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REGULAR AND IRREGULAR WEAR PATTERNS OF ELASTOMERS IN COMBINED ROLLING AND SLIDING
By
ALEXANDER ISAAC BENNETT
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2016
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© 2016 Alexander I. Bennett
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To my beloved sister Chelsea Ivanna Bennett. I love you
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ACKNOWLEDGMENTS
Foremost I would like to thank my wonderful, amazing, patient parents, William
and Rita Bennett for their love and support. I want to thank the two greatest advisors
and mentors I could ever ask for, Greg Sawyer and Tommy Angelini, I have learned
from you in ways I expected but some of the most important lessons I’ve ever received
were the ones you gave me that I didn’t expect. Dan Dickrell, I wouldn’t have
completed this work without your kind words and encouraging spirit. I would also like to
thank my fellow Tribolites – those who have moved on, those currently in the crucible,
and those just beginning – for their motivation, friendship, and feedback (read:
criticism). I would like to thank my friends who have stuck with me and provided
encouragement while also keeping me sane and humble. And lastly, I’d like to thank the
members of “Team Strange” for being there always and expecting nothing but the best
from me.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 9
LIST OF FIGURES ........................................................................................................ 10
ABSTRACT ................................................................................................................... 14
CHAPTER
1 INTRODUCTION .................................................................................................... 16
2 EXPERIMENTAL AND THEORETICAL BACKGROUND ....................................... 18
Material ................................................................................................................... 18 Long Chain Polymers ....................................................................................... 18 Mechanical Properties and Behavior ................................................................ 20
Elastic behavior of elastomers ................................................................... 20 Dynamic mechanical properties ................................................................. 21 Viscoelasticity ............................................................................................ 24 Glass transition temperature ...................................................................... 28
Contact Mechanics and Friction .............................................................................. 29 Contact Area .................................................................................................... 29 Adhesion .......................................................................................................... 31 Sliding Friction .................................................................................................. 31 Rolling Friction .................................................................................................. 35
Elastomer Wear ...................................................................................................... 37 Abrasive Wear .................................................................................................. 39
Intrinsic abrasion ........................................................................................ 39 Periodic abrasion ....................................................................................... 41
Wear Rate ........................................................................................................ 43 Combined Rolling-Sliding Wear ........................................................................ 44
3 SLIDING FRICTION AND FRICTIONAL HEATING OF ELASTOMERS................. 46
In Situ Thermal Measurements of Sliding Contacts ................................................ 46 Frictional Heating of Sliding Contacts ............................................................... 46 Tribometer Description ..................................................................................... 47 Image Acquisition and Temperature Analysis .................................................. 48 Force Measurements and Positioning .............................................................. 49 Materials ........................................................................................................... 50 Description of Loading and Sliding Experiments .............................................. 51 Results and Discussion .................................................................................... 53
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Dynamic In Situ Measurements of Frictional Heating on an Isolated Surface Protrusion ............................................................................................................ 56
Frictional Heating of Asperity Contacts ............................................................ 56 Asperity Contact Temperatures ........................................................................ 56
Instrumentation .......................................................................................... 56 Materials .................................................................................................... 59
Results ............................................................................................................. 59 Discussion ........................................................................................................ 61
L’Escargot Rapide .................................................................................................. 65 Soft Contacts at High Speeds .......................................................................... 65 Materials ........................................................................................................... 65 Experimental Methods ...................................................................................... 66
Tribometer .................................................................................................. 66 Experiments ............................................................................................... 67
Results and Discussion .................................................................................... 68
4 TRACTION AND WEAR OF ELASTOEMRS IN COMBINED ROLLING-SLIDING CONTACT ............................................................................................... 71
Combined Rolling and Sliding Contacts .................................................................. 71 Tribometer Description and Design ........................................................................ 72 Materials and Experiments ..................................................................................... 73
Materials and Sample Preparation ................................................................... 73 Traction and Wear Experiments ....................................................................... 74
Results and Discussion........................................................................................... 75 Traction Experiments ....................................................................................... 75 Wear Experiments ............................................................................................ 78
5 COMBINED ROLLING AND WEAR OF ELASTOMER MATERIALS ..................... 82
The Effects of various Material Properties on the Wear of Materials in Combined Rolling and Sliding .............................................................................. 82
Materials and Methods............................................................................................ 83 Natural Rubber ................................................................................................. 83 Polydimethylsiloxane ........................................................................................ 83 Counter Surface Modification ........................................................................... 84
Results and Discussion........................................................................................... 86 Natural Rubber ................................................................................................. 86 PDMS ............................................................................................................... 87
6 THE IRREGULAR WEAR OF ELASTOMERS IN COMBINED ROLLING AND SLIDING ................................................................................................................. 89
Irregular Wear ......................................................................................................... 89 Measurements of Onset and Development ............................................................. 90 Quantifying Irregular Wear Development ................................................................ 91 River Wear .............................................................................................................. 92
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7 A BINARY SYSTEM TO DESCRIBE IRREGULAR WEAR PATTERNS................. 98
A Simple Model for Irregular Wear .......................................................................... 98 Modeling ................................................................................................................. 99
Dimensional Problem Formulation ................................................................... 99 Non-Dimensionalization of the Model ............................................................. 102 The Conditional Bifurcation in Solutions for Steady-State .............................. 102 An Analytical Solution for the Time Constant ................................................. 106
Results and Discussion......................................................................................... 107
8 CONCLUSIONS ................................................................................................... 112
Friction of Elastomers ........................................................................................... 112 Frictional in Unidirectional Sliding ................................................................... 112 Frictional Heating ........................................................................................... 112 Asperity Heating of Elastomer Contacts ......................................................... 113 Friction of Elastomers in Combined Rolling and Sliding ................................. 113
Regular Wear of Elastomers ................................................................................. 114 Wear of Elastomers in Combined Rolling and Sliding .................................... 114
Irregular Wear of Elastomers ................................................................................ 115 Measurement and Quantification .................................................................... 115 Analytical Solution for Irregular Wear ............................................................. 116
APPENDIX
A MODEL FRICTIONAL HEATING DERIVATIONS ................................................. 117
Archard’s Method (1958) ...................................................................................... 117 Tian and Kennedy’s Method (1994) ...................................................................... 118
B COMBINED ROLLING AND SLIDING TESTING PROCEDURE .......................... 120
Sample Preparation .............................................................................................. 120 Step 1 ............................................................................................................. 120 Step 2 ............................................................................................................. 120 Step 3 ............................................................................................................. 121 Step 4 ............................................................................................................. 121 Step 5 ............................................................................................................. 122
Rolling-Sliding Instrument Preparation.................................................................. 122 Step 1 ............................................................................................................. 122 Step 2 ............................................................................................................. 123 Step 3 ............................................................................................................. 123 Step 4 ............................................................................................................. 124 Step 5 ............................................................................................................. 125 Step 6 ............................................................................................................. 126 Step 7 ............................................................................................................. 126 Step 8 ............................................................................................................. 127
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Rolling-Sliding Testing Procedure ......................................................................... 127 Step 1 ............................................................................................................. 127 Step 2 ............................................................................................................. 127 Step 3 ............................................................................................................. 129 Step 4 ............................................................................................................. 129 Step 5 ............................................................................................................. 130 Step 6 ............................................................................................................. 130 Step 7 ............................................................................................................. 130
C IRREGULAR WEAR QUANTIFICATION PROCEDURE ...................................... 131
Irregular Wear Analysis Procedure ....................................................................... 131 Step 1 ............................................................................................................. 131 Step 2 ............................................................................................................. 132 Step 3 ............................................................................................................. 132 Step 4 ............................................................................................................. 133 Step 5 ............................................................................................................. 135
D IRREGULAR WEAR QUANTIFICATION CODE ................................................... 136
LIST OF REFERENCES ............................................................................................. 141
BIOGRAPHICAL SKETCH .......................................................................................... 154
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LIST OF TABLES
Table page 3-1 Predicted and Measured Temperatures of the Protrusion .................................. 64
3-2 Measured values from the pin-on-disk testing of a filled natural rubber hemisphere on calcium fluoride. ......................................................................... 68
4-1 Rolling and sliding wear rates for different slip percentages and rolling velocities. ............................................................................................................ 79
5-1 Mechanical properties of Michelin Mix Compounds ............................................ 83
5-2 Wear rates for all Michelin tire compounds against smooth, rough, and smooth with third-body counter surfaces. ........................................................... 87
5-3 Wear rates for all PDMS formulations against smooth and rough counter surfaces. ............................................................................................................. 87
B-1 Reference values for prescribing slip% values ................................................. 129
C-1 A sample irregular wear analysis report for Mix 1 against a rough disk. ........... 135
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LIST OF FIGURES
Figure page 2-1 Stress vs extension ratio, λ, for the glassy, crystalline, and rubbery physical
states of a polymer. Adapted from Gent.29 ......................................................... 19
2-2 The stress response to an input strain for elastic, viscous, and viscoelastic materials. ............................................................................................................ 23
2-3 Time dependent stress response to strain of viscoelastic materials ................... 24
2-4 Elastic and viscous material behavior as represented by classic mechanical models ................................................................................................................ 25
2-5 The Maxwell and Kelvin-Voight viscoelastic material models ............................. 26
2-6 Physical state of a polymer vs. the temperature. ................................................ 28
2-7 Example schematics of contact area measurements with a sphere on flat indentation geometry as measured by Roberts and Kendall. ............................. 29
2-8 Archard’s surface roughness model. Roughness is represented as semicircles superimposed onto the surface. ....................................................... 30
2-9 The friction of an SBR rubber plotted against sliding speed. .............................. 33
2-10 Tangential displacement of a rubber surface due to a parabolically distributed force. .................................................................................................................. 33
2-11 Schallamach waves on an elastic hemispherical pin against a rigid glass flat. ... 35
2-12 Rolling friction as a function of the dimensionless relaxation time. ..................... 36
2-13 Schematic of the deformation of a rubber tire between a road surface and the tire carcass. ........................................................................................................ 37
2-14 Relations between the friction and wear mechanisms of rubbers. ...................... 38
2-15 Photoelastic snapshot of strain fields of a rubber flat sheared by the motion of a hard spherical slider. ................................................................................... 39
2-16 Cross sections and a micrograph of a rubber flat with developing periodic abrasion .............................................................................................................. 40
2-17 Material loss as a function of sliding distance for patterned abrasive wear and intrinsic abrasive wear of rubber. ........................................................................ 42
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2-18 Schematic illustrating the development of the abrasive wear pattern. ................ 43
3-1 Objective of infrared camera focused on the interface between a natural rubber half-sphere and a CaF2 countersample.10 ............................................... 48
3-2 The tribometer in profile. ..................................................................................... 49
3-3 Contact area and temperature as a function of speed and load. ........................ 51
3-4 Comparison of average contact temperature under forced and natural convection for a normal load of 500 mN and a sliding velocity of 750 mm/s. ..... 52
3-5 Evolution of contact shape and temperature as a function of applied load and sliding velocity. ................................................................................................... 53
3-6 Direct temperature measurements for 900 s tests compared to model predictions. ......................................................................................................... 55
3-7 Overview of the experimental method. ............................................................... 57
3-8 The SU-8 feature rotates with the disk and slides under contact once per revolution. ........................................................................................................... 58
3-9 Schematic of contact between pin, counter-surface, and positive feature. ......... 60
3-10 Average and maximum temperature rise of and within the region of interest of the positive feature at varying speeds. ........................................................... 61
3-11 Temperature rise of the feature as measured. ................................................... 62
3-12 Measured maximum and average temperature rises for each sliding speed. ..... 63
3-13 A schematic representation of the instrument .................................................... 66
3-14 Representative images from each experiment illustrating the variation in contact size and geometry. ................................................................................. 67
3-15 On the left, outlines of the stationary contact areas for a given load are superimposed to show their relative contact size. .............................................. 69
4-1 Schematic of the combined rolling–sliding tribometer. ......................................... 73
4-2 Friction coefficient data plotted against %slip for various normal loads, and linear velocities. .................................................................................................. 77
4-3 Wear data for natural rubber rolling and sliding samples. ................................... 78
4-4 Characterization of natural rubber wear in rolling and sliding. ............................ 80
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5-1 Indentation experiments were performed in order to estimate the Young’s Modulus of each of the PDMS samples .............................................................. 84
5-2 The combined rolling and sliding tribometer with magnetic vibrational feeder for application of kaolin powder. ......................................................................... 85
6-1 Irregular wear on a PDMS sample. ..................................................................... 89
6-2 Scanning white light interferometer sample mount. ............................................ 90
6-3 Profilometer scans of a natural rubber sample with irregular wear. .................... 91
6-4 Output from the irregular wear analysis code. .................................................... 92
6-6 The rolling-sliding tribometer geometry.152 ......................................................... 93
6-7 The pressure, %slip, and shear force distribution within the contact. ................. 95
6-8 Angled geometry for producing rivering irregular wear. ...................................... 96
6-9 Surface profiles of worn natural rubber (Mix 1) sample. ..................................... 97
7-1 A simple system comprised of two blocks coupled under a single normal load 100
7-2 Force partitioning on the two blocks is dependent on the force and slip on each block in the previous cycle. ...................................................................... 105
7-3 Using simulation constants ............................................................................... 108
7-4 Using simulation constants ............................................................................... 109
7-5 Dimensionalization of the time constant. .......................................................... 110
B-1 As received and trimmed rolling-sliding samples. Combined Rolling and Sliding Samples. January 19, 2016. ................................................................. 120
B-2 Slotted rolling sliding sample in SWLI sample mount. ...................................... 121
B-3 Schematic exploded view of the sample and sample holder assembly. ........... 122
B-4 The rolling and sliding tribometer code front panel. .......................................... 123
B-5 Selecting force zeros. The user should zero channels 0 through 6. ................. 124
B-6 Calibration coefficients for the rolling and sliding tribometer. ............................ 124
B-7 The combined rolling and sliding motion setup page. ....................................... 125
B-8 Combined Rolling and Sliding program test panel. ........................................... 127
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B-9 Rotary stage and z-height positioning stage. January 19, 2016. ...................... 128
B-10 Micrometer positioning stages for load and friction force adjustment. .............. 129
C-1 Vision software. A surface profile scan with line scan on the surface representation. .................................................................................................. 131
C-2 Example test information to be input into the irregular wear analysis code. ..... 132
C-3 Examples of proper selection for Step 3. .......................................................... 133
C-4 The fitted white light interferometer data. ......................................................... 134
C-5 Subtracted surface profile scans. ..................................................................... 135
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
REGULAR AND IRREGULAR WEAR PATTERNS OF ELASTOMERS IN COMBINED
ROLLING AND SLIDING
By
Alexander Isaac Bennett
December 2016
Chair: W. Gregory Sawyer Major: Mechanical Engineering
The unique mechanical, chemical, electrical, and tribological properties of
elastomers make them valuable engineering materials in a plethora of commercial,
industrial, and scientific applications. Since the discovery of the myriad uses of rubber in
~800 BC, the rubber industry has seen great scientific growth with the industrial
revolution, the invention of the tire, and both world wars all demanding a high quantity of
high quality natural and synthetic rubber.1 With rubber being used in tens of thousands
of products annually and the global production of rubber reaching over 12 million tons
annually, the need to study the life and wear properties of rubber has never been
greater.2 The aim of this work is to provide a robust examination of the tribological
properties of natural and synthetic rubber. Such an examination must cover the contact
mechanics, thermodynamics, sliding, rolling, and wear of these elastomeric materials.
Unidirectional sliding tests of rubber hemispheres on various counter-samples
were used to tease out the effects of sliding speed, contact pressure, and surface
roughness on the friction and wear of various rubber compounds. Through the use of
both brightfield and infrared imaging, the contact mechanics and morphology, as well as
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the frictional heating of these sliding contacts were studied. Observations of the contact
surface during and after sliding showed patterns and formations indicative of abrasive
elastomer wear.
In addition to pure sliding tests, a suite of combined rolling and sliding tests using
both natural and synthetic rubbers was conducted in order to examine their tribological
behavior in complex tribological systems. Wear rates for these samples were found to
be independent of rolling velocity but linearly dependent on the sliding, or slip,
conditions within the contact zone.
During high-cycle, low slip percent tests, a preferential wear development was
witnessed and hypothesized as related to the regions of slip within the contact. This
zone of localized, circumferential wear progressed at a faster rate than the global wear
of the sample. A simple binary model was devised to describe the causes for initiation,
and subsequent growth of this preferential, irregular wear zone.
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CHAPTER 1 INTRODUCTION
The benefits of the tribological advancement of elastomers is both far reaching
and of great value. Over the past century the production of natural rubber has grown
240x from 50,000 tons in 1900 to 12 million tons in 2013.2 The proliferation of rubber
products in such great quantities signifies a need to understand the means by which it is
used but also how and when it fails. Every year 5.17 million tons of rubber tire waste is
produced by end of life tires wearing out in the U.S. alone.3 While solid waste is indeed
unwanted, arguably the more detrimental aspect of waste tires on the environment is
the amount of extra fuel burned in engines to overcome the frictional “rolling resistance”
of rubber. It is then abundantly clear that understanding and improving the performance
and wear of elastomers, particularly rubber, is a topic that cannot be ignored.
A material’s tribological performance can be described as the materials friction
and wear behavior in a system of intimate contact and relative motion. For the specific
case of elastomers, much emphasis has been placed on the development of new
compounds with higher friction, reduced wear, and predictable viscoelastic response.4–
25 The complex coupling of these properties has confounded researchers whose best
efforts up until this point have resulted in the development of strict empirical rules and
large scale, prohibitive testing procedures. Development of an instrument capable of
economically and quickly testing the tribological performance of an elastomer in both
simple and complex system configurations has the potential to greatly advance material
development.26
The wear of rubber and rubber-like materials is a complex amalgam of the
contact mechanics, thermal behavior, surface chemistry, surface roughness, and
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system dynamics of the system. The proposed research is a multidimensional study on
the friction and wear of elastomers in both simple sliding and combined rolling-sliding
systems designed to emulate practical application. Characterization and analytical
methods will be used to gain a fundamental understanding of the effect of complex,
dynamic system configurations on the tribological behavior of rubber. This
understanding will then be used to develop robust tools for the design of rubbers.
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CHAPTER 2 EXPERIMENTAL AND THEORETICAL BACKGROUND
Material
Long Chain Polymers
Elastomers are long-chained molecules that typically form an amorphous rubbery
solid. These polymer compounds can have millions of atoms in each molecule thus
making their molecular weight incredibly high relative to most other compounds. The
polymer chains can be bound by either physical entanglements or chemical bonding
and are highly flexible, disordered, and intertwined. Most elastomers are composed
mainly of carbon and hydrogen (hydrocarbon) and can occur naturally or be produced
synthetically.27 Natural elastomers are processed using polyisoprene, rubber extracted
from various plants all over the world. Synthetic elastomers are synthesized from either
one (homo-polymer) or multiple (co-polymer) monomer constituents using either
solution or emulsion polymerization.
Natural and synthetic elastomers can be split into two groups, thermoplastics and
thermosets, based on the type of connections, or crosslinks, of their polymer chains.
The long, flexible molecules in thermoplastics are naturally entangled and while they will
flow and disentangle under an applied stress, these physical entanglements act as
mechanical crosslinks. Characteristically, these molecules contain few hydrogen-
bonding groups and exhibit strong intermolecular forces without chemical crosslinking.
Thermoplastics have definite melting points and can be reformed through heating and
pressure. They also have very low glass transition temperatures, Tg < 25 oC.
