Dissipative Processes in Tribology

DISSIPATIVE PR0CESSES IN TRIBOLOGY

TRIBOLOGY SERIES, 27 EDITOR: D. DOWSON

DISSIPATIVE PROCESSES IN TRIBOLOGY edited by

D. Dowson, C.M. Taylor, T.H.C. Childs, M . Godett and G. Dalmaz Proceedings of the 20th Leeds-Lyon Symposium on Tribology held in the Laboratoire de Mecanique des Contacts, lnstitut National des Sciences Appliquees de Lyon, France 7th-10th September 1993

ELSEVIER Amsterdam

London New York Tokyo

1994

For the Institute of Tribology, The University of Leeds and The lnstitut National des Sciences Appliquees de Lyon

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 21 1,1000 AE Amsterdam, The Netherlands

L i b r a r y o f Congress C a t a l o g i n g - i n - P u b l i c a t i o n

Data

Leeds-Lyons Symposium on T r i b o l o g y ( 2 0 t h : 1993 : I n s t l t u t n a t i o n a l des s c i e n c e s a p p l i q u i e s de L y o n ) D i s s i p a t i v e processes i n t r i b o l o g y : p r o c e e d i n g s o f t h e 2 0 t h Leeds -Lyon Symposium on T r i b o l o g y h e l d i n t h e L a b o r a t o l r e de mecanlque des c o n t a c t s , I n s t i t u t n a t i o n a l des s c i e n c e s a p p l i q u i e s de Lyon. France, 7 t h - 1 0 t h September 1993 1 e d i t e d by D. Dowson [et al.1. p. cm. -- ( T r i b o l o g y s e r i e s ; 27) I n c l u d e s b l b l i o g r a p h i c a l r e f e r e n c e s and i n d e x . ISBN 0-444-81764-6 ( a c i d - f r e e p a p e r ) 1. Trlbology--Congresses. I.Dowson. D. 11. U n i v e r s i t y o f Leeds. I n s t l t u t e of Tribology. 111. I n s t i t u t n a t i o n a l des s c i e n c e s IV. Tltle. V. S e r i e s . a p p l i q u e e s de Lyon. TJ1075.A2L43 1993 621.8'9--d~20 94- 18428 CIP

...

ISBN 0 444 81764 6 (Vol. 27)

0 1994 ELSEVIER SCIENCE B.V. All rights reserved. No part of this publication may be reproduced, stored i n a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred t o the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained i n the material herein. This book is printed on acid-free paper Printed i n The Netherlands

V

Proceedings of the 20th Leeds-Lyon Symposium on Tribology INTRODUCTION The twentieth Leeds-Lyon Symposium on Tribology was held at the Institut National des Sciences AppliquCes de Lyon from Tuesday 7th to Friday 10th September 1993. It discussed dissipative phenomena and particularly the origins of fiction at all scales, fiom all points of view coming fiom mechanics, physics and chemistry, encountered in all fields of Tribology, from thick film lubrication to dry friction. The Symposium opened on the Tuesday afternoon with two keynote lectures delivered by Dr Irwin Singer fiom the US Naval Research Laboratory, Washington DC, U S 4 on "Energy Dissipation during Friction : Interfacial Processes" and by Professor Kenneth Johnson from Cambridge University, UK, on "The Mechanics of Adhesion, Deformation and Contamination in Friction". They clearly showed the hndamental and pluridisciplinary aspects of dissipative processes in tribology. The meeting was attended by some one hundred and forty delegates from sixteen countries of North America, Asia and Europe and by fourteen accompanying persons. Thirty per cent of the delegates were from industry. It was again pleasant to welcome a large and active group from the University of Leeds, our sister institution. The Symposium Review Board had examined and selected the abstracts of more than eighty submitted papers. The organisers decided for the second time, instead of parallel sessions, to have classical sessions reserved for twenty minute paper presentations and hybrid sessions which included a five minute oral presentation followed by a poster session. The twentieth Symposium was dedicated to Professor Duncan Dowson, in honour of his retirement as a member of the staff at the University of Leeds. The Vice Lord Mayor of Lyon, hosted the delegates at a reception in the Lyon Town Hall to honour and recognise through the 20th LeedsLyon Symposium anniversary, the international scientific activity in the field of Tribology of the two co-founders of the Leeds-Lyon Symposia. Professor Duncan Dowson from the Institute of Tribology of the University of Leeds and Professor Maurice Godet, from the Laboratoire de Mecanique des Contacts of INSA of Lyon. The traditional Symposium banquet was held in the "Salle de la Corbeille du Palais de la Bourse" in Lyon. The dinner was prepared by one of the well known Chef of Lyon, Gilles Troump. On the Thursday evening, the delegates were invited the celebrate the 20th Leeds-Lyon anniversary in the Conservatoire National Superieur de Musique de Lyon. In this famous place, they discovered eight centuries of music in Lyon and the pleasure of listening to 500 years of fanfares played by the "Ensemble Cuivres et Percussion" of the "Orchestre National de Lyonll.

vi

The concert programme had been specially chosen by Maurice Godet to show how music can magnify science. It was a superb musical evening with Maurice. The usual Friday evening barbecue party was arranged by the Laboratory staff. The Saturday tour took some delegates on a trip through the Jura in Franche Comte to visit the fabulous Arc-etSenans Royal salt-works, to see the beautiful valley of the River Loue and test yellow wines. The success of the Symposium must be attributed to all members of the Laboratoire de Mecanique des Contacts. They are individually congratulated for their contribution and active participation in the Symposium organisation. Warmest thanks to all. Thanks are also due to the Direction des Recherches Etudes et Techniques, Direction GenCrale de I'Armement, Ministere de la Defense, Paris, France, for his financial support for the 20th LeedsLyon Symposium on Tribology. The topics covered by the Leeds-Lyon series of Tribology symposia are listed below: 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17.

18. 19. 20.

Cavitation and Related Phenomena in Lubrication SuperlaminarFlow in Bearings The Wear of Non-Metallic Materials Surface Roughness Effects in Lubrication Elastohydrodynamic Lubrication and Related Topics Thermal Effects in Tribology Friction and Traction The Running-In Process in Tribology The Tribology of Reciprocating Engines Numerical and Experimental Methods Applied to Tribology Mixed Lubrication and Lubricated Wear Global Studies of Mechanisms and Local Analyses of Surface Distress Phenomena Fluid Film Lubrication - Osborne Reynolds Centenary Interface Dynamics Tribological Design of Machine Elements Mechanics of Coatings Vehicle Tribology Wear Particles : From the Cradle to the Grave Thin Films in Tribology Dissipative Processes in Tribology

Leeds Lyon Leeds Lyon Leeds Lyon Leeds Lyon Leeds Lyon Leeds

1974

Lyon Leeds Lyon Leeds Lyon Leeds Lyon Leeds Lyon

1985

1975 1976 1977 1978 1979 1980 1981 1982 1983 1984

1986 1987 1988 1989 1990 1991 1992 1993

vii

Delegates to the Symposium and the international community of tribologists were deeply saddened by the news, so soon aRer the Symposium, of the untimely death of our dear friend and colleague Professor Maurice Godet on October 9th 1993. The 21st Symposium, to be held in Leeds from September 6th - 9th 1994 on the subject of "Lubricants and Lubrication", will be dedicated to the life of Maurice Godet. Gerard Dalmaz

...

Vlll

CONTENTS Introduction Session I

Session II

V

Opening Session Friction and Energy Dissipation at the Atomic Scale A Review I.L. SINGER The Mechanics of Adhesion, Deformation and Contamination in Friction K.L. JOHNSON

-

Liquid and Powder Lubrication A Rheological Basis for Concentrated Contact Friction S. BAlR and W.O. WINER On the Theory of Quaisi-Hydrodynamic Lubrication with Dry Powder : Application to Development of High-speed Journal Bearings for Hostile Environments H. HESHMAT The Influence of Base Oil Rheology on the Behaviour of VI Polymers in Concentrated Contacts P.M. CANN and H.A. SPIKES Temperature Profiling of EHD Contacts Prior to and During Scuffing J.C. ENTHOVEN and H.A. SPIKES Computational Fluid Dynamics (CFD) Analysis of Stream Functions in Lubrication D. DOWSON and T. DAVID Shear Properties of Molecular Liquids at High Pressures A Physical Point of View E.N. DIACONESCU

-

Session 111

Session iV

1 3 21 35 37

45 65 73 81 97

Surface Damage and Wear Magnetic Damage in Mn-Zn and Ni-Zn Ferrites Induced by Abrasion Y. AHN, R. HEBBAR, S.CHANDRASEKAR and T.N. FARRIS Effects of Surface Roughness Pattern on the Running-In Process of Rolling/Sliding Contacts J. SUGIMARA, T. WATANABE and Y. YAMAMOTO Influence of Frequency and Amplitude Oscillations on Surface Damages in Line Contact J. PEZDlRNlK and J. Vlz NTlN Effects of Surface Topography and Hardness Combination Upon Friction and Distress of Rolling/Sliding Contact Surfaces A. NAKAJIMA and T. MAWATARI Anti-Wear Performance of New Synthetic Lubricants for Refrigeration Systems with New HFC Refrigerants T. KATAFUCHI, M. KANEKO and M. IlNO

115

Miscroscopic Aspects A Molecularly-Based Model of Sliding Friction J.L. STREATOR Friction of Dielectric Materials : How is Energy Dissipated? B. VALLAYER, J. BIGARRE, A. BERROUG, S. FAYEULLE, D. TREHEUX, C. Le GRESSUS and G. BLAISE

171

117 125 139

151 163

173 185

ix

Friction Energy Dissipation in Organic Films B.J. BRISCOE and P. THOMAS Session V

193

Polymers Interfacial Friction and Adhesion of Wetted Monolayers J.-M. GEORGES, A. TONCK and D. MAZUYER Effect of Thickness on the Friction of Akulon A Problem of Constrained Dissipation L. ROZEANU, S. DIRNFELD and J. YAHALOM Interface Friction and Energy Dissipation in Soft Solid Processing Operations M.J. ADAMS, B.J. BRISCOE and S.K. SHINA The Effect of InterfacialTemperature on Friction and Wear of Thermoplastics in the Thermal Control Regime F.E. KENNEDY and X. TlAN

-

Session VI

Session VII

Session Vlll

Friction in Specific Applications The Relation Between Friction and Creep Deformation in Articular Cartilage K. IKEUCHI, M. OKA and S. KUBO Characteristics of Friction in Small Contact Surface Y. ANDO, H. OGAWA and Y. ISHIKAWA Sliding Friction in Porous and Non-Porous Elastic Layers: The Effect of Translation of the Contact Zone Over the Porous Material L. CARAVIA, D. DOWSON, J. FISHER, P.H. CORKHILL and B.J. TIGHE The Effect of Additive of Silane Coupling Agent to Water for the Lubrication of Ceramics K. MATSUBARA. S. SASANUMA and K. NAGAMORI The Origin of Super-Low Friction Coefficient of MoS2Coatings in Various Environments C. DONNET, J.M. MARTIN, Th. le MOGNE and M. BELIN

203 205 213 223 235 245 247 253 26 1 267 277

Coatings and Thin Films (Short Oral Presentations Associated with Posters) Characterisation of ElastioPlastic Behaviour for Contact Purposes on Surface Hardened Materials P. VIRMOUX, G. INGLEBERT and R. GRAS On the Cognitive Approach Toward Classification of Dry Triboparticulates H. HESHMAT and D.E. BREWE Surface Chemistry Effects on Friction of Ni-P/PTFE Composite Coatings E.A. ROSSET, S. MISCHLER and D. LANDOLT Transfer Layers in Tribological Contacts with Diamond-Like Coatings J. VIHERSALO, H. RONKAINEN, S. VARJUS, J. LIKONEN and J. KOSKINEN Surface Breaking Crack Influence on Contact Conditions. Role of Interfacial Crack Friction. Theoretical and Experimental Analysis M.-C. DUBOURG, T. ZEGHLOUL and B. VILLECHAISE

285

Macroscopic Aspects, Friction Mechanisms The Generation by Friction and Plastic Deformation of the Restraining Characteristics of Drawbeads in Sheet Metal Forming Theoretical and Experimental Approach E. FELDER and V. SAMPER

359

287 303 329 337

345

-

36 1

X

A Model for the Estimation Of Damping In Helical Strand Under Bending Vibration A. HADJ-MIMOUNE and A. CARDOU Energy Dissipation and Crack Initiation in Fretting Fatigue D. NOWELL, D.A. HILLS and D.N. DAI

Session IX

Session X

Session XI

Session XI1

373 389

Energy and Friction : Theoretical and Numerical Aspects Friction in Partially Lubricated Conjunctions 1.1. KUDISH and B.J. HAMROCK Third Body Theoretical and Numerical Behaviour by Asymptotic Method G. BAYADA, M. CHAMBAT, K. LHALOUANI and C. LICHT Thermomechanical State Near Rolling Contact Areas K. DANG VAN and M.H. MAITOURNAM

397

Thermal Power Dissipation in Machines Thermal Dissipation in Elliptical Bore Bearings M.T. MA and C.M. TAYLOR Material Dissipative Processes in Automotive Engine Exhaust Valve - Seat Wear Z. LIU and T.H.C. CHILDS Thermal Matching of Tribological Systems A.V. OLVER Power Loss Prediction in High-speed Roller Bearings D. NELIAS, J. SEABRA, L. FLAMAND and G. DALMAZ Power Dissipation in Elastohydrodynamic Traction Drives I.M. CIORNEI, E.N. DIACONESCU, V.N. CONSTANTINESCU and G. DALMAZ

429

General Aspects of Friction (Short Oral Presentations Associated with Posters) Frictional Heating of Elliptic Contacts J. BOS and H. MOES Soil-Structure Interface Friction in Reinforced Soils F. BAHLOUL, Y. BOURDEAU and V. OGUNRO Diagrams for Estimation of the Solidified Film Thickness at High Pressure EHD Contacts N. OHNO. N. KUWANO and F. HIRANO

489

Fatigue and Damage Fracture Modes in Wear Particle Formation A.A. TORRANCE and F. ZHOU The Influence of Lubricant Degradation on Friction in the Piston Ring Pack R.I. TAYLOR and J.C. BELL High Speed Damage Under Transient Conditions 0. LESQUOIS, J.J. SERRA, P. KAPSA and S. SERROR Incipient Sliding Analysis Between Two Contacting Bodies. Critical Analysis of Friction Law T. ZEGHLOUL, M.C. DUBOURG and B. VILLECHAISE

519

399 41 5 423

431 445 453 465 479

491 501 507

521 531 537 549

Written Discussion

559

List of Delegates

567

SESSION I OPENING SESSION Chairman:

Professor M Godet

Paper I (i)

Friction and Energy Dissipation at the Atomic Scale A Review

Paper II (ii)

The Mechanics of Adhesion, Deformation and Contamination in Friction

-

This Page Intentionally Left Blank

Dissipative Processes in Tribology / 1994 Elsevier Science B.V.

D.Dowson et al. (Editors)

3

Friction and energy dissipation at the atomic scale - a review I.L. Singer Code 6170, U.S. Naval Research Laboratory Washington DC 20375, USA ABSTRACT Discussions of energy dissipation during friction processes have captured the attention of engineers and scientists for over 300 years. Why then do we know so little about either dissipation or friction processes? A simple answer is that we can not see what is taking place at the interface during sliding. Recently, however, devices such as the atomic force microscope have been used to perform friction measurements, characterize contact conditions and even describe the "worn surface." Following these and other experimental developments, friction modelling at the atomic level -- particularly molecular dynamics simulations -- has brought scientists a step closer to "seeing" what takes place during sliding contact. With these investigations have come some answers and new questions about the modes and mechanisms of energy dissipation at the sliding interface. This paper will review recent results of 1) molecular dynamics and other theoretical studies that have identified modes of energy dissipation during friction processes and 2) friction experiments that have added to our understanding of dissipation processes and friction behavior. Finally, several approaches for addressing the questions of dissipation mechanisms will be presented. 1. INTRODUCTION.

Friction can now be studied at the atomic scale, thanks to developments in the past decade of a variety of experimental techniques [I]. The most well known techniques, often referred to as proximal probes, have been derived from Scanning Tunneling Microscopy (STM)(21; they include Atomic Force Microscopy (AFM) [3] and its sliding companion Friction Force Microscopy, (FFM) (4-61. These probes allow friction to be studied with atomic resolution in all three dimensions. Another proximal probe, generically known as a Surface Force Apparatus (SFA), affords atomic resolution only in the vertical direction, but allows direct measurement and/or control of micrometer-sized areas of contact in the lateral direction (7-121. A very recent technique, based on the Quartz Crystal

Microbalance (QCM), permits sliding friction processes to be studied at the Angstrom level and at time scales in the nanosecond range [13-

IS]. Although friction processes may originate at the sliding interface, the measurement of friction is usually performed by macroscopic devices -springs, levers, dashpots, etc -- often located far from the interface. In order to link measured frictional forces with theory, it has been necessary to examine the influence of mechanical parameters, such as stiffness, on friction measurements, not an unfamiliar problem to tribologists [I6,17]. Unlike traditional friction devices, however, proximal probes are sensitive to mechanical properties of the device at distances as close as the first atomic layer of the tip and as far away as the compliant lever arm (18-201.

4

The opening of experimental studies of friction at the nanometer and nanosecond scale has attracted theorists equipped to model physical and chemical processes at these scales. Surface scientists are now using sophisticated solid-state potentials to calculate mechanical interactions between surfaces [20]. Friction force calculations are performed either analytically or by molecular dynamics (MD) simulations [22,22/. MD simulation affords the added opportunity of using video animation to study friction processes. For the first time, scientist can "see what is happening" at the otherwise buried sliding interface. As you will read shortly, they see atomic and molecular excitation modes (vibrations, bending and rotations), electron-hole excitations, density waves and molecular interdigitation, to name a few. Once the modes are identified, physicists and chemists can address perhaps the most fundamental but least understood aspect of friction: energy dissipation processes. While it has been known for centuries [23,24] that most frictional energy dissipates as heat, neither the macroscopic nor microscopic mechanisms of energy dissipation have been fully explained. In a companion paper presented at this Leeds-Lyon conference, Ken Johnson discusses adhesion and deformation contributions to energy dissipation in friction. Here I review some recent theoretical and experimental studies of atomicscale friction behavior and modes of energy dissipation. Before launching into the studies, it is useful to review the thermodynamic criteria used to study energy dissipation. Clearly, if friction processes generate heat, they are irreversible and cannot be treated by classical thermodynamics. If, however, each step in a sliding interaction is executed with infinite slowness and with the two couples always in equilibrium (never an unbalanced force), then the process may be considered reversible. In such a quasi-static process, sliding can be achieved with zero

friction. Real systems can approximate such reversible, adiabatic processes so long as the rate at which each step is taken is much slower than the relaxation time of the system [25]. However, any instability in the mechanical system that leaves unbalanced forces will result in an irreversible process in which energy is lost (or, more precisely, unrecoverable). In a mechanical system, where force, F, acts over a distance, r, the fraction of energy, U, lost over a cycle is given by

AU

=

fF*dr.

In an atomic sliding calculation, a cycle can be a translation across some periodic distance of the lattice, e.g., one atomic spacing. In an experiment, one cycle can be a single pass over a surface, including the making and breaking of the contact. This paper is presented in the following sequence. The second section deals with theoretical approaches used to examine friction processes and friction measurements at the atomic scale. Conditions that can give zerofriction and finite friction in simple systems are presented. Two molecular dynamics simulations of more realistic yet "wearless" friction studies are described and modes of energy dissipation are identified. The third section presents two experimental approaches that have succeeded in identifying microscopic friction process; in both cases, energy losses are identified with hysteresis losses in the system. The fourth section summarizes our understanding of interfacial friction processes and energy dissipation mechanisms, and the fifth section considers ways that atomistic approaches can be used to solve practical problems in tribology.

5

Fig. I . Two representations of the motion of atom Bo in the independent oscillator model: atom-on-spring and atom-in-potential. Top and bottom rows depict strong and weak inter$acial interactions, respectively. i%e left-most diagrams display the relevant potentials (see text); subsequentpanels illustrate the response of Bo to progressive sliding of the lower layer of atoms. B, in the combined potential, V', is represented by a black dot plotted below each atom-on-spring diagram. From [26,27]. 2. THEORETICAL APPROACHES

2.1. Frictionless Sliding. Two analytical studies that identify conditions for zero friction behavior and the transition to finite friction are reviewed. The first investigates the potential seen by a moving atom, while the second examines the force experienced by the tip of a FFM. Although both present surface interactions at zero degrees Kelvin, they establish baseline criteria for zero friction systems and measurements of zero friction by which more general calculations can be evaluated. McClelland [26,27] describes the sliding friction behavior of a simple two-dimensional couple consisting of two substrates: the stationary upper substrate has an atom, $, attached to a spring while the moving lower

substrate consists of equally-spaced, rigid atoms (see Fig. 1). Because of the periodicity of the lower layer, the entire sliding behavior can be analyzed by following the motion of atom B, across one atomic spacing. According to this independent oscillator (IO) model, atom B, experiences forces exerted by a spring above and the atoms below; the forces are derived from potentials VBB and Vm, respectively. As the lower substrate moves, the combined interaction potential, Vs, changes. The changing potential and the atomic trajectory of atom B, are depicted in the five sketches running left to right across the figure. The upper sketches represent a "strong" interaction and the lower, a "weak" interaction. The atom's position on a spring is shown as an open circle and its position in the potential V, by a solid circle. In a "strong" potential, the atom is initially repelled to the

6

right; then beyond half an atomic spacing, it "snaps" back to the left, the atomic equivalent of "stick-slip." The snap back, or "plucking" motion, can be understood in terms of the evolving shape of potential Vs. In an adiabatic process, an atom must always sit at a minimum in the potential well. As can be seen, however, the position of the atom at (d) is only a local minimum, which disappears by position (e). At the moment the local metastable minimum "flattens out," there are unbalanced forces on atom B,; to regain an equilibrium state, the atom falls to the position of the stable minimum. Since this transition doesn't occur under equilibrium conditions, the process is irreversible and the energy lost in the fall cannot be reused to assist the sliding process. Stated in more physical terms, the strain energy put into stretching the spring at B, is not recovered locally; instead, it is converted into vibrational motion which dissipates into the substrate (as heat). This instability can be avoided by using a "weaker" interaction potential. As shown in the lower sequence of sketches in Fig. 1, the weaker potential does not develop a local minimum. The atom moves smoothly through a repulsive then an attractive force field, first being repelled by, then pushing, the lower substrate. Since the system remains in equilibrium throughout the cycle, no energy is dissipated and the friction force is zero. McClelland then gives more precise criteria for stability and shows that qualitatively similar friction behavior occurs with other more complex sliding couples. Tomanek, et al. [28,29] also describe conditions associated with frictionless sliding, emphasizing the mechanical properties of the apparatus as well as the

L

Fig. 2. Model of a Friction Force Microscope. External suspension M is guided along a horizontal suvace in the x direction. m e load Fa, is kept constant along the trajectory shown by arrows. The spring-tip assembly is elastically coupled to the suspension M in the horizontal direction by a spring of constant c. 7hefriction force FJ is related to the elongation x,-x, of the horizontal spring from its equilibrium value. From [28,29].

strength and shape of the interatomic potentials. They present two idealized models for a FFM tip interacting with an atomic surface, only one of which is presented here. Called a "realistic-friction microscope" by the authors, the model accounts for the elasticity of the FFM as well as the external load, F,,,, and the surface interaction potential. Fig. 2 shows a FFM, with a horizontal spring that pulls the spring-tip assembly along the interface potential of the substrate. The tip experiences the combined force of the interface potential and F,,,. As the tip moves across the surface, the horizontal spring elongates by an amount, xM - x,, which depends on the stiffness, c, of the horizontal spring. For a "hard" spring (c > ccriJ,both the tip trajectory and the Ff

7

-

1.5

'hardj

pfl"

1.o

0.5 0.0 c

Li -0.5 -1.0

1

-1.5 -2.0

9

0

AX XU

Fig. 3. 'Ihefriction force F, as a function of the FFM position x, and the average friction force for a hard and a sop spring. From (28,291.

3 00

200 A

7

E

z

v

U

100

0

2

6

8

10

Fig. 4. Contour plot of the average friction force < F,> per atom as a function of the load Fat and the horizontal spring constant, c. for monoatomic Pd tip on graphite, from /28,29])

curve are single-valued functions of x,; an example of the latter curve is labeled "hard" in Fig. 3. Since the positive and negative excursions of Ff are the same over a cycle, the average friction force C Ff> is zero. When c falls below ccri,,both the tip trajectory and the Ff curve are triple-valued functions of x,; an example of the latter is labeled "soft" in Fig. 3. In this "soft" spring case, the tip snaps forward at the point of "instability," as in the "strong" 0 model above, giving the spring case in the I asymmetric Ff vs X, curve shown by the curve with arrows. The average friction force, < Ff > , is therefore non-zero, as given by the dashed line. The energy dissipated by the collapse of the elongated spring is represented by the shaded area under a portion of the Ff vs X, curve. Fig. 4 gives the average friction for a range of values of horizontal spring constant, c, and Two results are apparent. external force, F,. First, the ability to measure zero friction depends on the value of c. Secondly, the friction coefficient p= < Ff> /Fea for atomic contacts is not independent of load. Further discussion of friction vs load behavior for atomic sliding is given elsewhere [28,29]. These simplified models of friction between atomic couples provide analytical criteria for transitions to zero friction, thus zero energydissipation conditions at T = OK. Zero friction requires "weak" interaction forces (low atomicscaled corrugations) for adiabatic motion and "hard" horizontal springs to measure the effect. An atomic "stick-slip" phenomena is predicted when these criteria are not met. The models are, in fact, consistent with atomic scale FFM measurements [4]. 2.2. Molecular Dynamics Simulations of Monolayer Films. Molecular dynamics (MD) simulations of friction behavior go beyond the simple analytical models. They use more realistic interaction potentials that exhibit anisotropy in two or three

8

dimensions; they account for effects of temperature and sliding velocity. Moreover, video animations of the simulations allow us to visualize the trajectories of the atoms at and near the interface. MD thereby gives us our "first look" at what happens at the buried interface during sliding contacts. Early examples of MD studies of frictional contact between solid surfaces depicted wear and transfer of material in the sliding contact (22,301. More recently, two studies of "wearless" friction have been reported; "wearless" friction, meaning that no atoms are lost from or transferred to either couple. In both cases, the solid surfaces were terminated with a monolayer of a simple hydrocarbon or hydrogen. This section presents these studies, which begin to address the role of surface films in friction processes. 2.2.1. monolayers of alkane chains McClelland and Glosli [27,31] have performed MD simulations of friction between two monolayers of alkane chains. The chains, six C atoms long, are initially ordered in a herringbone pattern. Chain bonds are allowed to bend and twist, but not stretch. C and H atoms on each chain interact with atoms of other chains by a Lennard-Jones potential of interaction strength, E,. The interaction strength at the interface, E , , is adjustable in order to study friction behavior as a function of interface interaction strength. The temperature of the layers is held constant by means of a heat bath; energy losses in the chains can thus be followed as heat losses in the layers. MD calculations are performed over a temperature range of 0 < T < 300K and sliding velocities up to v, = 204 mlsec. The friction force is calculated as the shear stress, T , averaged over several cycles of sliding. Fig. 5 shows two sets of friction vs temperature data for the case of strong ( E , / E , = 1.0) and weak ( E , / E , = 0.1) interactions at sliding speed of v = 20 m/sec. For both cases,

2.0

015 .

e A 6V t 0 ._

1.0

L

0 .L

0.5

0

1

2 3 Temperature T/E,

4

5

Fig. 5. Friction vs temperature data in normalized units for the case of strong (& = 1.0) and weak ( E ~ / C ~ = 0.1) inte@acial interactions at a sliding speed, v = 20 m/sec. Ihe solid line represents afit to the data using a thermal activation model. From [31]. the friction force exhibits different behaviors in the three (low, medium and high) temperature regimes; in addition, for the case of weak interactions, the friction vanishes at low and high temperatures. In order to interpret the friction behaviors and energy dissipation processes, McClelland and Glosli relied on video animations of the molecular dynamics and calculations of both shear stress and heat flow vs displacement curves. At low temperatures, the trajectories of the alkane chains are very much like that of the atom-on-a-spring in the I 0 model discussed earlier. For strong interactions, the ends of the chains are strained until the local minimum in the potential disappears. At that moment, the ends of the chains are released abruptly and the chains oscillate with a pivoted-hinge motion. As the two chains oscillate synchronously but in opposite directions, CH, groups on the backbone of the chains begin to vibrate but the chains themselves do not twist; the chains retain their initially ordered, herringbone pattern, indicating that vibration and twisting modes are decoupled at this temperature. Friction force vs distance

9

-22

3

4

5

0 0.25 0.5 0.75 1.0

Displacement (D/a)

Fig. 6. Shear stress (leji) and heat flow (right) vs sliding displacement for strong ( ~ J E , = , 1.0) and weak = 0.I ) intevacial interactions at T = 20K. From 1311. curves exhibit atomic stick-slip, as seen in the upper left of Fig. 6. At slip, strain,energy is released abruptly (25 psec), which caused the heat flow shown in the upper right of Fig. 6. Because the energy dissipation is associated with a plucking instability, the friction is expected to be independent of velocity; this was confirmed by MD calculations at several velocities for T = 20K. Plucking motion and energy dissipation occur until the interfacial interaction falls below E J E ~ = 0.4. Below this value, the friction force vs distance curves vary smoothly and symmetrically about Ff = 0 and there is no measurable heat flow (see lower left and right parts of Figs. 6 , respectively). Like the weakly0 model, the weaklyinteracting atom in the I coupled alkane chains exhibit harmonic motion as the two substrates slide past each other. As the temperature increases, the friction in both interaction ranges increases, reaching maximum values around T = 80K. At this temperature, video animations depict very complex dissipation modes. The strain energy released by the bent chain now couples into chain oscillations, CH, group vibration and, for the first time, a torsional mode in which chains

twist about their backbone. Torsional modes, which are "frozen" at lower temperatures, become excited at intermediate temperatures; this process is sometimes referred to as "rotational melting." With all three modes now anharmonically coupled, excitations damp out quickly by transferring energy to the lattice vibrations in the substrate. Hence, the increase in friction at intermediate temperature can be attributed to the excess energy associated with the unfrozen torsional modes. As the temperature increases above the rotational melting temperature, a third mechanism comes into play: molecules vibrate so actively in all directions that an increasing percentage of the chains can hop and slide over the opposite surface without introducing strain. This reduces the net frictional force in both weak and strong interaction cases; in fact, in the weak interaction system, the friction force goes to zero at highest temperatures. Glosli and McClelland then demonstrate that the friction behavior at high temperatures is consistent with classical behavior of polymeric films. MD calculations of strongly interacting couples showed increasing friction (sub-linear) with sliding velocity (not shown here) and decreasing friction with temperature (Fig. 5 ) . The latter curve was fitted to the Eyring thermal activation model that Briscoe and Evans /32] used to describe the shear behavior of thin polymeric films. The straight-line fit, drawn in Fig. 5 , required only one adjustable parameter, the activation energy, Q; it was determined to be Q = 70K. The shear strength (at T = 0) derived from Q was 7 = 32 MPa, about the same value obtained from experimental data for stearic acid (C18) and behenic acid (C22). In summary, MD simulations of friction behavior between monolayers of alkane chains show that the simple harmonic vs plucking friction behavior applies to more complicated systems; in addition, the simulations enumerate the modes of dissipation that arise with

10

increasing molecular complexity and increasing thermal activation. 2.2.2. H- and (H+C,H,,)-terminated diamond (111) surfaces

Harrison, White, Colton and Brenner /33-351 have simulated the friction behavior of H- and (H + C,H,)-terminated diamond (1 11) surfaces placed in sliding contact. The forces are derived from an empirical hydrocarbon potential capable of modeling chemical reactions in diamond and graphite lattices as well as small hydrocarbon molecules /36]. Two diamond (1 1 1) surfaces, each terminated in a (1 x 1) pattern, are placed in twin (mirror-image) contact; the distance of separation depends on the repulsive interaction potential and the load.- Sliding is performed-in two directions: the [ 1121 direction and the [ 1101 direction. In the [ 1121 direction, the opposing H atoms can make "head-on" contact at very high loads because their velocity vectors lie in a common plane perpendicular to the surface; wg'll call these "aligned trajectories." In the (1101 direction, the H atoms can only pass adjacent to each other (across each others' diagonals) because their velocity vectors do not lie in a common perpendicular plane. The lattice temperature is set at 300K, unless otherwise stated, and sliding velocities are 50 or 100 m/sec. Normal loads are varied up to 0.8 nN/atom, corresponding to mean pressures up to 20 GPa. Friction coefficients are averaged over a unit cell, and friction coefficient vs load data are presented for both sliding directions. Their first study examines two (1 x 1) Hterminated diamond (1 11) surfaces /33]. Along the [ 1121 direction, the friction coefficient begins near zero for lowest loads, increases nearly linearly with load up to 0.6 nN/atom, then levels out at p = 0. ( see Fig. 7). Video animation sequences of sliding along the "aligned" [ 1121 direction show different interaction mechanisms at low and high loads.

0.6

0.4

1

CH,CH,

1c

__

0.2 - J

a

A'

-

za

-.I I

I

-- dh I

1

1

I

Fig. 7. Averagefriction coeflcient a; a function of normal load for sliding in the I1121 direction at v = I k/psec and at T = 300K. Curves are for hydrogen, methyl, ethyl and n-propylterminated systems, respectively. From /35/. At lowest loads, opposing H atoms first repel each other backwards. Then as strain develops, they pivot sideways and revolve past each other at closest approach, finally pushing each other forward. With these trajectories, the net frictional force, averaged over a unit cell, nearly cancels and almost no energy is dissipated. At higher loads, atomic-level stick-slip occurs: instead of gently revolving by each other, opposing H atoms collide and momentarily become "stuck," then suddenly "slip" and revolve around one another. (Slip is definitely assisted by thermal motion, as will be made clear later.) The pivoting-then-revolving motion excites both vibrational and bending modes in the C-H bonds. These excitations are passed on to the lattice as vibrations (phonons) then heat. In this way, potential (strain) energy developed at high loads is transformed into kinetic energy

11

(mechanical excitations), leading to non-zero values of the friction coefficient. Friction coefficient vs load data for sliding along the [110] direction (not shown here) are about !a order-of-magnitude lower than along the [ 1121 direction. Animations show that H atoms, instead of meeting along aligned trajectories, "zig-zag" through channels of potential minima that run between adjacent rows of H atoms on the opposing surfaces. Since the opposing H atoms do not encounter each other directly, strain levels are lower and thus the frictional forces are lower. Hence, different sliding directions on identical surfaces lead to anisotropy in both atomic trajectories and friction coefficients. Perhaps these trajectories can account for the well-know frictional anisotropy of single crystals [37,38]. The friction coefficient of the H-terminated surface shows a temperature dependence that is qualitatively similar to that found with alkane chains. At a modest pressure of 3 GPa, the friction coefficient drops from p=0.4 at OK, to p=0.25 at 70K, then to p=0.15 at 300K. The friction coefficient is larger at low temperatures because opposing H atoms cannot rotate out of the way of aligned-trajectory collisions without the help of thermal motion. Harrison et al. [34] next examined the friction behavior of the same sliding configuration of H-terminated diamond but with two methyl groups substituted for two H atoms on one surface. Along the [ 1121 direction, the friction coefficient vs load data rise quicker than on the fully H-terminated surface, but reach lower steady-state values (p = 0.35 vs p = 0.4) (see Fig. 7). These differences can be attributed to the large volume occupied by a methyl group. Instead of easily "revolving" around the H-atoms during low-load, aligned-trajectory collisions, the methyl group gets stuck, then "slips" with a ratcheting motion. The ratcheting motion, analogous to the motion of a "turnstile," rotates the methyl group alternately clockwise then

counter-clockwise 120" around the C-C bond. The turnstile motion, accompanied by C-H bond excitations, is responsible for higher friction at low loads. At high loads, the size of the methyl groups keeps the two surfaces further apart than comparably-loaded H-terminated pairs. Harrison (391 speculates that the "flattening" out of the friction coefficient vs load data may be due either to screening of the interaction potential or to constraining the excitation modes. Sliding the methyl-substituted surface along the [110] direction (not shown) produces the same, strong loadd2pendent friction coefficients found in the [112] direction (see Fig. 7). Remarkably, the substitution of two CH, molecules for two H atoms produces an order of magnitude greater friction coefficient. Why? Unlike the terminal H atoms, which can "zigzag" freely through the adjacent H atom channels, the larger methyl groups exhibit "turnstile" rotations like those found in [ 1121 sliding. The rotations are accompanied by "zigzag" motion, which becomes more pronounced as the load increases. The increased energy expended by the larger molecules in this turnstile and zig-zag trajectory is responsible for the increase in friction coefficient with load. In their latest study, Harrison et al. [35] have substituted two ethyl and two n-propyl groups for two H atoms; friction coefficient vs load data are seen in the right two panels of Fig. 7. The larger, more flexible hydrocarbon groups reduce friction at high loads by a factor of 1.5 to 2 compared with the fully H-terminated diamond slider. Center-of-mass trajectories of the CH, portion of the ethyl groups, plotted on potential energy contour maps of a H-terminated diamond (1 11) surface, give insight into how the motion of an ethyl molecule affects the friction coefficient (not shown here). At low loads, the ethyl molecule bends over, lies down and is dragged almost straight across the repulsive potentials, like the trajectory a chain would have if one end were tied to the upper surface. At

12

high loads, however, the ethyl molecule uses its flexibility and length to "snake" (detour) around high potential energy barriers; this trajectory expends less energy and produces a lower friction coefficient at higher loads. In summary, Harrison et al. have identified several mechanisms which may account specitically for the friction behavior and energy dissipation of H- and hydrocarbon-terminated diamond surfaces, and, more generally, for boundary film lubrication. They have shown that, at 300K, H- and h ydrocarbon-terminations follow different trajectories when sliding along "hard" (aligned-trajectory) and "soft" (channels) directions of H-terminated diamond, and thereby explain the strong frictional anisotropy of Hterminated diamond pairs along selected lowfriction channels. They have cataloged numerous excitations modes (rotations, turnstiles,.. .) by which frictional energy is dissipated. In addition, they have shown that, at high contact stresses, larger hydrocarbon groups reduce friction even further because of size and steric effects.

changed by varying the atmosphere, temperature, velocity or related parameters. The main conclusion of the study is that the friction force does not correlate with the adhesion force (or adhesion energy, y), but rather with hysteresis in the adhesion force. As an example, Fig. 8 shows friction and adhesion measurements for loading and unloading two calcium alkylbenzenesulfonate (CaABS) monolayers at 25°C. The curves to the left (A and C) represent behavior of a liquid-like monolayer: quite a low friction coefficient and some hysteresis in the adhesion during the loading-unloading cycle. The curves to the right (I3 and D) show that after exposing the surfaces to decane vapor, the already low friction coefficient in dry (inert) air decreases even more and the adhesion hysteresis disappears. Previous studies [42,42] had shown that when hydrocarbon vapors condensed onto CaABS, the molecules penetrated the outer chain regions and fluidized the surface. Thus, it was hypothesized that if a monolayer were made more liquid-like, Inert Alr 10

A

Dacana Vaoor 10

I

3. EXPERIMENTAL APPROACHES

3.1. Friction and Adhesion Hysteresis Israelachvili and co-workers [40] have recently discovered a new relationship between adhesion and friction, based on experimental studies of surfactant monolayers. Experiments are performed with a SFA, in which both adhesion and friction are measured. Adhesion behavior is examined in contact radius vs load curves during loading-unloading cycles and in pull-off force measurements; friction behavior, in unidirectional and reciprocating sliding. The surfactant monolayers studied exhibit one of three phases: solid-like, amorphous or liquidlike. The amorphous state is a phase in between the solid-like and liquid-like state. Moreover, the phases of each of the layers could be

-L.

