubricated contact An example is the analysis carried out in

Chapter 4 on the contour of the system friction coefficient and that of the degree of

boundary protection in the operation space defined by the load and sliding velocity

These contour patterns may reveal insights into the tribo-instability of the system and the

124

safety of operation More insights may be gained into these two issues by conducting

similar parametric study with the model on different groups of factors In this way the

coupling effects and relative importance of each group of factors can be easily identified

The insights provided by the parametric study may help define the guidelines for

controlling the tribo-severity

The model also provides a framework which may be refined or extended in many

different ways This framework is developed with a flexible structure consisting of a few

interrelated modules The model may thus be improved at the asperity level andor the

system level by updating individual modules and refining their interaction For example

the current model assumes that the asperity contacts are independent of each other and

they are not affected by previous ones Thus one way to improve the asperity-level

modeling is to consider the mechanical and thermal interaction among neighboring

asperity contacts The other way is to consider the cumulative effects of consecutive

contacts on the asperity flash temperature and the effectiveness of boundary lubrication

In addition the competition between the formation and the rupture or removal of the

boundary films may be considered to refine the model For this purpose it is important to

include in the model the up-to-date and balanced information about the properties and

behavior of these films At the system level the surface plasticity index and the bulk

temperature are currently taken to be fixed parameters In reality they may either

increase or decrease during the contact process depending on the operation conditions

material properties and other factors Their evolution may significantly affect the

dominant deformation mode of contacting asperities and the state of boundary

125

lubrication Therefore a possible extension is to capture the trends of evolution by

modeling the global thermal balance and the progression of surface topography

The further development of the model may be related to its structure which is

characterized by the way to describe the surface topography The current model combines

the statistical surface descriptions with the ability to take account of interactive micro-

mechanical physicochemical and thermal processes involved in the contact This ability

is the core of the model and it may also be combined with the fractal or deterministic

types of surface descriptions to develop the corresponding surface contact models

Moreover a contact model of a totally new structure may be developed by viewing the

interfacial contact region as a network whose nodes are the asperity junctions From the

network point of view the system failure damage such as scuffing may be taken to be the

catastrophic collapse starting from a small number of nodes As summarized by Johnson

[166] many social artificial and natural networks crash in such a way These complex

systems have also been found to be similar in their structures and inter-node linkages

following some universal organizational principles The contact model of network

structure may open a new window to the boundary lubrication system and then lead to a

more insightful understanding of its failure mode and tribo-severity

126

Bibliography

1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18

2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk

Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225

3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary

Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849

4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications

Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the

Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14

6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin

Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I

Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films

of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148

9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)

331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their

Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334

11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430

12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE

Transactions 8 (1) 1-11

127

13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116

14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their

Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in

Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519

16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a

Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and

Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication

and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and

Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58

20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary

Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces

by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the

Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515

23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo

Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo

Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-

Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of

Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158

128

27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48

28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of

boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series

42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier

Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments

and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271

32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo

Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461

33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions

in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82

34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc

Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395

35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of

oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate

Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178

37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in

Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327

38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters

3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme

Pressure Lubricationrdquo Wear 108 (2) 185ndash199

129

40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article

in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999

ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203

42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP

Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces

with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship

between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on

Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628

46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength

measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore

bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156

48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145

49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating

Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411

50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics

Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films

under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122

52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA

1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198

130

53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-

mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157

54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical

Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47

55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and

nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162

56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive

Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis

of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81

58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple

Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat

Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-

geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417

61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-

Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of

Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76

63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness

Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216

64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications

to Engineering Surfacesrdquo Wear 180 (1) 17ndash34

131

65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11

66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal

Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial

Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822

68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial

Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832

69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal

Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless

Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough

Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic

Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for

Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as

Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74

75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough

Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough

Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact

Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250

78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the

Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42

132

79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded

Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in

Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65

82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on

Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an

Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model

Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93

85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the

contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of

Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface

Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo

Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined

Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393

90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and

Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of

the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the

Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402

133

93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742

94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear

Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of

Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499

96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface

Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121

97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough

Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a

Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible

and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying

Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664

101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying

Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of

Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90

103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between

surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational

Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140

134

105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545

106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual

Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235

107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding

Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)