Thermosets, however, will not flow when heated and cannot be reformed without
significant effort. This is due to covalent bonding between the elastomer molecules and
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a solvent chemical such as sulfur in a process called vulcanization. These vulcanized
rubbers maintain their elasticity even at high temperatures and, due to the addition of
sulfur in the interstitials, crystallize at retarded rates at extremely low temperatures
(Tg<0 oC). This combined with generally more desirable mechanical, chemical and
electrical properties make thermosets more commonly used in practical applications
than thermoplastics.28
Hydrocarbon rubbers, the most common chemical group of elastomers, contains
both natural and synthetic rubbers. Hydrocarbon rubbers include butadiene, styrene-
butadiene, polyisoprene, and isoprene rubbers. Other specific purpose rubbers have
been engineered to have a robust response to extreme forms of external input and
environments. Some such rubbers are acrylonitrile-butadiene (nitrile) rubber, butyl
rubber, and fluorocarbon rubber. Elastomers are unique in that they exhibit high
elasticity (100 % - 1,000 % elastic strain), soft, viscoelastic, have high or undefined
melting points, and low glass transition temperatures (Figure 2-1).29
Figure 2-1. Stress vs extension ratio, λ, for the glassy, crystalline, and rubbery physical
states of a polymer. Adapted from Gent.29
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Many additives and fillers can be added during processing to both natural and
synthetic elastomers in order to enhance the material properties of the rubber. The most
common and universal fillers are carbon black and silica which providing both
mechanical reinforcement, UV protection, and some electrical insulation. At a volume
fraction of 30% they can raise the elastic modulus of a rubber compound by a factor of
two to three. Reinforcing fillers are nanometers in diameter and exhibit good adherence
with the elastomer compound. Amines, phenols, and certain metals such as copper and
manganese can also be added in small amounts to inhibit degradation in the forms of
heat, UV light, oxygen, and ozone.29 Rubber by itself is a very versatile material and
with the addition of other materials its properties can be significantly altered in order to
fit an incredibly vast array of engineering needs.
Mechanical Properties and Behavior
Elastomers are extremely complex materials whose mechanical properties and
behavior do not adhere to any one fundamental material definition. They exhibit both
time-independent (elastic) and time-dependent (viscous) strain responses yet are
neither elastic nor viscous but lie somewhere in between. This “viscoelasticity” affords
elastomers a robust response to external input but requires a multifaceted approach to
understanding its fundamental behavior. The mechanical state of elastomers is a strong
function of the temperature and a large range of mechanical behavior can be observed
by altering the operating temperature.
Elastic behavior of elastomers
The mechanical behavior of a rubber at small strains and below the Tg is that of a
linear elastic solid. During such conditions the material can be described by
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fundamental elastic constants.29 The bulk modulus, K, is a measure of a material’s
resistance to hydrostatic compression. It is defined as the linear relation between an
applied pressure P and the ratio of the volumetric shrinkage –ΔV to the original volume
Vo.
𝑃𝑃 = 𝐾𝐾 −∆𝑉𝑉𝑉𝑉𝑜𝑜 (2-1)
The shear modulus, G, is the ratio of an applied shear stress, τ, to the amount of shear,
γ.
𝐺𝐺 =𝜏𝜏𝛾𝛾
(2-2)
Two more material parameters commonly used to describe elastic materials can
then be derived from Equations 2-1 and 2-2. The Young’s modulus, E, and Poisson
ratio, ν, are as follows:
𝐸𝐸 =𝜎𝜎𝜀𝜀
=9𝐾𝐾𝐺𝐺
3𝐾𝐾 + 𝐺𝐺 (2-3)
ν = 12
(3𝐾𝐾 − 2𝐺𝐺)3𝐾𝐾 + 𝐺𝐺
(2-4)
Under high strains, ε > 10%, rubbers behave as nonlinear elastic solids. Mooney
and Rivlin attempted to analytically define the elastic behavior material at high strains
but found poor experimental agreement with a traditional strain tensor approach.30 Gent
improved upon their original work with an equation that took into account decreasing
strain with % elongation.29
Dynamic mechanical properties
The pseudo-viscosity of these semi-elastic materials are the cause of an
interesting set of behaviors, including: stress relaxation, creep, and hysteresis. The
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nonlinear viscosity η can be described by the Arrhenius relationship for a range of
temperatures.
1𝜂𝜂
= 𝐴𝐴𝑒𝑒−𝑄𝑄𝑅𝑅𝑅𝑅 (2-5)
Where Q is the activation energy for relative molecular chain movement, R is the
universal gas constant, T is the temperature and A is a scaling coefficient obtained
experimentally.
An ideal elastic material’s behavior follows Hooke’s law where stress is directly
proportional to strain. The stress response to the input strain for elastic materials is in
phase, or it can be said that the phase shift is 𝜑𝜑 = 0. An ideal viscous material adheres
to Newton’s law where the stress is proportional to the strain rate and the stress
response and strain input are 90o out of phase (𝜑𝜑 = 90o). A viscoelastic material, as
mentioned previously, behaves somewhere in between an ideal elastic and ideal
viscous material, 0o < φ < 90 o.
Figure 2-2 shows the response of elastic, viscous, and viscoelastic materials. A
dynamic mechanical approach is used to study these materials, where a strain
frequency ω is applied to the sample. From Figure 2-2 we can write an expression for
the stress and strain of a viscoelastic material.
𝜀𝜀 = 𝜀𝜀𝑜𝑜sin (𝜔𝜔𝜔𝜔) (2-6)
𝜎𝜎 = 𝜎𝜎𝑜𝑜sin (𝜔𝜔𝜔𝜔 + 𝜑𝜑) (2-7)
Applying Hooke’s law allows us to define a set of moduli that make up the
dynamic modulus for viscoelastic materials. The tensile storage modulus, 𝐸𝐸′, and the
tensile loss modulus, 𝐸𝐸′′.
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𝐸𝐸′ = 𝜎𝜎𝑜𝑜𝜀𝜀𝑜𝑜 cos (𝜑𝜑) (2-8)
𝐸𝐸′′ = 𝜎𝜎𝑜𝑜𝜀𝜀𝑜𝑜 sin (𝜑𝜑) (2-9)
The storage modulus represents the stored energy in a material and accounts for
the elastic portion of the material response to an applied load. The viscous portion of
the material response is represented by the loss modulus which is a measure for the
energy dissipation caused by internal friction and physically manifested as heat
generation. A measure of a material’s viscous dissipation, the loss tangent, can then be
defined as
Figure 2-2. The stress response to an input strain for elastic, viscous, and viscoelastic
materials. The stress response to the strain for elastic materials is in phase, 𝜑𝜑 = 0, while the viscous stress response is directly out of phase to the strain input, 𝜑𝜑 = 90. The viscoelastic stress response is arbitrarily out of phase with the strain input 0 < 𝜑𝜑 < 90. Adapted from Gent, 2012.31
24
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝜔𝜔𝑡𝑡𝑡𝑡𝑡𝑡𝑒𝑒𝑡𝑡𝜔𝜔 = tan(𝛿𝛿𝜑𝜑) =𝑒𝑒𝑡𝑡𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒 𝑙𝑙𝐿𝐿𝐿𝐿𝐿𝐿𝑒𝑒𝑡𝑡𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒 𝐿𝐿𝜔𝜔𝐿𝐿𝑒𝑒𝑒𝑒𝑠𝑠
=𝐸𝐸′′
𝐸𝐸′ (2-10)
The loss tangent is the ratio of the maximum energy dissipated by a material per
cycle to the maximum energy stored by a material during the cycle.32,33
Viscoelasticity
A basic definition of viscoelasticity is a solid material’s time dependent response
to an input strain. The physical realization of this time dependency manifests in three
major ways creep, stress relaxation, and hysteresis (Figure 2-3).33 Not all elastomers
exhibit each characteristic behavior of viscoelastic materials but often a combination of
hysteresis and one of the other two.
Figure 2-3. Time dependent stress response to strain of viscoelastic materials. A)
Hysteretic loading and unloading of a viscoelastic material. B) Stress relaxation for a constant strain. C) A material exhibiting creep will have a constant stress state for increasing strain. Adapted from Gent, 2012.34
Hysteresis is the dependence of a material’s output based on the current and all
previous inputs. A viscoelastic material exhibiting hysteresis will lose a large portion of
the input energy as heat due to internal friction. In elastomers the disentanglement,
stretching, alignment, and then rearrangement of molecular chains is referred to as
creep. The stress state in a creeping material remains the same as the strain increases.
When a constant strain is applied to a viscoelastic material the stress within the material
will decrease over time. This process is called stress relaxation and is a function of time,
25
temperature and the initial stress level. Various models have been developed to predict
these behaviors in elastomers under a variety of conditions.35,36
The two most commonly used viscoelastic models are the Maxwell series spring-
dashpot model and the Kelvin-Voigt parallel spring-dashpot model.24,33,35,37–41 In these
models the linear elastic portion of a viscoelastic material is represented as a spring and
the nonlinear viscous portion of the material is represented by a dashpot, as shown in
Figure 2-4.
Figure 2-4. Elastic and viscous material behavior as represented by classic mechanical
models. A) Linear elastic behavior is represented by a spring. B) Linear, µ, and nonlinear, η, viscous behavior is represented by a dashpot (simple damper). Adapted from Gent, 2012.34
Maxwell proposed his model in 1867 and it assumes that the viscous and elastic
components occur in series Figure 2-5A. The model states that for a shear stress, τ, is
the same on both the elastic, τe, and viscous, τv, components, since the entire shear
stress is transmitted completely through the spring, once extended, to the dashpot
below or vice versa.42
τ = τe = τv (2-11)
26
The shear strain, γ, is the sum of the elastic strain, γe, and the viscous strain, γv,
since the total displacement is the sum of the displacement of the components.
𝛾𝛾 = 𝛾𝛾𝑒𝑒 + 𝛾𝛾𝑣𝑣 (2-12)
An expression for the time dependent shear strain can be obtained by
substituting equations Hooke’s law and Newton’s law into Equation 2-12 and
integrating.
𝛾𝛾(𝜔𝜔) =𝜏𝜏𝐺𝐺
+𝜏𝜏𝜂𝜂𝜔𝜔 (2-13)
Figure 2-5. The Maxwell and Kelvin-Voight viscoelastic material models. A) The
Maxwell model sets the elastic and viscous representation in series. B) The Kelvin-Voigt model places the representations in parallel. Adapted from Gent, 2012.34
The Maxwell model predicts permanent deformations due to flow at any applied
stress. At time zero when a stress is applied and the dashpot begins to flow while the
springs restorative forces act against the stress. When the stress is removed the spring
27
returns to zero displacement but the dashpot stays in its deformed state representing a
plastic deformation.
The Kelvin-Voigt model puts the elastic and viscous components in parallel with
each other as shown in Figure 2-5B. In this model, the stresses are additive and the
strain is distributed across the components.
𝜏𝜏 = 𝜏𝜏𝑒𝑒 + 𝜏𝜏𝑣𝑣 (2-14)
𝛾𝛾 = 𝛾𝛾𝑒𝑒 + 𝛾𝛾𝑣𝑣 (2-15)
Substituting Hooke’s law and Newton’s law into Equation 2-14 yields,
𝜏𝜏 = 𝐺𝐺𝛾𝛾 + 𝜂𝜂 𝑠𝑠𝛾𝛾𝑠𝑠𝜔𝜔 (2-16)
For a constant state of stress, 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 0, Equation 2-16 becomes,
𝛾𝛾 =𝜏𝜏𝐺𝐺1 − 𝑒𝑒𝑒𝑒𝑒𝑒
−𝐺𝐺𝜔𝜔𝜂𝜂 (2-17)
Under a constant stress, 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 0, the spring element stores part of the energy
while the dashpot dissipates the remainder. The elastic response is slowed by the
viscous element and the strain does not manifest immediately but increases to max
strain asymptotically. When the stress is released the elastic element acts to restore its
original shape but again is retarded by the dashpot. It again slowly proceeds to zero
strain asymptotically over time42. The time necessary to achieve an equilibrium state
after removal of stress is called the retardation time, 𝜔𝜔𝑟𝑟𝑒𝑒𝑑𝑑 = 𝜂𝜂𝐺𝐺. Substituting this into
Equation 2-17 yields
𝛾𝛾(𝜔𝜔) =𝜏𝜏𝐺𝐺1 − exp −
𝜔𝜔𝜔𝜔𝑟𝑟𝑒𝑒𝑑𝑑
(2-18)
28
Glass transition temperature
As we have discussed, delayed elastic response of rubbery materials to an
applied force is largely due to the internal viscosity between molecular chains. This is
attributed to the rate, φ, which these molecular segments move past each other as a
result of Brownian motion. This rate is strongly affected by the temperature of the
material.28,43
ln 𝜑𝜑(𝑇𝑇)𝜑𝜑𝑇𝑇𝑔𝑔
=𝐴𝐴𝑇𝑇 − 𝑇𝑇𝑔𝑔𝐵𝐵 + 𝑇𝑇 − 𝑇𝑇𝑔𝑔
(2-19)
Figure 2-6. Physical state of a polymer vs. the temperature. The polymer experiences a
sharp change in mechanical properties across the glass transition temperature moving from a glassy or crystalline behavior to an amorphous, rubbery behavior with increasing temperature.
29
The coefficients A and B are similar constants and Tg is the reference
temperature at which the material behaves as an amorphous glass. This behavior is
due to the molecular segments moving so slow, ~1x10-1 Hz, that their b1ehavior is near
that of a glassy solid. This temperature is referred to as the glass transition temperature
and is the largest contributor to the dynamic viscous behavior of rubbery materials.
Typical molecular rates of internal motion are much greater than the externally imposed
frequency, φ>>ʄ. As ʄ approaches φ, the dynamic modulus increases to the high
modulus of a glassy solid.
Contact Mechanics and Friction
Contact Area
In 1881 Heinrich Hertz proposed a contact model for two rigid elastic bodies that
is still relevant today. The model takes into account the modulus of both objects and can
predicts that the indentation depth is related to F2/3.44 Hertz based his model on the
assumptions that the surfaces were not adhesive and that the deformations were small
compared to the radius. Schallamach in 1957 tested the model on various rubbers and
found that the model performed well for large loads but predicted larger than expected
contact areas for lower loads.45
Figure 2-7. Example schematics of contact area measurements with a sphere on flat
indentation geometry as measured by Roberts and Kendall. A) Non-adhesive contact between sphere and flat. B) Adhesive contact between sphere and flat. Adapted from Kendall, 2007.46
30
Roberts (1968) and Kendall (1969) noted a similar phenomenon in their
dissertation work with smooth rubber spheres and smooth clean glass (Figure 2-7).
They noticed that contact area was larger than was predicted by Hertz. This led to the
Johnson, Kendall, Roberts adhesive contact model nearly a century after Hertz.47.
Around the same time, Bowden and Tabor emphasized the importance of roughness
between contacts through experiments with contact resistance.48
Figure 2-8. Archard’s surface roughness model. Roughness is represented as
semicircles superimposed onto the surface. A) The apparent area of contact is the 2D projection of 3D objects in contact. B) The real area of contact is a fraction of the apparent contact. Adapted from Archard, 1953.49
Their experiments led to the realization that the true area of contact is less than
the apparent area of contact as shown in Figure 2-8. Archard concluded that for all
surfaces, the contact area is proportional to the normal force 50. The real surface of
contact continues to be explored and our understanding continues to grow.12,15,48,51,52
31
Adhesion
Adhesion between macroscopic bodies, particularly rubbers, is a nuanced and
multifaceted topic.15,36,46,53,54 This unexpected behavior is the due to the very small real
contact area between rough bodies. Only when the real area of contact is large do the
forces of adhesion manifest on a macroscopic scale.15,55 Because of their viscoelasticity
and relative softness, the real area of contact between rubber and other materials is
relatively large making adhesion an important player in its tribological characterization.
Sliding Friction
While friction force is dependent on velocity for most materials, it has a
particularly interesting effect on rubber. Roth et. al. found that for rubbers friction force
increased with increasing velocity.56 After a certain velocity however, rubber does not
slide but instead translates through a stick-slip motion pattern. Schallamach came to a
similar conclusion but also noted that it was also temperature dependent, and
hypothesized that elastomeric friction was a function of both interfacial sliding and bulk
deformations.57 The formation and subsequent breaking of bonds at the interfacial
surface only accounted for part of the force and the viscoelastic molecular relaxation of
stressed molecules contributed to the frictional dissipation of energy.10,36,58–60
Greenwood and Tabor as well as Bueche and Flom also noticed that the energy
dissipation of a soft viscoelastic material is partially responsible for deformation of the
bulk.38,43,61 Separating the contribution of each of these dissipative processes to the
total friction of a sliding systems continues to be a challenge for researchers
today.37,51,62–66.
32
This growing realization the friction was at least partially a function of not only
surface but also bulk effects inspired Grosch to perform a systematic study of rubber
friction, temperature, velocity, roughness, and wear.67 He developed a tribometer that
could control the temperature, load and velocity of various rubbers sliding against three
different types of surfaces: rough silicon carbide paper, smooth flat glass with “gentle
wavy protuberances”, and silicon carbide paper coated in a magnesia powder. In the
initial set of experiments against smooth, wavy glass and silicon carbide paper Grosch
observed that friction had a nonlinear relationship with velocity. He also found that
friction decreased nonlinearly with temperature. What resulted was a set of friction-rate
isotherms which Grosch collapsed into a “master curve” using the Williams-Landel-Ferry
(WLF) transform shown in Figure 2-9. The WLF transform is a time-temperature
superposition tool that uses a reference temperature to relate rate dependent sets of
data, Equation 2-19.68
33
Figure 2-9. The friction of an SBR rubber plotted against sliding speed. A) Each line represents a temperature value. B) The isotherms are collapsed into one master curve using the WLF transform.67
Grosch then proceeded to conduct the same experiments on silicon carbide
paper with magnesia powder. By removing the adhesive component of friction (adding
magnesia powder) he hoped to find a frictional dependence on frequency relating to the
viscous loss contribution to rubber friction. He found that a similar set of curves could
be generated for friction vs frequency and temperature. The true value of this work was
that the velocity corresponding to maximum adhesive friction and the frequency
corresponding to maximum loss modulus, 𝐸𝐸′′, form a nearly constant ratio for the
various rubbers,
𝑉𝑉𝑠𝑠𝑓𝑓𝐸𝐸′′
= 𝜆𝜆 = 6 𝑡𝑡𝑛𝑛 (2-20)
Figure 2-10. Tangential displacement of a rubber surface due to a parabolically
distributed force. The majority of deformation occurs within a model space of the contact area. Adapted from Grosch, 1996.69
34
What Grosch believed was the interfacial adhesive process responsible for
friction is related to the intermolecular relaxation of polymer chains. This 6 nm value
represents a critical molecular length that could correspond to the length of molecular
jumps between two sliding bodies, or it could be the length scale at which viscoelastic
relaxation occurs.
Grosch’s publication prompted Schallamach, who still maintained that rubber
friction on smooth surfaces was dominated by an adhesive stick-slip process, to publish
the largest in situ rubber friction study up to that point.70 He proposed that within the
rubber contact a zone of shear is developed and the rubber deforms until a critical point
and then breaks free from the surface and makes a “jump” Figure 2-10. He conducted a
sliding experiment using a hard spherical slider on a flat slab of transparent rubber with
a grid pattern printed on it. As the sphere moved across the rubber surface images of
the contact were taken through the rubber. The grid pattern of the rubber bunched up
ahead of the contact in the sliding direction confirming his hypothesis. He then took
motion video of the experiment and found that above a certain velocity, waves of
detachment developed within, and moved through, the contact, Figure 2-11.