-La

Load, L (mN)

Fig. 8. (A and B) Friction traces of two CaABS monolayers at 25°C exposed to inert air and to air saturated with decane vapor. (C and 0) Adhesion energies on loading, unloading and pull-ofl measured under the same conditions as the upper friction traces. From [40].

13

Increasing load, organic vapors. Increasing chain fluidity, branching

Temperature, T

Longer, more saturated chains

("C)

Fig. 9. A schematic 'piction phase diagram" representing the trends observed in the friction forces of five different suvactant monolayer types studied. The curve also correlates with adhesion hysteresis of the monolayers but not with the adhesion per se. From [40]. the friction and adhesion hysteresis would be reduced. Many other correlations between friction and adhesion hysteresis and the phase of surfactant monolayers have been observed. Both friction and adhesion hysteresis increase when solid-like monolayers or liquid-like monolayers are made amorphous-like. Conversely, when amorphouslike monolayers are made more fluid-like or solid-like, both the friction and adhesion hysteresis decrease. Observed trends in friction and adhesion hysteresis behavior are summarized in the schematic "friction phase diagram curve" shown in Fig. 9. Maximum values of friction and adhesion hysteresis (but not adhesion values) are found around a "chain-melting"temperature, T,; this is the temperature at which the monolayer is in between the solid-like and liquid-like state -i.e., the amorphous-like state. Lower values of friction and adhesion hysteresis are found at temperatures above or below T,. It is seen that the amorphous-like state represents the highest friction and highest adhesiun hysteresis. Factors

that can change the phase state of monolayers, such as vapors, speed, etc.. . can effectively shift the curve in directions indicated by the arrows in Fig. 9. Israelachvili et al. give a physical basis for this behavior. Adhesion hysteresis is the "irreversible" part of the adhesion energy, and is related to the energy dissipated during the A likely loading and unloading process. molecular origin of adhesion hysteresis is the extent of interpenetration and subsequent ease of disentanglement of the molecules across an interface. If there is little interpenetration, as with solid-like layers, the friction is smooth and no additional energy is expended separating surfaces. If there is significant interpenetration, as with liquid-like layers, but also ease of disentanglement on separation, the friction is again low and little extra energy is expended A thermodynamic separating surfaces. description of the liquid-like case would conclude that the time to separate the chains is slower than the Telaxation time of the molecules and, therefore, that separation approximates a reversible process. In both the above cases, the systems are physically in similar states going into and coming out of contact. By contrast, with amorphous-like layers, there is significant chain interpenetration but separation occurs faster than the molecular relaxation time. Thus, amorphous-like layers will be in different states going into and coming out of contact, and consequently, more energy will be expended to separate them than would be needed for the two other phases. Thus, we add a new mechanism of friction in boundary film lubrication. This mechanism is associated with the time it takes a molecule to adapt its trajectory to the lowest possible interaction potential, relative to the timedependence of the potential. In term s of the MD studies of large hydrocarbon-terminated surfaces [35], the low friction achieved with npropyl groups at high loads requires that they

14

"follow" the lowest energy trajectories by following the contours of minimum energy. In monolayer film studies [40],we see examples of one state, liquid-like, in which the chains can follow minimum force trajectories; but a second state, amorphous-like, in which the slower moving chains cannot follow. The energy differences are seen, therefore, in both adhesion hysteresis and friction. 3.2. Friction and Slip of Monolayer Films Krim et al. [13-15] are studying the frictional forces for solid-like and liquid-like films adsorbed on conducting (metal) and insulating (oxidized metal) substrates. Their approach is quite novel. Films of gases such as Kr and Ar or C2H4 and C,H6 are condensed onto surfaces to thicknesses up to several monolayers. The thickness of the film is determined in a straightforward manner with a quartz crystal microbalance (QCM). In addition, the QCM monitors sub-Angstrom shifts in the vibrational amplitude caused by gas adsorption. These shifts are due to frictional shear forces between the condensing film and the oscillating surface. Krim et al [I31 have shown that the "slip time" 7 of a monolayer film can be determined with sub-nanosecond accuracy from these shifts. Note that the time and length scales, nanoseconds and Angstroms, makes these experiments unique in the field of tribology. Since slip is fundamentally an energydissipative process, the technique allows energy dissipation to be measured and the mechanisms of energy dissipation to be studied. Experiments with rare gas atoms have shown that (1) the slip times for monolayers physisorbed on smooth gold surfaces are on the order of nanoseconds and (2) solid-like films exhibit longer slip times than liquid-like films. These results are consistent with a frictional force proportional to the sliding velocity, indicating a viscous friction mechanism. Three models have been proposed to account

for energy dissipation of the sliding monolayers. The first two postulate that phonons carry away the energy. Sokoloff, using an analytical model of friction [43,44/, suggests that defects between the incommensurate monolayer-solid interface can account for the slip times. Robbins et al. [45] have used molecular dynamic simulations to determine the viscous coupling between a driven substrate and an adsorbed monolayer film. Requiring no arbitrary parameters, the model gives excellent agreement with many of the experimental observations; it gives the correct magnitude of slip time, 7 , a friction force proportional to 7 for physisorbed films, and less slip in liquid films than in solid films. They have also shown that their results agree with a simple analytic model, closely related to that of Sokoloff. The slip time is directly related to it is equilibrium properties of the film; proportional to the lifetime of longitudinal phonons and inversely proportional to the square of the density oscillations induced by the substrate. A third model, presented by Persson et al. (461, postulates energy dissipation by electron-hole scattering. It assumes that electrons in the metal substrate experience a drag force equal in magnitude to the force required to slide the adsorbed film. This force is estimated from measurements of the change in resistivity of metal films as a function of gas coverage. Slip times calculated from the forces are in good agreement with experimental values for adsorbed rare gases and hydrocarbon molecules. The electron-hole scattering model also predicts different slip times for C2H4and C2H6 adsorbate films, but only if electronic contributions are present e.g., with metals but not insulators. Krim's group [47] has recently tested this prediction by measuring slip times for C2H4 and c& on silver and on oxygen-coated silver. They find different slip times on Ag, but the sume slip time on oxygen-coated silver, consistent with the

15

predictions. Thus, based on these rather unique studies of friction, it appears that both phonon and electron mechanisms contribute to energy dissipation. 4. SUMMARY AND DISCUSSION

From studies just reviewed and others in the literature, our understanding of interfacial friction processes and energy dissipation mechanisms can be summarized as follows. Low friction, including zero friction, can be achieved at low loads, with weak interface interactions and with "small" atoms at the interface. The mechanical principle that explains this behavior follows from the simple, onedimensional, I0 model: the strain energy transmitted by interfacial atoms during the first half of the cycle is returned to them during the second half cycle. This behavior is also observed in the more realistic, three-dimensional MD simulations. The third dimension itself contributes an additional friction reduction channel; it provides the interfacial atoms an extra degree of freedom -- to move out of their common plane -- to escape "stick" events along aligned trajectories. For example, H-terminated atoms can rotate around each other in aligned collisions or "zig-zag" along potential minima channels; these trajectories are not available to atoms described in two dimensional models. In principle, many "low friction trajectories" can be found along selected directions in real crystals having anisotropic interaction potentials (corrugations); these possibilities are treated more quantitatively by Hirano and Shinjo [48'. In practice, however, too soft a spring in the measuring device can lead to friction force instabilities, resulting in measured friction forces that are higher than expected. Friction is increased by many factors. Strong interfacial interactions (corrugations), according to the simple I 0 model, give a finite static friction force, then stick-slip motion between atoms. Three-dimensional potentials also show

atomic-scale stick-slip processes, but the modes equivalent to twodimensional plucking, e.g. turnstile motion, are novel and more complex. Atomic-scale stick-slip processes have been seen in FFM measurements, but at present the modes responsible have not been identified because of the relatively slow response time of friction devices [49]. Anharmonic coupling of excited modes establishes multiple pathways for energy dissipation, thereby increasing friction coefficients. An example is the MD simulation of alkanes at intermediate temperature, where torsional modes become allowed, providing a new pathway for energy dissipation. Other excitation modes that enhance friction and dissipate energy are density oscillations, defects at interfaces and electron-hole coupling in metals. Commensurate lattices have been shown to increase friction forces by many orders of magnitude [44]. Friction force is expected to increase with increasing external force (load). However, as Zhong and Tomhek [SO] have shown, surface interactions can be perturbed over a selected load range, thereby lowering the friction coefficient as the load increases. Temperature can influence friction behavior in several ways. Thermal activation of an energydissipating mode, like rotational melting, increases friction. In contrast, thermalactivation can lower potential barriers and increase tunneling, thereby reducing friction. At high temperatures, thermal effects can dominate friction processes, giving liquidlike (viscous) friction instead of solid friction (finite static friction at all velocities). The MD simulation of alkane chain friction showed this transition from solid to viscous behavior with increasing temperature. The size and shape of molecules can also influence friction behavior. Small atoms or molecules may follow low-friction trajectories whereas larger atoms or molecules may not "fit" into the same channels, resulting in higher friction. The H- vs CH,-terminated diamond

16

along the [110] direction is such an example. By contrast, larger molecules can reduce friction more effectively than smaller molecules at higher loads if they have sufficient flexibility to spread across the surface; an example is the high load behavior of the hydrocarbon-terminated diamond surfaces. This steric accommodation, however, can increase friction if the molecule cannot follow the minimum-energy trajectory as fast as the surfaces move apart. Chain entanglement between amorphous-like films observed in SFA studies is an example of steric effects causing increased energy dissipation. A new concept of friction behavior has been demonstrated by Israelachvili et al. -- that energy dissipation is maximum when the time (and length) scales of contact (externally controlled) match the intrinsic time and length scales of molecular interactions. This concept is consistent with thermodynamic considerations of two bodies coming into contact. As mentioned in the introduction, the degree to which the contact process approximates a reversible, quasi-static process depends on the rate at which each step is taken compared to the relaxation time of the system. Put in terms of the driven oscillator analog, deviations from equilibrium and energy dissipation are maximum when time and length scales of the system and driver are matched. However, classical thermodynamics is not really suitable for the treatment of contacting surfaces let alone sliding surfaces. Even in the mildest contact circumstances, in which the two bodies retain their identity after separating, equilibrium was never achieved; at best, the two bodies reached metastable equilibrium. A more precise description of the thermodynamics of contacting surfaces is needed. Finally, these studies can give us some new insights into the role of surface films in friction processes. Generalizing the studies of Harrison et al, we see that films in which "small" atoms chemisorb one-to-one with the substrate lattice might provide the screening needed to prevent

interface "welding" and give low-friction trajectories along weak corrugation channels. Films made of larger molecules, with lower compressibility, might reduce friction at high loads by providing atomic screening as well as steric accommodation. 5. RECOMMENDATIONS

This review was meant to introduce tribologists to some of the more recent investigations of energy dissipation processes in interfacial friction. Many recommendations for future research in atomic-scale tribology can be made based on these preliminary investigations. For example, the remarks in the previous paragraph suggest two approaches for modeling boundary lubricant films: 1) gas or solutehdditive interactions with surfaces can be modeled with small, single-atom terminations and 2) "run-in" boundary films can be modeled by more complex molecular attachments. Many of the ideas in this paper were presented at a two week long NATO AS1 meeting held in Braunlage, Germany in August 1991. Considerable time was spent discussing "future issues" and suggested approaches for research in this field; these have been published in the Epilogue to the conference proceedings [52]. Here I summarize only a few of these items: How can atomistic modeling continue to make an impact on understanding friction? on understanding lubrication? Can algorithms (e.g. hybrid methods) be developed to simulate friction processes at time and length scales longer than can be treated in molecular dynamics calculations alone? e.g. that extend computational simulations from the nmjfemtosec scale to the pm/psec scale. Can lubricants be tailored to take advantage of the dynamic properties of certain fluids,

17

e.g., the "chemical hysteresis" of monolayer films Israelachvili, et al. o In practical machines, sliding is sustained on surface films -- whether organic lubricants, oxides or other solid films. Can molecular simulations help us to understand the chemistry of film formation, the mechanical properties of these films and how the films break down? One of the newest issues that tribology must deal with is the concept of matching time and lengths scales in friction studies. As we saw above, energy dissipation hence friction is intimately linked to time and length scales. Moreover, the atomidmolecular modes of interfacial interactions operate at time and length scales far shorter than traditional tribology measurements. In the section of the Epilogue [SlJ entitled New wavs of probing friction processes, we asked "How can we use the power of microscopic modeling to gain new insights into macroscopic friction processes and, ultimately, to solve technological problems?" Bill Goddard [52/ suggests that this can be done by progressing along the "chain-linked" ladder, illustrated in Fig. 10, from quantum-level studies to engineering design. His "hierarchy of modeling tribological behavior" unites atomistic models, which operate in very short length-time scales, with engineering models, which describe tribological behavior in length-time scales observable by more traditional measuring equipment. This approach "...allows consideration of larger systems with longer time scales, albeit with a loss of detailed atomic-level information. At each level, the precise parameters (including chemistry and thermochemistry) of the deeper level get lumped into those of the next. The overlap between each level is used to establish these connections. This hierarchy allows motion up and down as new experiments and theory lead to new understanding of the higher levels, and new

I

DISTANCE 1A

W)A

lOOA

!p

lcm

yofdt

Fig. 10. lime and length scales of present-day models and experiments in Tribology. From [51J.

problems demand new information from the lower levels." But where are the experimental approaches for investigating the "lower (short scale) levels?" As illustrated in Fig. 10, most "friction machines," including the proximal probe devices, are operated at long time scales. An abbreviated search of recent literature produced only three "tribology" tests and a fourth proximal probe method that come close to investigating friction behavior at short time and length scales. Labeled 1 through 4 in the Fig. 10, they are described briefly here: 1. Bair et al. I531 have used fast IR detectors to measure flash temperatures during high speed frictional contacts of asperities of length 10 pm and greater, with time resolution of about 20 psec. 2. Spikes et al. have developed real time optical techniques for investigating the physical behavior of EHL films down to 5 nm thick [54J and chemical processes occurring in contacts 10 pm wide by 80 nm thick (551. 3. Krim et al. [13,14] have used the quartz crystal microbalance experiments (described earlier) for probing atomic vibrations amplitudes between 0.1 to 10 nm and time

18

scales from 1W2 to sec. 4. Hamers and Markert /56] have shown that STM images are sensitive to the recombination of photo-excited carriers whose lifetimes are in the picosecond range. Clearly, innovative experimental approaches for measuring friction processes at short and intermediate time-length scales are needed to assist the modelers who are already there. Tribologists should seek out physicists and chemists working in these time-space domains and form collaborations to carry out tribologyoriented experiments. As I've tried to emphasize in this article, much of the progress in the field comes when experiments and theories can overlap on the same time-length scales. Finally, there is another time scale to consider, and that is "the question of time to translating this (fundamental) knowledge into engineering practice. Duncan Dowson, to whom the XXth Leeds-Lyon session on "dissipativeprocesses in Tribology" is dedicated, told us how this can be accomplished in the final part of the Epilogue [57]. I'

"We have heard quite a lot about the subject of friction from the atomic scale up to the macroscopic scale.. .. 7here are other aspects of scale I think we should reject on. One is the question of time in terms of translating this knowledge into engineering practice.. ..'Ihe time scales are generally enonnous. ... I think you should take note that it is going to be about A DECADE IF NOT A GENERATION before that impact will be seen fully in engineering. m i s makes it all the more importantfor ... physicists, chemists and engineers (to meet), so that the engineers can absorb by osmosis the concepts that you are revealing.. .. It is important that these ideasfeed into our consciousness so that we apply them sensibly in future developments. ...I hope equally that scientists will not be

toofrustrated by the fact we take 10 to 20 years to incorporate their bright new ideas in designing better skis or whatever it might be. Let us perpetuate this interaction between groups of people who all have one objective and that is to understand the laws of physics in order to apply them as eflectively as we possibly can for the good of society through the manufacture of reliable, eflcient and sensible products. " ACKNOWLEDGEMENTS. I am most grateful to the following colleagues for providing reprints and preprints and their willingness to engage in discussions of their results: J.A. Harrison, J.N. Israelachvili, J. Krim, G.M. McClelland, H.M. Pollock, M.O. Robbins, J.B. Sokoloff and D. TomiInek. I am also grateful to my hosts at the Laboratoire de Technologie des Surfaces, Ecole Centrale de Lyon, where this paper was conceived. REFERENCES 1.

2.

3. 4.

5.

6. 7.

8.

Fundamentals of Friction edited by I.L. Singer and H.M. Pollock (Kluwer Academic Publishers, Dordrecht, 1992). G. Binnig, H. Rohrer, Ch. Gerber and E. Weibel, Phys. Rev. Lett. 49 (1982) 57; Phys. Rev. Lett., 50 (1983) 120. G. Binnig, C.F. Quate and Ch. Gerber, (1986) 930. Phys. Rev. Lett., C.M. Mate, G.M. McClelland, R. Erlandsson and S. Chiang, Phys. Rev. (1987) 1942. Lett., G. Meyer and N.M. Amer, Appl. Phys. Lett., 5_6 (1990) 2100. E. Meyer, R. Overney, L. Howald, D. Brodbeck, R. Liithi and H.-J. Giintherodt, in Ref. 1, p. 427. D. Tabor and R.H.S. Winterton, Proc. Roy. SOC. London, A312 (1969) 435; J.N. Israelachvili and D. Tabor, Proc. Roy. SOC. London, A331 (1972) 19. J.N. Israelachvili, P.M. McGuiggan and

19

A.M. Homola, Science, 240 (1988) 189; A.M. Homola, J.N. Israelachvili, P.M.McGuiggan, and M. L. Gee, Wear, 136 (1990) 65. 9. Israelachvili, in Ref. 1, p. 351. 10. A. Tonck, J.M. Georges and J.L. Loubet, J. Colloid Interface Sci., 126 (1988) 150. 11. J-M. Georges, D. Mazuyer, A. Tonck and J.L. Loubet, J. Phys. Cond. Mat., 2 (1990) 399. 12. J-M. Georges, D. Mazuyer, J-L. Loubet and A. Tonck, in Ref. 1, p. 263. 13. J. Krim and A. Widom, Phys. Rev., B38 (1988) 12184. 14. J. Krim, D.H. Solina and R. Chiarello, Phys. Rev. Lett., 66 (1991) 181. 15. J. Krim and R. Chiarello, J. Vac. Sci. Technol., A9 (1991) 2566. 16. Ernest Rabinowicz, Friction and Wear of Materials (Wiley, New York, 1965) pp. 94107. 17. I.V. Kragelskii, Friction and Wear (Butterworths, Washington, 1965) pp. 208218. 18. J.B. Pethica and A. Sutton, J. Vac. Sci. Tech., 6 (1988) 2494. 19. J.R. Smith, G. Bozzolo, A. Banerjea and J. Ferrante, Phys. Rev. Lett., 63 (1989) 305. 20. J. Ferrante and G. Bozzolo, in Ref. 1, p. 437. 21. U. Landman, W.D. Luedtke and E.M. Ringer, Wear, 153 (1992) 3. 22. U. Landman, W.D. Luedtke and E.M. Ringer, in Ref. 1, p. 463. 23. J. Leslie, An Experimental Inauirv into the

G.

Nature and Propagation of Heat (Bell and Bradfute, Edinburgh, 1804). 24. Duncan Dowson, Historv of Tribology (Longman, London, 1979). 25. H.B. Callen, Thermodvnamics (Wiley, New York, 1960), p. 63. 26. G.M. McClelland, in Adhesion and Friction edited by M. Grunze and H.J. Kreuzer (Springer Verlag, Berlin, 1990) p. 1.

27. G.M. McClelland and J.N. Glosli, in Ref. 1, p. 405. 28. D. Tomdnek, W. Zhong and H. Thomas, Europhys. Lett., 15 (1991) 887. 29. D. T o m h e k , in Scanning Probe

Microscopy, edited by H.-J. Guntherodt and R. Wiesendanger (Springer-Verlag, Berlin, 1993). Chapter 11. 30. J. Belak and I.F. Stowers, in Ref. 1, p. 511. 31. J.N. Glosli and G.M. McClelland, Phys. Rev. Letts., 70 (1993) 1960. 32. B.J. Briscoe and D.C. Evans, Proc. Roy. SOC.London, A380 (1982) 389. 33. J.A. Harrison, C.T. White, R.J. Colton and D.W. Brenner, Phys. Rev., M (1992) 9700. 34. J.A. Harrison, R.J. Colton, C.T. White and D.W. Brenner, Wear, 168 (1993) 127. 35. J.A. Harrison, C.T. White, R.J. Colton

and D.W. Brenner, J. Phys. Chem., J. Phys. Chem. fl (1993) 6573. 36. J.A. Harrison, C.T. White, R.J. Colton and D.W. Brenner, Surf. Sci., 271 (1992) 57. 37. Donald H. Buckley, Surface Effects in

Adhesion. Friction. Wear. and Lubrication (Elsevier, New York, 1981). pp. 357-373. 38. M. Hirano, K. Shinjo, R. Kaneko and Y. Murata, Phys. Rev. Lett., 67 (1991) 2642. 39. J.A. Harrison, private communication, 1993. 40. H. Yoshizawa,

Y-L. Chen and J. Israelachvili, J. Phys. Chem., 97 (1993) 4128.

41. Y-L.

Chen, C.A. Helm and J.N. Israelachvili, J. Phys. Chem., 95 (1991)

10736. 42. Y.L. Chen and J.N. Israelachvili, J. Phys. Chem., p6 (1992) 7752. 43. J.B. Sokoloff, Phys. Rev., B42 (1990) 760; Wear, 167 (1993) 59. 44. J.B. Sokoloff, Thin Solid Films, 206 (1991) 208; J. Appl. Phys., 72 (1992) 1262.

20

52. "Tribology of Ceramics" National Materials Advisory Board Report 435, National Academy Press, 1988, Washington DC, p. 88. 53. S. Bair, I. Green and B. Bhushan, J. Tribology, 113 (1991) 547. 54. G.J. Johnston, R. Wayte and H.A. Spikes, communication. Tribology Transactions, 2 (1 99 1) 187. M. Hirano and K. Shinjo, Phys. Rev. !&l 55. P.M. Cann and H.A. Spikes, Tribology (1990) 11837. Transactions, 34 (1991) 248. I.L. Singer and H.M. Pollock, in Ref. 1, p. 56. R.J. Hamers and K. Markert, Phys. Rev. 582. Lett., 64 (1990) 1051. W. Zhong and D. TomBnek, Phys. Rev. 57. I.L. Singer and H.M. Pollock, in Ref. 1, p. Lett., B64 (1990) 3054. 586. I.L. Singer and H.M. Pollock, in Ref. 1, p. 569.

45. M. Cieplak, E. Smith and M.O. Robbins, submitted to Nature, 1993 46. B.N.J. Person, D. Schumacher and A. Otto, Chem. Phys. Letts., 28 (1991) 204; Phys. Rev. B, 44 (1991) 3277. 47. C.H. Mack, C. Daly and J. Krim, private 48. 49.

50. 51.

Dissipative Processes in Tribology / D. Dowson et al. (Editors) 1994 Elsevier Science I3.V.

21

The Mechanics of Adhesion, Deformation and Contamination in Friction K.L. Johnson 1, New Square, Cambridge CBl lEY, United Kingdom It is now half a century ago that Bowden & Tabor, first in the Laboratory for Tribophysics in Melbourne and then in the Cavendish Laboratory in Cambridge, revived an interest in friction as a respectable subject for scientific study. They rejected Coulomb's proposition that energy is dissipated in sliding by lifting the asperities of one surface over those of the other and identified adhesion at the interface, inelastic deformation of the solids and contamination of the interface as the principal factors influencing friction. In this paper the present status of these pervading concepts will be reviewed from a macroscopic point of view and, where possible, related to the present microscopic studies of surface interaction described by Singer in the accompanyingpaper. 1. THE COULOMB THEORY AND THE TIME FACTOR

The Coulomb theory of friction, in which the energy expended is attributed to lifting the asperities of one surface over those of the other, is normally discredited by the argument that the work done against the contact load in lifting is recovered when the surfaces move together. Although the model of an asperity encounter shown in Fig.1 is usually illustrated by rigid asperities, even if they are elastic there will be a normal impulse produced by the encounter which will generate wave motion and vibration. What then is the probable magnitude of this so called 'acoustic loss'?

circular area of the surface, of radius a, like a spring in parallel with a dashpot, in which the energy dissipated in the dashpot corresponds to the acoustic loss. The time constant T of this system is given by T

2a/c0

(1)

where co (= E/p) is the speed of longitudinal waves in the solid. Referring to Fig. 1, if an asperity encounter takes place in*a sliding distance L, the time of the encounter t = L/V, where V is the velocity of sliding. Assuming the impulse to be sinusoidal with a period 2t* , the ratio of the energy dissipated in the dashpot i.e. acoustic loss, to the maximum strain energy stored in the spring i.e. strain energy of deformation of the asperity, is given 3Y

V L co

2a.T - n--a -2t*

Figure 1. Interacting asperities in sliding contact. Interaction time 2t* = interaction distance L/sliding speed v. It is explained in ref.[ 13 that an elastic half-space responds to a sinusoidal impulse applied to a

The acoustic loss is therefore clearly negligible provided the sliding speed V is small compared with the elastic wave speed c,. Similar considerations of the time factor apply to other mechanisms of energy dissipation. Plastic deformation involves dislocation movement. The strain field and energy of a dislocation is unaffected by its speed of movement until the speed approaches the elastic wave speed c, . This means that plastic flow of asperities is fully dissipative and independentof rate effects provided V /ENGINEERING COMPONENTS

Is

i.e. when

V

= L/r

(3)

At sliding speeds either much greater than or much less than this value, the deformation will be elastic with modulus E, or E, respectively. This behaviour has been demonstrated very effectively in rolling friction by Greenwood & Tabor [2]. Many polymers, of course, have a more complex relaxation spectrum with more than one relaxation time. The 'time factor', discussed above in continuum terms in relation to the macroscopic interaction of sliding asperities, carries over to the microscopic studies of friction with the atomic force microscope or through molecular dynamics simulations. In this case the characteristic sliding distance is the atomic spacing: the characteristic time is the period of oscillation of the atoms about their minimum energy positions (e.g. McClelland & Glosi [3]). The map displayed in Fig.2 is adapted from Goddard [4]. For the practical range of sliding speeds, it shows the length and time scales of the different regimes of behaviour. Whether or not an event is dissipative depends upon the relation of the time of the event to the characteristic time of the dissipative process.

wf

('

o

n t 1n uu m

+

IN

1UUl

1-

lttlln

I m

LENGTH

Figure 2. Length and time scales in sliding contact at different speeds. We will start by considering the stationary contact of a sphere (radius R) with a plane, under a normal load, as a model of an asperity contact in an extended surface. According to the Hertz theory a load PI will result in a circular contact area of radius a given by

2. ADHESION AT THE CONTACT OF SOLIDS

a3 = 3RP1/4E*

Bowden & Tabor proposed that contaminant free surfaces form adhesive 'junctions' at the tips of the contacting asperities and that, during sliding, friction predominantly comprises the force to shear those junctions. Not surprisingly this proposition has met with considerable scepticism on account of the observation that such surfaces do not generally remain in adherence when the compressive load is removed. Some, but not all, of the difficulties involved in this proposition have been resolved by subsequent developments in contact mechanics, which have shown that in most practical

where 1/E* = [(1-vl2)/E1 + (1-~2~)/E2 ] ; E and v are Young's modulus and Poisson's ratio respectively. If the load is reduced to P(< PI), while the surfaces remain adhered together at the initial radius a , the pressure distribution takes the form shown in Fig.3a, having an infinite tension at the periphery. This result was used by me in 1958 [5] to explain why, in spite of initial adhesion, asperities would separate when the load was removed. But that was before fracture mechanics had taught us to live with stress singularities. In 1971 Johnson, Kendall & Roberts

(4)

23

[6] used the Griffiths' concept of balancing the rate of release of elastic strain energy with the change in surface energy w to obtain an equilibrium relationship between load P and contact radius a. Their results can be obtained more directly by using fracture mechanics concepts.

from which the graph of x(=nZ2)against P is plotted as curve I in Fig. 4. We note that when the load is reduced to zero the surfaces remain in contact with an area A, = 1 ~ 2 ~ They 1 ~ . snap apart when A has decreased to nn2I3, which requires a tensile (negative)force F, = -1 / 2 to pull them apart. i.e.

P, = 3nwR/2

(8)

Figure 3. Pressure distributions in adhesive contact. (a) JKR theory [6]; (b) DMT theory [12]. We hrst introduce non-dimensionalvariables:

P = P/(3mR) B = d(9wR2/4E*)lB and Pi = Z 3 The singularity in tension at the edge of the contact in Fig.3a can be expressed in terms of a stress intensity factor:

KI = P&d(m)

(3

where Pa is the effective force of adhesion. The strain energy release rate G is then given by

which, when equated to the surface energy w, gives

from which

Hence the net load F(= Fl - Fa) can be written

These results have been confirmed by many exueriments with low modulus elastic solids such as gehne [a]. The second term in eauation (7). i.e. (2F1)ln is Fa when &e iontact the effectiveforce of &ion size is a. It appears as the horizontal segment between curve I (JKR) and curve II (Hertz) in Fig.4. Thus the contribution of adhesion, expressed by the ratio of Pa to P1 ,decreases with the square root of the nominal Hertz load P1 . For metals the surface energy w has a value of about 1.0 J /m2. For an elastic sphere of radius 5mm, the ratio PJP1 will be less than 1.0% when the load P exceeds 10 N. The pull-off force P, is only 0.024 N. In these circumstances adhesion is clearly an effect confiied to light- load situations such as particle-particle interaction in colloidal suspensions. In the multi-

;

G = K~DE*

Pz= 8pwaE* = 6pwRP1

Figure4. Area - Load relationships in adhesive contact. I: JKR theory (reversible): 11: Hertz theory (no adhesion): III-IV: Loading-unloading cycle ABCD.shows effect of inelastic deformation.

(6)

24

asperity contact of rough surfaces,during unloading, the higher asperities progressively break the lower adhesive junctions until there are only a few junctions left and the final 'snap-off force is negligible [7]. It is not surprising that the presence of adhesion between contacting bodies is not readily detectable. There are circumstances,however, which lead to an enhanced effect of adhesive forces. First, if initial loading of the sphere (hard) against a plane (soft) produces a significant plastic indentation, the subsequent elastic recovery and pull-off force is governed by the relative radius of curvature R' between the sphere and the indentation [8], where

R = 4E* a/3xH = 4E*P1n/3(~H)3n (9)

P is the initial load and H is the hardness of the softer surface. Substituting R' for R in equation (8) gives the pull-off force Pd =2wE*(P/xH3 )1/2

G'= w + D'= w + a'G

(10)

where G is the elastic strain energy release rate during 'crack extension. In the perfectly elastic situation (the Griffith crack) G = w, which should

(1la)

With a closing crack (increasing contact area) the work done by surface forces w is reversed and the elastic strain energy increases so that in this case 4"= -W + D" = -W + a"fl

(1 1b)

from which the effective surface energy during opening wi may be written

wi =G'=w/(l-a')= and during closing

Large adhesion would be expected therefore when the hardness is small. This was found to be so by Tabor [9] in experimentsusing tin, lead and indium. Another, more subtle, effect acts to enhance adhesion. It was observed during experiments with rubber that the contact area varied with load according to equation (7) but that the effective work of adhesion during separation wi increased with velocity of separation and greatly exceeded that during compression wf. The reason for this effect lies in the infinite stress and strain predicted by the JKR theory at the edge of the contact area (Fig.3). Of course the actual stress will be finite, but it is still large, and equal to the ultimate strength of the material, of order E/30 for a crystalline solid. Consequently as the contact area changes there will inevitably be some internal dissipation viscoelastic or plastic - in this zone. The configuration of Fig.3a may be regarded as an external crack penetrating a ligament of material radius a. The stress singularity corresponds to a mode I stress intensity factor

KI= (2GE*)lR

be perfectly reversible, whether the crack is opening or closing, i.e. whether the contact area is shrinking or growing (Fig.4, curve I). If the zone of inelastic deformation is very local to the crack tip, i.e. small compared with a, we can use linear elastic fracture mechanics concepts. Assume that a fraction a of the elastic strain energy released G is dissipated, so that if D is the dissipation rate, with an opening crack (decreasing contact area) we can write

wi =G"=w/(l+a")=

k'w k " ~

Wb)

where k' > 1.0 and It'< 1.0. The correspondingcurvesof increasing (119 and decreasing (IV) area A as a function of load P are added to the non-dimensional plot in FigA When the surfaces first touch they snap into contact at point A. During compression the area increases, following curve I11 to pt.B, with w?= k" w. When the load is reduced, at first the contact area remains unchanged until, at C, the energy release rate is sufficient to peel the surfaces apart with an apparent surface energy wi = k' w. Peeling continues until the surfaces snap apart at D where the (tensile) load = -Pc. Greenwood & Johnson [lo] have analysed this behaviour for a linear viscoelastic solid and shown that k' is related to the ratio of the instantaneous elastic modulus E, to the relaxed modulus E,, which can have a value of two or three orders of magnitude, while k" is so small that the surfaces come together as though there were no adhesion. Since viscoelastic dissipation is rate dependent,k' is found to be an increasing function of peeling velocity. These conclusions are well supported by experiments using rubber [ll]. When the inelastic

25

dissipation is by plastic flow, as in metals, the conclusion is the same: when peeling the work of adhesion is enhanced by a factor k' (>LO), which is a function of ( & l o y )where 6 is the maximum adhesive stress in the adhesive zone and or is the yield stress of the material. This effect is well known in the fracture of metals, where the work of fracture greatly exceeds the surface energy of cleavage. When the length of the 'process zone' at the edge of the contact over which the adhesive forces act is no longer small compared with the radius of the contact a, the JKR theory becomes inappropriateand a better approximationis provided by the DerjaguinMuller-Toporov (DMT) theory [12]. In this model the deformed shape is assumed to be that given by Hertz and the adhesive forces act o_utside the contact area, as shown in Fig.3b. The P :Z relationship, equivalent to equation (7) of the JKR theory, is

F = F1 - 213 = Z3 - 213 Using the Dugdale model of elastic-plastic fracture, Maugis [13] has shown that the transition from JKR to DMT conditions depends upon the parameter

h=

26 ( 1 6 n ~ E/*9R)'I3 ~

where 6 is the peak adhesive stress. For the JKR model to apply h > 1.0. Taking 6 to be the yield stress this condition requires R to be greater than about 20 p m for a polymer sphere and about 2 pm for a metal sphere. 3. STATIC FRICTION AS MODE I1 FRACTURE

We shall start this section by accepting Bowden & Tabor's proposition that clean surfaces,

particularly in an out-gassed environment, adhere together with a strength (yield stress) approaching that of the bulk solids. An asperity contact is again modelled by a sphere pressed into contact with a plane by a normal force P, giving rise to a contact area of radius a, given by the JKR theory equation (7). The normal traction is that shown in Fig.3a. A monotonically increasing tangential force Q is

now applied, which gives rise to the tangential traction shown in FigSa:

The singularity at r = a corresponds to a mode I1 stress intensity factor KII= Q / (4m3 )In

(16)

in addition to the mode I factor KI given by equation (5). There has been considerable interest recently in mixed mode interfacial fracture in connection with debonding of composite materials. There it is usually assumed that an interfacial fracture will proPogate when

K: + W * I l2 = K,:

(17)

where KI, is the fracture toughness in tension and , found by experiment to be less than unity. When this criterion is applied to the adhesive contact of a sphere with a plane [14], it predicts that the application of a tangential force causes the surfaces to peel apart until the radius a and pressure distribution are Hertzian. This initial peeling, caused by the application of a tangential force to an adhesive contact, has been observed in rubber by Savkoor & Briggs 1141 but not by Israelachvili with mica surfaces [15]. It is difficult to be precise about what happens next. By analogy with a mode I crack, linear elastic fracture mechanics would say that Q could be increased until KII ,given by equation (16), reaches the value of the fracture toughness in shear KII,. whereupon the junction would shear in an unstable way (FigSa). But this picture ignores the fact that in mode I1 the crack faces remain pressed together behind the crack front ,which will resist slipping by a shear stress s. This situation, depicted in FigSb, has been analysed by Savkoor [161. If the tangential force is controlled, the junction fractures catastrophically when Q = Q,. With a large number of such junctions in an extended surface, each junction will be 'displacement controlled by the hinterland, thereby delaying unstable slip until the critical displacement6, is reached. The contact then slips until Q falls from Q, to Qs(= xa2s). This

p = KIJKII,

26

I

I

I F a - -

Q~= 2 a 6 a

KII~

ai

Figure 5. Static friction as mode I1 fracaUe. (a) Simple mode I1 fracture (unstable); (b) Savkoor's model (unstable if Qs< Qc); (c) stable sliding (Qs > Qc). stick-slip behaviour will only occur if the initial strength of the interface is greater than that after slip

Q,= rca2 > > Q ~= (4m

3 112 K~~~

hasoccurred.

Before accepting this model, we must consider the conditions in which elastic fracture mechanics is appropriate, i.e. that the process zone size at the crack tip should be small compared with the contact radius a. For this to be so,

i.e.

(18)

in which the maximum value of s is the yield stress k in shear of the softer material. There is little data

27

for KII, ,but if it is approximated by KI,,we have K1Jk for metals = 5-50 (pm)1/2;for polymers = 0.5-5Q~m)~”; and for ceramics = 0.04-0.1(pm)1~. The mean radius a of asperity contacts is estimated to be about 20 pm, from which it may be deduced that only ceramics clearly satisfy the condition (18) and metals do not. In the case of an eWtic-perfectlyplastic material (metal), with a strong adhesive bond (s=k), the sequence of events depends upon the intensity of the normal load. If the parameter (aE*/Ray)is less than about 2.0 the stresses due to the normal load will lie within the elastic limit [l]. The subsequent application of a tangential load causes a thin plastic layer to spread across the interface from the sides until complete ductile fracture occurs when Q = na2k (Fig.5~).At the other extreme, if (a E*/ Roy) exceeds about 30, the contact will be fully plastic. On first loading, the contact area increases as the tangential force is increased, until the contact pressure has dropped from about 5.6k (i.e. 2.8 ay) to k and the shear traction has increased to the value k. Shearing of the junction then takes place at a , agreement with coefficient of friction ~ 1 . 0 in experiments on clean ductile metals. This is the process of junction growth, proposed by Tabor [171 and analysed in plane strain by Johnson [181. If s < k this process is interrupted, and steady sliding takes place, when the mean shear stress Q/m2reaches the value s. The material for which the Savkoor model is appropriate, indeed for which it was conceived, is rubber, which has no yield strength in the conventional sense. The low elastic modulus enables it to mold itself to irregularities on the mating surface, so that the real area of contact approaches the nominal area. High adhesion at rapid peeling rates results in the high friction associated with dry rubber. Savkoor [161 conducted friction experiments with interacting model asperities in tangential motion. The surfacesadhered as the contact area grew, followed by large shear deformation. Frequently the rubber fractured in bulk before slip occurred at the interface. Low modulus rubber displays a unique mechanism of frictional ‘slip’ and energy dissipation, which depends directly upon its adhesion Properties described above. A rubber sphere in contact with a harder plane surface undergoes large deformations under the action of a tangential force. The region of singular compression just at

the rear of the contact buckles, as shown in Fig.6, to form a fold in the rubber surface. This then travels through the contact as a ‘wave of detachment’ (Schallamach wave [19]). Thus the sphere quires an apparent sliding velocity V without any actual slip at the interface. The wave moves through the contact area with velocity v by peeling at its leading edge and reattachment at its trailing edge. Thus the material is taken round the cycle shown in Fig.4; the difference in the work of adhesion during peeling from that recovered during reattachment accounts for the frictional energy dissipation. It follows that the correspondingfriction force Q is given by

Q = (k‘- k“) w v/XV

(19)

where 5 is the spacing of the waves and v >> V.