438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in

Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding

Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International

Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an

Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-

Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash

Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures

in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of

Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55

117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-

Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318

118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature

Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829

135

119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact

Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15

120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated

Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291

121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface

Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of

Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a

Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-

Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical

Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201

126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads

on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a

Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding

Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59

129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-

Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element

Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375

131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on

the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354

136

132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122

133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal

and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp

Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis

Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-

Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and

a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo

Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the

Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124

140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces

of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a

Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic

Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces

having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38

144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the

Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient

Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63

137

146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281

147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method

Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572

148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model

Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240

149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo

Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal

of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of

molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat

Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)

Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)

Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International

Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)

Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and

Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193

158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and

oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press

London

138

160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris

161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984

ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404

162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature

of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430

163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction

to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological

Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding

Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York

Times Company httpwwwracemattersorgcomplexsystemshtm

VITA

Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong

University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in

the School of Power and Energy Technology in Jiaotong University Xirsquoan

In August 1999 the author came to the Pennsylvania State University for the

PhD program in Mechanical Engineering He has been a Graduate Research Assistant in

the Tribology Group since then He also worked as a Graduate Teaching Fellow for one

semester

Huan Zhang is a student member of STLE (the Society of Tribologist and

Lubrication Engineers)

The thesis of Huan Zhang was reviewed and approved by the following

Liming Chang Professor of Mechanical Engineering Thesis Advisor Chair of Committee

Marc Carpino Professor of Mechanical Engineering

Seong H Kim Assistant Professor of Chemical Engineering Richard C Benson Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering

Signatures are on file in the Graduate School

iii

ABSTRACT















modeling






iv


















v

TABLE OF CONTENTS

List of Figures vii

List of Tables ix

Nomenclaturex

Acknowledgementsxii








24 Summary37





vi

34 Summary76







Bibliography 126

vii

List of Figures













80






viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



iii

ABSTRACT















modeling






iv


















v

TABLE OF CONTENTS

List of Figures vii

List of Tables ix

Nomenclaturex

Acknowledgementsxii








24 Summary37





vi

34 Summary76







Bibliography 126

vii

List of Figures













80






viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



iv


















v

TABLE OF CONTENTS

List of Figures vii

List of Tables ix

Nomenclaturex

Acknowledgementsxii








24 Summary37





vi

34 Summary76







Bibliography 126

vii

List of Figures













80






viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



v

TABLE OF CONTENTS

List of Figures vii

List of Tables ix

Nomenclaturex

Acknowledgementsxii








24 Summary37





vi

34 Summary76







Bibliography 126

vii

List of Figures













80






viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



vi

34 Summary76







Bibliography 126

vii

List of Figures













80






viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



vii

List of Figures













80






viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



viii


asperities 83


of contact 84





119



120

ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

121

Chapter 5








follows













122























123
























124























125


















126

Bibliography



















127



















128





















129




















130





















131




















132


















133


















134



















135




















136


















137


















London

138











VITA







semester



ix

List of Tables


x

Nomenclature






2

22

1

21 11

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus+

minusEEνν



















123

210002

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛sdotsdot

ϑπ

A

bb N

TmkTk


xi



















xii

Acknowledgements





















xiii











experience





years ago

Chapter 1

Introduction


















2




















3
























4


Review





















5























6



















sum+=j

njb

jPmicroττ 0 (12)



7






results














n

nrno t

RTQ

Ahf1

exp ⎥⎦

⎤⎢⎣

⎡∆sdot⎟

⎠⎞

⎜⎝⎛minus=∆ρ n = 1 or 2 (13)

8
























9













polyphosphates











10
























11
























12























13














contact spots










14
























15























16






















contact modeling


17






















18

125 Summary



















to the system load



19























20
























21







22

Chapter 2



21 Introduction









just a few









23

















extreme







24





capacity


















25






( ) ( )xpx microτ = (21)

