Shallamach’s experiments were the first to convincingly demonstrate
experimentally the theoretical surface traction force instability of rubber in sliding
contacts.45,70–75 He concluded that the number of detachment waves within a sliding
rubber contact is proportional to the sliding speed. He also noticed that their velocity
through the contact was related to the materials propensity for viscous damping.
Schallamach waves are still considered to be the primary mode of relative sliding
motion.5,10,49,76–82
35
Figure 2-11. Schallamach waves on an elastic hemispherical pin against a rigid glass
flat. A) Waves of detachment formed by a hard slider on a rubber flat. B) Initialization of a crack tip due to tensile stress imposed by the motion of the slider. C) A bulge behind the crack tip rises due to the viscoelastic nature of the material. D) The bulge comes into contact with slider and sticks effectively closing the crack. E) Reattached material continues to fill the crack and the process repeats.
Rolling Friction
The earliest studies of rolling friction were conducted by Coulomb in 1785 for the
rolling of wooden cylinders over a wooden plane. He deduced that rolling resistance
was inversely proportional to the diameter of the cylinder.83 Reynolds later conducted a
systematic study of rolling friction using elastic contacts.84 He found that due to the
deformation of the soft material a small amount of interfacial slipping occurs and that is
the origin of rolling friction. Heathcote noticed a similar phenomenon when he found that
slip was occurring in specific zones within the contact while studying ball bearings in
conformal grooves.85
Tabor changed the direction of rolling friction theory in 1955 by demonstrating
that adhesion and slip were not the major culprits of rolling friction. His experiments with
rubber rollers on various surfaces demonstrated two things: 1) rolling friction is largely
dependent on hysteretic losses and 2) the elastic work expended during rolling is the
same for a cylindrical or spherical roller.26,57,86,87 May later found that the maximum
36
rolling friction is velocity dependent. Using a modified Maxwell viscoelastic model he
was able to show that this dependence was related to the viscous relaxation property of
the bulk material.40
Figure 2-12 shows the rolling friction as a function of the dimensionless
relaxation time where τ is the ratio of nonlinear viscosity, η, to spring modulus, E, and T
is the time taken for the cylinder to move one contact half-width, a. Also to note from
Figure 2-12 is that the load required to maintain constant indentation depth increases
with velocity, a demonstration of the stiffening effect due to the dynamic modulus’
dependence on deformation frequency.
Figure 2-12. Rolling friction as a function of the dimensionless relaxation time.
The hysteresis effects on rolling friction continues to be analytically probed in order to
properly define the deformation losses for all materials and system configurations.18,88–96
While hysteresis commands a large portion of the frictional energy loss in rolling
contact, there are very few practical situations where the contact is devoid of sliding
friction. Combined rolling-sliding contacts are defined by the percentage of slip within a
37
tractive contact. When Tabor described his rubber ball in a conformal groove he made
note that the system was similar if not congruent with a deformed ball on a flat
surface.87 Svendenius improved upon the work of Pacejka by creating a model, Figure
2-13, of the complex deformations and loading in a combined rolling-rolling sliding
contacts.66,97 The model sections the material between the carcass of a tire and the
road surface into individual thin strips rigidly attached to the carcass as if they were
bristles in a brush. This analytical model was an early step in describing the complex
nature of rolling sliding contacts.
Figure 2-13. Schematic of the deformation of a rubber tire between a road surface and
the tire carcass. Due to a mismatch of linear velocity between the two surfaces a shear zone develops in the contact. The shear increases until the restoring forces of the rubber overcome the friction force and sliding occurs.
Elastomer Wear
Elastomer wear, like elastomer friction, is a function of its viscoelastic material
properties. Specifically, the adhesive and hysteretic components of friction have
analogous components of wear; abrasion and fatigue wear respectively, shown in the
flowchart in Figure 2-14 created by Moore.98 The abrasive wear of elastomers is
attributed to sliding friction and is a short duration tearing, cutting, or fracture of material.
Abrasive rubber wear is classified into two categories based on the sliding surface
characteristics of the system; dry or wet and periodic or intrinsic abrasion. While
38
abrasion is normally an acute and severe process, rolling wear, or fatigue wear, is a
mild continuous progression. In all sliding and rolling contacts there is a component of
both adhesive and fatigue wear. What becomes evident, after even a cursory glance, is
that the wear of an elastomer is a complex function of the combination of friction
mechanisms, surface morphology, and system dynamics.
Figure 2-14. Relations between the friction and wear mechanisms of rubbers.
39
Abrasive Wear
Intrinsic abrasion
Schallamach’s research in the 1950’s was one of the first documents to focus on
the abrasive wear of rubbers. He conducted tests by sliding a needle varying in tip
radius from 30 mm to 1 mm along rubber at various loads making sure to not pierce the
rubber specimen.99 He found that the damage was based on the mechanical strength
and elastic and frictional properties of the rubber sample. He hypothesized that tears
due to abrasion were related to normal load and roughness of the abrasive. He also
found that tearing from the abrader was due to stress concentrations at the rear of
contact using a photoelastic imaging procedure.
Figure 2-15. Photoelastic snapshot of strain fields of a rubber flat sheared by the motion
of a hard spherical slider. Adapted from Schallamach, 1968.99
40
The tensile stress concentration from a rigid point contact being slid across a
rubber surface shown in Figure 2-15 causes the development of both tensile and
compressive zones within the material. At this point, if failure occurs within the material
at the tensile zone, a tear will form. Successive abrader passes will cause the tear to
grow and eventually rupture and depart from the bulk. Schallamach found that the tear
width is proportional to the width and contact area of the abrasive particle. This then
lead him to develop expressions for abrasive wear in which the volume of rubber
removed is proportional to the cubed length of the original width of the tear. Zhang and
others have confirmed this result with their own single point abrasion tests.13,16,100–102
Figure 2-16. Cross sections and a micrograph of a rubber flat with developing periodic
abrasion A) Tongue formation due to unidirectional sliding of rubber flat against rough abrasive. B) Tongues begin to separate and lengthen from material as abrasive forces continue to apply tensile loads in the sliding direction. C)Periodic abrasion pattern of a natural rubber after being slid against rough carbide paper. Adapted from Schallamach, 1954.75
The single point abrasion test on rubbers attempts to mimic the single asperity
contact of rough sliding surfaces. In such a sliding contact the local stresses around the
asperity are high and tensile causing surface degradation in the form of crack opening
and tearing.103,104 For sliding tests conducted on roughened surfaces with “sharp”
41
asperities, this is the abrasion mechanic that dominates.105 Schallamach followed up his
single point abrasion tests with rubber sliding against abrasive paper.45 He found two
interesting things: 1) during unidirectional sliding a pattern developed which shall be the
subject of later discussion and 2) if the direction of sliding was changed, the total
abrasion could be measured in the form of a dimensionless measure of abrasion, δ(p),
where
𝛿𝛿(𝑒𝑒) =𝑠𝑠𝑒𝑒𝑒𝑒𝜔𝜔ℎ 𝐿𝐿𝑓𝑓 𝑡𝑡𝑎𝑎𝑒𝑒𝑡𝑡𝐿𝐿𝑎𝑎𝐿𝐿𝑡𝑡𝑠𝑠𝑎𝑎𝐿𝐿𝜔𝜔𝑡𝑡𝑡𝑡𝑑𝑑𝑒𝑒 𝐿𝐿𝑓𝑓 𝐿𝐿𝑙𝑙𝑎𝑎𝑠𝑠𝑎𝑎𝑡𝑡𝑡𝑡
(2-21)
Periodic abrasion
Periodic abrasion occurs when sliding rubber in a unidirectional system
configuration. In such a system a series of ridges form on the rubber surface referred to
as the abrasion pattern. These ridges run parallel to each other but perpendicular to the
sliding direction. This abrasion pattern first noticed by Schallamach, sometimes referred
to as Schallamach waves, is a characteristic sliding rubber wear pattern. Figure 2-16
shows these waves for various rubber vulcanizates sliding against tarmac and concrete
tracks. Schallamach found that the intensity of the pattern was directly related to
coarseness of the track and inversely related to the stiffness of the rubber.
Moore found that in a system where periodic abrasion dominates, the rate of
abrasion is greater than those systems with which intrinsic abrasion is the primary
abrasion regime (Figure 2-17).98 Schallamach hypothesized as to what the cause was
but it wasn’t until Southern and Thomas developed a system with which they could test
the mechanics of the abrasion pattern. This blade abrader consisted of a razor blade
pressed into contact with a rotating rubber disk. Using a fracture mechanics approach
they deduced that each wave was the result of crack openings on the surface.
42
Figure 2-17. Material loss as a function of sliding distance for patterned abrasive wear
and intrinsic abrasive wear of rubber. Adapted from Schallamach, 1954.75
As the blade passed each “tongue” it would strain the tongue and grow the rubber tear
as shown in Figure 2-18. Their theory relates the rate of abrasion to the crack growth
resistance of the rubber. For non-strain crystallizing rubbers, the theory holds but it is
even more pertinent for strain crystallizing rubbers such as natural rubber. Natural
rubber, which resists intrinsic abrasion due to its strain crystallizing nature, is particularly
susceptive to periodic abrasion.106 In such a way, the almost macroscopic tongues
developed in periodic abrasion will cause a greater wear rate than the micro-particles
generated in intrinsic abrasion.107
Though the blade abrader proved that the abrasion pattern was a definite wear
aggressor the mechanism of initiation was still undiscovered. Bhowmick in 1982 used a
scanning electron microscope to study the initiation of cracks and formation of the
abrasion pattern.108
43
Figure 2-18. Schematic illustrating the development of the abrasive wear pattern. As the
abrader moves across the surface it pull the tongue effectively applying a crack opening force. This force increases the length of the tongue and after a period of time causes the particle to separate from the body. Adapted from Schallamach, 1958.45
He found that rubber particles ejected from micro-tears would coalesce back into the
rubber surface forming a degraded rubber layer. Over time these particles would roll
and then thicken and fine ridges would form. Fukahori also studied these micro-tears
and micro-particles and found that they were generated during the high amplitude, high
frequency (~600 Hz) oscillations that occurred in the stick-slip motion of sliding
rubber.102,105,109 Uchiyama’s direct optical observations half a decade later found that
these fine ridges would fold over when stressed by a slider eventually evolving to large
bent over ridges, seen in Figure 2-16A and 2-16B.110,111
Wear Rate
It is clear that a way to measure the wear and wear rate of elastomers is
necessary. Wear of a material can be defined as the material loss generated due to one
objects effect on another. Initial efforts to study and quantify wear rate started with
Holm’s experiments on sliding electrical contacts. He postulated that the rate of wear of
two mated materials sliding across each other was related to contact area, load, and a
probabilistic material constant relating to a materials tendency to lose atoms.112 Burwell,
Rabinowicz, and Rhee all developed methods to measure a wear rate that was very
similar to Holm’s depending on contact area and surface mechanics. Because the
44
contact area and surface mechanics of elastomers are mercurial and ambiguous, these
models fall short when trying to predict the rate of wear. Archard in 1950 developed a
model for wear rate, Equation 2-22, that was proportional to the energy put into the
system but independent of any surface or bulk material mechanics.49
𝑊𝑊𝑒𝑒𝑡𝑡𝑒𝑒 𝑅𝑅𝑡𝑡𝜔𝜔𝑒𝑒 = 𝐾𝐾 =𝑣𝑣𝐿𝐿𝑙𝑙𝑣𝑣𝑛𝑛𝑒𝑒 𝑙𝑙𝐿𝐿𝐿𝐿𝐿𝐿
𝐴𝐴𝑒𝑒𝑒𝑒𝑙𝑙𝑎𝑎𝑒𝑒𝑠𝑠 𝐹𝐹𝐿𝐿𝑒𝑒𝑑𝑑𝑒𝑒 ∙ 𝑆𝑆𝑙𝑙𝑎𝑎𝑠𝑠𝑎𝑎𝑡𝑡𝑡𝑡 𝐷𝐷𝑎𝑎𝐿𝐿𝜔𝜔𝑡𝑡𝑡𝑡𝑑𝑑𝑒𝑒 (2-22)
Combined Rolling-Sliding Wear
The benefit of the Archard wear model is that it can then be applied to a complex
elastomer system with which the wear of the system is not only dependent on the
abrasion of rubber but also the fatigue experienced by the material such as elastomers
in combined rolling and sliding. Though this system is of great practical importance due
to its closeness to that of the system of a car tire, it has had relatively little experimental
effort devoted to it. In 1961 Grosch and Schallamach developed a large scale trailer
with which they sought to test the wear of tires in combined rolling and sliding. The
trailer was run against wet and dry road surfaces with varying slip percentages and
mass loss was recorded at intervals of sliding distance in order to measure wear. While
a good first effort, the test ultimately suffered from too many compounding variables
causing the data to be qualitative at best. The next true study of elastomer wear in
combined rolling sliding was Gerrard et. al in 2002 where they found that dynamically
changing the slip percentage by a large margin during rolling the wear rate was
substantially decreased.113 They hypothesized that the decreased wear rate was due to
intrinsic wear dominating the mechanics of the system because of the constant change
in the direction of abrasion. Recently, Xu developed a tribometer with which they could
study the rolling sliding characteristics of steel spheres placed in between a rotating
45
rubber flat and a steel plate with conformal grooves.114–116 Because the spheres were
unconstrained in the other axis, the mechanisms of wear could not be gleaned from
such tests. The opportunity to study the mechanisms of elastomer wear in combined
rolling and sliding will remain until a method to systematically test the imposed test
parameters individually.
46
CHAPTER 3 SLIDING FRICTION AND FRICTIONAL HEATING OF ELASTOMERS
In Situ Thermal Measurements of Sliding Contacts
Frictional Heating of Sliding Contacts
The effects of temperature on friction and wear are of significant practical
interest; for example, moving mechanical assemblies and the materials selected for
durable operation must survive not only the ambient conditions but also the thermal
conditions generated as a result of frictional sliding. Numerous studies have shown that
the real area of contact between two rubbing bodies is typically small compared to the
apparent area of contact.48,117–121 In these intimate contacts, friction and high contact
pressures frequently combine to generate substantial flash temperatures, and these
asperity contacts can have profound effects on the tribological operation.122
Characterization of the dependence of tribological properties on such temperatures and
pressures is an ongoing effort which has spanned analytical, numerical, and empirical
approaches.123 Here we expand these methods through precise micro-scale
experiments by (1) directly measuring the temperature profile of the apparent contact
area for small contacts, (2) correlating the measured temperature data with traditional
tribotesting (i.e. friction/wear) measurements, and (3) comparing these measurements
to the established models for frictional heating. These measurements not only provide
explicit temperature and pressure profiles within the contact, but can also provide data
for estimates of wear.
Frictional heating experiments have spanned a large range of methods starting
with embedded thermocouples, which then progressed to the use of dynamic and thin-
film thermocouples.124,125 The radiometric approach, which is considered to be the most
47
accurate because of its ability to sample at higher speeds and capture larger areas, has
been used recently but often at high temperatures (400°-500°C) and with large
variability.125–131 Major limitations associated with using a radiometric approach include
an incomplete knowledge of emissivity and real contact area of the contacting
bodies.126,130,131
This work was conducted on a new instrument that combines microtribological
probes and methods to perform sliding friction experiments while making high fidelity full
field surface temperature measurements of the contact through infrared thermography.
The instrument is capable of measuring normal and friction forces ranging from 10 µN to
over 2 N. This design facilitates synchronized measurement of externally applied
contact force, friction force, and in situ thermal imaging of the contact with a spatial
resolution limited by the diffraction limit (around 3 µm). Preliminary tribological tests with
in situ frictional heating measurements were performed between half spheres of filled
natural rubber and a flat calcium fluoride disk to directly obtain the temperature
distribution and average temperature rise within the contact and compare these to the
models previously set forth.
Tribometer Description
The in situ thermal micro-tribometer combines the ability to perform sliding
friction experiments while making full field surface temperature measurements of
tribological contacts through infrared thermography. Samples are mounted directly to an
instrumented cantilever that converts measured deflections to normal and frictional
forces. An open aperture rotary stage houses the calcium fluoride counter-sample which
is in-line with the camera. Manual micrometer stages control positioning and loading of
48
the sample. The 3x infrared objective is focused on the interface between the rubber pin
and counter-sample as shown in Figure 3-1.
Figure 3-1. Objective of infrared camera focused on the interface between a natural
rubber half-sphere and a CaF2 countersample.10 In Situ Thermal Tribometer. September 2, 2011. Courtesy of author.
Image Acquisition and Temperature Analysis
Energy dissipation in the form of frictional heating is caused by the relative
motion of the sample and countersample. Thermal radiation emitted by the surface of
the pin passes through the IR transmissive countersample and is picked up by the
detector in the camera. Calcium fluoride was chosen for its ability to transmit IR light
with little attenuation (92% transmissive in the 1-5 µm spectral range).126 All
considerations for accurate measurement of the temperature at the surface of the pin
have been accounted for. A further discussion of infrared thermography is given by
Volmer.132
Image acquisition occurred at 100 Hz by the FLIR SC7650. The camera acquires
with a 5 µm/px resolution with a field of view of 3.2 mm x 2.4 mm. Post processing in
Matlab™ is necessary to approximate the true area of contact and the temperature
49
distribution within said area. The steep thermal gradients caused by frictional heating
combined with intra and post processing provided sufficient contrast between areas in
and out of contact. Pixels within the contact area are then mapped to their
corresponding temperature value. Maximum and nominal temperatures and pressures
are obtainable after this mapping.
Figure 3-2. The tribometer in profile. The stage and counter-sample holder are cross-
sectioned to show the beam path. Radiation from sample surface is focused by the camera lens and measured by the camera detector. Adapted from Rowe, 2013.10
Force Measurements and Positioning
Normal and friction (tangential) forces are measure using an instruemented
cantilever assembly. The assembly consists of two sets of parallel flexures, offset by
90o, in series to constrain the cantilever to rectilinear motion (Figure 3-2, inset). The
normal and tangential flexures displace independent of each other allowing cantilever
stiffness can be tuned specific to the force direction. Independent force measurement
50
also serves to decrease alignment uncertainties.129 The cantilever free end provides
threads for mounting the sample as well as a conductive target for the displacement
sensing probes. The LION Precision capacitance probes monitor the normal and
tangential displacements of the cantilever. The load is then calculated using the
calibration constants of the probes (µm/µV) and stiffness (N/µm) of the cantilever.
Through cantilever design loads as low as 30 µN and as high as 2 N have been
measured on the instrument. The cantilever used for these experiments measures force
with an uncertainty of ± 0.015 mN (0.8% of applied load).
Providing rotational motion is an open aperature rotary stage that secures the
countersample holder and can spin at speeds between 1 and 1200 rpm (± 1 rpm). The
countersample is contained within a holder that reduces the total run-out of the system
to < 3µm. An optical read head attached to the stage contains a ring encoder which
measures the angular position of the stage within 0.0008o.
An program built in-house using LabView™ controls the equipment and data
acquisition. Data acquisition is provided by a 16-bit analog to digital card and all force
and position data was acquired at 1 kHz.
Materials
Hemispherical pins with radius 2 mm were used for the frictional heating study.
The pins were molded using a natural ruber that when vulacanized has a modulus of
approximately 4.2 MPa at 10% extension. The density of the sample rubber was 1,200
kg/m3 and a thermal diffusivity and conductivity common to filled natural rubber
compounds were used, 0.143x10-6 and 0.24 W/mK respectively. The surface
51
roughness, as measured through white light interferometry, was found to be
approximately 1 µm.