-

LI o Bulk wl.

v

I

Figure 6. Sliding of rubber: Schallamach waves of detachment. Wave speed v >> sliding speed V. 4. STEADY SLIDING : THE EFFECT OF

CONTAMINATION

In most practical situations, unless the sliding speed and/or temperatureare high, the conditions of high adhesion leading to seizure do not arise. This is the consequence of contamination, natural or artificial. Natural contaminants include oxide films on metals, adsorbed water vapour and organic matter. Artificial contaminants consist of lubricant films, surface layers and surface treatment. As we are always being reminded in this institution (INSA), in service the interfaceacquires debris from wear and from the environment. Modelling all this still has a long way to go. For lubricant films between smooth surfaces the frictional resistance to sliding (traction) is governed by the shear properties of the lubricant at the ambient conditions of pressure, temperature and shear rate (see Fig. 7). We note that, at a given

28

sliding speed, the shear rate increases inversely as the film thickness. This introduces a transition from the Newtonian relationship in classical hydrodynamic conditions, through the nonNewtonion equationsof EHL, to the equation which is found to govern boundary lubrication by organic liquids i.e.

s = so + up

I mm 10-3

+

1 7

(20)

where p is the pressure, and and u are constants. When the film thickness is reduced to a few molecular layers (order 1.0 nm) the properties of the fluid depart appreciably from those measured in bulk. There is a marked increase in viscosity and a tendency for organic surfactants to arrange themselves in a regular molecular pattern, i.e. to adopt the structure of a solid. The apparatus developed by Israelachvili [15], in which very thin films are sheared between molecularly smooth crossed cylinders of mica, are particularly revealing of the interaction of adhesion and fluid properties in sliding friction. During sliding in dry air at light loads, the presence of adhesion is apparent from the fact that the measured contact area A (= na2) follows the JKR relationship (7), rather than that of Hertz. By fitting equation (7) to the area measurements, values of the surface energy w and the effective elastic constant E* for the mica surfaces can be obtained. Friction measurements in this region give values which are directly proportional to the measured area, implyin a constant interfacial shear strength so = 2 x 10 N/m2 (p > 2). Due to the effect of adhesion, a friction force could be measured at a negative load (Fig.8). At a high load the surfaces became rough and s e w by flakes of wear debris: adhesion vanished and friction varied with load according to Amontons' law (p = 0.44). At high humidity, when a water film condensed on the surfaces of the mica, adhesion vanished and the friction became very low. This experiment illustrates the common features of unlubricated sliding: roughening of the surfaces and/or contamination, which destroys adhesion between them and replaces it by shear in the interfacial layer. A further effect of adhesion was found by Israelachvili [15]: with certain surfactant films the effective surface energy was found to be greater when the surfaces were being pulled apart (receding contact) than when they were coming together

1-

10-6

.-

ER L

( N e v to ni anI

-

T =

'ol

ti

e.9. Z

N on N ev ~

t.)

7(P>Tj.f(l)

T, sinh-'(TP/T.)

Bonndary Lubrication

104

I i

Enbancnd viscosilg, liquidlrolid transitions

Figure 7. Lubricant film thickness chart, showing regimes of behaviour. Note that the shear rate 9 is inversely proportional to film thickness leading to non-Newtonian response of thin films.

I

I

I

I

I

N

9

Normal Load. L (x 10N)

Figure 8. Friction experiments with crossed cylinders of dry smooth mica [HI.Undamaged surfaces: contact area follows JKR, interfacial stress sc = constant (p > 2). Damaged surfaces: no adhesion, friction follows Amontons' law (p= 0.4).

29

(advancing contact), much as described in 12 with viscoelastic solids. In this case however, the cause would appear to be different. The bond between the films on each surface strengthens with time through the process of 'interdigitation' of the organic molecules, which leads to an increase in adhesion energy with contact time. Returning to the more practical situation of high loads and thicker films, liquid, solid or both, it is clear that adhesive forces do not influence directly the deformation of the solid surface or the force of friction. From the point of view of contact mechanics it is sufficient to know or to idealise the shear strength of the film in the form s=f@,T,j)

(21)

where is the shear rate. Of course, adhesion forces may influence the details of the shear process such as slip at boundary or interaction between solid 'third body' particles. We shall hear a lot more about this central aspect of frictional behaviour during the remainder of the Symposium. 5. DEFORMATION LOSSES

In most circumstances losses through inelastic deformation of the sliding solids is appreciably less than the dissipation in the interfacial film. In fact deformation losses are probably more important in relation to wear and surface fatigue than they are in relation to friction. An exception is provided by a skidding tyre on a wet road, where the interfacial friction is low and deformation losses arise from ploughing of the tyre by the asperities on the road surface. Thus, in a tyre, high hysteresis is required in response to ploughing of asperities, combined with low hysteresis in the bulk deformation in normal rolling contact. Following the reasoning which led to equation(3), Bond [21] showed that a rubber could be formulated whose relaxation time would be close to the time of passage of the material through an asperity contact, but far from the time of passage through the bulk contact patch with the road. The discussion in the remainder of this section will be confined to materials in which inelasticity is due to plastic deformation in one or both surfaces. To assess the severity of plastic deformation when the asperities of a hard surface plough a softer one

we consider the map shown in Fig9 which refers to a rigid cylinder sliding perpendicular to its axis over the surface of an elastic-plastic solid. The abscissa of the map is the non-dimensional parameter (aE*/Rk), which has been shown to correlate the transition from elastic to plastic behaviour in a static indentation [l]. The ordinate is the friction factor f = s/k, where s is the interfacial shear stress and k is the yield stress in shear. At the left of the diagram the deformation is perfactly elastic and specifiedby the Hertz theory. At the right hand side in the 'fully plastic' zone - the stresses and deformation approximate to rigid-plastic models of the type proposed by Challon & Oxley for a sliding wedge [22,23]. In the intervening zone both elastic and plastic strains contribute to the overall deformation. The boundary between the elasticplastic and fully plastic zones has been sketched in and awaits further analysis in order to fix its position more precisely. For a random rough surface the Greenwood & Williamson theory gives the mean contact size a to be ( O $ I C , ) ~ ~ where 6, and K, are the r.m.s. height and curvature respectively. Thus for multi-asperity contact of a random rough surface, the parameter (a E*/Rk) becomes (E*/k)(o, IC,)'~.This will be recognised to be Greenwood & Williamsons' plasticity index (with H replaced by k = W6). The ranges of this parameter for surfaces of different materials and differentroughnesses are superimposed on the map. The region of the map which has received most attention m n t l y , on account of its importance for wear and surface fatigue, is the zone in the vicinity of the shakedown boundary. Surfaces in continuous sliding experience many repeated asperity interactions so that it is the steady 'run-in' state which is relevant. The running-in process has been analysed on the basis of shakedown theory. The hypothesis is made that those asperities which deform plastically change their profile in such a way as to carry the maximum load whilst not exceeding the shakedown limit. If such a state cannot be found then the surfaces will not shakedown, and continuous plastic deformation would be expected The unit event consists of an encounter in tangential motion of two individual asperities, as shown in Fig.10 [N]. Asperities of equal hardness can always reduce their height and curvature to reach a mutual shakedown profile but if one surface is hard the limit for the other is reached when it is completely flattened. Starting from the unit event,

30

of the model and the experiments of Williamson et al. [26]is remarkably good. The shakedown limit is reached when the soft surface is deformed into a flat (zero height deviation).

I

-

Load

3

.g LL

0.1 0.05

-

4-4 madm

--ELASTIC

t h c - p u s n c

',

I---+ Cerodcs Noh I

-A- ~ U YPMC

--

100

10

Figure9. Deformation map of sliding rough surfaces. os & K~ are mean summit height & curvature. Low values of Y : shakedown model [27]; high values of Y :slip-line models [2223].

Figure 11. Running-in by plastic shakedown of a random rough surface [23. Note reduction in height variation with increasing load. Theory compared with experiments of Williamson et al. [%I. tb)

um

II

Figure 10. Plastic shakedown in a repeated asperity encounter. Note height and curvature are reduced; the shakedown limit is reached when one surface is flattened.

the running-in of two nominally flat, randomly rough surfaces, one hard and one soft, has been modelled. If the soft surface initially has a Gaussian height distribution, this appears on 'normal' probability paper as a straight lineofgradient proportional to the standard deviation (Fig.11) [25]. After repeated sliding the height distribution becomes bi-modal; the plastically deformed asperities show a r e d u d variation in height (and curvature). The effect becomes more pronounced as the load is increased and the shakedown limit is approached. The agreementbeween the predictions

If the load is increased above this limit plastic deformation will take place with the passage of every asperity, causing plastic dissipation and eventual surface damage. The mechanics of accumulating plastic strain i.e. ratchetting, which occurs in these circumstances has been analysed by Bower & Johnson [27]. At friction coefficients in excess of 0.25 plastic deformation is confined to the near-surface layer. The stress cycle due to the frictionaltraction experienced by surface elements in sliding contact is shown in Fig.l2a&b. (The stresses due to the normal pressure p are hydrostatic, i.e. oxx= ozz = -p, and so do not appear). The plastic strain cycles are shown in Fig.12~. The direct strain increments, A E ~ :in the compression and tension sectors cancel out, but the shear increments A E are ~ cumulative, leading to the large plastic stnu observed in the tracks of sliding surfaces. The strain increment acquired in each cycle AeL , however, is small: only a few

31

s-'

Figure 12. Plastic ratchetting of a surface layer by a hard cylindricalasperity in sliding contact. (a) Elastic stresses at surface; (b) stress c cle OABCO at yield in am - ra stress space: (c) plastic strain cycles, showing accumulation of shear E, .

4

times the elastic yield strain. In the example shown

in Fig.12 the energy dissipated in this plastic deformation is less than 1.0% of the dissipation by friction at the interface.

(3)

With perfectly elastic solids wf = w = the thermodynamic work of adhesion (surface energy) which is reversible. Inelastic deformation, plastic or viscuelastc, in the zone of separation enhances wf during separation and attenuates it when the contact area is inclt-asing.

(4)

Clean smooth surfaces do adhere strongly. For the static friction of such surfaces to be modelled as a mode II fracture it is necessary for the contact size a >> (4/7C)(KIIdS)2 where s represents the interfacial shear strength s = f x yield stress k. This condition is satisfied for rubber and hard ceramics, but polymers and metals 'slip' by bulk shear. When s = k shear takes place in the solid, when s < k slip occurs at the interface.

6. CONCLUSIONS Acoustic losses are negligible at sliding speeds which are small compared with elastic wave speeds. Rate-dependent friction arises from rate dependent Properties of the solids or the interfacial film. Adhesion in Hertz contact increases the contact area and requires a tensile ('pull-off') force to separate the surfaces. It can be characterised by a work of adhesion wf or by a fracture toughness kIc = (2 wf E*)ln.

32

(5)

Fluid films of a few molecular layers may exhibit properties which differ from those in b u k enhanced viscosity and attachment to the substrate in an ordered way characteristic of a solid. Adhesion forces increase the contact area and give rise to a detectable friction force at negative loads. The adhesion effect becomes negligible at high load, i.e. P >> WfR.

(6) With thicker films between smooth surfaces friction is determined entirely by the shear strength of the film s . In boundary lubrication (10-100 nm), s = + up.

(7) The contribution to friction from inelastic deformation ,viscoelastic or plastic, of the solid surfaces is generally small compared with that arising from shear of the interfacial layer. (8)

In continuous sliding rough surfaces tend to run-in to a steady elastic (shakedown) state. In theory this is always possible with surfaces of equal hardness. If one suface is comparatively hard (under forming) a limiting pressure for shakedown exists; if this is exceeded plastic ratchetting occurs leading to large accumulatedplastic strains.

(9)

In attempting to relate molecular friction, measured by the atomic force microscope or modelled by molecular dynamics, to continuum models on the macroscopic scale, it must be recognised that there is an intermediate length and time scale (see Fig.2) with independentphysical phenomena. This is the scale of dislocations and microstructure: 10 - 103 nm.

REFERENCES 1. K.L. Johnson, Contact Mechanics. C.U.P. 11, 1985. 2. J.A. Greenwood & D. Tabor, The friction of hard sliders on lubricated rubber, Roc.Phys. Soc. 71 (1958 ) 989.

3. G.M. McClelland & J.N. Glosi, Friction at the atomic scale, in Fundamentals of Friction, NATO ASI, Ser.E, 220, (1992) 405 Khmer. 4. W. Goddard, Tribology of ceramics, Nat.Mat.Ad.Bd.Rpt.435, Nat.Acad.Press, Washington DC, (1988) 88. 5 . K.L. Johnson, A note on the adhesion of elastic solids, Brit.J.Appl.Phys. 9, (1958) 199. 6. K.L. Johnson, K. Kendall & A.D. Roberts, Surface energy and the contact of elastic solids, Proc.Roy.Soc.London, A 324 (1971) 301. 7. K.N.G. Fuller & D. Tabor, The effect of roughness on the adhesion of elastic solids, Proc.Roy.Soc. London, A 345 (1975)T 327. 8. K.L. Johnson, Adhesion at the contact of solids, Roc.Congress Th. & Appl. Mech. Ed. Koiter, North Holland. 1976. 9. D. Tabor, Hardness of Metals, Oxford 1951. 10. J.A. Greenwood & K.L. Johnson, The mechanics of adhesion of viscoelastic solids, Phi1.Mag.A 43 (1981) 697. 11. D. Maugis & M. Barquins, Fracture mechanics and the adherence of viscoelastic bodies, J.Phys.D. 11 (1978) 1989. 12. B. Derjaguin, V. Muller & Yu Toporov, Effect of contact deformations on the adhesion of particles, J.Col1. & Interface Sci. 53 (1975) 314. 13. D. Maugis, Adhesion of spheres: The JKRDMT transition using a Dugdale model, J.Coll. & 1nt.Sci. 150 (1992) 243. 14. A.R. Savkoor & G.A.D. Briggs, The effect of a tangential force on the contact of elastic solids in adhesion, Proc.Roy.Soc. A 356 (1977) 103. 15. J.N. Israelachvili, Adhesion, friction and lubrication of molecularly smooth surfaces, in Fundamentals of Friction, NATO AS1 Series E v 220,1992. 16. A.R. Savkoor, Dry adhesive friction of elastomers, Doctoral dissertation, T U Delft, 1970. 17. D. Tabor, Junction growth in metallic friction, Proc.Rov.Soc.A 251 (1948) 378. 18. K.L. Johnson, Deformation of a plastic wedge by a rigid flat die under the action of a tangential force, J.Mech. & Phys.Solids, 16 1968 395. 19. A. Schallamach,How does rubber slide?, Wear 30 (1971) 301.

33

20. A.D. Roberts & A.G. Thomas, Adhesion and friction of smooth rubber surfaces, Wear, 33 (1975) 45. 21. R. Bond, A new tyre polymer improving fuel economy and safety, Proc.Roy.Soc.A, 399 (1985) 1. 22. J.M. Challen & P.L.B. Oxley, The different regimes of friction & wear, Wear 53 (1979) 229. 23. J.M. Challen & P.L.B. Oxley, Slip line fields for polishing & related processes, 1J.Mech.Sci. 26 (1984) 403. 24. K.L. Johnson & H.R. Shercliff, Shakedown of 2-D asperities in sliding contact, 1nt.J.Mech.Sci. 34 (1992) 375.

25. A. Kapoor & K.L. Johnson, Steady state topography of surfaces in repeated boundary lubricated sliding, Proc. 19th.Leeds-Lyon Symp.on Tribology, Leeds, 1992. 26. J.P.B. Williamson, J. Pullen & R.T. Hunt, The shape of solid surfaces, in Ling F.(ed.) Surface Mechanics,New York, 1969. 27. A.F. Bower & K.L. Johnson, The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact, J.Mech. & PhysSolids, 37 (1989) 471.

This Page Intentionally Left Blank

SESSION II LIQUID AND POWDER LUBRICATION Chairman:

Professor J M Georges

Paper 11 (i)

A Rheological Basis for Concentrated Contact Friction

Paper 11 (ii)

On the Theory of Quasi-Hydrodynamic Lubrication with Dry Powder: Application to High-speed Journal Bearings for Hostile Environments

Paper 11 (iii)

The Influence of Base Oil Rheology on the Behaviour of V1 Polymers in Concentrated Contacts

Paper 11 (iv)

Temperature Profiling of EHD Contacts Prior to and During Scuffing

Paper 11 (v)

Computational Fluid Dynamics (CFD) Analysis of Stream Functions in Lubrication

Paper 11 (vi)

Shear Properties of Molecular Liquids at High Pressures a Physical Point of View

-

This Page Intentionally Left Blank

Dissipative Processes in Tribology / D. Dowson et al. (Editors)

37

CP 1994 Elsevier Science B.V. All rights reserved.

A RHEOLOGICAL BASIS FOR CONCENTRATED CONTACT FRICTION

Scott Bair and Ward 0. Winer Georgia Institute of Technology, School of Mechanical Engineering, Atlanta, GA 30332-0405 It has been observed in boundary lubrication and in some aspects of elastohydrodynamic lubrication that friction is nearly Coulombic in nature - the friction coefficient is only weakly dependent upon load and sliding velocity. In some instances the friction coefficient may be so similar in the boundary and EHD regimes that friction alone does not clearly discriminate the transition from one to the other. These attributes of liquid lubrication would seem enigmatic. However, recent observation of slip planes (shear bands) within a pressurized liquid film suggest that a low molecular weight viscous liquid possesses a "material internal friction coefficient" which is a material property of the lubricant and represents the ratio of shear stress to compressive normal stress at which slip within the film is incipient. The friction coefficient of the contact is then a consequence of and quantitatively related to the lubricant material internal friction coefficient. The Mohr-Coulomb failure criterion is introduced as a predictive method for slip. Mohr-Coulomb defines two possible orientations for stress induced shear bands. Both are experimentally observed in a highpressure flow visualization cell and the measure of the included angle between the types of bands is consistent with theory. The concept of a first normal stress difference (once the subject of much speculation in lubricated contact studies) must be introduced to account for the orientation of the shear bands with respect to the principal shear directions. 1. INTRODUCTION

In spite of the great progress made in understanding hydrodynamic lubrication and particularly the quantitative prediction of film thickness, the coefficient of friction in concentrated contact is still often dealt with as a disposable parameter in numerical analyses. Boundary and elastohydrodynamiclubrication have been recognized as separate regimes of concentrated contact lubrication since the EHD solution of Ertel-Grubin. However, the delineation of these regimes is neady always in practice based on the magnitude of the film thickness relative to surface roughness rather than a transition in friction. Friction in boundary lubrication is usually modeled as Coulombic with a coefficient of about 0.10 to 0.15. Friction in sliding EHL can also reasonably be modeled as Coulombic except at very high sliding velocity with a coefficient of about 0.03 to 0.12. In some experiments the friction coefficient varies continuously and smoothly as the lambda ratio is reduced from very high values to much less than unity. Recent EHD analyses

which address roughness (eg. Ref [l]) have pointed out that lubricant rheology is active in separating solid boundaries and transmitting shear force even when roughness interactions are important. Johnson [2] recognized boundary and EHD regimes as manifestations of the same lubricant rheological response. Much attention has been paid in the literature to the search for a general rheological constitutive equation for liquid lubricants at high pressure [3,4]. The task is further complicated by rhe observation of mechanically induced (not be confused with thermal or "adiabatic" shear bands which are simply a solution to the combined energy and momentum equations [S]) shear localization in liquids [a] under combined pressure and shear stress. Constitutive behavior by definition excludes localization. It is now apparent that much of the departure from Newtonian behavior which has been attributed to constitutive behavior is actually a result of localization in the form of shortlived shear bands. This paper will attempt to bridge the gap between constitutive modeling and recent observations of localization by offering a reasonable

38

rheological constitutive equation and introducing a failure criterion for predicting localization. 2. NEW OBSERVATIONS OF MECHANICAL SHEAR BANDS

The High-Pressure Flow Visualization Cell which we use to optically observe shear bands has been described in detail in Ref. [6] as well as the experimental procedures and preliminary results. It was shown that at a critical shear stress which marked the limit of linear response, an intermittent slip occurred across visible bands. Evidence has been introduced [7] which supports the existence of a mechanism within an operating EHD film, which scatters light in directions consistent with bands observed in flow visualization. In the same paper the existence of two types of bands with differing orientation was introduced. Figure 1 is reproduced from that paper to illustrate the orientations of the two types. New observations which give insight into the slip mechanism are presented in the next sections.

Figure 1. Two types of mechanically induced shear bands

Molecular orientation under stress gives rise to optical anisotropy - the refractive index, n, is different in different directions. Birefrigence observationscan be used to determine the principal normal stress difference, Ao, and the orientation of the principal stress axes if a suitable calibration of the birefrigence, An, can be obtained. Such a calibration is not possible with the flow cell. However, some generalizations regarding the stress distribution in the flow field may be obtained. The High-Pressure Flow Visualization Cell was placed between crossed polarizes See Figure 2. White light was filtered to produce a narrow band at 605 nm. The polyphenyl ether, 5P4E, was the model lubricant. Temperature was 23°C and pressure was 241 MPa. The shearing force was increased with time until shear bands were observed. The interference pattern shown in Figure 2 was obtained just before the shear localized. This fringe pattern disappeared at the time the shear bands formed. Similar experiments were performed with the crossed polarizers placed at eight different angles with respect to the shear direction equally spaced over a span of 180" in an attempt to determine the extinction angle. No effect of orientation of the polarizer/analyzer pair was noted on the contrast or geometry of the fringes or shear bands.

Figure 2. Flow birefringence in 5P4E at 23°C. 241 MPa.

2.1 Birefrigence

Flow birefrigence has been useful for stress analysis in flowing liquids in a manner analogous with photoelasticity in transparent solids [8].

Apparently, the high-pressurewindows which are sapphire are performing as fractional wave plates to eliminate the extinction angle effect [8] and so it

39

was not possible to obtain the principal stress directions. It is possible to obtain the stress distribution in a qualitative sense. If the stress-optic function is monotonic [9], then the fringe order is a representation of the relative magnitude of the principal normal stress difference (and the principal shear stress). Referring again to Figure 2, the zeroeth order fringe occurs in the liquid reservoir far to the right in the figure. The principal shear stress should be a maximum within the highest order fringe in the region marked A. Notice that the shear stress is uniform in the shearing gap at positions fartlier than about 3 times the film thickness into the gap. It was noted previously [lo] that the first band nucleates at the point A and runs to the point marked B in Figure 2. Two conclusions may be drawn from this observation. 1) The reason for the observation of the first bands near the entrance is that the shear stress is greatest there. 2) A band which nucleates within a region, A, of locally high stress (compared to the average stress within the field) will continue to run through a region of locally lower (than the average) stress. The significance of the latter is important. Once initiated the shear band defines a plane on which slip is more easily accommodated than in the bulk of the material - it becomes a "weak spot" and tlie slip is arrested only by pinning at the solid boundaries of the film. 2.2 Persistence of Shear Bands When mechanically induced shear bands are studied in solid amorphous polymers, the observations are usually made on sectioned specimens [lll long after the deformation responsible for the bands has ceased. This is possible because the image of the band, which is probably a result of damage or dilatation, persists. In the liquid lubricants this persistence time is shorter. In Figure 3 we have plotted tlie length of time for which a shear band was visible in 5P4E after shearing (persistence time) versus the viscosity at test conditions. If we define the mechanical shear relaxation time as the ratio of viscosity, 1.1, to shear modulus, G, then the straight line plotted in Figure 3 is the mechanical relaxation time for G = 1 GPa. We may define a characteristic time for thermal diffusion as h2/D where h is the thickness of the

band and D is thermal diffusivity. D is essentially independent of p and is typically about l o 7 m2/s for liquid lubricants. If a shear band is at most 2p.m thick, then the characteristic time for thermal diffusion is at most 40p. The correlation of persistence (of the order of seconds) with mechanical relaxation is much better than for thermal diffusion. Clearly the image of the shear band is a result of mechanical damage or dilatation and not temperature.

a

4 hhr.tlon 0 : 1 OP.

tima

-1

7.6

0.0

0.4 LOO

O.B

0.2

B.6

10.0

vlaco~ltv( h . * l

Figure 3. Time for which a shear band is visible. 2.3 Dilatation A related observation concerns the degree of contrast of the shear band images. Among the liquids in which we have observed shear bands is the perfhoropolyakyl ether, 143AD. In this material the bands appear so faint that it is difficult to reproduce them on a video print, The refractive index, n, is related to density, p, through the Lorenz-Lorentz equation.

1 n2-1 = constant P n2+2 From tliis we can obtain

where

E

is the dilatation. The refractive index of

40

143AD is 1.30 and the refractive index of 5P4E is 1.63. Inserting these values into eqn. (2) we find that the rate of change of n with respect to & is 0.3 for 143AD and -0A for 5P4E. So for the same dilatation the refractive index changes three times less for the perfluoropolyalkyl ether than it does for the polyphenyl ether. Thus the relative degree of contrast in shear band images is consistent with an argument that a persistent dilatation has occurred along the band.

-

reports that m = 0.027 for polycarbonate with shear bands. Such a low rate sensitivity (6%per decade) would possibly be undetectable in high-pressure rlieometers and disc machines; hence the limiting shear stress idealization. Glassy polymers are known to exhibit a pressure dependent shear strength which at high pressure follows the rule [ 113 ty = tyo+

Ap

(3)

2.4 Surface Roughness Effect It miglit be argued that the roughness of the

solid boundary provides a nucleation site for shear bands. The moving shaft which performs as one (the moving one) of the solid boundaries lins two sides each of which may be utilized in an experiment. One side was left as machined with an rms roughness of 1.Opm. The other side was polished to a roughness of 0.03 pn. The stationary surface has a roughness of 0.3 pn rms. Both sides of the moving shaft have been utilized. No difference in the character of the b'ands or die manner in which they developed was noted. We, therefore, cannot associate surface roughness, at least up to 1 pn, with shear localization in liquid films. 3. SLIP CRITERION

where 3iy varies from 0.1 to 0.25 and p is pressure. A similar rule is known to apply to the limiting shear stress of liquid lubricnnts, although with a lower proportionality constant. 3.2 Mohr-Coulomb Criterion The Mohr-Coulomb criterion predicts that slip will occur along any plane on which the ratio of shear stress, f0, to compressive normal stress, -oe, attains the magnitude of a material friction coefficient, T. Referring to Figures 4 and 5 , 78 and may be resolved from the shear stress, f, and normal stresses oxand o,,which are oriented along and perpendicular to the solid boundaries in a plane shear experiment. The mean mechanical pressure, p, is defined by p = -%(ox+ oJ and h = f/p.

3.1 Analogy to Glassy Polymers In a previous paper [6], tlie authors have

associated mechanically induced shear bands with the rate-independent shear stress which is observed in liquid lubricants under pressure in disc machines and rheometers. As a critical stress which is roughly proportional to pressure is approached, increasing amount of the deformation in an otherwise ratedependent matrix is accommodated by intermittent slip along inclined shear planes. When this critical (limiting) stress is reached, any increase of the apparent rate of shear can be accommodated by an increased production of sliear b'mds without changing the stress: the response is rate-independent. In practice, for glassy polymers, the strain rate sensitivity coefficient m = dPnf/dPnf, can be quite low when shear bands are operating. Here, z is shear stress and f is rate of shear. G'Sell [123

Figure 4. Definitions of angles and stresses.

41

which leads to

/ 0

/

0

Figure 5. Mohr's circle representation of slip criterion.

The two solutions for 8, are the shear band angles 8, and €I2for the fvst and second types respectively. The material friction coefficient is

In general, the nonnal stresses will not necessarily be equal and we will quantify the first normal stress difference, N, = ox - aY by 5 = N,/2p. The parameters h and 5 are dimensionless. We may resolve

(5) The slip criterion can be represented graphically on the Mohr's circle plot in Figure 5. The Mohr's circle of stress must fit within the envelope defined by the broken lines. These lines must curve away from the Q axis near p = 0 to allow a non-zero shear stress there. The included angle of he envelope is twice the material friction coefficient, q. Now, slip will occur when the circle of stress increases to tangency with the envelope and the orientation of the band will be such that I 7, / o0I is maximized. This is satisfied by the two radii of the circle drawn to the points of tangency. These radii represent the two shear band angles, 8, and 8, in Figure 4. Analytically, the shear band angles can be found by setting

If the first normal stress difference N, were zero, then the two shear band angles would be complimentary angles (sum to 90") defined by solutions of 8, = l/i sin-' h. Clearly the shear band angles in Figure 1 are not complimentary. It will be necessary to invoke the first normal stress difference to reconcile theory with observation. For example, from Figure 1 for the polyphenyl ether, 8, = 19" and O2 = 103". Then h = 0.089 and 5 = 0.056 are obtained from equation (7). This value of h is in agreement with the measured [13] limiting shear stress if h is applied to a linear equation like (3). The first normal stress difference obtained is a little more than the shear stress. This may seem excessive, since consideration of the normal stress difference has not been necessary to predict bearing load capacity; however, Tanner [141 showed that for typical lubrication flows the fractional increase in load capacity is unaffected by N, of the order of 2. The experimental measurement of N, at high pressure is challenging and has not previously been attempted - possibly because of the expected irrelevance to bearing load capacity. Measurement of N, could confm the applicability of Mohr-Coulomb and might be considered a research priority. In the example above, the material friction

42

coefficient, q = 0.106. Tlie value of q is simply tlie difference between d 2 and tlie included angle between the two types of band (in radians). Since this included angle is very nearly x/2, the determination of q (and h) from observations of bands is sensitive to tlie accuracy of the shear band angle measurement.

4. RHEOLOGICAL MODELS

In previous work [4] the authors showed from experimental measurements that the Maxwell Model, which sums elastic and viscous strain components, correctly describes tlie transient liquid response under pressure. The simplest incompressible fonn of tlie Maxwell model requires two rheological properties - for example: the limiting elastic shear modulus, G, and the limiting low shear viscosity, p. A f u l l rheological model for elastohydrodynamic lubrication was developed from observations of the liquid lubricanl response observed in high-pressure rlieometers. Here, 4, is the rate of deformation tensor, T,, is the deviatoric stress tensor and .re is the von Mises stress. In formulating this equation we adopted tlie Stokes' Condition - the mean mechanical pressure, p, being set equal to the tliemiodyn:unic pressure, p,.. However, in writing the full model in Ref [4] we also set the second coefficient of viscosity equal to zero atid this is inconsistent witli Stokes' Condition. We suggest tentatively tliat a knn be added to the right-h,md side of the equation so that tlie full model now reads

..3pK]

dij = d [r i j - 6r~dt 2G

;:

equation of state. Stokes' Condition is now ab'andoned 'and conipressional viscoelasticity is addressed explicitly. The second coefficient of viscosity is zero which sets the bulk viscosity equal to 2/3p. This choice is made so tliat the short time isothermal compressibility is 1/K and tlie long time compressibility is obtained from Uie state equation. The function F(T,~,T,)is an empirical rate relation. Equation (9) represents unfinished business in two respects. A first normal stress difference is not explicitly introduced. Tlie use of the Jaumann time derivative in (9) will result in N,/T of the order of T/G if F = 1 [15]. Thus the magnitude of shear stress, T, must be close to that of the shear modulus, G , to yield tlie first normal stress difference required by the Mohr-Coulomb theory and observed shear band angles. This is too great a value of T since z,/G is approximately 1/30 [2]. Secondly, we now know Chat tlie non-linear (in TJ form of F(T,p,z,) is at least in part due to shear localization. Constitutive behavior, by definition, excludes localization. Iliat is to say that only the behavior of the matrix between shear bnnds can be described by a constitutive law. For many simple, low molecular weight base stocks it may be most correct to set F = 1 and apply a slip criterion such as Molu-Coulomb. Then the non-linear behavior which was introduced previously through F is a consequence of the distribution of q through tlie material and tlie slip velocity. For simple cases where the flow is steady simple shear, an empirical stress equation such as we have advanced

t =

t,(l-e -CIflr')

+ -F(Tg,t,)

(9)

where K is now the bulk modulus of the glass and p,. is that pressure which yields tlie inst'antnneous density of the liquid when used in the equilibrium

is sufficient. Here the rate sensitivity coefficient, m, goes to zero as py/zL becomes large. For solid polymers where shear bands are operating, m is small but not zero. Also, recent measurements [ 131 indicate tliat the transition from Newtonian to "rateindepcndent" behavior for high molecular weight lubricants is broader in shear mte tlian Uiat which is described by equation (10). These deficiencies may be removed by using the Carreau-Yasuda form

43

Leeds-Lyon Symposium, (1993).

3.

Johnson, K. L. and Tevaarwerk, J. L., "Shear Behavior of Elastohydrodynaniic Oil Films," Proc. R. SOC. Lond., A-356, pp. 215-236, (1977).

The dimensionless exponent, a, controls the breadth of the transition and rate sensitivity coefficient, m, appears explicitly. Note that equation (11) is equivalent to the Elsharkawy and Hamrock [16] model when m = 0. For m very small (-0.01) but not zero, equation (11) fits experimental results well while removing Uie singularity which causes problems in numerical simulations.

4.

Bair, S. and Winer, W. O., "The High Pressure High Shear Stress Rheology of Liquid Lubricants," Trans. ASME, Journal of Tribology, Vol. 114, 1, pp. 1-13, (1992).

5.

Bair, S., Qureshi, F., and Khonsari, M., "Adiabatic Shear Localization in a Liquid Lubricant Under Pressure," ASME Journal of Tribology, 93-Trib 29, (1993).

5. CONCLUSIONS

6.

Bair, S., Qureshi, F., and Winer. W. O., "Observations of Shear Localization in Liquid Lubricants Under Pressure," ASME, Journal of Tribologr, 115, 3, pp. 507-514, (1993).

7.

Bair, S., Winer, W. 0. and Distin, K. W., "Experimental Investigations into Shear Localization in Operating Concentrated Contact," Proc. 19th Leeds-Lyon Symposium, (1992).

8.

Harris, John, Rheology and Non-Newtonian Flow, Longman Group, London pp. 63-64, (1973).

9.

Janeschitz-Kriegl, H., Polymer Melt Rheology and Flow Birefringence, Springer, Verlag, Berlin, p. 118, (1984).

Tlie Coulombic nature of concentrated conk?ct friction is apparently the result of an interniittent slip mechanism operating wilhin an otherwise linear viscoelastic liquid lihn under highpressure. A rigorous analysis of the response of lubricant films with a non-uniform stress field will require a Maxwell model for constitutive behavior coupled with a failure criterion for slip. Tlie MoluCoulomb Model is the appropriate criterion when a first normal stress difference is assumed.

6. ACKNOWLEDGEMENTS

This work was supported by a grant from the Office of Naval Research, Materials Division, Peter Sclunidt, Scientific Officer.

REFERENCES 1.

2.

10. Qureshi. F., "Kinematics of Shear Deformation of Materials Under High Pressure and Shear Stress," P1i.D. Thesis, Georgia Institute of Technology, (1992).

Cheng, L., Webster, M. N. and Jackson, A., "On the Pressure Rippling and Roughness Deformation in EHD Lubrication of Rough Surfaces," ASME Journal of Tribology, 115, No. 3, p. 44, (1993).

11. Bowden, P. B., "Yield Behavior of Glassy Polymers," Physics of Glassy Polymers, Wiley, New York, Edited by R. N. Haward, p. 313, (1973).

Johnson, K. L., "Non-Newtonian Effects in Elastohydrodyn,amic Lubrication," Proc. 19th

12. G'Sell, C., "Plastic Deformation of Glassy Polymers: Constitutive Equations and

44

Macromolecular Mechanisms," in Strength of Metals and Alloys, Pergramon Press, Oxford, p. 1955, (1986).

Contacts," Trans. ASME, Journal Tribology, 113, 3, p. 647, (1991).

of

13. Bair, S. and Winer, W. 0.. "A New HighPressure, High Shear Stress Viscometer and Results for Lubricants," STLE Tribology Trans., 36, 4, (1993).

FIGURES

14. Tanner, R. I., Engineering Rheology, Clarendon Press, Oxford, p. 237, (1985).

Figure 2. Flow birefrigence in 5P4E at 23"C, 241 MPa.

15. Hutton, J. F., "Theory of Rheology," Interdisciplinary Approach to Liquid Lubricant Technology, Edited by P. M. Ku, NASA, p. 206-207, (1972).

Figure 3. Time for which a shear band is visible.

16. Elsharkawy, A. A. and I-Ianuock. B. J., "Subsurface Stresses in Micro-EML Line

Figure 1. Two types of meclianically induced shear bands.

Figure 4. Definitions of angles and stresses. Figure 5 . Mohr's circle representation of criterion.

slip

Dissipative Processes in Tribology / D.Dowson e l al. (Editors) 0 1994 Elsevier Science B.V. All rights reserved.

45

On the Theory of Quasi-Hydrodynamic Lubrication with Dry Powder: Application to Development of High-speed Journal Bearings for Hostile Environments Hooshang Heshmat, Ph.D. Mechanical Technology Incorporated 968 Albany-Shaker Road Latham, New York 12110 USA

This paper describes a series of experiments aimed at the demonstration of the basic feasibility of developing a powder-lubricated, quasi-hydrodynamic (PLQH) journal bearing for high-temperature and hostile environments, where the use of liquid lubricants is impractical. A PLQH bearing has demonstrated operation at speeds to 2 x lo6 DN (58,000 rpm), and it may be the only bearing capable of meeting the ever-demanding tribological goals of a solid lubrication scheme for extreme environments. The work described exceeds the current state of the art (1.5-million DN) in solid-lubricated ceramic rolling element bearing technology, and there is great promise for integrating this technology in outer space systemdmechanisms and in other hostile-environment applications. Experimental evidence shows that powder lubricant films behave much as fluid films do, whereby mechanisms are provided that lift and separate bearing surfaces and cause side leakage. These mechanisms reduce the friction coefficient and, consequently, the heat generated in the bearings, which drastically reduces wear of the tribomaterials. Further, bearing side leakage provides a significant mechanism for heat dissipation because it carries away most of the heat generated by shear, reducing the heat to the critical bearing surfaces (see Figure 1). Experimental parametric studies have delineated the hydrodynamic effects of powder lubrication (MoS,) on bearing performance criteria, such as load, temperature, and power loss as a function of speed, including the effect of powder flow rate on bearing performance characteristics. Comparison with a liquid lubricant provides evidence for the continuum basis for the phenomenological unification of solid particulates and liquid.

- - v,

2v2

u v

Surface 1

I

1

L

U

"

i

= Velocity of Powder Lubricant = Velocity of Independent Bodies

6

= Surface Roughness

O(6) H h, -T Y

On the order of (6) = Heat Flow = Minimum Film Thickness = Total = Reference Coordinate =

!

911045

Figure 1. Quasi-Hydrodynamic Model for Powder Lubrication

46

wn

= 35 Ib (155.7 N)

N

= 8.87 rps

"0

Figure 2.

= 4.6 mlsec

Pivot Point 1, - 6

=

1 Volt

=

60% from Leading Edge of Pad

= 5sec

10013 psi (230 kPa)

Pressure Profiles for Powder-Lubricated Pivoted-Pad Thrust Slider

Wn = 155.68 N (35Ib)

m

a

Oil: SAE 10 wt

U

Figure 3.

