Modeling and Simulation of
Dry Sliding Wear for
Micro-machine Applications
Zur Erlangung des akademischen Grades eines
Doktors der Ingenieurwissenschaften
von der Fakultät für Maschinenbau der
Universität Karlsruhe (TH)
genehmigte
Dissertation
von
Vishwanath Hegadekatte M. Sc.
aus Hubli, Indien
Tag der mündlichen Prüfung:
Hauptreferent:
Korreferent:
22. März 2006
Priv.-Doz. Dr. -Ing. N. Huber
Prof. Dr. rer. nat. O. Kraft
Abstract
Study of wear in complex micro-machines is often accomplished with experimental methods, like pin-on-disc test, scratch test or atomic force microscopy.
Pin-on-disc tribometers are the most commonly used experimental technique to
study tribological performance of micro-machines due to their simplicity and cost
eectiveness. Such experiments attempt to mimic the contact conditions of the
micro-machine under study in terms of contact pressure and sliding velocity. Tribometry always allows for a qualitative study of the suitability of a particular
material combination, but is often not sucient for a prediction of the progress of
wear and eventually the useful life span of a micro-machine. A simple equation
used to compute volumetric wear in terms of applied normal load, sliding distance and the hardness of the softer material was given by Archard (1953). The
wear rate (e. g., from tribometry) is usually expressed as worn volume per unit
load and per unit sliding distance, which actually amounts to the application of
Archard's wear model.
In this dissertation, a very ecient incremental implementation of Archard's wear
model on the global scale (Global Incremental Wear Model - GIWM) for pin wear
and disc wear in a pin-on-disc tribometer is presented. It can be used to identify
Archard's wear coecient by tting the experimental results. It will be shown in
this work that GIWM can in fact predict pin-on-disc experimental results to a
limited extent. GIWM considers the elastic displacement in the normal direction
to the contact. However, the elastic displacement in the tangential direction
to the contact can have signicant eect on the computation of wear especially
in case of more compliant materials, in the early stages of sliding, or for very
low wear coecients. Therefore, two dimensionless parameters are introduced to
estimate the inuence of elastic deformation on the computation of wear.
iii
In order to apply the identied wear model including the corresponding wear
parameter to predict wear in a geometrically dierent tribosystem, the wear
model/parameters should be valid on the local scale. Therefore, a numerical
simulation scheme has been developed in this work. It represents a numerical approach of interpreting wear in a more general tribosystem. The approach consists
of a nite element based software tool, the Wear-Processor, which implements a
wear model based on Archard's wear law. The Wear-Processor works in association with commercial nite element package ABAQUS. In this tool, local wear is
computed and then integrated over the sliding distance using the Euler integration scheme. After every sliding distance increment the geometry is re-meshed to
correct the deformed mesh due to wear. The results from the simulation of the
pin-on-disc experiment for silicon nitride using the Wear-Processor are in good
agreement with that from the GIWM which in the rst place ts the experiments
favorably. Therefore it can be assumed that the identied wear coecient included in the wear model can be used to predict wear in a geometrically dierent
tribosystem (e.g. micro-machine) provided that the experiments were conducted
within the parameter space (in terms of contact pressure and sliding velocity) of
the micro-machine. It will also be shown in this dissertation that for the case
of more compliant materials (e.g., polytetrauoroethylene on steel), there is a
signicant eect from elastic deformation on the simulated wear values, which
results in an asymmetric wear on the contact surface. Such eects can only be
studied with a general simulation tool, such as the Wear-Processor.
iv
Kurzzusammenfassung
Das Verschleiÿverhalten von Werkstoen wird oft mittels experimenteller Methoden wie Stift-auf-Scheibe-Experimenten, Ritzversuchen oder Rasterkraftmikroskopie
untersucht. Stift-auf-Scheibe-Tribometer sind aufgrund ihrer Einfachheit und
Kostenezienz die am meisten benutzten Versuchsaufbauten um das tribologische
Verhalten von Materialien zu untersuchen. Mit diesen Experimenten werden nach
Möglichkeit die Kontaktbedingungen der zu untersuchenden Werkstopaarungen
im Hinblick auf Anpressdruck und Gleitgeschwindigkeit im Betrieb nachgebildet.
Ist eine exakte Abbildung der Betriebsbelastungen nicht möglich, ermöglicht
die Tribometrie dennoch immer eine qualitative Untersuchung der Tauglichkeit
einer speziellen Materialkombination. Insbesondere bei zeitlich veränderlichen
Gleitbedingungen ist die Tribometrie jedoch nicht ausreichend für eine Vorhersage des Verschleiÿes und somit der letztendlichen Lebensdauer zum Beispiel
einer Mikromaschine. Eine einfache Gleichung für die Berechnung des globalen
volumetrischen Verschleiÿes in Abhängigkeit der aufgebrachten Normalkraft, des
Gleitwegs und der Härte des weicheren Materials stammt von Archard (1953).
Die darin enthaltene Verschleiÿrate wird üblicherweise durch das abgetragene
Volumen pro Lasteinheit und Gleitweg ausgedrückt.
In dieser Dissertation wird eine eziente inkrementelle Implementierung von
Archards Verschleiÿmodell auf der globalen Skala (Global Incremental Wear
Model GIWM) für die Beschreibung des Stiftverschleiÿes beziehungsweise des
Scheibenverschleiÿes in einem Stift-auf-Scheibe-Tribometer dargestellt, das auch
elastische Deformationen im Kontakt berücksichtigt. Dieses Modell kann für die
Anpassung von Archards Verschleiÿkoezienten an experimentellen Ergebnisse
verwendet werden. Es wird gezeigt, dass das GIWM Messwerte von Stift-aufScheibe-Experimenten in einem gewissen Bereich vorhersagen kann. Das GIWM
v
ist allerdings nicht in der Lage, die elastischen Verformungen in der zum Kontakt
tangentialen Richtung zu berücksichtigen. Diese sind in der Regel vernachlässigbar, es ergibt sich jedoch insbesondere für nachgiebigere Materialien, bei sehr
niedrigen Verschleiÿkoezienten oder in der Anfangsphase des Gleitens ein sichtbarer Eekt. Es werden daher zwei dimensionslose Parameter eingeführt, die eine
Abschätzung des Einusses der elastischen Deformation auf die Berechnung des
Verschleiÿes erlauben.
Um das aus Tribo-Experimenten mit Hilfe des GIWM identizierte Verschleiÿmodell inklusive der zugehörigen Verschleiÿparameter für die Vorhersage von
Verschleiÿ in einem geometrisch anderen Tribosystem anwenden zu können, ist
dessen Gültigkeit für die lokale Anwendung zu überprüfen. Um diesen Nachweis
zu erbringen, wurde ein numerisches Simulationsprogramm entwickelt, mit dem
Verschleiÿ in einem allgemeineren Tribosystem berechnet werden kann. Dieser
sogenannte Verschleiÿprozessor ist ein Finite Elemente basierter Postprozessor,
der ein Verschleiÿmodell auf der Basis von Archards Verschleiÿgesetz beinhaltet.
Darin wird der lokale Verschleiÿ berechnet und anschlieÿend über dem
Gleitweg integriert. Nach jedem Iterationsschritt wird die aufgrund des Verschleiÿes veränderte Geometrie berücksichtigt.
Die Simulationsergebnisse für
Stift-auf- Scheibe-Experimente an Siliziumnitrid zeigen eine gute Übereinstimmung zwischen dem Verschleiÿprozessor, dem GIWM und den experimentellen
Daten. Daher kann angenommen werden, dass das Achard-Modell mit dem identizierten Verschleiÿkoezient innerhalb der durch die Experimente gegebenen
Grenzen geeignet ist, Verschleiÿ in einem geometrisch verschiedenen Tribosystem,
zum Beispiel in einem Mikrogetriebe, vorherzusagen. Es wird darüber hinaus
gezeigt, dass es für den Fall eines nachgiebigen Materials (z.B. Polytetrauorethylen auf Stahl) einen signikanten Eekt durch die elastische Deformation
gibt, der zu einem asymmetrischen Verschleiÿ des Polymer-Stiftes führt. Solche
Eekte können nur mit einem numerischen Simulationswerkzeug wie dem Verschleiÿprozessor untersucht werden.
vi
Acknowledgement
I am deeply indebted to many people over the last six years since I moved to
Germany for higher studies from a small town in southern India. I have beneted
both professionally and personally, from the numerous experiences that I have
had with these people, especially since I moved to Karlsruhe around four years
ago. At this time, I would like to acknowledge and thank everyone who have
contributed to my experience.
First of all, I would like to express my sincere gratitude to my advisor, Dr.
Norbert Huber, without whom this work would not have been possible. I would
specically like to thank him for giving me lot of freedom in conducting my
research and also for the countless, intense and motivating discussions that I
have had with him. Through his own example, he has instilled in me, the passion
for perfection and deep understanding. I am indeed thankful to Norbert for all
that he has done for me. It has been a real pleasure to work with, and learn
from him over the years. I would also like to gratefully acknowledge Professor
Oliver Kraft for nding time from his busy schedule to carefully read this thesis
and oer critical suggestions for improvement within a very short span of time. I
would also like to thank him for his advice and support during the course of this
research work. I would be profoundly ungrateful if I did not acknowledge and
thank Professor Dietrich Munz for showing condence in me and oering me this
job.
I would like to gratefully acknowledge Professor Karl-Heinz Zum Gahr, Dr. Johannes Schneider, Mr. Sven Kurzenhaeuser and Mr. Joerg Herz for supplying the
experimental data on silicon nitride used in this work. I would like to oer my appreciation for the valuable help and suggestions from Dr. Klaus Heiermann, Dr.
vii
Markus Scherrer, Dr. Joerg Buettner, Dr. Valentin Licht, Dr. Zhenggui Wang,
Miss Ulrike Mayer and Mr. Bernd Laskewitz. I would specially like to thank
my previous and present ocemates Dr. Vladimir Govorukha, Mr. Tzvetelin
Chehtov and Mr. Dong Wang for setting up a warm and cheerful atmosphere
at the oce. I need to also acknowledge the eorts of my fellow colleagues; in
particular I would like to mention Dr. Patricia Huelsmeier, Dr. Daniel Hofer,
Dr. Eduard Tyulyukovskiy, Mrs. Svetlana Scherrer-Rudiy, Dr. Sophie Eve, Dr.
Marc Kamlah, Dr. Dayu Zhou, Dr. Erica Lilleodden, Dr. Cynthia Volkert and
Dr. Ruth Schwaiger. I would like to specially thank Mr. Joerg Knyrim for the
German translation of the abstract of this dissertation.
I would like to thank Forschungszentrum Karlsruhe and the German Research
Foundation (DFG) for funding this work under project D4 - Wear Simulation
within the scope of the collaborative research center, SFB 499 - Design, production and quality assurance of molded micro parts constructed from metals and
ceramics.
Professor Manfred Kasper was responsible for introducing me to the eld of Mi-
crosystem Technology at Hamburg before I moved to Karlsruhe. I am grateful
for his tutelage and encouragement that inspired me to continue onto Karlsruhe.
I also need to acknowledge the Indian community at Forschungszentrum Karlsruhe for the wonderful time that I had with them especially during our many
cricket and curry sessions.
As far as personal acknowledgements, rst and foremost I want to acknowledge
my wife, Rashmi for her continuous support and encouragement. Thanks to
my parents-in-law, Mr. Gopalkrishna Padiadpu and Mrs. Devaki Padiadpu for
their support. I would like to dedicate this thesis to my mother Mrs. Indira
Hegadekatte and my late father Mr. Satishchandra Hegadekatte.
Vishwanath Hegadekatte
Karlsruhe, 21 Feb 2006
viii
Contents
Abstract
iii
Kurzzusammenfassung
v
Acknowledgement
vii
1 Introduction
1
1.1
Wear simulation strategy . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Overview of chapters . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Basic Principles and Literature Review
2.1
Basics of contact mechanics . . . . . . . . . . . . . . . . . . . . .
2.1.1
7
8
Formulation of the contact problem using the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . .
10
Formulation for the contact potential . . . . . . . . . . . .
16
Introduction to tribology . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.2
Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Wear modeling and simulation . . . . . . . . . . . . . . . . . . . .
28
2.3.1
Mechanistic wear modeling . . . . . . . . . . . . . . . . . .
28
2.3.2
Phenomenological wear modeling . . . . . . . . . . . . . .
30
2.4
Wear in micro-machines . . . . . . . . . . . . . . . . . . . . . . .
34
2.5
Objectives of this work . . . . . . . . . . . . . . . . . . . . . . . .
36
2.1.2
2.2
2.3
3 Global Wear Modeling
39
3.1
Tribometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2
Global Incremental Wear Models . . . . . . . . . . . . . . . . . .
41
ix
3.2.1
GIWM for pin wear . . . . . . . . . . . . . . . . . . . . . .
41
3.2.1.1
Comparison with experimental results . . . . . .
44
GIWM for disc wear . . . . . . . . . . . . . . . . . . . . .
45
3.2.2.1
Comparison with experimental results . . . . . .
48
3.2.3
Eect of elastic deformation on computation of wear . . .
50
3.2.4
Comparison of the GIWM with Kauzlarich and Williams
3.2.2
wear model . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.2.5
Study of the eect of friction coecient using GIWM . . .
54
3.2.6
Remarks on the GIWM
57
. . . . . . . . . . . . . . . . . . .
4 Finite Element Based Wear Simulation Tool
59
4.1
Wear-Processor Methodology
. . . . . . . . . . . . . . . . . . . .
60
4.2
Computation of surface normal vector . . . . . . . . . . . . . . . .
62
4.3
4.2.1
Normal vector for two dimensional topography of the surface 62
4.2.2
Normal vector for three dimensional topography of the surface 63
Computation of linear wear . . . . . . . . . . . . . . . . . . . . .
65
4.3.1
Computation of linear wear on the disc surface . . . . . . .
66
4.4
Re-meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.5
Sliding distance incrementation scheme . . . . . . . . . . . . . . .
70
5 Wear Simulation
5.1
5.2
73
Ring-on-ring tribosystem . . . . . . . . . . . . . . . . . . . . . . .
74
5.1.1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.1.2
Advantages of re-meshing . . . . . . . . . . . . . . . . . .
77
Pin-on-disc tribometer . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2.1
Finite element model of the pin-on-disc tribometer
80
5.2.2
Necessity for three dimensional nite element model for
. . . .
wear simulation in the pin-on-disc tribometer . . . . . . .
83
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.3
Asymmetric wear due to elastic deformation . . . . . . . . . . . .
89
5.4
Remarks on the Wear-Processor . . . . . . . . . . . . . . . . . . .
91
5.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.2.3
6 Summary
95
List of Figures
97
x
List of Tables
101
Bibliography
103
xi
Chapter 1
Introduction
"There is plenty of room at the bottom."
Richard P. Feynman
(1918-1988)
1.1 Wear simulation strategy
Micro-machines are subject to several modes of failure, which include fracture,
fatigue, wear, etc. Out of these modes, excessive wear is one of the most critical
mode of failure that limits, a micro-machine's life span substantially due to their
high operating speeds and surface area to volume ratio. Moreover, wear is least
predictable partially due to the imperfect knowledge of the appropriate wear rate
for the selected material pair to be used in calculating component life (Schmitz
et al., 2004). In-situ wear measurements are the best way to predict wear in
micro-machined components. But the manufacturing of prototypes followed by
the in-situ wear measurements to estimate the performance of micro-machines
involves extremely high costs both in terms of time and money. Therefore, the
wear rate is normally determined experimentally using a tribometer.
2
1.1. Wear simulation strategy
Pin-on-disc tribometry is the most commonly used experimental technique to
study the tribological performance of a material pair of interest due to its simplicity and cost eectiveness. In such an experiment, a pin is loaded against
a rotating at disc. The weight loss from either the pin or the counterface is
measured. Wear is usually expressed as worn volume per unit load and per unit
sliding distance which is also known as the dimensional wear coecient. This is
sometimes non-dimensionalised by dividing by the hardness of the softer surface
and is termed as the wear coecient. Dimensional wear coecient is the most
important parameter determined from such a tribometer. Such experiments attempt to mimic the contact conditions of the tribosystem under study in terms
of contact pressure and sliding velocity. The specimens used in the experiments
need to be manufactured with the same production/manufacturing processes and
therefore can then be considered to have the same microstructure as the micromachine itself. In any case, these experiments allow for a qualitative study of the
suitability of a particular material combination for a given application as shown
on the left hand side of Figure 1.1. However such experiments are not sucient
for a prediction of the progress of wear and eventually the useful life span of a
micro-machine since the geometry is dierent and the loading is time dependent
as unlike in the experiment. Therefore to evaluate and compare dierent designs
with respect to wear, a simulation strategy in association with the nite element
method can ll the gap between in-situ wear measurements on one hand and tribometry on the other hand. However, to perform such simulations, a wear model
has to be applied on the local scale and the parameters included in it have to be
identied from experiments as shown on the right hand side of Figure 1.1.
So far, very less eort has been made to identify a wear model including the corresponding wear parameters from the relevant experiments that can be applied
on the local scale or to at least check the suitability of an existing model like
Archard's wear model on the local scale. Archard's model is a basic formula for
relating overall wear rates to the normal load, the hardness of the softer component and the wear coecient. The identication of a local wear model/parameters
from experiments requires numerical simulations, since only global quantities can
be readily identied from tribometry. Global refers to the fact that the geometry
of the tribometer is implicitly included in the identied wear model/parameters
because the decay in the location dependent contact pressure on the surface as
1. Introduction
3
wear progresses is not considered. While, the advantage of a local wear model
is that it will only consider the material properties and the tribological parameters obtained from the tribometer and can therefore be used to simulate wear in
a geometrically dierent tribosystem, such as a micro-machine, where time and
location dependent contact conditions can be encountered.
Figure 1.1: Strategy for wear prediction in micro-machines. The nite element
contact analysis (right hand side top picture) on the micro gear (left hand side
top picture) was done by Stammberger (2003).
It can be seen from literature that researchers have often used the global wear
model identied from experiments to simulate wear assuming that the global
model is applicable on the local scale. Podra & Andersson (1999) identied the
wear coecient from experiments and showed that the local application of Archard's wear model using the identied global parameter yielded good results
in comparison to the experiments. Further, they used this identied parameter
for Archard's wear model to simulate wear in cone-on-cone and cone-on-torus tribosystems. In their work, the identication of the local wear model was done with
an axisymmetric nite element model of the pin-on-disc tribometer. One issue
4
1.2. Overview of chapters
presented in this work is that the asymmetric eects due to the friction between
the interacting surfaces are present and may play a role in the model/parameter
identication process. Also, it will be shown in this work that the eect of tangential elastic deformation on the computation of wear for interaction between
compliant materials would be signicant, thus indicating that assumption of axisymmetric nite element model for the pin-on-disc tribometer is not sucient in
any case.
The strategy for simulation of wear in a tribosystem (e.g., a micro-machine) is
depicted in Figure 1.1. The two fold strategy is, namely: (i) to identify a local
wear model/parameter with the help of a wear simulation tool from a pin-on-disc
tribometer conducted within the parameter space of the micro-machine, (ii) then
in the future to use this local wear model to predict wear in the micro-machines
with the goal to estimate its life span. Such a wear model would be restricted for
