PRE-SLIDING FRICTIONAL ANALYSIS OF A COATED SPHERICAL ASPERITY

PRE-SLIDING FRICTIONAL ANALYSIS OF A COATED SPHERICAL ASPERITY
A Thesis
Presented to
The Faculty of the College of Graduate Studies
Lamar University
In Partial Fulfillment
of the Requirements for the Degree
Master of Engineering Science in Mechanical Engineering
By
Akshay Pragneshbhai Patel
December 2017

AKSHAY PRAGNESHBHAI PATEL
Approved:
_____________________________
Ali Beheshti
Supervising Professor
_____________________________
Xuejun Fan
Committee Member
_____________________________
Jenny Zhou
Committee Member
_____________________________
Hsing Wei Chu
Chair, Department of Mechanical Engineering
_____________________________
Srinivas Palanki
Dean, College of Engineering
_____________________________
William E. Harn
Dean, College of Graduate Studies

© 2017 by Akshay Pragneshbhai Patel
No part of this work can be reproduced without permission except as indicated by the
“Fair Use” clause of the copyright law. Passages, images, or ideas taken from this work
must be properly credited in any written or published materials.

ABSTRACT
by
Akshay Pragneshbhai Patel
The contact and friction study of a single sphere has been the center of attention
for decades since several practical applications involve spherical contacts between
mechanical/biomechanical components, and more importantly, it represents micro-
contacts between asperities in rough surfaces interactions. Accordingly, spherical asperity
contact under the combined normal and tangential loadings with various materials and
interfacial conditions have been extensively investigated by several groups. Many of the
advanced applications require a small layer of coating on top of inherently rough surfaces
which can be modeled starting with a coated asperity in micro level. Notwithstanding, the
importance of coatings and their effects on contact and frictional behaviors, the study of a
coated asperity in contact with a rigid flat under combined loadings has not been
explored. In the current study, an asperity with a hard coating is modeled using finite
element model, and subsequently, its pre-sliding behaviors are studied. The sliding
criterion is defined by the maximum frictional shear stress for the contact region based on
the shear failure model. Upon verification and validation of available numerical and
experimental studies, a quantitative comparison is carried out for several coating
thickness ratios and coating-substrate material properties. The stress-strain field at the
contact region is studied with increasing tangential displacement, and the maximum

tangential force is recorded, and eventually, the static friction coefficient is obtained at
the sliding inception. It is found that higher Young’s modulus ratio of the coating to the
substrate along with higher coating thickness, increase the protective ability of coated
contact to sustain greater tangential load and results in higher friction coefficients.

iii
ACKNOWLEDGEMENTS
I am keen on expressing my honest gratitude to my mentor Dr. Ali Beheshti, for his
support, supervision and for providing me with this research opportunity. He has kindly
assisted me in putting together my presentation and improving my soft skills, thus, helping
me to develop professionally.
I, consider myself extremely fortunate, for having Dr. Xuejun Fan and Dr. Jenny
Zhou as my committee members, and I am thankful them for reviewing my work in detail
and attending my defense. I truly appreciate all the tidbits of knowledge, care, and
motivation provided by my colleagues at the Multi-Scale Tribology and Contact Mechanics
laboratory. Special mention of thanks to my friends for their love and affection, for making
me laugh and stay happy throughout this rigorous journey of my Master’s program.
Last but not the least, I take this opportunity to thank my family, who have always
been a tower of strength and for their never-ending kind support. Without them, I would
not be, who I am today.

iv
Table of Contents
List of Figures.................................................................................................................... vi
List of Tables ..................................................................................................................... ix
Nomenclature...................................................................................................................... x
Chapter Page
1. Introduction..................................................................................................................... 1
2. Literature Review ........................................................................................................... 8
2.1 Analytical Study of Homogenous Single Asperity (Uncoated) ...................8
2.2 Numerical Study of Homogenous Single Asperity (Uncoated).................11
2.3 Numerical Study of Coated Asperity (Hard coating on Soft Substrate) ....13
3. Empirical Formulations and Finite Element Model ..................................................... 16
3.1 Theoretical Model ......................................................................................16
3.1.1 Coated Spherical Contact under Normal Loading ......................17
3.1.2 Uncoated Spherical Contact under Combined Loading..............23
3.2 Finite Element Model.................................................................................26
3.2.1 Model Components .....................................................................26
3.2.2 Material Property ........................................................................28
3.2.3 Contact Interaction......................................................................28
3.2.4 Mesh............................................................................................30
3.2.5 Loading and Boundary Condition...............................................33

v
4. Results and Discussion ................................................................................................. 35
4.1 Model Verification and Validation. ...........................................................36
4.1.1 Homogeneous Model under Normal Loading.............................36
4.1.2 Coated Model under Normal Loading ........................................37
4.1.3 Homogeneous Model under Combined Loading ........................44
4.2 Coated Model under Combined Loading...................................................51
4.2.1 Stress and Strain..........................................................................52
4.2.2 Frictional Force Study.................................................................63
4.2.3 Static Friction..............................................................................68
5. Conclusion and Future work......................................................................................... 71
5.1 Conclusion..................................................................................................71
5.2 Future work ................................................................................................72
References......................................................................................................................... 74

vi
List of Figures
Figure Page
Figure 1 A coated rough surface in contact with rigid flat..................................................4
Figure 3.1 Contact of the coated spherical asperity with rigid flat....................................16
Figure 3.2 A deformable asperity pressed by a rigid flat before and after the loading.....17
Figure 3.3 Dimensionless critical load of the coated sphere as a function of dimension-
less thickness t/R, for different values of critical loads ratio (Goltsberg, Etsion, and
Davidi 2011) .....................................................................................................................19
Figure 3.4 Typical locations of yield inception in a sphere with hard coating compressed
by a rigid flat according to dimensionless coating thickness.............................................20
Figure 3.5 Deformable sphere in contact with rigid plate.................................................27
Figure 3.6 Coating-substrate system with designated partition and zones........................31
Figure 3.7 Finite element mesh of sphere..........................................................................32
Figure 3.8 Boundary condition of model...........................................................................34
Figure 4.1 Hertzian Validation, Load P vs. Interference δ................................................36
Figure 4.2 Hertzian Validation, P/E*
vs. δ/aH....................................................................36
Figure 4.3 Elastic normal loading, Load P vs. Interference ω for Young’s modulus ratio
𝐸𝑐𝑜/𝐸𝑠𝑢 = 4 and various thickness over radius ratio based on available FE results of
Goltsberg and Etsion (2015) and current FEM..................................................................38

vii
Figure 4.4 Elastic-plastic normal loading, Dimensionless contact load versus the
dimensionless interference, for thickness over radius ratio of 𝑡/𝑅 = 0.05 based on
available analytical results of Ronen et al. (2017) and current FE analyses for various
Young’s modulus ratios.....................................................................................................39
Figure 4.5 Elastic-plastic normal loading verification, Dimensionless contact area versus
the dimensionless interference, for thickness over radius ratio of 𝑡/𝑅 = 0.05 based on
Young’s modulus ratios.....................................................................................................40
Figure 4.6 Yield inception beginning in (a) coating and (b) substrate for 𝐸𝑐𝑜 =
2000𝐺𝑃𝑎, 𝐸𝑠𝑢 = 200𝐺𝑃𝑎, 𝑌𝑐𝑜 = 2000𝑀𝑃𝑎, 𝑌𝑠𝑢 = 200𝑀𝑃𝑎.........................................41
Figure 4.7 Parametric study for first δc1 and second δc2 critical interference contribution
in yielding of coating and substrate by using analytical equations....................................43
Figure 4.8 Dimensionless tangential force versus dimensionless tangential displacement
for various normal preloaded displacement (a) ω=0.5ωc, (b) ω=3ωc, (c) ω=12ωc, and (d)
ω=72ωc..............................................................................................................................46
Figure 4.9 Development of the von Mises stresses during the tangential loading under
different normal preloaded displacement...........................................................................47
Figure 4.10 Predicted friction coefficient for different displacement................................48
Figure 4.11 Predicted friction coefficient for different load..............................................49
Figure 4.12 Stress and Strain development in tangential loading under 𝛿 𝜔𝑐_𝑠𝑢 = 1⁄
for 𝑡/𝑅 = 0.005.................................................................................................................53

viii
for 𝑡/𝑅 = 0.005, 𝑡/𝑅 = 0.025 and 𝑡/𝑅 = 0.05...............................................................57
for 𝑡/𝑅 = 0.005, 𝑡/𝑅 = 0.025 and 𝑡/𝑅 = 0.05...............................................................60
for 𝑡/𝑅 = 0.005, 𝑡/𝑅 = 0.025 and 𝑡/𝑅 = 0.05...............................................................63
Figure 4.16 Dimensionless tangential load vs. dimensionless tangential displacement
under different normal displacement δ for 𝑡/𝑅 = 0.005..................................................64
under different normal displacement δ for 𝑡/𝑅 = 0.025..................................................65
under different normal displacement δ for 𝑡/𝑅 = 0.05.....................................................66
under normal displacement of 𝛿 = 𝜔𝑐_𝑠𝑢 for 𝐸𝑐𝑜/𝐸𝑠𝑢 = 2...............................................67
under normal displacement of 𝛿 = 𝜔𝑐_𝑠𝑢 for 𝐸𝑐𝑜/𝐸𝑠𝑢 = 4...............................................67
Figure 4.21 Predicted friction coefficient for load control model with different t/R
ratio (a) t/R = 0.005 (b) t/R = 0.025 (c) t/R = 0.05...............................................69

ix
List of Tables
Table Page
Table 4.1 Model input parameters of coated model..........................................................51
Table 4.2 Details about dimensionless critical interference parameters............................52

x
Nomenclature
𝐴 Contact are, 𝑚𝑚2
𝐸 Young’s modulus, 𝑀𝑃𝑎
𝐿 Load in stick, 𝑁
𝑃 Load in slip, 𝑁
𝑅 Radius of the spherical substrate, 𝑚𝑚
𝑅’ Radius of the coated sphere, 𝑚𝑚
𝑌 Yield strength, 𝑀𝑃𝑎
𝑡 Thickness of the coating layer on substrate, 𝑚𝑚
𝑣 Poisson’s ratio
𝛿 Interference in stick, 𝑚𝑚
𝜔 Interference in slip, 𝑚𝑚
Subscripts
𝑐 Critical value
𝑐1 First critical value
𝑐2 Second critical value
𝑐_𝑐𝑜 Critical value of the sphere made of coating material
𝑐_𝑠𝑢 Critical value of the sphere made of substrate material
𝑐𝑜 Coating
𝑝 Correspond to pick value
𝑠𝑡 Stick contact condition
𝑠𝑢 Substrate