Comparison of Lubricant Pressure Profiles for SAElO Oil and TIO, Powder vs. Extent of Pad

47

INTRODUCTION

Ideal rigid particles have been used in almost all attempts to build fundamental hypothesesdescribing the dynamics of powders. These particles have generally been assumed to be both smooth and spheroidal, when in fact actual media composed of nearly rigid particles rarely exist in such simple shapes, as evidenced by sand and many lubricating powders [6, 161. All these media are influenced by friction between the particles. The dynamic properties of media composed of ideal smooth particles in a high state of agitation have been the subject of many investigations. The first to mention particulate flow was Osbome Reynolds (1885) [ 11. Based on a series of other investigations [ 17-19], Heshmat conceptualized in 1988 [2] that the dynamic properties of a medium consisting of fine particulates of unrecognizable shapes, sheared in a narrow gap by forces transmitted through the medium, would be akin to fluid film properties. This unique property is called "quasi-hydrodynamiclubrication with dry triboparticulate matter" [2] because the fluid-like bulk property may change as a consequence of distortional strains or any disturbance that causes a change of volume, density, or temperature of the medium. A sound theoretical basis has now been established based on a series of systematic analytical and experimental investigations that demonstrate the similarity between the velocity, density, pressure, and temperature profiles produced by liquid-lubricated and powder-lubricated bearings [3,4, 5,241. Thus, a lubricant consisting of a fine powder either inserted deliberately or generated by the wear of the mating surfaces constitutes a viable lubricant that generates the required flows and pressures to prevent contact between the surfaces. RHEODYNAMICS OF POWDER LUBRICATION A recent series of investigations aimed at providing evidence of the quasi-hydrodynamic nature of a powder lubricant and which led to the first demonstration of a powder-lubricated journal bearing was completed in three stages. The first stage concerned the rheology and hydrodynamics of dry powder lubrication, in which Heshmat [2, 61 conceptualized the mechanism of powder flow that possesses some of the basic features of hydrodynamic lubrication. The

dynamics of the particles, provided they are of the proper size (1 to 10 pm), function not as aggregates or compacted individual bodies, but rather like a continuum with many of the velocity and shear characteristics analogous to hydrodynamic fluid films. While they also exhibit substantial differences, on the whole, they are closer to the nature of a lubricant film than to the behavior of an aggregate volume of discrete particles. The basic feature of the quasi-hydrodynamic flow is an in-situ, layered flow of the powder film, which is portrayed schematically in Figure 1. Thus, the hydrodynamic behavior of a powder lubricant can be described as a sheared layer that adapts itself to the adjacent layer, so as to cause the least possible discontinuity in the flow of the lubricant film. The basic model of powder lubrication (Figure 1) parallels other established tribological disciplines and permits an evaluation of powder lubrication performance in quantitative terms. Referring to Figure 1, in a sliding contact with powder lubrication, the film can be divided into regions of intermediate films and powder lubricant film. The thickness of the intermediate films is expected to be on the same order as the surface roughness O(6,) and O(6,). In fact, the surfaces of the sliders and counterfaces (surfaces 1 and 2) have been observed to have thin, adhered layers of extremely fine powder after a short period of testing with powder lubricants. This adhered layer plays a major role in terms of protecting the tribomaterial surfaces and dictating the magnitude of slip velocity in the boundaries. There were two main postulates resulting from this early work [2]: In the dry friction regime, the powder constitutes a lubricant, imparting to the interface many of the characteristicsand effects of a hydrodynamic film. Such a hydrodynamic regime holds only for a certain range of particle size (or wear debris) with respect to the nature of tribomaterial combinations. Based on these postulates, a theoretical, rheological model for quasi-hydrodynamiclubrication with dry triboparticulate matter was developed and some solutions were obtained [6, 8, 9, 16, 20-22, 241. The second stage of the investigations dealt with lubricant flow visualization studies conducted by Heshmat [4]. During these studies, a lubricant consisting of micron and submicron size powders was used that offered visual evidence of the theorized velocity

48

Tu

w $-u

-

Lubricant

i--i'P-.-I A(b)

I

1000

I

0,

C

c 0)

0

100

a:

10

-um 2

0

L

3

,

lo-'

Lubrication

t

\ \

\

/

\

\

/

1

I I

v)

(d) Hydrodynamic Lubrication

L

0,

C

z .-0 A=

1

+

-E

0.

.-

LL

. 5.0' '\

"

;I 4

0 Speed Parameter, p

--- Wear

---

Friction

Figure 4.

-

881604-1

Film Thickness/Surface Roughness

Traditional Stribeck Curve for Conventional Liquid-Lubricated Bearings

Conlrollable

WaIer-CooIeclI I . Support Splndk

Eleclric

Frictional Torque

Torque Arm

I

.

View A-A Tesl Load

Figure 5.

91889

Powder-Lubricated Hydrodynamic Journal Bearing Test Rig and Instrumentation

49

and shear characteristics of a powdered layer in the interspace of a journal bearing. The main results of this series of experiments conducted with a journal bearing having a D x L x C = 64.4 x 12.7 x 6.6 mm (2.5 in. x 0.5 in. x 0.26 in.) at an eccentricity ratio of E = 0.98 showed the following fluid-like behavior of the powder film: A boundary-layer-like flow occurs along the moving surface. A shear stress is distributed across the film. Compressive and tensile stresses are generated in the film due to journal rotation. The converging part of the film produces compressive stresses and forms fractures in the radial direction; the diverging region produces tensile stresses. Unique boundary conditions that are dependent on the nature, properties, and interaction between the powder and the walls prevail at the two solid surfaces. The originally circumferential flow of powder assumes an axial motion as minimum film thickness, hmin,is approached, indicating the formation of an axial pressure distribution and producing side leakage, analogous to hydrodynamic lubrication. Past hmin,a powder film shows chaotic motion akin to turbulent flow, or cavitation. The third stage in the investigations focused on experimentally determining the pressures generated in the powder film and their kinship with hydrodynamic pressure profiles associated with fluid lubricants. The experiments were conducted with a pivoted pad thrust bearing having an OD of approximately 200 mm (8 in.). Figures 2 and 3 show the nature of the pressure profile that was generated with a powder lubricant and compares it with a comparable oil film pressure profile [5]. As seen, the longitudinally skewed pressure profile generated with a powder film is typical of a hydrodynamic oil film. Following the series of powder lubrication efforts, an appropriate theoretical formulation of the quasihydrodynamic process of powder lubricants was established [6,16,24]. This theory was supported by experimental results, i.e., finding an adhered powder layer, observing the velocity and shear characteristics in a journal bearing, and measuring a pressure profile similar to that found with conventional liquid lubricants. Additional fundamental experiments investigating powder rheology [ 8, 9, 20-221 have been conducted which complement the work discussed above.

PROTOTYPE JOURNAL BEARING TESTING

Of the many items that typify hydrodynamic lubrication, e.g., a wedge-shaped film and the shape and magnitude of the pressure distribution, probably the most telling feature is the decreasing friction coefficient as a function of load parameter (Sommerfeld number). This is presented in classical hydrodynamic lubrication theory by the Stribeck curve, shown in Figure 4, which plots the friction coefficient versus the Sommerfeld number. The Stribeck curve is used to delineate the respective regimes of hydrodynamic, boundary, and dry lubrication as they apply to fluids or solids in direct contact. Having previously demonstrated powder flows and film pressures akin to liquid lubricants, it remained to demonstrate friction characteristics similar to those observed in classical hydrodynamic lubrication as shown on the Stribeck curve for a prototype powderlubricated journal bearing. An experimental program was conducted with the primary purpose of demonstrating the feasibility of powder-lubricated journal bearing operation in advanced turbine engines. These tests documented the friction performance of journal bearings under both oil and powder lubrication. The effects of bearing clearance and powder (MoS,) flow rate on bearing power loss as a function of load parameter were also investigated. Experimental Test Rig

A schematic and a photograph of the experimental setup are shown in Figures 5 and 6, respectively. The test journal bearing consisted of five equally spaced pads with the following dimensions: Bearing diameter: 34.1 mm (1.3438 in.) Bearing length: 20 mm (0.788 in.) Projected pad area: 682 mm2 (1.058 in.,) A novel powder-lubricated hydrodynamic journal bearing concept was designed and fabricated (Figure 7). The bearing pads are attached to the bearing cartridge via sets of adjustable compliant pad mounts that were designed to provide radial, pitch, and roll stiffness. The bearing diametral clearances of 0.1 and 0.2 mm (0.004 and 0.008 in.) were achieved by placing a shim of the proper thickness under the compliant mount elements and bearing cartridge. Special care was taken to maintain the designed structural stiffness of the bearing

50

Figure 6 .

Powder-Lubricated Journal Bearing High-speed Test Rig

I Powder B+

Figure 7.

Outlet

Powder-Lubricated Hydrodynamic Journal Bearing

911046

51

when the bearing clearance was altered. The instrumentation used consisted of the following: Fiber-optic-type sensor probe to monitor rotor speed. Strain gage load cell to measure frictional torque on the bearing. Thermocouples mounted on the back side of the bearing pads: one on the loaded pad, and the other on the unloaded pad at 250" with respect to the load vector. The location of the thermocouples, which were mounted on the back side of the bearing pads, was chosen to be at the centerline of the bearing (radially and axially). One thermocouple was mounted at the top of the pad (Pad No. 1) opposite the bearing load. The other thermocouple was mounted in the same manner on Pad No. 4 (pads were numbered in the counterclockwise direction starting from the top). The thermocouple junction was about 2.9 mm (0.116 in.) from the bearing pad contact surface. The supply powder flow temperature and bearing ambient temperatures were also monitored with additional thermocouples. Lubricant flow rates were calibrated prior to test and remained consistent during the test. Figures 5 and 6 show the instrumented test rig. The torque arm is bolted to the bearing cartridge with its centerline passing through the center of the bearing. The torque arm,in a vertical position opposite the bearing load direction, is restrained via a force transducer fixture. The test load was applied via graduated standard lab weights in various increments to a desired test load level through a cable and tray connected to the bearing cartridge. The bearing assembly, including torque arm,loading cable, and tray, weighed about 17.75 N (3.99 Ib). The test bearing cartridge and compliant mount elements were made of nickel-base alloy, Inconel 718. The test journal with a radius of 17.0 mm (0.669 in.) and bearing pads were made from bearing quality, M2 tool steel. The test journal and pad surfaces were ground after proper heat treatment and then lapped to obtain a surface finish better than 0.1 pm (4 pin.) rms. Test Lubricants Baseline performance data were obtained for liquid lubricant conditions. A petroleum base liquid lubricant spindle oil, Mobil Velocite No. 6, having kinematic

viscosity of 9.4 centistrokes at 40°C and 2.6 centistrokes at 100°C was used. The reference viscosity at the oil inlet temperature of 22°C was considered for data analysis; po = 9 cps, (1.305 preyn). The liquid lubricant was squirted through a nozzle at the beating end. The nozzle was aimed axially and directed at a gap, approximately at the leading edge of the load pad, between test bearing pad numbers 5 and 1. Molybdenum disulfide (MoS,), commercially available and suspension grade, was selected as the powder lubricant. This powder had a reported 50% cumulative particle size distribution of 1 to 2 pm with a purity level of 99.99%, a pour density of 1.125 g d c c , and a solid density of 4.8 g d c c [7]. The delivery of powder lubricant to the inlet zone of the pads was by means of a powder spray device, as shown in Figures 5 and 6. The powder spray nozzle discharged a (dryadpowder) mixture approximately 10 mm from the bearing pad axial end. Powder was blown through the bearing gaps axially via dry air which was supplied to the unit at 70 to 170 kPa (10 to 25 psi). The powder discharge flow rate was adjusted via a powder supply control system which was precalibrated to achieve 15 cc/min (0.28 gdsec) and 30 cc/min (0.56 gdsec). Experimental Results Preliminarv Runs. A number of runs was made with Velocite No. 6 oil lubricant at several loads and speeds in order to establish a reference set of bearing performance data and to check out the instrumentation and operation of the test assembly. This information could then be compared with known bearing inputs and, if necessary, adjustments made in the equipment used. The experimental data were analyzed and nondimensionalized for the generation of parametric plots showing both friction and power loss as functions of load parameter or bearing DN. Figure 8 is a plot of the measured coefficient of friction as a function of nondimensional load parameter, W = [p,, (N/P)(WC)* 1, for four different normal loads of 26.7 to 222.4 N (6 to 50 lb) at various journal speeds up to 20,000 rpm. The two curves on Figure 8 show that, within the limits of test rig operation, some deviation could be seen among the test data. The reference viscosity of the oil was kept constant at po = 1.3 preyn for all calculated nondimensional load parameters. Superimposing the

52

Powder Flow = 20 cc/rnin Bearing Clearance (C,)= 4 mil

1

0 Load- 61b

0.35 0.30 r

Load= 161b Load=36Ib Load=50Ib

Q Q

-

0.15

0.10 3

0.05

-

01

I

I

I

I

I

I

-

10-1

I l l

W = Po

I

0

-(-) N R

1

1

-

10‘

2

91801

P C

Figure 8.

Coefficient of Friction as a Function of Load Parameter @) for a Journal Bearing Lubricated with Oil

40 0

Lubricant: Oil

35 -

-

-

I

20 15 10 -

50

-’

Figure 9.

d mil

Lubricant Flow (0) = 20 cc/rnin Load: o -61b x 16 Ib 0 36 Ib a = 50 Ib

30 25

-

Bearing Claarance (C,)

I

I

I

I

I

I

Power Loss as a Function of Speed and Load for a Five-Pad Journal Bearing Lubricated with Oil

53

curves of Figure 8 on Figure 4, the dry friction, boundary, mixed, and hydrodynamic lubrication regimes become evident. Further data analysis of the nondimensional power loss as a function of speed and load (Figure 9) and comparison with the relevant data available in the literature confirmed the soundness of the test data acquisition procedure and instrumentation [25]. Powder Lubrication Tests. The remaining tests were conducted at what amounts to two different bearing diametral clearances and two powder flow rates, at speeds up to about 60,000 rpm (2 million DN) with applied normal loads up to 227 N (51 lb). The dry air supply delivered MoS, powder axially to the bearing from the outboard side. The powder was introduced prior to loading the bearing completely against the runner. The dry air supply and ambient temperatures were about 22°C. The tests were conducted with powder MoS,, a bearing diametral clearance of 0.102 mm (0.004 in.), and a constant powder flow rate of about 15 cc/min throughout the test operation. The reference viscosity (po)for MoS,, which was based on experimental data reported in Reference [6], was kept constant at 10 preyn for all parametric data analyses. This value was an order of magnitude higher than that of the test oil viscosity. Figure 10 is a plot of nondimensional power loss as a function of speed and load. Comparing Figure 10 with the data shown in Figure 9, a striking similarity with the conventional liquid lubricant power loss is seen. After completing the second series of tests, the bearing diametral clearance was increased from 0.1 to 0.2 mm (0.004 to 0.008 in.) in preparation for the thud test series. The powder flow rate and bearing loads were kept the same. The main purpose of the third test series was to study the effect of bearing clearance on the bearing operating performance. When the friction signatures from the second and third test series were compared, the coefficient of friction and, consequently, power loss, were found to be a weak function of bearing clearance. In the fourth test series, which was undertaken to study the effect of powder flow rate, Q, on bearing performance, the powder flow rate was changed to 30 cc/min. The remaining test conditions were kept the same as for the third series of tests. In general, the trend of measured bearing power loss as a function of

speed indicated an exponential relationship (Figure 1la and 1lb) as called for in classic hydrodynamic theory. COMMON TRIBOLOGICAL MECHANISMS

Figure 12 depicts the friction behavior of the powder-lubricated journal bearing as a function of normalized load parameter, G, for clearances of 0.10 mm (4 mil) and 0.20 mm (8 mil) and powder flow rates of 15 and 30 cc/min. In Figure 12, a set of curves is fitted to their upper and lower boundaries to represent the trend of friction coefficient versus load parameter. The most significant aspect of the frictional behavior of dry powder-lubricated bearings is that it closely resembles hydrodynamic behavior rather than the elastic behavior of dry contacts. Another example of the quasi-hydrodynamic behavior functioning in a powder-lubricated bearing is shown in Figure 13 which shows a drop in dimensionless power loss with increasing load parameter, (G). In particular, data obtained from the liquid lubricant tests are superimposed over the test powder data. As can be seen from this figure, excellent correlation has been achieved. This clearly is more direct proof of the quasi-hydrodynamic action of solid particles in a self-acting converging wedge. During the powder lubrication tests, the bearing pad temperature was measured at approximately 177°C (350°F). Considering the test speeds, this temperature was considerably lower than would be expected from solid lubrication with MoS, coating. Other key features of the powder-lubricated bearing test results presented in Table 1 are the range of measured bearing power loss, achieved DN value, total operating time, and wear distance. From this table, it is seen that operation above 1.3-million DN was achieved and maintained for up to 1080 sec. The low power loss and pad temperature rise are further evidence of the similarity of powders to oils and the ability of powder to carry heat away from the contact. One consequence of the friction in the powderlubricated hydrodynamicjournal bearing is the resulting heat generation. While the low friction coefficient provided by a powder lubricant film goes a long way to limiting the heat that is transmitted to the bearing journal and pads, without some mechanism to carry the heat away from the bearing, the local pad and journal temperatures would quickly rise and components would

54

.d."

Lgbricant: MoS, Bearirlg Clearance (C,) = 4 mil Lubricant Flow (0) = 15 a ' m i n Load: o = 61b i = 161b o = 32Ib

0

0.1

0.2

0.3

0.4

DN x lo6

0.5

0.6

0.7 91%

Figure 10. Power Loss as a Function of Speed and Load for a Five-Pad Journal Bearing Lubricated with Powder MoS,

55

melt. As with conventional hydrodynamic lubrication theory, the quasi-hydrodynamic powder lubrication theoretical model includes heat transfer via the lubricant. As shown in Figure 1, the generated heat is transmitted to the three elements comprising the bearing, namely the journal, the bearing pads, and the powder. The bulk of the heat is carried away from the bearing by the powder in a manner similar to liquid lubricant side leakage. Following an accumulated test time of 2730 sec under various loads and speeds, the test specimens were examined, and the bearing and shaft were observed to be in excellent condition. Figure 14 shows the posttest condition of the pads and test journal. A close examination of the surfaces revealed that a thin layer of powder about 2.5 to 7.5 pm (0.1 to 0.3 mil) had adhered to the surfaces of the test journal. This phenomenon has been observed following tests with various other types of powder, as reported in [8, 9, 14, 15, 20-221. The adhered layer is believed to play a major role in terms of protecting the tribomaterial surfaces and dictating the magnitude of slip velocity at the boundaries. The Quasi-Hydrodynamic Nature of Powder Lubrication

Two of the basic determinants of the mode of hydrodynamic lubrication are the pressure profile produced and the behavior of the coefficient of friction as a function of load (Stribeck curve). Based on the experiments conducted with the powderlubricated films, the results show a striking kinship of the powder film with that of a conventional liquid lubricant. Based on the friction data generated by an oil and a powder, Figure 15 shows that in all essential aspects the powder lubricant behaved as expected of a hydrodynamic bearing. The values of the coefficient of friction consistently showed that hydrodynamic effects are at work in all powder film interactions for all imposed loads and speeds. Thus, the analogy presented earlier that powders may be treated as continuum media in an analytical model based on continuum theory appears valid [2,6,24]. Therefore, a basis has been established which permits the engineering design of practical powder-lubricated bearings for hostile-environment applications. Further examination of the results strongly indicates that the mechanism of powder flow seems to

follow some of the basic features of hydrodynamic lubrication by exhibiting a layer-like shear, reminiscent of fluids. This shear is deemed responsible for the reduction in friction coefficient. The nature of the sheared flow causes the least possible discontinuity between the various laminae of the powder film. While a dry particulate film has no true viscosity as it is understood in liquids, it exhibits a strainstress relationship with a proportionality constant, po, which represents a resistance to flow or an "effective viscosity" [2, 5, 16, 19,231. The motion of the particles in this concept resembles molecules in a liquid. Thus, given an appropriate tribosurface geometry, the dry particulate film will generate a lift. Using po, friction and film thickness, and consequently wear, can be related to the group of variables [p,, (N/P)(R/C)*], much as the Sommerfeld number, S, is related to friction, film thickness, and wear in hydrodynamic lubrication. The results of the testing allow the conventional hydrodynamic lubrication Stribeck curve to be extended to include powder lubrication and limiting shear stress regimes. As shown in Figure 16, powder lubricants are included at both wings of the f-S spectrum, where the poin the Sommerfeld number is a properly defined viscosity equivalent of the powder. Figure 16 shows qualitatively the five operating regimes of lubrication, dry, boundary, mixed, hydrodynamic, and limiting shear stress. At low values of the speed parameter, there is intimate contact between the surfaces, and the conditions are essentially the same as those with dry friction, which results in high friction and wear. As the speed increases, a thin boundary layer of film is formed with a thickness of the same order as the composite roughness of the two surfaces. With a further increase in speed, the system moves into the mixed regime where the lubricant film thickness increases progressively to the level typically regarded as hydrodynamic and boundary lubricated. As the film increases, asperity contact declines rapidly, giving a significant reduction in both friction and wear. When full hydrodynamic lubrication is achieved, it is believed that the surfaces are completely separated by the lubricant film and that no wear due to asperity contact occurs. In the full hydrodynamic region, friction increases with speed due to viscous drag. For a finite value of an applied load, W, slip may occur at the

56

3 Powder Lubricant: MoS, C, = 0.2 m m (8 mil) L=20mm UD = 0.59 Wn = 71.2 N (16Ib)

h

a =u) 2

2

F

8

Y

-I

u) v)

ii

0 -1

:

a

1 5

0,

&!

.-C $ 1

.F

m

5

m

0 5

0 0.3

9.6 DN x

Figure 1 la.

I

61

d I

1.2

0.9

lo6

93330

Powder-Lubricated Journal Bearing Power Loss as a Function of DN and Powder Flow Rate

Lubricant: MoS, Bearing Clearafica (C,) = 8 mil Lubricant flow (Q) I 15 Wmin Load: o = 41b x o

-

l6lb

= 32ib 51 Ib

/

Lubricant flow (Q) = 30 d m i n Load: o 16 Ib + 32 Ib

/

/ /

/ /

/

/

/'

= 30 cc/min;

Q -,

W=161b

0.2

0.4

0.6

0.8

1.0

DN x 10'

Figure 1 lb.

0

a = 15 cc/rnin;

2

0

/1

1

I

I

I

1.2

1.4

1.6

1.8

2.0 QlQW

Journal Bearing Power Loss as a Function of Bearing DN

57

boundary or near the contact surfaces for increases in the relative sliding speed, U, beyond the full hydrodynamic region where the limiting shear stress may prevail. For powder, this phenomenon has been experimentally observed and reported by Heshmat [3, 8, 9, 20, 221, and for liquids, particularly in EHD lubrication, by Scott and Winer [ 10, 111, Tevaarwek and Johnson [12] and recently by Kaneta et al [13]. As a result of the slip between the film and the bearing surfaces, wear will increase rapidly and lubricant film thickness along with friction value will approach a limiting value [ 5 , 6, 161. Indeed, in the limiting shear region, wear of the tribomaterial is due to the tribological action taking place between the "lubricant" and the tribosurfaces, rather than asperityto-asperity or surface-to-surface contact. Note that the wear rate in Figure 16 is normalized with respect to the maximum wear rate believed to be occurring at the dry contact regime. SUMMARY AND CONCLUSIONS

Powder lubricant films behave much as fluid films do in their ability to generate lift, separating bearing surfaces and causing side leakage. In this way, the power loss or friction coefficient and, consequently, the heat generated in a bearing are reduced. Further, the side leakage provides the mechanism to carry the generated heat away from critical bearing surfaces. A series of experiments has demonstrated the basic feasibility of developing a powder-lubricated quasi-hydrodynamic bearing for advanced rotating machinery and extreme environments. In fact, based on the demonstrated operation of a powder-lubricated journal bearing at speeds to 58,000 rpm, these may be the only bearings capable of meeting and complementing the ever demanding tribological goals of a solid lubrication scheme for extreme environments. The specific conclusions of this investigation are as follows: A flexibly mounted five-pad journal bearing with a 60% pivot position capable of operating at speeds about 60,000 rpm (2 million DN) with dry MoS, powder lubricant was developed. Powder lubrication friction curves compare favorably with liquid lubrication friction curves, providing additional evidence for the validity of quasi-hydrodynamic theory.

The present work has extended the classical regimes of fluid and boundary lubrication in the conventional Stribeck curve to include the regimes of powder lubrication and limiting shear stress. NOMENCLATURE

4

Area under pressure curve Length of slider pad Radial clearance Diameter Speed parameter; D = mm; N = rpm fi Power loss H W [ npo L U, (WC)]; dimensionless power loss HP Bearing power loss (hp) L Bearing axial length P W/(LD); Bearing unit load (psi) Q Flowrate R journal radius S Sommerfeld number; { (po NAp/w,)(WC)2} U Linear surface velocity at R, wR W Bearing normal load \ij (poN/P)(R/C)*; dimensionless load W, Normal applied load e Journal bearing eccentricity f Friction force f Friction coefficient h Powder film thickness hmiominimum film thickness E e/C; eccentricity ratio po Reference viscosity o Angular velocity; radidsec B C D DN

REFERENCES

1. Reynolds, Osbome. "On the Diletancy of Media Composed of Rigid Particles in Contact - With Experimental Illustrations." London, Edinburgh and Dublin Phil. Mag and Journal of Science 55, 20, 127 (1885): 469-81. 2. Heshmat, H. "The Rheology and Hydrodynamics of Dry Powder Lubrication." STLE Tribologv * 34, NO. 3 (1991): 433-39. 3. Heshmat, H., 0. Pinkus, and M. Godet. "On a Common Tribological Mechanism Between Interacting Surfaces." 43rd Annual STLE Meeting, 912 May 1988, Cleveland, Ohio, STLE Transactions 32 (1989): 1, 32-43.

58

0.45-

o Bearing Clearance (C,) = 4 mil Lubricant Flow (a)= 15 d m i n

1o3

Dimensionless Load Parameter ( i )

91 1048

m, for a Powder-Lubricated

Figure 12. Friction Coefficient vs. Load, Hydrodynamic Journal Bearing

1o2

-

‘.r-

Lubricant: MoS,

v) v)

0 L I

Bearing Clearance (C,) = 4 mil Lubricant Flow (Q) = 15 d m i n

10’

a

3

W

0 a

Lubricant: MoS, Bearing Clearance (C,) E 8 mil Lubricant Flow (Q) = 15 W m i n

v) v)

-a L r.

.-0

!g

0

loo

E

B

Lubricant: MoS, Bearing Clearance (C,) = 8 mil Lubricant Flow (a) 30 d m i n

0

0

I

I

I

I

1

I

I

5

10

15

20

25

30

35

Dimensionless Load Parameter (fi)

Figure 13. Power Loss vs. Load Parameter for Oil- and Powder-LubricatedJournal Bearings

40 91863

59

4. Heshmat, H. "The Quasi-Hydrodynamic Mecha-

14. Heshmat, H. "Wear Reduction Systems for Coal-

nism of Powder Lubrication--Part I: Lubricant Flow Visualization." STLE 46th Annual Meeting, April-May 1991, Paper No. 9 1-AM4D- 1, Journal of STLE 48, No. 2 (February 1992): 96-104. 5. Heshmat, H. "The Quasi-Hydrodynamic Mechanism of Powder Lubrication: Part 11: Lubricant Film Pressure Profile." STLE/ASME Tribology Conf., St. Louis, Missouri, October 14-16, 1991, Journal of STLE 48, No. 5 (1992): 373-383. 6. Heshmat, H. "Development of Rheological Model for Powder Lubrication." NASA Lewis Research Center CR- 189043 (October 199 1). 7. Risdon, T. J. "Propertiesof Molybdenum Disulfide - MoS2 (Molybdenite)." Chemical Data Series Bulletin C-5C of Climax Molybdenum Co., a unit of AMAX Inc. (Aug. 1989). 8. Heshmat, H., and J. F. Walton. "The Basics of Powder Lubrication in High-Temperature PowderLubricated Dampers." International Gas Turbine and Aeroengine Congress and Exposition, Orlando, Florida, June 3-6, 1991, ASME Preprint Paper 9 1GT-248, ASME Transactions of the Journal of Engineering for Gas Turbine and Power 115, No. 2 (April 1993): 372-82. 9. Heshmat, H. "High-Temperature Solid-Lubricated Bearing Development: Dry-Powder-Lubricated Traction Testing." AIAA/SAE/ASME 26th Joint Propulsion Conference, Paper 90-2047 (July 1990), Journal of Propulsion and Power 7, No. 5 (199 1): 8 14-820. 10. Scott, B., and W. 0. Winer. "Shear Strength Measurements of Lubricants at High Pressure." ASME Journal of Lub. Tech. 101, No. 3 (1979):

Fueled Diesel Engines--Part I: The Basics of Powder Lubrication." 9th International Conf. on Wear of Materials, April 1993, San Francisco, California; Wear Elsevier Seauoia 162-164

25 1-57. 11. Scott, B., and W. 0. Winer.

"A Rheological Model for Elastohydrodynamic Contacts Based on Primary Laboratory Data." ASME Journal of Lub. Tech. 101, NO. 3 (1979): 258-65. 12. Tevaarwek, J. L., and K. L. Johnson. "The Influence of Fluid Rheology on the Performance of Traction Drivers." ASME J o u d of Lub. Tech. 101, NO. 3. (1979): 266-74. 13. Kaneta, M., H. Nishikawa, and K. Kameishi. "Observations of Wall Slip in Elasto-hydrodynamic Lubrication." Journal of Tribolom, Trans. ASME 112, NO. 3 (1990): 447-452.

(1993): 508-517. 15. Heshmat, H. "Wear Reduction Systems for CoalFueled Diesel Engines--Part 11: The Experi-

mental Results and Hydrodynamic Model of Powder Lubrication." 9th International Conf. of Wear of Materials, April 1993, San Francisco, California; Wear Elsevier S e 'w 162-164 (1993): 508-5 17. 16. Heshmat, H. and Brewe, D.E. "On Some Experi-

17.

18. 19.

20.

mental Rheological Aspects of Triboparticulates." Proceedings of the 18th Leeds-Lyon Symposium on 'WEARPARTICLES: From the Cradle to the Grave, ' Lyon, September 3-6, 199 1, Elsevier Science Publishers, Tribology Series 18 (1992). Berthier, Y., Vincent, L. and Godet, M. "Velocity Accommodation Sites and Modes in Tribology" Eur. J, Mech.. NSolid 11, No. 1, (1992): 35-47. Berthier, Y."Experimental Evidence for Friction and Wear Modelling." Wear 139 (1990): 77-92. Heshmat, H.,Godet, M. and Berthier, Y."Technical Surveillance on the Role and Mechanism of Dry Triboparticulate Lubrication." STLEIASME Joint Trib. Conf, 1992, submitted for publication as an RCT paper in ASME Transactions. Heshmat, H., and Dill, J. F. "Traction Characteristics of High-Temperature, Powder-Lubricated Ceramics (Si,N,/aSiC)." ASWSTLE Tribology Conference 1990, STLE Transactions 35, No. 2

(1992): 360-366. 21. Heshmat, H. and Walton, J. F. "High-Tempera-

ture, Powder-Lubricated Dampers for Gas Turbine Engines." AIAA/SAE/ASME 26th Joint Propulsion Conference Proceedings,Paper No. 90-2046, (July 19901, U P r O P U l sion and Power 8, No. 2, Mmh-April(1992): 449-456. 22. Heshmat, H. "Rolling and Sliding Characteristics of Powder-Lubricated Ceramics at High-Temperature and Speed." 47th Annual STLE Meeting, Philadelphia, Pennslvania, May 1992; to be published by STLE Lub. E w (1993).

60 Table 1 Summary of Powder-Lubricated Hydrodynamic Journal Bearing Performance with MoS,

M2 Tool Steel, Pad Dimensions:L = 20,6 = 17.8,D = 34 mm; Total Number of Pads = 5 Pads

1 DN Average (mm x rpm) x 1o6

Average Tlme (set)

1.3to 2.0

FxD (lb-In.) x 10

1020

1.37 0.31 15.70 3.34 0.15

2730

20.87'

0.27

Maximum Power Loss (hP)

Maximum Pad Temperature

0.24 0.23 2.5 1 .o 0.16

146 188 250 to 300 275 to 350 200 to 250

(OF)

1

-

a1- 193-1

'Total work done on bearing = 1.75x

lo6 ft x Ib z 0.655kWh.

61

23. Walton, O.R. and Braun, R.L., "Viscosity, Granular-Temperature, and Stress Calculations for Shearing Assemblies of Inelastic, Frictional Disks." John Wiley & Sons, Inc., Journal of Rheology 30(5), (1986): 949-980. 24. Heshmat, H. "The Quasi-Hydrodynamic Mechanism of Powder Lubrication: Part 111: On the Theory and Rheology of Triboparticulates." STLE Annual Meeting, Calgary, Canada, May 17-20, 1993; submitted for publication in STLE Transactions.

25. Heshmat, H. "Starved Bearing Technology: Theory and Experiment." Ph.D. thesis, Mechanical Engineering Department, Rensselaer Polytechnic Institute, Troy, New York, December, 1988.

62

Figure 14. Post-Test Condition of Journal Bearing Showing Thin Adhered Layers of MoS, Film

63 0.55

0.50 0.45 -

Lubricant: Oil Bearing Clearance (C,) = 4 mil Lubricant Flow (a)= 20 d m i n Lubricant: MoS, 0 Bearing Clearance (C,) = 4 mil

Figure 15. Experimental Data Correlation Between an Oil- and a Powder-Lubricated,Five-Pad Journal Bearing

64

1

z

s U Y

? I

2

o3

1

Speed Farameter, po

-

T:(

(UP)

(b;

Figure 16. Extension of the Stribeck Curve to Include Powder Lubrication

901686-1

Dissipative Processcs in Tribology / I). I>owson et al. (Editors) 1994 Elscvier Science I3.V.

65

The Influence of Base Oil Rheology on the Behaviour of VI Polymers in Concentrated Contacts P.M.Cann & H.A.Spikes Tribology Section, Department of Mechanical Engineering, Imperial College, London SW7 2BX Whereas elastohydrodynamic (EHD) film thicknesses for simple base stocks can be predicted with some confidence from their bulk properties this is not so for polymer-containing fluids. The behaviour of such fluids in an EHD contact is extremely complex and can include elements of shear thinning, viscoelasticity and boundary properties. In this paper polymer solution behaviour in a concentrated contact has been investigated through detailed EHD film thickness measurements. A series of model polymers; polyisoprenes in the molecular weight range 27-86,000, have been studied in two different basestocks, the intention being to examine the effectof base stock rheology and solvation properties on polymer behaviour. 1. INTRODUCTION

The rheology of a simple basestock in the EHD inlet is usually assumed to be Newtonian even at the extreme strain rate conditions found in this region. Film thickness can therefore be predicted generally from low shear viscosity measurements. This is not the case for polymer-containing fluids which do not obey Newtonian rheology, particularly at high strain rates. It is difficult to measure viscosities at the required strain rates and it is not possible to predict polymer solution behaviour from the usual inelastic flow models. Estimation of EHD film thicknesses therefore for polymer containing fluids is currently impossible. The rheology of these fluids is extremely complex. At low shear rates ( < l d ~ polymer-~) containing fluids are essentially Newtonian. As the strain rate increases viscosity loss occurs. This is usually attributed to alignment or orientation of the polymer coils within the shear field and is reversible once shearing has stopped. A second Newtonian rcgion is observed at very high shear rates (>lo6,l). Rhcological expressions have been developed to describe such behaviour and they are usually in the form of power (1) or doubly truncated power law (2) relationships. Only one (3) is based on a physical model of polymer behaviour. However these models have never been fully investigated at the strain rates associated with an EHD contact, which are often greater than 1 0 V . The viscosity enhancement due to the polymer depends upon both polymer and base stock

properties. Polymer solution viscosity increases with molecular weight and concentration. The solvation properties of the base stock influence the polymer coil size in solution. The larger the coil size the greater the viscosity enhancement. However this is counter-balanced by a greater susceptibilityto shear alignment since the shear stress experienced by the polymer is proportional to coil area. At high stress levels; polymer molecular rupture takes place resulting in permanent viscosity loss. This is thought to occur in the centre of the contact (4), in the high pressure region, and is not the primary cause of viscosity loss in the inlet. It is not therefore considered in this paper. Polymer solutions are also thought to possess viscoelastic properties which, by the generation of forces normal to the direction of shear, will increase film thickness. This has not been measured directly in an EHD contact and has only been inferred from anomalous results for polymer containing fluids in hydrodynamic bearing tests (5). There has also been speculation that polymer containing fluids form boundary layers and, again, this has been inferred from engine wear tests (6). No direct evidence has been offered probably because such films are too thin for normal detection methods. EHD film thickness measurements have been used extensively to investigate polymer solution behaviour (7)(8). Most workers conclude that the film thicknesses are significantly less (60-701(8))

66

than predicted by low shear rate viscosity measurements. Hirata (8) also reports that they are 20% less than predicted from high shear rate measurements. Polymer solutions therefore, within an EHD contact, can exhibit viscosity loss, viscoelasticity and boundary layer formation. All of these will influence EHD film formation. In this paper, detailed film thickness measurements have been made for a series of simple polymers in hydrocarbon solution. The film thickness range studied was 2-200 nm; the intention being to examine the contribution of the polymer to both boundary and hydrodynamic performance. The base stocks were chosen to give a range of pressure viscosity coefficients since this determines the shear stress level experienced by the polymer in the inlet region. In addition the polarity and, hence, solvation properties of the fluids will differ. The low shear rate viscosities of the solutions were measured on a cone-on-plateviscometer. These values were used to calculate the predicted EHD film thickness for each of the base stocks and solutions. These results could then be compared to the measured EHD films. In this way it was possible to estimate the viscosity contribution of the polymer in the inlet region. 2 EXPERIMENTAL

EHD film thicknesses were measured using thin film interferometry,a technique described in detail in earlier papers (9). This method measures EHD film thicknesses down to 2 nm with a resolution of 2 nm. A simple bearing contact simulation device was used (9). The contact is a steel ball driven in pure rolling by a rotating glass disc. The load was 16N giving a maximum Hertzian pressure of 0.47 GPa All tests were run at 22°C. The polymers were monodisperse cis-cis polyisoprenes. Their molecular weights are listed below in Table 1. Table 1 Polymer properties Polymer Molecular weight A 27000 B

c

63000

m

The base stocks used were a synthetic cycloaliphatic hydrocarbon traction fluid ( S A N 40), and a poly-a-olefin hydrocarbon (SHC). Their viscosities and pressure-viscosity (a)values are given in Table 2. The polymer solutions were prepared as 2% weighvweight concentration. Table 2 Lubricant basestocks Basestock Viscosity Pas woe)

0.0485 0.048

S A N 40 SHC

a Value GPa-1 (22°C) 36 13

3 RESULTS & DISCUSSION

The viscosities of the base stocks and solutions were measured on a cone-on-plate viscometer at 22°C. The results for the polymer solutions are shown in Table 3 for a shear rate of 8500 s-l. Table 3 Viscosity results for 2% polymer solutions (Pas) Polymer SAN40 SHC A 0.073 0.064

B

0.1009

C

0.117

2*4

0.071 0.080

1 SAN40 0

SHC

0

1.4

1.2

0

n

20000 4oooO

60000 8oooO 1OOOOO

Molecular weight Figure 1 Relative viscosity as a function of molecular weight These results are plotted in figure 1 in a different form: as viscosity relative to the base oil ie viscosity (solution)/viscosity (baseoil). In this way

67

the different viscosity enhancement for each of the base stocks can be seen. This is larger for the traction fluid which probably reflects the latter’s better solvation properties and hence a greater polymer coil size. EHD film thickness was measured with increasing rolling speed for each of the base stocks and solutions. Representative results are shown plotted in log-log form in figures 2 and 3 for both the base stocks and 2%C solution. The film enhancement due to the presence of the polymer can clearly be seen, particularly at low rolling speeds.

The second effect can be seen in figure 2 where the result for 28C appears to approach an asymptote at low speeds indicating the existence of a speedindependent, boundary film. This result is typical for polymers in the ply-a-olefin solution. It was not found with the traction fluid (see figure 3 below).