Calcium fluoride was chosen as the countersample for these experiments due to
its hardness and infrared transmissive properties. Calcium fluoride transmits 92-95% of
infrared wavelengths between 0.2 and 6.5 µm. The countersamples were 50 mm in
diameter and 3 mm in thickness with modulus of 75 GPa and surface RMS roughness
of 6 nm.
Description of Loading and Sliding Experiments
Figure 3-3. Contact area and temperature as a function of speed and load. Adapted
from Rowe, 2013.10
Two sets of experiments were conducted in order to determine the geat
generation due to friction of natural rubber. Long time duration (900 s) testing was
conducted to observe the effects of load and velocity on the average temperature
increase of the contact. A set of forced convection tests determined that the finite
52
geometry of the system was preventing the sample from returning to the initial
temperature, To. This prompted a third set of tests which were performed for short
durations (~25 s) to better match analytical models.
Tests were performed at prescribed speeds and loads for short durations (25 s)
and long durations (900 s). Images were acquired before, during, and after sliding at a
rate of 80 Hz for short tests and 0.5 Hz for long tests. Tests were performed at 100,
250, 500, 750, and 1000 mN normal loads with varying sliding velocities of 200, 500,
750, and 1000 mm/s. Representative stills are shown in Figure 3-3.
Figure 3-4. Comparison of average contact temperature under forced and natural
convection for a normal load of 500 mN and a sliding velocity of 750 mm/s. A true steady-state is only achievable through application of forced convection. Adapted from Rowe, 2013.10
Forced convection experiments were performed for 900 s where forced
convection was applied using a steady flow of laboratory air was directed at the sliding
interface for a period of approximately 400 s and then removed. The temperature was
53
measured during both the periods of forced and natural convection. Steady state
temperature was achieved during the period of forced convection. When the airflow was
removed the temperature climbed to a psuedo steady-state as shown in Figure 3-4.
Eventually a true steady-state may have been achieved but was not obtainable due to
gross wear of the sample material.
Results and Discussion
Figure 3-5. Evolution of contact shape and temperature as a function of applied load
and sliding velocity. Increasing velocity and normal load distort the contact significantly from its original, circular, shape. Adapted from Rowe, 2013.10
54
The measured contact temperature as a function of applied load and sliding
velocity is shown in Figure 3-5. Average temperature rises ranged from ~3 °C, at the
lowest load and sliding velocity, to ~26 °C, coinciding with the highest sliding velocity of
1,000 mm/s and maximum applied normal load of 1,000 mN. The maximum single point
temperature was measured to be ~ 51 °C. Table 2 shows the measured values for
each pin-on-disk experiment.
The present experimental configuration allows for the direct measurement of
contact temperature in addition to providing an estimate of the shape and area of
contact. This creates a unique situation where all of the variables required for thermal
modeling are either known a priori or measured in situ.
Both Blok and Jaeger hypothesized that a fraction, α, of the heat flux generated
(per unit time over the area of contact) passes into body 1 and the remaining fraction,
(1-α), passes into body 2. The partitioning coefficient, α, can be obtained by equating
the interfacial contact temperature due to a moving heat source, with heat flux qα going
into body 1, to the contact temperature due to a stationary heat source, with q(1-α)
going into body 2. (Here the disk is taken as body 1 and the pin as body 2). Laraqi et
al131 showed the partitioning coefficient remains approximately constant for Peclet
numbers greater than 30.
Figure 3-6 compares the frictional heating models of Jaeger, Archard, and Tian
and Kennedy to the measured average contact temperature rise values. These model
derivations are also given in Appendix A. The measured temperature rise in the short
duration (~ 25 s) sliding experiments were very close to those predicted by the model
over the range of normal forces, velocities and resulting Peclet numbers tested.
55
However, long duration (~ 900 s) sliding experiments were observed significantly higher
temperature rise than predicted by any of the models. The models neglect convection of
heat away from the disk and do not account for the residual heat in the disk that passes
back through the contact with each revolution. For pin-on disk geometries with Pe > 20
a distinct tail of residual heat is developed.131 The residual head is responsible for
artificially increasing the contact initial temperature resulting in a higher than predicted
equilibrium contact temperature. The use of forced convection allows disk material
exiting the contact to cool back down to ambient temperatures before reentering;
mimicking the effect of a sufficiently low Peclet number for frictional heat to dissipate.
Figure 3-6. Direct temperature measurements for 900 s tests compared to model
predictions. The long duration sliding measurements exceed the model temperatures do to residual head contained within the countersample from the previous cycle. Adapted from Rowe, 2013.10
56
Dynamic In Situ Measurements of Frictional Heating on an Isolated Surface Protrusion
Frictional Heating of Asperity Contacts
Building on the work done by Rowe et. al. an experiment has been designed in
which the temperature rise due to a prescribed geometry moving through contact is
measured.10 Local temperature rises of the feature are appreciable over the
temperature rise of the global contact even at sliding speeds as low as 10 mm/s. The
maximum and average temperature of the feature follows the classical thermal heating
models of Jaeger. Additionally, the range of sliding speeds used may have crossed two
frictional heating regimes which, with further investigation, may lead to a better
understanding of sliding contact frictional heating relative to surface roughness.
Asperity Contact Temperatures
Instrumentation
A pin-on-disk, in situ thermal micro-tribometer was used for all tests conducted in
this experiment where contact is made between a rubber hemisphere on a 3 mm thick
cylindrical optical window.10 The rubber, Viton A, is pressed into contact with the
calcium fluoride disk and the sample is held stationary while the disk is rotated in a ring
bearing stage, driven by a servo motor. Loading and positioning of the sample are
controlled by manual micrometer stages and held constant during testing. Two
capacitance probes attached to a rectilinear cantilever transduce displacements into
voltages which are then converted into force measurements.
The work done to overcome the friction between the rubber pin and the rotating
calcium fluoride (CaF2) disk causes heat to be generated within the contact. The
infrared radiation emitted by the sample is then measured by an IR camera (FLIR
57
SC7650) and related to temperature. The indium antimonide detector is sensitive to
radiation wavelengths between 3 and 5 µm where the CaF2 counter-sample transmits
up to 92% of incident IR. At quarter frame resolution (128 x 160 pixels at 5 µm/pixel) the
camera is capable of acquiring at 870 Hz. LabVIEW™ was used for instrument control
and data acquisition. Post processing of data was performed in Matlab®.
Figure 3-7. Overview of the experimental method. A) Schematic of contact between the
pin, counter-surface, and positive feature. As the feature moves into contact it displaces the Viton A pin. B) Data analysis scheme of the feature as it moves through the global contact. The position of the pin is measured from the center of apparent contact of the pin with the counter surface to the center of the feature. C) The temperature development as the feature moves under contact shows a correlation with the pressure distribution in the contact. Highest temperatures manifest near the center of contact where the highest local pressures exist. Adapted from Bennett, 2014.133
Tests were conducted at a prescribed load of 200 mN and contact pressure of
~620 kPa. One sliding speed was used for each test and varied from 10 mm/s up to
58
200 mm/s. Each test lasted ~22 seconds and was imaged at 870 Hz. Frames were
acquired for each pass of the feature through the field of view of the camera Figure 3-
7C. The asperity was tracked frame by frame using an algorithm, designed in house,
that assigns a region of interest to the cylindrical asperity for all frames in which it was
visible between the pin and the counter-sample Figure 3-7B.
Figure 3-8. The SU-8 feature rotates with the disk and slides under contact once per
revolution. IR radiation from the sliding contact is measured by the 3X lens which then focuses the radiation on the indium antimonide detector. A) Surface feature before sliding experiment. B) Surface feature after a sliding experiment. Adapted from Bennett, 2014.133
For every video the frames where the feature appeared were separated from the
rest of the video. Image editing tools in Matlab were applied to each of the frames which
isolated the feature from the rest of the image using a region of interest overlay. As the
frames progressed the region of interest moved according to predicted values of the
59
features location. The dimensions and location of the ROI was then applied to the
temperature data of the image. The average and maximum temperatures of the isolated
regions were recorded. The distance, s, is measured from the center of the image to the
center of the ROI.
Materials
Calcium fluoride was used as the counter-sample. Similar to other ionic solids,
CaF2, is transmissive in the infrared spectrum and exhibits 92-95% transmission
throughout the 0.2-6.5 µm waveband; this range encompasses the 3-5 µm range in
which the FLIR SC7650 is capable of measuring. Due to its hardness and thermal
conductivity of 9.71 W/m-1K-1, as well as optical transmission, CaF2 is an ideal choice
as a counter-sample.
To produce the positive feature on the surface of the counter-sample an 18 µm
layer of a common photoresist epoxy, SU-8, was spun coat onto the CaF2. A
photomask was then applied for UV curing in order to produce patterns on the disk. The
desired pattern was a single cylinder, 18 µm tall and 150 µm in diameter at a disk radius
of 15.725 mm. After curing the excess uncured photomask was dissolved away using
Propylene glycol monomethyl ether acetate. The deposition process left an optically
smooth positive feature on the CaF2as seen in Figure 3-8. SU8 has a thermal
conductivity of 0.2 W/m-1K-1 which matches well with the thermal conductivity of the
pin.
Results
The average temperature rise of the surface feature (above ambient) ranged
from 0.5 °C, for the slowest sliding speed test, to 7 °C for the fastest. The maximum
60
temperature rise followed the same trend with temperature rises from 1 to 18 °C. A
characteristic image for each test is displayed in Figure 3-9. Ambient and global contact
temperature, friction coefficient, and load did not change appreciably during the
experiments. Maximum and average temperatures of the feature for each test are given
in Table 1.
Figure 3-9. Schematic of contact between pin, counter-surface, and positive feature. As
the feature moves into contact it displaces the Viton A pin. The position of the pin is measured from the center of apparent contact of the pin with the countersurface to the center of the feature. Adapted from Bennett, 2014.133
The center of the features position, s, as measured from the center of global
contact, was recorded for all frames where the feature is in view. Position data was then
binned in 25 µm increments; the standard deviation of the temperature for each bin is
plotted in Figure 3-10. In some tests two distinct peaks form near s = 0. It has been
documented that increased speeds can cause rubber pins in tribological sliding
experiments to bifurcate into two zones separated by a section of non-contact.10 This
61
could explain the drop in feature temperature and also bolsters the correlation between
local contact pressure and temperature rise.
Figure 3-10. Average and maximum temperature rise of and within the region of interest
of the positive feature at varying speeds. Adapted from Bennett, 2014.133
While rubber wear debris is characteristic of sliding tests with completely
roughened surfaces, no wear debris was observed or recovered during these tests.
Neither the pin nor the raised feature showed appreciable degradation as observed by
white light interferometer scans.
Discussion
The average and maximum temperature is plotted at a particular feature position
in Figure 3-11A and Figure 3-11B respectively. Position (s) is measured from the center
of global contact to the center of local contact; each temperature point corresponds to a
locus of positions 25 µm wide. As the feature first enters contact it begins to heat; for
62
low speed tests the temperature throughout global contact plateaus but at higher speed
test there are distinct temperature peaks around s = 0. This trend alludes to a
correspondence between the maximum temperature and the location of the highest
pressure under the contact. The parabolic pressure distribution in circular contacts has
been studied before and the findings correlate well with the data of this experiment.123
Figure 3-11. Temperature rise of the feature as measured. A) Average temperature rise
of positive feature region of interest as it moves through global contact. B) Maximum temperature rise within the positive feature region of interest as it moves through global contact. The position of the 150 µm feature was determined from the center of global contact between the pin and disk. Adapted from Bennett, 2014.133
The local feature temperature distribution is more likely related to the Peclet
number. Because the feature width is small compared to the contact width and because
the pin material can accommodate large strains pressure across the feature can be
assumed to be constant. As the sliding speed increases, the distribution of heat shifts
from the leading edge of the feature to the trailing edge. The thermal gradient at low
speeds is fairly shallow but as speed increases the gradient across the feature can
realize half of the total temperature rise due to the feature moving through contact.
63
An analytical model for the average and maximum temperature rise in sliding
contacts has been solved for and verified experimentally.134 The model partitions the
heat between the two sliding bodies based on their individual thermo-mechanical
properties. In the performed experiments the feature becomes a third body and must be
accounted for in the heat partitioning equations. According to the models the lowest
temperature rise will be realized when contact consists of only the pin and counter-
sample which is the state of lowest contact pressure. The highest contact pressure
would then be found when the load is applied only to the contact area of the feature.
Applying these two conditions to Jaeger’s model provides bounds for the measured
data shown in Figure 3-12 with values given in Table 3-1.
Figure 3-12. Measured maximum and average temperature rises for each sliding speed.
Jaeger’s solution for average and the modified solution for maximum temperature rises within contact provide bounds for the data. Adapted from Bennett, 2014.133
64
Table 3-1. Predicted and Measured Temperatures of the Protrusion Predicted ΔT Measured ΔT Sliding Speed Avg. Max Avg. Max
mm/s °C °C °C °C 10 0.7 4.3 0.8 1.7 25 1.4 7.2 1.9 4.0 50 2.0 10.0 2.9 5.6 75 2.2 10.5 2.8 5.5
100 3.3 16.1 5.6 12.9 150 4.2 19.1 6.7 14.2 200 5.0 22.4 8.0 16.7 The analysis process predicts feature position fairly well; however, the final
judgment on feature position lies with the experimentalist. A small portion of frames
exist in which the feature is obscured and the experimentalist must decide, based on
prior frames and localized heating in the current frame, where the feature is located.
This contributes to the error in Figure 3-10.
65
L’Escargot Rapide
Soft Contacts at High Speeds
Schallamach was the first to show that rubber can move without true interfacial
sliding. He observed “waves of detachment” where frictional contact was lost between
rubber and the counter-surface when an elastic instability formed at the front of contact
and propagated through.70 These phenomenological occurrences have been the focus
of much of the literature regarding the sliding of soft unfilled rubber.9,17,21,25,36,55,78,81,135–
139 However, Schallamach waves are not always responsible for interfacial movement,
as ridges may appear when soft filled rubber slides on clean smooth glass but do not
propagate.25 These ridges most likely form due to elastic instability within the contact
but unlike the waves of detachment these form and persist. Modeling has previously
shown that such an instability is predicted to exist under certain conditions.10,129 The
velocity and forces necessary to develop these ridges are examined herein.
Schallamach waves have been observed at sliding speeds of 40 to 800 µm/s
between a stationary sheet of vulcanized rubber and a moving glass probe.136 The
literature focuses on these velocities and is largely silent regarding higher velocities.78
These higher velocities will be examined thusly to determine their effect on contact
area. In these experiments a hemispherical rubber pin was in a persistent state of
contact on a rotating transparent counter-surface vice the historic setup of a glass probe
on a rubber counter-surface.25,70,78,135,136
Materials
Carbon black filled natural rubber hemispheres of 2 mm radius were used in this
study. The rubber had an elastic modulus of approximately 6.5 MPa. The average
66
surface roughness (Ra) and root mean square roughness (RMS) of the molded rubber
were determined to be ~800 nm and ~1µm, respectively, by scanning white light
interferometer (Veeco Wyko NT9100).
Calcium fluoride optical windows were used as the counter-surface. The disks
were 50 mm in diameter and 3 mm thick with an RMS roughness of ~ 6.5 nm. Calcium
fluoride is comparable to borosilicate glass with regards to transmission of light in the
visible spectrum (greater than 90% transmission of light between 400-700 nm
wavelengths).
Experimental Methods
Tribometer
Figure 3-13. A schematic representation of the instrument configuration showing the
cantilever load head assembly, the rotating counter-sample and the microscope objective. Images of the contact interface between an elastomer pin and calcium fluoride were acquired at 125 fps. Adapted from Schulze, 2014.140
A pin on disk tribometer with in situ imaging capabilities was used in this study,
Figure 3-13.141 A microscope objective was focused on the interface made between a
67
rotating transparent disk and a stationary pin.142 The pin was mounted to a
displacement-based cantilever load head assembly that measured normal and
tangential (friction) forces. Normal load was applied by a displacement controlled linear
air bearing and spring assembly that also held the cantilever load head. The disk was
mounted in an open aperture rotary stage that was driven by a digitally controlled servo
motor.
Experiments
Prior to sliding, the sample was loaded against a clean calcium fluoride disk to a
prescribed force. An image of the loaded contact area was taken before sliding.
Unidirectional sliding experiments were then performed at prescribed velocities (0.001,
0.01, 0.1, and 1 m/s) and loads (0.1 and 1 N) for 35 s. Images were acquired during
sliding at 125 fps.
Figure 3-14. Representative images from each experiment illustrating the variation in
contact size and geometry. Velocities beyond 0.01 m/s, and independent of load, the contact separates into two distinct regions; a high pressure zone and low pressure zone. These two zones were accompanied by a region where there was no observable contact. Adapted from Schulze, 2014.140
68
Results and Discussion
At low velocities (0.001 and 0.01 m/s) the contact area remained uniform while at
higher velocities (0.1 and 1 m/s) the contact area bifurcated into two distinct regions,
reminiscent of a snail’s foot. This occurrence was independent of the applied normal
load. The only observable effect of increased applied load was an increase in the
overall area of contact both before and during sliding. However, the ratio between the
stationary contact area to the contact area during sliding was qualitatively the same
between all tests at a given velocity Figure 3-15.
Table 3-2 Measured values from the pin-on-disk testing of a filled natural rubber hemisphere on calcium fluoride. Contact areas are nominal and correspond to the stationary contact area before sliding (Astationary), the continuous contact area during sliding (Afront), and the sporadic contact area during sliding (Arear). These areas are schematically illustrated in Figure 3-15.
v Fn µ Astationary Afront Arear m/s N mm2
0.001 0.14 2.30 0.22 0.15 - 0.99 1.60 0.89 0.73 -
0.01 0.11 2.90 0.22 0.15 - 1.03 2.20 0.92 0.67 -
0.1 0.14 2.40 0.22 0.05 0.10 0.99 2.30 0.98 0.22 0.39
1 0.14 3.50 0.20 0.05 0.11 0.99 2.60 0.95 0.23 0.30
An overview of the contact areas resulting from the pin-on-disk sliding
experiments is shown in Figure 3-14 and relevant measured values are given in Table
3-2. Each image is representative of the contact area observed during each experiment.
The contact area reached its steady state geometry after a short sliding distance (~2 x
the contact width). During sliding no Schallamach waves were observed.
69
Figure 3-15. On the left, outlines of the stationary contact areas for a given load are
superimposed to show their relative contact size. Shown on the right are the stationary and moving contact areas for each test normalized by stationary contact area. The ratio between the stationary contact area to the contact area during sliding was qualitatively the same between all tests at a given velocity. Adapted from Schulze, 2014.140
The contact area at velocities greater than 0.01 m/s was separated into two
distinct regions. At the front of the contact there was a continuous, and most likely high
pressure region, followed by a trailing region with a discontinuous profile (sporadic
contact) which would support less of the applied load. A region was considered
continuous if the pixel intensity remained constant over the prescribed region or
discontinuous if the pixel variability was high over the same region within the image.
This observation is supported by previous data obtained with thermal imaging
which showed higher contact temperatures at the front of contact, suggesting higher
pressure, and lower temperatures at the exit, suggesting lower pressure.141 It is also
possible that the low pressure zone is only in optical contact with the disk which would
be difficult to validate with either of these methods.
70
The region of noncontact behind the high pressure front is most likely a fold in
the material induced by high strain within the contact. It is most likely energetically
favorable to create this tensile zone/ fold at high sliding velocities. It is possible that by
having a patterned surface this material response could be mitigated.