simulation of wear on the particular tribosystem in whose parameter space the
experiments were conducted. This approach applies both to the macro- as well as
the micro-scale systems. However, of special interest to the micro-scale systems is
if the high contact pressures and sliding velocities encountered in micro-machines
would result in high wear rates in the relevant experiments, thus increasing the
numerical diculties of simulating large amounts of wear in short spans of time.
1.2 Overview of chapters
Chapter 2 starts with an introduction to the basics of contact mechanics and the
formulation of the contact problem using the nite element method followed by
an introduction to the topics of friction and wear. Later a comprehensive survey
of the literature on modeling and simulation of wear with a special emphasis to
its relevance to micro-machines is presented. A global incremental wear model
for pin and disc wear in a pin-on-disc tribometer is presented in Chapter 3.
Two dimensionless parameters for estimating the eect of elastic deformation
on wear are also presented in this chapter. In Chapter 4, the nite element
based wear simulation tool, Wear-Processor is described in detail. The results
from the wear simulation applied to a ring-on-ring tribosystem and a pin-on-disc
tribometer using the Wear-Processor is presented in Chapter 5. The discussion
1. Introduction
5
on wear simulation in the pin-on-disc tribometer corresponds to the experiments
conducted by Herz et al. (2004). To further study the eect of elastic deformation
on wear, simulation of a compliant pin on a stier disc is presented in this chapter.
Finally a summary of this work is provided in Chapter 6.
Chapter 2
Basic Principles and Literature
Review
"The most direct, and in a sense the most important, problem which our conscious
knowledge of nature should enable us to solve is the anticipation of future events."
Heinrich Hertz
(1857-1894)
In this chapter, the general concepts and denitions for the study of tribology
from the computational standpoint will be briey presented. The whole process
of simulating wear in any general tribosystem consists of two main tasks, namely,
solution of the contact problem and using this solution to compute wear. The
most widely used and simplest wear model is the Archard's wear model, which
relates linear wear to the contact pressure and the sliding distance. The normal
surface traction (contact pressure) used as input for the computation of wear
is one of the quantities calculated during the solution of the contact problem.
The other solved quantities include the tangential surface tractions, surface displacements and the contact area. The rst section of this chapter gives a brief
introduction to contact mechanics and the formulation of the contact problem
with the nite element method. Then, some basic concepts of friction and wear
8
2.1. Basics of contact mechanics
are introduced. Later a comprehensive survey of the literature on modeling and
simulation of wear with a special emphasis to its relevance to micro-machines is
presented.
2.1 Basics of contact mechanics
Contact phenomena are abundant in everyday life and play a very important
role in engineering structures and systems. They include friction, wear, adhesion
and lubrication, among other things; are inherently complex and time dependent;
take place on the outer surfaces of parts and components, and involve thermal,
physical and chemical processes. Contact Mechanics is the study of relative
motion, interactive forces and tribological behavior of two rigid or deformable
solid bodies which touch or rub on each other over parts of their boundaries
during lapses of time. However, the contact between deformable bodies is very
complicated and it is not yet well understood. This is mainly due to the fact that
it is very dicult to measure directly the evolution of surface quantities during
contact processes (Shillor, 2001). Barber & Ciavarella (2000) have made a brief
survey on the recent developments in the eld of contact mechanics.
The contact theory was originally developed by Hertz (1882) and it remains the
foundation for most contact problems encountered in engineering. It applies to
normal contact between two elastic solids that are smooth and can be described
locally with orthogonal radii of curvature such as a toroid. Further, the size of the
actual contact area must be small compared to the dimensions of each body and
to the radii of curvature (non-conforming contact). Hertz made the assumption
based on observations that the contact area is elliptical in shape for such three
dimensional bodies. The equations simplify when the contact area is circular such
as with spheres in contact. At extremely elliptical contact, the contact area is
assumed to have constant width over the length of contact such as between parallel cylinders. The Hertz theory is restricted to frictionless surfaces and perfectly
elastic solids. Later Johnson (1985) gave an elaborate treatment to mechanics
of contacts between non-conforming surfaces, contacts involving inelastic solids,
sliding contacts, rolling contacts and also contact between rough surfaces. Hertz
contact solutions circular contact area and rectangular contact area will be in-
2. Basic Principles and Literature Review
9
troduced in Chapter 3. The basic assumptions governing contact mechanics for
application to any arbitrary geometry or with materials that deviate signicantly
from the elastic properties will be discussed in the following, however, some of
these assumptions are included in the Hertz theory discussed above. With these
assumptions, the solution of contact problems can be more easily implemented
within a nite element program.
Geometrically two bodies are said to be in unilateral contact if they are contiguous, impenetrable but separable (Curnier, 1999). The interaction between
the contacting surfaces consists of two components (i) normal to the surface and
(ii) tangential to the surface. The normal interaction in a mechanical contact
between two surfaces is modeled by three conditions
• A geometric in-equality condition of impenetrability on the contact gap,
g≥0
(2.1)
• A static inequality condition of intensility on the contact pressure (contact
pressure has a negative sign as it is compressive in nature),
p≤0
(2.2)
• An energetic equality condition of complimentarity on the gap-contact pressure pair
g·p=0
(2.3)
All together, these three conditions (Equations (2.1), (2.2), (2.3)) form the HertzSignorini-Moreau law of unilateral normal contact. As seen above, unilateral
contacts are governed by a non-smooth multi-valued contact law relating the
normal pressure to the normal gap between the two bodies. Being non-smooth,
contact problems are inherently nonlinear. In the commercial nite element program, ABAQUS, used in this work, the contact constraint is applied when the
gap between the surfaces becomes zero. In the present work, a hard contact
formulation is employed where there is no limit on the magnitude of the contact
pressure that can be transmitted between the surfaces (see Figure 2.1 (a)).
When surfaces are in contact, they usually transmit shear as well as normal
forces across their interface. Thus, the analysis may need to take frictional forces,
10
2.1. Basics of contact mechanics
(a)
0
Contact clearance, g
(b)
Shear Stress
Slipping
tcrit
Contact pressure, p
Sticking
Slip
Figure 2.1: (a) Contact pressure-clearance relationship for "hard" contact; (b)
Frictional behavior at the contact.
which resist the relative sliding of the surfaces, into account. The laws of friction
were introduced by Amonton (1699) which were further developed by Coulomb
(1785). Leonardo da Vinci also had observed the above phenomena (see Meyer
et al. (2002) for more details). Coulomb friction (as it is commonly termed) is
a popular friction model used to describe the interaction of contacting surfaces,
which is also used in the present work.
The Coulomb friction model characterizes the frictional behavior between the
surfaces using a coecient of friction, µ. The tangential motion is zero until the
surface traction reaches a critical shear stress value, τcrit , which depends on the
normal contact pressure, according to the following equation (see Figure 2.1 (b)):
τcrit = µ · p.
(2.4)
2.1.1 Formulation of the contact problem using the Finite
Element Method
The simulation of wear to be discussed in this thesis involves the solution of
the contact problem using a commercial nite element code, ABAQUS. In this
subsection, the mathematical formulation for the solution of a general contact
2. Basic Principles and Literature Review
11
problem using the nite element method will be discussed.
In nite element analysis, the contact conditions are a special class of discontinuous constraint, allowing the forces to be transmitted from one part of the
model to another. The constraint is discontinuous because it is applied when
the two surfaces come in contact. When the two surfaces separate no constraint
is applied thus making the problem nonlinear. Contact problems range from
frictionless contact in small sliding to contact with friction in nite sliding and
involving interaction between rigid and deformable bodies or interaction between
two or more deformable bodies. If the absolute and hence the relative displacement undergone by the contacting bodies are small in comparison with the body
sizes, then the contact principles are simple to establish and most of the novelty
and the complexity comes from the tribological laws. When on the contrary the
two bodies undergo large relative motions, the general principles become more
complicated whereas the tribological laws remain virtually the same from the
computational standpoint. The formulation of contact conditions is the same in
all these cases, the solution of the nonlinear problems can in some analysis be
much more dicult than in other cases.
In the following, the solution of the contact problem using the nite element
method will be presented (Bathe (1996); for a summary also see Huber et al.
(1994)). The general form of the governing equilibrium equations is formulated.
Then the terms in the equilibrium equations are manipulated to include the
contact condition. The theoretical considerations in the following are:
1. Arbitrarily shaped body,
2. Small deformations,
3. Linear elastic material property,
4. Static problem,
5. External forces include volume forces, surface forces and concentrated forces
(which includes the contact forces).
Assuming a linear elastic continuum, the total potential is given by :
12
2.1. Basics of contact mechanics
Z
1
Π(u, λ) =
εT DεdV
2 V
|
{z
}
Strain Energy
Z
−
uT f V dV
| V {z
}
Work done by body forces
Z
T
− uS f S dS
| S {z
}
Work done by surface forces
X T
−
uC FC
|C {z }
Work done by concentrated forces
X T
T
λk (Qk uk − ∆k ),
+
(2.5)
k
|
{z
}
Work done by contact forces
(Lagrange multiplier method)
where, ε is the strain tensor, D is the stress-strain matrix for the linear elastic
material, u is the displacement vector, f V is the body force vector, f S is the vector
of surface forces, FC is the vector of concentrated forces, λk
T
is the vector of
k
contact forces for contact node k (see Subsection 2.1.2), Q is the contact matrix
for contact node k (see Subsection 2.1.2), ∆k is the penetration for contact node
k (see Subsection 2.1.2), V is the volume, S is the surface, ()C is the superscript
for concentrated forces and the respective displacements and ()k is the superscript
for contact node k .
The governing equilibrium equations can be formulated by invoking the stationarity of the total potential, Π (principle of minimum potential energy), i.e, δΠ = 0.
The contact constraints are imposed using the Lagrange Multiplier Method and
using that D is symmetric, we obtain:
2. Basic Principles and Literature Review
Z
δΠ =
13
Z
Z
T
T
T V
δε DεdV −
δu f dV − δuS f S dS
V
V
S
X
X
T
T
kT k
kT
CT C
[δλ (Q u − ∆k ) + λk Qk δuk ].
−
δu F +
C
(2.6)
k
To evaluate Equation (2.6), the displacements should satisfy the geometric (essential/Dirichlet/displacement) boundary conditions. Hence, any variations on
the displacements that satisfy the geometric boundary conditions and their corresponding variations in strains are considered. In nite element analysis, the
continuous body is approximated as an assemblage of discrete nite elements
with the elements being interconnected at nodal points on the element boundaries. Therefore the equilibrium equations that correspond to the nodal point
displacements of the assemblage of nite elements is written as a sum of integrations over the volume and areas of all nite elements; i.e.,
δΠ =
XZ
XZ
M
−
(M )
Dε
dV
(M )
−
V (M )
M
−
δε
(M )T
X
C
XZ
M
δuS
(M )T
S
T
δuC FC +
fS
(M )
X
δu
(M )T
fV
(M )
dV (M )
V (M )
dS (M )
T
T
T
T
[δλk (Qk uk − ∆k ) + λk Qk δuk ].
(2.7)
k
The displacements measured in a local coordinate system within each nite element are assumed to be a function of the displacements at the nite element
nodal points. Therefore for element M we have
u(M ) (x, y, z) = H(M ) (x, y, z)Û,
(2.8)
where H(M ) is the displacement interpolation matrix, Û is the vector of global
displacements at all nodal points. The variation of u(M ) is given by:
δu(M ) (x, y, z) = H(M ) (x, y, z)δ Û.
(2.9)
With the assumptions on displacements in Equation (2.8), the corresponding
14
2.1. Basics of contact mechanics
element strains are given by:
ε(M ) (x, y, z) = B(M ) (x, y, z)Û,
(2.10)
where B(M ) is the strain displacement matrix. The variation of ε(M ) is given by:
δε(M ) (x, y, z) = B(M ) (x, y, z)δ Û.
(2.11)
Substituting for the element displacements and strains we obtain:
δΠ =
XZ
V (M )
M
−
T
δ ÛT B(M ) D(M ) B(M ) ÛdV (M )
XZ
M
−
XZ
M
−
X
δ ÛT HV
(M )T
fV
(M )
dV (M )
V (M )
δ ÛT HS
(M )T
fS
(M )
dS (M )
S (M )
i
Xh
T
T
k
T
δ λ̂ (Q Û − ∆ ) + δ Û Qλ̂ .
δ Û F +
T
C
(2.12)
k(M )
C (M )
Applying the principle of the stationarity of the total potential we get:
δ ÛT
"
XZ
M
#
T
B(M ) D(M ) B(M ) dV (M ) Û
V
T
= ∂ Û
(M )
XZ
M
+∂ ÛT
T
HB(M ) f B(M ) dV (M )
V (M )
XZ
T
HS(M ) f S(M ) dS (M )
S (M )
M
−∂ ÛT
X
k(M )
−
X
Qλ̂ + ∂ ÛT
X
FC
(M )
C
h
i
δ λ̂T QT Û − ∆k ,
(2.13)
k(M )
where F is the vector of all the concentrated forces, λ̂ is the vector of all the
contact forces (see Subsection 2.1.2), Q is the complete contact matrix of the
assemblage (see Subsection 2.1.2), ∆ is the vector of all the penetrations (see
2. Basic Principles and Literature Review
15
Subsection 2.1.2). Since, for a linear elastic continuum, the principle of minimum
potential energy is identical to the principle of virtual displacement, the unknown
nodal point displacements can be obtained from Equation (2.13) by imposing unit
virtual displacements in turn at all displacement components. In this way we have
δ Û = I.
The equilibrium equations of the element assemblage corresponding to the nodal
point displacements is given by:
#
# "
#"
# "
"
Û
R
Rk
K Q
+
=
∆
QT 0
λ̂
0
where,
K=
XZ
(2.14)
T
B(M ) D(M ) B(M ) dV (M )
V (M )
M
R = RB Z+ RS
X
T
=
HB(M ) f B(M ) dV (M )
V (M )
M
+
XZ
M
T
HS(M ) f S(M ) dS (M )
S (M )
Rk = −Qλ̂T .
For nonlinear problems (e.g., contact problems), the system of equations is formulated in an appropriate way and is solved for equilibrium using the Newton-
Raphson Method as shown in the following:
"
t+∆t
K(i−1)
t+∆t (i−1)T
Q
t+∆t
Q(i−1)
0
#"
∆Û(i)
∆λ̂(i)
#
"
t+∆t
=
"
+
R
0
t+∆t
#
"
t+∆t
−
(i−1)
Rk
t+∆t (i−1)
∆
#
,
F(i−1)
0
#
(2.15)
where ∆Û(i) is the vector containing the nodal displacements in the ith iteration
(= Û(i) − Û(i−1) ), ∆λ̂(i) is the vector containing the nodal contact forces in the ith
16
2.1. Basics of contact mechanics
iteration (= λ̂(i) − λ̂(i−1) ) (see Subsection 2.1.2),
t+∆t
R is the force vector at time
t + ∆t,
is the vector of the contact force on the contact nodes after
th
the i − 1 iteration (see Subsection 2.1.2), t+∆t ∆(i−1) is the penetration of the
contact node after the i − 1th iteration (see Subsection 2.1.2), t+∆t K(i−1) is the
t+∆t
(i−1)
Rk
2.1.2) and
t+∆t
F(i−1)
t+∆t
T
Q(i−1) is the contact matrix (see Subsection
is the force from the stress distribution after the i − 1th
stiness matrix of the element,
iteration.
2.1.2 Formulation for the contact potential
The discussion in the following corresponds to two dimensional four node elements undergoing nite sliding in the contact zone. The contact forces can be
determined by applying the impenetrability condition in Equation (2.1). The
intensility condition in Equation (2.2) determines whether the contact is taking
place or not. When surfaces come in contact they usually transfer both shear
(tangential to the interface) and normal forces. The sticking and slipping condition of the contact is determined by the tangential component of the contact
force.
Figure 2.2 shows a contact zone between two bodies. The surface of body 1
(slave surface) is modeled with internal contact elements and the surface of body
2 (master surface) is dened by a slide line. The surface segments along the slide
line are obtained by linear interpolation between the surface nodes. Contact takes
place between the slave surface nodes and the master surface (slide line).
For a slave surface node, k to come into sticking state, there are two possibilities
1. At the end of i − 2th iteration, the node k is not in contact with the counter
body (body 2). After the i − 1th iteration, the slave surface may penetrate the counter body. And in the ith iteration the contact forces may be
adjusted so that the node k is placed on the slide line.
2. The node k may switch its state from slipping to sticking state.
In Figure 2.3, the above discussed rst possibility after i − 1th iteration is shown.
From the schematic representation, the following quantities can be determined:
2. Basic Principles and Literature Review
17
Contact zone
1
Slave surface
k-1
k+1
k
B
A
Slide line
(Master surface)
2
Figure 2.2: Contact zone between two bodies modeled with two dimensional, four
node nite elements.
Penetration of node k is given by:
(i−1)
∆k(i−1) = t+∆t xk
(i−1)
− t+∆t xC
(2.16)
.
Two mutually perpendicular unit vectors can be dened on each segment of the
slide line. nr and ns are the unit vectors tangential and normal to the slide line
segment respectively. The length of the surface segment, j :
(i−1)
dj
= nTr
h
t+∆t (i−1)
xB
(i−1)
− t+∆t xA
i
(2.17)
.
The determination of the parameter β depending on the position of the contact
point C on the surface segment, j :
t+∆t (i−1)
xC
(i−1)
= t+∆t xA
(i−1)
⇒ β (i−1) dj
(i−1)
+ β (i−1) dj
(i−1)
nr = t+∆t xC
nr
(i−1)
− t+∆t xA
.
Using Equation (2.16),
(i−1)
β (i−1) dj
h
nr =
t+∆t (i−1)
xk
− ∆k
(i−1)
(i−1)
− t+∆t xA
i
.
18
2.1. Basics of contact mechanics
C
A
Slave surface
k
B
Slide line
(Master surface)
t+
tX (i-1)
k
y
t+ tX (i-1)
A
t+ tX (i-1)
B
t+ tX (i-1)
C
j
i
x
A
nr
ns
C
i-1)
) (
(i-1 d j
b
k
1)
k(i-
(i-1)
B
dj
Figure 2.3: A schematic of a slave surface penetrating the master surface after
the i − 1th iteration.
(i−1)
Multiplying by nTr and dividing by the segment length dj
β (i−1) =
nTr ht+∆t
(i−1)
dj
and denoting nr =
β (i−1) = n̄Tr
h
nT
r
(i−1)
dj
(i−1)
xk
− ∆k
(i−1)
(i−1)
− t+∆t xA
, we get:
i
,
, we get:
t+∆t (i−1)
xk
− ∆k
(i−1)
(i−1)
− t+∆t xA
i
.
(2.18)
At the i − 1th iteration, for the contact forces at node k , the following condition
2. Basic Principles and Literature Review
19
should be valid;
t+∆t (i−1)
λk
(i−1)
(i−1)
= t+∆t λkx i + t+∆t λky
t+∆t (i−1)
λk
applied
t+∆t (i−1)
t+∆t (i−1)
λA
and
λB
(2.19)
j.
The contact force,
at node k of body 1 is associated with the
nodal forces
of body 2 and thus for equilibrium we have
(see Figure 2.4):
t+∆t (i−1)
λA
¡
¢
(i−1)
= − 1 − β (i−1) t+∆t λk
t+∆t (i−1)
λB
= −β (i−1)t+∆t λk
(i−1)
(2.20)
(2.21)
.
t+ tl
nr
t+ tl (i-1)
k
t+ tl (i-1)
k
t+ tl
(i-1)
A
A
ns
C
(i-1)
i-1)
) (
(i-1 d j
B
b
B
(i-1)
dj
Figure 2.4: Free body diagram of a contact surface segment showing the contact
forces.
(i)
(i)
(i)
In the ith iteration, the displacement increment ∆uk , ∆uA , ∆uB are such that
the penetration ∆k
(i−1)
is zero. Therefore the node k comes in contact with the
contact point C (sticking) and the parameter β (i−1) is a constant (β i = β (i−1) ).
The contact potential Wk is given by:
³
´
k(i−1)
(i)T
(i)
(i)T
(i)
(i)T
(i)
Wk = t+∆t λk
∆uk + ∆
+ t+∆t λA ∆uA + t+∆t λB ∆uB , (2.22)
20
2.1. Basics of contact mechanics
where
t+∆t (i)
λk
(i−1)
= t+∆t λk
(i)
(2.23)
+ ∆λk .
Substituting Equations (2.16), (2.20), (2.21) in Equation (2.22) we get:
Wk =
(i)T
+∆λk
h³
´
i
¡
¢
(i)
(i)
− 1 − β (i−1) ∆uA − β (i−1) ∆uB
h³
´ ¡
i
¢
(i−1)
(i)
(i)
(i)
∆uk + ∆k
− 1 − β (i−1) ∆uA − β (i−1) ∆uB .
T
t+∆t (i−1)
λk
(i)
∆uk + ∆k
(i−1)
(2.24)
The virtual work for sticking contact done by the contact elements is −
P
Wk ,
which will go into the functional (total potential), Π. Multiplying both the sides
of Equation (2.24) by −1 and reorganizing it we get:
´ h (i−1)T
i
³
(i−1)
(i−1)T
(i)T
Qk
∆ − ∆k
,
(2.25)
−Wk = t+∆t λk
+ ∆λk
where
Qk