.001

.o 1

.1

1

Rolling Speed ( 4 s ) Figure 3 EHD film thickness results for SAN40 and 2%C

.oo1

.01 .1 Rolling Speed (mh)

1

Figure 2 EHD film thickness results for SHC and 2%C Figure 2 shows that, whilst the measured gradients for the base stocks show reasonable agreement with the predicted value of 0.7 (lo), values for the polymer solutions were significantly lower, often less than 0.5. This can be seen in the curve for 2%C which approaches that of the base stock at high rolling speeds. Two factors may cause this; (i) shear thinning of the fluid occurs so that the effective viscosity in the inlet is progressively reduced with increasing speed. (ii) a boundary layer forms at the solid surfaces. This results in an anomalously thick film at low rolling speeds.

It is probable that both shear thinning and boundary film formation occur within an EHD contact. The relative magnitude of these effects and the resulting EHD film thickness will depend on polymer size and inlet shear rate. The raw data of the EHD tests was therefore analysed in the following way to try and understand these two effects. Most tests were repeated twice and a large number of data points generated. Some scatter was observed in the results and to aid interpretation the data was curve fitted using a high order polynomial. Typical results generated by such curve fitting are shown in figures 4 and 5. The film thicknesses predicted using EHD theory (10) are also shown. In both figures the fitted results are shown as the solid line, the predicted result as the dotted line. For both additive-freebase fluids (bold lines) the curve fitted experimental film thicknesses are close to film thicknesses predicted from EHD theory. For the polymer solutions there are however striking

68

differences. Polymer solutions in SHC gave film thicknesses greater than predicted from the viscosity of the polymer solutions. For SAN40, by contrast, the measured film thicknesses were lower than predicted.

1 .001

.o 1

.1

1

.01 .1 Rolling Speed (m/s)

It can be seen that polymers in SHC form boundary films of approximately 13 - 16 nm, but the same polymers in SAN40 form much thinner films. In an earlier paper (1 1) these boundary fdms have been ascribed to the adsorption of polymer molecules at the solid surfaces. Similar results have been presented for polyisoprenes from surface force measurements (12). The thickness of the film increases with molecular weight and was related to the polymer coil dimensions (11)(12). It is noteworthy that polymers in SAN40 solution give only thin boundary films. The existence of a high viscosity surface layer is supported by the observation that when motion of the disc is halted that a separating film persists in the contact. This was not observed for the base stock tests. Residual film thickness is seen to decay with time in the stationary contact. A typical result is shown in figure 6. Table 4 Boundary film thickness (in nm) from EHD tests Test Fluid SAN40 SHC Basestock 2 3 + 2%A 8 13 +2%B 0 16 + 2%C 2 13

Rolling Speed (mh) Figure 4 Generated and predicted film thickness curves for SHC and 2%C solution

.oo1

made for both base stocks and polymer solutions. These results are shown in table 4.

1

Figure 5 Generated and predicted film thickness curves for SAN40 and 2%Csolution 3.1 Boundary Film Component

By extrapolating the curve fits to zero speed then an estimate of the boundary film thickness can be

0

200 400 600 800 lo00 Time (secs)

Figure 6 Residual film decay at zero speed for 2%C in SHC

69

The low shear rate viscosity results suggest that in the traction fluid polyisoprene adopts a more open conformation and has a relatively large coil size. This should result in a thicker residual film however this is not observed in the film thickness results for S A N 40.

and C in SAN40 do not form boundary films so that the effect of shear thinning can be seen directly.

From these results it is also possible to estimate the “effective viscosity” of the polymer solution relative to that of the base oil in the inlet regiona. Effective relative viscosity is calculated from the measurd film thickness results where:

3.2 EHD Film Component To deduce the hydrodynamic contribution of the polymer solutions it is necessary to remove the effect of the boundary layer. The generated film thickness curves have been replotted with the calculated boundary film thickness subtracted. This is shown in figure 7 which is a linear film thickness/speed plot for 2%C in SHC. Curve A corresponds to the original curve-fitted film thickness results, B when the boundary film (13 nm) has been subtracted. Curve C is the theoretically predicted film result calculated from the low shear rate viscosities. Even with the boundary film subtracted,thepolymer solution still appears to give a thicker EHD film than its low shear rate viscosity would suggest.

effectiverelative viscosity = (hsolutionhbase stock) 1P.67 h is the film thickness generated from the curve fit

Any adsorbed boundary film would distort the relative viscosity results calculated this way. Relative viscosity has therefore been calculated both with ((i)) and without ((ii)) the boundary film present. These are plotted as a function of rolling speed for both fluids in figures 8 and 9. The relative viscosity for the solution measured on the cone-onplate is also shown as (iii),

n

E

: 100 3

8

E f

50

0

0.00 0.05 0.10 0.15 0.20 0.25 Rolling Speed ( 4 s ) Figure 7 Film thickness curves for 2%C in SHC effectof boundary layer The film thicknesses for the polymers in S A N 40 are lower than predicted, although the agreement between predicted and measured for the base stack was very good. This would suggest that significant shear thinning is occuring in the inlet. Polymers B

0.00

0.05

0.10

0.15

0.20

0.25

Rolling Speed ( 4 s ) (a)2%A Figure 8 Effective relative viscosity for polymer solutions in SHC against rolling speed

I

0 k ' ul

in

t

I

in

8

h

.3

0

8

LA

c 0

w

.

I I

z

I

2

.

I

Effective Relative Viscosity

- 0

c

c

z

LA

!Q

Effective Relative Viscosity

P

w N L

- 0

Effective Relative Viscosity Effective Relative Viscosity

Figure 8 (cont.) Effective relative viscosity for polymer solutions in SHC against rolling speed

Figure 9 Effective relative viscosity for polymer solutions in SAN40 against rolling speed

71

3

x

.I

2*6 viscometer

00

1.o 1 : 0.00

.

I

I

.

0.05

0.10

.

1

0.15

'

l

.

I

0.25

0.20

Rolling Speed ( 4 s )

EHD result

1

o.ooe+o

1.OOe+6

2.OOe4

Shear Rate (s-1) Figure 11 Effective relative viscosity plotted against shear rate for 2%Cin SAN40

(c) 2%C Figure 9 (cont.) Effective relative viscosity for polymer solutions in SAN40 against rolling speed The effective viscosities of both SAN40 and

SHC solutions clearly decrease with increasing rolling speed. These results can also be plotted as a function of maximum inlet shear rate (7) as illustrated in figures 10 and 11 where the results from viscometer measurement is also shown.

For polymer solutions in SHC at low shear rates (low rolling speeds) the solutions are showing effective relative viscosities greater than their low shear viscosities determined by a cone-on-plate viscometer. As the shear rate is raised,the effective relative viscosity falls presumably due to shear thinning The effective relative viscosity results for S A N 40 solutions are very different to those seen for

SHC. Only A formed a significant boundary film and the relative viscosity results are thus greater than expected when this is present. For polymers B and C the results are far lower signifying considerable viscosity loss in the inlet. 0

8

EHDresult

result

O.OOe+O

1.OOe+6

2.OOe+6

3.OOe+6

S h a Rate (s-1) Figure 10 Effective relative viscosity plotted against shear rate for 2%C in SHC

The film thickness results presented have shown that polymer-containing fluids do not inevitably give lower EHD films than predicted. At low rolling speeds far greater films are seen, even when the boundary films have been taken away a substantial viscosity enhancementcan be seen. This effect is greatest for the lower molecular weight polymers. It is interesting that SAN40 does not appear to form boundary films under rolling conditions, or at least they do not survive. It is possible that the high shear stresses associated with the traction fluid at the surface prevent the formation of a stable adsorbed layer. The behaviour of polymer solutions in SAN40 is quite different. From the lowest attainable shear rate the effective relative viscosity of these fluids are

12

less than measured in the cone-on-plate viscometer. This is attributed to shear thinning of the solutions which is more likely in SAN40 than SHC due to the following; (i) greater polymer coil size due to better solvation properties (ii) greater shear stresses in inlet due to higher avalue. The additional ftlm thickness enhancement over and above that predicted by the adsorbed layer is intriguing and it is interesting to speculate on its origins. This effect is only seen when the layer is present, this observation would suggest that the adsorbed polymer coils at the metal surface induce a localised change in the fluid structure immediately above the boundary layer. This might take the form of local ordering or structuring of base fluid and polymer or increase in the polymer concentration. This would result in an increase in viscosity or a viscoelastic effect due to the loss in mobility of the polymer molecules in the ordered region. Both of these effects would contribute to an EHD film enhancement. 4. CONCLUSIONS

The film thickness results presented have demonsmted the complexity of polymer solution behaviour in EHD contacts. Depending upon the film thickness range studied this behaviour can be dominated by either a boundary or rheological response. This study has demonstrated, through direct measurement, that polymer solutions contribute to EHD film formation in the following manner; (i) Polymer solutions form boundary surface films in rolling contacts and these maintain separation even at zero speed. (ii) These films, probably of adsorbed polymer molecules, have viscoelastic properties which further enhance EHD film thickness. The degree of the enhancement represents a balance between increasing polymer size and the resultant increasing susceptibility to shear alignment.

(iii) Both these effects are dependent upon the solvation and rheological properties of the basestocks.

REFERENCES

1. Whorlow, R.W., “‘RheologicalTechniques.” 2nd

edition Ellis Horwood. 2. Wu, C.S., Melodick, T., Lin, S.C., Duda, J.L. and Klaus, E.E., “The Viscous Behavior of Polymer-Modified Oils Over a Broad Range of Temperature and Shear Rate.” J. Trib., 112, pp417425, (1990). 3. Cross, M.M., J. Colloid Sci., 20, pp417-437, (1965). 4. Walker, D.L., Sanborn, D.M.and Winer, W.O., “Molecular Degradation of Lubricants in Sliding Elastohydrodynamic Contacts.” ASME Trans. J. Lub. Tech., 97, ~~390-397, (1975). 5. Bates, T.W., Williamson, B., Spearot, J.A. and Murphy, C.K. “The Importance of Oil Elasticity.“, Ind. Lub. & Tech., 40, pp4-19, (1988). 6.0krent, E.H. “The Effect of Lubricant Viscosity and Composition on Engine Friction and Bearing Wear.’’ ASLE Trans., 4, ~~257-262, (1961). 7. Foord, C.A., Hamman, W.C. and Cameron, A. “Evaluation of Lubricants Using Optical Elastohydrodynamics.” ASLE Trans. 11, pp 31-43, (1968). 8. Hiram, M. and Cameron, A. “TheUse of Optical Elastohydrodynamics to Investigate Viscosity Loss in Polymer-thickened Oils.” ASLE Trans. 27, pp 114-121,(1984). 9. Johnston, G.J., Wayte, R. and Spikes, H.A. “The Measurement and Study of Very Thin Lubricant Films in Concentrated Contacts.” STLE Trans. 34, pp. 187-94, (1991). 10. Hamrock, BJ, and.Dowson, D.D.. “Ball Bearing Lubrication. The Elastohydrodynamicsof Elliptical Contacts.” Pub. John Wiley & Sons (1981). 11. Cann, P.M. and Spikes, H.A., “The Behavior of Polymer Solutions in Concentrated Contacts: immobile Surface Layer Formation.” STLE Preprint 93-TC-1B-1 12. Georges, J-M., Millot, S.. Loubet, J-L. and Tonck, A., “Drainage of Thin Liquid Films Between Relatively Smooth Surfaces.”, J. Chem. Phys., 98, ~~7345-7360, (1993). 13. Bair, S. and Winer, W.O. “Shear Rheological Characterisation of Motor Oils.” STLE Trans. 31, pp 316-323, (1988).

Dissipative Processes in 'I'ribology / D. Dowson et al. (Editors) 1994 Elsevier Science B.V.

73

Temperature Profiling of EHD Contacts prior to and during Scuffing. J C Enthoven & H A Spikes

In this paper a novel "nodding mirror" infra-red line scanner is used to study the effect of additives on contact temperatures and hence scuffing in a sliding point contact. The "nodding mirror" line scanner is capable of taking temperature profiles across the contact in a very short time (less than 30 msec). The objective is to capture the temperature history across the contact just prior to and during scuffing using lubricants with and without additives. Tests were carried out in a device consisting of a steel ball loaded and sliding against a stationary sapphire window. The lubricant basestock used in this study was purified hexadecane. This was chosen as it is a simple, low viscosity lubricant that forms a negligible EHD film under the conditions of these tests. This was important as one of the aims was to study boundary additive response to contact temperatures and their effectiveness in postponing or preventing failure. Temperature profiles prior to and during scuffing have been taken in tests with pure hexadecane and with hexadecane containing (i) 1.0 wt% dibenzyldisulfide, an EP additive, and (ii) 0.1 wt% stearic acid, a friction modifier. 1. INTRODUCTION Practical components in which a high degree of sliding exists, such as gears, and cams and tappets, often fail duc to scuffing. Although this catastrophic failure mode has been the subject of research for many years, the exact mechanisms are still not clearly understood. Many scuffing theories have been proposed which try to predict the onset of scuffing [1,2]. The best known are Blok's critical temperature hypothesis [3,4], the failure of elastohydrodynamic lubrication as proposed by Dyson [5,6], and the frictional power intensity model [7]. In each of these models a different thermal criterion is used to predict the onset of scuffing. Blok postulated that the transition from smooth running to scuffing would occur when the contact temperature exceeds some critical value. This critical tempcrature is assumed to be independent of load, sliding speed and test history. His postulate has been the subject of much testing and is generally considered to only be valid for un-doped mineral oils. In Dyson's theory, failure is based upon the breakdown of the main elastohydrodynamic lubrication film in sliding contacts. According to this modcl scuffing will occur when the lubricant inlet viscosity is insufficient to generate the large pressures needed for successful operation of the main and micro EHL films. Since the inlet viscosity is dependent upon the inlet tcmpcrature, Dyson's criterion can be related to a critical inlct tcmpcrature.

In the Frictional Power Intensity model, the amount of frictional heat generated in the contact area is assumed critical. Despite our current lack of detailed understanding of its origin and mechanism, the occurrence of scuffing is generally regarded as resulting from thermal feedback. Under a particular combination of load, speed and friction, a critical temperature is reached somewhere in the vicinity of the contact. At this temperature the lubricant film weakens, resulting in an increase in asperity contact friction. If this increase in friction causes a still higher temperature then the consequence can be the collapse of the lubricant film. The magnitude of the temperatures within and in the vicinity of the contact depends upon the amount of heat generated and dissipated. It is apparent from the above that for a complete analysis of scuffing, information about contact temperatures is vital. However traditionally, scuffing tests are conducted using actual gears or discs. In these experiments, contact temperatures are either measured using embedded or trailing thermocouples, or calculated theoretically from the contact conditions at which scuffing takes place. The transition to scuffing occurs very suddenly and the actual contact temperatures just prior to failure can not be found in this way. In previous work [8] infrared temperature profiles were taken across the point contact formed between a steel ball loaded and sliding against a stationary sapphire window. In these tests purified hexadecane

74

was used as a lubricant. The profiles were taken by moving an infrared microscope across the contact area using a x-y table controlled by a stepper motor. Some temperature profiles taken by this means are shown in Figure 1. In this figure, the temperatures across the contact area at each load stage up to scuffing can be seen. Scuffing occurred at a maximum contact pressure of 1.77 GPa, shortly after the last temperature trace had been taken. The authors compared their data with Blok’s critical temperature hypothesis and with Dyson’s theory but no agreement was found. The authors did find that scuffing between a steel ball and sapphire window was similar to that observed between a steel ball and a steel flat. 250

I

-

d

-

I

-

-

-

I

i * 200 ..................... .;.............. r... i.77GPa

:

_ - -

1

-

-

-

I

-

under the conditions of these tests. This was important as one of the aims was to study the response of boundary additives to contact temperatures and their effectiveness in postponing or preventing failure. 2. EXPERIMENTAL APPARATUS

A schematic diagram of the sliding test rig used is shown in Figure 2. A 25.4 mm diameter AISI 52100 steel ball is loaded against a 2 mm thick sapphire window. The loading mechanism is such that the plane of the applied load passes through the centre of the nominal point contact. A strain gauge beam is then used to monitor the friction force accurately.

- -

Hexadecane Tbulk=6OOC “s = 1 m’s Pscuff = 1.i7 GPa ”’:

.w

n 150

3

I

’=

.e5

100 50

c -600

-400

-200

0 200 X-AS, micron

400

600

Figure 1 Temperature traces for a scuffed run, from reference [8]. One of the limitations of the work reported in [8] was the amount of time needed to take one temperature trace, which was about 30 seconds. Since the transition from “smooth” running to scuffing occurs very suddenly, it was clear that in order to be able to study boundary additive response, the time required to take one trace needs to be reduced considerably. This has been achieved in the work describcd in this paper by converting an infrared microscope into a ‘nodding mirror’ infrared line scanner. This is capable of taking successive temperature profiles across the contact in less than 0.03 seconds. The objective is to capture the temperature history across the contact just prior to and during scuffing. Tests were carried out in a point contact device consisting of a steel ball loaded and sliding against a stationary sapphire window. The lubricant basestock used in this study was purified hexadecane. This was chosen as it is a simple, low viscosity lubricant that forms a negligible EHD film

Figure 2 Schematic diagram of sliding test rig. The infrared microscope is mounted in a fixed position above the ball. The microscope focuses the radiation from a 36 pm diameter spot from the contact zone through a x15 reflecting objective onto a liquid nitrogen cooled indium antimonide detector. The detector has a spectral response of 1.8 to 5.5 pm. The microscope is also fitted with a parfocal channel, with a lox eycpicce to permit simultaneous viewing of the area being studied. Positioned between the microscope objective and the sapphire window is the ‘nodding mirror’ assembly. This is mounted on a solid steel block to limit vibrations. The ‘nodding mirror’ is driven by a

75

rotating steel shaft, which is connected to a gear box and a high speed electric motor. A more detailed picture of the workings of the ‘nodding mirror’ is shown in Figure 3. Two reflective mirrors can be seen. One 17 mm square mirror is mounted on the x15 reflecting objective which is attached to the infrared microscope. A smaller, 7 mm square mirror is attached to the nodding mount. This mount consists of a thin piece of carbon fibre tubing with a 45’ flat face on which a very thin and lightweightmirror is glued. The mount can pivot about a 1 mm diameter shaft marked “0” as seen in Figure 3. A horizontal slot is machined in the carbon fibre tubing and the cam on the rotating shaft fits in this slot.

c)

Figure 3 Nodding mirror assembly. The principle behind the approach is that it is easier to deflect an infrared beam rapidly using a low inertia mirror, than to move either the test rig rapidly beneath the microscope, or the microscope over the test rig. The radiation from the 36 pm diameter focal point is reflected off the small nodding mirror on the mirror attached to the objective. The objective in turn will then focus the radiation onto the detector. By rotating the steel shaft the cam inserted in the mirror mount will pivot the mirror backwards and forwards about the shaft marked “0”.This in turn will move the focal point from left to right in a horizontal plane as shown in Figure 3. Thus by “nodding” the small mirror, the infrared microscope can scan the surface and therefore now operates as an infrared line scanner. The distance the focal point will move depends on the eccentricity of the cam on the rotating shaft. Different s h a h were made but the one used in this work moved the focal point by about 1200 mm, which is well across the contact zone. Shown very schematicallyin Figure 2 is the optical pick-up mounted on the rotating shaft. This triggers data acquisition at the beginning of the movement of

the focal point from left to right, i.e. from the contact inlet to outlet. The microscope takes 150 data points at a frequency of loo00 Hz when scanning. No data is taken whilst the focal point moves back to the inlet. In the current study, experiments were carried out with the shaft rotating at lo00 rpm. This means that it takes 0.03 seconds to complete one trace. In this time, all 150 data points need to be first measured by the microscope, then converted into digital values and finally stored in a buffer of the microcomputer. To do this, a fast analogue to digital converter was needed, and a 12 bit linear converter which can sample analogue voltages at rates up to 160000 Hz was employed. The result of each conversion was stored in a buffer, sited on the A D board of a microcomputer. The host computer then accessed these intermediate results simultaneouswith the ADC starting the process of converting the next analogue sample. In this work a clock speed of 10000 Hz was used for collecting the detector data. In order to allow for fast data acquisition a buffer was created on the microcomputer large enough to contain 50 traces each containing 150 data points (50 traces x 150 points x 4 bytes = 30 kbytes). After 50 traces had been taken, the 5tst trace was written over the 1st trace, the 52nd trace stored in place of the 2nd trace, etc. At the end of the experiment, when the space bar was touched, the buffer was written to an array and stored onto the hard disc. This way a series of 50 temperature profiles over the last 3 seconds of any test were available for subsequent analysis. 3. CALIBRATION

In these experiments, the infrared microscope was used in “transient mode”. In this mode, the incoming radiation is not chopped as in earlier work [8], but instead the signal is amplified and demodulated inside the microscope housing before being send directly to thc microcomputer. Two probIems arise in the calibration procedure. Firstly, since in AC mode the measured radiance is not compared with the background radiation of the chopper, the microscope electronics positions the incoming signal around a floating base line rather than relative to the chopper background value. This is done in such a way that the area below the base line equals the area above, as can be seen in Figure 4.The second problem is that the base line drifts. This means that

76

there is no reference point or line which is present when the microscope is used in DC mode. Both problems are a result of the electronics inside the microscope housing and could not be changed easily.

I

4

I

time I Figure 4 Input voltage versus time in AC or transient

I

mode. The drift was measured and found to be insignificant in the first 7 minutes of the test, but considerable after 150 minutes. This means that the drift did not affect the results obtained over 50 uaces or 3 seconds. In order to convert the measured radiance values into temperatures, a new baseline needed to be determined. This was achieved by measuring the track temperature of the ball with the microscope in chopped mode in a separate experiment. Once the track temperature was known, the radiance traces could be converted into temperature profiles following the calibration procedure used in [8]. The reader is referred to this paper for a complete radiation analysis. The error in determining the track temperature for the transient traces is about f 15 %. Due to the shape of the calibration curve, the resulting error in the maximum contact temperature will be less than this. 4. EXPERIMENTAL RESULTS

All experiments reported in this work wcre carried out with polished stcel balls having a roughness of 0.016 RMS.A new ball was used for each test. Each sapphire window was used several times but the window was mounted off-centre in its stainless steel holder so that rotation of the holder enabled a new region of the sapphire surface to be cmploycd for each run. Prior to a test, the lubricant bath, ball and sapphire window were thoroughly cleaned with toluene and then analytical grade acetone. The oil bath was filled with lubricant so that the ball was just over halfimmersed. The rig was then allowed to heat up to the desired bulk oil temperature. During this time, the stcel ball was lightly loaded against the window at a contact pressure of 0.80 GPa whilst sliding at 1.0 m/s.

In all the tests reported in this work, the operating temperature was 80 ‘Cand the sliding speed equalled 1.99 mls. After a test was started, the load was increased in steps of two minute intervals until scuffing occurred or until the 500 N load limit of the rig was reached. The starting load was 50 N and the load was increased in 50 N stages. The surface was scanned continuously during a test but, as was mentioned above, only the temperature history of the last 50 traces or last 3 seconds was saved. This meant that the results presented in this paper only show the temperature traces taken at the last load stage. Figure 5 shows ten consecutive temperature traces taken with the infrared line scanner. The nodding mirror drive shaft rotated at a speed of 1000 rpm, thus each trace in Figure 5 was taken at an interval of 0.03 seconds. Every profile contains 150 data points, sampled at a frequency of 10,000 Hz.

400

u

350

150 100

50

inlet

outlet

Figure 5 Tempcrature traces taken at 0.03 seconds interval up to scuffing. The horizontal axis in Figure 5 shows the position, with the inlct on the left and the outlet on the right hand side. The scanning distance is approximately 1200 pm. The vertical axis shows the stecl ball surface temperature in ‘C. The lubricant in this experiment was purified hexadecane and scuffing occurred at a contact load of 250 N. The traces shown in Figure 5 were all taken at this load and show the transition from “smooth” running to scuffing. Within the 10 traces, or 10 x 0.06 = 0.6 seconds, failure occurred, and in this time the maximum contact temperature increased from about 170 ‘C to more than 420 ‘C. No temperatures could be measured above 420 ‘C since the voltage input to the

77

computer was then outside the range of the A-to-D converter. In Figure 6 the maximum contact and inlet temperature is shown for the 50 saved traces. The horizontal axis shows the time in seconds. Time equals zero and 3 seconds do not correspond to the beginning and end of the test, respectively, but merely to the first and 50th trace, respectively. 450 400

P

350

150

100 50

Time, seconds

Figure 6 Maximum contact and inlet temperature at last load stage, from Figure 5. Looking at Figure 6, one can again see the sudden transition in both maximum and inlet temperature. Up until scuffing the inlet temperature is about 125 ‘C and the maximum contact temperature is about 170 ‘C, except for the peak at 1.5 seconds. This trace, and the traces before and after this peak are shown in Figure 7. 250

to be influenced by this event. The trace taken before and after the peak are very similar in shape. The behaviour seen in Figure 5 and 6 was typical for scuffing test with pure hexadecane. The transition occurs very suddenly (within one second) and without any warning, and this emphasises the need for a very rapid measuring technique, in order to follow the development of scuffing failures. A different response was seen in experiments carried out with hexadecane + 1.0 wt% dibenzyldisulor phide ( C ~ H ~ C H ~ S S C H ~ CDibenzyldisulfide ~HS). “DBDS”, is a sulphur-based anti-wear additive, and its addition increased the failure load of hexadecane considerably. Scuffing occurred at a contact pressure of 2.31 GPa. Figure 8 shows the maximum and inlet temperature at the last load stage. Looking at this figure, one can see that both temperatures vary by large amounts from trace to trace. Until the onset of scuffing, differences of almost 100 “C in maximum contact temperature were found between traces. This behaviour stands in contrast to what was seen in experiments with pure hexadecane, where up until the transition, the maximum contact temperature remained fairly constant. . . . . , - - - . I . ..

. I

.... I ....

400

[

350 300

8 5 c 250

2

...............................

200 150

‘

’

t . . . . ....I .... .... I . . . . I . . . .

0

0.5

1

1.5

2

2.5

3

Time, seconds

..................................................................... inlet

outlet

Figure 7 Temperature traces at around 1.5 seconds. Looking at this figure one can see that the maximum contact temperature suddenly increased to more than 220 ‘C. The inlet ternperaturc does not seem

Figure 8 Maximum and inlet temperatures for a test with hexadccane with 1.0 wt% DBDS. In Figure 9, six temperature profiles are shown from the results of a test with hexadecane + 1.0 wt% DBDS. Five of the traces are marked from A to E and one trace is marked “scuffed”. Up until scuffing, the shape of the profiles rapidly changed from A to E and back to A again. It can be seen in Figure 9 that in between traces, the temperatures changed by large amounts. Profile B is very irregular of shape, with a maximum temperature of 400 “C.On the other hand,

78

450

_.....................................I..,~.,,.,.,,.,..,..,..,.~....................~..~..~..~..~~~ 450 ...................................................................................................

400

_............. .....................................................................................

350

...................................................................................................

100

450

400

350

100

450 400

,

.

1

inlet

.

1

.

'

-

1

inlet

ouuet

A

_..................................................................................................

450

outlet

B

..........................................................................................

...............

~.,,.,,.,,.,...............(............,,,,...,....,,,,,,,..,....,.,,,.....,,......

_...............................................,,.,.,...,..,,..............,.....................

l

inlet

.

l

.

C

I

.

1

-

I

l

inlet

outlet

_..................................................................................................

450

.

I

.

l

.

(

oudet

D

-................,....(................~.....,......................................................

...................................................................................................

1

inlet

E

outlet

inlet

.

1

.

1

SCUFFED

-

1

outla

Figure 9 Six temperature profiles from a test with hexadecane + 1.0 wt% DBDS.

-

1

79

the next race taken 0.03 seconds later is very smooth and the contact temperature is about 80 ‘Clower. This rapid changing of shape of the temperature profiles was increasingly seen at load stages above the failure load of pure hexadecane. It appears that the addition of 1.0 wt% DBDS to hexadecane increases the failure load by responding to incipient failure, allowing for recovery. When the conditions become too severe, no recovery takes place and the scuffing occurs, as seen in the trace marked “scuffed” in Figure 9. In the last experiment to be discussed, 0.1 wt% stearic acid was added to the hexadecane. Stearic acid (CH3.(CH2)16,COOH) is a boundary additive which adsorbs onto the surface. The molecules are thought to desorp from the surface at a certain temperature depending upon the contact pressure and additive concentration [91. Stearic acid proved to be a very effcctive additive in the conditions of this test. In an experiment with hexadecane + 0.1 wt% stearic acid no scuffing occurred, the test was terminated when the maximum load of the rig was reached instead. At this load, the maximum contact pressure equalled 2.39 GPa. Shown in Figure 10 are the maximum and inlet temperature at the last load stage. Looking at this figure one can see that the maximum contact temperature equals about 220 ‘C,with every fifth trace having a contact temperature about 30 ‘C higher. The inlet temperature averages about 150 ‘C. 300

~

I.............

250

P

$ 225

- .

- .~.

J ...............6 .......... 6 I...

. ..........I....

-.

.. .

......

i

I,......

-. t .............

appears around the point of the maximum Hertzian contact pressure. The inlet temperature does not seem to be affected. The reason for the observed cyclic behaviour shown in Figure 10 is not yet clear. The effect was not seen in another experiment in which thc nodding mirror line scanner was used. The failure load for hexadecane + 0.1 wt% stearic acid is much higher than for pure hexadecane and stearic acid as an additive seems to outperform DBDS as well. 250 220

P

4

I90

er.

8 160

b

130

---

inlet

OUtlCl

Figure 1 1 Three temperature profiles for hexadecane + 0.1 wt% stearic acid. This is surprising. However, one plausible explanation is that besides the absorbed layer, stearic acid might react with the sapphire (AI203) to form (basic) aluminium stearate, which is a metal soap. This in turn might form a very protective layer covering both the steel and sapphire surface.

5. CONCLUSIONS

Temperature profiles prior to and during scuffing have been taken at 0.03 seconds intervals across the 175 contact area. A single basestock with two boundary b 150 additives has been studied. In scuffing tests with pure hcxadecane, the 125 transition from smooth running to scuffing occurred 1 M L : : i : : : : !::::!::::!:::A:::-! very suddenly (within 0.6 seconds). This emphasises 0 0.5 I 1.5 2 2.5 3 the nced for a very rapid measuring technique, in order Time, seconds to follow the development of scuffing failure. Figure 10 Maximum and inlet temperature for The temperature profiles taken in experiments with hexadecane + 0.1 wt% stearic acid. hcxadccane + 1.0 wt% dibenzyldisulphide were very irregular of shape, with large fluctuationsin maximum Figure 11 shows thrce successive tcmperature profiles temperature. The addition of DBDS to hexadecanc taken with the infrared line scanncr. It can be seen in seems to increase the failure load by responding LO this figure that the increase in maximum tcmperature

E$ m

80

incipient failure allowing recovery. This possibly provides further insight into the mechanisms of action of EP and anti-wear additives. No failure could be obtained with hexadecane + 0.1 wt% stearic acid. Stearic acid might react with the sapphire to form a metal soap, which protects the surfaces very effectively.

REFERENCES 1 Dyson, A., "Scuffing - a review, part l", Tribology

International,April 1975, pp. 77-87. 2 Dyson, A,, "Scilffing - a review, part 2",Tribology International,April 1975, pp. 108-122. 3 Blok. H., "Surface temperatures under extreme pressure conditions",Congres Mondial du Peuole, pp. 471-486, Paris, 1937. 4 Blok, H., (1969), "The postulate about the constancy of scoring temperature", Interdisciplinary Approach to the lubrication of Concentrated Contacts, Troy, N.Y.. Vol. SP-237, pp. 153-224. 5 Dyson, A., "The failure of elastohydrodynamic lubrication of circumferentially ground discs", Proc. Inst. Mech. Engrs., 190,52/76, 1976. 6 Dyson, A., "Elastohydrodynamiclubrication of rough surfaces with lay in the direction of motion", 4th Leeds-Lyon Symp., pp. 201-209.1977. 7 Bell, J. C., Dyson, A., "The effect of some operating variables on the scuffing of hardened steel discs", Roc. Inst. Mech. Engrs., Symp. on EHL, 1972, Paper C11P2, pp. 61-67. 8 Enthoven, J. C., Cann, P. M., Spikes, H. A., "Temperature and Scuffing", Tribology Transactions, 36, 1993, p. 258-266. 9. Spikes, H. A., "Boundary lubrication and boundary films", XIXth Leeds-Lyon Conference, 1992, Leeds.

Dissipative Processes in Tribology / D. Dowson CI al. (Edilors) 0 1994 Elscvier Scicnce R.V. All rights reserved.

81

Computational Fluid Dynamics (CFD) Analysis of Stream Functions in Lubrication D Dowson and T David Department of Mechanical Engineering The University of Leeds, Leeds LS2 9JT, United Kingdom In fluid mechanics streamlines are widely used in association with ideal fluid flow, but less frequently for slow viscous flow. They can nevertheless indicate details of lubricant behaviour of considerable value to the tribologist and the purpose of this paper is to demonstrate the way in which streamline solutions for simple bearing forms can be derived and interpreted. It is now possible to apply versatile computational fluid dynamics (CFD) software to bearing problems as an alternative procedure for the generation of streamline solutions and this is demonstrated in the present paper. 1. INTRODUCTION

The basic equation of fluid film lubrication is a second order partial differential equation in pressure formed from an amalgamation of the equation of continuity and the Navier-Stokes equations of motion. Analytical and numerical solutions to this equation have provided the foundations for studies of bearing performance and design ever since it emerged from the classical analysis by Osborne Reynolds [ 11 in 1886. Once the pressure distribution in a bearing has been established, the cross film variation of lubricant velocity parallel to the entraining surface (u) can readily be derived to reveal important illustrations of fluid motion, as shown by Reynolds in his most effective illustrations. Likewise, contours of constant fluid velocity in the lubricating film (isotachs) and of constant pressure (isobars) can be constructed to reveal further details of the field of viscous flow. Michell [2] presented isotachs for the plane inclined or tilting pad bearing in his book published by 1950. Whereas streamlines are normal features of solutions to ideal fluid flow problems, they are less frequently represented in viscous flow analysis. They can, nevertheless, convey important information on the nature of slow viscous flow in fluid-film lubricated bearings, particularly in relation to cross film velocity components, perpendicular to the entraining bearing surfaces and

one of the purposes of this paper is to focus attention on the merit of recording streamlines in many lubrication problems. Some authors have adopted this approach to good effect for both hydrodynamic and elasto-hydrodynamic conjunctions. Two notable and early solutions to bearing problems were recorded in the 1950's by Wannier [3] and Milne [4], who analysed infinitely long journal and plane inclined thrust bearings respectively. Jeffery [5] provided a classical solution for stream functions between submerged rotating cylinders as early as 1922. This was later extended to circumstances more akin to journal bearings by Diprima and Stuart [6], while Kamal [7] examined flow separation in such configurations. Simuni [8] and Putrie [9] used numerical methods to reveal stream functions in stepped parallel slider bearings while Jinn-An Shieh and Hamrock [ 101 considered the stream function in elastohydrodynamicconjunctions.

For the slow viscous flow of an isoviscous, incompressible fluid, the stream function (y) is governed by the bi-harmonic equation, as is the stress function (0) in plane strain elasticity problems. For this reason, solutions to the biharmonic equation have attracted considerable attention and in some cases analytical solutions exist for simplified bearing configurations. In general, however, numerical solutions are required for realistic bearing configurations and initially finite-difference representations and relaxation procedures were adopted. Some early solutions based upon this approach will be noted. When

82

inertia terms in the equations of motion are considered, numerical solutions are invariably required, but the influence of flow reversal and circulation within lubricating films has nevertheless been explored through this approach. In recent years advances in computational fluid dynamics (CFD) have led to the development of robust software for the solution of a wide range of fluid flow problems. In the present paper we demonstrate how one such CFD package can be adapted to solve plane-flow lubrication problems. Simple solutions for plane-inclined and Rayleigh step bearings are compared with earlier finitedifference results, but the CFD package is then used to investigate features of lubricant flow into a bearing groove or pressurised recess. 2. THEORETICAL BACKGROUND

The plane flow of an incompressible, isoviscous Newtonian fluid is governed by the equations of continuity and momentum (NavierStokes).

It follows at once from equations (1,2) that if inertia can be neglected, the pressure in the field of flow is governed by the Laplace equation.

(4)

- (v)automatically satisfies the continuity equation (1).

Furthermore, the stream function ( y ) satisfies the biharmonic equation,

Solutions to equation ( 5 ) for the stream function (v), with boundary conditions appropriate to a particular bearing geometry and operating conditions, will thus satisfy the requirements of both mass and momentum conservation. Whereas a limited number of analytical solutions to the biharmonic equation ( 5 ) exist which are of interest in the analysis of fluid-film bearings (Jeffrey [ 5 ] , Jaeger [ 1l]), numerical solutions have been obtained by both finite difference and finite element procedures. When inertia is considered, numerical solutions to equations (1) and (2) are inevitably required. Such solutions enable Reynolds number effects to be explored and although these do not often influence the overall performance of bearing systems, they can materially influence the local behaviour of lubricant within a bearing. 3.

In addition, if a function ( y ) exists such that,

NUMERICAL COMPUTATION

ANALYSIS

AND

Early numerical solutions for stream functions in lubricating films were based upon finitedifference representations of the governing equation. More recent approaches have adopted finite element formulations in general CFD solvers. 3.1 Finite-Difference Solutions Much has been written about the use of finite difference representations to the governing

83

equations and the adoption of relaxation methods in studies of fluid-film bearings. In general there are familiar problems of mesh sizing; the failure of grids to coincide with realistic bearing boundaries; the selection of over and under relaxation factors and convergency problems, particularly when reverse flows are encountered. Since this is a long established procedure it is not necessary to dwell on the technique at this stage. 3.2 CFD Approach based upon Finite Element

Formulation

Once again the approach to numerical solutions of general fluid flow problems for incompressible, isoviscous fluids and steady state

:(

=0)

conditions is based upon the discretisation of the conservation equations for mass and momentum (equations (1) and (2)).

I u.vu = - v p + - v*u

Re

FIDAP uses the Galerkin form of the method of weighted residuals [13], allowing the residuals to be orthogonal, in an integral formulation, to the interpolant functions of each element. A matrix equation results from the assembly of the element formulations. This matrix may be solved in a number of ways, but is usually accomplished, as was the case for the solutions presented here, by a Newton iterative scheme. Inspection of equations (7) and (2) shows that for the nondimensional equation set FIDAP can accept a density of unity and a viscosity proportional to the inverse of the Reynolds number characteristic of the flow regimes. Hence, in specifLing the input data, various dimensionless flow solutions may be obtained readily for a variety of Reynolds numbers by simply changing the adopted value of viscosity. 4.

For the results presented in the next section we have adopted a finite element formulation utilising a commercially available solver, FIDAP. The fluid domain is sub-divided into discrete elements fixed in space and mesh generation schemes can be adopted to represent the domain geometry in the most effective manner for discretisation purposes. Within each element the dependent variables of pressure and velocity are interpolated by functions of known order whose coefficients are to be determined. For a valid and robust scheme the order of interpolation is such that the pressure interpolant needs to be one order lower than that of the velocity interpolant, the so called LBB condition [121.

By choosing a characteristic velocity, normally that of the moving surface if onc of the bearing solids is stationary, and a characteristic length scale, normally the minimum or outlet film thickness, the conservation equations can be reformulated into their non-dimensional equivalents and written in vector form as; v.u = 0

(6)

(7)

RESULTS

Initially solutions for streamlines in lubricating films were obtained analytically for simple configurations, or numerically by means of finite difference representations and the use of relaxation methods for realistic configurations. A range of numerical solutions obtained in this way will be presented for simple bearing forms, before the application of CFD solvers to this field is discussed. 4.1 The inlet region between rigid rotating cylinders An illustration of the streamlines in the inlet region to the nip between two rotating rigid cylinders with parallel axes obtained by Oteri [14] was included in a paper [15] presented to the first Leeds-Lyon Symposium on Tribology. It is reproduced have as Figure l(a), together with additional information in the form of contours of uniform velocity (isotachs) and uniform pressure (isobars) in Figures I@) and l(c).