71
CHAPTER 4 TRACTION AND WEAR OF ELASTOEMRS IN COMBINED ROLLING-SLIDING
CONTACT
Combined Rolling and Sliding Contacts
The dynamic combined rolling and sliding contact involves a wide variety of
conditions which can have profound effects on friction, wear and lubrication. As such, it
has been widely studied for a multitude of applications ranging from high performance
bearing lubrication to automotive tires.26,60 Typically, these materials will experience
millions of cycles under complex loading and slip conditions. Elastomers are ideal
materials for rolling elements due to their good fatigue resistance, highly elastic and
hysteretic behavior, and varied friction coefficient.67,143 Because of their wide use as
rolling elements a comprehensive understanding of how soft and highly elastic materials
respond under these circumstances is necessary. Numerous studies on the contact
mechanics and friction behavior of these materials in pure sliding5,12,14,25,63,67,73,129,144,145,
rolling6,38–40,54,84,87,93,143,146,147 and combined rolling and sliding20,69,89,148,149 have been
rigorously performed for a variety of materials, surfaces, contact conditions and
geometries. However, the wear of soft materials in combined rolling and sliding has
primarily focused on abrasive surfaces, large slip angles and high slip velocities.
11,13,20,69,98,115,116,148,150,151
Iwai et al. performed well controlled combined rolling and sliding traction
experiments with spherical elastomer specimens on a rotating glass disk.31 In their
investigation they found that changes to the areas of the sticking and slipping regions in
the rolling contact scaled directly with the imposed slip percent, with each area
contributing to the measured traction response. Xu et al. performed combined rolling
72
and sliding experiments with a metal sphere rolling on various elastomer flats.115,116,148
They quantified the amount of material removed due to pure rolling conditions without
imposing global slip within the contact.
Here, a simple and unique instrument was designed for the tribological testing of
materials in com- bined rolling and sliding contact. In this configuration a driven sample
is in a persistent state of rolling contact with a smooth rotating disk. Using this
instrument, the tribological performance of a carbon black-filled natural rubber
compound was evaluated with an emphasis on the load and velocity dependence of the
traction coefficient as well as the effects of slip on wear rate.
Tribometer Description and Design
Combined rolling and sliding experiments were performed on a ball-on-flat
tribometer shown in Figure 4-1. The tribometer was comprised of a rotary stage and a
machine spindle whose rotational axis are aligned perpendicularly to each other. The
Bell Everman rotary stage is the same model as the one used for the thermal in situ
microtribometer and as such has the same specifications. Force measurement is
accomplished through a six-channel load cell mounted directly under the spindle. The
spindle is mounted on the load cell and aligned in a manner to make all moments zero
when the sample is stationary and not in contact. By isolating the spindle on the force
transducer, it is possible to inferentially move the sample to the pure rolling point by
monitoring the force and moment channels without recourse to complex alignment
schemes and precision machining. Such a point occurs only when the losses due to
rolling friction are balanced with the losses due to sliding friction (slip), resulting in a net
zero force within the plane of contact. This inference is possible since all forces acting
73
on the sample must go through the load cell and are therefore known. LabVIEW™ was
used for experimental control and data acquisition. A 16-bit analog to digital acquisition
device externally conditioned all force and position measurements; data was acquired at
1000 Hz.
Figure 4-1. Schematic of the combined rolling–sliding tribometer. Since all forces acting
upon the sample must pass through the load cell, it is possible to simply move the specimen to the pure rolling point without a priori knowledge of the contact location. The pure rolling point is identically where all forces within the plane of contact cancel, resulting in a net-zero force. Adapted from Rowe, 2015.152
Materials and Experiments
Materials and Sample Preparation
Injection molded carbon black-filled natural rubber spheres of 20 mm diameter
were used in this study. The molded rubber was conventionally vulcanized and had a
composition similar to that of a generic truck tire compound. This material had an elastic
modulus of approximately 4.2 MPa at 10% extension and a density of 1200 kg·m-3. The
average surface roughness (Ra) and root mean square roughness (RMS) of the molded
rubber were determined to be ≈800 nm and ≈1000 nm, respectively, by scanning white
light interferometer (Veeco Wyko NT9100). A machined 6061 aluminum disk with an
74
anodized coating, approximately 10 μm thick, was used as the counter surface. An
alumina coating was used to minimize the effects of surface chemistry on the wear and
friction morphologies of the elastomer sample. The anodized disk had an average and
root mean square roughness of 460 nm and 590 nm, respectively.
The elastomer specimens were initially cleaned with a mild detergent and
deionized water, and allowed to dry for 24 h before testing. A five-step solvent cleaning
process was used to clean the disk before each test.153
Traction and Wear Experiments
Traction and wear experiments were conducted by affixing the sample to the
spindle collet and loading the sample to the desired normal force. The sample and disk
were set to the same rotational velocity based on the measured track and sample radii;
the sample was then adjusted to the pure rolling point. This was accomplished by
moving both the spherical sample in x and y, as well as the rotational stage in z, and
monitoring the forces along the load cell x-axis and z-axis as well as the moment about
the y- axis. Where these forces and the moment simultaneously cross zero denotes the
pure rolling point. The velocity of the disk was then set to the desired slip velocity (Vball -
Vdisk) and the z-stage was adjusted to minimize cross slip within the contact (forces
along x-axis).
Traction tests were performed for a range of slip velocities from pure rolling to
±10% of the ball velocity. This was repeated for four different ball velocities, Vb, of 100
mm/s, 300 mm/s, 1000 mm/s, and 3000 mm/s at a 10 N normal load, as well as at
different normal loads of 2 N, 6 N, 10 N and 18 N for an average ball velocity of 1000
75
mm/s. During each test the disk velocity was changed but the ball velocity remained
constant.
Wear rates were determined by mass loss measurements taken at regular rolling
distance intervals. The mass loss of the sample was converted into a volume loss and
from this the wear rate of the sample obtained.153 Wear experiments were performed at
1%, 3% and 5% slip for a ball velocity of 1000 mm/s; and at 3% slip for ball velocities of
300 mm/s and 3000 mm/s.
Results and Discussion
Traction Experiments
The effects of ball velocity on the traction response, μ, of this material at various
slip velocities are shown in Figure 4-2A. These tests were performed for one normal
load of 10 N, resulting in a nominal contact pressure of ≈0.47 MPa. A marked change in
the traction response was observed when the ball velocity was decreased by an order
of magnitude (e.g. 1000 mm/s to 100 mm/s) even though for all experiments the highest
tested slip velocity corresponded approximately to the same disk to ball velocity ratio.
This data is plotted again in Figure 4-2B where the slip velocity has been recast as a
slip percentage (Equation 4-1), which is commonly expressed as a fraction and referred
to as a slide-roll ratio (SRR) in literature. Using slip percent instead of slip velocity
resulted in the collapse of the traction data onto one curve that described the material’s
traction response independent of rolling and slip velocities. There was also no
observable traction maximum over this large range of slip values, and at the tested
velocities (both ball and slip) and corresponding rotational frequencies (≈5–50 Hz) the
traction coefficient remained unaffected, only being a function of slip percent.
76
% 1001 ( )2
ball disk
ball disk
V VslipV V
−= ×
+ (4-1)
The dependence of the traction coefficient on slip ratio is attributed to changes
within the contact during rolling with sliding. As the amount of imposed slip increases
the contact divides into an increasingly slipping region and a decreasingly sticking
region, an effect which has been experimentally verified.88 In this experimental
configuration rolling losses have been cancelled through the zeroing of the instrument,
but contributions to friction from material deformations are still present. These include
the shearing of material in the sticking region of contact as well as asperity deformations
of the material in the slipping region of contact. Increasing slip within the contact would
result in an increase in the sliding distance and velocity of the slipping portion of the
material as well as increase the amount of strain within the sticking portion. Both of
these would alter the traction response, whether by energy dissipation due to asperity
induced material deformations and oscillations or due to the shearing of the non-slipping
area.
Changing the applied normal load, and thus mean contact pressures, over a
range of slip percent- ages, did have an effect on the measured traction coefficient.
Increasing the normal force resulted in a shallower traction response (Figure 4-2C);
however, increasing the normal force beyond a certain point (10 N) had little effect. This
suggests that the traction coefficient of rubber is closely related to contact area in
combined rolling and sliding. For small slips (micro-slip region) contact mechanics
77
Figure 4-2. Friction coefficient data plotted against %slip for various normal loads, and
linear velocities. A) Traction coefficient versus slip velocity for a carbon black filled natural rubber sphere for different ball and disk velocities. B) The same traction data as in 2a collapses onto a single curve when plotted against percent slip. C) Effects of increasing normal load on the traction response of the filled rubber for various slip percentages. D) In the linear region of 2c (≈ ±4% slip) the traction coefficient scales with contact area, or fn1/3. Adapted from Rowe, 2015.152
predicts that the slip (s) in the contact is proportional to the tangential force (FT) over the
square of the contact radius (a), which gives the friction coefficient the form of Equation
4-2.
2
T
n N
F saF F
µ = ∝ (4-2)
78
From Hertzian contact theory the normal force is proportional to the contact radius to
the one-third power (FN/a1/3). Using this relationship in Equation 4-2 predicts that the
friction coefficient is proportional to the normal force to the negative one-third power
(Equation 4-3).
1
3NFµ−
∝ (4-3)
Within the ±4% slip region of the data plotted in Figure 4-2C the traction
response is approximately linear for all normal loads considered in this investigation.
Applying the relationship given in Equation 4-3 to the traction data from Figure 4-2C
produced a single curve with a constant slope (Figure 4-2D); beyond this linear region
the traction response no longer obeys this relationship. This agreement provides
reasonable evidence that with this material, and within the range of normal loads and
slip percentages employed here, the traction coefficient is proportional to the real area
of contact under combined rolling and sliding conditions.
Wear Experiments
Figure 4-3. Wear data for natural rubber rolling and sliding samples. A) Volume loss as
a function of rolling distance (d) for a ball velocity of 1000 mm/s at 1, 3 and 5% slip. B) Taking into account the sliding distance by including slip, the wear data collapse onto a single line. Adapted from Rowe, 2015.152
79
The volume lost as a function of the product of the applied normal load, FN, and
rolling distance, d, for three different slip percentages (1%, 3% and 5%) is shown in
Figure 4-3A. For a given rolling distance, higher slip percentages resulted in a greater
volume of material removed; in fact, the slope of these curves linearly increased with
slip percentage. However, when the slip within the contact, or sliding
Table 4-1. Rolling and sliding wear rates for different slip percentages and rolling velocities.
Vb Krolling u(Krolling) Ksliding u(Ksliding) Slip % mm/s (mm3/Nm) x 10-6
1.0 1000 1.2 0.0 97.0 8.7 3.0 1000 3.2 0.1 100.0 5.2 5.0 1000 5.5 0.4 100.0 8.0 3.0 300 2.4 0.1 75.0 6.6 3.0 3000 5.1 0.2 110.0 4.7 3000 mm/s resulted in sliding wear rates of 7.5 × 10-5 mm3/Nm and 1.1 × 10-4
mm3/Nm, respectively. These relationships may not hold true for larger slip percentages
where the traction response is no longer linear and thermal effects may become
dominant.
The predicted energy dissipation in a rolling contact should scale with the applied
normal force to the two-thirds power. In these experiments, the majority of material
removal was considered to be due to the relative sliding between the elastomer ball and
the rotating disk. It is for this reason that the wear data was presented as a sliding wear
rate, and thus, a function of the normal force to the first power. Further investigations of
the effects of normal force on volume loss would show whether or not the worn volume
scales with the predicted energy dissipation.
Representative wear debris, generated during these experiments, is shown in
Figure 4-4A and 4-4B. This debris is most likely the result of small particles liberated
80
from the sample surface that either stick to the disk and smear or agglomerate together
and become entrained in the contact. Each successive pass of the sample over this
material rolls the debris and causes more particles to become entrained, either from the
disk or the rolling sample. These agglomerates eventually reach a critical size and are
ejected from the contact, appearing on the edges of the wear track. This debris was
observed to range in size from a few micrometers up to a few millimeters in length.
Figure 4-4. Characterization of natural rubber wear in rolling and sliding. A) Scanning
electron micrographs showing wear debris between 1-2 mm. B) Scanning electron micrograph showing wear debris between 10-150 µm. C) Optical profilometer scans across the ~5 mm wide wear track on the ball specimen show that material removal was biased to the left side of the wear track (smaller disk radius). Adapted from Rowe, 2015.152
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Wear of the rolling sample was found to preferentially occur over a small portion
of the contact width (~2 mm across the ~5 mm wide contact), and its location on the ball
did not change with any of the different driving slip conditions, only its magnitude.
Representative curvature subtracted line scans show material removal biased to the left
of the sample center (smaller disk radius) (Figure 4-4C). For this material removal to
occur there must exist a normal pressure and a sliding distance, in this case slip, at this
location. It is here that slip must be maximum, resulting in this locally increased material
removal. Although the data presented here is for a wear test at 3% slip it is
representative of all the wear tests in this investigation.
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CHAPTER 5 COMBINED ROLLING AND WEAR OF ELASTOMER MATERIALS
The Effects of various Material Properties on the Wear of Materials in Combined Rolling and Sliding
The combined rolling and sliding tribometer was used to test the dependence of
wear rate on the sliding velocity, load, and slip percent within the contact. This was
achieved through subjecting a 20 mm diameter natural rubber rolling sliding sample to a
variety of system conditions and monitoring the wear rate over 1 million cycles of
rotation (~60 km). Besides verifying the efficacy of the instrument, it was also
determined that the friction of the natural rubber compounds tested exhibited a
dependence on both slip velocity and contact area. This was to be expected as it
follows the same trend for friction coefficients of elastomers under dry sliding. Wear was
found to be affected by the amount of global slip within the contact (i.e. greater amounts
of slip resulted in larger volumes of worn material for a given rolling/sliding distance).
While sliding speed, load, and %slip do effect the friction and wear of materials in
combined rolling and sliding, they are system parameters and not material properties.
The wear rate, however, is also dependent on material properties such as the stiffness
and rigidity, the dynamic viscoelastic response, the shear force within the area of
contact, as well as many other parameters.47 The ability to measure the wear rate of
these parameters will help both experimentalist and designers choose which materials
will be appropriate for each specific application.
To test the instruments capability to measure differences in wear rate due to
changes in material properties a suite of materials and counter surfaces will be
examined in combined rolling and sliding. Michelin has provided three natural rubber
83
samples which have been specially made each to have a different shear modulus while
keeping other material properties like modulus, work of adhesion, and loss modulus
constant. Another set of PDMS rolling sliding samples have been fabricated at UF using
Sylgard® 184 in order to test the effect of modulus and work of adhesion on the wear
rate. Additionally a smooth aluminum surface, a roughened aluminum surface, and a
smooth aluminum surface with Kaolin added as a third body were used to test wear rate
dependence on counter surface condition.
Materials and Methods
Natural Rubber
The three samples provided by Michelin (Mix 1, Mix 3, and Mix 4) are molded
carbon black filled natural rubber which has been vulcanized. Each sample is 20 mm in
diameter and attached to a steel taper in the same manner as the samples in Chapter 4.
The samples were prepared in the same fashion as in Chapter 4 and an in depth
procedure of sample preparation and testing can be found in Appendix B. Material
testing was conducted at Michelin and the values for elastic modulus, density, and
shear rigidity are provided in Table 5-1.
Table 5-1. Mechanical properties of Michelin Mix Compounds
Material Density Shear
Rigidity g/cm3 MPa
Mix 1 1.109 1.8 Mix 3 1.117 1.3 Mix 4 1.132 1.6
Polydimethylsiloxane
PDMS rolling sliding samples of similar geometry were also tested in a similar
manner as the natural rubber samples. PDMS samples were molded at three different
concentrations of base polymer to crosslinker (10:1, 20:1, 30:1). Through a series of
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JKR style indentations the elastic modulus of each PDMS formulation was derived as
shown in Figure 5-1.
Figure 5-1. Indentation experiments were performed in order to estimate the Young’s
Modulus of each of the PDMS samples. Plots of the contact radius versus force reveal strongly hysteretic behavior against the lower modulus samples. A weighted non-linear least squares method was used to fit the JKR model (orange points) to the loading portions (closed colored points) of the experimental data. Open points indicate the unloading portions of each experiment. Experiments were performed over a ramped loading and unloading profile that was symmetric and 2,000 seconds in duration.
Counter Surface Modification
Each of the three mixtures was tested against a counter surface that was
smooth, rough, and smooth with an additional third body in the contact. The substrates
were all aluminum (Al 6061) and approximately 10 mm thick. The smooth sample had
an average roughness of approximately 400 nm as measured by a scanning white light
interferometer (SWLI) (Veeco Wyko NT9100). Mass loss and profile measurements
were made every 200,000 cycles.
85
The rough counter surface, which better represents a road surface, was
prepared. The rough substrate was identical to the smooth substrate in physical
dimensions but the testing surface was roughened with a sand blasting procedure to
give an average roughness of ~5.2 µm as measured by the SWLI. An increase in oily
wear debris accumulated on the surface of the substrate during these tests. It is
believed that this decreased the wear rate due to the sample only coming into contact
with poissage debris on the wear track after a period of time. To account for this, a stiff
nylon wire brush was placed in contact with the substrate in the region of the wear track
away from the sample contact as shown in Figure 5-2. Doing this reduced the
accumulated poissage to an acceptable amount.
Figure 5-2. The combined rolling and sliding tribometer with magnetic vibrational feeder
for application of kaolin powder. A vacuum below the sample collects extra kaolin. The poissage brush is pressed against the disk away from contact. Combined Rolling and sliding Tribometer. July 2, 2015. Courtesy of author.
The final substrate testing condition consisted of the smooth disk augmented with
a constant stream of a micron sized third body particulate (kaolin). Kaolin, a clay
material, was chosen due to it being used in tribological tests at Michelin Americas. The
86
kaolin was administered by placing a preset amount, 2 grams, into a magnetic vibration
feeder set to a low feed rate. The amount of kaolin was chosen based on the lowest
feed setting and the amount necessary for nearly constant flow of particulate per
measurement interval. The kaolin was fed over the top of the sample into and through
contact. The third body administration is shown in Figure 5-2. A similar accumulation of
wear debris manifested in initial tests using kaolin as a third body. Subsequent tests
were augmented with the nylon wire brush to prevent build-up.
Results and Discussion
Wear rate was calculated as the volume lost divided by the product of the applied
normal load, rolling distance, and slip percent. Mass measurements were taken at
200,000 cycle intervals and the density of the material was used to calculate the volume
loss. Normal force per cycle is recorded during testing and the data is processed by a
Matlab® code. A Monte Carlo simulation is then conducted on the data in order to find
the best match for the wear rate. Combined standard uncertainty is used to calculate
the error within the measurement. A more thorough description of wear rate and
uncertainty calculation is given in Chapter 4.
Natural Rubber
The wear rate was calculated for each material and substrate pair. Table 5-2
shows the wear rates collected. Wear rates are highest for the tests with third body
particulate and lowest for the tests ran against smooth aluminum. The wear rate data
shows that dependence of the wear rate on the shear rigidity was low but the
dependence on the surface condition was high.
87
Table 5-2. Wear rates for all Michelin tire compounds against smooth, rough, and smooth with third-body counter surfaces.
Material Smooth Rough Third Body mm3/Nm (106)
Mix 1 0.089 2.060 85.200 Mix 3 0.210 1.030 103.000 Mix 4 0.144 1.020 72.700
PDMS
While the softer PDMS sample experienced higher wear rates than the more rigid
sample, an interesting behavior of the wear rate was observed when comparing the
wear rate against the smooth surface verses the rough surface. The 10:1 and 20:1
mixtures of PDMS had lower wear rates against the rough substrate than against the
smooth substrate. However, the wear rate for 30:1 PDMS against the rough surface
was approximately 2x as large as the wear rate against the smooth surface. The wear
rates for each of the PDMS samples against the smooth and rough counter sample
surfaces is shown in Table 5-3.