(i−1)
and
−1
0


0
−1

 1 − β (i−1)
0

=

0
1 − β (i−1)

 β (i−1)
0

0
β (i−1)

(i)
∆ukx


 ∆uk(i) 
y 

 ∆u(i) 

Ax 
=
.

 ∆u(i)
A
y 

 ∆u(i) 

Bx 
(i)
∆uBy
k











(2.26a)

∆uk
(i)
(2.26b)
For sliding contact, the friction force within an iteration remains constant. Any
change in the contact force takes place only in the normal direction. The change
2. Basic Principles and Literature Review
21
(i)
in the contact force ∆λk in the ith iteration is given by:
(i)
(i)
(2.27)
∆λk = −∆λS nS ,
where the minus sign on the right hand side results from the direction of the local
(i)
(i)
unit normal vector nS . In addition to the displacements increment ∆uk , ∆uA ,
(i)
∆uB , there is a change in the parameter β i due to the sliding.
Substituting β (i−1) in Equations (2.20) and (2.21) for β i and then substituting
them in Equation (2.22), we get:
h
i
¡
¢
(i−1)
(i)T
(i)
(i)
(i)
Wk = t+∆t λk
∆uk + ∆kk
− 1 − β (i) ∆uA − β (i) ∆uB ,
(2.28)
where
β (i) = β (i−1) + ∆β (i) .
(2.29)
Equation (2.18) can be linearized as follows (Chaudhary & Bathe, 1985):
h
i
¡
¢
(i−1)
(i)
(i)
(i)
∆β (i) = n̄Tr ∆uk + ∆kk
− 1 − β (i−1) ∆uA − β (i−1) ∆uB
(2.30)
With the assumption that the change in β i , ∆β (i) is negligible and then substituting Equations (2.23) and (2.27) into Equation (2.28), the virtual work for
sliding contact is given by:
Wk =
i
¡
¢
(i)
(i)
− 1 − β (i−1) ∆uA − β (i−1) ∆uB
h³
´ ¡
i
¢
(i−1)
(i)
(i)
(i)
(i)
−∆λS nS ∆uk + ∆k
− 1 − β (i−1) ∆uA − β (i−1) ∆uB .
T
t+∆t (i−1)
λk
h³
(i)
∆uk + ∆k
(i−1)
´
(2.31)
Analogous to Equation (2.25), Equation (2.31) can be rearranged as follows:
³
´ h (i−1)T
i
(i)
(i−1)
(i−1)T
(i)T
−Wk = t+∆t λk
− ∆λS nS Qk
∆uk − ∆k
,
(2.32)
where Qk
(i−1)
and ∆uk
(i)
are as dened earlier (see Equations (2.26a) and (2.26b)).
The nite element formulation discussed above is used for carrying out the rst
of the two main tasks for simulating wear in any general tribosystem, i.e., the
solution of the contact problem. Its advantage is that any general geometry can
22
2.2. Introduction to tribology
be solved for contact pressure, which is the main result on which we base the computation of wear in this thesis. It will be elaborated later in Subsection 2.3.2 that
some researchers have in fact implemented wear models within nite element formulations while other researchers have preferred to use the nite element contact
results as input for the computation of wear within a post-processor. Before this,
a brief introduction to various aspects of tribology will be given in the following
section.
2.2 Introduction to tribology
Tribology is the study of adhesion, friction, wear and lubrication of surfaces in
relative motion. It remains as important today as it was in ancient times, arising
in the elds of engineering, physics, chemistry, geology and biology (Urbakh et al.,
2004). Tribology is crucial to modern machinery which uses sliding and rolling
surfaces and more so to micro-machines where the surface to volume ratio is very
high and are operating at very high sliding velocities. Study of tribology is also
motivated from the economics point of view - according to some estimates, the
amount of money saved per year by better tribological practices is around $ 16
billion in the U.S., ¿ 500 million in the UK and similar numbers in other advanced
economies. Of special importance to microsystems, $ 10 billion alone are wasted
at the head-medium interfaces in magnetic recording. This loss is mainly due
to frictional wear, frictional heat generation which causes the softening of the
contacting surfaces and the possible damage of the contacting surfaces. Therefore,
the accurate prediction of the evolution of frictional contact processes and their
control is of major economic importance (Bhushan, 1999).
2.2.1 Friction
The fact that friction exists between any two sliding surfaces in relative motion has been known to mankind from prehistoric times (Suh, 1986). Frictional
behavior is aected by the following factors:
1. Kinematics of the surfaces in contact
2. Basic Principles and Literature Review
23
2. Externally applied load and/or displacements
3. Environmental conditions such as temperature, humidity etc.
4. Surface topography
5. Material properties
This list of important factors that control friction indicates that the coecient
of friction (Equation (2.4)) is not a simple material property. The coecient
of friction under normal sliding conditions is independent of normal load and
sliding speed to a rst approximation. However, at very high values of normal
load and sliding speed, the friction coecient decreases and at any intermediate
range, later the friction coecient can reach a peak value. Such phenomenological
observations of the frictional behavior of all materials were explained in terms
of the adhesion model which assumes that, as the surface consists of asperities,
the interface is made up of asperity contacts where the real area of contact is
much lower than the nominal (apparent) area of contact as shown in Figure
2.5. According to the adhesion theory of friction, when a relative motion is
FN
Nominal area of contact
FT
Real area of contact,
AR = Sum of
individual asperity
contact area
Individual asperity
contact area
Figure 2.5: Schematic of a typical interface when two nominally at surfaces
come in contact.
imparted at the interface, by applying a tangential force, each pair of contacting
asperities weld together and shear to accommodate the relative motion. With
some mathematical manipulation of Equation (2.4), it can be shown that the
24
2.2. Introduction to tribology
coecient of friction, in case of metallic contacts,
FT
AR · τ
=
FN
FN
FN · τ
1
τ
=
=
≈ ,
H · FN
H
6
µ =
(2.33)
where µ is the friction coecient, FT is the frictional force, FN is the normal load,
AR is the real area of contact, τ is the shear strength of each junction which can
be assumed to be the shear strength of the weaker bulk material and H is the
hardness of the weaker material. The ratio of the shear strength to the hardness
in Equation (2.33) yields a value of 1/6, since for metals, the Mohr's circle for
yield in uniaxial tension gives a value for the shear strength as half of the yield
strength in tension and the hardness is nearly equal to 3 times the yield strength
in tension. The above predicted value of 1/6 is much smaller than the typical
values observed under steady state sliding conditions as shown in Table 2.1.
Table 2.1: Friction coecients for various metals mating with itself (Roberts,
2005).
Material
Gold
Copper
Chromium
Silver
Magnesium
Lead
Friction Coecient
2
0.7 − 1.4
0.4
1
0.5
1.5
This disparity is often attributed to work hardening at the asperity contact and
junction growth (Sarkar, 1980; Roberts, 2005). In order to improve the correlation between the experimental results and the adhesion theory, Rabinowicz
(1965) considered the surface energy of adhesion in the calculation of the contact
area. Kragelskii (1965) considered plowing and the adhesion contribution to the
frictional force. More recently Straelini (2001) has given a simplied approach
to the adhesive theory of friction encompassing the concept of work of adhesion
and the growth of the real area of contact. Suh & Sin (1981) have given an
elaborate explanation on the genesis of friction, especially its time dependent
2. Basic Principles and Literature Review
25
behavior considering the eects of wear debris, environmental conditions, surface topography, interface temperature, asperity interaction, history of sliding
etc. Developing equations for the friction coecient based on similar ideas for
multi-asperity contacts has picked up in the recent past (Pollock & Chowdhary,
1982; Zhang et al., 1991a,b).
With the advances in computational methods and the possibility to implement
frictional laws within a nite element program for the solution of contact problems, several eorts have been made to develop incremental general constitutive
friction laws. Such a law based on the ow-theory in plasticity for large deformation analysis between a deformable and a rigid body (Cheng & Kikuchi, 1985)
and between elastically and plastically deformable bodies have been developed
(Gläser, 1992). These laws include the Coulomb friction as a special case. These
strategies use the unilateral contact constraints for the contact pressure - gap relationship (see Equations (2.1), (2.2), (2.3)). Oden & Pires (1983) have proposed
a nonlocal and nonlinear friction law. It assumes that the impending motion at a
point of contact between two deformable bodies will occur when the shear stress
at that point reaches a value proportional to a weighted measure of the normal
stresses in a neighborhood of the point. Karpenko & Akay (2001) have given a
computational method to calculate the friction force between three dimensional
rough surfaces. This model assumes that friction is due to elastic deformation
and the shear resistance of adhesive junctions at the contact areas.
Most available models for friction, use a very limited number of parameters (Heilmann & Rigney, 1981). With the exception of a yield stress or hardness parameter, basic material parameters tend to be neglected. Tribologists do not yet agree
on which properties of sliding materials and which additional physical parameters
are essential and should be included in a model for friction.
2.2.2 Wear
In the German industrial standard, DIN 50320, wear is dened as the progressive
loss of material from the surface of the solid body due to mechanical action,
i.e., the contact and relative motion against a solid liquid or gaseous counter
body (Zum-Gahr, 1987). Wear of materials occurs by many dierent mechanisms
26
2.2. Introduction to tribology
and is aected by various parameters. They include material parameters, the
environmental and operating conditions, and the geometry of the wearing bodies.
Suh (1986) classies wear mechanisms into two groups, those dominated by mechanical behavior of solids and those dominated primarily by the chemical behavior of materials. The mechanical behavior dominated wear mechanism is further
classied as:
1. Sliding wear: This mechanism occurs when two materials slide against each
other. It is characterized by plastic deformation, crack nucleation and propagation in the subsurface. Such a mechanism is commonly observed in
journal bearings, gears, cams, sliders, often in micro-machines due to design restrictions.
2. Fretting wear: This mechanism is encountered when interfaces undergo
small oscillatory motion. The early stages of fretting wear are the same
as sliding wear but depend on relative amplitude. The entrapped wear
particles can have signicant eect on wear. The relative displacement
amplitude is important in this case. e.g., In press t parts with a small
play.
3. Abrasive wear: This mechanism comes into play when hard particles or
hard surface asperities plow and cut the wearing surface in relative motion.
e.g., Earth moving equipment.
4. Erosive wear: It occurs when solid particles impinge on the wearing surface.
This mechanism is characterized by large sub-surface deformation, crack
nucleation and propagation. Sometimes the surface is cut by solid particles
impinging at shallow angles. e.g., Turbines, helicopter blades etc.
5. Fatigue wear: This mechanism of wear is dominant in cyclic loading conditions. It is characterized by fatigue crack propagation usually perpendicular
to the surface without gross plastic deformation. e.g., Ball bearings, roller
bearings etc.
The chemical behavior dominated wear mechanism is classied as
1. Solution wear: Solution is formed between the materials in contact to de-
2. Basic Principles and Literature Review
27
crease the free energy. This mechanism is characterized by formation of
new compounds, atomic-level wear process and high temperatures. e.g.,
Carbide tools in high speed cutting.
2. Diusive wear: It occurs when there is diusion of elements across the
interface. e.g., High speed steel tools.
3. Oxidative wear: At increasing sliding speeds and low loads, thin, patchy
and brittle oxide lms are established. At much higher velocities, the oxide
lm becomes thicker and continuous, covering the entire surface. Under
these conditions the frictional heating is considerable, the metal is partly
insulated by the oxide lm, but the oxide itself is considerably hot so that
it can ow plastically or even melt. e.g., Sliding surfaces in oxidative environments.
4. Corrosive wear: This mechanism is characterized by corrosive grain boundaries and formation of pits. e.g., Sliding surfaces in corrosive atmosphere.
In many wear situations, there are many mechanisms operating simultaneously,
however, there will usually be one primary rate determining mechanism which
must be identied to deal with the wear problem.
The correlation between the friction coecient and the wear rate is not straightforward, the interrelationship can change with time (Blau, 2001). Consider the
case of dry sliding between metallic contacts, as the sliding speed increases, the
wear rate increases initially to a maximum and then decreases before peaking
again at high sliding speeds, a further fall is seen after the second peak. The
initial rise in wear rate is probably due to the high wear rate observed till the
surfaces conform to each other, then the wear rate will decrease steadily as the
sliding speed increases, this probably is due to the rise in temperature associated
with speed, which in turn favors the formation of an oxidation layer over the surface that inhibits wear. As the sliding speed increases further, the temperature
increases further and thus the material starts to soften increasing the wear rate.
Whereas, the coecient of friction decreases as the sliding speed increases, since
the materials get stronger at higher strain rates associated with higher speeds
and probably also due to the fact that the oxide layer acts as a lubricant. But as
the sliding speed increases, the interface temperature increases thus the material
28
2.3. Wear modeling and simulation
softens and the oxide layer begins to fail and fresh metal surfaces come in contact,
now the coecient of friction decreases due to melt lubrication even though the
wear rate increases as more metal melts at high sliding velocities.
In Subsection 2.3.2, the most widely used and simplest wear model, the Archard's
wear model will be introduced along with its improvement to include the eect of
friction. Several other attempts to improve existing wear models by including the
eect of friction in order to more accurately with correlate experimental observations have been made (Liu & Li, 2001; Huq & Celis, 2002). In the next section,
a comprehensive review of the available literature on modeling and simulation of
wear is presented.
2.3 Wear modeling and simulation
Wear modeling has been a subject of extensive research over the past in order
to derive predictive governing equations. The modeling of wear found in the literature (Meng, 1994; Meng & Ludema, 1995; Hsu et al., 1997; Blau, 1997) can
be broadly divided into two main approaches, namely, (i) mechanistic models,
which are based on material failure mechanism and (ii) phenomenological models, which often involve quantities that have to be computed using principles of
contact mechanics.
2.3.1 Mechanistic wear modeling
Peigney (2004) presented a mechanistic model, where the steady state wear in
fretting is estimated by minimizing the energy dissipated in the wear process,
however, this method is approximative in essence and wear on only one of the
interacting surfaces could be computed. This model interprets wear as phase
transformation taking place on the interacting surface where wear particles are
created from the parent material and are eliminated as soon as they are created.
Williams (2004) has discussed two important mechanistic wear models for cases
with repeated loading namely, (i) under certain loading condition there will be
plastic deformation in the early stages of the repeated contact, and then, the
2. Basic Principles and Literature Review
29
load will be supported entirely elastically, which is referred to as shakedown
(Johnson, 1992; Kapoor et al., 1994) and (ii) under severe loading conditions,
an increment of plastic deformation occurs from each successive load cycle and
the failure (wear) occurs when the accumulated strain exceeds a critical value,
which is referred to as ratchetting (Kapoor & Johnson, 1994; Kapoor, 1997).
Franklin et al. (2001) applied the ratchetting theory for wear in a computer
simulation scheme for ductile anisotropic materials, where they considered a two
dimensional layer model and a two dimensional brick model of the sample. The
layer model allowed for the normal variation of the material property and assumed
that the failed sub-surface layers (which usually occurs at low friction coecient)
had to wait till the surface layers were removed. The brick model addressed
this shortcoming by allowing for the lateral variation of the material property
as well. It allowed for weak bricks to be created, which could be regarded as
potential sites for crack initiation, but articial roughening of the worn surface
occurred. Stalin-Muller & Dang (1997) applied the ratchetting theory based
mechanistic wear models to simulate wear in a pin-on-disc tribometer where the
Hertz (1882) solution was modied to approximate the contact parameters for
the wearing process, thus making the simulation scheme specic to the particular
case of a spherical tipped pin on a at disc. Christodes et al. (2002) made a
qualitative prediction of the wear of coated samples in a pin-on-disc tribometer
by implementing the ratchetting failure based wear prediction model within a
two dimensional nite element based computational scheme. Yan et al. (2002)
proposed a computational approach for simulating wear on coatings in a pin-ondisc tribometer, which showed good agreement with experimental results. Their
approach employed the ratchetting based failure criterion to predict wear in a
two dimensional periodic unit cell model of the disc where the applied normal
tractions and the scaling parameters for the unit cell were extracted from a three
dimensional non-sliding deformable-rigid nite element contact analysis. The
initial contact pressure distributions were used throughout the wear simulation,
which meant that the contact geometry did not evolve due to wear. Such an
assumption could be made in this case since the aim was to predict wear in
coated discs where the amount of wear is small and so there is no appreciable
change in the geometry. Also for the same reason, wear on only the disc surface
was computed. The asymmetric eects coming from the friction between the
30
2.3. Wear modeling and simulation
contacting surfaces were not considered in the three dimensional nite element
contact analysis, but were added in the two dimensional periodic cell model.
The ratchetting based model relies on the fact that the wear on the surface
of a component depends on the existence and development of the sub-surface
plastic zones in the contacts, but Rigney (1997) raised the issue of inconclusive
evidence for the origins of sub-surface cracks responsible for generating wear
debris. Additionally due to the complex structure of the engineering surfaces
(see Persson (1998), ch. 4), tribochemical reactions at the surface (Krause, 1971),
presence of contaminants and wear debris, formation of nanocrystalline layers on
metallic surfaces during wear (Shakhvorostov et al., 2005) etc., make it unrealistic
to expect that a single wear equation can cover all cases and be truly deterministic
in nature.
2.3.2 Phenomenological wear modeling
Phenomenological models oer promising hope to address the shortcomings of
mechanistic models discussed earlier. Archard (1953) proposed a phenomenological model to describe sliding wear. It assumes that the critical parameters in
sliding wear are the stress eld in the contact and the relative sliding distance
between the contacting surfaces. The classical form of this wear model is:
V
s
FN
H
= kD · FN ,
= k·
(2.34)
where V stands for the volume of material removed, s is the sliding distance,
FN is the applied normal load, H is the hardness, k is the dimensionless wear
coecient and kD is the dimensional wear coecient. In Equation (2.34), the
hardness (ratio of load over projected area) is that of the softer material. By
dividing both the left hand side and the right hand side of Archard's wear law
in Equation (2.34) by the real area of contact (see Figure 2.5), one can get the
2. Basic Principles and Literature Review
31
relation for linear wear,
h
= kD · p,
s
(2.35)
where h is the linear wear. This equation is quite important as the amount of wear
is a more important measure compared to the wear volume. Further, it is quite
dicult to accurately measure the wear volume since the boundaries of the wear
scar are established subjectively (Kalin & Vizintin, 2000). The wear coecient
has been interpreted in various ways; as the probability that an asperity contact
generates a wear particle, as the fraction of asperities yielding wear particles,
as the ratio of the volume worn to the volume deformed, as a factor inversely
proportional to a critical number of wear cycles, and as a factor reecting the
ineciencies associated with the various processes involved in generating wear
particles (Rigney, 1994). Hardness is the only material property used in the
model explicitly and the eects of all other material and tribological properties
are assumed to be included in the wear coecient. Even though the wear model
was originally intended for adhesive wear, it has been applied to a wide variety
of wear conditions e.g., (i) Kapoor & Franklin (2000) applied the ratchetting
failure based approach to simulate delamination wear proposed by Suh (1973,
1977) where Archard's dimensionless wear coecient was determined based on the
material removal rate. (ii) Quinn (1971) used Archard's wear model for deriving
a wear equation for mild oxidational wear. The above two varied examples show
the popularity and importance of Archard's model in wear modeling.
As discussed earlier, the correlation between the coecient of friction and the
wear rate is complicated. However, Sarkar (1980) has given a wear model that
relates the friction coecient and the volume of material removed. This model
is an extension of the Archard's wear model and is given as:
FN p
V
=k·
1 + 3µ2 ,
s
H
(2.36)
where µ is the friction coecient and all other nomenclature remain the same as
before. If we assume all the variables as constants except µ and V , then there
is always some wear even in the absence of friction. This is contradictory since
zero friction would mean no physical contact and so there should be no wear.