Patel 1
Chapter 1
Introduction
Mankind has been the most intelligent and dominating creature on the planet earth
since they evolved. They kept on continuing towards growth and advancement in
technological inventions. The spectacular discovery of the fire was probably the first
application of friction, rubbing of two stones. Whenever surfaces of two objects come in
to contact, friction comes into play. Friction is advantageous and vital in many of the day
to day applications such as lighting fire, rolling, stopping and giving direction to the
wheels, or, even in the most mundane activities such as walking, running, climbing,
sitting or even connecting to someone sitting thousands of miles away with one swipe of
the finger on a smartphone. Just as a coin has two sides, friction has been the curse of the
human’s efficiency since the era of wheel invention. The most common difficulties of
friction have been its resistance to motion, production of noise and heat, necessitating
additional power to run the equipment.
Humans as per their inherent nature have tried to solve the drawbacks of friction
by breaking through the ceiling of technological inventions by usage of mechanisms like
wheels, pulley, ball bearings, roller bearings and air cushion, and lubricants like oil,
water, grease, graphite, talc, etc. During the operation of the machines, 70% of the
equipment lose their usefulness due to the surface degradation, i.e., lubrication breakage,
corrosion, and wear which are tightly linked to friction. Furthermore, each year
approximately one-third of the world's primary energy consumption is attributed to
friction (“Tribology on the US Economy” 2017). Therefore, it makes sense even to a

Patel 2
layman to gain further control to minimize losses because of friction, thus extending the
life expectancy of the systems and have greater efficiency and productivity of the
mechanical resources. This quest to control friction created an advanced subgenre in
mechanical engineering and material science named ‘Tribology', a collective and
comprehensive study of friction, wear, and lubrication.
Friction with many of its advantages and disadvantages remains a significant and
extremely complex physical phenomenon for the human mind to understand it
thoroughly. Scientists and researchers over the ages have tried to simplify the concept of
friction by taking simpler assumptions and modularizing its various effects. One of the
common classifications being, categorizing friction into pre-sliding friction (static) and
sliding friction (kinetic). In pre-sliding friction, the applied tangential force is less than
the maximum static frictional force. Pre-sliding friction is the maximum resistance
offered by a body in rest to an external force being applied to it, thus body resist the
change of state from rest to motion. When the tangential force acting on the body
becomes greater than or equal to the static frictional force, the body moves into a state of
motion, and, sliding friction comes into the picture.
In several engineering applications, the mechanical components or parts of the
equipment are in contact and relative sliding. These components, often, experience
friction, wear, adhesion, high temperature, and other environmental factors like exposure
to moisture, corrosion in natural conditions or, even in vacuum and space (Vanhulsel et
al. 2007). For example, in case of cutting tools, MEMS devices, optical micro switches,
magnetic disk drive, electrical circuits, automotive engines as well as biomedical
prosthetics, the operating standards are extremely critical, and therefore, it is desirable to

Patel 3
gain the most optimum output from these tools/equipment. Coatings on the interacting
surfaces has proven to be a highly effective technology to ensure the maximum
utilization of the tribological potential of these devices by reduction of wear and friction,
enhancement of thermal and electrical conductivity, increase in resistance to plasticity
and many more enhanced tribological improvements.
Selection of the coating-substrate system for the tribological application has been
centered on the basis of various constitutive factors of the material such as the thickness
of the coating, Young's modulus and Poisson's ratio of the coating and substrate, yield
strength and tensile strength of coating and substrate, hardness, composition, coating to
substrate adhesion and roughness of the contacting surfaces (Holmberg, Matthews, and
Ronkainen 1998). The most common classification of the coatings are based on
stiffness/hardness of the coating and substrate. Accordingly, coatings are categorized to
soft coating (compliant-soft coating on the hard substrate) and hard coating (stiffer-hard
coating on the soft substrate) system. This type of classification encompasses both the
soft and hard coatings with a thickness typically in the range of 0.1-50 µm.
Irregardless of the extensive application of the coatings in numerous industries, in
absence of frictional study, the selection of the coating thickness and mechanical
properties are yet mostly chosen by the trial and error system for the best performance
(Goltsberg, Etsion, and Davidi 2011). In order to curb this trial and error method, so, as
to select the most optimum of the coating as per the required application, the pre-sliding
frictional behavior of the coatings need to be studied and analyzed.
Every surface, however, it may look smooth to the naked eye or feel smooth to
touch is actually rough at a microscopic level, because of the random distribution of

Patel 4
asperities with varying geometry. This phenomenon is also observed in the case of coated
surfaces as well when the coating thickness is in the same order of the substrate micro-
features, and thus it follows the texture of the substrate. Shown in Figure 1 is the
schematic of the interaction between coated rough surfaces which occurs at the junction
of the peaks of the contacting asperities (Greenwood and Williamson 1966)(Chang,
Etsion, and Bogy 1987)(Goltsberg, Etsion, and Davidi 2011).
Figure 1 A coated rough surface in contact with rigid flat

Patel 5
A wise approach to understand the interaction between asperities, is to initially
analyze the behavior of a single coated spherical asperity in contact with a rigid flat
surface. This approach may be extended to extrapolate the contact model behavior of the
rough-coated surface by assuming a statistical distribution of asperities throughout the
surface. Researchers in the past years following this approach have noted different
behaviors of the asperities under external load such as elastic, elasto-plastic and fully
plastic deformation in their analysis of contact loading. The important parameters that
have a strong impact on the deformation of asperities are the material constitutive
properties and geometry. The applied external load is, also, one of the defining
components to understand the effect of friction on the coated surface. It is necessary to
simplify the total load into directional components such as normal and tangential, to see
the significant effects in pre-sliding coated interaction.
It was proven by researchers that the best approach to understand the frictional
contact behavior of a single coated spherical asperity is by adopting any of the two
known paths, i.e., indentation and flattening of the surface (Jackson and Kogut 2006). A
single asperity can either indent a contacting surface (indentation) or be flattened by it
(flattening). Many of the studies on coatings have tried to assimilate the results of
indentation of coated flat substrates by an uncoated spherical indenter and present it in a
coherent form to the interested reader. The main intention of these studies is the
characterization of the mechanical properties such as hardness, modulus of elasticity and
shear modulus of the coatings. In indentation model, when the tangential load is applied,
indentation of rough surfaces is interrelated with high abrasive friction and wear that
result from plowing in the substrate. In contrary, the moderate adhesive friction and wear

Patel 6
can be more effectively described by flattening of the asperities, and this is why
flattening of the coated asperity is preferable to investigate the pre-sliding frictional
behavior of the contact.
An accurate grasp on the static coefficient of friction is attained by understanding
the beginning of sliding in two contacting surfaces using the flattening approach. This is
beneficial in the selection of coatings instead of the trial and error method discussed
earlier for wide range applications. So far studies have been done to conduct the frictional
analysis of uncoated surfaces by considering an elastic-plastic uncoated spherical asperity
under rigid flat with combined normal and tangential loading to find out stress, strain,
tangential force, normal force and contact area throughout the tangential loading. These
studies were focused on some physical based criteria to capture the pre-sliding frictional
behavior, e.g. the local yielding criterion of the KE model (Lior Kogut and Etsion 2004),
the tangential stiffness criterion of BKE model (V. Brizmer, Kligerman, and Etsion 2007)
and maximum frictional shear stress criterion of WSP model (Wu, Shi, and Polycarpou
2012). Yet, comprehensive research has not been attempted for the coated spherical
asperity to fathom the nature of the onset of sliding.
The fundamental goal of this research is an attempt to explore pre-sliding
frictional characteristic for the coated spherical asperity under combined normal and
tangential loading. A spherical contact model with the hard coating is proposed and is
verified and validated with available FE models and experimental results. The stress,
strain and tangential force during the tangential loading are investigated for the coated
model and compared for the various Young’s modulus ratios of the coating to the

Patel 7
substrate along with the different coating thickness ratios. Moreover, the effect of the
coating parameters is observed for static friction coefficient.

Patel 8
Chapter 2
Literature Review
Hard coatings are extensively used to enhance tribological performance in many
engineering applications consisting of parts in contact and in relative sliding. Selection of
coating thickness and material properties as discussed earlier is mostly done by trial and
error method, owing to the non-existence of a comprehensive frictional theory. A detailed
study of the onset of sliding for coated asperity can shed light on the unknown and dark
aspects of the problem. The elastic or elastic-plastic behavior of uncoated spherical
contact has been under the critical attention of researchers in contact mechanics for a long
period of time. The onset of sliding has been studied mostly by the analytical and
numerical (finite element) method.
2.1 Analytical Study of Homogenous Single Asperity (Uncoated)
Previously, a frictionless (slip) contact of two non-conforming perfectly elastic
spheres of different radii under normal compression was solved analytically by Hertz. He
also developed analytical relations between contact pressure, contact area, and
interference (Hertz 1882). Goodman extended the Hertz theory to address stick contact
interface of two elastically dissimilar spheres pressed normally together and introduced
an analytical solution for the tangential stress over the contact area by considering the
normal Hertzian contact pressure distribution (Goodman 1962). A more accurately stick
contact condition under monotonically normal loading condition explained by Spence, at
the interface of punch indenter on elastic half-space. He analytically solved pressure
distribution, shear stress and compressive load over the contact area (Spence 1975).

Patel 9
Under combined normal and tangential loading contact of two identical elastic
spheres was systematically studied by Mindlin and Deresiewicz (Mindlin 1949) (Mindlin
and Deresiewicz 1954) in continuation to work done by Cattaneo in 1938. According to
the analytical solution by Mindlin, the contact area made up of middle stick region is
enclosed by an annular slip zone. As soon as the tangential load increases, the central
stick zone slowly weakens till it finally disappears completely. At this instant, full sliding
initiates and satisfies classic coulomb law of friction which says the tangential load is
equal to specific static friction coefficient times normal load. In this piece of research,
when the tangential load is applied in the presence of a frictionless normal load, the area
of contact and the pressure distribution follows the Hertzian solution. Mindlin also
computed shear stress distribution for full stick and partial slip conditions. In the event of
a full stick condition, he found infinite shear stress around the edge of contact which is
unrealistic in a real-world scenario. To eliminate this improbable scenario, he defined
Coulomb’s friction law as the upper boundary of local shear stress. When the calculated
shear stress goes on the upper limit, the computed shear stress is replaced by the local
shear stress limit, thus marking the beginning of local slip. Full slip sliding inception
takes place when the shear stresses equal the upper limit over the complete contact area
and central stick region wipe out completely.
Similar to Mindlin’s approach, Keer et al. (Keer, Ahmadi, and Mura 1984)
extended the model to the tangential contact between different elastic sphere to find the
contact region for complete sliding. The von Mises yields condition were applied as a
failure criterion.