Table 5-3. Wear rates for all PDMS formulations against smooth and rough counter surfaces.
Material Modulus Smooth Rough PDMS E, kPa K, (mm3/Nm) x 10-4 10 to 1 2120 3.93 2.83 20 to 1 749 4.12 2.34 30 to 1 241 6.09 12.80
As the ratio of base polymer to crosslinker is increased there is more polymer
that has not experienced crosslinking. This serves to decrease the modulus but also
increases the adhesion. The peculiar behavior of the wear rate could stem from the
ability of the softer PDMS to accommodate large strains. With higher adhesive forces
against the rough surface due to larger surfaces areas, the PDMS can reach higher
88
strains before a slip event occurs. With fewer slip events occurring in the contact the
10:1 and 20:1 PDMS samples could achieve a lower wear state. While this explains the
wear for the first two formulations of PDMS, it does not explain the behavior of the 30:1
PDMS. It’s possible that the increased adhesion and
89
CHAPTER 6 THE IRREGULAR WEAR OF ELASTOMERS IN COMBINED ROLLING AND SLIDING
Irregular Wear
The irregular wear phenomenon, also known as river wear, is a preferential zone
of wear that develops in tires that experience relatively low slip in the contact patch over
long distances. The current way to measure irregular wear is by outfitting long range
trucks with experimental tires which must be brought back in to the test facility to
conduct measurements. These tires see a wide range of environments over their life
and determining what causes irregular wear to initiate and grow is nigh impossible. A
laboratory scale test that can replicate irregular wear is then invaluable as it reduces the
cost and time required to produce the phenomenon.
Figure 6-1. Irregular wear on a PDMS sample. A) An example of the wear track and
irregular wear zone on a PDMS rolling-sliding sample. B) Scanning white light interferometer profile scan of the PDMS sample. with and without a simple curve subtraction applied.
The first step was to determine whether it was possible to use the rolling-sliding
tribometer to reproduce irregular wear in the laboratory samples. Profilometer scans
were taken at each measurement interval for the tests conducted in Chapter 5. Irregular
wear was first noticed on a PDMS sample as a circumferential band which visually
looked different than the rest of the wear track. Scanning the surface of this material on
90
the scanning white light interferometer gave us a profile which showed that in the area
which we assumed was irregular wear there was an increased area of material loss as
shown in Figure 6-1. The next step was to see if we could produce the irregular wear
ring on natural rubber compounds.
Measurements of Onset and Development
Figure 6-2. Scanning white light interferometer sample mount.
The surface profile scans made at each interval give incremental surface profile
data. This makes it possible to monitor the development and growth of irregular wear.
The surface scans were made with a 20x scanning objective. The scan area was
approximately 7 mm by 0.5 mm across the wear track. Each scan takes approximately 1
hour to complete. While the zone of irregular wear is a preferential area of increased
wear rate, it is free to develop at any point of the wear track. In these experiments we
found the irregular wear to be isolated to one side of the wear track. While not a true
case of irregular wear, this prescribed wear zone created an ideal situation for the
development of a tool to monitor the development and growth of irregular wear.
91
Figure 6-3. Profilometer scans of a natural rubber sample with irregular wear. A) surface
profile scans made at 200,000 cycle intervals. B) surface profile scans with a simple third order polynomial fit subtracted from the scan dataset.
To make sure that the same portion of the irregular wear zone was measured a
sample mount was made for the scanning white light interferometer. This mount, shown
in Figure 6-2, increased the repeatability of the surface profile measurements but there
is still room for significant improvement.. A characteristic example of irregular wear
development is shown in Figure 6-3A and 6-3B.
Quantifying Irregular Wear Development
After establishing the ability to measure irregular wear in rolling and sliding
samples it was necessary to develop a method to quantify the irregular wear zone. A
Matlab® code was designed that was able to isolate the irregular wear zone and,
through curve fitting and subtraction, give values for the irregular wear area in line
scans. This area could then be used to produce useful metrics such as mass/volume
loss due to irregular wear, irregular wear zone depth and irregular wear zone width.
92
Examples of the code output are given in Figure 6-4 A-C. A detailed description of the
quantification method is given in Appendix C, the Matlab® code is given in Appendix D.
Figure 6-4. Output from the irregular wear analysis code. A) The raw SWLI data plotted
against a simple visual aid for analysis. B) The raw SWLI data with a third order polynomial fit of the sample curvature. C) The raw data subtracted from the third order polynomial.
River Wear
The final step was to reproduce the rivering effect seen in irregular wear on long
range tires. As previously stated, the irregular wear seen in samples is preferentially
isolated to a specific zone on the sample. This is hypothesized to be due to the
perpendicular instrument geometry of the rolling-sliding tribometer. The current
geometry where the rotational axis of the spindle is perpendicular to the rotational axis
93
of the stage, shown in Figure 6-6, causes a state in which the shear field within the
contact patch is not symmetric across the patch. This is due to a variation in velocity
and %slip across the contact patch.
Figure 6-6. The rolling-sliding tribometer geometry.152
The linear velocity of the ball and the disk within the contact are as follows:
b b bV Rω= (6-1)
d d dV Rω= (6-2)
The slip velocity is given as the difference between the ball velocity and the disk
velocity, Equation 6-3, and the rolling velocity is the average of the ball and disk
velocity, Equation 6-4. The percent slip within the contact is then defined as the ratio of
sliding to rolling or the slip velocity to the rolling velocity, Equation 6-5.
s b b d dV R Rω ω= − (6-3)
94
2
b b d dR
R RV ω ω+= (6-4)
% s
R
VSlipV
= (6-5)
Plugging Equations 6-3 and 6-4 into Equation 6-5 gives the percent slip as a function of
the rotational speed and the rolling radii of the ball and disk.
( )
% 12
b b d d s
b b d d
R RSlipR R
ω ω
ω ω
−=
+ (6-6)
Using Equation 6-6 it is possible to solve for the %slip across the contact patch.
Assuming a Hertzian pressure distribution it is also possible to solve for the pressure
distribution within the contact, Equation 6-7.44 Finally, using the pressure values found
in Equation 6-7, the probability for a wear event is given as the %slip multiplied by the
pressure, Equation 6-8.154 The pressure, %slip, and wear probability distribution across
the contact for a perpendicular rolling-sliding geometry is shown in Figure 6-9.
1
*2 3
3 2
6o
PEpRπ
=
(6-7)
( )
1*2 3
3 2
612
b b d d s
b b d d
R R PERR R
ω ωρπω ω
−
= ⋅ +
(6-8)
In all samples that develop irregular and were tested on the combined rolling and
sliding tribometer with the perpendicular spindle/stage geometry, irregular wear was
isolated to a specific portion of the contact patch. This section of the contact patch
corresponds to the area where the %slip as well as the shear force shift from negative
to positive.
95
Figure 6-7. The pressure, %slip, and shear force distribution within the contact. A)
Contact patch analysis of perpendicular system geometry. B) Contact patch analysis of angled rolling-sliding contact geometry.
Rivering irregular wear is not constrained by such a condition and is free to
develop laterally across the contact patch. In order to prevent such a constraint, a
rotation stage was added to the base of the spindle on the combined rolling and sliding
tribometer so that we could prescribe an angle to the sample. This angle changes the
velocity profile across the contact patch so that it is constant instead of increasing with
track radius. This instrument geometry approximates a cone rolling on a disk shown in
Figure 6-8. In this geometry the rotational axis of the cone intersects the surface of the
disk at the axis of rotation of the disk. When the spindle is angled the radial distance
from the center of the disk, x, corresponds to a ball sample radius given by Equation 6-
9. The angle which sets the spindle axis to intercept the rotational axis of the disk is
then given obtained in Equation 6-10 through simple trigonometry. By forcing this
geometry on the sample/disk system, every point of contact between the ball and disk
are at equal linear velocity.
96
( ) sin( )br x x θ= ⋅ (6-9)
1 1 1sin sin sinb b b
d d d
V xV x
ω ωθω ω
− − −= = = (6-10)
Figure 6-8. Angled geometry for producing rivering irregular wear. The prescribed angle
is dependent on the linear velocity in the contact and the radius of the ball. For a linear velocity of 1000 mm/s and a slip velocity of 3% slip the angle is ~11o.
Applying the new velocity profile within the contact to Equations 6-6, 6-7, and 6-8
gives the %slip, pressure, and shear force profiles within the contact for the angled
rolling sliding system, shown in Figure 6-7.
A rolling sliding test was conducted with the angle applied to the system. The Mix
1 natural rubber had the greatest tendency to develop irregular wear and as such was
chosen for the initial test to produce rivering irregular wear. The sample completed
500,000 cycles on the rough disk at 3% slip. After the experiment, a 360o surface scan
was made of the wear track, shown in Figure 6-9. Line scans at 60o intervals were
isolated to show the development of irregular wear around the circumference of the ball.
The sample exhibited significant irregular wear which was not constrained to a specific
portion of the track nor to a constant width like previous samples. This was the first
sample to exhibit true irregular wear.
97
Figure 6-9. Surface profiles of worn natural rubber (Mix 1) sample. A) Regular wear
profile of a sample that exhibits no irregular wear. B) Directed irregular wear profile developed when the combined rolling and sliding tribometer was aligned in the perpendicular orientation. C) Irregular wear profile developed with an angular alignment of the combined rolling and sliding tribometer.
98
CHAPTER 7 A BINARY SYSTEM TO DESCRIBE IRREGULAR WEAR PATTERNS
A Simple Model for Irregular Wear
Advances in tribology have yielded a better understanding of wear and have
guided the creation of many wear-resistant materials (e.g. metal alloys, polymer
composites, coatings, and elastomers). The design of rolling elements and sliding
systems has progressed to a point where low and ultra-low wear rates are routinely
achievable. While crucial, a material’s intrinsic wear resistance may only partly account
for the success or failure of a design or component. Each sliding system is unique and
in complex mechanical assemblies seemingly volatile, and often surprising, behaviors
emerge. Wear and the consequences of changes in shape have been shown to
influence system dynamics and kinematics, and result in a coupled evolution of
performance and wear. In many cases, the end of useful life is dictated by a loss of
function - not a complete loss of material.
A perfect machine has been described by some designers as a system in which
every component would wear-out or fail simultaneously at the designed end-of-life. In
this manuscript we demonstrate that this may be nearly impossible to realize even for a
simple binary system. Some single component wearing systems, such as an
eccentrically mounted circular cam-follower, adjust to wear in a predictable and
monotonic fashion.155 In other systems, such as dynamic multi-body mechanisms, both
the wear and driving dynamics may increase during use.156,157 Finally, for systems with
multiple materials, the shape may evolve and alter the contact mechanics during
use.158,159 To capture the complexities of real tribological systems, which are
confounded by numerous material and environmental variables, numerical and finite
99
element models are more frequently applied.160–170 All of these systems are complex
and challenging, with few yielding closed-form analytical solutions.
A curious finding in multi-body systems is that one of the components appears to
wear out long before the others, and this single component failure renders the entire
system inoperable. Irregular, uneven, and runaway wear events have been observed in
systems that undergo relatively simple repetitive motions (e.g. tires, teeth, train wheels
and rails, artificial hip joints159,171–175). In this manuscript, a simple binary system176 of
blocks coupled under a single normal load is modeled in order to describe a possible
source of instabilities.
Modeling
Dimensional Problem Formulation
In this problem formulation Archard’s model is used to predict the recession of
material as a function of the wear rate, K , the slip distance, S , the total applied load,
TF , and the apparent area of contact, A .49 The binary system modeled here is shown
in Figure 7-1, and is comprised of two blocks that are independently attached to ground,
and loaded through a pair of springs in parallel with constant applied normal load to the
pair. The motion of the slider is treated as a periodic single stroke of constant distance,
L . The formulation is identical to a reciprocating contact without the complexity of
direction changes in the solution. The model assumes steady friction coefficient, µ ,
and assumes that there is slip, S , between the blocks and the slider during the first
cycle. Additionally, the model assumes the there is some minute difference in the wear
experienced between the blocks due to either differences in wear rate, spring constants,
100
friction coefficient or undeformed spring lengths; such differences are the norm, not the
exception.
Figure 7-1. A simple system comprised of two blocks coupled under a single normal
load ( TF ) is imagined. Each block experiences a portion of the normal load through its normal spring with spring constant Nk , the deflection of which depends on the wear heights of the individual blocks and the magnitude of the partitioned normal load. Lateral springs with spring constant Lk attached to each block simulate resistance ( X ) to slip ( S ) against the cyclic slider distance ( L ) in proportion to the normal load and friction coefficient. Adapted from Harris, 2016.177
The lateral spring deflection, X , in Equation 7-1 is solved for each block
independently (the superscripts preceding each symbol denote the block to which they
refer, and bold terms denote that this parameter is assumed constant for the analysis),
and is determined by the cycle by cycle normal load, NF , on each block, the friction
coefficient, µ , and the respective spring constants, Nk and Lk . During a single cycle,
defined by a total slider displacement, L , the blocks travel with the slider as far as the
101
frictional deflection of the lateral springs allows, X , and slip a remaining distance, S
(Equation 7-2). Each block wears hδ as a result of slip according to Equation 7-3.
, ,
, Na b a b
a b ii
µ FX ⋅=
Lk (7-1)
, ,a b a bi iS X= −L (7-2)
, ,
, Na b a b
a b i ii
F Shδ ⋅ ⋅=
AΚ (7-3)
The total wear recession h of each block is then defined as the sum of the incremental
wear from all previous cycles (Equation 7-4).
, ,
1
na b a b
i ii
h hδ=
=∑ (7-4)
In this system, any differences in wear between the two blocks alters the partitioning of
the normal load, Equation 7-5, and subsequently the slip on the following cycle,
Equation 7-6. oS , Equation 7-7, is the slip that occurs on the initial cycle.
( ), , ,1 1
12
Na b b a a b
i i iF h h− − = + − T NF k (7-5)
( ),
, , ,1 12
a ba b a b b a
i i iµS h h− −
⋅= + −
⋅N
oL
kSk
(7-6)
,
2
a bµ ⋅= −
⋅T
oL
FS Lk
(7-7)
The complete expression for the incremental recession of material from each block for
an individual cycle is then given by Equation 7-8.
( ) ( ),
, , , , ,1 1 1 11 1
2 2
a ba b b a a b a b b a
i i i i iµh h h h hδ − − − −
⋅ ⋅ ⋅= + − + − ⋅ ⋅ ⋅
T o N N
T o L
K F S k kA F S k
(7-8)
The total recession of material ( ,a bih ) for each block is then the sum of the height
recessions accumulated over all preceding cycles as calculated above.
102
Non-Dimensionalization of the Model
Dimensionless expressions are developed in which an asterisk denotes a
dimensionless variable. The non-dimensional incremental material recession, , *a b Hδ , is
defined in Equation 7-9 and is a result of normalization by the maximum possible height
difference between the two blocks, which is equal to the deflection of one of the normal
springs under the total applied load, TF .
,
, *a b
a b ii
hH δδ ⋅= N
T
kF
(7-9)
Substitution of Equation 7-9 into Equation 7-8 and collection of terms creates two
additional dimensionless groups as given by Equation 7-10 and Equation 7-11, which
represent the non-dimensional wear potential and the non-dimensional slip fraction
respectively. On a cycle-by-cycle basis, the non-dimensional recession in height can be
given by Equation 7-12.
* ⋅ ⋅= NK k L
AΩ (7-10)
*o =
oSSL
(7-11)
( ) ( )( ), * , * , * , * , *1 1 1 1
1 1 12
a b b a a b a b b ai i i i iH H H H Hδ − − − −
= ⋅ + − + − − * * *
o oS SΩ (7-12)
The Conditional Bifurcation in Solutions for Steady-State
The Janus Blocks are found to drive monotonically to two different steady-state
conditions in which there is either zero wear on the blocks or a partitioning of load that
guarantees the wear is equal across the blocks. The system becomes stable for either
condition. Assume that the wear rates of the two blocks is the same, and that the
incremental amount of wear between the two block during a given cycle is equal,
(Equation 7-13). This is possible when the product of the force and slip distance on
103
each of the blocks is also equal (Equation 7-14). In this case, during the following cycle
the force on each block will not be altered, and thus neither will the slip distance.
Thereafter, the blocks continue to wear indefinitely at a steady rate and constant force.
Interestingly, a most unusual situation is also predicted: a situation in which one block
runs to zero slip and the other to zero load. Here, we follow the analytical solution to
determine which parameters dictate whether the system moves to this extreme
condition or if the system will find a steady partitioning of forces that are non-zero.
* *a bi iH Hδ δ= (7-13)
N Na a b bF S F S⋅ = ⋅ (7-14)
Taking Equation 7-14 and substituting Equation 7-7 for the slip as a function of normal
load, stroke length, lateral spring constants, and friction coefficient gives Equation 7-15
in an expanded form.
( ) ( )N NN N
a a b ba bµ F µ FF F⋅ ⋅
⋅ − = ⋅ −L L
L Lk k
(7-15)
Recognizing that the sum of the two forces on the block is equal to the total applied
force, Equation 7-15 can be rewritten in terms of the normal force aNF and the total
force TF as shown in Equation 7-16.
2 2 22 = 2N N N Na b a a a aF µ F F µ F − ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ + + ⋅ T L T TF L k F F (7-16)
For this problem we have made the following additional assumptions. First, we
have assumed that the friction coefficient bµ is greater than the friction coefficient aµ , (
b aµ µ> ). Second, as a logical result of the first assumption, the force bNF is always
greater than aNF , ( b a
N NF F> ). Non-dimensionalization of aNF , is normalized by the total
applied force TF , and following the assumptions of this problem definition * 10 2a
NF< < .
104
Additionally, the friction coefficients can be written in terms of bµ , where a bµ µ µ= −∆ ,
and µ∆ is a difference in the friction coefficients. These substitutions provide a useful
method to simplify the analysis of stability and to recast the Eq. 16 in a slightly simpler
form, Equation 7-17.
* * *21 2 = 2 (2 )N N Na b b a b aF µ µ F µ µ F − ⋅ ⋅ − ⋅ ⋅ + ⋅ − ∆ ⋅ ⋅
T
L
FL k
(7-17)
This expression, Eq. 17, can be further simplified by recognizing that the fraction
/ ⋅T LF L k is itself a dimensionless group. To further simplify the analysis, assume that
the system is designed such that when 1µ = , both of the lateral springs will be stretched
the length of the stroke without the blocks slipping ( /= ⋅T LL F 2 k ), which confines the
problem to situations where 1µ ≤ . For this analysis the ratio / ⋅T LF L k is then equal to 2.
With this substitution, Equation 7-17 can be rearranged into Equation 7-18, which is
given in the form of a classic and familiar quadratic expression in *aNF .
* 2 *(2 ) ( ) (1 2 ) ( ) ( ½) 0N Nb a b a bµ µ F µ F µ⋅ − ∆ ⋅ + − ⋅ ⋅ + − = (7-18)
Examination Equation 7-18 reveals solutions for *a
NF are real, positive, and between 0
and ½ for values of 10 2bµ< < . The system will always reach this intermediate stable
condition when the friction coefficient for block ‘b’ is less than ½ (Equation 7-19).