However, it will be shown in this thesis that this model can be used to t pinon-disc experimental results for silicon nitride and also in prediction for higher
32
2.3. Wear modeling and simulation
loads.
With the advent of modern high performance computers, considerable computational eorts have been made towards wear simulation, especially using Archard's
phenomenological wear model. Strömberg (1999) presented a nite element formulation for thermoelastic wear based on Signorini contact (see Equations (2.1),
(2.2), (2.3)) and Archard's wear model. de Saracibar & Chiumenti (1999) presented a numerical model for simulating the frictional wear behavior within a
fully nonlinear kinematical setting, including large slip and nite deformations.
This model was implemented into a nite element program, where the wear was
computed using Archard's wear model. Molinari et al. (2001) implemented a
modication of Archard's wear model where the hardness of the softer material
is allowed to be a function of temperature; other features like surface evolution
due to wear, nite deformation thermo-plasticity and frictional contact were also
included. Due to the computational expense, only a simple contact problem of a
block sliding/oscillating over a disc was simulated.
As a faster and ecient approach, post-processing of the nite element contact
results with a suitable wear model to compute the progress of wear for a given
time interval/sliding distance has started to gain in popularity. The wear simulations by Podra (1997), Podra & Andersson (1999), Öquist (2001), Ko et al.
(2002), McColl et al. (2004), Ding et al. (2004), Gonzalez et al. (2005), Kónya
et al. (2005) are based on Archard's wear model and are implemented as a nite
element post-processor. The assumptions in the above works are, (i) simplication to two dimensions, (ii) lack of a viable re-meshing technique which limits
the maximum wear by the surface element height and (iii) either determining the
wear on only one of the interacting surfaces as the nite element contact results
are available only on one of the surfaces or using the computationally expensive
method of switching the role of the contact pairs (symmetric contact) to get the
nite element contact results for all the contacting surfaces (ABAQUS, 2004).
Sui et al. (1999) and Homann et al. (2005) implemented a re-meshing scheme
for geometry update. Kim et al. (2005) has also included a three dimensional
nite element model and a re-meshing scheme for simulating wear on a block on
a rotating ring tribosystem and have also shown that their results compare favorably with experimental results. The computational costs in such nite element
2. Basic Principles and Literature Review
33
based approaches are mainly from to the computation of the contact stresses,
which requires the solution of a nonlinear boundary value problem. Therefore
to reduce the computational eort, Podra & Andersson (1997), Jiang & Arnell
(1998), Dickrell & Sawyer (2004), and Sawyer (2004) make use of elastic foundation method (see pages 104 - 106 of Johnson (1985)) for the computation of the
contact pressure and are also based on Archard's wear model. The elastic foundation method for contact pressure computation does not consider the eects of
shear deformation in the contact, which, however, may be considerable for higher
values of the friction coecient and compliant materials. Therefore, any alternate method to solve the contact problem, like the elastic foundation method will
have to take this eect into account for satisfactorily describing the evolution of
the worn surface. Liu et al. (1999) have developed a numerical technique based
on the variational approach for minimizing the contact energy to compute the
contact stress distribution in three dimensional contact models of computer generated real surfaces which eliminated the additional iteration for determining the
contact area and also studied the role of friction and stress distribution in the
wear process.
In order to provide guidance and a sound theoretical framework for the modeling
eort, wear maps for specic materials (Steel: Lim & Ashby (1987), Ceramics:
Hsu & Shen (1996), Magnesium alloy: Chen & Alpas (2000)), that establish the
wear mechanisms for particular operating conditions have been developed. Hsu &
Shen (2004), based on their earlier work on ceramics wear maps, have developed
phenomenological wear models capable of predicting wear of ceramics within one
order of magnitude using material property and operating parameters. Cantizano
et al. (2002) developed and implemented a user-dened contact and wear nite
element that activated the appropriate predominant wear mechanism from the
wear maps of steel on steel depending on the sliding velocity and the normal load.
Apart from the phenomenological models, which have a physical background, a
class of wear models called the empirical models can be found in the literature. In
empirical models, the wear resistance of the mechanical components is considered
to be a property that can be calculated by empirical equations. Such equations
are constructed by tting experimental data using power law formulas (Kumar
et al., 2002). Therefore they are usually valid only within the experimental range.
34
2.4. Wear in micro-machines
Since wear is a complex function of various material and operating parameters,
often in order to generalize the results, dimensional analysis is performed (e.g, see
Vishwanath & Bellow (1995), Rajesh & Bijwe (2005)) where the main advantage
is that with a few dimensionless parameters it is possible to qualitatively predict
the mathematical relationship between the physical variables and the parameters
of that particular system. Empirical models by far are the easiest, fastest and
cheapest way to obtain the wear volume in a tribological test.
2.4 Wear as a reliability issue in micro-machines
Microsystems and micro-machines in particular are a rapidly emerging technology, nding a wide variety of applications. Tribology is expected to play a strong
role in enabling microsystems technology because surface forces dominate body
forces (Tichy & Meyer, 2000). As the size of a structure is reduced, surface phenomena (van der Waals, electrostatic, capillary forces) dominate the behavior and
interaction of structures compared to volumetric eects (inertia and gravitational
forces). These minute scales create unique problems. Several primary factors in
determining micro-machine's performance and reliability, concern characteristics
of adhesion, friction, wear, lm stress, fracture strength and fatigue (Romig et al.,
2003).
Wear has been recognized as a life-limiting failure mechanism for micro-machines
since Gabriel et al. (1990) observed wear in a silicon surface micro-machined gear
spun on a hub at high speed using an air jet. Various other investigations on
micro-machines (Bhushan, 1999; Maboudian et al., 2002) show that the tribological behavior plays a key role in the performance of micro-machines. Williams
(2001) has shown that 1 minute of life for a micro-machine represents a degree
of wear and degradation equal to more than 10 years for a well designed watch
bearing. Wear being a surface phenomenon is identied as a critical factor, which
can limit the life span of such micro-machines due to their high surface to volume ratio and high operating speeds. A majority of research on modeling and
simulation of wear is mainly for applications in the macro world. Comparatively
less eort has been made to study wear in micro-machines due to the obvious
diculties at the micro dimensions. At present, in-situ wear measurements are
2. Basic Principles and Literature Review
35
the most realistic methods to predict wear in micro-components. Tanner et al.
(2000) made a detailed in-situ study of the tribological performance on polysilicon micro-engines. Gee & Buttereld (1993) and Patton & Zabinski (2002)
have observed that environmental eects like humidity play a major role in the
tribological performance of micro-machines. Moreover, the manufacture of prototypes is highly expensive both in terms of time and money for such in-situ
studies. Therefore experimental techniques like pin-on-disc test, scratch test,
atomic force microscopy etc. are used to characterize the tribological properties
of various materials used in microsystems technology.
Further, the choice of materials for fabricating micro-devices is mostly restricted
to the materials used in semiconductor technology because of their well developed
processing methods. If a micro-machine is to generate useful mechanical work
output, low wear materials have to be used for bearing surfaces, with a value
of kD of the order of 10−8 mm3 /N · m . The tribological designs need to take
into account the fact that surfaces will wear and geometries change and should
not be based on the assumption that elastic or Hertzian loading conditions will
be maintained. Recently, new fabrication methods (see the volumes edited by
Löhe & Hauÿelt (2005)) for microsystem technology have been developed for employing various wear resistant materials like ceramics. Such technologies have
increased the size of the material palette, thus giving larger room for microsystem designers. However, the ability to predict wear and life-span is still essential
for the development of reliable micro-machines. Pfu & Brocks (2001) and Huber & Aktaa (2003) (also see Huber & Aktaa (2001)) discussed the design and
production of a micro-pump where one of the outcomes was that there exists no
adequate predictive method for wear in micro-machines with continuously evolving contact conditions. Therefore, a simulation tool is essential to close the gap
between in-situ wear measurements, standard tribological experiments and the
actual operation of a micro-machine.
Due to design restrictions in micro-machines, sliding wear is the most commonly
encountered type of wear. Moreover, at these small dimensions, there is little
scope for improving the surface quality and adjusting the tolerances for favorable
tribological performance. Therefore the achievable tolerances between the mating
surfaces with the available fabrication technique can be one of the prime factors in
36
2.5. Objectives of this work
deciding the tribological performance of micro-machines. Sliding wear is mechanistically more complex than certain other forms of wear because it not only
involves the cutting and plowing included in abrasive wear but also the adhesion
of asperities, third bodies, sub-surface crack initiation and growth, the transfer
of material to and from the mating surfaces, subtle changes in surface roughness
during running in, tribochemical lm formation and many other processes (Blau,
1997). However, for simulating wear in microsystems, the Archard's wear law
is the most popular model as discussed by Williams (2001), where it is used to
predict wear in rotating pivots for moving micro mechanical assemblies. Zhao
& Chang (2002) have developed a micro-contact and wear model for predicting
the material removal rate from silicon wafer surfaces during chemical-mechanical
polishing, where the developed equation is a representation of the Archard's wear
law. Sawyer (2004) used a simulation scheme based on the Archard's wear model
for the surface shape and contact pressure evolution during copper chemicalmechanical polishing. At a more fundamental level, Bhushan et al. (1995) studied
the atomic processes occurring at the interface of two materials when brought in
contact, separated or moved with respect to one another with scanning probe
microscopy and molecular dynamics simulations. Such methods will greatly aid
in the fundamental understanding of the interactions at an interface.
2.5 Objectives of this work
The main objective of this work is to develop a methodology for simulating wear
and predict the life-span of micro-machines made from ceramics. Towards achieving this goal, simulation of the pin-on-disc experiment, conducted within the
parameter space of the micro-machine is necessary. With such a simulation, it
would be possible to establish the relevant parameters in the Archard's wear
model. However, the wear model has to be applied on the local scale and if the
simulation results compare favorably with the experimental results then the wear
model can be applied for simulating wear in micro-machines in the future. A
nite element based post processing approach is best suited for this task.
In order to achieve this objective, in this thesis, a nite element based wear
simulation tool will be developed that can simulate wear on both two dimensional
2. Basic Principles and Literature Review
37
and three-dimensional nite element models. It will be possible to simulate wear
on both the interacting surfaces because a method will be implemented in the
Wear-Processor to compute the contact pressure on both the surfaces from the
stress tensor received as a nite element result and the computed surface normal
vectors, thus avoiding the necessity to use computationally expensive symmetric
contact. A re-meshing strategy will be implemented in the wear simulation tool
for not being restricted by the surface element height to simulate a pre-determined
sliding distance, especially considering the long term goal of simulating wear in
micro-machines, which can involve large number of operating cycles.
Chapter 3
Global Wear Modeling
". . . it doesn't matter how beautiful your theory is, it doesn't matter how smart you
are - if it doesn't agree with experiment, it's wrong."
Richard P. Feynman
(1918-1988)
In this chapter, a brief description on pin-on-disc tribometry will be given. Then
a global incremental wear model for pin and disc wear in a pin-on-disc tribometer
will be presented. The t and prediction with this modeling scheme will be discussed in detail along with the scheme's salient features. Later, two dimensionless
parameters will be derived to study the signicance of elastic deformation on the
computation of wear. Finally, a study on the eect of friction on the computation
of linear wear with an improved Archard's wear model implemented within the
global incremental wear model for pin wear will be presented.
40
3.1. Tribometry
3.1 Tribometry
Unidirectional sliding tests were conducted within the project D3 of the collaborative research center, SFB 499 by the Institute of Materials Science and
Engineering II, University of Karlsruhe. A micro pin-on-disc tribometer with a
spherical tipped silicon nitride pin and a disc of the same material was used for
the tests (see left hand side bottom picture in Figure 1.1). The tests were carried
out over a sliding distance of 500 m in ambient air and in water at normal loads
of 200, 400 and 800 mN and a sliding speed of 400 mm/s (Herz et al. (2004);
Schneider et al. (2005)). The disc specimens had the dimensions of ® 8 × 1 mm2
and the polished pin specimens had a diameter of 1.588 mm. The specimens were
purchased from Saphirwerk Industrieprodukte. The disc surface was ground to
an average surface roughness values of Ra = 0.11 µm. The polished pin specimens had a surface roughness values of Ra = 0.07 µm. The parameters used in
the experiments were supplied by the Institute of Product Development, University of Karlsruhe on the basis of a system analysis of the micro-turbine and the
-planetary gear train (Albers & Marz, 2005a,b) used as demonstrator for SFB
499 (see left hand side top picture in Figure 1.1).
Normal force, FN and the friction force FT were continuously measured with
the help of strain gages during the tests. The sum of linear wear on both the
pin and disc was also continuously measured capacitively with a resolution of
±1 µm. After the tests, the total wear volume was calculated by measuring
the contact diameter on the worn ball specimens using an optical microscope as
well as optical and tactile surface prolometry of the disc specimens. The wear
coecient (see Equation (2.34)) is calculated by dividing the calculated worn
volume (from the measured wear scar) with the product of the applied normal
load and the maximum sliding distance. The wear coecient identied from
such experiments implicitly includes the geometry specic to the tribometer and
also the decay of the contact pressure as the wear progresses is not considered.
Thus, the wear coecient identied from tribometry is global in nature. Apart
from the studies of the changes in the microstructure of the worn surface, results
from pin-on-disc tribometry are typically summarized with the help of a bunch
of plots like the linear wear and coecient of friction over sliding distance and
normal load, plot of wear coecient over coecient of friction for various material
3. Global Wear Modeling
41
combinations.
In the following section, a global incremental wear modeling scheme to t and
predict experimental data like the one above discussed will be presented.
3.2 Global Incremental Wear Models (GIWM)
The term global is used to indicate that this wear modeling scheme considers
only the global quantities, such as the average contact pressure and not the
location specic quantities e.g., local contact pressure. However, the average
contact pressure is updated at the end of each sliding distance increment due to
the resulting increase in the contact area, thus the term incremental is used. In
the next two sub-sections a global wear modeling scheme adopted specically for
computing wear on pin and for the case of wear on disc will be presented.
3.2.1 GIWM for computing wear on pin
In this subsection, the GIWM implemented for the case of a comparatively softer
spherical tipped pin sliding over a harder at disc will be presented. In such
a situation it can be assumed that most of the wear occurs on the pin, while
negligible wear occurs on the disc. The GIWM for computing wear on pin is
based on the idea of successively computing the contact radius and thus the
contact area due to the attening of the spherical tipped pin.
The ow chart of this scheme is shown in Figure 3.1 (a), where p is the contact
pressure, FN is the applied normal load, a is the contact radius due to elastic
displacement and wear, h is the total displacement at the pin tip, RP is the radius
of the pin as sketched in Figure 3.1 (b), he is the elastic displacement, hw is the
current wear depth, kD is the dimensional wear coecient, ∆s is the interval
of the sliding distance, smax is the maximum sliding distance, i is the current
wear increment number and EC is the elastic modulus of the equivalent surface
calculated using the following equation (see Johnson (1985), page 92):
2
1 − νP2
1 − νD
1
=
+
,
EC
EP
ED
(3.1)
42
3.2. Global Incremental Wear Models
(a)
(b)
w
h0 = 0, s0 = 0
a0 = 3
3 FN R p
PIN
Rp
4 Ec
FN
h0 =
2 E c a0
e
si +1 = si + Ds
FN
2
pai
hiw+1 = hiw + kD pi Ds
e
hi +1 = hi + hi +1
i = i+1
w
ai +1 = 2 Rp hi +1 - hi2+1
hi +1
e
ai +1 = 2 Rp hi +1 - hi2+1
FN
=
2 Ec ai +1
si + 1 < smax
hi+1
pi =
YES
NO
END
Figure 3.1: (a) Flow chart for the global incremental wear model (GIWM) for
computing pin wear; (b) computation of the contact radius.
3. Global Wear Modeling
43
where EP and ED are the Young's modulus of the pin and disc materials respectively and the Poisson's ratio of the pin and the disc materials is represented by
νP and νD respectively.
The global wear modeling scheme begins with the computation of the initial
contact radius, a0 using the Hertz (1882) solution for circular contact area,
r
3 · F N · RP
(3.2)
a0 = 3
4 · EC
and the elastic deformation normal to the contact using the Oliver & Pharr (1992)
relation given below:
hei+1 =
FN
.
2 · EC · ai+1
(3.3)
Then the following quantities are calculated for each increment of sliding distance
as shown in Figure 3.1 (a) till the maximum sliding distance is reached:
• Average contact pressure based on the applied normal load and the current
contact radius using,
pi =
FN
.
π · a2i
(3.4)
• Integral of the linear wear increment (Equation (2.35)) is computed using
the Euler explicit scheme as given below :
w
hw
i+1 = kD · pi · ∆si + hi .
(3.5)
• The current contact radius is calculated (see Figure 3.1 (b)) based on the
sum of the linear wear and the elastic deformation normal to the contact
as given below:
e
hi+1 = hw
i+1 + hi+1 ,
q
ai+1 = 2 · RP · hi+1 − h2i+1 .
(3.6a)
(3.6b)
As seen from Equation (3.4), the average contact pressure is used in the computation of wear in Equation (3.5). Alternatively, the Hertzian maximum pressure can
also be used in the wear modeling scheme above. The maximum pressure is then
computed as 1.5 times the average pressure as it is the case for the initial Hertzian
contact. However, such a computation of the maximum pressure may only be
44
3.2. Global Incremental Wear Models
applicable in the initial stage of sliding as long as the contact remains Hertzian.
It will be shown in Section 5.2 that for large sliding distances the pressure in the
contact will approach a attened distribution. Thus, the global incremental wear
model considering the average pressure gives a better approximation for larger
amounts of wear in the pin.
3.2.1.1 Comparison with experimental results
The GIWM using averaged contact pressure was used to t the results of the
pin-on-disc experiment on silicon nitride with 200 mN normal load, discussed
in Section 3.1 ( also see Herz et al. (2004); Schneider et al. (2005)). In this
work, kD was identied to be 13.5 × 10−9 mm3 /N · mm. For the following
FN
FN
FN
50
h
w
[µm]
40
Expt. (
= 200mN)
Expt. (
= 400mN)
Expt. (
= 800mN)
GIWM
30
20
10
0
kD
3
=13.5E-9 mm /Nmm
0
100
200
s
300
400
500
[m]
Figure 3.2: Results from the GIWM in comparison with the experimental results
(Herz et al., 2004) from the pin-on-disc tribometer at two dierent normal loads
(200 mN and 400 mN )
discussion, the material properties for silicon nitride were chosen from Callister
(1994), page 768, with Young's Modulus, EP = ED = 304 GP a and Poisson's
Ratio, νP = νD = 0.24. Later, the identied wear coecient was used to predict
3. Global Wear Modeling
45
the 400 mN experiment. It can be seen from the graph of the t for 200 mN and
prediction for 400 mN in Figure 3.2 that the results from the GIWM are in good
agreement with the experiments. However, the GIWM over estimates the 800 mN
normal load experiment as shown in the same gure. Therefore, one should be
careful in predicting the wear behavior when the experimental range is exceeded
by a factor of about 2 in the applied normal load for a given geometry. Further
experimental validation of this modeling scheme is necessary. As it can be seen
from Figure 3.2, the amount of linear wear after 500 m of sliding is only slightly
higher for 800 mN compared to the 400 mN normal load. This dierence in the
behavior could come from the activation of a dierent dominant wear mechanism