Patel 10
After thorough analysis, Hamilton (Hamilton 1983) gave the pioneering research
by implementing the Hertz contact pressure distribution and Mindlin’s shear stress
distribution in full slip as a limit condition on the contact area, to derive explicit
equations for the stresses underneath a sliding spherical contact. He pointed out that the
beginning of yielding can be on or underneath the contact area subject to the pre-defined
coefficient of friction. Nowell et al. (Nowell, Hills, and Sackfield 1988) investigated
shear stress distribution of contact area at the center of two elastically different cylinders
under combined loading. The tangential load had been applied in partial slip condition
with pre-selected frictional coefficient after the frictional normal preload.
In the following years of the numerical investigation for elastic-plastic contact
under combined loading by many researchers has gained the acceptance of Mindlin’s
approach of a local coulomb friction law for the pre-sliding initiation; assuming a pre-
defined friction coefficient of friction to be put at the upper bound of the local shear
stress.
An innovative analytical approach for the sliding initiation was developed by
Bowden et al. considering material properties to understand the failure mechanism
(Bowden, Tabor, and Palmer 1951). The normal load was determined by the contact area
multiplied material hardness. While the tangential load at the onset of sliding was the
contact area multiplied by the shear strength of the material. Under these assumptions,
the maximum shear stresses at the contact interference were found to be absolutely
independent of the normal stresses, and the static friction coefficient takes the form of
shear strength divided by the hardness of the material. Later on, Tabor (Tabor 1959)
found out that adhesion is crucial in friction between the metals. The concept of “junction

Patel 11
growth” can be explained by keeping a constant von Mises stress at contact points with
fully plastic yielding. “Adhesion friction theory” claimed that sliding can occur if and
only if the contact area at the junction, irrespective of, its strength undergoes shear
failure.
The CEB model (Chang, Etsion, and Bogy 1988) expanded the adhesion failure
theory of Bowden and Tabor by adopting the stress field of Hamilton (1983) to measure
the allowable maximum tangential load absorbed by a single asperity prior to the
occurrence of plastic yield on or below the contact interface. After that, the total
tangential force of the entire population of rough surface’s contacting asperities was
determined by statistical summation method. Agreeing to CEB model, sliding between
the contact asperity interface initiated from the break down of the small junctions. Under
a specific normal preload, the maximum static friction force is the tangential force after
local yielding is complete. Summarizing, sliding begins as local yielding starts.
2.2 Numerical Study of Homogenous Single Asperity (Uncoated)
Even after several assumptions to simplify the model, it is still difficult to get an
analytical solution of frictional interaction problems. This is not just because of inelastic
basic nature of the material but also due to the complexity of boundary condition and
contact interface. So, the finite element method (FEM) is an excellent choice for analysis
of elastic-plastic frictional contact complications.
An elastic-plastic sphere contact with the rigid flat under normal load was
examined by Kogut and Etsion (Kogut and Etsion 2002). They demonstrated the
evolution of plastic deformation using different region at the contact interface of the

Patel 12
asperity in terms of critical interference ωc. Based on normal contact KE model, (Kogut
and Etsion 2003) gave the semi-analytical approximate solution for the sliding beginning
of an elastic or elastic-plastic sphere contact with rigid flat under combined normal and
tangential loading. Like CEB model (Chang, Etsion, and Bogy 1988), sliding initiation
was considered as plastic yield failure mechanism by the use of the von Mises yield
criterion. The model also assumed that the contact area, the contact pressure distribution
and the interference due to frictionless normal preload are kept constant throughout the
further tangential loading condition. In simpler words, when local yielding occurs elastic
material surrounding the plastic zone in contact area can sustain the even more external
tangential load. As a result, CEB and KE model undervalues the maximum static friction
force.
Another FEM study by Brizmer et al. (V. Brizmer, Kligerman, and Etsion 2007)
investigated the contact parameters, for instance, junction tangential stiffness, contact
pressure distribution, static friction force and static friction coefficient for the elastic-
plastic spherical contact under combined loads in full stick condition of contact. Sliding
inception criterion was governed by the gradual decrease of tangential contact stiffness
till it reaches its pre-determined minima. Theoretically, whenever the tangential contact
stiffness drops to its minimum value, i.e., 0 or 0.1, sliding of contact begins and the
corresponding tangential force is equal to the maximum static friction force. This type of
condition is hard to establish and maintain in the real-life scenario. However, (A.
Ovcharenko et al. 2006) on the basis of the in situ and real-time optical experiment
investigations, Overcharenko found out FE results offered good co-relation with the
experimental results. BKE model is more appropriate for a range of high load because

Patel 13
BKE model does not allow any slip between the contact interface for a low normal
interference. It overestimates the friction force for low load range, thus, making it more
apt for the high load range.
Partial slip model by treating the shear failure mechanism as sliding inception
criterion was presented by Wu et al. (Wu, Shi, and Polycarpou 2012). In this model, the
shear strength of the weaker material was fixed as the critical frictional shear stress. As
soon as the shear stress reaches the critical value in the contact area, local sliding occurs
at that point. When all points in contact region slide, at that moment gross sliding occurs.
This is the reason the partial slip model is close to KE model for low load and close to
BKE model for high load, and it can be utilized for full load range application as a
common model, which carry out the transition from KE to the BKE model with
increasing normal interference. It was found that the Coulomb model and WSP model
expect a lower tangential force at an assigned tangential displacement loading than the
fullstick BKE model.
2.3 Numerical Study of Coated Asperity (Hard coating on Soft Substrate)
The effect of elastic-plastic indentation of metallic coated rough surfaces explored
using finite element method was analyzed by several researchers (Komvopoulos 1988)
(Kral, Komvopoulos, and Bogy 1995). The numerical FE investigation for the flattening
of single coated asperity was initiated by Goltsberg et al. (Goltsberg, Etsion, and Davidi
2011a); they inspected the plastic yield inception of coated spherical asperity contact
compressed by a rigid flat under the slip condition for the case of hard coating. A critical
value of the coating thickness parameter to achieve the maximum resistance to the onset
of plasticity was identified, and it was understood that the yielding initiation can take

Patel 14
place in three different locations based on the above discussed coating thickness
parameter. Because of this, probable weakening effect of the substrate was noticed at
very small coating thickness lower than the critical value. Same kind of behavior was
seen for the plastic yield inception with indentation of coated flat by a rigid sphere (Song
et al. 2012). The weakening effect reviewed in detail with reference to the dimensionless
transition thickness, indicating the limit of the weakening zone (Goltsberg and Etsion
2013).
The universal model for frictionless elastic coated spherical contact under normal
loading with thin coating thickness was proposed by Goltsberg and Etsion (Goltsberg and
Etsion 2015b). They observed the individual participation of the coating and the substrate
to the total interference and obtained the unique transition point, where these
contributions become equal. Normalizing the contact parameters by their corresponding
transition value allow to develop a general expression for the relation between the various
dimensionless contact parameters (Goltsberg and Etsion 2015a). Another universal model
was presented by Chen et al. for an elastic-plastic coated sphere with moderate to large
coating thickness, and the improved equation for the critical dimensionless coating
thickness was set up for the plastic yield evolution (Z. Chen, Goltsberg, and Etsion 2017).
Chen et al. (Zhou Chen, Goltsberg, and Etsion 2016a) investigated the elastic-
plastic contact of coated spherical asperity pressed by a rigid flat under the slip contact
interaction. This was accomplished for relatively thick coating thicknesses, where the
first onset of plasticity occurred in the coating and second on the substrate side of the
interface. The empirical expression was developed for the critical interference of the first
and second inception of yield as a function of the coating thickness and material

Patel 15
properties. It confirmed that a combination of a thick coating and lower ratio of Young’s
modulus of the coating to the substrate is to prevent substrate yielding.
A recently published article explains research of elastic-plastic single coated
asperity flattened by rigid flat under the stick contact conditions (Ronen, Goltsberg, and
Etsion 2017). They acquired the results for contact parameters like interference, load and
contact area for the stick condition and compared with their earlier results under the slip
condition of contact. It was understood that the impact of contact conditions on the
parameters in the examination of plasticity yielding evolution is negligible. The above-
mentioned study was a precursor to the study intended to be done by the authors by
incorporating combined normal and tangential loading to develop coated spherical
asperity contact model in the near future.
There is substantial research available in the area of flattening of coated asperity
but those, mostly, pertaining to normal loading, which restricts the selection procedure of
coating through trial and error method to accomplish the most optimal tribological
performance. Also, as it can be noticed from the above literature review, several studies
have been done to look into sliding initiation phenomena of uncoated single spherical
asperity contact under combined loading. However, a similar type of comprehensive
studies for the pre-sliding of elastic-plastic coated asperity and its frictional behaviors are
missing in the literature. The present study seeks to analyze pre-sliding of elastic-plastic
coated spherical contact under combined normal and tangential loadings. It is attempted
to co-relate static friction coefficient with material properties of the substrate and coating
and coating thickness.

Patel 16
Chapter 3
Empirical Formulations and Finite Element Model
3.1 Theoretical Model
The current study models a 3D coated deformable elastic-plastic spherical
asperity in contact with a rigid flat and under combined normal and tangential loading.
The loading process starts with the application of normal load and then is followed by
tangential loading. Cross section of the coated asperity contact in the x-y plane, prior to
the normal loading is schematically represented in Figure 3.1, where 𝑅 is the radius of the
substrate sphere and 𝑡 is the thickness of the coating layer on a substrate with the overall
radius of 𝑅′
= 𝑅 + 𝑡. When the coating material is identical to the substrate, Figure 3.1
aptly describes a homogeneous uncoated asperity contact under loading.
Figure 3.1 Contact of the coated spherical asperity with rigid flat

Patel 17
3.1.1 Coated Spherical Contact under Normal Loading
3.1.1.1 Homogeneous Contact under the Slip and Stick Condition
As pointed out when the coating material is similar to the substrate material, it can
be considered as a homogeneous uncoated spherical asperity.
Figure 3.2 A deformable asperity pressed by a rigid flat before and after the loading
The critical interference 𝛿 𝑐 and critical load 𝐿 𝑐 at yielding inception under full
stick contact condition for normal loading are given by Brizmer et al. (Victor Brizmer et
al. 2006) as functions of their corresponding values (𝜔𝑐 and 𝑃𝑐) in perfect (full) slip, in
the following formulas:
𝛿 𝑐
𝜔𝑐
= 6.82𝑣 − 7.83(𝑣2
+ 0.0586) (3.1)
𝐿 𝑐
𝑃𝑐
= 8.88𝑣 − 10.13(𝑣2
+ 0.089) (3.2)

Patel 18
where 𝑣 is the Poisson’s ratio. It should be noted that full stick contact refers to the
condition where contacting points do not have any relative motion once they come into
contact. The critical parameters under slip condition 𝜔𝑐 and 𝑃𝑐 are presented by:
𝜔𝑐 = [𝐶 𝑉
𝜋(1 − 𝑣2)
2
(
𝑌
𝐸
)]
2
𝑅 (3.3)
𝑃𝑐 =
𝜋3
𝑌
6
𝐶 𝑉
3
[𝑅(1 − 𝑣2) (
𝑌
𝐸
)]
2
(3.4)
where 𝐸 is the Young modulus, 𝑌 denotes yield strength and 𝐶 𝑉 = 1.234 + 1.256𝑣 is the
maximum dimensionless contact pressure at yielding inception under full slip. The
critical contact area, for stick and slip conditions are given as 𝐴 𝑐 = 𝜋𝛿 𝑐 𝑅 and 𝐴 𝑐 =
𝜋𝜔𝑐 𝑅, respectively.
It is shown that for ductile materials 0.2 < 𝑣 < 0.5 , the yield inception always
occurs at single point on the axis of symmetry (Victor Brizmer et al. 2006). For 𝑣 < 0.32,
the stick critical parameters 𝛿 𝑐 and 𝐿 𝑐 are considerably smaller than their corresponding
slip contact parameters. This is due to higher tangential stresses at the stick contact
interface. However, for higher value of Poisson’s ratio, 𝑣 > 0.32, the tangential stresses,
under stick contact decreases and the critical parameters become comparable to those in
slip.
3.1.1.2 Coated contact under the slip and stick conditions (hard coating on soft substrate)
As shown in Figure 3.3, for slip condition the highest resistance to the onset of
plasticity for each curve is associated with a certain value of the dimensionless coating
thickness (𝑡/𝑅) 𝑝, which depends on material properties.