An interesting feature of the solutions shown in Figure 1 is the reverse flow in the inlet region and the formation of a free boundary between the oil and the air. Oteri [ 141 was able to show that surface

84

tension played a negligible role in determining the form and location of this free surface in the case considered. 4.2 The Rayleigh step bearing

Details of the viscous flow in a step bearing were considered in detail by Toyoda [16] through the use of finite difference and relaxation methods. An important issue was the specification of the velocity and pressure boundary conditions at entry to and exit from the bearing. The Dirichlet (no slip) boundaq~conditions on velocity were adopted on each of the solid bearing surfaces, while the von

(

:-I

Neumann condition on pressure p = - - 0 was applied across the inlet and outlet sections to the lubricating film. In reality, the pads in a thrust bearing ring will experience the formation of a ram pressure effect, but this can readily be taken into consideration if the external flow field upstream of the bearing pad is also analysed. The well known linear pressure profile for the complete step bearing is readily revealed by the numerical solutions as shown in Figure 2(a). Perturbations to the pressure in the vicinity of the step were revealed by Toyoda [16] employing finite difference methods and isobars are portrayed in Figure 2@) for a Rayleigh step bearing with the step located half way along the bearing and Reynolds numbers of R, = 10 and R, = 100. It is evident that cross film presusre variations are of greatest magnitude in the vicinity of the step. A solution based upon the CFD code FIDAP revealed the crossfilm pressure variation shown in Figure 2(c) for a step located mid-way along the bearing and a step height ratip = h, / h o ) of 1.8. It should be noted, however, that the pressure variation across the film is only about 2 percent.

(6

Figure 1Finite difference solutions for the flow in the inlet region to two rigid, rotating cylinders (a) Streamlines (b) Isotachs (c) Isobars

It is, however, the details of the flow in the vicinity of the step which generally attracts attention in the analysis of the Rayleigh step bearing. This can be analysed by either finite difference and relaxation procedures or by the application of a CFD solver. The influence of step height ratio

85

0.3

P

0.2

- U0.1

0

I

U-

p

I

liil

i

0.5

(PI

1

Figure 2(a) Linear pressure profile for Rayleigh Step Bearing ( n = 0.5, 6 = 1.5))

(h, - hi/h,)

upon the flow in the vicinity of the

step, as revealed by the CFD solver FIDAP, is shown in Figure 2(d). Most of the lubricant entering the step bearing is drawn into the region of minimum film thickness beyond the step if the step height is small, albeit with a pocket of recirculation in the comer of the step in most cases. For larger step height ratios it is essentially only the lubricant in the lower part of the inlet film which is transported beyond the step, while the remainder forms a reverse flow stream in the inlet land region. Such details, which are readily revealed by the stream functions, are particularly important when viscous dissipation and thermal balances are considered.

Figure 2(b) Isobars in the vicinity of the step in a Rayleigh step bearing (n = 0.5, (i) R, = 10, (ii) R, = 100) While numerical solutions readily reveal the overall features of viscous flow in standard bearing configurations, such as plane inclined and Rayleigh step bearings, it is for the less standard lubrication problems that CFD solvers are likely to find greatest appeal. In order to illustrate this point we turn to the age-old problem of lubricant entering a bearing via a supply slot and a groove in the next section. 4.3 Lubricant flow in a supply slot and groove.

The general configuration of the problem considered is shown in Figure 3, Lubricant enters a relatively deep groove from a supply slot with pressure (pS)before being entrained into the bearing clearance by the surface moving with velocity 0. Much of the lubricant circulates within the groove, with only a small amount being drawn into the clearance space. Details of the flow patterns, which are influenced by Reynolds number in the groove, are readily revealed by FIDAP solutions to the problem.

86

0

0.-3

0.6

0.9

I.?

1.5

14

1hlh.l

Figure 2(c) Cross film pressure distribution at the step in a Rayleigh step bearing ( n = 0.5, h, = 1.8) The finite element mesh for the groove domain in the vicinity of the supply slot, groove inlet and groove outlet regions are shown in Figure 4. A summary of the conditions considered is given in Table 1.

Depth of supply groove Chamfer radius in groove Groove width (L) Supply pressure (P,) Lubricant viscosity (q) Lubricant density (p) Surface velocity cu)

2mm 1 mm 128 mm 3.5 MPa 0.037 Nm2/s 880 kg/m3 0 - 21.21 m/s

\

I

Figure 2(d) Streamlines for various step height (Ki = hi/h,) and the ratios development of reverse flow in a Rayleigh step bearing (FIDAF').

87

n.Vu = 0

n.Vu = 0

/ -

\

u-

'I

I'

Figure 3 Lubricant supply configuration.

slot

and

groove

The Reynolds number (RJ was defined in relation to the surface velocity 0, the groove width (L) and the lubricant properties (q, p) as; Re = -P

a

q Streamlines in the vicinity of the central lubricant supply slot are shown for Reynolds numbers of 0,50, 100 and 200 in Figure 5 .

The streamlines for the same range of Reynolds numbers, but in the regions of the upstream inlet and downstream exit from the groove, are shown in Figures 6 and 7. 5. DISCUSSION

The results obtained for the Rayleigh step bearing have shown that current CFD tools can not only predict overall features of fluid film bearing performance characteristics, but also reveal fine detail of the flow patterns, velocity distributions and cross film pressure variations. A range of solutions for different step height ratios has been used to illustrate the recirculation that occurs in the vicinity of the step and the substantial recirculation in the inlet zone under certain conditions. The normality of one particular package, FIDAP, has been demonstrated by considering the flow conditions in

a lubricant supply slot and groove for various Reynolds numbers. Analytical solutions do not exist for the geometry and flow conditions considered, but the CFD package enables the effect of operating conditions upon flow parameters to be explored with ease. Since the overall geometry has been assumed to be fixed in the case considered, the solutions for various Reynolds numbers represent different sliding speeds 0. When both the bearing surfaces are stationary flow through the supply slot and groove into the bearing films is symmetrical and dominated by the supply pressure (Pa. As the bearing surface starts to move with velocity 0,all the lubricant entering the groove from the supply slot initially flows upstream in the reverse flows shown in Figures 5@), (c) and (d). At the higher Reynolds numbers of 100 and 200 a recirculation of lubricant takes place within the supply slot. As the reverse flow in the upstream part of the groove approaches the inlet region, it is entrained by the rapidly flowing lubricant adjacent to the moving bearing surface. Recirculating flows are in evidence in the outlet regions of the groove and it is interesting to note that much of the lubricant recirculates, while a substantial proportion of the fluid drawn into the groove by the moving surface is 'curried out' into the lubricating film formed beyond the groove. Once again, the importance of this feature of the flow becomes even more significant when thermal aspects of the problem are considered and the phenomenon of 'hot oil carry over' is an important issue in thrust bearing design. The role of the Reynolds number upon the flow patterns is significant, but it was found that changes in the supply pressure had only a modest impact upon the streamline patterns.

6. CONCLUSIONS

The major objective of the paper has been to draw attention to the value of the additional information revealed by stream function solutions to lubrication problems, rather than the conventional solutions to the Reynolds equation. This has been illustrated by examples from the flow at the inlet to lubricated rigid cylinders; the Rayleigh step bearing and the lubricant supply to a bearing through a slot

88

I

Figure 4 Finite element mesh for lubricant supply groove domain. (a) Central supply slot (b) groove inlet (c) groove outlet

89

U-

P

Figure 5 (a), (b) Streamlines in the vicinity of the central lubricant supply slot Ps= 3.5 MPa (a) & = O (b) & = S O

90

Figure 5 (c), (d) Streamlines in the vicinity of the central lubricant supply slot Ps= 3.5 MPa (c) = 100 (d) R,= 200

91

u=o

u-

Figure 6(a), (b) Streamlines near the upstream inlet into the lubricant groove P,= 3.5MPa (a) % = O (b) % = 5 0

I

92

I

p

u-

P

U-

Figure 6(c), (d) Streamlines near the upstream inlet into the lubricant supply groove Ps= 3.5 MPa (c) % = 100 (d) %=200

I

93

u =o

Streamlines near the downstream exit from the lubricant supply groove Ps= 3.5 MPa (a) R, = 0 @) R, = 50

94

Figure 7(c), (d) Streamlines near the downstream exit from the lubricant supply groove P S = 3 . 5 M P a (c) % = 100 (d) %=200

95

and groove. Furthermore, it has been shown that current CFD solvers, such as FIDAP, can be used effectively to solve realistic bearing problems.

Basic Engineering, Series D, Vol. 88, No. 4, 717-724.

8. L M Simuni, The numerical solutions of some

problems in the flow of a viscous fluid, Inzhen. Zhur. (1964), 4 446-450.

ACKNOWLEDGEMENTS

The authors are pleased to acknowledge the numerical solutions to lubrication problems by former PhD students (Oteri [14], Toyoda [16]) and by recent MSc project students (Y M Zhou (1991) and K Roberts (1992)).

9. H A Putre, Computer solution of unsteady

Navier-Stokes equations for an infinite step bearing (1970), NASA TN D-5682. 10. Jinn-An Shieh and B J Hamrock,

REFERENCES 1. 0 Reynolds, On the theory of lubrication and its

application to Mr Beauchamp Tower's experiments including an experimental determination of the v i s p i t y of olive oil, Phil. Trans. Roy. SOC.(1886), 177, 157-234. 2.

A G Michell, Lubrication Its Principles and Practice, (1950), Blackie, London and Glasgow, I

1-317. 3.

G H Wannier, A contribution to the hydrodynamics of lubrication, Quarterly of Applied Mathematics, (1950), Vol. 8, No. 1, 132.

Stream

functions in fluid film lubrication, US/Taiwan Joint Symposium, on Tribology, (1989), Proceedings, 69-78. 11. J C Jaeger, Elasticity, Fracture and Flow, with

Engineering and Geological Applications, (1956), Methuen & Co Ltd., London 1-152. 12. I Babushka and A K Aziz, Lectures on the

Mathematical Foundations of the Finite Method, Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, (Ed. A K Aziz), Academic Press, New York, pp 135, (1972). 13. FIDAP Manuals 1-4 (Version 6.03). pub.: Fluid

Dynamics International Inc. 4.

A A Milne, A contribution to the theory of

hydrodynamic lubrication. A solution in terms of the stream function for a wedge shaped oil film, (1957), WEAR, Vol. 1,32. 5.

G B Jeffery, The rotation of two circular cylinders in a viscous fluid, Proc. Roy. SOC. (1922), A, 101.

6. R C Diprima and J T Stuart, Flow behveen

eccentric rotating cylinders, Trans ASME, Journal of Lubrication Technology, (1972), Series F, Vol. 94, No. 3, pp 266-274. 7. M M Kamal, Separation in the flow between eccentric rotating cylinders, ASME Journal of Basic Engineering, Series D, Vol. 88, No. 4, 717-724. 7. M M Kamal, Separation in the flow between

eccentric rotating cylinders, ASME Journal of

14. B J Oteri, A study of the inlet boundary

condition and the effect of surface quality inb certain lubrication problems, (1972), PhD thesis, The University of Leeds. 15. D Dowson, The inlet boundary condition, Proceedings of the 1st Leeds-Lyon Symposium

on Tribology, 'Cavitation Phenomena', Mechanical Publications, pp 143-152.

and Related Engineering

S Toyoda, A study of the effect of fluid inertia and lubricant starvation upon fluid film lubrication, (1977), PhD thesis, The University of Leeds.

16. D Dowson and

This Page Intentionally Left Blank

Dissipative Processes in Tribology / D. Dowson et al. (Editors) 1994 Elsevier Science I3.V.

97

Shear properties of molecular liquids at high pressures - a physical point of view E.N. Diaconescu

University of Suceava, 1 University Street, 5800, Romania A liquid at high pressures is regarded as an overheated solid having a Lennard-Jones-London molecular interaction potential. Jump frequency of a molecule, established by using the concepts of disturbed and activated lattice oscillators, is applied to derive the net frequency of molecular jumps along an applied shear stress. Two molecular displacements, elastic and viscous, are found to take place as it is implied by a Maxwell rheological model. The solid like behaviour of liquid state is then justified. A general expression for intrinsic viscosity of a liquid is derived. This is found to reduce, in certain circumstances, to various known formulae for viscosity. Then shear viscosity is deduced, which is found to follow a hyperbolic sine law. An expression for piezoviscous coefficient is worked out by taking into account the effect of pressure upon interaction potential. It is shown that intrinsic viscosity changes exponentially after a pressure step and, if molecules are plane or linear, it decreases with increasing shear duration as a result of molecular alignment. Tangent shear modulus is found to decrease with shear strain and to depend nearly linearly on pressure. Limiting shear stress occurs at a critical value of shear strain which changes little with pressure. A linear dependence of this stress on pressure can be accepted as a satisfactory approximation of actual variation. 1. INTRODUCTION

It is nowadays widely accepted that, at moderate shear rates, the lubricant exhibits non-linear viscoelasticity inside an EHD oil film, [l-41. The response of an oil film to shear depends on viscosity, 7 , usually a function of shear rate, i ,and on elastic shear modulus, G. A non-linear Maxwell model is often used to describe this behaviour, [1,2,4-61:

z being the applied shear stress. At large shear rates, increasing shear stress under increasing shear rate is limited by solid like shear strength of lubricant, z - ~ , [ 1-1 11. Various relations between viscosity and shear rate or shear stress have been proposed, widely acccpted being Eyring hyperbolic sine law, [ 1,4,6], and Winer logarithmic law, [5-121. In the former law, a reference shear stress of liquid.?,, called Eyring stress, and an initial viscosity of the fluid, qo,defined for a shear rate tending towards zero are used. The most widely used dependence of viscosity

on pressure is expressed by Barus equation, which is based on a piezoviscous coefficient, a. Up to date, shear modulus is determined experimentally by high frequency oscillatory shear, [I 1,13,14], light scattering, [15,16], stress-strain investigations, [2,5,8] and traction tests, [4,11,17251. Most results suggest a linear dependence of this modulus on pressure:

where Go=(O...0.8)GPa i s the valuc of modulus at zero pressure and A=0.5...5 is a dimensionless coefficient, [1,2,13,15,19,26]. Traction experiments systematically yield smaller values of Go and A than physical investigations. Limiting shear stress, measured by various procedures, such as static shear in especially devised apparatus, [2,3,8,27], pressure shear plate impact experiments, [9,28], dynamic shear by a jumping ball, [3,29], temperature mapping of an EHD contact by infrared radiometry, [30,31], or traction experiments, [ 1,7,10,11,18-251, seems to vary linearly with pressure:

98

where rto is the shear strength of the fluid at p=O and p is the slope d r , / d p . Jacobson, [3,27], found values of 15 MPa for rto in the case of mineral oils and of 0.026 ... 0.14 for p . The ratio of shear modulus to limiting shear stress depends on oil, pressure, temperature and on actual measuring procedure. Traction experiments yield lower values of this ratio, of up to 12.5, whereas physical results show values of 26 to 4 1. Excepting some works upon viscosity, [32-351, and simple attempts to explain shear behaviour of liquids, [36,37], no other physical explanation has been advanced for shear properties of lubricants. Present work aims to improve physical knowledge on the subject by taking into study, for a start, only simple molecular liquids. 2. SIMPLE MOLECULAR LIQUIDS

A simple molecular liquid results from the melting of a simple molecular crystal. Spherical or nearly spherical molecules are placed in the lattice sites of such a crystal. These are bound by weak, non directional van der Waals' forces. The stability of a crystalline structure held by weak forces requires it to bc close packed, [38-413. If the molecules are spherical, the lattice cell is facecentred cubic. A similar arrangement is preserved if the molecules are not spherical, but have comparable dimensions on all three directions. The forces a molecule exerts upon its neighbours are supposed to have the same origin, called centre of interaction of that molecule. At equilibrium, these centres are placed in the lattice sites. Due to dimensional homogeneity of the molecules, molecular crystals contain mainly Shottky point defects, called vacancies or holes, and dislocations. At the melting temperature, T,, a critical value of vacancy concentration is reached, the holes having a homogenous distribution over the volume. Latent heat of hsion generates nm additional vacancies which destroy the stability of the crystal and confer fluidity on the substance. New holes are formed at even higher temperatures. Regarded instantaneously, a molecular liquid maintains locally the original close packed arrangement of the originating crystal, but the long range order is disturbed by the presence of vacancies. The molecules oscillate around the lattice sites for very short times, until they gain

enough energy to jump into neighbouring holes, leaving vacancies in the initial positions. These jumps confer a great mobility on the molecules and, consequently, produce the fluidity of the substance. This fluidity does not allow the persistence of dislocations. The concentration of vacancies in the liquid state is given by a Bose-Einstein distribution, [37,42]:

(4) n being the number of vacancies in a mole of liquid, N the Avogadro's number, k the Boltzmann's constant, T the absolute temperature and dg, the Gibbs free energy of a hole formation in the liquid state, dcduced in [42]. A mole of liquid substance containing N molecules is studied further down. It contains n vacancies and, consequently, N + n lattice points. The average molecular volume is w = V/N , where V denotes the molar volume of the liquid. In fact, a molecule placed in one of the lattice sites, surrounded by other molecules, occupies a smaller volume, w,. This is the molecular volume in a hypothetical perfect solid lattice of the same substance, subjected to the same pressure and temperature as the liquid is. The volume of ideal crystalline component of the liquid is V, = N u , , and V f represents the free molar the difference V -V, volume. The free volume per molecule is w / = V / / N = w -w , . Above it has been assumed that although locally, in the neighbourhood of a vacancy, the crystal latticc is strained, this strain vanishes at a few intermolecular distances. Consequently, a vacancy increases both, the free and the total volumes by the same amount w,, although its actual volume is slightly smaller. The following relations hold between these volumes

Owing to a rather solid like structure of a liquid subjected to such high pressures as encountered inside EHD films, thc oscillations of molecules around lattice sites can be assessed by applying the theory of solid state. According to this theory, the oscillatory motions of individual molecules are replaced by a collective movement of all lattice elements. A system of coupled harmonic oscillators

99

is formed, which can be de coupled by conveniently transforming the equations of motions and by using generalised co-ordinates, called normal coordinates. Thus, in a mole of substance there are 3N independent harmonic oscillators, called normal oscillators of the lattice. They belong either to an acoustic or to an optical vibration mode of the lattice, each of these being longitudinal or transverse. Because the reciprocal lattice of a face centred cubic molecular lattice contains a single molecule, acoustic vibration modes, longitudinal and transverse, occur only. Acoustic vibrations of low frequencies, having a maximum wave length twice the length of the crystal, exist at a very low temperature. As the temperature rises, acoustic vibrations of higher frequencies occur. When the temperature reachcs a characteristic value, called Debye temperature of the substance, T, , two neighbouring molecules oscillates in counterphase. This vibration mode defines the smallest length of wave excited in a crystal. Corresponding maximum frequency of acoustic oscillations is called Debye frequency of that substance, v,. The following equation holds between v, and 7‘’ : kT,

= hv,,

h being Planck’s constant. If the temperature rises above T,, no new vibration modes occur but the additional energy increases the amplitude of existing oscillations. The above defined Debye parameters, T,, and v D , are valid for a perfect lattice. In a real lattice, in immediate neighbourhood of a vacancy, the stiflness of the lattice is considerably lowered and the vibration frequency is diminished. According to Mott and Gurney, [43], this effect is significant only for molecular vibrations along directions joining the hole centres with neighbouring, occupied lattice sites. Therefore, a hole lowers the Debye frequency of normal oscillators which contain molecules neighbouring a hole, to a value i, 4 v,. Such oscillators are called disturbed oscillators and is called disturbed Debye frequency of that substance. This is another characteristic frequency of observed substance. The normal oscillators, attached to lattice vibrations, can be regarded as quantum oscillators. From this point of view, the energy of an oscillator,

which is identified by a wave vector in the reciprocal lattice, g , and by a vibration branch, s, takes discrete values only, E + , called energy levels:

where v+ is the frequency of considered oscillator and nis is an integer, known as a quantum number. The smallest value of energy level, cGS= 0.5hvqS, is called zero energy, and it represents the vibration energy at absolute zero. Energy difference between two adjacent levels corresponds to a change of quantum number by unity, and it is Asqs = hvis. It represents the quantum of vibration field of a crystalline lattice, and bears the name of phonon. The phonon is a quasi- particle because it has no mass and, unlike ordinary particles, it cannot appear in vacuum since it needs a material medium to appear and to exist. Like the vibration mode, q s , to which a phonon is attributed, the latter can be acoustic or optic, and longitudinal or transverse. Between phonons there no interactions because they stem from harmonic and independent oscillators. The phonons travel through the lattice with the phase velocity of the corresponding wave. Consequently, the lattice oscillations can be represented as a gas of phonons, confined within the limits of the crystal. The interactions between the crystalline lattice and phonons can be visualised as simple collisions between phonons and molecules. If a molecule belongs to a normal oscillator having a frequency vqs,equal to or less than the frequency of the phonon, the latter is absorbed by oscillator and the energy of the oscillator increases by /lvis. The difference / I (vp - vqs), vp being the frequency of the phonon, is freed as a lower frequency phonon. If the frequency of the phonon is less than that of the oscillator, the phonon is elastically reflected without energy exchange. At thermal equilibrium, the phonons, having no spin, obey a Bose-Einstein distribution with zero chemical potential, the average number of phonons in the state qs being:

100

Therefore, the average energy of the oscillator is: (9)

The Debye temperature of a molecular crystal is less than 100 K. Thus, practical operating temperatures are much higher than TD. Therefore, the argument of hyperbolic cotangent function is much smaller than the unity, and by using the approximation coth x s If x , when x +*1 , the average energy of an oscillator becomes:

The potential energy of interaction bctween two molecules in a molecular solid or liquid can be accepted to be given by a Lennard-Joncs-London potential, of 6-12 kind:

L

J

where r is the distance between interaction centres of adjacent molecules, cr is that value of r which yields q ( r ) = 0 and E is the minimum valuc the potential reaches at equilibrium. Values of E and cr, taken from [44], can be seen in Table 1. The intermolecular force is equal to the negative

Table 1 Potential paramcters and calculated piezoviscous coefficients for sevcral hydrocarbons Substance Molecular weight a, (Angsfr) E/ k, ( O K ) i -C,H,, 58.12 5.311 3 13 n - C5HI2 72.15 5.769 345 n -C,H,, 86.17 5.909 413 n - C8H18 114.22 7.451 320 Cyclohexnne 84.16 6.093 324 C6 H6 78.11 5.270 440

a,,( l / G P a ) 21.55 27.15 29.20 58.50 32.00 20.70

derivative of thc potcntial: F ( r ) = --

dr

I sL

A

This force becomes equal to zero at equilibrium, when the potential energy is at a minimum. In this situation the intermolecular distance reaches the equilibrium value v = ro = 21’6 0. A structural unit, as shown in Figure 1, consisting of four neighbouring nioleculcs is considered. The centres of interaction of these four molecules are the vcrtices of a regular tetrahedron. If a pressure p is applied to the liquid, a forcc F, directcd towards the centre of the tetrahedron acts upon each molecule. As a result, the equilibrium intcrniolccular distance r,, decreases to a valuc r, I r , in order to produce the repulsion required to As the area balance thc prcssurc force F,. attributcd to each niolccule in a plane pcrpcndicular on I;, is (&/2)vi, the force F, takes the following value:

Equilibrium intermolecular distancc under pressure, r,, is, therefore, the solution of equation F(r,) = F, which can be expressed as:

Figure 1 Molecular structural unit under pressure

101

Table 2 Dependence of solid like shear properties on pressure for a simple liquid p=pd/E F=r,/o G=Gd/E r1 = r,d / E 0 1.1224620 24.00 1.99 5 1,0941643 36.87 2.91 10 1,0764083 48.32 3.72 25 1.0441497 79.35 5.91 50 1.0142835 126.41 9.19 100 1.0441497 213.38 15.25 297.27 20.99 150 0.9608 17 1 200 0.94708 10 378.46 26.56 300 0.9250444 536.87 37.38 691.95 400 0.9100070 47.95 500 0.8983152 844.86 58.34 Dimensionless distances 7 = rp/ o,computed from equation 14, are given in Table 2 for different values of dimensionless pressure p = p d / E .

3. MOLECULAR JUMPS

Each vacancy is surrounded by z neighbouring molecules, z being the co-ordination number. Consequently, a vacancy introduces z disturbed Debye oscillators. This means that in a mole of liquid there is a number Nd = nz of disturbed oscillators, which vibrate on a diminished Debye frequency, vD. As the total number of oscillators is 3 N , the probability that a given oscillator is disturbed is given by

Pd = N d / 3 N = n z / 3 N .

(15)

According to equation 10, the average energy of a disturbed oscillator equals k T . A disturbed oscillator can change energy only when interacts with phonons having a frequency equal or superior to t i , that is only with phonons of frequencies vD and vD. The absorption of such phonons increases the oscillator energy. Once a certain level of this energy, called jump activation energy, s J , is reached, a molecule of this oscillator is able to jump into a neighbouring vacancy. This activation energy corresponds to an activation number of phonons, tiJ

After each jump, the surplus energy is freed as heat,

YCI.

0.189 0.171 0.164 0.155 0.150 0.145 0.143 0.142 0.140 0.139 0.138

that is as phonons of frequency GD, freely moving through the lattice. As shown by Planck, [45], the quantum number obeys a Boltzmann distribution. Consequently, the number of activated oscillators, N,, having an energy superior to E, , is

N,

=

3 Nexp(-gj / kT) .

(17)

The activation probability for a given oscillator can be written as:

The jump probability, defined as the probability that a given molecule is performing a jump, is the product of both independent probabilities that the considered oscillator is disturbed, Pd, as well as activated, p0,:

3N Obviously, the jump is performed when the oscillator elongation is at a maximum. This situation repeats itself with a frequency v,. Consequently, the jump frequency of a molecule is given by the following equation:

znvD E. v . = P . v =-exp(---?-). ’ ” 3N kT I

In a liquid at rest, these jumps appear with the same frequency on all directions. As a result, no

102

definite flow occurs. If the components of these jumps on three rectangular directions are considered, then the jump frequency along one of these directions is:

given instant, i.e. flowing at that instant, can be estimated by using the following obvious equation: NI , = N P

"

v

'J

- znvD

3

9N

= _ -

zn =_

18

&

exp(--L)sinh(-).7Ws

kT

2kT

(24)

&

-exp( -1)

kT

Usually, a liquid is not at rest, but flows on certain surfaces, along certain lines, under the action of applied external forces. A shear stress t acts in any point of a flow line. This generates a shear rate i . Under the action of this shear stress, a molecule that jumps performs, by displacing itself to the potential barrier, a work w = 0 . 5 ~ This ~ ~ . work is subtracted from activation energy for the jumps along the flow and is added to it for the jumps against the stress. The jump frequencies along the flow, denoted by +, and against it, denoted by -,become different:

The factor 1/2 multiplying the frequency v, takes into account the equal jump frequencies on both directions on a straight line in a liquid at rest. The net frequency of molecular jumps along the shear stress is:

In a sheared liquid, any molecule oscillates around its equilibrium position in a time interval equal to the period of jump, P = 1 1 v,, and then jumps in a new position, where it oscillates again. The displacement of a molecule along the flow can only be performed during a jump. For the duration P the molecule behaves as in a solid state. The effect of a shear stress depends on its duration, t . If t 4 P , the mean position of the observed molecule is displaced elastically only, by a very small distance A,. If t + P , the molecule jumps integer [ t / P ] intermolecular distances R. along the flow, that is it flows, and then, in the new position, as wcll as in each of the intermediate positions, becomes elastically displaced by A, . The number of molecules performing the jump at a

Because NJs44 N , it can be stated that, at any instant, the great majority of molecules behaves in a solid like manner. The elastic shear strain, y e , has the value ye = Arctun(A, / A,), A, being the distance between two adjacent molccular layers, parallel to the shear plane. Net flow or viscous displacement of a molecule during a time interval t is S, = [ t / P ] A . At large values of t it increases nearly linearly in time because an integer part becomes practically equal to its argument. Elastic and viscous displacements add together to give the total displacement of a molecule. This justifies the use of a Maxwell model for shear behaviour of molecular liquids. If shear duration is large in comparison to jump period, the solid like deformation is overwhelmed by viscous flow and it cannot be observed. Although unnoticeable in the presence of a large viscous flow, this solid like behaviour is of utmost importance because the increasing shear stress under increasing shear rate can reach the shear strength of the instantaneously "solid component" of the liquid and the latter shears as in a solid state. Therefore, the limiting shear stress of the liquid is the shear strength or the ultimate shear stress of its instantaneously "solid component", consisting of Nss = N - N,, molecules. As a consequence, the solid like properties of a liquid can be determined if the viscous flow is disregarded and the instantaneous liquid lattice, made of N , molecules, is seen as being "frozen".

4. VISCOSITY

4.1. Intrinsic viscosity As stated above, in a liquid at rest there is not a net flow. Nevertheless, there is an internal mobility of molecules, caused by molecular jumps. Jump frequency of a molecule on a certain direction is L;, given by equation 29. An average energy kT is available for jump on that direction. This can be visualised as generating, at a molecular level, an average shear stress 7,, directed along the jump. If

103

intermolecular interaction is elastic, r, yields from rows= 2kT and it is I , = 3 k T / w , . This average shear stress z, .available in a liquid at rest on the direction of a molecular jump, is in fact the Eyring stress, a liquid parameter. Introduced this way, Eyring stress acquires a physical meaning. As shown by Tabor, [46], an average velocity gradient, grad u = RV, / Ro, occurs between a molecule performing a jump and its neighbours. By definition, the viscosity is the ratio of applied shear stress to the velocity gradient it produces. Accordingly, the intrinsic viscosity of a liquid, a measure of the resistance a fluid at rest opposes to a molecular jump, is expressed by the equation:

The jump activation energy, E ~ ,is, under constant temperature and pressure, a measure of useful work done against intermolecular forces. Consequently, it represents the free Gibbs energy of activation of a molecular jump, that is E, = Ag,, and the intrinsic viscosity becomes:

Ro 18kT

17, ==

4,

exp -. R w,zdvD kT

-~

By using expression 4 of vacancy concentration, this equation takes the form:

which is the empirical formula proposed by Batschinski, but with an explicit constant. If w, is replaced by w / ( l + d ) and d by its relation 4 in equation 26, the latter takes the following form:

2, 18kT R wvD

qi = --exp-,

As, kT

where dga = Ag, +dg, is the Gibbs free energy of flow activation, introduced by Eyring, [32]. By taking into account that Eyring used a molecular vibration frequency v = kT/h instead of vD and that (182, / z R ) e l , the above expression transforms into a well known formula for viscosity, deduced by Eyring:

4% h qi = -exp-, w kT As found experimentally, [32], molecular Gibbs free energy of flow activation is a fraction of latent heat of vaporisation,h, , or of lattice energy, E, :

By regarding the latent heat of vaporisation as work done against internal pressure, hv = copi,and by using van der Waals' expression for thermal pressure, p, = kT / w then, at low pressures, when p 44 p, , p, z p , and equation 29 takes the form: f ,

R 18kT

w

R zwv,

wf

11, = o-exp(c-).

Equations 26 and 27 take into account both, the activation energy for a jump and the free volume of a liquid, and therefore they can be considered as general equations of viscosity of simple liquids. Under certain circumstances they reduce to other well known viscosity equations. For instance, if Ag, is only marginally superior to the average energy k T , that is, kT S A g J 1 1 . 1 5 k T , then the obvious approximation exp(dgJ / k T ) z e A g J / kT holds with less than 1% error. This transforms equation 26 as into:

This is a formula proposed experimentally by Doolittle, [47], and deduced theoretically by Cohen and Turnbull, [48], but having now an explicit constant. Other known formulae, such as those of Williams-Landel-Ferry, Fulcher-Tamman, [48], or Vogel, [49], can easily be derived. 4.2. Shear viscosity

The viscosity of a sheared liquid depends on applied shear stress or shear rate. At zero value of this rate, shear viscosity, denoted by q,, is called initial viscosity. The name of final viscosity, q,, is used when shear rate tends towards infinity. As the flow proceeds along a shear stress, t , the

104

net jump frequency along direction of shear, v,, is given by equation 23 and velocity gradient becomes grad u = Av, / A,. Consequcntly, the shear viscosity can be written as:

E! =

(38)

1.0147pri + 8 . 6 1 ~ .

If one assumes w go, in equation 29 and takes into account equations 31 and 38, the following expression for viscosity at high pressures is obtained:

(33) which is the hyperbolic sine Eyring law. When r+O, q + i i 0 and if r j a , q + q , = O , that is initial viscosity is equal to intrinsic viscosity of that liquid and final viscosity is zero. Equation 33 can be rewritten as:

'

7/ 7,

'lo sinh( P/

70)

(34)

This formula prcdicts a non-Ncwtonian behaviour of a simple liquid, its shear viscosity decreasing with increasing shear rate. This behaviour is usually called a "shear thinning effect". 4.3. Effect of static pressure upon viscosity A pressure p applied to a liquid reduces the intcrmolccular distances from ro to 1 ; . A force F,, given by equation 13, acts upon each molecule towards the centrc of tctrahedron shown in Figure 1. This generates an intermolecular repulsive force:

By dividing this equation by v,(O),by assuming that z, vD,and A,(p) / A ( p ) do not depend on pressure and by replacing w , ( p ) w,(O) = (rp/ r0)3, the following final exprcssion is found for intrinsic viscosity at high pressures:

A direct comparison of equation 39 with Barus equation yields the following expression of piezoviscous coefficient:

where c has been considered to be 0.408. This equation can be expressed in terms of Lennard-Jones parameters as follows:

"1

a(p)=The force hy stems from an intcrmolccular potential which is modified by applied pressure:

Thc solution of this differential equation under initial condition qp= -E when p = 0. is: J2 pP = --prP 12

&

0.414(-) rp 3 -+-InE 3 Q kT

ro] , rP

(41)

where ji = pa / E is the dimensionless pressure. The second term in right hand members of equations 40 and 41 is much smaller than the first. This leads to a simple, but sufficiently precise formula for piczoviscous coefficient:

a ( p ) = O . 4 1 d Lr3 kT

--E.

(37)

For a 6-12 Lennard-Jones potential, molecular lattice = 8 . 6 1 ~ . By energy under zero pressure is assuming a similar proportionality under pressure, the following cspression can be written:

and justifies the assumption w z 0,. Piezoviscous coefficient decreases with incrcasing prcssure due to decreasc of intcrmolecular distance. Variation of dimensionless piezoviscous coefficient, E = a ( p ) / a(O), computed by using equation 42 and Table 2, is shown in Table 3.

105

Table 3 Variation of piezoviscous coefficient with pressure 4

0

5

25

50

a

1

0.926

,805

0.738

-

100 0.668

Computed values of piezoviscous coefficients at zero pressure for several simple hydrocarbons can be seen in Table 1. 4.4. Viscosity under transient pressure A vacancy is formed when a molecule from volume jumps on interface and vanishes by a reverse jump. The frequency of vacancy formation, vht , is equal to frequency of molecular jumps from volume on interface, whereas the frequency of hole vanishing, vh- , equals the frcquency of jumps from interface into volume. Total frequency of jumps that influence the number of vacancies is vhr= vht + vh-, whereas net frequency of hole formation is vh = vh+ - vh-. At equilibrium, the number of vacancies is

constant, that is net frequency of vacancy formation is zero, 4h = O, Or 4h+ = 4h-> and the frequency a molecule performs jumps is given by equation 20. Total frequency of jumps, on and from the interface, is N , times larger than vJ, N , being the number of holes sited on interface: znN, vD 3N

A

4, = -exp( -A). kT

Equilibrium frequencies

4h t = 4 h-

=

(43)

43, M pit : 42 3, S pit : 40.5 2

d

m

N

-

154

cosity v , 66.3mm2/s at 40"C, 8.9mm2/s at lOO"C, pressure viscosity coefficient a ; 16.6GPa-', specific gravity 15/4"C : 0.877) was supplied at a flow rate of 15cni3/s. The oil temperature was kept at 45°C and the corresponding oil viscosity v was 52.5inm2/s. The state of oil formation between discs was continuously monitored by means of an electric resistance method. The voltage of 150mV was impressed between discs (the resistance of about 1 . l l k R was connected in parallel in the measuring circuit), and the variation of the voltage during operation was observed. When the oil film is developed fully, the voltage Eab recorded on a chart reaches 15mV. The frictional force between discs was measured using strain gauges stuck 011 the driving shaft (via slip rings), and the actual surface temperature on the track was also measured successively using trailing thermocouples. The theoretical oil flni thickness hmin[6] was calculated using the oil viscosity a t the actual temperature of the disc surfaces. For reference, the value for the viscosity at the inlet oil temperature (45°C) was also calculated. 3. RESULTS AND DISCUSSION 3.1.F'riction and surface temperature Experiments were all carried out at a constant normal load giving a maximum Hertzian stress O f pH = 1.2GPa and a t a designated slip ratio of s = -3.6% or -19.2%. The testing machine is equipped with an automatic stopping device which is worked by the vibration induced. In the present experiments, each test was continued up to N=106 cycles unless any serious surface damages occurred. The main results are summarized in Table 1. Figs.3 to 5 show some results of the friction measurements. On the whole, the frictional force had a tendency to be large immediately after the start of ruiiiiing, then decrease monotonously and settle down. However, if examined in detail each result, considerably different behaviors were recognized depending on the combination of discs and

0.10

Hv: 760/760,760/320

\

0.08

Axial/P

A-3

8-2

8-5

A

A

c-l

c-2

-1

---

Oblique/P---

; I

B'

c 0

.r

A-1

0.06

.r

YL

+0

1

c, C

.r

.,U 0.04 Y'c 0) V 0

0.02 S=

I

0 104

-19.2%

I

I

I I I l l 1

I

105 Number o f cycles

I

1

1

1

1

lo6 N

Fig.3. Changes in coefficient of friction ( S = -19.2%) the running condition. Fig.3 shows the results of pairs of a relatively rough D disc and a smooth F disc (tested at s = -19.2%). As is apparent from the figure, the circumferentially ground discs showed a higher friction than both the obliquely ground discs and the axially ground discs. Furthermore, a t the initial stage of running, the discs of different hardness combination (760/320Hv) showed a higher friction compared with the pairs of equal hardness (760/760Hv). Especially, in Exps.A-3 and A-4, where circumferentially ground discs were used in the different hardness combination, considerably high friction, high surface temperature and severe wear were observed. Fig.4 shows the results of axially ground rough discs tested a t s = -19.2%. Although two surfaces were equally rough and the value of A was less than 0.5, the friction in

155

I

0.06

c

:

1 01

1 o4

I

1 I 1 1 1 1 1

I

a)

8-1, a / a

IZI 8-3, a / a

I

1

I 1 1 1 1 1

I

A-7, c l p

A C-3, O/P

1

s = -19.2%

I

I

I

I I I I I I

I

I

105 Number o f c y c l e s N

I l l l l l l

1 O6

Fig.4. Changes in coefficient of friction ( S = -19.2%) Exp.B-1 of equal hardness combination was not so high, and there was hardly change during operation. On the other hand, the friction in Exps.B-3 and B-4 of different hardness combination was high at first, then dropped gradually in the same manner as Exp.B-5 shown in Fig.3. The results tested at s = -3.6% are shown in Fig.5. As seen from the data in Table 1, the coefficient of friction increased to some extent with the change of the slip ratio from s = -19.2% to -3.6% in the combination of smooth discs (compare Exps.A-2 and A-5 with Exp.A-6). In Exps.A-7, B-7 and C-3, where relatively rough D disc and smooth F disc were combined ( A = 1.6 2.4), the difference in the friction among discs with different topographies became very small in contrast with the case of s = -19.2%. Figs.6 to 8 show the surface temperatures of discs during operation. The temperatures were measured on the track of D disc side with a faster peripheral velocity. Except for a few cases, the variation of surface temperature during operation was relatively small in each test, and the temperatures were in the range of approximately 100 130°C with rough discs and 75 95°C with smooth discs when tested at s = -19.2%. Of course, some differences were observed depending on the

0 104

105 Number o f c y c l e s

106 N

Fig.5. Changes in coefficient of friction (S = -3.6%)

-

-

“I

104

5x104 lo5 Number o f c y c l e s

5x105

lo6

N

Fig.6. Temperature of disc surfaces (S = -19.2%)

N

surface topography, and the axially ground discs generally showed a lower temperature

156

150

Y 01

a +.' m

01

8100 #

c,

q 9

01 U

I I

c-1

S=

a

5

0

1

a)

A-7

A

C-3

UI 8-7

c, 3

m W L

El00 42 #

.

aJ

c

-19.2%

v)

B-2

50 104

1

01 L

m

Y-

V

5x104 l o 5 Number o f c y c l e s

I I 5x105

v) 3

104

N

Fig.7. Temperature of disc surfaces (S 1-19.2%) than the obliquely ground discs or the circumferentially ground discs. Among the rest, the temperature was extremely high when the circumferentially ground rough D disc with higher hardness and the F disc with lower hardness were combined. In fact, in Exps.A-4 and A-3, the maximum temperature reached easily even 200°C 225°C immediately after the start of running as shown in Fig.6. In the experiments tested at s = -3.6%, however, the temperature increased a little but 75"C, and the differremained about 55 ence among discs became smaller or negligible as shown in Fig.8.

