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 su cient 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 e cient 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 coe cient 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 signi cant eect on the computation of wear especially in case of more compliant materials, in the early stages of sliding, or for very low wear coe cients. Therefore, two dimensionless parameters are introduced to estimate the in uence of elastic deformation on the computation of wear. iii
In order to apply the identi ed 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 identi ed wear coe cient 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., polytetra uoroethylene on steel), there is a signi cant 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 Kostene zienz 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 e ziente 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 Verschleiÿkoe zienten 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 Verschleiÿkoe zienten oder in der Anfangsphase des Gleitens ein sichtbarer Eekt. Es werden daher zwei dimensionslose Parameter eingeführt, die eine Abschätzung des Ein usses der elastischen Deformation auf die Berechnung des Verschleiÿes erlauben. Um das aus Tribo-Experimenten mit Hilfe des GIWM identi zierte 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 identi zierten Verschleiÿkoe zient 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. Polytetra uorethylen auf Stahl) einen signi kanten 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 bene ted 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 speci cally 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 con dence 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 o cemates Dr. Vladimir Govorukha, Mr. Tzvetelin Chehtov and Mr. Dong Wang for setting up a warm and cheerful atmosphere at the o ce. 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 coe cient 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 coe cient. This is sometimes non-dimensionalised by dividing by the hardness of the softer surface and is termed as the wear coe cient. Dimensional wear coe cient 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 su cient 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 identi ed 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 coe cient. The identi cation of a local wear model/parameters from experiments requires numerical simulations, since only global quantities can be readily identi ed from tribometry. Global refers to the fact that the geometry of the tribometer is implicitly included in the identi ed 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 identi ed from experiments to simulate wear assuming that the global model is applicable on the local scale. Podra & Andersson (1999) identi ed the wear coe cient from experiments and showed that the local application of Archard's wear model using the identi ed global parameter yielded good results in comparison to the experiments. Further, they used this identi ed parameter for Archard's wear model to simulate wear in cone-on-cone and cone-on-torus tribosystems. In their work, the identi cation 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 identi cation 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 signi cant, thus indicating that assumption of axisymmetric nite element model for the pin-on-disc tribometer is not su cient 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 di culties 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 de nitions for the study of tribology from the computational standpoint will be brie y 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 di cult 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 signi cantly 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 coe cient 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 di cult 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
ˆ u(M ) (x, y, z) = H(M ) (x, y, z)U,
(2.8)
ˆ is the vector of global where H(M ) is the displacement interpolation matrix, U displacements at all nodal points. The variation of u(M ) is given by:
ˆ δu(M ) (x, y, z) = H(M ) (x, y, z)δ U.
(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:
ˆ ε(M ) (x, y, z) = B(M ) (x, y, z)U,
(2.10)
where B(M ) is the strain displacement matrix. The variation of ε(M ) is given by:
ˆ δε(M ) (x, y, z) = B(M ) (x, y, z)δ U.
(2.11)
Substituting for the element displacements and strains we obtain:
δΠ =
XZ
T
V (M )
M
−
ˆ T B(M ) D(M ) B(M ) UdV ˆ (M ) δU
XZ M
−
XZ
−
(M )T
fV
(M )
dV (M )
V (M )
ˆ T HS δU S (M )
M
X
ˆ T HV δU
ˆ T FC + δU
(M )T
fS
(M )
dS (M )
i Xh T T ˆ k T ˆ ˆ ˆ δ λ (Q U − ∆ ) + δ U Qλ .
(2.12)
k(M )
C (M )
Applying the principle of the stationarity of the total potential we get:
ˆT δU
" XZ M
# T
ˆ B(M ) D(M ) B(M ) dV (M ) U V
ˆT = ∂U
(M )
XZ M
ˆT +∂ U
T
HB(M ) f B(M ) dV (M ) V (M )
XZ
T
HS(M ) f S(M ) dS (M ) S (M )
M
ˆT −∂ U
X
k(M )
−
X
ˆT ˆ + ∂U Qλ
X
FC
(M )
C h i ˆ T QT U ˆ − ∆k , δλ
(2.13)
k(M )
ˆ is the vector of all the where F is the vector of all the concentrated forces, λ 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 ˆ = I. δU The equilibrium equations of the element assemblage corresponding to the nodal point displacements is given by: # # " #" # " " ˆ U R Rk K Q + = ˆ ∆ QT 0 λ 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 )
ˆT . Rk = −Qλ 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
#"
ˆ (i) ∆U ˆ (i) ∆λ
#
"
t+∆t
= " +
R
0 t+∆t
#
"
t+∆t
− (i−1)
Rk t+∆t (i−1) ∆
# ,
F(i−1) 0
#
(2.15)
ˆ (i) is the vector containing the nodal displacements in the ith iteration where ∆U ˆ (i) is the vector containing the nodal contact forces in the ith ˆ (i) − U ˆ (i−1) ), ∆λ (= U
16
2.1. Basics of contact mechanics
ˆ (i) − λ ˆ (i−1) ) (see Subsection 2.1.2), iteration (= λ
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 de ned 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 de ned 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 =
¯ Tr β (i−1) = n
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) = n ¯ Tr ∆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 de ned 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 coe cient of friction (Equation (2.4)) is not a simple material property. The coe cient 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 coe cient decreases and at any intermediate range, later the friction coe cient 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
coe cient 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 coe cient, 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 coe cients for various metals mating with itself (Roberts, 2005). Material Gold Copper Chromium Silver Magnesium Lead
Friction Coe cient 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 simpli ed 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 coe cient 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 de ned 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) classi es 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 classi ed 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 signi cant 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 classi ed 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 identi ed to deal with the wear problem. The correlation between the friction coe cient 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 coe cient 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 coe cient 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 coe cient) 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 arti cial 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 modi ed to approximate the contact parameters for the wearing process, thus making the simulation scheme speci c to the particular case of a spherical tipped pin on a at disc. Christo des 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 coe cient and kD is the dimensional wear coe cient. 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 di cult to accurately measure the wear volume since the boundaries of the wear scar are established subjectively (Kalin & Vizintin, 2000). The wear coe cient 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 re ecting the ine ciencies 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 coe cient. 