26
























27









contact














28




plasticity



















29






















30


contact




222 Hp =+ατ (22)
















31























32





















region

33























34























35









plastic flow















36























37




24 Summary



















38





















39

Rigid flat

δ




40



35

0 02 04 06 08 1 0

5

10

15

20

25

30

35

40 δ1δ10

δ2δ10 (a)

0 02 04 06 08 1 10 -1

10 0

10 1

10 2

δ1 δ10 δ2 δ10

Crit

ical

nor

mal

app

roac

hes

(b)

Crit

ical

nor

mal

app

roac

hes


41



(a)

(b)

Contact width


Rigid flat

Asperity

42



(a)

(b)

Contact width

Friction force

43

(a)



Contact width

(b)

44



(b)

Contact width (a)

45

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


46

-3 -2 -1 0 1 2 3 0

05

1

15

micro=10

micro =07

micro =038


Contact size

Fric

tion

coef

ficie

nt


47

0 02 04 06 08 10

05

1

15

2



Con

tact

var

iabl

es


48

Chapter 3


Friction

31 Introduction


















49
























50
























51



32 Modeling

321 Model Structure







asperity

dz minus=δ (31)





mm Pmicroτ = (32)



52






















53











then given by [79]

( )21

34 ⎟

⎠⎞

⎜⎝⎛=

REPm

δπ




==




( ) HP m ==0

micro


54







( )( ) 2121

αmicro

microδ+







( ) ( )1020

10

lnlnlnln

δδδδ

δminusminus







55

( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnlnminus









RAl ==0

10δδ le (313)



given by


RAl 20=

= 20δδ ge (314)





RAl )231( 320


2010 δδδ lele (315)


56






=micro











Or





range to give

57

( ) ( )[ ] ( )( ) ( )microδmicroδ


12

1

lnlnlnln11minus







( )( ) 2121

αmicro

micromicroδτ+

=H















58









approximated by

( )w

ibl

αsin=∆ (321)



( ) iw

bl 2sin2

11 ⎥⎦

⎤⎢⎣

⎡+=

α (322)


( )wAlk

αsin2110 += (323)




59











( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

geminusminus

sdotminus+


cmc

cmAlcAlAlc

cmc

mAlc

mAl

kkk

kk

ττττττ

ττττ

τ

00

011 (324)




( )( )

( )( ) ( )

( )( )⎪⎪

⎩

⎪⎪

⎨

⎧

ge+minus

+minusminus+

ltle+

minus+

=

c

c

cAlcAlAlc

c

c

Alc

Al Hkkk

Hk

kmicromicro

αmicroττ

αmicroτmicro

micromicroαmicroτ

micro

micro

2120

212

0

212

1

1

01

11

(325)


60

212)(

minus

⎥⎦

⎤⎢⎣

⎡minus= α

τmicro

cc

H (326)















3

2

2YJ = (327)



61

( ) ( ) ( )[ ] 222222


σσσσσσ+++




YkP YmY = (329)



==


REYkY

2

2

10 43

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

πδ (330)






Tij


where Nijσ and T






62








approach

( )R

EZYXJJ Lδ

πmicro

2

22 34 ⎟⎟

⎠

⎞⎜⎜⎝





( ) ( )3

34 21

2

max2 YR

EJ L =⎟⎟⎠

⎞⎜⎜⎝

⎛ microδπ

micro (335)


( ) ( ) REY

J L

2

max2

1 43

⎟⎠⎞

⎜⎝⎛=π

micromicroδ (336)


63


max2

max21

0

L

L

JJ

=prime (337)










402 asymplowast

YRaE (338)



( ) 21202 2 δRa = (339)


by

REY 2

20 800 ⎟⎠⎞


64














( )( )

( )( )

11

2

1

2

=

=micro

microδmicroδ








65

( )

( )

( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ge+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast

microδδαmicro



microδδδπ

microδ

2212

2212

1

1

21

1

lnlnlnln

34

H

PPP

RE

P mYmFmYm

(343)



repeated below

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (344)

and

( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln11

01

Al

AlA

k

kk (345)