-

-

3.2. Changes in surface As shown in Table 1, some experiments of the pairs with different hardness were discontinued before N = lo6 cycles owing to the occurrence of surface damages such as rippling (i.e. corrugation), pitting and severe wear. In Exps.A-3 and A-4 of circumferentially ground discs, the disc surfaces became discolored noticeably, possibly owing to the extremely high temperature and the ensuing oxidation, arid it is not unimaginable that scuffing caused severe wear. In Exps.B-3, B-4 and B-5, where axially ground discs were used in the different liardness coinbination, pitting occurred a t an earlier stage on the slower F disc with lower hardness. This is consistent with previous work on the effect of asperity interaction

I

50

lo6

5x10" 1 Number o

5x105 cycles

I

10'

N

Fig.8. Temperature of disc surfaces ( S 1 -3.6%) pitting[7] which shows that the asperity interactions accompanied with sliding become severe when the directions of sliding and machining marks cross each other, and consequently the pitting life of axially ground discs is generally short compared with that of circumferentially ground discs. In the experiments where the same high hardness discs were combined, neither serious daniage like pitting nor scuffing occurred except for a slight discoloration on the track. AHv in Table1 shows the increase in microVickers hardness of disc surfaces after running. In Exps.A-3, A-4, B-3, B-4, B-5, B-6 and C-2 tested at s = -19.2%, work hardening was remarkable on the F disc surfaces with lower hardness. In the experiments tested at s = -3.6%, however, the increase in hardness was only a little as shown in Exps. A-7, B-7 and C-3. Figs.9, 10 and 11 show some of the profile curves, the three dimensional views and the photographs of contacting surfaces, respectively. When there is a large difference in the hardness between two surfaces, the roughness of the harder surface remains almost 1111changed and plays a dominant role on the running-in or the occurrence of surface damage and wear of the mating surface with lower hardness. Some results are shown in the figure (Exps.A-4, B-3, C-2, etc.). On the other hand, in the case of almost the same hardness

on

157

rD Before runnning

L

After runnning EXp.A-1

Before runnning

[r

rD

,

-

Before runnning

Exp.B-1

,

.

.

.

. . . . , . . _ _ . .

After runnning

-

I

,

Before iunnning

-

Exp.B-2

After runnning

Before runnning

-

Before runnning 7

After runnning Exp.B-5

L‘

-

Exp.B-3

Before runnning

!

After runnning

Before runnning

E ~ ~ . c After - ~ runnning D///////// F m Axial direction

:L

c,

l.Ornm

Fig.9. Profile curves of mating surfaces

Exp.B-5

After runnning

Circumferential direction

ET

158

EXp.B-1

EXP.B-3

Fig.10. Three dimensional views of contacting surfaces after riinning

.-.

1W m r n y 0 . 2 ~ ~

159

Fig.11. A pair of disc surfaces after running (Exp.A-4)

Fig.12. Contour maps of contacting surfaces (D disc)

I60

combination, the roughness has a tendency t o diminish on both sides when two surfaces are equally rough. However, when two surfaces with a large difference in the roughness were combined, the decrease in the roughness of the rougher surface becomes modest[8]. For example, in Exp.B-1 of equally rough discs (ground axially) , the asperities became blunt and the roughness diminished to some extent on both surfaces. However, in Exps.A-1, B-2 and C-1, where a relatively rough disc and a smooth disc were combined, the magnitude of surface profile curves of rougher disc hardly changed but only the tips of asperities flattened slightly. In order t o understand easily such a slight change in surface, the authors drew contour maps of contacting surfaces based 011 the digital data of surface profile measured threedimensionally. A few examples of them are shown in Fig.12, where depth levels are classified by shading (the original one was displayed by color). We can see the increase or the extension of higher flattened areas after running.

3.3. States of oil film formation Fig.13 shows some examples of the progress of oil film formation during operation. In most cases, immediately after the start of running, the voltage between discs showed nearly zero or a very low value owing to the

104 15

r

5x104 lo5 5x10' Number o f cycles N

lo6

5x1 0'

1 O6

f4 B-1, a / a

1 o4

5x104 lo' 5x10' Number o f cycles N

1 Number o f cyc l e s N

Fig.13. Changes in voltage between discs (Eab=OmV:contact, 15mV:separation)

161

metallic contacts, then the oil film came to be built up and the voltage rose gradually. However, as shown in the figure, the processes of oil film formation were considerably different according to the combination of discs and the running condition. In the combination of rough and smooth discs with equally high hardness, the voltage rose rapidly when they were tested at s = -19.2% (Exps.A-1, B-2 and C-1). Among the rest, the circumferentially ground one (Exp.A-1) showed a most rapid increase contrary t o the order expected from the friction measurements. Such a result may be explained partly by the formation of oxide film, possibly accelerated during operation. Also in the combination of equally rough discs, the progress was rather slow but the oil film came to be built up steadily (Exp.B-1). As shown in Fig.9, the total roughness depth (peakto-valley height) after running was still large enough although the crests of asperities flattened out, and thus it is supposed that the micro-EHL action[9] plays an important role in the oil film formation between such rough surfaces. Meanwhile, in the combination of discs with different hardness, the voltage rise was hardly observed or showed a very slow progress when the harder disc was rough and tested at s = -19.2% (Exps.A-3, A-4, B-6, C-2, etc.). Especially, in Exps.A-3 and A-4 of circumferentially ground discs, the voltage remained almost zero during the operation, and this corresponds well to the results of friction measurements . When a rough harder disc and a lower hardness disc are rotated in company with sliding, asperities on the harder surface penetrate into the lower hardness surface and plastic deformation due to the indentation and ploughing is repeated there, although the condition of asperity contacts varies with work hardening of the lower hardness surface and ensuing deformation of asperities on the harder surface. That is possibly the main cause for the high friction and the poor oil film formation in the different hardness combination. However, when they were tested at s =

-3.6%, owing to the less surface temperature rise, the states of oil film formation were improved evenly although some differences among discs were still observed ( Exps. A-7, B-7, C-3, etc.). 4. CONCLUSIONS

Using carburized alloy steel discs (760Hv) and thermally refined carbon steel discs (320Hv) ground circumferentially, axially, and obliquely, the effects of surface topography and hardness combination on the behavior of friction and the occurrence of surface distress were investigated under the lubricated rolling with sliding conditions of two fixed slip ratios (s = -3.6% and -19.2%). The main results are summarized as follows:

(1) Generally, the friction had a peak at first, then decreased gradually and settled down at a certain value. However, significant differences in the behavior were recognized depending on the combination of discs and the running condition. (2) The circumferentially discs showed a higher friction than the axially ground discs, and the obliquely ground discs showed an intermediate value between these two when they were tested a t s = -19.2%. (3) And then the discs of different hardness combination (760 / 320Hv) showed a higher friction compared with those of equal hardness combination (760/760Hv) at the initial stage and showed a gradual decrease with the progress of running. Especially, considerably high friction, high surface temperature and sever wear were observed with circumferentially ground discs. (4) By changing the slip ratio from s = -19.2% to -3.6%, the friction increased t o some extent with smooth discs. Although a little more higher friction was obtained with relatively rough discs, the effect of difference in the topography became very small owing to the formation of thick oil film.

162

(5) The mechanisms which caused such different behaviors in the friction were discussed based on the observed surface temperatures, the changes in hardness, the surface profiles, the states of oil film formation, etc. (6) Even when the friction decreased and the oil film came to be built up significantly, the changes in the magnitude of roughness were relatively small in the present experiments. Therefore, in order to understand such a situation, it is necessary t o take into account microtopographical changes of asperities which engage in contact.

REFERENCES 1. P. H. Dawson, J . Mech. Engng. Sci., 4 , 1

(1962) 16. 2. T. E. Tallian, ASLE Trans., 10, 4 (1967) 418. 3. S. Bair and W. 0. Winer, Trans. ASME, J. Lub. Tech., 104, 3 (1982) 382. 4. C.R.Evans and K.L.Johnson, Proc. Inst. Mech. Eng., 201, C2 (1987) 145. 5. K. L. Johnson and D. I. Spence, Tribology Int., 24, 5 (1991) 269. 6. D. Dowson, Proc. Inst. Mech. Eng., 182, Pt 3A (1968) 151. 7. K. Ichimaru, A. Nakajima and F. Hirano, Trans. ASME, J.Mech.Des., 103, 2 (1981) 482. 8. A. Nakajima and T. Mawatari, Proc. 6th Int. Cong. on Tribology, Budapest, Vo1.5 (1993) 290. 9. M.Kaneta and A.Cameron, Trans. ASME, J. Lub. Tech., 102, 3 (1980) 347.

Dissipative Processes in l'ribology / D. Dowson e[ al. (Editors) 0 1994 Elsevier Science B.V. All rights reserved.

163

Anti- wear Performance of New Synthetic Lubricants for Refrigeration Systems with New HFC Refrigerant. T.Katafuchi'

,

M. Kaneko'

,

M. Iinob

'Lubricants Research laboratory, Idemitsu Kosan Co. Ltd. 24-4 Angasaki-kaigan, Ichihara, Chiba, Japan 'Lubricating Oils Division, Idemitsu Kosan Co. Ltd. 3-1-1 Marunouchi, Chiyoda, Tokyo, Japan

The influence of new synthetic lubricants for use with HFC134a on the wear between aluminum-alloy and steel, which are used as mechanical parts in the compressor of automotive air conditioning systems refrigerated by HFC134a under severe lubricating conditions, was studied using a flat ring on flat ring friction apparatus equipped with a newly developed in situ wear sensor. It was clarified that a PAG type lubricant had excellent anti-wear properties compared with an ester type lubricant during the break-in stage for practical compressors with HFC134a..

1. Introduction The charge of refrigerant in automotive air-conditioning systems from CFClZ t o HFC134a is underway in compliance with the CFC phase-out program[l. 2,31. The main problem to be solved is the reduction of wear of mechanical parts in compressors in order to make this replacement. A sensor was developed for in situ monitoring of

the wear between aluminum-alloy and steel which are used asmechanical parts in compressors. The wear characteristics of ester and PAC type lubricants, which were recently commercialized as refrigeration oils for HFC134a, were evaluated using a flat ring on flat ring friction apparatus[41 equipped with the in situ wear sensor.

164

2. Test apparatus and lubricants 2. 1 Test apparatus

A rotating flat ring on a stationary flat ring type friction apparatus was used for the evaluation. This friction apparatus was adopted in order to reproduce the lubricating conditions in practical compressors, because the flat face to face contact keep the contact pressure constant during the wear test. The upper rotating ring was made of bearing steel, and the lower stationary ring was aluminum-alloy(Si:14%). The design of these rings is shown in Fig. 1.

contact pressure at 500 rpm with a continuous supply of gaseous HFC134a. The oil tempreature was kept at 50 "C. The applied load, frictional force, electric resistance between sliding surfaces under 15mV of applied voltage and 100 ohm of parallel resistance, and wear traced by the in situ wear sensor were measured during the wear test. 2.2 In situ wear sensor. Wear particles of aluminum-alloy, which are stably dispersed in the test lubricants, increase with test duration. Therefore, it is possible to trace the wear continuously by measuring the light transmittance of the test lubricant during the wear test, which changes with the concentration of dispersed wear particles.

Lower stationary ring Upper rotating ring aluminum - Alloy) (Bearing steel)

Fig. 1 The design of Test Ring

The test conditions were as follows. After gaseous HFC134a had been supplied to the test lubricant at 54!/hr for 30 minutes through a hole drilled near the center of the stationary flat ring, the wear test was conducted under the selected

i 6 t Fig. 2 The cornfiguration of the detector part of the wear sensor

The newly developed in situ wear

165

monitoring sensor utilizing an optical fiber cable was based on this concept. The configuration of the detector part of the wear sensor, which is set near the test specimen in the test lubricant during the wear test, is shown in Fig.2. 2.3 Test lubricants. Commercially available ester and PAG type refrigeration oils for automotive air conditioning systems with HFC134a were used for the evaluation. The properties of these lubricants are shown in Table 1. The base fluid of the PAG type lubricant is an end-capped polyalkyleneglycol. The ester type lubricant is composed of a di-pentaerythritol carboxylic acid

ester. Both lubricants can be completely dissolved in liquid HFC134a in compressor systems at 50 "C. I t was difficult to evaluate the difference i n lubricity between the ester and the PAG type lubricants from the load carrying capacity using the Falex Test, because these two lubricants gave almost the same failure load with the supply of gaseous HFC134a as shown in this table.

3. Resu I t s and discuss ion 3 . 1 Performance of the newly developed

in situ wear sensor. The relationship between aluminum content in the test lubricants analyzed by

Table 1 The properties of synthetic test lubricants Ester type Kinematic Viscosity

@ 40"C

PAG type

80.88

49.99

10.40

10.34

112

214

Total Acid Value (mgKOH/g)

0.01)

0.01)

Saponification Value (mgKOH/g)

337

5)

- 50)

- 50)

(mm2/s>

@ 100°C

Viscosity Index

Pour Point ("C)

Phase Separation Temp. ("C) HFC134aAubricant = 80/20 Failure Load by Falex Test (lbs.) with supply of gaseous HFC134a at 52! /hr

-4O),

80 (

1250

- 40 )

, 80

1080

I66

an ICP emission spectrometer and the transmittance measured by the developed wear sensor is shown in Fig. 3. .00

80

R

=-

0.926695

3 . 2 Test results under stepped-up load

conditions. Average friction coefficient, electric resistance and light transmittance versus contact pressure are shown in Fig.4, 5 and 6, respectively, The wear tests for both PAG and ester type lubricants were

60 40

lKt

20 0 ' 0

I

I

3

6

I

I

9 12 A1 content (ppm)

I P

Fig. 3 Relationship between A1 content of test lubricants and light transmittance

0

A good correlation was obtained as shown in this figure. It can be seen that the n e w l y d e v e l o p e d w e a r s e n s o r is satisfactory for practical use.

1.5 3.0 Contact Pressure (MPa)

Fig. 5 Relationship between average electric resistance (R) and contact pressure 100 -

0.2 -

T (%)

,u

50 -

0.1 -

1 PAG 0. 0

I

Fig. 4 Relationship between average friction , ) and contact pressure coefficient ( u

I

0

I

1.5

Fig. 6 Relationshiup between average light transmittance (T) and contact pressure

I

3.0

167

conducted by stepping up the load every 3 minutes. The ester type lubricant decreased the electrical resistance and light transmittance with an increase of the friction coefficient at low contact pressure compared with the FAG type lubricant. These test results were noticeably different from those from the Falex test. The load carrying capacities of these two lubricants by Falex test were almost the same, as mentioned before. 3. 3 Test results under constant

load conditions. Wear tests were conducted at 1.2 MPa for the PAC type lubricant and 0.75 MPa for the ester type lubricant f o r 3 minutes, respectively.in order to determine the wear behavior at an early stage of the wear test at low contact pressure. The selected load was the load at which the wear started to be observed and the average light transmittance of the test lubricants decreased to about 85% of the new oil. These test conditions were considered to be reasonable to evaluate the wear characteristics of refrigeration oils at the break-in stage of practical compressors. The relationship between friction

OL OL

o w * 0

.

I

Fig. 7 Relationship between friction cient ( D O , electric resistance light transmittance (T),and duration with the PAC type

0

I

1 Test duration 2 (mid3

1

2

coeffi (R) and test lubricant.

3

Test duration (min)

Fig. 8 Relationship between friction coeffi cient (D),electric resistance (R) and light transmittance (T),and test duration with the ester type lubricant

coefficient, electric resistance and light

168

transmittance, and test duration for each test lubricant is shown in Fig.? and 8. The P A G type lubricant slightly decreased the light transmittance, in other words, the wear was slightly increased with decrease of friction coefficient and rapid increase of the electrical resistance after 1 minute from the start of the wear test. The wear did not increase after that and the friction coefficient became stable at about 0.03, which indicated a regime of fluid lubrication. Meanwhile, the electrical resistance reached a maximum, which meant that the sliding surfaces were electrically insulated by the oil film. On the other hand, the ester type lubricant continuously increased the wear with a slight increase of friction coefficient from 0.12 to 0.14 over the test durat ion. The electrical resistance stayed around zero, which meant that metal contact continously occurred between the sliding

(1)

-.. '

I

.. . ,:

New Specimen

'-"-

.

..

.

. .

. ...... .

i

.

.

.

- 1

(2) Tested with PAC type lubricant

'- T .I

-

:

1

.._

-- .

'(3) Tested with &ter type lubricant

Fig. 9 Surface roughness of test specimens

surfaces. In order to clarify the difference in wear behavior between PAC and ester type lubricants, the surface roughness of the aluminum-alloy test specimen after the wear test at constant load for 3 minutes were measured in comparison with that of a new specimen, as shown in Fig.9. Micro scuffing was partially observed on the sliding surfaces tested by ester type

Table 2 Elements on the sliding surfaces

I Smoothly worn surface I

Micro scuffed surface

A1 on steel

1.5

1.4

F e on Al-alloy

2.8

2.8

A1 on steel

1.5

1.7

101.1

F e on A1 - alloy

4.7

3.0

12.5

PAG type lubricant ~~~

107.4

Ester type lubricant 15.2

I

169

lubricant. On the other hand, the PAG type lubricant did not generate micro scuffing, the surface being perfectly smooth like a new specimen. Furthermore, the concentration of related elements on the sliding surfaces were analyzed using EPMA. The analytical results are shown in Table 2 as a ratio of the element on the sliding surface to that of a new specimen. A larger amount of metal transfer between the micro scuffed surfaces was observed, compared with smoothly worn surfaces. I t indicated that the PAG type lubricant had excellent anti-wear performance at the break-in stage of practical compressors for HFC134a compared with the ester type lubricant.

4. Conc I us i on

The summary and conclusions obtained from these test results are as follows: (1) The newly developed in situ wear sensor has a good performance for practical use. (2) The in situ wear behavior of refrigeration oils at the break-in stage of practical compressors for HFC134a can be precisely traced by using a flat ring on flat ring friction apparatus equipped with the sensor.

(3) The PAG type lubricant is considered to have excellent anti-wear performance compared with the ester type lubricant at the break-in stage of practical compressors for HFC134a.

Reference

1. Montreal Protocol on Substances that Deplete the Ozone Layer. 1987 2. Helsinki Declaration on the Protection of Ozone Layer. 1989 3.Copenhagen Agreements and Amendments to a Phase-out of Ozone Depleting ubstances. 1992 4. Subcommittee on Wear, Lubrication Fundamentals Committee, ASLE (eds.) . Friction and Wear Devices. 1976. 223 5.H.A.Hanna and F. Shehata. J. of Lub. Eng. Vo1.49. No.6. 1993. 473

This Page Intentionally Left Blank

SESSION IV MICROSCOPIC ASPECTS Chairman:

Dr I L Singer

Paper IV (i)

A Molecularly Based Model of Sliding Friction

Paper IV (ii)

Friction and Dielectric Materials : How is Energy Dissipated?

Paper IV (iii)

Friction Energy Dissipation in Organic Films

-

This Page Intentionally Left Blank

Dissipative Processes in Tribology / D. Dowson el al. (Editors) 0 1994 Elsevier Science B.V. AU rights reserved.

173

A molecularly-basedmodel of sliding friction J. L. Streator

G. W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332-0405 Kinetic friction is well known to be a dissipative process. The key to understanding friction, then, is to discover the mechanisms by which mechanical energy is dissipated when one body slides upon another. Some recent reports in the literature have discussed the source of dissipation in the context of several models of atomic interaction, including the Frenkel-Kontorova(FK) and the Independent Oscillator (10) model. In the current work, aspects of the FK and I 0 models are incorporated to model the interaction between a solid asperity and an adsorbed interfacial film. As predicted by the I 0 and FK models, calculations indicate that the existence of an appreciable dissipative component of friction depends on the occurrence of an instability in the configurations of the molecules. It is also found that this dissipative component is influenced by whether or not there exist mechanisms for energy to continually propagate away from the interface. When such avenues exist, the friction force is found to be significantly greater than under conditions where the energy can return to the interface. To provide additional insight into the mechanism of dissipation, an analogy is made between the discrete, molecular model and that of a string on an elastic foundation. The conditions governing the propagation of waves with the continuous model suggest that significant dissipation requires a frictional instability. 1. INTRODUCTION Kinetic friction has long been recognized as a dissipative process, but the mechanism by which mechanical energy is lost remains a point of investigation [l]. It is known that the kinetic friction between two bodies results from interactions that occur on the atomic level. It follows, therefore, that the elucidation of frictional dissipation should result from consideration of atomic and/or molecular interactions. Probably the first atomic account of kinetic friction was due to Tomlinson [2]. Tomlinson proposed that when an atom of one body moves tangentially past an atom of another body, the lateral component of force may be different between approach and separation. The non-symmetry in this lateral force arises because, at a certain point, the atoms attain a position of unstable equilibrium. This condition leads to the sudden motion of the atoms toward the nearest stable equilibrium configuration. Tomlinson assumes that this "jump" phenomenon irreversibly transforms all of the stored energy into molecular kinetic energy or heat, thereby accounting for

mechanical losses. Thus the mechanism involves the nonadiabatic motion of atoms. Tomlinson's mechanism appears in the context of the Independent Oscillator (10) model [3, 41, shown schematically in Fig. l(a). Here the atoms of one surface are modeled by masses which are independent of one another but are harmonically coupled to a rigid base. The influence of the opposing surface is modeled by a sinusoidally varying force which translates quasistatically in a direction parallel to the interface. Although drawn vertically, both the interaction force and springs act tangential to the interface. Above a certain magnitude of interaction force the model atoms are displaced by the force in the direction of sliding until they reach a point of unstable equilibrium. Further translation of the upper surface causes the atoms to suddenly spring back. Assuming that this release of energy is irreversible, the instability in the I 0 model accounts for the Occurrence of friction. If, however, the interaction force is low, it can be shown that the molecules remain in positions of stable equilibrium throughout the sliding process. In this case the average tangential force vanishes so

174

L

k

/// Figure 1 : Previous models of dissipation. (a) Independent Oscillator (10) model. (b) Frenkel-Kontorova (FK) model. that the dissipcltive component of friction force is zero [3.4]. Similar effects can be seen in the FrenkelKontorova (FK) [S] model, shown schematically in figure I@). In this model, the atoms are modeled In the by a set of harmonically coupled masses. classical treatment, the system is analyzed for static configurations which lead to instability in the positions of the model atoms. It has been shown that this instability occurs at a critical value of the interaction force between the two surfaces. In the quasi-static analysis, the occurrence of this instability has heen identified as the source of friction. Since the masses of the surface are mutually coupled, the system behavior is influenced by their average spacing as compared to the wavelength of the translating force [ 6 ] , which is taken to be the lattice spacing of the upper surface. It has been found that when the ratio a/b is irrational, yielding incommensurate lattices, the friction force is considerably less than when a/b is a rational number [3.4].

In both the (quasistatic) FK and I 0 models. the Occurrence of friction depends on the existence of unstable equilibrium positions during the translation of one body over another which leads to sudden displacement or "plucking" of the atoms. In contrast. Hirano and Shinjo (1990) [7] performed calculations based on the Morse potentials for selected metals and found that, for these materials, such an instability should not occur. In a separate investigation [8] the authors analyzed a one-dimensional FK model with kinetic energy terms and found that friction could arise under dynamic conditions in cases where the quasistatic analysis would predict no instabilities of the atoms. The authors associated the Occurrence of friction with the transformation of macroscopic kinetic energy into internal motions of the body. Sokoloff (1992) [9] studied the interaction of a translating sinusoidal force with a layered crystalline lattice, modeled by a collection of springs and masses. In this investigation, the author emphasized the importance of internal damping on the generation of friction force for any finite-sized crystal. The Same author also compared frictional forces between commensurate and incommensurate sliding and found the former values to be 12 orders of magnitude greater than the latter [ 101. In the present work. we investigate a simple model of surface interaction which incorporates aspects of the both the 10 and FK models. I n contrast to previous studies, however, we consider directly the role that energy propagation plays i n influencing the kinetic friction. The motivation for this investigation comes from the observation that. when two macroscopic bodies slide. there is an appreciable amount of energy which propagates away from the interface in the form of acoustic waves. The question arises how this mechanism of energy loss affects the magnitude of friction. 2. MOLECULAR MODEL The physical system to be modeled is schematically shown in Fig. 2a. The upper surface is a hard asperity which slides against an adsorbed film of an opposing surface. The mathematical model is shown in Fig. 2b. Since the molecules of the film are anchored to the surface, the masses of

175

vo 0

(a)

*

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

into the contacting bodies, supporting structures. etc.

0

2.1 Governing Equation

With reference to Fig. 2b. the equation of motion for each of the masses of the model is written

with

Q(x, ,I ) = [H (X, - x, - V,I)- H ( X, - (X, + L ) - V,I )] x

Figure 2: Model of interface. (a) Asperity sliding on adsorbed film. (b) Mathematical model. the model (Fig. 2b) are connected to springs which resist absolute displacement relative to the base. The set of molecules are also assumed to prefer a certain equilibrium spacing which is modeled by the coupling between the masses. Since the upper surface is assumed to be that of a hard solid, it is modeled by a rigid sinusoidal force. All of the couplings are harmonic in nature. Following Shinjo and Hirano (1993) in their application of the FK model [8], we integrate the dynamical equations of motion for each of the masses. To directly investigate the role of energy propagation. the extent of the adsorbed film is taken either to be ( I ) just large enough to cover the contact length over the sliding distance, or (2) effectively unlimited in each direction. In practice, this latter condition means that the surface extent is sufficiently large that, over the duration of the sliding. elastic waves propagate outward and do not return. Physically, this feature is used to model what happens when real 3D bodies experience sliding Contact. In this case, acoustic waves generated at the Contacting interfaces propagate away from the interface region

where m is the mass, k and ke are the spring constants (Fig. 2b), ui is the displacement of the ith mass from its unforced static equilibrium position, xi is the absolute position of the ith mass, Q(xi.1) is the interaction force on the ith mass. Qo is the force amplitude, V, is the sliding velocity, xo is the initial position of the left end of the contact region. L is the length of the contact, a is the wavelength of the periodic force. The function Hc.>is the unit step function. From consideration of the arguments of the step function. it is seen that the interaction force translates from left to right with speed Vo. We can nondimensionalize the governing equation by selecting appropriate length. time and force scales. The parameter b, which gives the equilibrium spacing between the molecules when there is no interaction force (Fig. 2a) is chosen as the length scale, b/Vo is chosen as the time scale, and kb is chosen as the force scale. We also make the following definitions ,41

=by,

t=--'S b

v,

co = b

g

The parameter co has the dimensions of velocity and is nominally the velocity of sound for the medium [ 111. Using the above definitions we have

176

=i'

2.2 Simulation Procedure Equation (2) governs the motion of the ith particle. To study the frictional force we integrate equation (2) for all of the particles in the system using the classical 4th-order Runge-Kutta method [12]. The integration is begun from a position of static equilibrium, which itself is determined by iterative relaxation from an assumed configuration. Starting from this initial static equilibrium position, we sum the interaction force. Q. over all of the particles at each time step. This summation provides the friction force exerted by the translating force field upon the lower surface. The time step was selected to be 0.02 b/co As a check on computational accuracy. we compared the power input to the stored energy of the system. The rate of work performed on each particle is calculated by forming the product of the interaction force on the particle and the particle velocity. After summing over all of the particles in the system at each time step. the power was Integrated in time using Simpson's Rule to provide the total work done by the interaction force at a given instant. The total energy of the system was calculated by summing the kinetic energies of all of the system particles along with the potential energies stored in each of the springs. It was found that there was good agreement between the calculated work done and the energy stored in the system. At the end of the simulation. the relative difference in the two quantities was consistently found to be less than 0.1% .

3.0 RESULTS AND DISCUSSION The governing equation (2) was integrated for a number of parameter combinations. The following values are applicable to all of the calculations to be discussed: a/b = 1, Vdco= 0.02, k& = 0.8. The ratio a/b defines whether or not the interface is commensurate, and the ratio Vdco is the ratio of sliding speed to the nominal acoustic wave speed. With the value of this ratio set to

d

W

OS3

v3

y

d b rn I J

l1 -

-I

Qo/kb=0.05 L h - 5 Qo/kb = 0.1 Qo/kb=0.2

0.2 0.1

d

z 0 F

0.0

u

U

-0.1

-0.2 0

1

2

TIME (Vot/b)

3

Figure 3: Frictional stress vs. time for three force amplitudes. 0.02, the sliding process is slow compared with the speed of relaxation in the interface. 3.1 Effect of Interaction Force Figure 3 shows the (dimensionless) frictional stress vs. (dimensionless) time for three values of the interaction force amplitude and for a (dimensionless) contact length of L/b = 5 . The extent of the model surface film was taken to be unlimited in each direction. The frictional stress is defined to be the friction force, F, per dimensionless contact length L/b, normalized by kb. Thus frictional stress has the form F/kL. Since there is one particle per b spacing. the frictional stress gives a measure of the force per particle in the conlact region. Similarly, one unit of dimensionless time corresponds to the force field translating the equilibrium distance. b, between the molecules. As observed in Fig. 3. For Q&b = 0.05, the frictional stress is nearly sinusoidal being slightly distorted due to some displacement of the masses in the direction of sliding. Moreover, the

177

n

d

w

-

Qo/kb=0.05 ~ b - 5 - Qo/kb=O.l 0.03 - - Q$b ~ 0 . 2

*

W

u g 0.02 zw

u

p

I

-

0.01 -

2

2

0.00

I

I

Figure 4: Kinetic energy within contact for three force amplitudes. amplitude of the frictional stress (0.05) is equal to the amplitude of the normalized interaction force. Since the force profile is nearly symmetrical with respect to the origin, the average frictional stress over the sliding distance is quite small (.OOOOl). When the normalized force amplitude, Q&b, is doubled to 0.1, the friction trace shows a much greater deviation from sinusoidal behavior. Once the frictional stress reaches its peak, there is a faster drop to the minimum value after which some high-frequency oscillations are observed. The average frictional stress remains small but increases to 0.001. When the interaction force is increased to 0.2 kb. there is a sudden drop in the friction force followed by substantial oscillations in the force. In addition, the average frictional stress increases to 0.03. The degree of excitation in the interface can be measured by the amount of kinetic energy generated. Fig. 4 shows the amount of kinetic energy generated in the contact as a function of time. The energy is normalized by kb2 and the dimensionless sliding length L/b. For the case of

Q@b = 0.2, substantial kinetic energy is produced after each drop in the force (see Fig. 3). When the interaction force is reduced to 0.1 kb, the amount of kinetic energy generated is more than order of magnitude less. When Q$b = 0.05. the amount of kinetic energy produced is negligible, as indicated by the horizontal line at the origin. The differences in kinetic energy observed in Fig. 4 compared with the frictional profiles of Fig. 3, indicate the role of frictional instabilities in exciting the interface.

3.2 Effect of Surface Extent Figure 5a shows the frictional stress vs. time for the case of Q&b = 0.5 and L/b = 0.5. Here the surface is unlimited in extent (in each direction). In other words, a sufficient number of surface particles is selected to insure that. during the duration of the simulation, no waves originating from the interface can return to the interface upon reflection from the surface boundaries. As observed in the figure. the friction force demonstrates the plucking instability identified in Fig. 3. Again there is a sudden drop in the friction force followed by substantial oscillations. Fig. 5b shows the (normalized) energy delivered to the surroundings-that is the amount of energy, potential plus kinetic. possessed by the particles ourside of the contact region. The energy is normalized in such a manner that the graph provides the amount of energy lost per particle in the contact region. The figure shows that the energy in the surroundings accumulates with time. As the contact region translates to the right (see Fig. 2b), the particle at the left edge of the contact eventually leaves the contact region while, at the same time, a particle enters the contact from the right. If the particle leaving the contact has been excited by plucking. then its departure will generally result in a greater loss of energy from the contact than is gained from the particle which enters. The key observation here is that once the energy is lost from the contact, it does not return: the energy accumulated outside of the contact increases almost monotonically in time. This steady increase in the energy of the surroundings indicates that energy is continually propagated away from the interface. We compare the foregoing results to those of Fig. 6. In Fig. 6a, frictional stress is computed for

178

the same conditions as those in Fig. Sa. except thal thc surface extent is limited to 10 particles. This number of particles is just enough to cover the contact region for the duration of the simulation (4 units of time). As observed in the figure. the friction reaches an initial peak as in Fig. 5a, but the "stick-slip'' cycle is not repeated to the same degree. Instead. following the initial slip. the force essentially oscillates between positive and negative values. with an average near zero. In fact if we compare the average friction forces in Fig. Sa and Fig. 6a, after time = 1 (i.e.. after the initial stickslip). we find a substantial difference: Fig. 5a corresponds to an average frictional stress of 0.194. while Fig. 6a gives a negative value of -0.03 Clearly, then. the surface extent has a profound effect on the friction force. Consideration of the energy explains the foregoing friction behavior. Figure 6b shows the energy accumulated in the surroundings for the sliding conditions of Fig. 63. In this figure it is observed that the energy of the surroundings initially rises but afterwards does not appreciably increase. This result contrasts that of Fig. Sb i n which the energy lost to the surroundings steadily increases with time. When the surface is unlimited in extent. the energy is continually lost by propagation to the particles surrounding the contact, thus providing a source of dissipation. On the other hand. when the extent of the system is limited. energy does not continually flow out of the contact and dissipative effects are small. 3.3 Effect of Contact Length The role of energy propagation can also be seen when the contact length is varied. Figure 7a shows the amount of energy, per particle in thc contact region. that goes to the surroundings for several values of the contact length: L/b = 5. 10.20, and 40. The surface is of infinire extent. As the contact length increases. the amount of energy lost (per particle in the contact region) decreases, indicating that there is proportionately more energy delivered to the surroundings when the contact length is small than when it is large. This result is expected since, for a shorter contact, a greater percentage of the particles are at the boundary of the contact region. The longer contact length also affects the nature of the friction force. as observed in Fig. 7b.

-0.5 0

3

1

4

TIME~V 0t/b) Figure 5a: Frictional stress vs. time for Q J k b = 0.5 and L/b = 5. n

stf!

0.6

0.5

Pi

d

3

0.4

rn 0

b

cl

0.3

0.2

$(

u

g 0.1 z W

I

0

1

I

I

I

4

T I M E ~ 0Vt/b; Figure 5b: Energy to surroundings for QJkb = 0.5 and L/b = 5 .

179

This figure provides the friction trace for the same interaction force amplitude as i n Fig. Sa. but with a dimensionless contact length. L/b = 40 instead of L/h = 5 . Initially the friction peaks. as in Fig. Sa. but after the sudden drop, the friction force remains low. Since there is a greater proportion of energy which remains in the contact, the amount of dissipation is less and the average friction is small. Figure 8 shows the time average frictional stress as a function of contact length for several amplitudes of the interaction force. The average is taken over the duration of the sliding which is 3 units of time. In each case the frictional stress is observed to decrease with increasing contact length. The consistency of the trend provides further support for the previous conclusion--namely that the magnitude of the average frictional stress is influenced by the tendency for energy to propagate away from the contact. With a smaller contact region, the energy is more readily lost to the surroundings and the average frictional stress is greater.

3.4 Continuous Approximation Inspection of the governing equation (2) reveals that an analogy can be made between the model of Fig. 2b and that of a string on an elastic foundation. In this regard, we introduce u(x.1) as the transverse string displacement and make the following associations:

0.7

3b Y

R

0.3

b m A

0.1

< z 0 i?

-O.l

W

2

-0.3 -0.5

I

I

I

I

0

1

2

3

TIME(V0t/b)

Figure 6a: Frictional stress vs. time for finite surface extent.

0.5

d

at-

m

3

0.4 L

0

b 0.3 b

-

-

0.2

-

-

8 A

We also let

surface extent: 10 particles

0.5

m

a4

ii, e 7

2 Q#b=0.5 L/b=5

Using the above definitions, equation (2) becomes

a h c, ?+ a% pu = (I

--

at?

as-

0.0

0

(3)

1

3

4

TIME t V 0 t/b) Figure 6b: Energy to surroundings for finite surface extent.

I80

Equation (3) is recognized as the governing equation for a string on an elastic foundation. There are two characteristic homogenous solutions to this equation, given by [ 131:

I

sw

d

I

1

0.4

d

3

cn

0.3

0 where w is the frequency of vibration, y is the wavenumber. and 7 is magnitude of the wavenumber when the wavenumber is imaginary. Equation (4a) represents a propagating solution and applies when w2 > p. Conversely. equation (4b) represents a non-propagating solution and the applies when w2 5 p. Recalling that 0 = conditions on w indicate that energy will propagate away from the contact region when frequencies are excited that exceed some critical value characteristic of the system. This result indicates that continuous energy loss to the surroundings requires some excitation at the interface. We expect therefore that a plucking instability should be associated with the Occurrence of dissipation and should increase the friction force. We observed this effect in Fig. 3. When Q&b = 0.05 and there is no plucking. the average frictional stress is very low (.ooOOl). On the other hand. with Q&b = 0.2 there was a clear instability and the average frictional stress was markedly higher (.03). Thus a fourfold increase in the interaction force increased the average frictional stress by more than three orders of magnitude.

urn.