Patel 19
Figure 3.3 Dimensionless critical load of the coated sphere as a function of dimensionless
thickness t/R, for different values of critical loads ratio (Goltsberg, Etsion, and Davidi
2011b)
The associated dimensionless coating thickness (𝑡/𝑅) 𝑝 is given by:
(
𝑡
𝑅
)
𝑝
= 2.824 (
𝐸𝑠𝑢
𝑌𝑠𝑢
)
−1.014
(
𝑃𝑐_𝑐𝑜
𝑃𝑐_𝑠𝑢
)
0.536
(3.5)
where,
𝑃𝑐_𝑐𝑜
𝑃𝑐_𝑠𝑢
= (
𝐶 𝑉_𝑐𝑜
𝐶 𝑉_𝑠𝑢
)
3
(1 − 𝑣𝑐𝑜
2 )
(1 − 𝑣𝑠𝑢
2 )
(
𝑌𝑐𝑜
𝑌𝑠𝑢
)
3
(
𝐸𝑐𝑜
𝐸𝑠𝑢
)
2
(3.6)
Here, 𝑃𝑐_𝑐𝑜 and 𝑃𝑐_𝑠𝑢 are the critical loads at yielding inceptions for homogeneous
sphere, made of coating and substrate material respectively. When the coating and
substrate material have the identical Poisson’s ratio, Eq. (3.5) can be modified as (Zhou
Chen, Goltsberg, and Etsion 2016b):
(
𝑡
𝑅
)
𝑝
= 2.824 (
𝐸𝑐𝑜
𝐸𝑠𝑢
)
0.536
(
𝐸𝑐𝑜
𝑌𝑐𝑜
)
−1.608
(
𝐸𝑠𝑢
𝑌𝑠𝑢
)
0.595
(3.7)

Patel 20
It should be noted that equations (3.5) and (3.7) were obtained for a reasonably
small range of 𝐸𝑐𝑜 ∕ 𝐸𝑠𝑢 values, between 1 and 4.5.
Figure 3.4 Typical locations of yield inception in a sphere with hard coating compressed
by a rigid flat according to dimensionless coating thickness
It is found that for very thin coatings, yielding initiates within the substrate
(location 1 in Figure 3.4) when 𝑡/𝑅 is much smaller than (𝑡/𝑅) 𝑝. Yield inception occurs
on the substrate side of the interface (location 2 in Figure 3.4) if 𝑡/𝑅 has higher values
compared to former case, but still lower than (𝑡/𝑅) 𝑝. When 𝑡/𝑅 is higher than (𝑡/𝑅) 𝑝, it
happens slightly below the contact area inside the coating (location 3 in Figure 3.4)
(Goltsberg, Etsion, and Davidi 2011). Although the analysis is done for = 0.32, it is

Patel 21
indicated that under slip condition for a coated spherical contact, the Poisson’s ratio has
negligible effect.
To prevent possible failure of substrate by the weakening effect as discussed in
Figure 3.4 for location 1 and 2, Zhou et al. (Zhou Chen, Goltsberg, and Etsion 2016b)
analyzed coated contact for 𝑡/𝑅 > (𝑡/𝑅) 𝑃. As the interference increases beyond (𝑡/𝑅) 𝑃,
the first yield happens in the coating at the first critical interference 𝜔 𝑐1, and a second
yielding happens in the substrate at second critical interference 𝜔 𝑐2. The empirical
expressions for these two critical interferences correspond to 𝜔𝑐_𝑐𝑜 (the critical
interference at yielding inception for homogeneous sphere made of coating material) are
shown as follows:
𝜔 𝑐1
𝜔𝑐_𝑐𝑜
= 1 + 3.78 (
𝑡
𝑅
)
−1.29
(
𝐸𝑐𝑜
𝐸𝑠𝑢
− 1)
0.811
(
𝐸𝑐𝑜
𝑌𝑐𝑜
)
−1.3
(3.8)
𝜔 𝑐2
𝜔𝑐_𝑐𝑜
= 0.25 (
𝑡
𝑅
)
1.34
(
𝐸𝑐𝑜
𝐸𝑠𝑢
)
−0.14
(
𝐸𝑐𝑜
𝑌𝑐𝑜
)
2
(
𝐸𝑠𝑢
𝑌𝑠𝑢
)
−0.66
(3.9)
It is found that under stick contact condition, similar to slip, for coating thickness
𝑡/𝑅 < (𝑡/𝑅) 𝑝_𝑠𝑡 (in case of thin coating) the first yield inception each time happens in
the substrate and for 𝑡/𝑅 ≥ (𝑡/𝑅) 𝑝_𝑠𝑡, yield inception happens in the coating (Ronen,
Goltsberg, and Etsion 2017) (please note that subscript “st” symbolizes stick contact
condition). In addition, the discrepancy between the two contact conditions is more
evidently seen nearer to the region of the contact area, where the stress under the stick
condition is much higher than that under slip condition. As a result, for the slip case,
yield inception occurs at the substrate side of interface, whereas, for the stick case it
occurs within the coating, slightly below the contact area.

Patel 22
The analysis is performed for varying magnitudes of different input parameters
and subsequently an empirical relation for optimum contact thickness in stick (𝑡/𝑅) 𝑝_𝑠𝑡
is suggested by curve fitting as follow:
(
𝑡
𝑅
)
𝑝_𝑠𝑡
= 3.95𝑣0.724
(
𝐸𝑐𝑜
𝐸𝑠𝑢
)
0.39
(
𝑌𝑐𝑜
𝐸𝑐𝑜
)
1.41
(
𝑌𝑠𝑢
𝐸𝑠𝑢
)
−0.45
(3.10)
Further examination of the elastic-plastic regime under the stick condition is
restricted to dimensionless thickness 𝑡/𝑅 > (𝑡/𝑅) 𝑝_𝑠𝑡. This applied constraint ensures
that the first yielding occurs inside the coating and eliminates the probable failure due to
coating delamination or the weakening effect initiated by yield inception at the
substrate/coating interface or within the substrate. As the loading increases the first
yielding occur within coating at the critical interference 𝛿 𝑐1. However, as interference
increases second yield happens in the substrate at the critical interference 𝛿 𝑐2 . The
empirical expressions for 𝛿 𝑐1 and 𝛿 𝑐2 are obtained by curve-fitting as follows:
𝛿 𝑐1
𝜔𝑐_𝑐𝑜
= [6.82𝑣 − 7.83(𝑣2
+ 0.0586)]
[1 + 0.007 (
𝐸𝑐𝑜
𝐸𝑠𝑢
− 1)
0.646
(
𝑌𝑐𝑜
𝐸𝑐𝑜
)
0.244
(
𝑡
𝑅
)
−1.21
]
(3.11)
𝛿 𝑐2
𝜔𝑐_𝑐𝑜
= (
𝑡
𝑅
)
1.17
(
𝐸𝑐𝑜
𝐸𝑠𝑢
− 1)
−0.09
(
𝑌𝑐𝑜
𝐸𝑐𝑜
)
−1.93
(
𝑌𝑠𝑢
𝐸𝑠𝑢
)
0.89
(3.12)
Using the similar approach, from the curve fitting of numerical results, expression
for other second critical parameters like load and contact area are obtained corresponding
to 𝑃𝑐_𝑐𝑜 and 𝐴 𝑐_𝑐𝑜 respectively:

Patel 23
𝐿 𝑐2
𝑃𝑐_𝑐𝑜
= (
𝑡
𝑅
)
1.8
(
𝐸𝑐𝑜
𝐸𝑠𝑢
− 1)
−0.27
(
𝑌𝑐𝑜
𝐸𝑐𝑜
)
−2.3
(
𝑌𝑠𝑢
𝐸𝑠𝑢
)
0.78
(3.13)
𝐴 𝑐2_𝑠𝑡
𝐴 𝑐_𝑐𝑜
= (
𝑡
𝑅
)
1.6
(
𝐸𝑐𝑜
𝐸𝑠𝑢
− 1)
−0.2
(
𝑌𝑐𝑜
𝐸𝑐𝑜
)
−2
(
𝑌𝑠𝑢
𝐸𝑠𝑢
)
0.7
(3.14)
It must be noted that goodness-of-fit for equations (3.11), (3.12), (3.13) and (3.14)
is higher than 0.93, 0.97, 0.95 and 0.99 respectively. After performing the comparable
investigation for slip contact, it is concluded that for the entire range of the inputs, the
effect of contact consideration on the secondary critical parameters is insignificant and
the inception of plasticity is similar. They also put forth the relation for contact load and
area in elastic-plastic regime for the range 𝛿 𝑐1 < 𝛿 < 𝛿 𝑐2 :
𝐿
𝐿 𝑐2
= (
𝛿
𝛿 𝑐2
)
1.25
(3.15)
𝐴 𝑠𝑡
𝐴 𝑐2_𝑠𝑡
=
𝛿
𝛿 𝑐2
(3.16)
It is well established that perfect slip condition does not capture real tangential
stresses in the contact area (Victor Brizmer et al. 2006). Therefore, the stick contact
conditions and its expressions (3.10-3.16) are considered in this study for the further
investigation of coated spherical contact under the combined normal and tangential
loading.
3.1.2 Uncoated Spherical Contact under Combined Loading
Here, we focus on the pre-sliding frictional behavior of the contact between the
uncoated elastic-plastic spherical asperity and rigid flat under combined normal and
tangential loadings based on specific sliding criteria established by different researchers.

Patel 24
Plastic yield failure mechanism was adapted as the sliding inception for the
shearing of the small junction between the two contacting asperities of the rough surfaces
in the CEB model (Chang, Etsion, and Bogy 1988). Under an applied normal load 𝑃, the
maximum static frictional force 𝑄 𝑚𝑎𝑥 is the tangential force as local yielding occurs, in
layman’s terms, sliding is initiated after the occurrence of local yielding. The KE model
(Lior Kogut and Etsion 2003) pointed the flaws in the CEB model which underestimated
the maximum static friction force and, hence is not considered reliable for the analysis of
the static friction. They concluded that when local yielding occurred, the elastic zone
surrounding the plastic zone in contact area can still sustain further tangential load.
Considering the full stick condition between contacts of the elastic-plastic sphere
and rigid flat, tangential stiffness criterion was introduced as the sliding inception in so
called BKE model (V. Brizmer, Kligerman, and Etsion 2007). In their FE model, a
normal load is first applied, followed by a tangential ramp loading. At any step 𝑖 the
tangential contact stiffness (𝐾 𝑇)𝑖 can be calculated as:
(𝐾 𝑇)𝑖 = (
𝜕𝑄
𝜕𝑢 𝑥
)
𝑖
≈
𝑄𝑖 − 𝑄𝑖−1
(𝑢 𝑥)𝑖 − (𝑢 𝑥)𝑖−1
(3.17)
(𝐾 𝑇)𝑖
(𝐾 𝑇)1
≤ 𝛼 (3.18)
where 𝑄 is the tangential force, and 𝑢 𝑥 is the tangential displacement of the rigid flat.
Theory of sliding inception states that whenever the tangential contact stiffness
(𝐾 𝑇)𝑖 drops to a small predefined number (e.g., 𝛼 = 0 𝑜𝑟 0.1, see Eq. 3.18), sliding
initiates and the corresponding value of 𝑄𝑖 is the maximum static friction force 𝑄 𝑚𝑎𝑥.