66


















( ) ( ) ( )



( ) ( ) ( ) ( )



67







( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(348)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (349)





[138]

πα

σηβ sR3

481

== (350)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (351)

68

( ) 21

4

ssz

πα=lowast (352)


Rσ

33 Result Analysis















( )20τα H= (353)

69














( ) 2110δσψ a= (354)







70











characteristics









given by

71

( )( )int

intinfin

+

=

10

d

d

de

dzzf

dzzfδ

φ

(355)






(354)













72























73




( ) ( ) ( )zfWzW

zft

lW

micromicro

= (357)


















74




much increased










system level by

( ) ( )( )micro

microφ

0

dAdAdA

t

ttAj

minus= (358)







75
























76













34 Summary









77








contact severity














78



79



0 02 04 06 08 1 10

-1

10 0

10 1

10 2

Fully plastic

Elastic deformation


( ) 101 δmicroδ

micro

10δδ

δ


h sz

dz


80



αw αw

P

F


(b) 2bi

αw αw

P

Q


(a)

∆l

81



0 02 04 06 08 101

05

1


micro

0 02 04 06 08 110-2

10-1


micro

δ2δ20

δ1δ10

82

0 2 4 6x 10-4

05

1

15

2

0 2 4 6 8x 10-4

05

1

15

2

0 02 04 06 08 1

x 10-3

05

1

15

2



(a) ψ = 066

nt AEW lowast

(b) ψ = 093

nt AEW lowast

micro = 00 ~ 02

micro = 04

micro = 07

micro = 03

micro = 0 ~ 01

σh

(c) ψ = 186

micro = 07

micro = 05

σh

σh

83



-4 -2 00

01

02

03

04

05

(d+δ10)σa

I II III

f(zσa)

2 4 dσa

zσa

-1 0 1 2 3 4 5 6 70

02

04

06

08

Wf

az σ

30=micro

00=micro

70=micro

od

84

0 2 4 6x 10-4

0

005

01

015

02

025

0 2 4 6x 10-4

0

005

01

015

02

025

0 02 04 06 08 1x 10-3

0

005

01

015

02

025


Ajφ

nt AEW lowast

nt AEW lowast

nt AEW lowast

Ajφ

Ajφ


micro = 07

micro = 04

micro = 07

micro = 02

micro = 04

micro = 07

(a) ψ = 066

(b) ψ = 093

(c) ψ = 186

micro = 03

85



lowasth

ψ 05 075 10 15 20

066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367

86

Chapter 4


41 Introduction


















87
























88








model of surfaces










42 Modeling





89























90



1=++ nra SSS (41)




whole into this set
















91





[ ]

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

minusge

+

ltltminus

minusminus+

le⎟⎠⎞

⎜⎝⎛

=

lowast




P

l

l

ll

ll

llmYlmFlmY

l

lm

)(

1

)()()(ln)(ln

)(lnln)()()(

)(3

4

)(

2212

21

12

1

121

microδδ

αmicro



microδδδπ

microδ

(42)








by

( ) ( )⎪⎩

⎪⎨

⎧


le=

=

20

201032

10

0

2231


δδδπmicroδ

micro

RR

RAl (44)

92



( )( )

( )[ ] ( )( ) ( ) ( ) ( )

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

ge

ltltminus

minusminus+

le

=

llAl

llll

llAl

l

lA

k

kk

microδδmicro


microδδmicro

microδδ

microδ

2

2212

1

1

lnlnlnln

11

01

(45)


detail in Chapter 3














93




)(

0)(

ij

nji

jP ⎥⎦

⎤⎢⎣


















94











⎟⎟⎠

⎞⎜⎜⎝

⎛primeminus

prime=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆

a

a

lc

am S

STR

HPb

1exp0 (48)


numerical method







[155]

95

⎟⎟⎠

⎞⎜⎜⎝

⎛ minus∆=

lc

erf TR

Htt

γσexp0 (49)












mse C τσ = (410)