½bµ < (7-19)
Fundamentally this condition makes physical sense, and can be described as
follows. Given a design in which a friction coefficient of 1µ = , and the load of /TF 2 is
sufficient to stretch the spring to a condition of no slip, then for any friction coefficient
less than 12
bµ = there is insufficient traction to pull the spring to the zero slip condition,
105
even if all of the load is on the block. Thus, the bifurcation is entirely dependent on
whether or not at full load on a single block there is sufficient traction to go to a zero slip
and thus zero wear condition. If so, the system will bifurcate and focus all of the slip on
one of the blocks and the entire load on the other. If not, then the system will find a
steady wearing configuration that partitions the load as given by Equation 7-15. This is
shown schematically in Figure 7-2, where the solid lines describe the behavior of block
b in each of the two cases, and the dashed lines describe the conditions of block a.
Figure 7-2. Force partitioning on the two blocks is dependent on the force and slip on
each block in the previous cycle. In this system, when 1 2bµ < , one cycle will always eventually occur in which * *a bh h∆ = ∆ , in which case the forces on each block will stabilize before the blocks reach the zero slip/zero force condition and the blocks will continue to wear together indefinitely. When
1 2bµ > , the blocks wear independently until a condition of zero slip (block b) and zero load (block a) is achieved. The dashed lines describe the two cases for block a, and the solid lines describe the two cases for block b. Adapted from Harris, 2016.177
106
An Analytical Solution for the Time Constant
In the spirit of bounding this problem and defining a time constant for the
evolution of wear to a steady-state condition a time constant has been derived
analytically. A few modifications to the model are necessary in order to arrive at an
analytical solution for the dimensionless recession of material for each block. One block
is artificially constrained such that it does not slip, and therefore does not wear. The
analytical solution, Equation 7-20, describes the differential recession of material, a hδ ,
of the other block as a function of the wear rate, the normal spring stiffness, the stroke
length, the initial slip of the wearing block, and the apparent contact area.
2 2 2
aa a µ hh hdiδ ⋅ ⋅
= − + ⋅ T N
N oL
F kK k SA k
(7-20)
The total wear height, ah , is reached by integration and is given by Equation 7-
21, in which a new variable, λ , is defined by Equation 7-22. The non-dimensional
expression for the wear height of a block is given by Equation 7-23. A time constant for
the system, Nτ , describes the number of cycles the simulation progresses before
reaching 63.2%, or (1-1/e), of the final height difference between the two blocks.
1
1 1
ia
i
ehe λ
− ⋅
− ⋅
− = + −
T
N
o
Fk L
S
λ
(7-21)
⋅ ⋅=
⋅NK k L
2 Aλ (7-22)
2
*
2*
1
1 1
ia
i
o
eHe
Ω− ⋅
Ω− ⋅
−=
+ −
1S
(7-23)
* **
2 ln( ) ln( 1 )o oN eτ = − − − + S SΩ
(7-24)
107
Results and Discussion
A representative set of results from a complete numerical simulation using the
dimensionless formulation discussed in section 2.2, and the problem formulation
followed in section 2.3 are shown in Figure 7-3. For this simulation we used 1000 N for
TF , 100 N/mm for Nk , 10 N/mm for Lk , and 50 mm for L . These plots reveal the
progression of wear versus cycle number over a billion passes. *Ω , given in Equation 7-
10, contains the wear rate, and the fiction coefficients along with the difference in friction
coefficient can be seen to drive the separation in the two solutions. What is clearly
apparent in this figure is that as the wear depths diverge between the two blocks so too
does that normal load. Additionally, the numerical solution reveals that the criterion of
1 2bµ = as the critical point is correct. The result of this numerical simulation is typical,
and illustrates the salient features of the model. Namely, even a simple binary system
is capable of spontaneously generating an instability that bifurcates the behavior driving
one block to a zero wear condition as a result of wearing out of contact, while the other
achieves zero wear because it ends up carrying all of the load without slip. While such
a process appears to be an advantageous behavior producing zero wear, this is often
orthogonal to the designed purpose of the system, which may be to provide traction,
damping, coordinated movement, or distribution of stress. These Janus blocks are so
named because of the opposite terminus processes by which they stop wearing, and
because one of the blocks appears to run-away in terms of wear while the other
monotonically drops to zero.
108
Figure 7-3. Using simulation constants such that a bµ µ> and *1 b
oµ− = S and * 45 10−= ⋅Ω , the friction coefficient of block b, bµ , and the difference in friction coefficient between block a and b, µ∆ , are varied. A) The cyclic evolution in wear height of each block. B) The cyclic evolution of the force partition across the blocks is shown. Adapted from Harris, 2016.177
The progression of this evolution in wear follows the mathematical form of
geometric growth, which is consistent with the physics of integrating prior states that
drive future states. This seemingly divergent behavior is seen to be the expected
109
outcome for such lightly coupled systems. Interestingly, the most effective way to
eliminate this process is by reducing the friction coefficient and thus ensuring that slip
must occur in the contact and that a condition of complete stick is impossible. Such a
system, though robust to fluctuations in input variability, is oddly stiff, low friction, and
high wear (i.e. this of Janus process is more likely to be seen in soft, high friction, and
low-wear systems). The difference in the wear depth at steady-state is shown to simply
be ( /T NF k ), which demonstrates a finding - that stiff systems in the normal direction will
process smaller differences in the overall wear progression.
Figure 7-4. Using simulation constants such that a bµ µ< , and *1 b
oµ− = S , 0.95bµ = and0.05µ∆ = , the wear rate, K , is varied. Decreasing the wear rate increases
the number of cycles before the blocks reach the zero wear condition. Adapted from Harris, 2016.177
In many applications that involve a sacrificial tribological part it is desirable to
consume the material in a gradual and predictable fashion. The plot in Figure 7-4
reveals that as predicted by the simple models, the wear rate itself is not sufficient to
alter or arrest the bifurcation process – it simply delays the time for onset. This plot was
110
generated over a billion cycles and despite being plotted on log-vs-log scaling shows
that the final wear depths are nearly identical over six orders of magnitude variation in
wear rate. The analytical solution for the time constant to steady-state behavior is
plotted in Figure 7-5 A-C, and clearly reveals that the wear rate is the dominant
determinant in the time to steady-state (Figure 7-5C). The wear rate however is not
necessarily the largest contributor to the total amount of wear in the system. An
understanding of both wear rate and material properties is necessary for a complete
picture.
Figure 7-5. Dimensionalization of the time constant. A) The dimensionless differential
wear height of block a with block b constrained to zero wear as is necessary in order to provide an analytical solution. B) The dimensionless wear heights of block a as it evolves when block b is constrained to zero wear. C) the effect of varying *
oS and *Ω on the time constant, Nτ . Adapted from Harris, 2016.177
111
What we have shown in this simple model is that such a process is inherently a
risky design philosophy, because of variability in components. Fundamentally, these
variabilities lead to minute differences in the first cycle slip events. These slip events
drive differences in wear across surfaces that were notionally designed to wear together
as a system. From a design perspective, the precision required to maintain infinite
stability is practically unachievable. The rate with which the system diverges is directly
proportional to the accumulated errors that result in the first cycle slip differences
between the blocks. For many practical applications the emergence of preferential wear
on a single object in a multi-body dynamic system may not imply a material difference at
all; rather, it may be the result of a simple kinematic coupling of load transfer as
illustrated here. This model may be useful as a basis for more complex future
simulations that seek to study the results of other coupling methods between the blocks
that may influence the design of systems susceptible to irregular or runaway wear.
112
CHAPTER 8 CONCLUSIONS
Friction of Elastomers
Frictional in Unidirectional Sliding
The experiments conducted on the unidirectional sliding of elastomers created a
basis for further study of the tribological properties of elastomers in sliding and
combined rolling and sliding. Frictional heating within elastomeric, specifically carbon
black filled natural rubber, contacts is a strong function of sliding speed, contact
pressure in un-lubricated contacts. Counter surface roughness between 6 nm and 1,600
nm was not a significant contributor to the measured friction coefficient of these
materials. It is likely that the weak response of friction to surface roughness is due to a
combination of material transfer and transfer film development causing psuedo-
lubrication and the absence of low frequency variation in counter surface height. Studies
from literature suggest that third body particulate added to the contact zone has been
useful in preventing semi-permanent material transfer.
Frictional Heating
An in situ thermal micro-tribometer was developed to measure the full field
temperature distribution two bodies in sliding contact. Unidirectional sliding experiments
were conducted using carbon black filled natural rubber hemispherical samples and
smooth calcium fluoride counter surfaces in order to observe the effect of normal load,
sliding speed, and friction coefficient on frictional heating within the contact. The method
allowed for measurement of nominal contact temperature rise up to 26 oC and single
point temperature rises of approximately 50 oC. Measured temperatures matched well
113
with predicted values from frictional heating models of Jaeger, Archard, and Tian and
Kennedy proving the efficacy of this radiometric approach.
Asperity Heating of Elastomer Contacts
A method to measure the input of heat, due to a single perturbation into the
surface of sliding elastomeric contacts was developed. This method provides a way to
study the effects of frictional heating on a simple and fundamental level. A positive,
cylindrical SU-8 micro-feature was cured onto a calcium fluoride counter-surface. The
rubber pin and modified counter-sample were tested in a pin on disk configuration and
sliding experiments were conducted at various sliding speeds.
The feature created an average of 8 °C and maximum of 16 °C temperature rise
over the bulk sliding temperature. Feature temperature rises increased with increasing
sliding speeds. The measured average and maximum feature temperature was
compared to predicted temperature values predicted by Jaeger’s heating model for
sliding contacts. The predicted average and maximum temperature values provided
lower and upper bounds, respectively, to the measured data.
Friction of Elastomers in Combined Rolling and Sliding
A simple instrument was created for the tribological testing of elastomers in
combined rolling and sliding contact. Mounting the sample spindle directly to the force
transducer made it possible to inferentially move the sample to the pure rolling point
instead of relying on standard metrological methods. Traction response and wear
behavior of carbon black filled natural rubber and polydimethylsiloxane spherical
samples were tested using experiments where load and slip conditions were varied.
114
Similar to the friction coefficient of elastomers under dry sliding, the traction
coefficient of the tested elastomers under combined rolling and sliding exhibited a
dependence on both slip velocity and contact area. For a given normal force,
decreasing the ball velocity resulted in a steeper traction response over a range of slip
velocities. A single traction curve could be used to describe this effect independent of
the ball and slip velocities when slip percent was used instead of slip velocity. The
traction coefficient was also shown to be proportional to the contact area of the sample
(μ ~ FN-1/3). Increasing the normal force resulted in a decreasing slope of the traction
curve, but after a certain point this effect was minimal. These experiments proved that
the tribometer was capable of measuring differences in wear rates when system
parameters are changed. From here it is possible to test the material dependence of
elastomer wear in combined rolling and sliding.
Regular Wear of Elastomers
Wear of Elastomers in Combined Rolling and Sliding
The wear resistance of elastomers in combined rolling and sliding was tested as
a function of %slip within the contact, normal load, and sliding speed. Wear was found
to be affected by the amount of global slip within the contact, with increasing amounts of
slip resulting in a greater volume of worn material, for a given distance. However, it was
found that a single parameter could be used to describe the material’s wear response
independent of the slip condition when sliding distance, instead of rolling distance, was
used to compute the wear rate. This was the case over a range of slip percentages (1%
– 5%) as well as rolling velocities (300 mm/s – 3000 mm/s). Even though this material
has a high sliding wear rate the conditions under which it is used are mild, resulting in
115
low wear. This means that less wear resistant materials, although possessing other
desirable tribological qualities (friction coefficient, corrosion resistance, etc.), may not be
precluded from service as long as the contact conditions under which they are used are
appropriate.
The ability of the tribometer to measure differences in wear rate of materials with
different material properties was tested. Three natural rubber samples as well as three
formulations of PDMS were tested at 1% slip on aluminum counter surfaces. The PDMS
samples increased in wear rate with decreasing young’s modulus. The adhesion of
these samples also increased with decreasing modulus so the highest wearing sample
was also the softest and most adhesive sample. The natural rubber samples were
custom made to have identical mechanical properties except for the shear modulus.
These samples were found to have increasing wear rates with decreasing values of
shear rigidity.
Three aluminum counter surface conditions were used to determine if a surface
roughness dependence on the wear resistance of samples Lowest wear rates were
recorded on the smooth counter surface while the highest wear rates occurred on the
smooth sample with a third body within the contact. This result suggests that the ability
remove wear debris from the track is critical to measuring the true wear rate of
elastomers in combined rolling and sliding.
Irregular Wear of Elastomers
Measurement and Quantification
Surface profile measurements were made on all rolling sliding samples in order
to capture the initiation and development of irregular wear. The irregular wear zone can
116
be observed by tracking the curvature deviation in the sample surface line scans. A
Matlab® code was written which fit the regular wear portion of the line scan and
subtracted the curves in order to isolate and measure the irregular wear. In this way the
irregular wear was able to be quantified on a per-sample basis.
One issue which arose was the directed development of irregular wear on a
specific area in the contact patch for all samples. Due to a prescribed geometry, slip
and velocity were not constant across the contact region for the sample. When the
geometry was changed to create constant slip across the contact patch, the irregular
wear was able to develop freely across the contact patch.
Analytical Solution for Irregular Wear
This analysis of Janus blocks reveals that under conditions of low wearing
materials, slight differences in any of the parameters (load, spring stiffness, wear rate,
or geometry) will lead to a situation in which one of the blocks appears to run away and
incur all of the wear while the other monotonically moves towards a condition of
complete stick and zero wear. The critical parameters to suppress this are to design a
system in which sliding is always ensured even if all of the load is carried by a single
component. Systems that cannot go to a complete stick and complete slip scenario will
continuously wear, and will do so with a partitioning that entirely determined by the
variations in the friction, wear, and stiffness across the design
117
APPENDIX A MODEL FRICTIONAL HEATING DERIVATIONS
Archard’s Method (1958)
Starting from Archard’s solution for the average temperature rise due to a
stationary circular source with a rate of heat supply, Q, from area A (using Archard’s
notation) and the solution due to a fast moving circular source (Equation A-2) it is
possible to calculate an overall heat partitioning coefficient, α.
4
Bm
B
QaK
θ = (A-1)
120.31 C C
mC
QK a Va
χθ =
(A-2)
A portion of the heat generated in the contact, α, goes into the body experiencing the
moving source and the remaining fraction, (1-α), goes into the body experiencing the
stationary source. Equating the partitioned average temperatures of the contact
(Equation A-3) α is easily obtained (Equation A-4).
( )121 0.31
4B C c
B C
Q QaK K a Va
α α χ− =
(A-3)
0.25
0.31 0.25
C
CB C
K
K KaV
αχ
=+
(A-4)
From this result the average contact temperature rise due to a circular source
with partitioning of heat being accounted for is given by Equation A-5 and shown in a
non-dimensional form in Equation A-6. Here K is a ratio of thermal conductivities (
C BK K K= ), χ is the thermal diffusivity, a is the source half width ( 2A aπ= ), and Pe is
the Peclet number; a dimensionless speed parameter ( Pe Va χ= ).
118
0.0775
(0.31 0.25 )
C
CB C
m
qaV
a K KaV
χ
θχ
=+
(A-5)
2
12
10.0775
(0.31 0.25 )B
mK qaq
Pea Pe K
φ πθ−
−= =
+ (A-6)
Tian and Kennedy’s Method (1994)
Tian and Kennedy derived an equation for the average contact temperature rise
over the entire range of Peclet numbers for uniform (Equation A-7) and parabolic
(Equation A-8) circular heat sources (using Tian and Kennedys’ notation).
( )0
0.612 0.6575
aveave
T KRq Pe
φπ
= =+
(A-7)
( )0
0.7322 0.874
aveave
T KRq Pe
φπ
= =+
(A-8)
Using Blok’s postulate, equating the maximum interfacial temperatures with a
fraction of the heat flux, qα , going to the body experiencing the moving source and the
remaining fraction, (1 )q α− , going into the other, the average contact temperature rise
may be calculated taking into account the partitioning of heat between the two
contacting bodies. The average contact temperature rise for both uniform (Equation A-
11) and parabolic (Equation A-12) sources was calculated in this fashion using
Equations A-9 and A-10 in the derivation ofα .
(uniform circular source)
( )
0max
21.273
RqTK Peπ
=+
(A-9)
( )
0max
2.321.2344
RqTK Peπ
=+
(A-10)
119
1
1 1 1 2 2
0.7767 1.2730.6575(1.1284 1.273 1.1284 1.273)
Pe qRT
Pe k Pe k Pe+
=+ + + +
(A-11)
1
1 1 2 21
0.4660.874 (0.56419 1.2344 0.56419 1.2344)
1.2344
qRTPe k Pe k PePe
=+
+ + ++
(A-12)
In non-dimensional form the equations for uniform (Equation A-13) and parabolic
(Equation A-14) circular sources are:
1
1 1 2
0.7767 1.2730.6575(1.1284 1.273 1.1284 1.273)m
PePe K Pe Pe
θ+
+ + + += (A-13)
1
1 21
0.4660.874 (0.56419 1.2344 0.56419 1.2344)
1.2344
m Pe K Pe PePe
θ =+
+ + ++
(A-14)
It should be noted that the results are essentially the same whether using Tian and
Kennedys’ solutions for the entire range of Peclet numbers or the respective individual
equations for slow and fast moving sources of heat.
120
APPENDIX B COMBINED ROLLING AND SLIDING TESTING PROCEDURE
The combined rolling and sliding tribometer code was written in LabVIEW® at
the University of Florida Tribology Lab to control the Combined Rolling and Sliding
Tribometer. Due to the nature of the LabVIEW® programming environment, the code
cannot be given in an appendix. It will be included as an attachment. The following is a
complete explanation of how to prepare and test a rolling sliding sample.
Sample Preparation
Step 1
Figure B-1. As received and trimmed rolling-sliding samples. Combined Rolling and
Sliding Samples. January 19, 2016. Courtesy of author.
Michelin samples are received with the flash still attached from the mold. The first
step is to cut away the flash with scissors or a razor. An example of the as received
sample and trimmed sample are shown in Figure B-1.
Step 2
For our surface profile scan we use a sample mount that attaches to the
scanning white light interferometer shown in Figure 4. The sample mount as an
121
alignment cylinder, 1/8” in diameter, which helps the user scan the same area. An
alignment slot must be cut into the sample taper to match the alignment pin on the
sample mount. We use a 1/8” endmill to cut a slot approximately 5 mm long into the
taper as shown in Figure B-2. The sample is then placed on the sample mount and
scanned.
Figure B-2. Slotted rolling sliding sample in SWLI sample mount. Combined Rolling and
Sliding Sample Mount. January 19, 2016. Courtesy of author.
Step 3
After scanning the sample is cleaned with a non-scented dish soap and lightly
scrubbed with gloved hands. The sample is then placed by the rolling-sliding tribometer
to dry. The sample is given at least 3 hours to dry. Often multiple samples are cleaned
at the same time and it is infrequent that a sample has not been allowed to dry for more
than 24 hours.
Step 4
The aluminum counter surface is cleaned with isopropyl and disposable wipes. If
there is debris on the substrate from a previous test it may be cleaned with a stiff
122
polymer wire brush. The combination of brush, isopropyl, and wipes removes an
acceptable amount of material debris.
Step 5
The last step before testing is a mass measurement. Because the sample mass
can detectably fluctuate over hours of time, the sample must be weighed immediately
before testing and immediately after for a proper mass loss measurement.