Patel 25
𝛼 = 0 is an extreme case so for convenience a value of 𝛼 = 0.1 was selected. After the
tangential stiffness drops to this value it is assumed that entire contact begins to slide.
Another method for monitoring the sliding inception is the maximum shear stress
criterion. Wu et al. presented a partial-slip model in which they assumed that there is
some local slip even though gross slip does not happen (Wu, Shi, and Polycarpou 2012).
The critical frictional shear stress 𝜏 𝑐 was set by the shear strength of weaker material. The
shear strength is 𝜏 𝑐 = 𝜎𝑠 /√3 where 𝜎𝑠 is the yield stress under uniaxial tension. i.e.,
once the frictional shear stress in contact area reaches the shear strength, local sliding
occurs at that point. When all the points covered in contact area reach critical shear stress,
the entire interface starts sliding. The corresponding tangential loading, at this moment is
the maximum tangential loading or maximum static friction force 𝑄 𝑚𝑎𝑥.
A static friction coefficient 𝜇 𝑠 can be defined as the ratio of the maximum
tangential load 𝑄 𝑚𝑎𝑥, required to trigger sliding, to the current normal load 𝑃.
𝜇 𝑠 =
𝑄 𝑚𝑎𝑥
𝑃
(3.19)
As put forward by experimental results, it’s difficult to establish and maintain full
stick contact in practical applications. In general, full slip and full stick, both are the
extreme cases. Hence, an alternative partial slip model is more suitable. The partial slip
sliding inception model which is based on shear failure criterion can be utilized to model
the coated spherical contact and to find out the frictional behavior of the contact
interface.

Patel 26
3.2 Finite Element Model
In this study, a 3D contact problem between a rigid flat plate and a deformable
spherical coated asperity under the combined normal and tangential loading is
investigated. In order to compute numerical simulations, commercial FE simulation
package ABAQUS 6.14 is employed with ABAQUS/Explicit quasi-static scheme.
ABAQUS/Explicit uses dynamic finite element formulation, and is generally applied to
solve the problems which are transient and dynamic in nature or quasi-static problems
where large deformation and excessive geometry and meshing exist. Displacement
control approach is adapted to apply load on rigid flat, which has higher computational
efficiency compared to the force control loading. The coating and substrate, both are
assumed to be in the form of elastic-plastic homogeneous material, where the coating is
completely bonded to the substrate.
3.2.1 Model Components
Since the frictional stress result of full sphere model is in complete coherence
with the model of half-half sphere model, a spherical asperity contact model with a rigid
flat can be modeled as half-half of the asperity in contact with a rigid flat (Wu, Shi, and
Polycarpou 2012). As shown in Figure 3.5 a deformable spherical solid part is partitioned
to create coating layer covered on the substrate. The total radius of the coated sphere
is 𝑅′ = 𝑅 + 𝑡. The radius of substrate sphere 𝑅 = 10𝑚𝑚 is set for this study, whereas
thickness of the coating is controlled by varying the dimensionless coating thickness 𝑡/𝑅.
The value of 𝑡/𝑅 is selected from wide range 0.002 ≤ 𝑡/𝑅 ≤ 0.05 by keeping 𝑡/𝑅 >
(𝑡/𝑅) 𝑝_𝑠𝑡 to protect the substrate by the weakening effect. In the contact of deformable
substrate-coating sphere, rigid flat plate is modeled having the dimensions 3𝑅 × 𝑅. The

Patel 27
rigid flat part is demonstrated using analytical rigid feature, since the analytical rigid
surface results in the smoother surface description in contact with curve reducing contact
noise and providing a better approximation to the physical contact constraint.
Furthermore, it has the computational cost advantages over the discrete element based
rigid surfaces. A rigid body reference point is created as shown in Figure 3.5 which
transmits the motion of the entire rigid body. The rigid flat is assembled to the sphere by
using coincident constrain between the reference point of rigid flat and topmost node of
the spherical asperity.
Figure 3.5 Deformable sphere in contact with rigid plate

Patel 28
3.2.2 Material Property
The coating and the substrate are defined as isotropic elastic-perfect plastic
homogeneous solid sections. Properties of the substrate and coating can be suitably
defined by providing value of Young’s modulus, yield strength, and Poisson’s ratio,
𝐸𝑠𝑢, 𝑌𝑠𝑢, 𝑣𝑠𝑢 and 𝐸𝑐𝑜, 𝑌𝑐𝑜, 𝑣𝑐𝑜 respectively. The Poisson’s ratio of the substrate and
coating are maintained equal 𝑣 = 𝑣𝑠𝑢 = 𝑣𝑐𝑜 = 0.32 for simplification. Young’s
modulus of the substrate material is assumed to be 200 𝐺𝑃𝑎 for all cases. Wide range of
the material properties ratios is covered to investigate its effect on the onset of
sliding; 2 ≤ 𝐸𝑐𝑜 𝐸𝑠𝑢 ≤ 10⁄ , which describes the mismatch between the coating and the
substrate Young’s modulus; Young’s modulus to yield strength to ratio for substrate and
coating, 𝐸𝑠𝑢 𝑌𝑠𝑢 =⁄ 𝐸𝑐𝑜 𝑌𝑐𝑜 = 1000⁄ is taken.
3.2.3 Contact Interaction
Interaction module in ABAQUS is used to define the behavior of the contact
condition between the rigid flat and spherical asperity. The interaction condition is
initiated only when the two surfaces are in contact. The “surface-to-surface” approach
provides inherent smoothing of the surfaces leading to better convergence and often
improves the accuracy of contact stresses due to a better distribution of contact forces
among the master nodes. Therefore, surface-to-surface contact type interaction is selected
to define contact pair, sliding formulation, and contact interaction property.
3.2.3.1 Contact Pair
ABAQUS uses master-slave contact algorithm for contact pairs. The rigid body in
the contact pair is always the master surface in a contact interaction, and slave surfaces
should always be attached to deformable bodies. Accordingly, in this analysis, the rigid

Patel 29
flat plate is defined as the master surface, and the coating surface of the deformable
coating-substrate sphere system is defined as the slave surface.
3.2.3.2 Sliding Formulation
In sliding formulation, there are two options available to choose ‘finite sliding’ or
‘small sliding’ that specifies the expected relative tangential displacement of the two
surfaces. In this analysis, finite sliding is used to control the behavior of two contacting
surfaces. Finite sliding contact formulation requires that ABAQUS continually tracks
which part of the master surface is in contact with each slave node. This is very complex
calculation but comparatively less complex for contact interface between a deformable
body and a rigid surface.
3.2.3.3 Contact Interaction Property
The stick contact condition is implemented in the normal direction loading before
the initiation of tangential loading. Sliding inception occurs for the tangential loading
when critical shear stress limit is achieved. The following mechanical contact interaction
property is selected for contact region in the finite element model to define specific
condition in normal and tangential directional loading:
(i) Normal Loading
During the normal loading, penalty constraint enforcement with default stiffness for
“hard” contact pressure-overclosure relationship is used for normal behavior. Separation of
the contact is not allowed after the surfaces come into contact, and tangential behavior is
selected as rough to create fully adhere contact.

Patel 30
(ii) Tangential Loading
Penalty friction formulation is used to define nearly infinite (1000) local
coefficient of friction between the contacting surfaces, which prevents any relative
sliding motion. The corresponding normal direction constraints are kept active, and at
same time frictional shear stress limit is set as 𝜎𝑠/√3 for contact sliding inception
according to model given by Wu et al for considered coating material.
Using this combination of surface interactions ensures that the surfaces remain
fully bonded together (no separation and no tangential sliding) once they are in contact,
and local sliding takes place for the points where the shear stress limit reaches the defined
upper bound value. When all points of contact region reach the maximum shear strength,
gross sliding takes place.
3.2.4 Mesh
Pre-sliding simulation for the proposed model is performed with the 3D stress
finite element explicit library mesh. To have the time cost-effective, and better accuracy
of the results, the half-half sphere is divided into different zones. The coated sphere of
radius 𝑅’ is partitioned in substrate and coating section to allocate two different properties
simultaneously. Coating with thickness 𝑡 is split on the substrate sphere of radius R as
displayed in Figures 3.5 and 3.6. It should be mentioned here that the coating partition is
shielded on only 20% of the radius of substrate sphere for the computational efficiency as
it reduced unnecessary meshing. The area is chosen large enough to simply provide
identical results compare to whole cover substrate coating for all combined loading

Patel 31
ranges. An optimum number of mesh elements should be in the contact area of two
contacting surfaces to get mesh independent results.
Figure 3.6 Coating-substrate system with designated partition and zones

Patel 32
Figure 3.7 Finite element mesh of sphere

Patel 33
The six different zones, as labeled in Figure 3.6, are generated on substrate-
coating partition system to mesh the sphere in an intuitive manner. The different mesh
density regions I, II, III, IV, V, and VI are planned in terms of t and R with the size of
0.2t, 0.7t, 0.05R, 0.1R, 0.2R and 0.22R respectively. For a different radius of substrate
sphere and various thickness of the coating, zones and partitions vary according to the
choice of radius and thickness.
Zone entities I, II, III, IV, V are meshed with structural control mesh to have a good
visualization of the stress behavior in the contact region. In other zones, free type of control
mesh is utilized to decrease the total number of mesh elements. Zone I to V are meshed with
linear hexahedral elements of type C3D8R while zone VI and rest of the sphere are
meshed with quadratic tetrahedral elements of type C3D10M as shown in Figure 3.7.
Mesh element size is gradually increased as the mesh grows further away from the
contact region. For the sphere with different coating-substrate thickness ratio, the element
size is maintained in order of 0.0003R at contact region with around 400K to 800K total
number of elements. Flawless implementation of mesh development is especially
challenging for cases involving curved partitions.
3.2.5 Loading and Boundary Condition
Boundary conditions are given to confine the motion of the nodes. The
deformable hemisphere is fully constrained by ENCASTRE condition at the bottom to
prohibit both transverse and rotational motion in all directions. The symmetry boundary
condition 𝑍𝑆𝑌𝑀𝑀(𝑈𝑧 = 𝑈𝑅 𝑥 = 𝑈𝑅 𝑦 = 0) are applied in the symmetry plane XY of
coated sphere to obtain comparable results with the full sphere. A displacement control
loading technique is applied to the reference point of the rigid flat at the topmost node of

Patel 34
spherical contact in normal direction 𝑌 and then a ramp displacement loading is applied
in the tangential direction 𝑋 as displayed in Figure 3.8.
Figure 3.8 Boundary condition of model

Patel 35
Chapter 4
Results and Discussion
The mesh convergence study is conducted by changing the mesh density at the
contact region to optimize the element sizes. It is further refined till any further
refinement stops having any considerable effect on the results. Generally, the mesh
element size does not have much influence on the results of normal loading. However, to
satisfy the combined normal and tangential loading mesh element size is selected
0.0003R near the contact region, with no element size larger than 0.006R being allowed
for the effective stress contour zone.
Since ABAQUS/EXPLICIT solver is utilized, upon the completion of the mesh
convergence study, it is essential to calibrate the loading rates to make sure that results of
simulations are, indeed, quasi-static. The pre-sliding frictional analysis is a certified
quasi-static non-linear problem which includes large deformation. As described by Wu
and Shi (Wu and Shi 2013) and based on their study for the variety of the displacement
loading rates, the loading rate of 0.1 𝑚/𝑠 with smooth step amplitude function shows
good agreement with static implicit results. Also, it is checked to ensure that the ratio of
kinetic energy to internal energy throughout the analysis stays below 5%. In the current
study, for most of the cases, it stays below 2.6%. Thus, the quasi-static assumption is also
satisfied from the energy point of view.
In order to precisely analyze pre-sliding behavior, it is absolutely essential to
verify and validate the model.