or formation of protective tribological layers reducing the wear rate. However,
up to the rst 100 m of sliding, the GIWM is still in good agreement with the
experimental data. Further experiments at 600 mN could give some insight into
this behavior but data are not yet available. However, the GIWM was successful
in predicting the results of pin-on-disc experiment when the normal load was
doubled.
3.2.2 GIWM for computing wear on disc
In this subsection, the GIWM will be extended to the case of a comparatively
harder spherical tipped pin sliding over a softer at disc. For such a case, it can
be assumed that most of the wear occurs on the disc and negligible wear on the
pin. The GIWM for computing wear on disc assumes the evolution of an elliptical
contact area (Sarkar, 1980) where the contact length (minor axis of the contact
ellipse), 2aH progressively decreases while the wear track width (major axis of
the contact ellipse), 2a progressively increases over sliding as shown by the dotted
ellipse in Figure 3.3. There is a net increase in the contact area and thus the
contact pressure decreases over sliding. The ow chart of this scheme is shown
in Figure 3.4, where r is the radius of the wear track and all other nomenclature
remains the same as before (see Subsection 3.2.1).
The global wear modeling scheme begins with the computation of the initial
contact radius, aH0 using the Hertz (1882) solution given in Equation (3.2) where
a circular contact area results. The initial elastic deformation normal to the
46
3.2. Global Incremental Wear Models
2 ai +1
2 ai
2 a0
2aH 0
2aH i
2 aH i+1
Wear Track
Direction of Sliding
Initial Contact Area
(before wear)
Figure 3.3: Schematic for the computation of the evolution of the real contact
area for disc wear.
contact is computed using Equation (3.3). Then, the following quantities are
calculated iteratively for each increment of sliding distance (one revolution of the
pin over the disc) till the maximum sliding distance is reached as shown in Figure
3.4:
• Integral of the linear wear increment (Equation (2.35)), is computed using
the Euler explicit scheme as given below:
w
hw
i+1 = 2 · kD · pi · aHi + hi ,
(3.7)
where the sliding distance increment is given by the contact length, 2 · aH ,
since each material point on the disc wear track comes in contact with the
pin only once per revolution
• Average contact pressure based on the applied normal load and the elliptical
contact area is calculated using,
pi =
FN
π · ai+1 · aHi
.
(3.8)
• The elastic deformation normal to the contact is computed using the Oliver
3. Global Wear Modeling
47
w
h0 = 0, s0 = 0
aH0 = a0 = 3
h0 =
FN
2 Ec a0
p0 =
FN
pa02
e
3FN Rp
4 Ec
si +1 = si + 2pr
hiw+1 = hiw + 2kD pi aH i
e
hi +1 = hi + hi +1
w
ai +1 = 2 Rp hi +1 - hi2+1
pi =
i = i+1
FN
pai +1aH i
hi +1 =
FN
2 Ec ai +1aH i
aH i = 2
FN RP
pai +1 pEc
e
si + 1 < smax
YES
NO
END
Figure 3.4: Flow chart for the global incremental wear model (GIWM) for computing disc wear.
48
3.2. Global Incremental Wear Models
& Pharr (1992) relation adopted for the assumed elliptical contact area as
given below:
hei+1 =
FN
,
√
2 · EC · aHi · ai+1
(3.9)
where the equivalent contact radius for the elliptical contact area is given
√
by aHi · ai+1 (obtained by equating the elliptical area to the circular area).
• The semi contact length in the direction of sliding is computed using the
Hertz (1882) solution for rectangular contact area (assuming a plain strain
condition at the center of the wear track) using:
s
F N · RP
aHi =
.
(3.10)
ai+1 · π · EC
However, an elliptical contact area was assumed in Equation (3.8) and
Equation (3.9). Therefore, the contact width, ai+1 , in Equation (3.10) has
to be corrected by equating the elliptical and rectangular contact area.
Hence, it has to be multiplied by a correction factor of π and the modied
equation given below is used to calculate the contact length.
s
F N · RP
aHi =
.
π · ai+1 · π · EC
(3.11)
3.2.2.1 Comparison with experimental results
The GIWM for computing disc wear is used to t the experimental results obtained by Jiang & Arnell (1998) where they used a spherical tipped pin made of
polished tungsten carbide with a diameter of 6.35 mm sliding on a diamond like
carbon coated tool steel disc. The thickness of the coating was around 1.4 µm.
They conducted the experiments at room temperature in dry air with a sliding
speed of 50 mm/s and the normal loads used were 20 N and 40 N . The GIWM
was used to t the results of the 20 N normal load experiment (Figure 3.5 (a)
and (b)), where kD was identied to be 21 × 10−11 mm3 /N · mm. Figure 3.5 (a)
shows a comparison between the cross section prole of the wear track obtained
from the experiment and that used in the GIWM (based on the radius of pin
and calculated linear wear). In Figure 3.5 (b) the graph of linear wear at the
3. Global Wear Modeling
49
(b)
GIWM
-0.6
FN = 20 N
= 20 N
0.3
0.2
Jiang and Arnell (1998)
GIWM
-0.8
300
FN
0.4
-0.2
-0.4
Jiang and Arnell (1998)
0.5
0.0
h [µm]
Wear Track Depth [µm]
(a)
350
0.1
400
450
500
kD
0
Wear Track Width [µm]
500
(c)
FN
0.6
h [µm]
Wear Track Depth [µm]
[m]
GIWM
FN = 40 N
-0.4
= 40 N
0.4
0.2
-0.6
350
1500
Jiang and Arnell (1998)
0.8
GIWM
-0.2
-0.8
300
s
1000
(d)
Jiang and Arnell (1998)
0.0
3
= 21E-11 mm /Nmm
0.0
400
450
500
kD
3
= 21E-11 mm /Nmm
0.0
0
500
Wear Track Width [µm]
s
1000
1500
[m]
Figure 3.5: Comparison of the cross section prole for disc wear between GIWM
and experimental data for an applied normal load of 20 N (a) and 40 N (c); Graph
showing comparision between GIWM and experimental data for the progress of
maximum linear wear over the sliding distance for the 20 N (b) and 40 N (d)
normal load experiment.
center of the wear track over sliding distance both from the above mentioned
experiment and the t using the GIWM is shown. For the t, the material properties (Young's Modulus, EP = 669 GP a; ED = 180 GP a and Poisson's Ratio,
νP = 0.2; νD = 0.3; subscripts P and D denote pin and disc respectively) for
tungsten carbide were chosen from other literature and the material properties
for diamond like carbon were chosen from the above mentioned article. The identied kD from the above t was used to predict the 40 N normal load experiment
presented in the same article (Jiang & Arnell, 1998). In Figure 3.5 (c) and (d) it
can be seen that the results from the prediction are in good agreement with the
experimental values at large sliding distances (s > 1000 m).
50
3.2. Global Incremental Wear Models
It should be noted that the t shown in Figure 3.5 (b) is for the experimental
data point at larger sliding distances, correspondingly, the prediction from this t
is in good agreement for the experimental data point at larger sliding distances
as shown in Figure 3.5 (d). The limitations of the GIWM can be seen from the
deviations between the GIWM and the experimental data in Figure 3.5 (b) and
(d) at lower sliding distances. Since the GIWM implements a phenomenological
wear model, it cannot account for the mechanistic evolution that takes place in
the contact. Jiang & Arnell (1998) discuss in their article about the formation and
destruction of a wear-protective transfer layer that leads to wear rate transitions
from severe to mild wear as observed in their experimental data (Figure 3.5
(b) and (d)). Technically, a criterion for the wear rate transition can be easily
implemented within the GIWM, but the determination of a suitable criterion is
an issue which will be the scope for further research.
3.2.3 Eect of elastic deformation on computation of wear
The GIWM for the computation of pin wear discussed in Subsection 3.2.1 assumes an axisymmetric pressure eld. Therefore the results from the GIWM
are satisfactory when wear on sti material is considered. But, if wear on more
compliant material is to be considered, the results from the GIWM can be misleading, since one would expect that due to the elastic deformation on the sliding
pin there would result an asymmetric wear on the pin surface (higher wear on
the front side of the pin compared to the back side). This eect would be more
pronounced in the early stages of sliding when there occurs maximum elastic deformation, since the contact is more Hertzian. Also this eect occurs for very low
wear coecients for the very same reasons as will be elaborated in Subsection
3.2.4.
The study of the eect of elastic deformation on the computation of wear can
be accomplished by comparing some dimensionless quantities for various experiments found in the literature. We can derive two such dimensionless parameters
as shown in the following (the nomenclature remains the same as described in
Subsection 3.2.1):
3. Global Wear Modeling
51
The dierential form of Equation (2.35) is given by (also see Section 4.3):
dhw
= kD · p.
ds
(3.12)
Substituting for p using Equation (3.4) and further substituting for the contact
√
radius, a ≈ 2 · RP · hw from Equation (3.6b) (also see Figure 3.1 (b)) into
Equation (3.12), we get:
dhw
FN
=
kD ,
ds
2 · π · RP · hw
(3.13)
Integrating hw with respect to s we get:
r
FN
w
· kD · s.
h =
π · RP
(3.14)
Further dividing Equation (3.3) by Equation (3.14) and maintaining the dimensionless form on either side of the equation, we get:
FN
√
he
2·EC · 2·RP ·hw
q
,
=
FN
hw
·k ·s
π·RP
D
h
1√
√ w
=
π
2
h · RP
e
(3.15a)
s
EC2
FN
.
· RP · kD · s
(3.15b)
The dimensionless parameter on the left hand side is a measure of the extent
of the elastic deformation and is termed as dimensionless elastic deformation
Q
( e ) and the dimensionless parameter on the right hand side includes all the
tribological and material parameters used in the wear model and is termed as the
Q
dimensionless system parameter ( s ).
Figure 3.6 shows a plot of the
Q
e
vs.
Q
s.
The data points shown in Figure
3.6 are obtained from the GIWM for computing pin wear after tting the experimental results found in the literature as described in Subsections 3.2.1.1 and
3.2.2.1. The data points in Figure 3.6 for steel on steel correspond to the t for
the experimental values found in Podra & Andersson (1999) and the data points
for silicon nitride on silicon nitride correspond to the t for the experimental
values found in Herz et al. (2004) (also see Section 3.1). Further, the data points
for polytetrauoroethylene on steel and diamond like carbon on tungsten carbide
correspond to Khedkar et al. (2002) and Jiang & Arnell (1998) respectively. The
data points for diamond like carbon in the latter case were obtained by identifying
52
3.2. Global Incremental Wear Models
3
PTFE/Steel, k
D
3
DLC/WC, k
1
= 21E-11 mm /Nmm
D
3
Steel/Steel, k
1/2
D
= 12E-8 mm /Nmm
Si N /Si N , k
3
4
3
4
D
3
= 13.5E-9 mm /Nmm
A
0.01
B
1E-3
e
: h
e
w
P
/ (h R )
0.1
= 92E-8 mm /Nmm
1E-4
s
1E-5
1E-5
1E-4
= 0.009148
1E-3
: (F
s
s
0.01
0.1
2
N
= 0.349
1
1/2
/ (E R k s))
C
P
D
Q
Figure 3.6: Graph Q
of dimensionless elastic deformation, e vs. dimensionless
system parameter, s for studying the eect of elastic deformation on computation of wear. Points A and B will be discussed later in Subsection 5.3.
the wear coecient using the GIWM for computing disc wear and then using the
identied parameter in the GIWM for computing pin wear assuming the pin and
disc material have been interchanged. This inversion of the pin and the disc material was necessary for the comparison with the other data in a single graph. It
√
can be seen from Figure 3.6 that the slope of 21 π according to Equation (3.15b)
approximates the calculated data points very good. The deviation between the
√
slope of the data points and the straight line with a slope of 12 π is coming from
Q
Q
the derivation presented above. In the derivation of e and s , the dierentiation of Equation (2.35) does not consider the elastic deformation while the linear
wear calculated from GIWM does include the elastic displacement. Since linear
wear is a history dependent quantity, any error accumulated at the beginning of
sliding will persist till the end.
The data points for the silicon nitride pin sliding on a disc of the same material
are located in the lower left part of the graph (see Figure 3.6), indicating that the
eect of elastic deformation can be signicant only in the initial stages of sliding.
But, the data points for the polytetrauoroethylene pin sliding on a steel disc
3. Global Wear Modeling
53
fall on the upper right part, indicating that the eect of elastic deformation is
pronounced even for large sliding distances and, therefore, should be taken into
account in any wear modeling/simulation scheme.
The GIWM does consider the elastic deformation in the normal direction to the
contact. However, it does not consider the elastic deformation tangential to the
contact. A realistic way to consider the elastic deformation both normal and
tangential to the contact (thus, accounting for the asymmetric wear on the pin)
is to apply the wear model on the local scale using a nite element based wear
simulation tool, which will be described in Chapter 4 and 5 .
3.2.4 Comparison of the GIWM with Kauzlarich and
Williams wear model
GIWM
Kauzlarich and Williams (2001)
kD
3
= 2.0 E-06 mm /Nmm
1E-4
kD
h
w
[mm]
0.1
3
= 2.0 E-13 mm /Nmm
1E-7
1E-10
0
100
200
s
300
400
500
[mm]
Figure 3.7: Comparison between the GIWM and the wear model of Kauzlarich
& Williams (2001) for FN = 20 N , EC = 109.18 GP a and Rp = 1.5 mm.
As described in the previous subsection, the GIWM considers the elastic deformation (normal to the contact) of the pin, which is not included in the global
wear model of Kauzlarich & Williams (2001). The graph in Figure 3.7 shows the
54
3.2. Global Incremental Wear Models
comparison between the above two models at two dierent values of kD diering
by several orders of magnitude. It can be seen that for very low value of kD , the
two models give dierent results. For low kD values, the increase in the contact
area due to elastic deformation is signicant compared to the increase in the
contact area due to wear, and thus, both models show dierent results. However
at higher values of kD , both models give the same results since the increase in
the contact area due to wear becomes dominant and elastic deformation can be
neglected.
3.2.5 Study of the eect of friction coecient using GIWM
In order to investigate the eect of the coecient of friction, µ, on the evolution
of linear wear for the pin, Archard's wear model implemented in the GIWM was
modied according to Equation (2.36) (Sarkar, 1980). The coecient of friction
in Equation (2.36) was based on an exponential t as shown in Equation (3.16):
µ = A · e−s/c + b,
(3.16)
where s is the sliding distance and A, b, c are the parameters dening the change
of µ with increasing s. The t was made using the measured coecient of friction
by Herz et al. (2004) as a function of the sliding distance. The graph showing the
t and the values for the parameters from the t for the 200 mN , 400 mN and
800 mN normal load experiment are given in Figure 3.8 (a), (c), (e) and Table
3.1 respectively.
Table 3.1: Values for the parameters in the exponential t (Equation (3.16)) for
the coecient of friction, µ, as a function of the sliding distance, s, at three
dierent normal loads (200 mN , 400 mN and 800 mN ).
Parameters
A
b
c
200 mN
0.226
0.496
32.877
400 mN
0.244
0.396
607.837
800 mN
0.284
0.404
81.704
As discussed in Subsection 3.2.1, the modied GIWM for pin wear was used to
t the 200 mN normal load experiment (see Section 3.1). The identied value of
3. Global Wear Modeling
55
(a)
(b)
0.8
[mm /Nmm]
2.0x10
1.8x10
-8
3
1.6x10
Sarkar (1980)
-8
0.4
1.4x10
-8
kD
µ [-]
0.6
Expt.
-8
0.2
Expt
1.2x10
FN=200 mN
Fit
0.0
-8
FN=200 mN
-8
0
100
200
300
s [m]
400
1.0x10
500
0.45
0.50
(c)
[mm /Nmm]
2.0x10
0.60
0.65
1.8x10
1.6x10
Expt.
-8
Sarkar (1980)
-8
-8
3
0.6
0.4
1.4x10
-8
kD
µ [-]
µ [-]
(d)
0.8
0.2
Expt
0
100
1.2x10
FN=400 mN
Fit
0.0
200
300
s [m]
400
-8
FN=400 mN
-8
1.0x10
500
0.45
0.50
(e)
0.55
µ [-]
0.60
0.65
(f)
0.8
[mm /Nmm]
1.6x10
0.6
0.4
1.4x10
1.2x10
Expt.
-8
Sarkar (1980)
-8
-8
3
µ [-]
0.55
kD
0.2
Expt
FN=800 mN
Fit
0.0
0
100
200
300
s [m]
400
1.0x10
8.0x10
6.0x10
-8
-9
FN=800 mN
-9
500
0.40
0.45
0.50
µ [-]
0.55
0.60
Figure 3.8: Graph showing the exponential t on the measured µ as a function of
s for (a) 200 mN , (c) 400 mN and (e) 800 mN normal load; Comparison of kD as
a function of µ from the experiments on silicon nitride (Herz et al., 2004) and from
the modied wear model of Sarkar (1980) with kD = 10.2 × 10−9 mm3 /N · mm
for (b) 200 mN , (d) 400 mN and (f) 800 mN normal load.
56
3.2. Global Incremental Wear Models
Expt. (
Expt. (
Expt. (
FN
FN
FN
kD
= 800 mN)
= 400 mN)
= 200 mN)
GIWM (
50
3
=13.5 E-9 mm /Nmm)
Sarkar (1980) (
kD
3
=10.2 E-9 mm /Nmm)
h
w
[µm]
40
30
20
10
0
0
100
200
s
300
400
500
[m]
Figure 3.9: Results from the modied Archard's wear model (Equation (2.36))
implemented within the GIWM for pin wear in comparison with the experimental results (Herz et al., 2004) from the pin-on-disc tribometer at three dierent
normal loads ( 200 mN , 400 mN and 800 mN ).
kD using Equation (2.36) was 10.2 × 10−9 mm3 /N · mm, where the wear model
takes the friction between the surfaces into consideration. The identied kD was
p
now slightly lower (by a factor 1 + 3 · µ2 for µ ∼ 0.45) when compared to that
in Subsection 3.2.1 while the resulting curves for the linear wear over sliding
distance were in exact agreement. The material properties for silicon nitride
were the same as used in Subsection 3.2.1.1. Figure 3.8 (b), (d) and (f) shows
kD as a function of µ calculated from the 200 mN , 400 mN and 800 mN normal
load experiments in comparison to that from Equation (2.36) given by Sarkar
(1980) implemented in GIWM. The dierence in the results between the data
from Equation (2.36) given by Sarkar (1980) and the experimental data can be
3. Global Wear Modeling
57
seen in the prediction of the 400 mN normal load experiment as shown in Figure
3.9. It can be seen that, the prediction for 400 mN normal load is an under
estimate when compared with the experimental results. This under estimation
becomes clear when Figures 3.8 (b) and (d) are compared. It can be seen that
p
the eective kD value (kD · 1 + 3 · µ2 ) is lower than that calculated from the
experiments for the 400 mN normal load data as compared to the corresponding
data of the 200 mN normal load experiment. As in the case of Archard's wear
model discussed in Subsection 3.2.1, the Equation (2.36) given by Sarkar (1980)
implemented in GIWM over estimates the 800 mN normal load experimental
data. Figure 3.8 (f) makes this dierence clear as the eective kD value is very
much higher than that compared with the data calculated from the experiments.
For comparison, the results presented in Subsection 3.2.1.1 (Figure 3.2) are also
included in the Figure 3.9. The dierence between the curves using Archard's
wear model and Equation (2.36) given by Sarkar (1980) is due to the eect of
friction which is included in the latter. However, the linear wear over sliding
distance plot in Figure 3.9 shows that the eect of µ has marginal inuence on
the wear behavior for silicon nitride. The modied Archard's wear model given
by Sarkar (1980) can satisfactorily describe the trends seen in the experiments
considering the scatter in the measured data.
3.2.6 Remarks on the GIWM
The GIWM can be very handy in tribometry, where the specimen geometry is
simple (e.g., spherical tipped pin and at disc). Additionally, the results presented
in the earlier subsections conrm that the GIWM can be used successfully to
predict pin-on-disc experiment within a limited range. The extent of the limits
of this prediction is an open research topic. In any case, the GIWM can be used
to quickly interpret the tribological performance of a given material pair in a pinon-disc experiment when some experimental parameters (normal load and sliding
distance) are changed. It was shown in Subsection 3.2.2 that the GIWM can be
used to predict wear in a pin-on-disc tribometer when the materials of the pin
and the disc are interchanged. In Subsection 3.2.5, it was shown that the GIWM
can be used to test existing simple wear models (or their improvements) for their
58
3.2. Global Incremental Wear Models
suitability.
GIWM assumes a constant average pressure over the worn surface in any sliding distance increment and also the frictional eects are not considered in the
deformation of the pin. Particularly in the case for pin wear (Subsection 3.2.1),
the pin is assumed to be axisymmetric. The worn out pin surface is assumed to
be always at and wear on pin alone is considered. Further for the case of disc
wear (Subsection 3.2.2), the worn out disc surface always has the curvature of
the pin and only the wear on the disc is computed. These assumptions made in
the GIWM limit its usage to certain geometries and material combinations. The
GIWM can be used to make a rst guess for the local wear model, which can
then be implemented in a nite element based wear simulation tool, such as the
one to be discussed in Chapter 4 and 5. If this more general tool also gives satisfactory results for the pin-on-disc tribosystem using a local wear model, then the
included local wear model can be used to predict wear in a geometrically dierent
tribosystem (e.g., micro-machines).
Chapter 4
Finite Element Based Wear
Simulation Tool
"In mathematics you don't understand things. You just get used to them."
Johann von Neumann
(1903-1957)
In this chapter a nite element based wear simulation strategy will be presented.
The working of the wear simulation tool (Wear-Processor) will be explained in
detail with the help of a ow chart. The features of the Wear-Processor are: (i)
it employs a three dimensional nite element model of the pin and the disc, (ii)
wear on both the surfaces are computed, (iii) it employs an ecient re-meshing
technique to avoid severe deforming of the mesh, (iv) it uses Coulomb friction,
where the friction coecient is taken from the experimental measurements.
60
4.1. Wear-Processor Methodology
4.1 Wear-Processor Methodology
It is important to note that the wear simulation tool, the Wear-Processor does not
aim to simulate the entire sliding process with nite element analysis but instead
solves a general deformable-deformable contact problem a number of times at different intervals of sliding with the surface updated at each stage to be as realistic
as possible. The ow chart for the Wear-Processor is shown in Figure 4.1 and in
the following its working is explained in detail (Hegadekatte et al., 2005c,b,a). As
it can be seen from the ow chart, the entire wearing process is discretized into
a nite number of sliding distance increments (outer loop; thick arrows). During any particular sliding distance increment (inner loop), the contact conditions