Rolling-Sliding Instrument Preparation
Step 1
Figure B-3. Schematic exploded view of the sample and sample holder assembly. The
tapered sample mount (4) fits into a machine collet mounted to the spindle. The removal nut (3), which has left handed threads, is fastened inside of the tapered sample mount (4). The taper mounted rubber sample (2) is then inserted into the corresponding tapered sample mount (4) and secured with the draw bolt nut and alignment washer (1). The draw bolt (right handed threads) threads into the removal bolt (3). TO remove the sample, the draw bolt (1) is unscrewed and a hex key is inserted through the sample (2) and into the removal bolt (3). The bolt is unscrewed and as it is unscrewed it forces the tapered sample (2) out of the sample mount (4).
The first step is to mount the sample in the tribometer. A custom sample holder
was designed to accommodate the rolling-sliding samples. Figure B-3 shows the parts
of the sample holder. The tapered sample mount is usually left in the spindle collet
between tests. The removal bolt is screwed into the tapered sample mount. The top of
the removal bolt has a female hex pattern which accepts a 3/16” allen key. Inside hex
pattern is a threaded hole that accepts a 1/8” screw. The sample is then pressed into
123
the tapered sample mount. The draw bolt and washer are passed through the sample
and screwed into the removal bolt using a 5/32“ allen key.
Step 2
After the user mounts the sample, they should move the sample close to the
aluminum counter surface but not in contact with it by actuating the linear manual
positioning stages underneath the load cell. The reason for this is that when the test
starts the spindle and rotational stage begin to move at slightly different times. If the
sample is in contact at this point, a brief moment of pure sliding will occur between the
sample and counter surface.
Step 3
Figure B-4. The rolling and sliding tribometer code front panel.
The user is now ready to start the code. The run arrow (shown in Figure B-4)
starts the code. The spindle will start the homing function which causes it to slightly
124
rotate back and forth. The user now must choose the zero values for the test by clicking
the circles next to the forces they wish to zero. When the forces are selected, the circle
will change from white to red as shown in Figure B-5. When the force zeros have been
selected, the user must click the accept button to move the code forward.
Figure B-5. Selecting force zeros. The user should zero channels 0 through 6.
Step 4
Figure B-6. Calibration coefficients for the rolling and sliding tribometer.
125
A calibration coefficients page is shown. These are set when the instrument is
installed and should be the same as in Figure B-6. The user can confirm the coefficients
and then click accept to move to the next step.
Step 5
Figure B-7. The combined rolling and sliding motion setup page.
On the motion setup tab, shown in Figure B-7, the user must input the following test
parameters:
• Track Radius (mm): An estimate of the distance between the center of the disk and the center of the contact patch on the sample. Can be measured with a millimeter incremented ruler.
• Track Velocity (mm/s): The user prescribed linear velocity of the track. This is the velocity of the track at the Track Radius.
• Ball Radius (mm): An estimate of the radius of the sample. At UF this is measured using digital calipers with a soft touch around the circumference of the sample.
• Ball Velocity (mm/s): The user prescribed linear velocity of the sample. This is the velocity of the sample, in the center of the contact patch.
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• Revolutions per Cycle: The number of ball revolutions to count as a cycle.1
• Cycles to Complete: The number of cycles to complete minus 1. If the user wants to test 10,000 rotations with a Revolutions per Cycle of 100 they would input 99.2
• Velocity (rev/s): This is an indicator which shows the revolutions per second of the sample. The user cannot modify this field.
• Sample Rate (ms): The sampling rate of the data acquisition device. 5 milliseconds ( 200 Hz) is a suitable acquisition rate for this data. If a higher acquisition rate is required the user may modify this field.
• Dialog Box: The user may enter relevant information about the test as reference. This dialog will be appended to the data file.
When the user has finished entering the testing parameters, they can click accept to
move on.
Step 6
The user may choose to save or not save the data. If they choose to save the
data, a file setup window will appear. The user can should save the data with a .xls or
.csv filename (i.e. “Data.xls” or “Data.csv”). The code is moved forward when they
choose to save the or not save the data.
Step 7
The next window is for verification of the normal load. If the sample is not in
contact with the counter surface then the force should be close to zero. The user can
move the code forward by clicking the accept button.
1 The program displays force data for every cycle as the test runs. Displaying all data for every cycle will quickly consume the computing power of the computer running the instrument which will cause dropped cycles. To alleviate this, the data from multiple revolutions are averaged for each cycle and then displayed.
2 This is because the code counts the 0th cycle. 99 cycles to complete is 99 + 0 or 100 cycles.
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Step 8
The start window is the last window before the test. The user can check to make
sure that everything is ready before clicking the start button to begin the test.
Rolling-Sliding Testing Procedure
Step 1
Once the test has started the user should click on the Fn vs Cycle tab and apply
load to the sample by moving the sample towards the disk and pressing the sample into
contact with the disk. The user must drive the sample to the desired load using the load
micrometer positioning stage(Figure B-10). The scale of the Fn axis can be manipulated
for more precise load control.
Step 2
Figure B-8. Combined Rolling and Sliding program test panel. The forces graphs (top-
left), moments graphs (top-right), and data tabs (bottom).
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Once the load has been applied, the user must steer the amplitude of the x-force
(Fx) and the friction coefficient (mu) to zero. To do this, the user can look at the force
axes graph, shown in Figure B-8 and adjusting the z-height stage (Figure B-9) that the
rotary stage is mounted to. Acceptable values are between 0.1 and -0.1. When Fx is
within this range, the user should adjust mu to be close to zero. The user can do this by
adjusting the friction micrometer stage (Figure B-10). Acceptable values for mu are
between 0.1 and -0.1.
Figure B-9. Rotary stage and z-height positioning stage. January 19, 2016. Courtesy of
author.
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Figure B-10. Micrometer positioning stages for load and friction force adjustment.
January 19, 2016. Courtesy of author.
Step 3
The user can then check the Fx and Fn to make sure they are at the target
values. If they have drifted then the user can adjust them back to the target values.
Once Fx, Fn, and mu are at the target values the user is ready to prescribe a slip %.
Step 4
To prescribe a slip percentage, the user must change the Track Velocity value.
The slip % is given in Eq. A-1. A set of quick reference values for Track Velocity are
given in Table B-1. Once the desired Track Velocity has been entered, the user must
click the update button and the value will be updated at the next cycle. Once the value
is updated, the user must bring Fx back to zero.
Table B-1. Reference values for prescribing slip% values
Slip % Ball Velocity
Track Velocity
Ball Velocity
Track Velocity
mm/s mm/s 1 1000 1010 1000 990 3 1000 1030 1000 970 5 1000 1060 1000 940
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Step 5
The test is now running properly and the user only needs to monitor that Fx stays
within the target range (0.1 to -0.1). Some tests have a significant amount of drift while
others have very little drift. Fx drift tends to be greatest on new samples and less once a
sample has been run for multiple intervals.
Step 6
The user should take into account the time remaining in the test. This value can
be found in the Time Remaining tab. At the end of the test, the sample and counter
sample stop at slightly different times. To prevent a pure sliding event at the end of the
test, the user can move the sample out of contact right before the test ends.
Step 7
A mass measurement must be taken immediately (within ~5 minutes) after the
test finishes as the sample mass may change in the time after the test completes. A
sample scan should also be taken after every test but it is not a time sensitive
measurement.
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APPENDIX C IRREGULAR WEAR QUANTIFICATION PROCEDURE
A code was created to analyze irregular wear data from profilometer scans and
obtain meaningful metrics which can be used to design materials which are resistant to
irregular wear. The code takes a line scan from the profilometer data and fits a curve
which follows the curve of the sample ignoring irregular wear. The line scan with
irregular wear is then subtracted from the fitted curve. The user then isolates the area of
irregular wear and the code analyzes the irregular wear area. A report is then generated
containing information about the irregular wear development for the test analyzed. The
analysis procedure follows.
Irregular Wear Analysis Procedure
Step 1
Figure C-1. Vision software. A surface profile scan with line scan on the surface
representation.
The scanning white light interferometer (Veeco Wyko NT9100) uses the
Vision™ software provided with the instrument to make surface scans. Each scan was
made as a 7 mm x 0.5 mm stitch with a scanning depth of 1 mm. The rectangular area
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scanned crosses the entire wear track. The profilometer has a resolution of
approximately 10 nm with the 20x objective. This resolution is likely not necessary and a
scan made with 5-10 µm of resolution would be acceptable. The software gives an
overview of the surface scan with horizontal and vertical line profiles along the center of
the scan as shown in Figure C-1. The scan can be exported to a .csv file which we
name in numerical order corresponding to the scan interval, starting at zero. For
example, the initial scan made before testing would be “0.csv” and the scan
corresponding to the first mass measurement at 200,000 cycles would be “1.csv”. The
user must then go in and remove the first 3 columns of the .csv file. All files are put into
a single folder and then are ready for processing. The user also has the option add a
path which will direct the code to the folder containing the test data.
Step 2
Figure C-2. Example test information to be input into the irregular wear analysis code.
Next the user must input the test specific information into the code. The user
must input the filename for the report that is generated at the end of the code, wear rate
in mm3/(Nm), test intervals, sample radius at 0 cycles, applied normal force in newtons,
slip percent (1% = 0.01), and the density of the sample. An example is given in.
Step 3
Once all necessary information is entered the user can run the code. The user
will be prompted to select the .csv files to be analyzed. The user must now select the
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data which the code will use to generate a curve that fits the data but ignores the
irregular wear zone. To do this the user should select at least two areas. The first area
must capture a good portion of the curvature of the sample. The second area should
include the area immediately after the irregular wear zone. The user may select multiple
areas to accomplish this. The user should not select any portion of the irregular wear
zone. Examples are shown in Figure C-3.
Figure C-3. Examples of proper selection for Step 3. A) The user can select two sets to
represent the data. The last point of irregular wear needs to be selected. B) Multiple small sections can be selected to represent the curve.
When the user starts the code a pop-up appears displaying a graph of the first
line scan shown in Figure C-3. The data from the profile scan is represented by a blue
line and the red line is a visual aid to help the user select the proper data. By clicking
and dragging the user can select an area of data. If the user wants to select multiple
areas they must hold down SHIFT. Once the areas to be analyzed have been selected
the user must left-click inside the command window and press SPACEBAR.
Step 4
Two more pop-up windows will appear. The first (Figure C-4) displays three
graphs for examination. The first graph is the same as in Figure C-3. The second graph
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is the third-order polynomial fit generated by the selected areas and the data, the
second graph displays the data subtracted from the polynomial fit. The second figure
(Figure C-5a) displays the data subtracted from the polynomial fit.
Figure C-4. The fitted white light interferometer data.
These windows may be resized as necessary. The user is now prompted to select the
irregular wear area on Figure C-5a. The user does this in the same was as before by
clicking and dragging over the areas to be selected. The user must select the irregular
wear zone to be analyzed. While it is important to be accurate, the code is built to
analyze the user selected area and choose where the irregular wear area begins and
ends based on the user’s selection. Examples of proper selections are given in Figure
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C-5b. Once the irregular wear zone has been selected, the user must left-click inside
the command window and press SPACEBAR
Figure C-5. Subtracted surface profile scans. A) Surface profile data subtracted from
third order polynomial fit. B) Properly selected irregular wear area.
Step 5
The next dataset is loaded and the process begins repeats. Once all samples
have been analyzed a report is generated in the current folder Matlab® is operating
out of. This text file contains the following parameters for each analyzed dataset: full-
width, half-max , max depth, and area of the irregular wear zone, the total sliding
distance, volumetric and mass due to total wear, and the volumetric and mass loss due
to irregular wear as well as the percent mass loss of irregular wear to total wear. A
sample report is shown in Table C-1
Table C-1. A sample irregular wear analysis report for Mix 1 against a rough disk. MIX 1 FWHM Max_Depth Irr_Area D_Slide Wear_Reg Wear_Reg Wear_Irr Wear_Irr
n mm mm mm2 m mm3 mg mm3 mg 0 0.187 0.004 -0.866 0.000 0.000 0.000 0.000 0.000
200000 0.604 0.004 -1.596 126.543 1.071 1.187 0.090 0.100 400000 0.259 0.009 -3.181 253.087 2.141 2.374 0.179 0.198 600000 0.500 0.006 -2.826 379.630 3.212 3.562 0.159 0.176 800000 0.602 0.010 -4.561 506.173 4.282 4.749 0.256 0.284
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APPENDIX D IRREGULAR WEAR QUANTIFICATION CODE
close all clear all clc rho1 = 1.109; rho3 = 1.117; rho4 = 1.132; filename = 'Mix_3_Test_Smooth_Data_Summary_8_3_2016.txt'; K = 8.91e-7; % n = [500000 600000 700000 800000 900000 1000000]; % n = [000000 100000 200000 300000 400000 500000 600000 700000 800000 900000 1000000]; n = [000000 200000 400000 600000 800000 1000000]; r = 9.8; F = 10; SlipPercent = 0.01; density = rho1; default_location='C:\Users\abenn_000\Desktop\Michelin SWLI Data\Rough Disk Testing\Mix 3\Mix 3 SWLI Rough\Excel Files Edited'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% Load Excel CSV SWLI Scan Data %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [FileName,PathName,FilterIndex] = uigetfile('.csv','Select Multiple SWLI Files','MultiSelect','on',default_location); addpath(default_location); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% Picking Irregular Wear Portion %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MLT = 0; j=1; for i = 1:length(FileName) file = xlsread(FileName1,i); file(isnan(file))=0; line = zeros(size(file,1),1); cycles = n(i); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% Curve Clean Up %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x = file(:,1); % Cycle Values X y = file(:,2); % Cycle Values Y
137
Xo = (max(x))/2; % Fit X center Yo = -9.85; % Fit Y center r = 10.055; yo = sqrt(r^2-(x-Xo).^2)+Yo; % figure(1) % p = plot(x,y); % drawnow prompt = 'Start Cleaning PRocess? (y,n): '; ansr = input(prompt,'s'); % ansr = 'y'; if strcmp(ansr,'y') disp('Choose data for fitting') while strcmp(ansr,'y') figure(1) p = plot(x,y,'Marker','none'); hold on plot(x,yo,'Marker','none','LineWidth',1); hold off title(FileName1,i) legend('visual aid', 'linescan') brush on % cont = uicontrol; pause xd = get(p, 'XData')'; yd = get(p, 'YData')'; b = get(p,'BrushData')'; % linkdata on brushed_x = NaN(length(xd),1); brushed_y = NaN(length(xd),1); line1 = zeros(length(xd),1); brushed_x(b==1) = xd(b==1); brushed_y(b==1) = yd(b==1); brushed_x(b==0) = []; brushed_y(b==0) = []; weights = ones(size(brushed_x)); fl = floor(brushed_x); fi = find(fl==max(fl)); weights(min(fi):min(fi)+100) = 100; fo = fitoptions('poly3', 'Normalize', 'on', 'Robust', 'LAR','Weights', weights); f = fit(brushed_x,brushed_y,'poly3', fo); Fit = feval(f,x); subt = y-Fit; figure(2) subplot(3,1,1)
138
plot(x,y,'Marker','none'); hold on subplot(3,1,1) title('SWLI Scan and Basic Circle Fit') plot(x,yo,'Marker','none','LineWidth',1); legend('SWLI Data','Circle Fit') subplot(3,1,2) plot(x,y,'Marker','none'); hold on subplot(3,1,2) title('SWLI Scan and Poly3 Fit') plot(f); legend('SWLI Data','Poly3 Fit') subplot(3,1,3) plot(x,subt) hold on subplot(3,1,3) title('Fit Subtracted from Poly3 Fit') plot(x,line1) axis([0 7.2 -0.1 0.1]) xlabel('Scan Length, x (mm)') ylabel('Scan Height, y (mm)') legend('SWLI Data (subtracted)','Baseline') hold off drawnow % prompt = 'Satisfactory? (y,n): '; % ansr = input(prompt,'s'); disp('Brush irregular wear area') j=j+1; posi = find(subt>0); a = 5; b = [0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1]; subt1 = filter(b,a,subt); figure(3) p = plot(x,subt1,'r'); hold on % plot(x,subt1,'g') % subplot(3,1,3) title('Fit Subtracted from Poly3 Fit') plot(x,line1) axis([0 7.2 -0.1 0.1]) xlabel('Scan Length, x (mm)') ylabel('Scan Height, y (mm)') legend('SWLI Data (subtracted)','Baseline') brush on pause
139
xd = get(p, 'XData')'; yd = get(p, 'YData')'; b = get(p,'BrushData')'; hold off brush_x = NaN(length(xd),1); brush_y = NaN(length(xd),1); line = zeros(length(xd),1); brushed_x(b==1) = xd(b==1); brushed_y(b==1) = yd(b==1); brushed_x(b==0) = NaN; brushed_y(b==0) = NaN; line = zeros(length(brush_x),1); data = line; data(b==1) = brushed_y(b==1); saveas(figure(2),strtok(FileName1,i,'.'),'svg') ansr = 'n'; end brush_x = NaN(length(xd),1); brush_y = NaN(length(xd),1); line = zeros(length(xd),1); else hold off X = x; Y = y; Xshow = x; Yshow = y; end halfmax = min(data)/2; width = find(data < halfmax); fwhm(i) = (x(width(end),1)-x(width(1),1)); % mm depth(i) = abs(min(data)); % mm z = find(data < 0); A(i) = sum(data(z,1))*9.1282e-04; % mm^2
140
D_slide(i) = SlipPercent*2*pi*r*cycles/1000; % m W_reg(i) = K*D_slide(i)*F; % mm^3 W_irreg(i) = A(i)*2*pi*r; % mm^3 W_M_reg(i) = W_reg(i)*density; W_M_irreg(i) = W_irreg(i)*density; Mass_Loss(i) = W_M_irreg(i)/W_M_reg(i); if Mass_Loss(i) == -inf MLT = 0; Mass_Loss_Total(i) = 0; else Mass_Loss_Total(i) = MLT+Mass_Loss(i); end MLT = Mass_Loss_Total(i); line = zeros(size(file,1),2); if i == 1 fileID = fopen(filename,'w'); fprintf(fileID, '%1s %10s %10s %10s %10s %10s %10s %10s %10s %10s %10s\r\n','n', 'FWHM(mm)', 'Max_Depth(mm)',... 'Irr_Area(mm^2)', 'D_Slide(m)', 'Wear_Reg(mm^3)','Wear_Reg(mg)', 'Wear_Irr(mm^3)', 'Wear_Irr(mg)',... '%Mass_Loss(cycle)','%Mass_Loss(total)'); fprintf(fileID, '%5f %4f %4f %4f %4f %4f %4f %4f %4f %4f %4f\r\n', n(i), fwhm(i), depth(i), A(i), D_slide(i), W_reg(i),... W_M_reg(i), W_irreg(i),W_M_irreg(i),Mass_Loss(i), Mass_Loss_Total(i)); fclose(fileID); else fileID = fopen(filename,'a'); fprintf(fileID, '%5f %4f %4f %4f %4f %4f %4f %4f %4f %4f %4f\r\n', n(i), fwhm(i), depth(i), A(i), D_slide(i), W_reg(i),... W_M_reg(i), W_irreg(i),W_M_irreg(i),Mass_Loss(i), Mass_Loss_Total(i)); fclose(fileID); end close all end
141
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BIOGRAPHICAL SKETCH
Alex Bennett began his research career when he joined the Tribology at the
University of Florida while completing his Bachelor of Science degree. With the
encouragement of Greg Sawyer and Dan Dickrell he decided to pursue his doctorate in
mechanical engineering with a specialization in Tribology and instrument design. He
has worked primarily on the friction and wear of elastomers but has also contributed to
the transition of the Tribology Laboratory to a Soft Matter Research Center. Upon
receiving the degree of Doctor of Philosophy Alex hopes to begin a career in the
biomechanical industry as a research scientist focused on improving the lives of people
through advances in surfaces in the body.