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 coe cient 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 coe cient of friction and the wear rate is complicated. However, Sarkar (1980) has given a wear model that relates the friction coe cient 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 coe cient 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 modi cation 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 e cient 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) simpli cation 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 coe cient 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 speci c 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-de ned 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 identi ed 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 di culties 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 & Butter eld (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 signi cance 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 pro lometry of the disc specimens. The wear coe cient (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 coe cient identi ed from such experiments implicitly includes the geometry speci c to the tribometer and also the decay of the contact pressure as the wear progresses is not considered. Thus, the wear coe cient identi ed 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 coe cient of friction over sliding distance and normal load, plot of wear coe cient over coe cient 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 speci c 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 speci cally 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 coe cient, ∆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 identi ed 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 identi ed wear coe cient 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 modi ed 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 identi ed to be 21 × 10−11 mm3 /N · mm. Figure 3.5 (a) shows a comparison between the cross section pro le 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 pro le 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 identi ed 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 coe cients 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 polytetra uoroethylene 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 coe cient using the GIWM for computing disc wear and then using the identi ed 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 signi cant only in the initial stages of sliding. But, the data points for the polytetra uoroethylene 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 signi cant 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 coe cient using GIWM In order to investigate the eect of the coe cient of friction, µ, on the evolution of linear wear for the pin, Archard's wear model implemented in the GIWM was modi ed according to Equation (2.36) (Sarkar, 1980). The coe cient 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 de ning the change of µ with increasing s. The t was made using the measured coe cient 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 coe cient 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 modi ed GIWM for pin wear was used to t the 200 mN normal load experiment (see Section 3.1). The identi ed 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 modi ed 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 modi ed 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 identi ed 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 in uence on the wear behavior for silicon nitride. The modi ed 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 con rm 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 e cient re-meshing technique to avoid severe deforming of the mesh, (iv) it uses Coulomb friction, where the friction coe cient 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 in nitesimal 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 in nitesimally 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 coe cient. 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 coe cient.
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 modi ed 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 e ciency, 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 (in nitesimal 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 coe cient, 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 de nite 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 su cient 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 identi ed 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 satis ed.
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 coe cients 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 coe cients 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 e cient 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 signi cance both from the point of view of the computational expense and more importantly on the stability of the analysis. Such di culties 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 arti cial 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 coe cients (also see Hegadekatte et al. (2005c,b,a)), however, this strategy for real values of the wear coe cient resulted in a dent on the pin surface in the contact region. For a contact pressure pro le as shown in Figure 4.6, the calculated wear pro le would look like the dashed line in the same gure for high values of ∆si (dent formation). The dent is formed due to the insu cient frequency of updating of
4. Finite Element Based Wear Simulation Tool
71
the contact pressure due to wear. Such a wear pro le 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 coe cient, ∆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 arti cial 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 coe cient identi ed 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 coe cient for brass (Lancaster, 1962) is around four
5. Wear Simulation
75
times higher than that for steel (Podra & Andersson, 1999). The dimensional wear coe cient 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 coe cient 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 coe cients, 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 coe cient 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 coe cients 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 coe cient, 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 identi ed 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 in nitesimal 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 coe cient 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 e cient 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 pro le and (b) σZZ pro le 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 in nitesimal 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 coe cients; (b) Deviation of the contact pressure for various friction coe cients related from the frictionless case. clari ed 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 coe cient (Hegadekatte et al., 2005b). The curve for the contact pressure distribution shifts slightly in the direction of sliding with increasing friction coe cient 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 coe cient 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 coe cient, 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 pro les 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 pro le after various intervals of sliding; (f) σZZ pro le 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 pro le at the contact edge. However, since the integral of the contact pressure pro le 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 pro les 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 con guration, 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 con guration.
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 su ciently 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 coe cients). 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 polytetra uoQ 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 coe cient 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 polytetra uoroethylene 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 e cient 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 coe cients. 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 arti cial 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 su ciently 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 arti cial 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 arti cial 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 e ciency 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 di culty, a computationally e cient 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 e cient 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 con rm 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 identi cation 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 coe cient. It was shown in this chapter that for stier materials like silicon nitride, the GIWM can be used for the Q identi cation of the wear coe cient 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., polytetra uoroethylene 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., polytetra uoroethylene 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 coe cient 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 coe cient identi ed 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 identi cation from the relevant experiment can be done with the GIWM. In the future, the identi ed wear coe cient 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 pro le 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 modi ed 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 modi ed 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 pro le and (b) σZZ pro le 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 coe cients; (b) Deviation of the contact pressure for various friction coe cients 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 pro le after various intervals of sliding; (f) σZZ pro le 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 polytetra uoroethylene on steel and silicon nitride on silicon nitride respectively. . . . . .
90
List of Tables 2.1
Friction coe cients for various metals mating with itself (Roberts, 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
24
Values for the parameters in the exponential t (Equation (3.16)) for the coe cient 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