Patel 36
4.1 Model Verification and Validation.
4.1.1 Homogeneous Model under Normal Loading
Figure 4.1 Hertzian verification, Load P vs. Interference δ
Figure 4.2 Hertzian verification, P/E*
vs. δ/aH
0
2
4
6
8
10
12
0 0.0002 0.0004 0.0006 0.0008 0.001
Load,P(N)
Interference, δ (mm)
Hertz Analytical
FEM
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 0.002 0.004 0.006 0.008 0.01
P/E*
δ/aH
Hertz Analytical
FEM

Patel 37
The accuracy of the finite element model is tested through comparison of the
current model with the elastic Hertz analytical solution, for the homogeneous example of
the sphere (same substrate and coating material) under perfect slip contact condition. The
elastic homogeneous solid model is validated under normal displacement loading by
utilizing following parameters: radius of sphere 10.09𝑚𝑚, Young’s modulus
74000𝑀𝑃𝑎, yield strength 325𝑀𝑃𝑎, and Poisson’s ratio 0.32.
The analytical results are plotted against the homogeneous asperity FE results in
Figure 4.1 and in Figure 4.2, respectively, where, 𝐸∗
is equivalent Young’s modulus and
𝑎 𝐻 is semi Hetzian contact radius. The results generated from the simulation of the
current model for force versus displacement, and for contact pressure versus contact area,
show close agreement with the analytical data and the error is approximately 1% for the
maximum normal load of any coating-substrate system.
4.1.2 Coated Model under Normal Loading
Similar to the Hertz contact, Goltsberg and Etsion (Goltsberg and Etsion 2015)
represented the universal model for the load-displacement relation in elastic spherical
contact with hard coating under the perfect slip condition. The current coated model can
be compared for elastic normal loading. According to Goltsberg and Etsion work,
multiple simulations are done for current model with preselected radius of substrate
sphere 𝑅 = 10𝑚𝑚, Young’s modulus of substrate 𝐸𝑠𝑢 = 200𝐺𝑃𝑎 and coating to
substrate Young’s modulus ratio 𝐸𝑐𝑜/𝐸𝑠𝑢 = 4 , and the coating thickness ratios are varied
within specified range. The results for the current coated model are represented besides
their FE data in Figure 4.3. It displays good harmony within 2.2% error for the maximum
normal load for the range of coating thickness ratios.

Patel 38
Figure 4.3 Elastic normal loading, Load P vs. Interference ω for Young’s modulus ratio
𝐸𝑐𝑜/𝐸𝑠𝑢 = 4 and various thickness over radius ratios based on available FE results of
Goltsberg and Etsion (2015) and current FEM.
Similar to elastic normal loading for the coated model, the current coating-
substrate spherical model can be also verified for the elastic-plastic normal loading under
the stick contact condition for different coating thickness ratios and material properties
based on Ronen et al. (Ronen, Goltsberg, and Etsion 2017).
The critical parameters for interference, load and contact area associated with first
and second yielding 𝛿 𝑐1, 𝛿 𝑐2, 𝐿 𝑐2 and 𝐴 𝑐2 𝑠𝑡
can be calculated analytically by using
equations (3.10) to (3.16) in the Chapter 3 of this study. Following values are selected for
the analytical calculation and simulation: the coating thickness ratio, 𝑡/𝑅 = 0.05,
Poisson’s ratio, 𝑣 = 𝑣𝑠𝑢 = 𝑣𝑐𝑜 = 0.32, Young’s modulus substrate, 𝐸𝑠𝑢 = 200𝐺𝑃𝑎,
Young’s modulus over yield strength ratio for substrate and coating both 𝐸𝑠𝑢 𝑌𝑠𝑢 =⁄
0
1
2
3
4
5
6
0 0.00005 0.0001 0.00015 0.0002
Load,P(N)
Interference, ω (mm)
Goltsberg and Etsion (2015)
t/R=0.009 FEM
t/R=0.007 FEM
t/R=0.005 FEM
t/R=0.003 FEM
t/R=0.001 FEM

Patel 39
𝐸𝑐𝑜 𝑌𝑐𝑜 = 1000⁄ , and Young’s modulus ratio of the coating to substrate is varied in the
range of 2 ≤ 𝐸𝑐𝑜 𝐸𝑠𝑢 ≤ 10⁄ , i.e., 2, 4, 6, 8 and 10. The displacement controlled FE
simulation is performed up to the end of second critical interference 𝛿 = 𝛿 𝑐2. Results are
obtained for the contact area and load, and are normalized with corresponding critical
values.
A relation between the dimensionless contact load versus dimensionless
interference, and dimensionless contact area versus dimensionless interference for
dimensionless coating thickness ratio, 𝑡/𝑅 = 0.05 and for different Young’s modulus
ratios of coating to substrate are plotted in Figures 4.4 and 4.5, for the current FE results
against empirical results by Ronen et al. for comparison purposes.
Figure 4.4 Elastic-plastic normal loading, Dimensionless contact load versus the
dimensionless interference, for thickness over radius ratio of 𝑡/𝑅 = 0.05 based on
Young’s modulus ratios.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
DimensionlesscontatcLoad,L/Lc2
Dimensionless Interference, δ/δc2
t/R=0.05
Ronen et al. (2017)
Eco/Esu=10 FEM
Eco/Esu=8 FEM
Eco/Esu=6 FEM
Eco/Esu=4 FEM
Eco/Esu=2 FEM

Patel 40
Figure 4.5 Elastic-plastic normal loading verification, Dimensionless contact area versus
the dimensionless interference, for thickness over radius ratio of 𝑡/𝑅 = 0.05 based on
Young’s modulus ratios.
Similar results for different ratios of 𝐸𝑐𝑜 𝐸𝑠𝑢⁄ are observed for all of the coating
thickness ratios. This observation holds true since the dimensionless values are used for
the comparison between analytical and FEM results making them independent from
Young’s modulus ratio.
Ronen et al. (Ronen, Goltsberg, and Etsion 2017) established equations for the
critical parameters by fitting the curves from numerical data of FE results, and those
comparisons have limited 𝑅2
goodness-of-fit, which explains the discrepancy in the
results. Results of 2D simulations of coated asperity examined by them indicate marginal
erroneous results in dimensionless comparison. But in case of 3D simulations, these
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
DimensionlesscontatcArea,Ast/Ac2_st
Dimensionless Interference, δ/δc2
t/R=0.05
Ronen et al. (2017)
Eco/Esu=10 FEM
Eco/Esu=8 FEM
Eco/Esu=6 FEM
Eco/Esu=4 FEM
Eco/Esu=2 FEM

Patel 41
results after the comparison of the FE simulation results with the empirical calculations
are slightly elevated, but are found to be within the permissible limit.
Figure 4.6 demonstrates the typical yielding process in the coating and the
substrate. Once the equivalent von Mises stress becomes equal to the yield stress of the
relevant material, yield inception occurs in the coating and in the substrate. Here, it
should be noted that for the selected range of dimensionless coating thickness (𝑡/𝑅 >
(𝑡/𝑅) 𝑝_𝑠𝑡), upon application of the normal loading, yield inception occurs first inside the
coating at first critical interference 𝛿 𝑐1 and after that, it penetrates to the substrate at
second critical interference 𝛿 𝑐2 under stick contact condition.
Figure 4.6 Yield inception beginning in (a) coating and (b) substrate for 𝐸𝑐𝑜 =
2000𝐺𝑃𝑎, 𝐸𝑠𝑢 = 200𝐺𝑃𝑎, 𝑌𝑐𝑜 = 2000𝑀𝑃𝑎, 𝑌𝑠𝑢 = 200𝑀𝑃a
(a)
(b)

Patel 42
The comprehensive parametric study is carried out analytically by using the
equations (3.11) and (3.12) provided by Ronen et al. to understand the relations of the
first and second yielding inceptions regarding the normal interference for a range of
dimensionless coating thickness ratios and Young’s modulus ratios. As it is observed in
Figure 4.7 (a), (b) and (c), the first and second critical interference parameters are
absolutely influenced by both the coating thickness ratio and Young’s modulus ratio.
0
2
4
6
8
10
12
0 0.01 0.02 0.03 0.04 0.05
δc1/ωc_su
t/R
Eco/Esu=2
Eco/Esu=4
Eco/Esu=6
Eco/Esu=8
Eco/Esu=10
0
5
10
15
20
25
30
35
40
45
0 0.01 0.02 0.03 0.04 0.05
δc2/ωc_su
t/R
Eco/Esu=2
Eco/Esu=4
Eco/Esu=6
Eco/Esu=8
Eco/Esu=10
(a)
(b)

Patel 43
Figure 4.7 Parametric study for first 𝛿 𝑐1 and second 𝛿 𝑐2 critical interference contribution
in yielding of coating and substrate by using analytical equations
For the lower range of coating thickness ratio up to 𝑡/𝑅 ≤ 0.005 and higher
Young’s modulus ratio of the coating to the substrate 𝐸𝑐𝑜 𝐸𝑠𝑢 = 6, 8 𝑜𝑟 10⁄ , calculated
first critical interference is higher as compared to the second critical interference which
promotes weakening effect by first yielding the substrate and delaying in the yielding of
the coating. However, as coating thickness ratio increases and the ratio of Young’s
modulus decreases, the dominance of weakening effect wanes down. In simple words, as
coating thickness ratio increases and the ratio of Young’s modulus decreases it favors the
yielding of coating before the yielding of substrate which protects the substrate from the
delamination and weakening effect. That’s why for the further investigation of the current
model under combined loading, the range of coating thickness ratio 0.005 ≤ 𝑡/𝑅 ≤ 0.05
is considered. The normal displacement range is selected as 0 ≤ 𝛿 𝜔𝑐_𝑠𝑢 ≤ 60⁄ based on
the evaluation of first and second critical interferences in Figure 4.7. This selected range
0
5
10
15
20
25
30
35
40
45
0 0.01 0.02 0.03 0.04 0.05
δc2/δc1
t/R
Eco/Esu=2
Eco/Esu=4
Eco/Esu=6
Eco/Esu=8
Eco/Esu=10
(c)

Patel 44
ensures full exploration of the normal loading influence in its entire range, i.e., (i) before
first critical interference, (ii) between the first and second critical interferences and (iii)
beyond the second critical interference.
4.1.3 Homogeneous Model under Combined Loading
Before delving into the results for the coated asperity under combined loading, it
is impractical that the model is verified for the additional tangential loading. As discussed
in Chapter 3, maximum shear stress criterion is adopted for the sliding inception based on
the approached provided by Wu et al. (2012). In this model, contact interface condition is
assumed to be frictionless during the normal loading in both directions. Similarly, during
tangential loading, contact interface condition is considered frictionless in the normal
direction. However, the contact is under full stick in a tangential direction while the
tangential shear stress is limited to critical shear stress and sliding happens when the
shear stress at all contacting points reaches its threshold. When the coating and substrate
share the same material, the current homogeneous model with sphere radius 𝑅’ =
10.5𝑚𝑚 can be compared with the WSP model under the combined normal and
tangential loading.
Material properties for the simulation of the current model are congruent with the
WSP model elastic-perfect plastic where Young’s modulus, Poisson’s ratio, and yield
strength are 74000𝑀𝑃𝑎, 0.33, and 325𝑀𝑃𝑎, respectively. The simulations are
completed for different normal displacement preloaded case and are compared to their FE
results of dimensionless tangential force versus dimensionless tangential displacement.
Figure 4.8 displays the tangential frictional force 𝑄 vs. the tangential displacement 𝑢 𝑥 for
shear failure sliding under the full range of preloaded normal displacement (a) ω=0.5ωc,