are assumed to be constant. The processing of wear begins with the solution
of a three dimensional static contact analysis between deformable bodies with
innitesimal sliding to include the asymmetric eects coming from the friction
between the sliding surfaces. The solution of this boundary value problem is accomplished with the commercial nite element code, ABAQUS. The stress eld,
the displacement eld and the element topology are then extracted from the nite
element results le using an interface program, FE-Post.
For each sliding distance increment, the contact pressure for each of the surface
nodes on each of the interacting surfaces is calculated by rst computing the
traction vector using the following equation:
tj = σij · ni ,
(4.1)
where tj is the traction vector, σij is the stress tensor, ni is the inward surface normal vector at the corresponding surface node, the subscripts i and j correspond
to the tensor components in three dimensional space. The contact pressure, p at
each surface node is calculated using:
p = tj · nj
(4.2)
In the following, the computation of the surface normal vector will be discussed
for two and three dimensional nite element models of any tribosystem.
END
END
YES
ss ³³ ssmax
max
•Inward
•Inward Surface
Surface Normal
Normal
•Contact
•Contact Pressure
Pressure
•Local
•Local Wear
Wear Model
Model
(Archard’s
(Archard’s Wear
Wear Law)
Law)
•Wear
•Wear Depth
Depth
Wear-Processor
Wear-Processor Core
Core
Interface
Interface (FE-Post)
(FE-Post)
Finite
Finite Element
Element Simulation
Simulation
(ABAQUS)
(ABAQUS)
NO
Max.
Max. Allowable
Allowable Wear
Wear to
to
Flatten
Flatten the
the Triboelement
Triboelement
OR
OR
Wear
Wear Depth
Depth
³³ d×Surfaced×SurfaceElement
Element Height
Height
(current
(current wear
wear step)
step)
NO
Wear-Processor
Re-meshing
Re-meshing
•Linear
•Linear Wear
Wear as
as Boundary
Boundary Condition
Condition
•Linear
•Linear FE
FE Simulation
Simulation
•Extract
•Extract Node
Node Co-Ordinates
Co-Ordinates
YES
Figure 4.1: Flow chart of the Wear-Processor. The notations, utotal is the total displacement, which is the algebraic sum
of the elastic displacement (uelastic ) and the displacement computed due to wear (uwear ), s is the current sliding distance
and smax is the maximum sliding distance. PATRAN is a commercial nite element pre- and post-processor.
Viewing
Viewing the
the
Wear-Processor
Wear-Processor
Results
Results (PATRAN
(PATRAN))
uutotal
= u elastic ++ uuwear
total = uelastic
wear
•Geometry
•Geometry
•Contact
•Contact Definition
Definition
•Material
•Material Model
Model
•Boundary
•Boundary Condition
Condition
4. Finite Element Based Wear Simulation Tool
61
62
4.2. Computation of surface normal vector
4.2 Computation of surface normal vector
For the computation of the contact pressure at each surface node of the nite
element model of a tribosystem, the surface normal vector at the corresponding
nodes has to be calculated. The unit inward surface normal vector at each of
the surface nodes is computed based on the element topography (two or three
dimensional) as will be discussed below.
4.2.1 Normal vector for two dimensional topography of the
surface
Figure 4.2 shows a part of a contact surface in a two dimensional nite element
model. To calculate the inward surface normal vector for the surface node number
1, the following nomenclature is used: ri1 is the right hand side edge vector
connecting the right hand side surface node number 2 and its normal vector is
denoted by n1i . Similarly, the left hand side edge vector and its corresponding
normal vector is denoted by ri2 and n2i respectively.
The surface normal vector (inward or outward) at the surface node number 1 can
be calculated by applying the following equation:
rik · nki = 0
∀k,
(4.3)
where the subscript i corresponds to the tensor components in two dimensional
space and the superscript k denotes the right and left hand side vectors described
above. The determination of the inward surface normal vector is done by enforcing the following condition:
nki · si ≥ 0
∀k,
(4.4)
where si is the subsurface vector formed by connecting to the node at the interior
(node number 4) of surface node number 1. The unit average normal vector from
the above computed inward right and left hand side normal vectors is determined
from the following relation:
ni =
(n1i + n2i )
.
kn1i + n2i k
(4.5)
4. Finite Element Based Wear Simulation Tool
63
4.2.2 Normal vector for three dimensional topography of
the surface
The computation of the inward surface normal vector at each of the surface nodes
is slightly more complicated than the previously discussed two dimensional case.
Figure 4.3 shows a part of a contact surface in a three dimensional nite element
model. For calculating the inward surface normal vector for the surface node
number 1, the following nomenclature is used: the right, back, left and front side
edge vector are denoted by ri1 , ri2 , ri3 and ri4 respectively. The normal vector to
the surface element face 1, 2, 3 and 4 is denoted by n1i , n2i , n3i and n4i respectively.
The normal vector, npk , for each of the surface element face described above is
calculated by taking a vector cross product of the edge vectors belonging to a
particular surface element using,
npk = εijk riq rir
∀p, q, r,
(4.6)
where εijk is the permutation symbol, the subscripts i, j and k correspond to the
tensor components in three dimensional space, the superscripts q and r denotes
the right, back, left and front vectors described above and the superscript p is
used to denote the inward or outward normal vector. The determination of the
inward surface normal vector is done by enforcing the condition explained earlier
in Equation (4.4) using the subsurface vector formed by connecting to the node
at the interior (node number 6) of surface node number 1. The unit average
inward surface normal vector for each surface node is computed in a similar way
as shown in Equation (4.5), except that in this case, four inward surface normal
vector will be used in the computation.
The unit inward surface normal vector is calculated for each of the surface nodes
for each sliding distance increment with one of the above methods depending on
whether the tribosystem is modeled in two or three dimensions.
64
4.2. Computation of surface normal vector
ni
4
si
1
2
3
y
ni2
ni1
ri
2
2
1
i
1
r
x
Figure 4.2: Calculation of inward surface normal vector for a two dimensional
nite element model. Numerals enclosed within the circle indicate the element
number and the plain numerals indicate the node numbers of the nite element
mesh.
3
ri 2
2
1
ri
3
1
4
ni4
2
1
i
r
ni3
ri 4
6
1
i
si ni2 n
ni
4
3
5
y
x
z
Figure 4.3: Calculation of inward surface normal vector for a three dimensional
nite element model. Numerals enclosed within the circle indicate the element
number and the plain numerals indicate the node numbers of the nite element
mesh.
4. Finite Element Based Wear Simulation Tool
65
4.3 Computation of linear wear
In order to simulate the evolution of the contact surface topography due to wear,
it is necessary to determine the wear depth locally as a function of the location
of the contact node in the nite element model. Therefore, for an innitesimally
small apparent contact area, 4A, the increment of linear wear, dhw , associated
with the increment of the sliding distance, ds, is determined for each surface
node. This can be obtained by applying Equation (2.34) locally (for each surface
node) to the area 4A and for the increment of sliding distance, ds:
dV
fN
=k·
,
ds
H
(4.7)
where V stands for the volume of material removed, fN is the part of the applied
normal load that that has to be considered for the computation of the contact
pressure on 4A, H is the hardness (ratio of load over projected area) and k is the
dimensionless wear coecient. Then, dividing both sides by 4A, the following
equation is obtained:
dV
fN
=k·
.
ds · 4A
H · 4A
fN
4A
is the local contact pressure, p.
(4.8)
dV
4A
is the resulting increment of local linear
wear, dhw , noting that hw is a function of both the location and the total local
sliding distance, s. The following equation is thus obtained for the prediction of
the increment of local linear wear:
dhw
= kD · p,
ds
where the quantity
(4.9)
k
H
is replaced by kD which is the dimensional wear coecient.
As a simplifying assumption an average value of kD across the complete contact
area is used. The implication of Equation (4.9) is that the incremental linear wear
at a given point in the contact is proportional to the local contact pressure, p, and
the local increment of sliding distance, ds. The incremental form of Equation (4.9)
is integrated with respect to the sliding distance using an explicit Euler method
as shown in Equation (3.5) and Equation (4.10).
In Chapter 5, Equation (3.5) will be used to compute wear on a ring-on-ring
tribosystem (two dimensional, axisymmetric nite element model) and pin-on-
66
4.3. Computation of linear wear
disc (three dimensional nite element model) tribometer. Equation (3.5) can
directly be used to compute wear on both the ring surfaces in the ring-on-ring
tribosystem and also on the pin surface in the pin-on-disc tribosystem, since
the pin surface is always in contact with the disc. In the following subsection,
Equation (3.5) is used in a modied form for the computation of wear on the disc
surface in the pin-on-disc tribosystem.
4.3.1 Computation of linear wear on the disc surface
In a pin-on-disc tribometer, each surface node on the disc surface within the wear
track comes in contact with the pin surface only once in one complete revolution
of the pin over the disc. Therefore, the computation of wear on the pin and on
the disc surface have to be considered separately.
For computational eciency, only a small representative portion of the disc is
considered. The computation of wear in the wear track on the disc should consider the fact that each of the disc surface nodes in the wear track sees the pin
approaching it, come directly above it and then slide away. It is during this period
when the wear takes place on each of these nodes, depending on the contact pressure that the considered node experiences. The approach adopted in this work is
not to perform a nite element contact simulation for the entire sliding process
of the pin over the disc, which is computationally very expensive if not impossible in view of the large number of operating cycles that needs to be simulated.
Instead the contact simulations are reduced to the pin pressing the disc and then
sliding over the disc by one element length in the contact region to include the
frictional eects (innitesimal sliding). In order to calculate the wear on the disc
surface nodes, we integrate the contact pressure over the circumferential length
at a given radius within the wear track, r, of the disc using:
Z 2π
w
hi+1 = kD
pi · r · dϕ + hw
i ,
ϕ=0
(4.10)
w
where pi is the contact pressure, hw
i and hi+1 is the cumulative wear depth up to
the previous and current sliding distance increment respectively, kD is the dimensional wear coecient, ϕ is the circumferential coordinate, and i is the current
wear increment number. Since the nite element model of the disc is only a very
4. Finite Element Based Wear Simulation Tool
67
small representative portion of the actual disc, as will be discussed in Subsection
5.2.1, the wear calculated at each radius should be representative of the wear on
any point in the wear track of the real disc. The integral in Equation (4.10) can
be computed with relative ease if the nite element surface nodes are located
exactly along the circumference of the various streamlines within the wear track.
This would apply restrictions on the way the model is meshed. However, after
testing various meshing strategies for the pin and the disc, it was found that
such a restriction on the nite element mesh does not yield good results. This
restriction on the mesh is overcome by interpolating the contact pressure onto
a grid of points that are located at denite pre-determined radii of the circumferential streamlines along the forming wear track (Hegadekatte et al., 2005c).
Figure 4.4 shows the grid of points superimposed on a plot for the stresses in the
Figure 4.4: A grid of points superimposed on the plot for the distribution of
normal stress on the top surface of the disc where the pin is engaged.
y-direction (normal to the contact). As our quantity of interest is the contact
pressure, it is sucient if we consider the face of the three dimensional surface
element that forms the contact surface, which is basically the same as a four-node
two dimensional element. To interpolate the contact pressure computed at the
surface nodes (Equation (4.2)) onto each of the points of the grid, the concerned
68
4.3. Computation of linear wear
point has to be rst identied with the three dimensional surface element face
that it is located on. Figure 4.5 enunciates the method of detection of the three
x
3
ri3
z
ri
4
qi4
4
ri1
1
ni4
ni3
qi3
qi2
2
q
qi1
ni2
ri2
ni1
Figure 4.5: Detection of the surface element on which the grid point is located.
dimensional surface element face that contains a particular point of the grid. To
determine if a grid point, q , is located inside a particular surface element face
or not, three vectors for each of the surface nodes that make up the surface element face are constructed. The three vectors include the edge vector, rik , its
normal vector nki and the vectors connecting the surface node belonging to an
element and the point, q , of the grid, qik . The normal vector to the edge vector
is calculated by applying the following equation:
rik · nki = 0
∀k,
(4.11)
where the subscript i corresponds to the tensor components in two dimensional
space and the superscript k denotes the vectors shown in Figure 4.5. If the
grid point is located within the considered surface element, then the following
condition should be satised.
nki · qik ≥ 0
∀k.
(4.12)
Once the grid point is determined to be contained within a particular three dimen-
4. Finite Element Based Wear Simulation Tool
69
sional surface element face, the previously computed contact pressures (Equation
(4.2)) at the surface nodes have to be interpolated onto the grid point. Since the
location of the grid point in the element local coordinate system is not known
but its location in the global coordinate system is known, a polynomial function
of the type given in Equation (4.13) has to be solved for the coecients α1 , α2 ,
α3 and α4 for the four nodes using the Gaussian elimination method.
pi = α1 + α2 · xi + α3 · zi + α4 · xi · zi ,
(4.13)
where pi is the respective contact pressure of the respective surface nodes making
up the surface element face, xi and zi are the global coordinates of the surface
nodes that make up the surface element face and the subscripts i correspond
to the surface nodes belonging to the three dimensional element face. Once
the coecients in Equation (4.13) are calculated, the contact pressure can be
interpolated onto the grid point, q , using Equation (4.13) by substituting for the
global coordinates of the grid point. Such a computation is performed for every
point on the imaginary grid. The linear wear for the disc surface nodes in the
wear track is computed by integrating the contact pressure with respect to the
circumferential coordinate (Equation (4.10)) along various streamlines that make
up the wear track as indicated by the curved arrow in Figure 4.4.
4.4 Re-meshing
The calculated wear from the Archard's wear model is used to update the geometry by repositioning the surface nodes based on the calculated wear at that
node. But, in this way we will be limited by the surface element height, which
means that the surface elements have to be meshed in such a way that they have
enough height to accommodate the entire sliding distance that is planned to be
simulated. Otherwise there will be nite element degeneracies occurring very
early. However, such a strategy for meshing will aect the accuracy of the nite
element results. It is not aordable to have a coarse mesh in the contact region
unless the accuracy of the results is compromised. In order to achieve the highest
possible accuracy in the nite element results and at the same time accommodate
the wear on the surface for the entire planned sliding distance, the nite element
70
4.5. Sliding distance incrementation scheme
model of the tribosystem has to be re-meshed at the end of each sliding distance
increment. This is indicated as Re-meshing in Figure 4.1. Inspired by the
structural shape optimization scheme (Mattheck, 1998), an ecient re-meshing
technique is implemented in the Wear-Processor, that makes use of the boundary displacement method, where a linear system of equations are solved using
ABAQUS. Thus the obtained new reference geometry is used in the next sliding
distance increment to get the updated stress distribution, which in turn is used
to compute the updated contact pressure distribution. At the end of each sliding
distance increment the total displacement (sum of the elastic displacement and
linear wear) for each of the surface nodes of the interacting surface elements is
written to an ABAQUS compatible le for viewing with PATRAN (a commercial
pre- and post-processor).
4.5 Sliding distance incrementation scheme
The choice of a suitable value for the sliding distance increment, ∆si , corresponds
to the decision on when to start the re-meshing step. This is of great signicance
both from the point of view of the computational expense and more importantly
on the stability of the analysis. Such diculties have been reported by Podra
(1997), Podra & Andersson (1999), Öquist (2001), McColl et al. (2004) and Kim
et al. (2005). If the chosen value for ∆si is high, then articial roughening occurs
in the initial stages resulting in very erratic results and if a lower value for ∆si
is used, then the computational costs become exorbitant.
Two strategies were tried in the Wear-Processor for choosing an optimum value
for ∆si . In the rst strategy, when in a particular sliding distance increment,
the linear wear at any surface node reached a certain percentage (denoted by δ
in Figure 4.1) of the corresponding surface element height the re-meshing step
was activated. This strategy yielded good results for very low wear coecients
(also see Hegadekatte et al. (2005c,b,a)), however, this strategy for real values of
the wear coecient resulted in a dent on the pin surface in the contact region.
For a contact pressure prole as shown in Figure 4.6, the calculated wear prole
would look like the dashed line in the same gure for high values of ∆si (dent
formation). The dent is formed due to the insucient frequency of updating of
4. Finite Element Based Wear Simulation Tool
71
the contact pressure due to wear. Such a wear prole in any particular sliding
distance increment, leads to a severe distortion of the mesh. This problem of dent
formation was tackled by the second strategy which employed a more frequent
updates of the contact pressure distribution.
Pin
Contact Pressure
profile
Dhmaxi
y
x
Surface nodes in
contact
Figure 4.6: Optimal value for the maximum allowable wear.
A strategy is implemented in the Wear-Processor to determine the optimal value
of the maximum allowable wear or in other words the optimal value for attening
the pin. It involves the detection of the surface nodes located on the contact edge
as shown in Figure 4.6. The dierence between the y coordinates (normal to the
contact) of this node and the center node is determined and then the optimal
value of ∆si is calculated using:
∆si =
² · ∆hmaxi
,
kD · pi
(4.14)
where pi is the contact pressure at the center node in the ith sliding distance
increment, kD is the dimensional wear coecient, ∆hmaxi is the maximum allowable wear as shown in Figure 4.6. Some fraction, ², of ∆hmaxi is taken in the
calculation of ∆si . A value of ² = 0.15 was used in the current wear simulations.
The problem of denting is pronounced especially in the early stages of sliding
when the drop in the contact pressure due the increase in the contact area is very
drastic. Once contact pressure distribution becomes attened, denting is a smaller
issue and ∆si can be increased to speed up the wear simulation. This strategy
72
4.5. Sliding distance incrementation scheme
is contrary to the method used by Öquist (2001), where a coarse sliding distance
increment was used in the initial stages of sliding and a ne wear increment was
used in the later stages of sliding in order to smoothen the articial roughness
from the initial stages. Therefore it should be noted that depending on the
tribosystem a suitable strategy has to be adopted.
Chapter 5
Wear Simulation
"If the facts don't t the theory, change the facts."
Albert Einstein
(1879-1955)
In this chapter, the results of the application of the Wear-Processor to simulate
wear in an axisymmetric dry sliding contact, namely the ring-on-ring tribosystem
will be discussed rst. Then the advantages of re-meshing implemented in the
Wear-Processor will be presented. The last two sections pertain to simulation
of wear in a pin-on-disc tribometer, where rstly, the Wear-Processor is used to
test the suitability of a local wear model (Archard's wear model) using the wear
coecient identied with the GIWM by tting the linear wear results for silicon
nitride from Herz et al. (2004) discussed in Chapter 3. A study of the eect of
elastic deformation on wear will be presented in the last section.
74
5.1. Ring-on-ring tribosystem
5.1 Simulating wear in a ring-on-ring tribosystem
- an axisymmetric dry sliding contact
The simplest of the wear simulation problems are the ones that can be reduced
to a static axisymmetric nite element model (Hegadekatte et al., 2005c). The
reason for their simplicity is the fact that the surfaces of the interacting bodies which are initially in contact will remain in contact throughout the wearing
process and the calculation of the surface normal vector at each surface node
is simpler owing to the two dimensional geometry and so is the computation of
wear. Here we take the example of a hemispherical brass ring rotating on a at
steel ring (ring-on-ring) in dry sliding contact for computing the progress of wear
using the Wear-Processor. The geometrical details and the material properties
used in the simulation are shown in Figure 5.1, where E is the Young's modulus
and ν is the Poisson's ratio.
3 mm
F 20 mm
5 mm
y
x
10 mm
5 mm
Brass
Steel
E = 110*10 3 N/mm2
E = 200*103 N/mm2
n = 0.35
n = 0.29
Simulation
Geometry
Figure 5.1: Hemispherical brass ring rotating over a at steel ring in dry sliding
contact.
The upper hemispherical ring is loaded with a pressure of 3000 N/mm2 and the