96








⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛ ∆minus=prime

infinrarrinfinrarr

f

c

n

f

c

n

n

fnr

tt

ntt

ttS

exp

1lim1lim (413)










97






is given by





lbl TTT ∆+= (418)








is given by

98

2211 874087408260

ecec

ml PKPK

VaT

+++=∆






2211 658065806880

ecec

ml PKPK

VaT

+++=∆







is given by

( ) ( )( ) ( ) 2211 eTceTc

mTl PGKPGK

VaDT

+++=∆

δδτδ

δ (421)





99

( ) ( )( ) ( ) ( )TeTp

ll

lTeT DDDD minus

minusminus

+=microδmicroδ

microδδδ

12


and

( ) ( )( ) ( ) ( )TeTp

ll

lTeT GGGG minus

minusminus

+=microδmicroδ

microδδδ

12

















using Eq (421)

100




(417)
















η (424)

101




minus=η (426)





ttt WF=micro (427)





layer are given by

( ) ( )

( )intint

infin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (428)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (429)

( ) ( )

( )intint

infin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (430)

102


by

( ) ( )

( )intint

infin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (431)



( ) ( ) ( ) ( )



( ) ( ) ( ) ( )



( ) ( ) ( )








( ) ( )( )int

intinfin

infinminus

=

d

d n

ntdzzf

dzzfdzSS (435)

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d r

rtdzzf

dzzfdzSS (436)

103

( ) ( )( )int

intinfin

infinminusprime

=prime

d

d a

atdzzf

dzzfdzSS (437)

( ) ( )( )int

intinfin

infinminus∆

=∆

d

d l

ldzzf

dzzfdzTT (438)



( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

50exp2

1

aa

zzfσσπ

(439)


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝


2

50exp21 zzf

aa σσ

σσ

π (440)





πα

σηβ sR3

481

== (441)

21896801

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

sa α

σσ (442)

104

( ) 21

4

ssz

πα=lowast (443)







43 Result Analysis




velocity









105























106



( ) 2110δσψ a= (444)



















107



















instability




108
























109
























110























111























112
























113







44 Summary









in the following




load and speed



114





















115


116

|error| lt ε

End


Start









τm = τm


117

ψ = 066

ψ = 093

ψ = 186

ψ = 255

0 02 04 06 08 1

x 10-3

0

02

04

06

08


( =V 10ms and =bT 50˚C)

tmicro

nt AEW lowast

118

hσ = 05

hσ = 10

hσ = 20 0

005

01

015

02

-1 0 2 4 60

01

02

03

04

05


( =V 10ms and =bT 50˚C)

z

nm ττ

nm ττ

0

02

04

06

08

1

-1 0 1 2 3 4 5 60

01

02

03

04

05

zσ

(b)

(a)

nm ττ

f(zσ)

Asperity height

Shea

r stre

ss

Shea

r stre

ss

Dis

tribu

tion

dens

ity

(c) au

119

0 02 04 06 08 1x 10-3

08

082

084

086

088

09

092

094

096

098

1

0 02 04 06 08 1x 10-3

05

1

15

2

0 02 04 06 08 1x 10-3

0

002

004

006

008

01

012


(ψ = 186 and =bT 50˚C)

σh

No-sliding

=V 10ms

=V 40ms

nt AEW lowast

nt AA

No-sliding =V 10ms

=V 40ms

(b)

(c)

nt AEW lowast

rtS prime

=V 10ms

=V 40ms

(a)

nt AEW lowast

120

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

01

01

01

01

02

02

02

03

03

03

04

04

05

06

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9x 10-4

099

099

095

095

095

09

09

09

085

085

08

08

075

07



(b) rtS prime

(a) tmicro

nt AEW lowast

V (ms)

V (ms)

nt AEW lowast

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Chapter 5








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London

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VITA







semester



A DETERMINISTIC-STATISTICAL MODEL FOR TRIBO-CONTACTS …

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