Patel 45
(b) ω=3ωc, (c) ω=12ωc, and (d) ω=72ωc respectively. The tangential force is
dimensionalized by the critical normal load 𝑃𝐶 and tangential displacement by critical
interference 𝜔𝑐.
0
0.1
0.2
0.3
0.4
0 0.2 0.4 0.6 0.8 1
Q/Pc
ux/ωc
ω=0.5ωc
Wu et al. (2012)
FEM
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
Q/Pc
ux/ωc
ω=3ωc
Wu et al. (2012)
FEM
(a)
(b)

Patel 46
Figure 4.8 Dimensionless tangential force versus dimensionless tangential displacement
for various normal preloaded displacement (a) ω=0.5ωc, (b) ω=3ωc, (c) ω=12ωc, and (d)
ω=72ωc
The results of the current model for the homogeneous case are in close proximity
of WSP model FE results, and it can be considered as reasonably accurate having the
maximum 5.6% error for any case of combined loading.
0
3
6
9
12
0 3 6 9 12 15
Q/Pc
ux/ωc
ω=12ωc
Wu et al. (2012)
FEM
0
20
40
60
80
100
0 10 20 30 40 50
Q/Pc
ux/ωc
ω=72ωc
Wu et al. (2012)
FEM
(c)
(d)

Patel 47
𝑄 = 0 𝑄 = 0.8𝑄 𝑚𝑎𝑥 𝑄 = 𝑄 𝑚𝑎𝑥
𝜔 = 0.5𝜔𝑐
𝜔 = 3𝜔𝑐
𝜔 = 12𝜔𝑐
𝜔 = 72𝜔𝑐
Figure 4.9 Development of the von Mises stresses during the tangential loading under
different normal preloaded displacement
In Figure 4.9, the von Mises stress distribution is displayed for the FE results of
the current homogeneous model simulation, which mimic close behavior with Wu-Shi
model performance. Before the tangential loading, at 𝑄 = 0, the stress distribution is
axisymmetric and the maximum stress is underneath the surface. With the increase of the
tangential loading, the stress distribution changes to asymmetric and the maximum stress

Patel 48
moves to the contact surface and reaches the well-defined yielding stress limit. A further
increase of tangential displacement loading results in a larger yield area under the
constant stress value, which covers the whole contact region as shown in Figure 4.9. The
related force at this moment is the maximum static friction force 𝑄 𝑚𝑎𝑥.
The static friction coefficient 𝜇 𝑠 can be expressed as the ratio of the maximum
tangential frictional force 𝑄 𝑚𝑎𝑥 to the normal preload 𝑃. By utilizing the maximum
frictional shear criterion, Wu et al. provided an empirical friction coefficient for
homogeneous sphere asperity as:
𝜇 𝑠 = 0.3 coth (0.57 (
𝜔
𝜔𝑐_𝑠𝑢
)
0.41
) (4.1)
Figure 4.10 Predicted friction coefficient for different displacements
0
0.3
0.6
0.9
1.2
1.5
1.8
-5 5 15 25 35 45 55 65 75
Staticfrictiuoncoefficient,µs
Dimensionless normal loading, ω/ωc_su
Wu et al. (2012)
Current FEM, Eco/Esu=1

Patel 49
The proposed model where the coating is made of identical substrate material can
also be verified for the static friction coefficient using the analytical equation proposed
by Wu et al., as shown in Figure 4.10. Homogeneous asperity proves good conformity
with the Wu et al. under displacement control. Using the same criterion for the sliding
inception, this can be of course expected. However, the comparison ensures the accuracy
of the model to capture sliding inception.
The validity of the current frictional FEM model can be furthered examined with
the available experimental results of Ovcharenko et al. (Andrey Ovcharenko, Halperin,
and Etsion 2008). Simulations with the proposed model under load control are performed,
and the results are plotted in Figure 4.11 along with the experimental results.
Figure 4.11 Predicted friction coefficient for different loads
0
0.3
0.6
0.9
1.2
1.5
1.8
-5 5 15 25 35 45 55 65 75 85 95 105
Dimensionless normal load, P/Pc_su
Cu D=5mm, Ovcharenko et al. (2008)
Current FEM, Eco/Esu=1

Patel 50
It can be seen from the Figure 4.11 that the current FEM homogeneous model
predicts friction coefficients very close to the experimental results especially at high
loads.
Based on the presented results, it can be concluded that the results of the current
model study show good conformity with 1. elastic normal loading (Hertz 1882), 2. elastic
normal loading with different coating thickness ratio and constant coating to substrate
properties (Goltsberg and Etsion 2015), 3. elastic-plastic normal loading with different
material property and constant coating thickness ratio (Ronen, Goltsberg, and Etsion
2017), and 4. WSPmodel (Wu, Shi, and Polycarpou 2012) for combined normal and
tangential loading of the homogenous elastic-perfect plastic sphere. This actually verifies
the capability of the current coated model to capture accurately tangential loading effect,
to analyze pre-sliding for different coating thickness ratio and coating-substrate material
properties, and it can be utilized for further frictional analysis.

Patel 51
4.2 Coated Model under Combined Loading
In this study, the coated spherical asperity contact is investigated for different
input parameters under the combined normal and tangential loadings and compared with
the homogeneous asperity made of the related substrate material.
Required input parameters for the proposed model are listed in Table 4.1. By
choosing the different combinations of the available options, the coated contact can be
simulated under different preloaded normal displacement, to find out the effect of these
parameters on sliding inception. In the proposed coated model sliding inception is treated
as the shear failure mechanism by selecting the coating shear strength as the upper
boundary.
Table 4.1 Model input parameters of coated model
Input Parameters of Coating-Substrate model
Radius of substrate sphere, 𝑹 10 𝑚𝑚
Dimensionless coating thickness, 𝒕/𝑹 0.005, 0.025, 0.05
Young’s modulus of rigid surface, 𝑬 ∞
Young’s modulus of substrate, 𝑬 𝒔𝒖 200 𝐺𝑃𝑎
Coating to substrate ratio, 𝑬 𝒄𝒐/𝑬 𝒔𝒖 1,2,4
Young’s modulus to yield strength to ratio for
substrate and coating, 𝑬 𝒔𝒖/𝒀 𝒔𝒖 = 𝑬 𝒄𝒐/𝒀 𝒄𝒐
1000
Poisson’s ratio, 𝒗 0.32

Patel 52
4.2.1 Stress and Strain
At given normal displacement loading, the stress and strain development with the
tangential ramp loading are investigated for the all chosen combinations. Here, the
preloaded normal displacement 𝛿 is selected in the range 0 ≤ 𝛿 𝜔𝑐_𝑠𝑢 ≤ 60⁄ to observe
the stress-strain development close to the first and second critical interference parameters
of substrate-coated asperity, 𝛿 𝑐1/𝜔𝑐_𝑠𝑢 and 𝛿 𝑐2 /𝜔𝑐_𝑠𝑢 respectively. The critical
interference parameters are dicussed earlier for the parametric examination in Figure 4.7
(a) and (b), however, the calculated values of the critical parameters are displayed in
Table 4.2 to understand the behavior thoroughly.
Table 4.2 Details about dimensionless critical interference parameters
𝒕/𝑹
𝜹 𝒄𝟏/𝝎 𝒄_𝒔𝒖 𝜹 𝒄𝟐/𝝎 𝒄_𝒔𝒖
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒
𝟎. 𝟎𝟎𝟓 1.65 2.40 2.68 2.43
𝟎. 𝟎𝟐𝟓 1.03 1.13 17.60 15.95
𝟎. 𝟎𝟓 0.97 1.01 35.88 39.61
Figure 4.12 shows the comparison of the von Mises stress distribution and plastic
strain, for different Young’s modulus ratio of the coating to the substrate for constant
thickness ratio 𝑡/𝑅 = 0.005, when the given normal preload displacement is
𝛿 𝜔𝑐_𝑠𝑢 = 1⁄ . The distribution of stress is axisymmetric before the tangential loading,
and with increasing loading the stress distribution turn into asymmetric. Once the
maximum stress area moves to the surface, reaching the yield stress, covering the whole

Patel 53
contact area gradually. At this moment the static tangential force becomes the maximum
tangential force 𝑄 𝑚𝑎𝑥. Further tangential displacement loading results in a larger yield
area, and same kind of behavior can be seen for all combinations.
𝑸 = 𝟎 𝑸 = 𝑸 𝒎𝒂𝒙 𝑸 = 𝑸 𝒎𝒂𝒙
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟏
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒
Figure 4.12 Stress and strain development in tangential loading under 𝛿 𝜔𝑐_𝑠𝑢 = 1⁄ for
𝑡/𝑅 = 0.005
The applied normal displacement 𝛿 𝜔𝑐_𝑠𝑢 = 1⁄ is not enough to affect the
substrate sphere covered by higher coating thickness ratio, hence, in that situation, the
stress distribution developed on coating, can be understood similar to homogeneous
sphere. However, small thickness ratio of coating displays the stress contour development
over the substrate, as shown in Figure 4.12. During normal displacement loading

Patel 54
𝛿 𝜔𝑐_𝑠𝑢 = 1⁄ there is no yielding on coating and substrate. Yielding occurs only on the
top surface of the coating during the tangential loading since the applied normal
displacement is lower than first critical interference 𝛿 𝑐1 /𝜔𝑐_𝑠𝑢. Accordingly, The plastic
strain development grows directlty with the sliding of rigid plate in tangential direction
and covers the whole contact area gradually as shown in Figure 4.12 at maximum static
friction force.
The normal displacement loading 𝛿 𝜔𝑐_𝑠𝑢 = 10⁄ is higher than the first critical
interference 𝛿 𝑐1 /𝜔𝑐_𝑠𝑢 for all combinations, hence, yielding of the coating is occured
during the normal loading. However, lower coating thickness ratio 𝑡/𝑅 = 0.005
experiences the yield of substrate with increasing normal displacement preload, because
applied normal loading exceeds the second critical interference 𝛿 𝑐2 /𝜔𝑐_𝑠𝑢. The stress and
strain contours before applying the tangential load and after its application are shown in
the Figure 4.13. It is found that when the normal interference is between first and second
critical interference 𝛿 𝑐1 < 𝛿 < 𝛿 𝑐2 yielding initiates and propogates in the coating
during the normal loading and covers the whole contact region with the increment of
tangential loading as displayed for 𝑡/𝑅 = 0.025, 𝑎𝑛𝑑 0.05. When the interference is
higher than the second critical interference 𝛿 > 𝛿 𝑐2 yielding take place first in the
coating followed by substrate side of coating-substrate interface and keeps on growing
occurs at coating side of the interface during the normal loading, and with the increase of
tangential loading yield inside the substrate grows till it occupies the whole coated
contact regeion as displayed for 𝑡/𝑅 = 0.005. Additionally, It is found that for applied
normal displacement load 𝛿 < 𝛿 𝑐2 , higher coating thickness ratio, with higher ratios of
Young’s modulus of the coating to the substrate, resists the yield failure of coating.