bottom surface nodes of the lower ring are xed in space. It can be found in
the literature that the wear coecient for brass (Lancaster, 1962) is around four
5. Wear Simulation
75
times higher than that for steel (Podra & Andersson, 1999). The dimensional
wear coecient for the steel ring was arbitrarily chosen as 2 × 10−13 mm3 /N ·
mm and that for the brass ring was four times higher than that for steel at
8 × 10−13 mm3 /N · mm. The very low wear coecient for steel was chosen in
order to simplify the data handling, so that the minimum possible sliding distance
increment was more than a revolution (preferably few thousand revolutions) of
the upper ring. An elastic material law and a deformable-deformable contact
were used in the nite element simulations. It can be expected that the initially
non-conformal contact (Hertzian contact) begins to conform with sliding due to
wear.
5.1.1 Results
The result from the wear simulation is presented in Figures 5.2 and 5.3. In Figure
5.2 (a) - (f), the stresses normal to the contact are plotted on the upper and the
lower ring in the contact region at dierent stages of sliding. It can be observed
from Figure 5.2 (a) - (f) that, with sliding, the contact area increases due to
wear taking place. The two triboelements begin to increasingly conform to each
other, and as a result, the contact pressure progressively decreases. It should
be noted that the plots have the same range for the spectrum. The absolute
value of the stresses can be obtained from Figure 5.3 (a). In any sliding distance
increment, ∆si was determined by the sliding distance performed by the upper
ring till the linear wear at any surface node reached a certain percentage (δ as
explained in Subsection 4.5) of the corresponding surface element height in the
current sliding distance increment. The curves in Figure 5.3 (a) are plotted on
the surface nodes of the lower ring in the contact region. The progress of wear
over the sliding distance (number of rotations) on the upper and lower ring is
shown in Figure 5.3 (b). It can be seen from Figure 5.3 (b) that the wear on
both surfaces progresses according to Archard's wear law, where the parameters
involved are: their respective wear coecients, the local contact pressure and
the sliding distance. The slope of the linear wear curve for both the rings is
steadily decreasing, owing to the fact that, as the contact widens, the contact
pressure decreases steadily, and therefore, linear wear per unit sliding distance
continuously decreases. This is termed as running-in in the literature. The
76
5.1. Ring-on-ring tribosystem
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.2: σY Y plotted on a zoomed section of the contact between the upper
hemispherical brass ring and the lower at steel ring at dierent stages of sliding:
(a) initially when contact takes place; (b) after 9.9 × 104 rotations; (c) after
2.2 × 105 rotations; (d) after 3.5 × 105 rotations; (e) after 4.7 × 105 rotations; (f)
after 6.0 × 105 rotations. The range of the spectrum for all the plots is the same
(Spectrum Range: −10000 to −100 N/mm2 ).
5. Wear Simulation
77
(a)
(b)
x [mm]
5
6
7
8
9
10
11
0.3
12
13
14
15
0
h [mm]
-2
-4
-6
-8
-10
Lower Flat Ring (Steel)
0.2
0.15
0.1
After 0E+00 Rotations
Hertzian Contact Pressure
-12
Upper Hemispherical Ring (Brass)
0.25
0.05
After 6E+05 Rotations
-14
*103
0
0
100
200
N
300
400
500
600
*103 [-]
Figure 5.3: (a) Graph of σY Y versus x-coordinate in the contact region on the
lower at ring; (b) Progress of wear on the upper (brass) and lower (steel) ring.
upper brass ring wears out faster than the lower steel ring because of the higher
wear coecient chosen for brass than that for steel. This demonstrates that
the Wear-Processor is able to handle wear of two components made of dierent
materials.
5.1.2 Advantages of re-meshing
In this section, the advantages of implementing a re-meshing scheme will be
presented. Figure 5.4 (a) and (b) shows the initial and worn-out mesh for the ringon-ring problem. The elastic deformations of the contacting rings are included
in both the Figures 5.4 (a) and (b). It can be seen from Figure 5.4 (b) that the
wear on the upper hemispherical brass ring is more than the wear on the lower
at steel ring due to the dierence in the wear coecients for the two materials
as discussed in the previous subsection. Additionally, due to the re-meshing after
each sliding distance increment, the mesh is fairly uniform in the contact region
after accommodating wear compared to the un-worn mesh in Figure 5.4 (a).
One of the highlights of the Wear-Processor from the view of computational
expense is the implementation of a re-meshing scheme. In Figure 5.5, it can
be seen that the number of contact simulations required with and without remeshing is nearly half for a value of δ (explained in Subsection 4.5) less than 50
%. In order to make the comparison, the maximum sliding distance that can be
78
5.1. Ring-on-ring tribosystem
(a)
(b)
No. of Contact Simulations [-]
Figure 5.4: Finite element mesh of a hemispherical ring on a at ring before (a)
and after (b) 6 × 105 rotations; elastic deformations are included in the gures.
20
With Re-Meshing
15
Without Re-Meshing
10
5
0
0
25
50
d
75
100
[%]
Figure 5.5: Comparison of the number of contact simulations required for dierent values of δ with re-meshing and without re-meshing for performing 6 × 105
rotations.
5. Wear Simulation
79
set (smax in Figure 4.1) should not result in a wear that is more than the surface
element height. Only in a scenario when δ is small, and at least a few contact
simulations are needed to attain smax , the obvious advantage of re-meshing can
be seen. When re-meshing is used, the allowable increment of sliding distance is
higher, and thus, requires less number of contact simulations to simulate smax .
Whereas, when re-meshing is not used, the surface elements are continuously
worn-out and so the allowable increment of sliding distance is comparatively
lower and is continuously decreasing, it is clear that more contact simulations are
required for performing the same smax . For larger values of δ , the entire smax can
be simulated in two sliding distance increments, thus we will not be able to see
the advantages of the re-meshing scheme. The real advantage of re-meshing is
the seemingly limitless smax that can be set. Therefore, wear simulation can be
speeded up by using the re-meshing scheme which helps in reducing the number
of contact simulations, since the solution of the contact problem takes around 50
% of the computation time for each sliding distance increment.
5.2 Simulating wear in a pin-on-disc tribometer
- a three dimensional quasi-static dry sliding
contact
As discussed in Chapter 1, the main task of this work is to apply a wear model
(Archard's wear model) on the local scale and identify the included parameter
by tting the pin-on-disc experiment (see Section 3.1) conducted with in the
parameter space of the micro-machine (e.g., SFB 499's demonstrator: Microturbine and -planetary gear train (see left hand side top picture in Figure 1.1)).
Therefore, in order to identify the wear coecient, the pin-on-disc experiment
has to be simulated with the Wear-Processor. If the simulation results are in
good agreement with the experimental results, then the identied parameter can
be used to simulate wear in the micro-machine. In the succeeding subsections,
the three dimensional nite element model of the pin-on-disc tribometer used
in the wear simulation will be presented. The reason for modeling the pin-ondisc tribometer in three dimensions, in spite of its computational expense will be
80
5.2. Pin-on-disc tribometer
discussed next followed by a discussion of the results.
5.2.1 Finite element model of the pin-on-disc tribometer
The geometrical details of the pin-on-disc tribometer setup is shown as a
schematic in Figure 5.6 (a) and the values for the various parameters (FN is
the applied normal load, RP is the radius of the pin, RW T is the radius of the
wear track, RD is the radius of the disc and tD is the thickness of the disc) are
given in Table 5.1 (Hegadekatte et al., 2005a). The geometry inside the dashed
line is used for the FE simulation and subsequently the Wear-Processor (see Figure 5.7 for the nite element model). The results from a contact simulation can
depend on the discretization of the nite element model if it is not generated in
the right way. The nite element model of the pin was generated from a two
dimensional template of the contact zone, which was swept in two directions as
shown in Figure 5.6 (b) to get the complete model of the pin. A similar strategy was used for the disc. It was found that, with such a uniform mesh in the
contact zone, avoids the usage of wedge elements around the center line of the
nite element model of the pin and the disc. The results were in good agreement
with the solutions of contact stresses for a Hertzian circular contact area (also
see Figure 5.8).
Table 5.1: Geometrical parameters for the pin-on-disc tribometer (see Figure 5.6
(a)).
Parameter
Pin Radius
Disc Radius
Wear Track Radius
Disc Thickness
Normal Load
Value
RP = 0.794 mm
RD = 4 mm
RW T = 3 mm
tD = 1 mm
FN = 200 mN
Figure 5.7 shows two nite element models of the pin-on-disc tribometer. The
nite element model is built as a very small slice of the pin-on-disc tribometer
shown within the dashed circle in Figure 5.6 (a). A normal load was applied to
the top surface of the pin as an equivalent distributed load (pressure over top
5. Wear Simulation
81
(a)
(b)
2·RP
FEM
Model
y
FN
tD
z
x
Wear
Track
2·RWT
2·RD
Figure 5.6: (a) Model of pin-on-disc tribometer in dry sliding contact; (b) Finite
element model of the pin at an imtermediate stage of its construction showing
the two sweep directions for completing the model.
(b)
0.18 mm
1.43 mm
Rp = 1.5 mm
No. of Elements = 39072; No. of Nodes = 44287
0.755 mm
Rp = 0.794 mm
0.0954 mm
(a)
No. of Elements = 78144; No. of Nodes = 86550
Y
X
Z
1.059 mm
0.0794 mm
2 mm
0.15 mm
1
Figure 5.7: Full (a) and Half (b) nite element model of the pin-on-disc tribometer
(cut along the direction of sliding).
82
5.2. Pin-on-disc tribometer
surface area of the pin). In order to hold the top surface of the pin parallel to
the disc surface at all times during the loading, a linear multi-point constraint
was applied on all the nodes at the top surface of the pin. The bottom surface
of the disc was xed in all the three perpendicular directions. An innitesimal
sliding was applied to the pin to include the asymmetric eects coming from the
friction between the sliding surfaces. An elastic material law and a deformabledeformable contact was used. The friction coecient was supplied to the nite
element simulation based on the average value determined from experiments (see
Subsection 3.2.5). The solution of the contact problem using the two nite element models in Figure 5.7 (a) and (b) was carried out using the commercial nite
element program, ABAQUS on an IBM RS 6000 Power 4 supercomputer with
4 processors in parallel. The rst nite element model shown in Figure 5.7 (a)
was solved using a main memory of 10 GB and a computation time of 8.5 hrs.
Making use of the symmetry in the contact problem (pin-on-disc tribometer),
this full nite element model of the pin and the disc was cut into a half along
the direction of sliding (Figure 5.7 (b)) and then simulated. Figure 5.8 (a) and
(b) shows that the contact stresses are in good agreement with each other. The
parameters used in the simulation were: FN = 4.5 N , EP = ED = 207 GP a,
νP = νD = 0.29, RP = 1.5 mm (subscripts P and D denote pin and disc respectively). With this improvement, the main memory usage and computation time
reduced to 1.5 GB and 2 hrs respectively, thus making it ecient and without
loss in accuracy. Therefore the half model (Figure 5.7 (b)) was used in the wear
simulations.
(a)
(b)
0.00
-0.1
-0.05
0
0.05
0.1
-0.50
-1.00
Hertz (1882)
-1.50
-2.00
FEM - Half Model
2
3
szz *10 [N/mm ]
2
sYY *10 3 [N/mm ]
z [mm]
2.5
Huber et al. (2005)
1.5
FEM - Half Model
FEM - Full Model
0.5
-0.5-0.20
z [mm]
-0.10
0.00
0.10
0.20
-1.5
-2.5
FEM - Full Model
Figure 5.8: Graph showing the (a) σY Y prole and (b) σZZ prole for the two
nite element models of the pin-on-disc tribometer in comparison with the Hertz
solution.
5. Wear Simulation
83
5.2.2 Necessity for three dimensional nite element model
for wear simulation in the pin-on-disc tribometer
In this subsection, the necessity for the computationally expensive three dimensional nite element contact simulation will be discussed. It can be recognized
that the solution of the contact problem for the wear simulation is computationally very expensive. In such a case, a question arises if the pin-on-disc tribometer
can be modeled as an axisymmetric problem. It turns out that if the pin-on-disc
tribometer were to be modeled as an axisymmetric problem, then the asymmetric
eects in the direction of sliding due to the friction between the sliding surfaces
cannot be modeled accurately.
(a)
(b)
Direction of sliding
1500
ZZ
2
[N/mm ]
1000
500
0
-500
Direction of Sliding
3D
-1000
-1500
-0.04
Axisymmetric
-0.02
0.00
z
0.02
0.04
[mm]
Tensile Stresses
Compressive Stresses
Figure 5.9: (a) Comparison between the surface stresses in the direction of sliding
on the disc for a two dimensional and three dimensional calculation; (b) Distribution of σZZ stresses on the top surface of the disc (Spectrum Range: −500 to
100 N/mm2 ).
It can be seen from Figure 5.9 (a) that if axisymmetry were to be assumed for the
pin-on-disc tribometer, then the stress distribution in the assumed direction of
sliding would follow an axisymmetric curve. However, if the tribometer is modeled
in three dimensions with innitesimal sliding as shown in Figure 5.7, then the
asymmetric eects due to friction can be captured as described by the asymmetric
curve shown in Figure 5.9 (a) (also see Figure 5.8 (b)). This eect is further
84
5.2. Pin-on-disc tribometer
(a)
(b)
Sliding direction
z [mm]
400
-0.03
0.03
0.08
2
p [N/mm ]
-500
-1000
-1500
-2000
µ=0
µ=0.5
µ=0.75
µ=1
Deviation, d [N/mm 2]
0
-0.08
300
200
100
0
-100-0.05
z [mm]
-0.03
0.00
0.03
0.05
-200
-300
-400
µ=0.5
µ=0.75
µ=1
Figure 5.10: (a) Comparison between the contact pressure distributions in the
direction of sliding for various friction coecients; (b) Deviation of the contact
pressure for various friction coecients related from the frictionless case.
claried in Figure 5.9 (b) where σZZ (tangential stresses in the direction of sliding)
is plotted on the disc surface and the top view is shown. The parameters used
for the simulation were: FN = 200 mN , EP = ED = 304 GP a, νP = νD = 0.24,
RP = 0.794 mm (subscripts P and D denote pin and disc respectively). It can be
seen from the same gure that on the front side of the location of the pin there
are compressive stresses and on the back side, there are tensile stresses. When
such a stress distribution is used in the calculation of the contact pressure (see
Equation (4.1) and (4.2)) and further in the computation of wear, there is bound
to be higher wear on the front side of the pin compared to the back side of the
pin, which we term as the leading edge eect. To understand this leading edge
eect further, it is important to observe how the contact pressure distribution
gets aected due to the sliding of the pin. Therefore, contact simulations on
a nite element model for the experiment in Podra & Andersson (1999) were
conducted. The geometrical details of the nite element model are as shown in
Figure 5.7 (a) and the material properties for steel were chosen from Callister
(1994), page 766, as Young's Modulus, EP = ED = 207 GP a and Poisson's
Ratio, νP = νD = 0.29 (subscripts P and D denote pin and disc respectively).
The normal load applied on the pin was 4.5 N . In Figure 5.10 (a) the distribution
of the contact pressure on the surface nodes (in the direction of sliding) is plotted
as a function of the friction coecient (Hegadekatte et al., 2005b). The curve
for the contact pressure distribution shifts slightly in the direction of sliding with
increasing friction coecient causing a deviation from the frictionless condition,
5. Wear Simulation
85
which is plotted in Figure 5.10 (b). It can be seen from this graph that the
pressure is around 60 % higher on the leading edge and around 80 % lower on
the trailing edge of the contact when compared with the frictionless condition
at the highest points of the deviation curve. This leading edge eect leads to
higher wear rates and mass loss on the front side of the pin compared to the back
side. The above simulations were carried out on the un-worn geometry. To study
such eects during wear, a three dimensional model for the wear simulation is
necessary. In addition, it can be expected that asymmetric eects will be more
pronounced with the formation of the wear track on the disc surface. Therefore,
the three dimensional contact simulations are necessary in order to (1) accurately
simulate wear in a pin-on-disc tribometer, (2) to study the leading edge eect
and (3) to study the eect of the friction coecient on wear which would not be
possible with an axisymmetric model.
5.2.3 Results
The results from the Wear-Processor for the pin-on-disc experiment for silicon
nitride on silicon nitride at 200 mN normal load (Herz et al., 2004), will be
presented in this subsection. The value of the dimensional wear coecient, for the
wear simulation using the Wear-Processor was kD = 13.5 × 10−9 mm3 /N · mm for
which the results are presented in Figure 5.11 and Figure 5.12. In Figure 5.11 (a)
and (c) σY Y , the stresses normal to the contact before wear and after 71.36 mm
of sliding is shown on the cross section of the pin and the disc. It can be seen
from these plots that there is a drastic reduction in the contact pressure in the
contact region due to wear. To aid the comparison, the range of the spectrum
for both the pictures is the same at −300 to 0 N/mm2 . In Figure 5.11 (e) a
graph of the contact pressure proles is plotted on the surface nodes of the pin
on the symmetry edge at various stages of sliding where the continuous decay in
the contact pressure distribution can be seen more clearly. The contact pressure
distribution progressively approaches a attened distribution as the contact area
widens due to wear.
For the sake of comparison, the curve computed using the Hertz (1882) solution
for circular contact area is also plotted in the same graph. The σY Y stress dis-
86
5.2. Pin-on-disc tribometer
(a)
(b)
(c)
(d)
(e)
(f)
Huber et al. (2005)
0
1200
-200
-400
-600
-800
-1000
-1200
-0.03
-0.02
-0.01
z
0.00
0.01
[mm]
s =
0.00 mm
s =
11.83 mm
s =
36.35 mm
s =
60.64 mm
s =
71.36 mm
0.02
0.03
600
2
[N/mm ]
Hertz (1882)
s
300
s =
0.00 mm
s =
11.83 mm
s =
36.35 mm
s =
60.64 mm
s =
71.36 mm
0
-300
ZZ
p [N/mm
2
]
900
s
-600
-900
-1200
-0.03 -0.02 -0.01
z
0.00
0.01
0.02
0.03
[mm]
Figure 5.11: Contact stresses plotted on the cross section of the pin and the disc
(a) σY Y stresses before wear (Spectrum Range: −300 to 0 N/mm2 ); (b) σZZ
stresses before wear (Spectrum Range: −500 to 100 N/mm2 ); (c) σY Y stresses
after 71.36 mm of sliding (Spectrum Range: −300 to 0 N/mm2 ); (d) σZZ stresses
after 71.36 mm of sliding (Spectrum Range: −500 to 100 N/mm2 ); (e) Contact
pressure prole after various intervals of sliding; (f) σZZ prole after various
intervals of sliding.
5. Wear Simulation
87
tribution in the contact from the nite element solution is within 10 % of the
Hertz (1882) solution. The dierence is mainly due to the usage of coarse linear
elements near the contact edge, which makes the nite element solution unable to
describe the sharp transition in the contact pressure distribution at the contact
edge. Therefore, there is a llet like behavior in the contact pressure prole at
the contact edge. However, since the integral of the contact pressure prole from
the nite element solution and the Hertz (1882) solution has to be the same, the
value of the maximum contact pressure is around 10 % lower than the maximum
Hertzian contact pressure.
Similarly in Figure 5.11 (b) and (d), the σZZ stresses (surface traction) for the
same stages of sliding are shown. Also a graph of the σZZ proles is plotted
on the surface nodes of the disc on the symmetry edge after various intervals of
sliding. The σZZ stress distribution for the unworn conguration, computed from
the Design Tool developed by Huber et al. (2005), is also plotted in the same
graph to verify the nite element results for the unworn conguration.
0.30
Pin
Disc
0.25
0.20
kD
0.15
3
=13.5E-9 mm /Nmm
h
w
[µm]
GIWM
0.10
0.05
0.00
0
10
20
s
30
40
50
60
70
80
[mm]
Figure 5.12: Graph of progress of wear over sliding distance for pin and the disc
in comparison with the GIWM for silicon nitride on silicon nitride.
In Figure 5.12, linear wear over sliding distance graph is plotted for both the
88
5.2. Pin-on-disc tribometer
(a)
(b)
x [mm]
5
6
7
8
9
10
11
12
13
14
z
15
-0.04
0
-0.02
[mm]
0.00
0.02
0.04
0.00
-0.02
-0.04
-0.01
[µm]
-0.06
-0.08
-0.03
u
YY
-0.1
-0.02
-0.12
-0.04
-0.14
-0.16
After 0E+00 Rotations
-0.18
After 6E+05 Rotations
s =
0.00 mm
s =
71.36 mm
-0.05
Figure 5.13: (a) Graph of uY Y versus x-coordinate at surface nodes in the contact
region on the lower at ring (ring-on-ring); (b) Graph of uY Y versus z-coordinate
at the surface nodes in the contact region of the disc (pin-on-disc).
pin and the disc along with the results from the GIWM for the early stages of
sliding. In the initial stages of sliding, the slope of the curve from the WearProcessor results for the pin is higher compared to that from the GIWM. This
dierence is due to the fact that in the GIWM, an average contact pressure
over the contact surface is considered while in the initial stages of sliding the
contact is more Hertzian which is inherently considered by the Wear-Processor.
Therefore, due to the higher pressures, there will be higher wear computed by
the Wear-Processor. However, in the later stages of sliding the slope of both the
curves becomes nearly the same. The results from the Wear-Processor are in good
agreement with the GIWM. The dierence of approximately 25 nm between the
two results are well below the measurement accuracy of the experiments which
is around 1 µm as shown in Figure 3.2. It should also be noted that the wear
on the disc in the wear simulation is negligible compared to the pin, which was
also observed in the experiments. Therefore Archard's wear model serves as a
suciently accurate model both in its global as well as local implementation for
this particular material combination.
Figure 5.13 (a) and (b) shows the comparison between the total displacement
in the y-direction (normal to the contact) for the ring-on-ring and pin-on-disc