Patel 55
𝑸 = 𝟎 𝑸 = 𝑸 𝒎𝒂𝒙
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟎𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟎𝟓
Cont.

Patel 56
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟐𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟐𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟓
Cont.

Patel 57
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟓
𝑡/𝑅 = 0.005, 𝑡/𝑅 = 0.025 and 𝑡/𝑅 = 0.05
Also, it can be seen that plastic strain shrinks in the coating for applied normal
loading as the coating thickness ratio and the ratio of Young’s modulus of the coating to
the substrate increases.
Similar kind of deviations in the stress and strain and the yield failure can be seen
in substrate and coating for thickness ratio 𝑡/𝑅 = 0.005, 𝑎𝑛𝑑 0.025, for the tangential
sliding, under the normal preloaded displacement 𝛿 𝜔𝑐_𝑠𝑢 = 30⁄ in Figure 4.14.
However, for the case of higher thickness ratio 𝑡/𝑅 = 0.05 given normal preload is lower
than the second critical interference, thus, yielding occurs in the coating and occupies the
entire coated contact region with tangential displacement loading. Here, It can be clealy
noted down that as the coating thickness ratio increases, it reduces the risk of substrate
yielding.

Patel 58
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟎𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟎𝟓
Cont.

Patel 59
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟐𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟐𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟓
Cont.

Patel 60
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟓
𝑡/𝑅 = 0.005, 𝑡/𝑅 = 0.025 and 𝑡/𝑅 = 0.05
The given normal preloaded displacement 𝛿 𝜔𝑐_𝑠𝑢 = 60⁄ , is larger than second
critical interference parameter for all combinations, therefore, yielding of coating
followed by yielding of substrate occurs during the normal loading and with the
increment of ramp tangential loading yield region grows in substrate and simultaneously
covers the entire coated contact region . The development of the von Mises stress
distribution and plastic strain, for different Young’s modulus ratios of the coating to the
substrate for various thickness ratio, is displayed in Figure 4.15. It is found clearly that,
for applied normal displacement preload 𝛿 > 𝛿 𝑐2 , higher coating thickness ratio in
combination with lower ratio of Young’s modulus resists the yield failure of substrate.

Patel 61
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟎𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟎𝟓
Cont.

Patel 62
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟐𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟐𝟓
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟐 for 𝒕/𝑹 = 𝟎. 𝟎𝟓
Cont.

Patel 63
𝑬 𝒄𝒐/𝑬 𝒔𝒖 = 𝟒 for 𝒕/𝑹 = 𝟎. 𝟎𝟓
𝑡/𝑅 = 0.005, 𝑡/𝑅 = 0.025 and 𝑡/𝑅 = 0.05
4.2.2 Frictional Force Study
The dimensionless tangential force 𝑄/𝑃𝑐_𝑠𝑢 versus the dimensionless tangential
displacement 𝑢 𝑥/𝜔𝑐_𝑠𝑢 under different normal displacement loads (𝛿 = 1𝜔𝑐_𝑠𝑢, 𝛿 =
10𝜔𝑐_𝑠𝑢, 𝛿 = 30𝜔𝑐_𝑠𝑢 and 𝛿 = 60𝜔𝑐_𝑠𝑢) are investigated for the different dimensionless
coating thickness ratios and Young’s modulus ratios. Here, the critical load 𝑃𝑐_𝑠𝑢 is
calculated by using the equation (3.4) for the homogeneous sphere made of the substrate
material.
It is found that at the beginning of the tangential loading, all ratio of Young’s
modulus 𝐸𝑐𝑜/𝐸𝑠𝑢 display the same start point after normal preloaded displacement. With
the growing of displacement loading in tangential direction, all combinations show
softening of the tangential stifness. The results of the tangential ramp displacement

Patel 64
confirm that, as ratio 𝐸𝑐𝑜/𝐸𝑠𝑢 increases, it delays the stiffness softening. The results of
the different coating-substrate material property are compared to the homogeneous sphere
plotted in Figures 4.16, 4.17 and 4.18 for coating thickness ratio 0.005, 0.025 and 0.05,
correspondingly.
Figure 4.16 Dimensionless tangential load vs. dimensionless tangential displacement under
different normal displacement δ for 𝑡/𝑅 = 0.005
The significant dissimilarity between the tangential force with an increment of
tangential displacement demonstrates the characteristic behavior of the coating material
during pre-sliding. The much higher load is required for the coated asperity to shear the
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
Q/Pc_su
ux/ωc_su
δ=1ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
5
10
15
20
25
30
35
0 2 4 6 8 10
Q/Pc_su
ux/ωc_su
δ=10ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
20
40
60
80
100
120
0 5 10 15 20 25 30
Q/Pc_su
ux/ωc_su
δ=30ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
50
100
150
200
250
0 10 20 30 40 50 60
Q/Pc_su
ux/ωc_su
δ=60ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4

Patel 65
entire contact under applied normal interference as compared to the homogeneous
spherical asperity. In simpler words, after the constant normal preloaded displacement,
coated asperity can sustain much higher loads before the onset of sliding. The ability to
sustain higher tangential load or static frictional force at contact region increases with
further increment of Young’s modulus 𝐸𝑐𝑜/𝐸𝑠𝑢 ratio as presented in the results.
It should be mentioned that since the ratio of the yield stress to Young’s modulus
kept constant, in all of the cases same ratios for the yield stress of the coating to the
substrate are considered similar to Young’s modulus ratios.
under different normal displacement δ for 𝑡/𝑅 = 0.025
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
Q/Pc_su
ux/ωc_su
δ=1ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
5
10
15
20
25
30
35
0 2 4 6 8 10
Q/Pc_su
ux/ωc_su
δ=10ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
20
40
60
80
100
120
0 5 10 15 20 25 30
Q/Pc_su
ux/ωc_su
δ=30ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
50
100
150
200
250
0 10 20 30 40 50 60
Q/Pc_su
ux/ωc_su
δ=60ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4

Patel 66
under different normal displacement δ for 𝑡/𝑅 = 0.05
The effect of coating-substrate material property ratio is noticeable as shown in
the above comparison. But, to observe the effect of coating thickness ratio, it is required
to plot tangential friction force for different thickness ratios for the constant material
property. The overall performance of a change in coating thickness ratio and its influence
on the static friction force throughout the pre-sliding under the equal normal preloaded
displacement δ = 1𝜔𝑐_𝑠𝑢 and the identical coating material is shown in Figure 4.19 and
Figure 4.20.
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
Q/Pc_su
ux/ωc_su
δ=1ωc_suEco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
5
10
15
20
25
30
35
0 2 4 6 8 10
Q/Pc_su
ux/ωc_su
δ=10ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
20
40
60
80
100
120
0 5 10 15 20 25 30
Q/Pc_su
ux/ωc_su
δ=30ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
50
100
150
200
250
0 10 20 30 40 50 60
Q/Pc_su
ux/ωc_su
δ=60ωc_su
Eco/Esu=1
Eco/Esu=2
Eco/Esu=4

Patel 67
under normal displacement of 𝛿 = 𝜔𝑐_𝑠𝑢 for 𝐸𝑐𝑜/𝐸𝑠𝑢 = 2
under normal displacement of 𝛿 = 𝜔𝑐_𝑠𝑢 for 𝐸𝑐𝑜/𝐸𝑠𝑢 = 4
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
Q/Pc_su
ux/ωc_su
Eco/Esu=1
Eco/Esu=2, t/R=0.05
Eco/Esu=2, t/R=0.025
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
Q/Pc_su
ux/ωc_su
Eco/Esu=1
Eco/Esu=4, t/R=0.05

Patel 68
These results provide lucid clarification in relation to the effect of the material
property and the coating thickness ratio under the normal preloaded displacement δ =
1𝜔𝑐_𝑠𝑢. It is found that the higher ratio of coating to substrate Young’s modulus is more
suitable for protection of the coated contact during the higher load applications with same
coating thickness ratio, however, the arrangement with higher coating thickness presents
noteworthy tribological performance of coating on the substrate asperity.
4.2.3 Static Friction
The static friction coefficient results for different Young’s modulus ratio of the
coating to the substrate and coating thickness ratio are investigated and are compared
with homogeneous asperity by simulating the current model under the different normal
pre-loads.
0
0.3
0.6
0.9
1.2
1.5
0 10 20 30 40 50 60 70 80 90 100
t/R=0.005 Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
(a)

Patel 69
Figure 4.21 Predicted friction coefficient for load control model with different 𝑡/𝑅 ratio
(a) 𝑡/𝑅 = 0.005 (b) 𝑡/𝑅 = 0.025 (c) 𝑡/𝑅 = 0.05
As can be seen in Figure 4.21, the higher Young’s modulus ratio of the coating to
the substrate predicts slightly more friction coefficient which explains the effect of the
0
0.3
0.6
0.9
1.2
1.5
0 10 20 30 40 50 60 70 80 90 100
t/R=0.025 Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
0
0.3
0.6
0.9
1.2
1.5
0 10 20 30 40 50 60 70 80 90 100
t/R=0.05 Eco/Esu=1
Eco/Esu=2
Eco/Esu=4
(b)
(c)

Patel 70
coating material properties. Also with the increase of the coating thickness ratio
from 𝑡/𝑅 = 0.005 to 𝑡/𝑅 = 0.05 the deviation among the predicted the friction
coefficient for different Young’s modulus ratios increases as displayed in Figures 4.21
(a), (b) and (c).

Patel 71
Chapter 5
Conclusion and Future work
5.1 Conclusion
A 3D finite element model of elastic-plastic coated spherical asperity in contact
with a rigid flat under combined normal and tangential loading is developed by
considering the maximum frictional shear stress criterion for sliding inception. The
proposed model is verified for the different coating thickness ratio and several material
properties in normal displacement loading, and is also validated and verified against
experimental findings and numerical results for combined loading with the sliding
criterion for homogeneous asperity.
Upon verifications and validations, the von Mises stress and plastic strain
distributions are studied for coated contact, when tangential displacement loading is
added to the initial normal preloaded displacement. The process of contact failure is
explored for the pre-sliding of a proposed coated asperity by changing material properties
and coating thickness ratios. At the applied normal interference 𝛿 < 𝛿 𝑐1 , 𝛿 𝑐1 < 𝛿 <
𝛿 𝑐2 and 𝛿 > 𝛿 𝑐2 under combined loading, three different failure mechanism in coating
and substrate are observed and it is found that:
(a) For 𝛿 < 𝛿 𝑐2 , higher 𝑡/𝑅 with higher 𝐸𝑐𝑜/𝐸𝑠𝑢 shows good resistance against
the yield failure of coating.
(b) For 𝛿 > 𝛿 𝑐2 , higher 𝑡/𝑅 with lower 𝐸𝑐𝑜/𝐸𝑠𝑢 shows good resistance against
the yield failure of substrate.

PRE-SLIDING FRICTIONAL ANALYSIS OF A COATED SPHERICAL ASPERITY

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to PRE-SLIDING FRICTIONAL ANALYSIS OF A COATED SPHERICAL ASPERITY

Similar to PRE-SLIDING FRICTIONAL ANALYSIS OF A COATED SPHERICAL ASPERITY (20)

Recently uploaded

Recently uploaded (20)

PRE-SLIDING FRICTIONAL ANALYSIS OF A COATED SPHERICAL ASPERITY