tribosystem respectively. The total displacement (elastic + wear) after wear is
smaller than the corresponding total displacement before wear in the central region of the contact for the pin-on-disc case (Figure 5.13 (a)) which is qualitatively
5. Wear Simulation
89
contrary to the behavior observed in the ring-on-ring case (Figure 5.13 (b)). The
reason for this qualitative dierence comes from the fact that for the pin-on-disc
tribosystem, the displacement due to wear is just 6.5 % of the total displacement while for the ring-on-ring case the contribution from wear is around 42 %
of the total displacement. This dierence in the contribution from wear to the
total displacement is due to the fact that the two triboelements are in contact
throughout the sliding duration in the ring-on-ring case, while in the pin-on-disc
case, each point on the disc in the wear track comes in contact with the pin only
once per revolution. In any wearing non conformal contact, the contact begins
to conform with sliding, but in the pin-on-disc tribosystem, the amount of linear
wear on the disc is much less compared to that on the pin (also see Figure 5.12).
However, the contribution from the elastic displacement to the total displacement
is progressively reducing with the conforming of the contact, thus the total displacement on the disc surface is progressively reducing with sliding. While in the
ring-on-ring case, the wear on both the upper and the lower ring is comparable to
each other (any dierence is only due to the dierence in the wear coecients).
Therefore, a reduction in total displacement is not observed.
5.3 Asymmetric wear due to elastic deformation
In this section, a study of the leading edge eect discussed in Subsection 5.2.2
will be discussed. Such an eect can be very eectively studied with the WearProcessor. The Wear-Processor was used to simulate wear in a polytetrauoQ
roethylene pin on steel disc tribometer which corresponds to point A ( s = 0.349)
in Figure 3.6. At point A, according to the graph in Figure 3.6, the leading edge
Q
eect can be quite considerable due to a high value of e (∼ 0.3). The location
of point A was determined by the used sliding distance of 0.55 mm in the wear
simulation. The wear coecient of kD = 92 × 10−8 mm3 /N · mm was used in
the wear simulation. A normal load of FN = 200 mN was applied on the pin
while the dimension of the pin and the disc was the same as that used in the wear
simulation discussed in the previous section (see Figure 5.7 (b)). The results at
point A are compared to point B (see Figure 3.6) which corresponds to the wear
simulation discussed in the previous subsection (silicon nitride on silicon nitride)
90
5.3. Asymmetric wear due to elastic deformation
Q
for a sliding distance of 8.6 mm. At point B, the value of s is 0.009148 and the
Q
corresponding value for e is 0.00846. Thus it can be expected that at point B,
the leading edge eect is comparatively much less.
In Figure 5.14, the results for the wear on the pin along the center line of the pin
in the direction of sliding for the two points A and B in Figure 3.6 is compared.
It can be seen from the graph in Figure 5.14 that for the results corresponding
to the point A, there is a higher wear on the front side of the pin compared to
the back side. However, this asymmetric eect is not observed at point B (in
Q
Figure 3.6), which corresponds to a one order of magnitude lower value for e
when compared to that at point A. Therefore for tribosystems with higher values
Q
of e , the Wear-Processor is better equipped to describe the topography of the
Q
worn surface satisfactorily. While in cases where the value of e is low, GIWM
can give satisfactory results.
Front side of Pin
0.08
Back side of Pin
B:
s
= (F
N
2
1/2
/ (E R k s))
C
P
D
= 0.009148
0.04
h
w
[µm]
0.06
A:
s
= (F
N
2
1/2
/ (E R k s))
C
P
D
= 0.349
0.02
0.00
0.00
0.01
0.02
0.03
0.04
0.05
|z| [mm]
Figure 5.14: Distribution of wear on the pin along
Qthe center line of the pin in
the direction of sliding for two dierent values of s (points A and B in Figure
3.6). A and B corresponds to polytetrauoroethylene on steel and silicon nitride
on silicon nitride respectively.
5. Wear Simulation
91
5.4 Remarks on the Wear-Processor
The Wear-Processor was used to simulate wear for both two dimensional and
three dimensional nite element models. A method was implemented in the
Wear-Processor to compute wear on both the surfaces from the contact pressure
calculated from the stress tensor (received as a nite element result) and the
computed surface normal vectors. Computation of the contact pressure avoids
the necessity to use computationally expensive symmetric contact. The WearProcessor employs an ecient re-meshing technique, thus the surface element
height does not limit the simulation of a pre-determined sliding distance. It was
also shown that the Wear-Processor can handle the interaction of triboelements
made of dierent materials, which have dierent wear coecients. It also allows
for a user determined nite element discretization of the tribosystem, thus removing the restriction of having to mesh with slender elements at the surface.
Also, the use of an articial grid for integrating the contact pressure over the
sliding distance practically allows for any type of mesh for the disc.
On the numerical aspects of the wear simulation tool, it was observed that the
accuracy of the simulation of dry sliding wear is very sensitive to the size of
the sliding distance increment and also the nite element discretization of the
contacting bodies. A good discretization and suitable boundary conditions have
to be found for other contacting geometries than the pin-on-disc so that the
results are suciently accurate and at the same time the computation time is
minimized. Decreasing the size of the sliding distance increment, increases the
number of computationally expensive nite element contact simulations. While
increasing the size of the sliding distance increment can lead to articial roughening/denting of the surface. An automatic sliding distance incrementation scheme
was implemented in the Wear-Processor for determining an optimal value for the
sliding distance increment (see Section 4.5) in order to avoid articial roughening/denting of the surface. It was observed that the solution of the contact
problem takes 50 % of the computation time of a sliding distance increment.
There is a scope for further improving the computational eciency either in the
solution of the contact problem or the Wear-Processor itself.
92
5.5. Discussion
5.5 Discussion
The main objective of this work was to develop a strategy for simulating wear
and predicting the life-span of micro-machines made from ceramics. As discussed
in Chapter 1, the rst step in implementing this strategy involves simulating pinon-disc experiments, conducted within the parameter space (in terms of contact
pressure and sliding velocity) of the micro-machine to identify the relevant parameters in the wear model. In Chapter 4, a nite element based post-processor, denoted as Wear-Processor, was presented. In the current chapter the results from
the wear simulation was presented. The salient features of the Wear-Processor
include; application of a wear model on the local scale, its ability to simulate wear
on three dimensional nite element models and its scope for handling arbitrary
geometry of tribosystems that could be made of dierent materials. However it
was seen in the current chapter that the Wear-Processor was computationally
expensive and therefore has to be used only when it is absolutely necessary for
satisfactorily describing the evolution of the worn surface.
In order to overcome this diculty, a computationally ecient incremental implementation of Archard's wear model on the global scale (Global Incremental
Wear Model - GIWM) for pin wear and disc wear in a pin-on-disc tribometer was
presented in Chapter 3. This ecient method of wear simulation can be very
handy for tribologists to quickly interpret their measured data from pin-on-disc
tribometers for most material combinations encountered in practical applications.
Additionally, the results presented in Chapter 3 conrm that the GIWM can
be used successfully to predict pin-on-disc experimental results within a limited
range. GIWM is indispensable for use as an easy tool in fast identication and
validation of wear models (e.g., Equation (2.36) given by Sarkar (1980)) or more
sophisticated wear models that include the dependency of velocity, temperature
etc., which however, is practically not possible with the Wear-Processor.
The results from the simulation of the pin-on-disc experiment for silicon nitride
using the Wear-Processor were shown to be in good agreement with that from the
GIWM, which, in the rst place tted the experiment favorably. Evaluation of the
Q
dimensionless parameters, the dimensionless system parameter ( s ) and thus the
Q
dimensionless elastic deformation ( e ) presented in Chapter 3, help to answer
5. Wear Simulation
93
the question as to when the Wear-Processor is necessary for simulating the pinon-disc experiment in order to identify the wear coecient. It was shown in this
chapter that for stier materials like silicon nitride, the GIWM can be used for the
Q
identication of the wear coecient on account of their lower s . As the WearProcessor works in association with a commercial nite element package for the
solution of the contact problem, it has the advantage that with some extension,
it can handle wear simulation in any geometrically dierent tribosystem.
A detailed summary on the ndings of this work will be presented in the following
chapter. However, in order to put the outcomes from this research work in the
right perspective, it should be noted that, two strategies for simulating wear
were developed during the course of this work. They include GIWM and the
Wear-Processor. A criteria based on the evaluation of the two dimensionless
Q
Q
parameters ( s and e ) was given for choosing the best strategy based on the
material combination and system parameters. With these strategies, it would be
possible to interpret experimental data and validate existing wear models and
also aid in the development and validation of new wear models for the correct
simulation of wear in micro-machines.
Chapter 6
Summary
"If I have seen further than others, it is by standing upon the shoulders of giants."
Isaac Newton
(1642-1727)
In this work, an important rst step towards simulating dry sliding wear in a tribosystem (micro planetary gear train) was presented. The strategy makes use of
experimental results from a pin-on-disc tribometer and nite element based wear
simulation. A nite element based software tool, denoted as Wear-Processor
was developed. It implements Archard's wear model on the local scale, which
is a rst and most simple wear model that can be assumed. The tool was used
to test the suitability of Archard's wear model for describing linear wear for silicon nitride from a pin-on-disc tribometer. Two useful dimensionless parameters,
Q
namely, the dimensionless elastic deformation ( e ) and the dimensionless sysQ
tem parameter ( s ) were introduced. These parameters were used to determine
the relative importance of the eect of elastic deformation on the computation
of wear. Evaluation of these parameters can assist in the decision on when a
computationally expensive nite element based wear simulation method has to
be used in order to realistically describe the evolution of the worn surface.
96
The Wear-Processor was used to simulate wear in a compliant pin on a stier disc
(e.g., polytetrauoroethylene on steel) for studying the asymmetric wear due to
elastic deformation coming from the friction between the contacting surfaces. It
Q
was shown that for a higher value of s (e.g., polytetrauoroethylene on steel),
there is considerable asymmetric pin wear, while, for a lower value of the same
parameter (e.g., silicon nitride on silicon nitride), the linear wear on the pin is
in fact axisymmetric. For use in the latter case, a computationally inexpensive,
incremental implementation of Archard's wear model on the global scale, Global
Incremental Wear Model (GIWM) for computing pin wear and disc wear in a
pin-on-disc tribometer was presented. The GIWM was used to identify the wear
coecient from a pin-on-disc tribometer and also it was shown that GIWM can be
used in a limited way to predict such experiments. The wear coecient identied
using the GIWM was used in the simulation of wear of the same experiment with
the Wear-Processor, where the results showed a good agreement with each other
and the deviations between the two remained well within the uncertainties in the
experimental measurement. It can be concluded that Archard's wear model is
valid both at the global and local scale in the particular case of silicon nitride on
Q
silicon nitride where s is low and thus, the eect of elastic deformation is negligible. In such cases, the parameter identication from the relevant experiment
can be done with the GIWM. In the future, the identied wear coecient in the
Archard's wear model can be used to predict wear in the micro planetary gear
train in whose parameter space the experiments were conducted. To achieve this
goal, the Wear-Processor needs to be generalized towards simulating wear in two
dimensional transient models of tribosystems, which is however the goal of future
research.
List of Figures
1.1
Strategy for wear prediction in micro-machines. The nite element
contact analysis (right hand side top picture) on the micro gear
(left hand side top picture) was done by Stammberger (2003). . .
2.1
(a) Contact pressure-clearance relationship for "hard" contact; (b)
Frictional behavior at the contact. . . . . . . . . . . . . . . . . . .
2.2
23
(a) Flow chart for the global incremental wear model (GIWM) for
computing pin wear; (b) computation of the contact radius. . . . .
3.2
19
Schematic of a typical interface when two nominally at surfaces
come in contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
18
Free body diagram of a contact surface segment showing the contact forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
17
A schematic of a slave surface penetrating the master surface after
the i − 1th iteration. . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
10
Contact zone between two bodies modeled with two dimensional,
four node nite elements. . . . . . . . . . . . . . . . . . . . . . . .
2.3
3
42
Results from the GIWM in comparison with the experimental results (Herz et al., 2004) from the pin-on-disc tribometer at two
dierent normal loads (200 mN and 400 mN ) . . . . . . . . . . . .
3.3
Schematic for the computation of the evolution of the real contact
area for disc wear. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
44
46
Flow chart for the global incremental wear model (GIWM) for
computing disc wear. . . . . . . . . . . . . . . . . . . . . . . . . .
97
47
98
LIST OF FIGURES
3.5
Comparison of the cross section prole for disc wear between
GIWM and experimental data for an applied normal load of 20 N
(a) and 40 N (c); Graph showing comparision between GIWM and
experimental data for the progress of maximum linear wear over
the sliding distance for the 20 N (b) and 40 N (d) normal load
3.6
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q
Graph of dimensionless elastic deformation, e vs. dimensionless
Q
system parameter, s for studying the eect of elastic deformation
49
on computation of wear. Points A and B will be discussed later in
Subsection 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7
52
Comparison between the GIWM and the wear model of Kauzlarich
& Williams (2001) for FN = 20 N , EC = 109.18 GP a and Rp =
1.5 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8
53
Graph showing the exponential t on the measured µ as a function
of s for (a) 200 mN , (c) 400 mN and (e) 800 mN normal load;
Comparison of kD as a function of µ from the experiments on
silicon nitride (Herz et al., 2004) and from the modied wear model
of Sarkar (1980) with kD = 10.2 × 10−9 mm3 /N · mm for (b)
200 mN , (d) 400 mN and (f) 800 mN normal load. . . . . . . . .
3.9
55
Results from the modied Archard's wear model (Equation (2.36))
implemented within the GIWM for pin wear in comparison with
the experimental results (Herz et al., 2004) from the pin-on-disc
tribometer at three dierent normal loads ( 200 mN , 400 mN and
800 mN ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
56
Flow chart of the Wear-Processor. The notations, utotal is the
total displacement, which is the algebraic sum of the elastic displacement (uelastic ) and the displacement computed due to wear
(uwear ), s is the current sliding distance and smax is the maximum
sliding distance. PATRAN is a commercial nite element pre- and
post-processor. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
61
Calculation of inward surface normal vector for a two dimensional
nite element model. Numerals enclosed within the circle indicate the element number and the plain numerals indicate the node
numbers of the nite element mesh. . . . . . . . . . . . . . . . . .
64
LIST OF FIGURES
4.3
99
Calculation of inward surface normal vector for a three dimensional nite element model. Numerals enclosed within the circle
indicate the element number and the plain numerals indicate the
node numbers of the nite element mesh. . . . . . . . . . . . . . .
4.4
64
A grid of points superimposed on the plot for the distribution of
normal stress on the top surface of the disc where the pin is engaged. 67
4.5
Detection of the surface element on which the grid point is located. 68
4.6
Optimal value for the maximum allowable wear. . . . . . . . . . .
5.1
Hemispherical brass ring rotating over a at steel ring in dry sliding
contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
71
74
σY Y plotted on a zoomed section of the contact between the upper
hemispherical brass ring and the lower at steel ring at dierent
stages of sliding: (a) initially when contact takes place; (b) after
9.9 × 104 rotations; (c) after 2.2 × 105 rotations; (d) after 3.5 × 105
rotations; (e) after 4.7 × 105 rotations; (f) after 6.0 × 105 rotations.
The range of the spectrum for all the plots is the same (Spectrum
Range: −10000 to −100 N/mm2 ). . . . . . . . . . . . . . . . . . .
5.3
76
(a) Graph of σY Y versus x-coordinate in the contact region on the
lower at ring; (b) Progress of wear on the upper (brass) and lower
(steel) ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
77
Finite element mesh of a hemispherical ring on a at ring before (a)
and after (b) 6 × 105 rotations; elastic deformations are included
in the gures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
78
Comparison of the number of contact simulations required for different values of δ with re-meshing and without re-meshing for performing 6 × 105 rotations. . . . . . . . . . . . . . . . . . . . . . .
5.6
78
(a) Model of pin-on-disc tribometer in dry sliding contact; (b)
Finite element model of the pin at an imtermediate stage of its
construction showing the two sweep directions for completing the
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7
81
Full (a) and Half (b) nite element model of the pin-on-disc tribometer (cut along the direction of sliding). . . . . . . . . . . . .
81
100
LIST OF FIGURES
5.8
Graph showing the (a) σY Y prole and (b) σZZ prole for the two
nite element models of the pin-on-disc tribometer in comparison
with the Hertz solution. . . . . . . . . . . . . . . . . . . . . . . .
5.9
82
(a) Comparison between the surface stresses in the direction of
sliding on the disc for a two dimensional and three dimensional
calculation; (b) Distribution of σZZ stresses on the top surface of
the disc (Spectrum Range: −500 to 100 N/mm2 ). . . . . . . . . .
83
5.10 (a) Comparison between the contact pressure distributions in the
direction of sliding for various friction coecients; (b) Deviation
of the contact pressure for various friction coecients related from
the frictionless case. . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.11 Contact stresses plotted on the cross section of the pin and the
disc (a) σY Y stresses before wear (Spectrum Range: −300 to
0 N/mm2 ); (b) σZZ stresses before wear (Spectrum Range: −500
to 100 N/mm2 ); (c) σY Y stresses after 71.36 mm of sliding (Spectrum Range: −300 to 0 N/mm2 ); (d) σZZ stresses after 71.36 mm
of sliding (Spectrum Range: −500 to 100 N/mm2 ); (e) Contact
pressure prole after various intervals of sliding; (f) σZZ prole
after various intervals of sliding. . . . . . . . . . . . . . . . . . . .
86
5.12 Graph of progress of wear over sliding distance for pin and the disc
in comparison with the GIWM for silicon nitride on silicon nitride. 87
5.13 (a) Graph of uY Y versus x-coordinate at surface nodes in the contact region on the lower at ring (ring-on-ring); (b) Graph of uY Y
versus z-coordinate at the surface nodes in the contact region of
the disc (pin-on-disc). . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.14 Distribution of wear on the pin along the center line of the pin in
Q
the direction of sliding for two dierent values of s (points A and
B in Figure 3.6). A and B corresponds to polytetrauoroethylene
on steel and silicon nitride on silicon nitride respectively. . . . . .
90
List of Tables
2.1
Friction coecients for various metals mating with itself (Roberts,
2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
24
Values for the parameters in the exponential t (Equation (3.16))
for the coecient of friction, µ, as a function of the sliding distance,
s, at three dierent normal loads (200 mN , 400 mN and 800 mN ). 54
5.1
Geometrical parameters for the pin-on-disc tribometer (see Figure
5.6 (a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
80
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Resume
Personal Details:
Name:
Vishwanath Hegadekatte
Address:
Hirschstrasse 16, D-76133, Karlsruhe, Germany
Date / Place of birth:
16 Jun 1974 / Hubli, India
Marital status:
Married since 24 May 2002 to Rashmi Padiadpu
Education:
1980 - 1987
Primary School
1987 - 1990
High School
1990 - 1992
Pre-University College
1992 - 1996
Bachelor of Engineering (Mechanical), Karnatak
University, India
1999 - 2001
Master of Science (Mechatronics), Technical University of Hamburg-Harburg, Germany
Experience:
1996 - 1997
Engineer - Purchase, Kinetic Honda, India
1997 - 1998
Engineer - Marine, Mitsui OSK Lines, Japan
Feb 2002 - Sep 2003
Associate Scientist, Institute for Materials Research II (IMF II), Forschungszentrum Karlsruhe
GmbH, Germany
Oct 2003 - Feb 2006
Associate Scientist, Institute for Reliability of
Components and Systems (IZBS), University of
Karlsruhe, Germany
Karlsruhe, 21 Feb 2006