3.5 Relation to Other Mechanisms It was shown in previous sections that the magnitude of the dissipative component of friction is influenced by the efficiency with which energy is propagated away from the contacting interface. We now discuss this result in light of other mechanisms of dissipation identified in similar studies. Sokoloff (1992) [HI finds that internal damping within a (finite) crystal lattice is critical for the Occurrence of frictional losses at the interface. In fact, for a purely elastic model. the dissipative component of friction is found to be virtually zero. When damping is present, the elastic waves which emanate from the contact return to the interface

EE-

rn 0 cl

0.2

*u

d 0.1

W

zW 0.0 0

T~ME(V 0t i )

3

Figure 7a: Energy to surroundings for variable contact length.

m

cn

0.3

cn




,

6 0.0 0

I

10

1

,

20

I

30

I

I

40

CONTACT LENGTH (L/b) Figure 8: Average frictional stress vs. contact length for several force amplitudes. difference between the two forms of work, therefore, reveals the component of friction which is not associated with energy changes in the system. Hence, there are two components of dissipation: one component of dissipation depends on the flow of energy away from the contact to the rest of the system, while the other occurs even when no net energy is lost. Our concern here is upon which of these mechanisms is predominant. In Fig. 9a. it is seen that the bulk of the frictional loss (i.e.. the macroscopic work) is associated with the increase in the energy of the system (i.e., the net work). The same result is observed in Fig. 9b, where the extent of the surface is limited to 10 particles. Hence, while frictional losses do not necessitate the flow of energy from the contact region, they are greatly enhanced by it. 4.0 CONCLUSIONS A simple model of frictional contact was presented which focuses on the energetics of sliding at the molecular scale. The model. which

182

macroscopic work

0.9

2 0.8

-

9

0.7

-

0.6

-

z0 0.5

-

w

W

n 0.4 d 0

*

contains aspects of the I 0 and FK models. was used simulate the frictional interactions belween a rigid asperity and an adsorbed film. The rigid asperity was modeled as a translating sinusoidal potential. while the adsorbed film was modeled by a set of harmonically coupled masses which were attached by springs to a rigid base. The frictional behavior of the system was determined by numerical integration of the equations of motion for each of the model molecules. Three key observations werc made:

-

0.3

-

0.2

I I

0

I

I

I

1

4

3

1

T I M E ~ 0Vt/b)

Figure 9a: Macroscopic work vs. total work done on system. 0.5

~

~- 1

- 1

1

1

e W

W

These three observations can hc attributed to the same mechanism: energy propagation away from the region of conhct. First, an instability in the motions of the molecules is needed to excire the molecules in the interface. Second. for significant dissipation to occur, an avenue is required through which energy can be lost from the contact. Third. when the contact length is large, there is less availability for energy to propagate away from the interface, yielding lower frictional stress. Although the results presented here correspond to a limited number of cases investigated by simple model, they provide evidence that frictional losses depend, to a great extent. on the way in which energy propagates away from the contact.

Q0/kb = 0.5 L/b=5

9 -

0.3

z n 0.2

surface extent: 10 particles

0

d 0

*

0.1

0 .o 0

I

I

1

2

1

TIME(V 0t/b)

The average frictional stress was found to hc much higher when the surface was unlimited in extent than when it was finite. The average frictional stress decreased as thc lengrh of contact increased.

- net work - - macroscopic work

2 0.4

An appreciable dissipative component of friction was found to occur only when the interaction force was of sufficient magnitude. This effect corresponded with the onset of an instability in the positions of the model particles.

1

3

Figure 9b: Work comparison for surface of finite extent.

4

ACKNOWLEDGMENT The author would like to thank the Narional Science Foundation for support of his work.

183

REFERENCES 1. Tabor, D., in Fundamentals of Friction: Macroscopic and Microscopic Processes, I. L. Singer and H. M. Pollock, eds., NATO AS1 Series, vol. 220, Kluwer Academic Publishers, The Netherlands, p. 3 (1992).

2. Tomlinson, G. A., Phil. Mag., vol. 7, n. 46, p 905 (1929). 3. McClelland, G. M., in Adhesion and Friction, M. Grunze and H. J. Kreuzer, eds., Springer Series in Surface Science, vol. 17, Springer Verlag, p. 1 (1989). 4. McClelland, G. M., and Glosli, I. N., in Fundamentals of Friction: Macroscopic and Microscopic Processes, I. L. Singer and H. M. Pollock, eds., NATO AS1 Series, vol. 220, Kluwer Academic Publishers, p. 405 (1992). 5. Frenkel Y. and Kontorova T. ,Zh. Eksp. Teor. Fiz., vol. 8, p. 1340 (1938).

6. Peyrard M., and Aubrey, S., J. Phys. C: Solid State Phys., vol. 16, p. 1593 (1983).

7. Hirano, M. and Shinjo, K., Phys. Rev. B. , vol. 41, n. 17, p. 11837 (1990). 8. Shinjo, K., and Hirano, M., Surface Science, vol. 283, p. 473 (1993). 9. Sokoloff, J. B., J. Appl. Phys., vol. 72, n. 4, p. 1262 (1992). 10. Sokoloff, J. B., Phys. Rev. B., vol. 42, p. 760 (1990). 11. Tabor, D., Gases, Liquids, and Solids, 3rd edition, Cambridge University Press, p. 181 (1990). 12. Gear, W. C., Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, New Jersey, p. 47 (1971). 13. Graff, K. F, Wave Motion in Elastic Solids, Ohio State University Press, p. 51 (1975).

This Page Intentionally Left Blank

Dissipative Processes in Tribology / D. Dowson et al. (Editors) 0 1994 Elsevicr Sciencc U.V. AU rights reserved.

185

FRICTION OF DIELECTRIC MATERIALS: DISSIPATED ?

HOW IS ENERGY

B. Vallayera, J. Bigarrea, A. Berrouga, S. Fayeullea, D. Treheuxa, C. Le Gressusb, G. Blaisec Laboratoire Matiriaux-Mtcanique Physique URA CNRS 447 Ecole Centrale de Lyon, 69131 Ecully FRANCE

a

CEA-DAM, Centre d'Etudes d e Bruykres-le-Chatel BPI 2 91 680 Bruykres-le-Chatel FRANCE

b

Laboratoire de Physique du Solide Universitt Paris Sud 91405 Orsay FRANCE Abstract

Storage and dissipation of energy in a dielectric material are discussed using the results of the space charge physics. In this model, polarization of the medium is related to trapping of electrical charges in the lattice. This allows the storage of high amount of energy (5eV or more per charge) which, when dissipated, can lead to transfer to the phonon bath, creation of defects, exoemission or even fracture. During friction of dielectrical materials, electrical charges and sites of trapping are created. Therefore, the space charge physics can be applied to explain some aspects of t h e tribological behavior of materials such as ceramics or polymeres. Examples are given using experimental results obtained with sapphire.

1. INTRODUCTION It has been guessed for a long time that electrical and mechanical properties of die lect rical m ate rials are close 1 y related. Experimental evidences have been provided

to correlate electrical breakdown and fracture of insulators. The role of electrical Phenomena has also been underlined during friction of ceramic or polymer mate rials: t riboe lectrification and electrical component of the adhesion force

186

have been partly recognized. From a macroscopic point of view, the theory of elasticity in which forces acting at a point are determined by local deformation, is generally not applicable in dielectrical crystals as soon as it is necessary to take into account the polarization of the medium (i.e. the electrical fields acting on particles) in determining the equilibrium of a point. Therefore, a microscopic approach can provide useful guidelines to understand some mechanical behavior. In order to understand the mechanisms of energy dissipation in dielectrics, we propose to apply the new concepts of the space charge physics (1,2). The mechanisms of storage and of dissipation of energy are related to electrostatic interactions: polarization of the material and displacements of electrical charges. Polarization is quickly achieved and increased during friction because of the built-up of a space charge in the material owing to trapping of charge carriers. This trapping on defects already present in the material before testing or created during friction allows the storage of very high amount of energy (5 eV or more per charge) which, when dissipated, can lead to catastrophic failure. Experimental results obtained on single crystal alumina slid against itself in dry conditions are used to illustrate the discussion of t h e d i e l e c t r i c a l mechanisms. X-ray irradiation is used to modify the dielectrical properties of the superficial layers. Results are discussed in view of the space charge physics model. 2. STORAGE AND DISSIPATION OF ENERGY IN A DIELECTRIC MATERIAL: SPACE CHARGE P H YS I C S

A major characteristic of a dielectric material is that electrical charges can be trapped in some sites of the lattice, leading to the appearance of local intense electrical fields. Therefore, polarization of the medium can be induced locally around these charges. Its value can be calculated owing to microscopic determination of the local electrical fields (1,2). Generally, trapping of charges is achieved in a crystal as soon as the polarisability (from a microcoscopic point of view) or the permittivity (from a continuum point of view) of the medium is modified. This variation can result from the presence of defects (vacancies, interstitials, chemical impurities, dislocations, grains -boundaries...) (1). The possibility of self trapping of electrons by their own potential (polarons) has also to be considered (3-4). The potential energy of these traps has been estimated to range from a few meV to some eV. The total energy of polarization stored around a trapped charge is much greater and can reach 5 eV or more (2). This energy i s mainly stored as mechanical energy in the lattice all around the trapped charges (displacement of the ions from their equilibrium site). Since storage of energy is linked with trapping of charges, detrapping leads to dissipation. Relaxation of t h e mechanical energy associated with trapped charges occurs through a two step process. First, charges are detrapped. This does not release a lot of the stored energy. This detrapping can be obtained for example if a critical number of trapped charges (i.e. a critical value of the local electrical field) is reached. Then, the lattice around the location of t h e charges g e t s depolarized, i.e. comes back to its equilibrium configuration. The

187

relaxation process has been theoretically described owing to a many-body universal model involving the whole dielectric material (5-7) and implies a transfer of the polarization energy to the phonon bath. Dissipation of the mechanical energy corresponding to this transition can lead to breakdown of the material through flashover with treeing, thermal shock waves or exoemission. 3. APPLICATION OF THE SPACE CHARGE PHYSIC TO FRICTION AND WEAR PROCESSES

In order to apply the space charge physics to study the tribological behaviour of dielectric materials (ceramics, polymeres ...), it is necessary to show that electrical charges and sites of trapping do exist during friction experiments. 3.1 Electrical charges On figure 1 is shown the contact area after friction of a diamond tip against a quartz sample. The micrograph has been obtained in a SEM with low voltage. Electrical charges are clearly detected, leading to a potential of several hundreds of volts (8). This kind of phenomena, i.e. the charging of originally uncharged materials when brought into contact has been recognized for a long time and called triboelectrification. It has been assumed to be due to charge transfer by tunneling from delocalized states in the metal or from extrinsic states in the forbidden gap of the dielectric to extrinsic states of the counterface dielectric (9). This has been widely studied for various materials (1 0-14). During tests performed with polymers in contact with metals, surface charge density of some 10-7 C.cm-2 has been

measured (1 0,ll).Electron transfer has also been observed during contact between ceramics (mica-silica ): charge density of 10-6 C.cm-2 has been detected (1 5). When contact and friction are achieved in a polar environment, electrification can result from the formation of an electrical double layer in the interface (1 6,17).This has been observed even when the interfacial polar film is very thin, e.g. adsorbed water layers on surfaces (18). When surfaces in contact are separated, electrical charge of about 10-7C.cm-2 has been measured on various polymeric materials (1 9,20). When sliding occurs, further electrification can result from the exoemission of charged particles. Emission of electrons and ions (positively and negatively charged) has been detected in addition to t ri bo lu mi nescence (emission of photons). This phenomenon is much more intense on ceramics than on metals (21-23). It depends on the type of ceramics, atmosphere, temperature, speed and illumination (24).

3.2 Sites of trapping In order to characterize the polarization properties of a material and to determine if trapping sites exist in the lattice, an experimental method is needed. Such a method, called the mirror method, has been recently developped in which the possibility to create trapped electrical charges in a given sample was measured experimentally owing to a Scanning Electron Microscope. In this method which has been fully explained previously (25,26), samples were bombarded using the electron beam of the microscope at high voltage Vo (usually 30 kV). If defected areas (where polarisability is changed) exist

I88

Figure 1 : Electrical charges observed on a quartz sample after friction with a diamond tip.

Figure 2: The mirror method: image of the chamber of the microscope and of the diaphragm

189

in the superficial layers, the incoming electrons were trapped, leading to the formation of a space charge in the material. This charge was characterized by measuring the distribution of the elect rostatical potential around it. This was achieved by performing further observations of the sample in the SEM with low voltage beam (300-1000 V): in this case, image of the microscope diaphragm was formed because the electron beam of lower voltage was reflected by the potential of the trapped charges (figure 2). The radius R of the observed equipotential was calculated from the diameter D of the diaphragm image and the curve 1/R = f(V) was then plotted. This curve gave information on the charging capacity of the material: the lower the curve, the higher the charging capacity (more precise measurement can be deduced from this kind of curve but this will not be discussed in this paper). Indeed, it has been clearly shown with this kind of experiment that charges are trapped on the intrinsic traps of the material, not on defects created by the electron bombardment. Moreover, in order to remove contamination layers such as adsorbed water that could affect charging capacity, samples were heated in SEM at 200°C before measurement at room temperature. Using this method, it is easy to show that traps exist in most insulators materials as soon as exist grains boundaries, impurities ... In these cases, the role of friction of trapping is more difficult to characterize. More interesting is to study material without such sites of trapping. For example, highly pure sapphire (single crystal alumina) samples are free of such sites and no charging effect is detected at room temmperature in the mirror method before friction. But after a short friction

test (five cycles in very low load conditions), charging is observed in and out of the contact area (figure 3) revealing that sites of trapping now exist everywhere in the sample, that is that storage and relaxation of polarization energy has already been achieved in the bulk material, not only in the surface films. 4. DISSIPATION ENERGY IN MATERIALS

OF FRICTION DIELECTRIC

Since electrical charges and traps are associated with friction and wear processes, the results of the space charge physics can be applied to understand the mechanisms of dissipation of energy. For example, the dielectric constant is proportionnal to the inverse of the trap energy and thus is a major parameter to characterize a material in the space charge physics. Therefore, it should have a major role in friction. One way to modify the dielectric constant of sapphire is to irradiate samples with X rays (ref). Figure 4 shows the variation of the friction coefficient versus time for pure and irradiated sapphire tested in dry conditions (with very low load). The coefficient of friction of irradiated samples is always higher than the one for non irradiated samples. This can be explained by an easier dissipation of energy in the non irradiated samples. Indeed, at room temperature, pure sapphire does not charge in the mirror experiment: electrical charges are not trapped and move easily in the lattice. Parallely, in the friction experiment, the material is easily polarized and depolarized when the strain field is moving through the sample (displacement of the pin on the flat sample fo'r example). Only small

Figure 3: charging effect before friction (curve a) and after a 5cycle friction test in (curve b) and out (curve c) of the friction track

Figure 4 : Friction coefficient of pure (curve a) and X-ray irradiated (curves b) sapphire samples

191

amount of the friction energy is stored as trapped charges and most of it is diss i pated t hrough "soft" mec hanis ms (transfer to the phonon bath, exoe missio n.. .). On the contrary, irradiated samples are more easily charged in the mirror experiment: traps are deeper. During friction, the energy is no more dissipated but is stored as trapped charges in the material: polarization and depolarization are much more difficult and electrostatic forces due to electrical charges are increased: the friction coefficient is higher. In both cases, the steady state period during which the friction coefficient is constant corresponds to an equilibrium between storage and dissipation. During all this stage, defects are created in the samples because of dissipation and the material is progressively weakened. Finaly, when a critical threshold is reached, macroscopic wear is observed. This means that dissipation occurs now through more violent phenomena such as cracking. The exact nature of the threshold is not yet well known but is probably related to a critical amount of trapped charges in the lattice (i.e. a critical amount of energy stored in the material) or to a critical level of degradation of the lattice (i.e. a critical state of the weakened material). It leads to detrapping of a lot of charges, that is to a very high amount of energy (5 eV per charge) in the material: failure of the sample is observed. 5. CONCLUSION

Results of the space charge physics have been used to provide new insights in the friction and wear behavior of insulators. The main results can be summarized as follows:

- Storage of energy in dielectrics is related to the trapping of electrical charges on some sites of the lattice. Relaxation occurs when the charges are detrapped. The amount of energy can reach 5eV or more per charge and its relaxation can be soft (phonon, defects...) or hard (fracture, breakdown, ...) - During a friction test, defects are created everywhere in the samples, i.e. inside and outside the wear tracks. Formation of these defects is explained by the relaxation of polarization energy which is a many body phenomenon. These defects can be detected owing to the mirror method. - X-ray beam irradiation of sapphire samples increased both charging capacity and friction coefficent. This has been related to the fact that energy of traps after irradiation is higher. Charges are thus less mobile and dissipation of energy is more difficult. - Wear (i.e. detachment of particles) is interpreted as the result of the relaxation of high amount of polarization energy stored in the lattice during friction. It depends on the energy and number of traps.

REFERENCES 1. G. Blaise, in Dielectrics, Properties , characterization and Applications (Societe FranGaise du Vide, Paris, 1992) 1 2. G. Blaise, in Dielectrics, Properties, characterization and Applications (Societe FranGaise du Vide, Paris, 1992) 417 3. C. Kittel, Introduction to Solid State Physics (John Wiley, New York, 1976) 4. I.G. Austin and N.F. Mott, Adv.

192

Phys. 18 (1969)42 5. L.A. Dissado and R.M. Hill, Nature 279 (1979) 685 6. L.A. Dissado and R.M. Hill, Phil. Mag. 841 (1980) 625 7. A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1983 8. A. Berroug, these de doctorat, Ecole Centrale de Lyon, 1993 9. T.J. Lewis, IEEE Trans, El. 19 (1984) 210 10. D.K. Davis, in Static Electrification, Institute of Physics, London, 1967 11. D.K. Davis, in 1972 Annual Report Conf. Electr. Insul. and Diel. Phenom. (NAS, Washington, 1973) 12. J.J. Ritsko, in Electronic Properties of Polymers Eds J. Mort, G. Pfister (Wiley, New york, 1982) 13 13. K. Ohara, J. Electrostatics 15 ( 1 984)249 14. W.R. Harper, Contact and Frictional Electrification , Oxford University Press, London, 1967 15. R.G. Horn and D.T. Smith , Proc. Int. Conf. on Electronic Structure, of Bonding a n d Properties Ceramics, Florida, Oct. 1991 16. B.V. Derjaguin, N.V. Churaev and V.M. Muller, Surface Forces, Plenum Press, New york, 1987 17. B.V. Derjaguin, V.M. Muller, N.S. Mikovich and Yu. P. Toporov, J. Colloids Interface Sci. 118, 2 (1987) 553 18. B.V. Derjaguin and N.V. Churaev, Colloids and Surfaces 41 (1989) 223

19. B.V. Derjaguin, Yu. P. Toporov and I.N. Akinikova, J. Colloids Interface Sci. 54 (1976)59 20. B.V. Derjaguin, V.M. Muller, Yu. P. Toporov and I.N. Akinikova, Powder Technol. 37 (1984) 87 21. A.J. Walton, Adv. Phys. 26 (1977) 887 22. Y. Enornoto, Proc. Eurotrib 85 (Elsevier, 1985) paper 5.1.11 23. K. Nakayama, H. Hashimoto, Wear, 147 (1991) 335 24. S. Sasaki, Proc. 18th Leeds Lyon Symp. Ed. D. Dawson, M. Godet (Elsevier, 1991) 25. C. Le Gressus, in Dielectrics, Properties, characterization and Applications (Societe FranGaise du Vide, Paris, 1992) 149 26. A. Berroug, S. Fayeulle, B. Hamzoui, D. Treheux, C. Le Gressus, IEEE Trans. Elec. Insul. 28, 4 (1993) 528 27. K.H. Oh, C.K. Ong, B.T.G. Tan, C. Le Gressus in Dielectrics, Properties, Characterization and Applications (Societe FranGaise du Vide, Paris, 1992) 320

Dissipative I'rocesscs u1 Tribology / D. Dowson ct al. (Editors) 0 1994 Elsevicr Science U.V. AU ngh& reservcd.

193

Friction Energy Dissipation in Organic Films B.J. Briscoe and P.S. Thomas Department of Chemical Engineeringand Chemical Technology, Imperial College of Science, Technology and Medicine, London, SW7 2BY.

The paper speculates upon the molecular relaxationand dissipation mechanisms which may be responsible for the interfacerheological characteristicsof thin organic films. The molecular structureand interfacerheology of a homologous series of poly(n-alkyl methacrylate)s are characterised for this purpose. These rheological properties are identified by the measurement of the interface shear strength, z as a function of the contact pressure. 'Ihe molecular structure is deduced using vibrational spectroscopy. The Eyring model for molecular plastic flow is then applied and a correlationis drawnbetween the rheological and structural properties which are discussed in terms of possible mechanisms for the dissipation of the frictional energy.

1. Introduction Credible microscopic mechanisms for energy dissipation in friction processes have been sought for at least the past four centuries. Several models have been derived and proposed to explain the mechanisms of frictional motion from the gross surface ratcheting action offeredby Coulomb to the apparently more sophisticated molecular models of the present day [1-91. All of these models essentially describethe relative motion of entities between, or over, physical barriers, but few have truly adchessed the nature of the energy dissipation mechanism itself. A short, but selective, history is shown in Table 1. The present paper concentratesmainly on the discussion of molecular pathways for molecular rearrangement in plastic flow induced by shear deformationin the context of friction processes.The types of mechanisms involved in the dissipation of frictional energy are also discussed. For these purposes, the application of two diverse techniques to the characterisation of thin polymer films ~ I C reported. The rheological macroscopic properties of thin polymer films are characterised by the measurement and rationalisation of the interface shear strength. The intrinsic molecular architectures are characterised directly by using infrared light as the molecular probe to monitor the vibrational spectrum of a variety of thin polymer films using Fourier Transform Infiared (FTIR) spectroscopy. A correlation between the macroscopic and the microscopic responses is then made using a

molecular model which introduces a molecular scale interpretation into the macroscopic contact mechanical parameterswhich are used to describethe friction experiment.

1.1 Rheology The rheological properties of thin poly(n-alkyl methacrylate) films have been characterisedby the measurement of the interface shear strength, 2, which is defmedas the frictional force, F, per unit area, A, of contact. It is synonymous to the magnitude of the energy dissipatedin sliding friction per unit areaof contact per unit sliding distance.z is a strong function of several contact mechanical variables including: the mean contact pressure, P, the temperature, T, and the sliding velocity, V, or the contact time. These functionalities have been observed to have, within experimental error, the following approximate forms [4-6]:

z=zo+aP z = io exp -

(2) (3)

where zo, T ~ ' ,and zO",are material constants, 01 is

194

Table 1. A brief chronology of the advances in the modelling of the friction process.

da Vinci Amontons

1470 1699

Greasy surfaces Low Friction

F=pW

Coulomb

1785

ShipsNood

Leslie

1804

Friction Energy Dissipation: subsurfacedeformation

Hady/Doubleday

1922

Boundary Friction: molecular friction

Deryagin

1934

Adhesion Model: interface (bulk) shear

ratchet friction

P=

P=*

Bridgman

1935

Anvil Technique: bulk shear bands

Bowdeflabor

1950

Asperity Contact Model: molecular shear

Bailey/Courlney-Pratt

1955

Mica Cross Cylinders: methyl friction

Bowers/Zisman Pooleynabor Briscoe/Amuzu Evans SutcliffdCameron

1954 1972 1976 7976 1972

Molecular TopographicalModel Molecular TopographicalModel Eyrins Approach Dislocation Model Methyl Isomerism

BowerdTowle

1972

Anvil Technique

BriscoelScruton Briscoe/Tabor Briscoe/Smith

1973 1977 1983

Model Contact Contact Time Effects

Briscoe/Smith

1984

BriscoeDbomas Israelachvili/Klein/ Grannick

1993

Solvent Effects: ductilehrittleresponse Molecular Environment Static Molecular Topography

the pressure coefficient, Q is an 'activation energy', R is the gas constant and 8 is the velocity index. These functionalitiesare observedto be quite general for polymers [1,4-81, although there are some exceptions which have been fully discussed elsewhere [7,8]. In addition to the contact variable dependencies,z is also known to be a function of the morphological properties of polymer films which are governed by the various processing

1 chain length F + PO)

F = kWn

z=zO+aP

z = X : non-linear period where the tangential force increases until an equilibrium value leading to a low friction coefficient (= 0.01) independent of the applied normal load.

209

Due to the very low tangential compliance -6 of the apparatus ( C $ = 2 x l O m/Nl and the large displacement resolution, the tangential compliance of the film itself and the variations of the tangential force F'x can be detected. Indeed, the measured tangential compliance is

(K'

Id ~ I x-0 x

= 4.5x

lodm/N.

Besides this compliance is given by :

as long as v does not exceed vc = 5nmls. The friction force reaches a stabilised value after a sliding distance of about 9 nm (Figure 5. b). When the sliding speed is increased by step (Figure 5. c-d-e), aRer a small and fast increase, the friction force reaches an equilibrium all the more rapidly because the speed is high. This suggests that the relevant parameters to describe that speed-dependent behaviour (parts b-c-d-e in Figure 5.) of the interface are a distance and time.

(3) w

I/

I

0

10 20 30 40 SLIDING DISTANCE X (nm)

50

0

10 20 30 40 SLIDING DISTANCE X (nm)

50

I

I

I

where C,f and C?" are the tangential A

A

compliances of the thin film and the Hertzian He and c;e are contact respectively. As Cx related [151:

cp

PI

2.4 x cFe = 3.2x 10-6

L d N

(4)

tangential compliance is assumed to be :

where Gf is the elastic shear modulus of the film in the XOY plane and is equal to 3 x lo6 N/m2, according to relation (5). Therefore, for small tangential forces, (part a in Figure 4.), the tangential compliance of the thin layer leads to a linear relation between the tangential force and the displacement X : FX = ; c a 2 G f x X D

As the tangential displacement exceeds a critical value X* = 0.4 nm, the tangential force Fx is no longer linearly related t o sliding distance X indicating a sliding process and now depends on the sliding speed v = X

Figure 5. Influence of the sliding speed on the evolution of the tangential force FX related to the variations of the thiclmess of the interface. The friction experiment is carried out for 4 successive sliding speeds (b-c-d-el. The increase in speed causes, first, a steep increase of f i and then a slow decrease to a limiting value. The interface accommodate the sliding process by small variations of its thickness (= nm) correlated to the variations of the fictional force. These speed effects (b-c-d-e)are characterised by a "length constant" of about 4.5 nm.

210

Besides, the friction force variations due to s l i k (part b in Figure 5 . ) or to change in the slidmg speed (part c-d-e in Figure 5 . ) are accommodated by small but sigmficant film thickness variations AD deduced fkom the electrical sphere-plane 'apacitance, assuming no variation of the contact area at constant normal load Fz. When the tangential force increases, a decrease of the thickness of the film is observed (see Figure 5 . ) and vice versa. For every sliding speed, the thickness of the interface reaches a stabilised value in the same time as the fictional force with the same "length constant" of about 4.5 nm. If the very low values of AD are supposed to be due t o a variation of the orientation of the stearic acid molecules relative to the tangential plane of the contacting surfaces, a variation AD = 0.01 nm of the thickness of the film would correspond to an angle of 4" between the normal of the solid surface and the principal axis of the molecule. As the sliding speed exceeds the critical value vc, the tangential force becomes constant and is completely determined by relation (6) in which x = x * . It is interesting to consider the transit time t which is the time taken by a C a group to overcome another CH3 group on the opposite surface, t

3 :

X* . In this experiment, V

there is one critical transit time & under which the molecules are not sensitive to the speed :

X*

tc=-=80ms

(7)

VC

This critical time tc is long compared to the relaxation of a molecule, suggesting the sliding process involves a set of molecules. Indeed, complementary experiments for different applied normal loads show, first, that the distance X* is independent of the normal load FZ and therefore of the mean contact pressure p and second, that,, according to relation (61, the shear modulm,

Gf is proportional to the mean contact pressure p:

Therefore, the friction coefficient p defined as the ratio of the tangential stress t to the pressure p is simply given for a sliding speed v E vC:,by the relation : p =$=1~-=0.009 G X*

P

D

It can be deduced h m relation (9) that p is also independent of the normal load. The value of X* is close to 0.45 nm which corresponds to the average distance between two CH3 p u p s in a monolayer in crystalline condensed state [161. 6. CONCLUSION Our results show that the frictional behaviour of the stearic acid monolayer presented, in this paper, is similar to that of solid-like monolayers observed in previous experiments [l]. In particular, for sliding speed v < vc, the fiction force decreases as v increases. The low compliance of our apparatus leads to friction experiments without stick-slip phenomenon and permits a better description of relaxation processes. The studies of these effects which are comparable to that observed for adsorbed polymers [3], may provide a new approach for relating the chain-like aspects of surfactants or polymer molecules to the macroscopic theory of lubrication like elastohydrodynamic lubrication.

6. ACKNOWLEDGEMENTS The authors are indebted to Shell Research Limited for financial support. We also thank the C. N. R. S. and the members of the GDR 936 "mesure de forces de surfaces en milieu liquide" for helpful discussions.

211

REFERENCES

9.

M. Jacquet and J. M. Gorges, J. Chimie Physique (Paris), 11(1974) 1529

1. H. Yoshizawa, Y. L. Chen and J. N. Israelachvili, J. Chem. Phys., 97, 2 (1993)4128. 2.

J. Van Alsten and S. Gmnick, Phys. Rev. Lett.,61 (1988) 2570

3.

J. Klein, D. Perahia and S. Warburg, Nature, 352 (1991) 143

4.

M. L. Gee, P. J. Mac Guiggan and J. N. Israelachvili, A. M. Homola, J. Chem. Phys., 93, 3 (1990) 1895

5.

A. M. Homola, J. N. Israelachvili, M. L. Gee and P. J. Mac Guiggan, Tribology, 111(1989) 675

6.

A. Tonck, D. Mazuyer and J. M. Georges, to be published

7.

A. Tonck, J. M. Georges and J. L. Loubet J. of Coll. and Inter. Sci.,126, 1 (1988) 1540

8.

J. M. Georges, S. Millot, J. L. Loubet and A. Tonck, J. Chem. Phys., 98, 8 (1993) 7345

10. E. L. Smith,C. A. Alves, J. W. Anderegg, F. Porter and M. D. Siperko, to be appeared Langmuir (1993) 11. A. Tonck, F. HouzB, L. Boyer, J.L. Loubet and J. M. Georges, J. Phys. Condensed Matter 3 (1991) 5195 12. A. S. Akmatov, Molecular Physics of Boundary Lubrication, Israel Prog. for Sci. Trans., Jerusalem, 1966 13. B. V. Derjaguin, V.M. Muller and Y. P. Toporov, J. of Coll. and Inter. Sci., 53 (1975) 314 14. J. N. Israelachvili, Intermolecular and Surface Forces, 2nd Edition, Academic Press, 1992 15. R. D. Mindlin, J. of Appl. Mechanics, 16 (1949) 289 16. D. Tabor in Microscopic Aspects of Adhesion and Lubrication, J. M. Georges (Ed..),Tribology Series 7, Elsevier, 1982

This Page Intentionally Left Blank

Dissipative I'roccsscs in Tribology / 1). I)owsoii ct al. (Editors) 1994 Elsevier Science R.V.

213

EFFECT OF THICKNESS ON THE FRICTION OF AKULON A PROBLEM OF CONSTRAINED DISSIPATION L.Rozeanu, S.Dirnfeld and J.Yahalom Dept. Materials Eng., Technion, Haifa, ISRAEL

This paper describes friction experiments with steel balls sliding on Akulon disks of various thicknesses. It is found that the friction torque does not vary monotonously with the specimen thickness. This unusual friction behavior is assumed to be due to a mechanical constraint which prevents the free motion of the entangled molecular chains and to a thermal constraint which prevents free cooling. When these constraints are active the material strength increases, there is less volume work performed and the friction torque decreases.

1. INTRODUCTION The friction of polymers has been extensively studied and it is difficult to find a practical detail or a functional feature which has escaped the attention of researchers. The fast initial R&D progress, has followed the usual trend toward a plateau as regards application know-how; competent review papers written 20 years ago [l] are still useful to-day. The more recent progress aims rather at building up a consistent theoretical frame for the facts already known and at designing new materials. For this reason the finding that the thickness of Akulon does not affect monotonously the friction wass a surprise. The results reported in the present paper show that the resistance to friction follows an initial zigzag variation with the specimen thickness: at least in certain working conditions (loads of 0.2-0.8 Kgm, velocities of 0.5-2.5 m/s and temperatures of 30-60 "C): the friction is maximum in a narrow thickness range of about 1 to '3 n m . In order to explain this it is assumed that the macropolymer is subject to two constraints: A mechanical constraint (the bond to the metallic base) which interferes with the free motion of the entangled chains under stress as if the bulk strength has increased. a thermal constraint (the greater thickness

of the specimen). This slows down the heat flow out of the system promoting adiabatic behavior, less strain for the Same stress.

1.1 NOTATIONS A Free Work (Helmholz) F Applied force G Free Energy (Gibbs) L Nominal length of reference system P Pressure R Radius of thermally driven agitation S Entropy T Temperature V Volume (other notations are explained in Fib. 6) 2. GENERAL

Usually the friction behavior of plastic materials is considered a generic attribute of the material or of the material couple. The sliding interaction between the members of the friction couple is assumed to be confined to the surface and the surface is considered an indistinguishable part of the bulk, the two acting as a single entity. If two factors act in the Same direction, but the contribution of one is very small, its omission is justified, even necessary. Indeed

214

many details can be omitted without harm. The omission of some details, however, can penalize the research. One such detail in friction experiments is the thickness of the specimen which, in certain working conditions, affects substantially, both mechanically and thermally, the friction behavior of plastic materials. The conclusions to be presented refer to one material, Akulon. There are good reasons to believe that other plastic materials behave similarly and that the thickness must be specified except when the friction behavior presents no interest. The fact to retain is that the variation of the friction force of Akulon in sliding interaction with steel does not vary monotonously with specimen thickness: it is low for thin and thick specimens and high in the middle thickness range ( 5 1 to 2 3 rnm). Load, sliding velocity and temperature affect the friction sensitivity to thickness but at a smaller scale. 3. EXPERMENTAL SET-UP

The testing unit (Fig. 1) consists of a rotating disk driven by a variable speed electric motor. On top of this disk is mounted another disk with complete axial and radial degree of freedom, provided with means for applying the load and for ineasuring the friction torque. On the lower disk are fixed 3 steel balls (6 mrn diameter) at 120" and at 17.5 mrn from the center. On the upper disk, made of aluminium, are fixed the Akulon specimens with a cyano-acrylic glue, few p m thick when dry. The Akulon disks were brought to the final thickness by grinding following an adhoc but strictly respected procedure. The active part of the apparatus was enclosed in a thermostatic container were it was kept for 30 min., for thermal conditioniiig prior to each test. The loads (up to 785 gm) and velocities (0.5 to 2.5 m/s) were in the medium range generally encountered in service. 4. RESULTS

The results of over 100 tests are presented in FIG. 2 as averages of torque-vs.disk thickness, indiscriminately, over all loads, velocities and temperatures adopted.

THICKNESS , m m

-

Fig. 2 Torque-vs.-Thickness Averages over all L,T and v

Fig. 1 'I'ES'I'ING U N I T

This rough presentation indicates that the torque for the disks in the thickness range of about 1 to 4 mm is almost twice as high as those outside this range.

215

More information is provided by the following bar graphs which show tile effect of each experimental condition individually (load, temperature and velocity) within each thickness range. Fig. 3 shows the effect of load; for each thickness; as expected, the torque increases with load. The disks of medium thickness continue to give the higher torque results.

In the case of the medium thickness disks the experimental errors (no attempt was made to discard them) or other factors lead to apparently abnormal results without altering the conclusions emphasized by Fig. 2. 400

l250rpm

B600rpm

I1000rpm

Fo (Fig. 6c left). In the adiabatic case the tensile stress rises fast for a small strain, as if passing in Fig. 6 from T=T1, F=Fo to T3 >T1, F1> Fo (Fig. 6c right) the material warms up (if polymer), and the strain continues at constant stress until it reaches the iso-thermal end point (Fig. 7). The horizontal strain segment is completed during a time depending on the heat exchange conditions, Effective thermal isolation of the stressed specimen will inhibit the strain.

218

5.2 Prevailing constraints The thermodynamic picture is essential inasmuch as it indicates the possible paths by which a system can recover equilibrium but is unable to specify the rate at which a change will occur. Moreovere it places the change under the control of the prevailing constraints. The experimental conditions adopted can be redefined such as to take into account the unusual properties of rubber-like p;astics in terms of constraints and degrees of freedom. The applied stress is a constraint. Without a path (degree of freedom) for stress relies (for example by strain) the materid will remain under that stress constraint for ever just as humans live under the " 1 Atm" pressure constraint for ever. Ignoring the contribution of surface interaction, the friction of polymers can be described by the sequence: perform volume work, produce heat, dissipate heat. If the process occurs in cycles and at the end of each cycle all intensive properties return to the values they had the previous cycle, the system has reached steady state friction. Next the thickness of the material is added as supplementary specification. At first sight this does not alter the reasoning based on the indicated sequence. Then an additional statement is made: the polymer specimens are attached to an aluminium plate with a hard glue. The entangled molecular chains are anchored at various points to the surface. This is a mechanical constraint for the free motion of the chains like crosslinking or vulcanization. The result will be less strain for the same stress, less volume work and smaller friction torque. But the chains have a limited space occupancy and the effect of the "wall" constraint decays until it becomes negligible. The connection with the thickness is obvious. The plate supporting the specimen can be considered a good "heat sink" and the glue, although a poor heat conducting material, too thin to affect the heat transfer. Now add the statement: The volume work is performed near the surface (like a hardness test), the surface temperature is high due to friction interaction and the heat produced by volume

work can be dissipated only through the metallic support. The heat transfer efficiency depends on the distance between heat source and heat sink. As long as the distance is short the system can develop a temperature gradient adequate for thermal steady state. If the distance exceeds a certain limit thermal steady state cannot be reached. The specimen thickness had become a thermal constraint. As a result the temperature, in the upper layer where volume work is performed, goes up arid the stressed polymer behaves as indicated by the adiabatic path in Fig. 7. Comments on surface Contribution. The interpretation of the results in terms of constraints and degrees of freedom for volume work is based on the assumption that the surface contribution to the friction torque is constant or not very sensitive to the same service variables, Load, Temperature, Velocity. In order to get a clearer picture, after the friction torque experiments the disks of Akulon were subjected to hardness measurements using the Durometer Hardness Tester (ASTM C 2240-85). An indenter produces its imprint at a specified speed and the hardness is read on a 0-100 scale The measurements were made inside the grooves (produced by the steel balls during friction) and outside the grooves, on the free surface of the disks . The depth of the groove is so small that the nominal thickness of the disks inside and outside the groove could be considered the same. The difference between the results for the same disk would indicate the effect of the changes produced by friction. There were no significant differences; as can be seen in Fig. 8, the same hardness values were found on the same disk, wherever they were measured. As regards hardness variation with disk thickness, it followed the friction torque variation: high for thin and thick disks, small for the intermediate range disks. Of interest is the fact that the change of hardness just above 1 mm thickness is very rapid showing an almost step-like decay of the mechanical constraint.

.

219

5.1 The mechanical constraint,

HARDNESS v s THICKNESS

"I-

*FLAT

79

0''

I

2

3

4

I

OPATH

5

6

7

8

9

THICKNESS,mm

Fig.8 Hardoess-vs.-Plate 7liickiiess iriside arid outside grove. Average and iiidiviuduai points

6. DISCUSSION In the particular case of metal sliding on Akulon (Nylon 6.6) plates glued on metallic supports: - The friction is not a monotonous function of thickness, showing a maximum in a narrow range, not less than about 1 mm and not more than about 3 mm. - In the same thickness range the material hardness is the lowest. The correlation between friction and hardness was expected according to the experimental results but was not necessarily evident. It follows that the thickness of plastic materials is a design detail which should not be neglected. - Although there is no experimental evidence to justify extension of the conclusions reached for Akulon to other plastic materials, there are good reasons to believe that the same functional interactions will control the behavior of all polymeric materials obeying the same thermodynamic relations. In the present paper it was assumed that the friction work involves a surface interaction independent of disk thickness, and a volume work subject to constraints, some of them controlled by the thickness of the polymeric material. Two such constraints were discussed:

- The attachment of polymer chains to the hard interface glue; this prevents them from yielding freely under the applied stress. The strain is completely prevented at the wall and gradually less away from the wall until the constraint ceases to operate at the depth reached by the volume work. The depth of THIS work can be calculated knowing the width of the groove created by friction and the radius of the sphere which has produced it. In the present case it is found that the depth of the groove does not exceed 0.2 mm. Therefore there is plenty of space available for volume work and the higher strength of the polymer is due to the decreased mobility of the entangled chains anchored to the wall. Next it is the case to ask if there is a property related to the friction behavior of thin polymeric materials. There are two aspects to consider: - the rate at which the wall effect decreases with the distance from the wall, and - how far in depth spreads the volume work. It is helpful at this stage to refer to the Poisson Coefficient. If its value is 0.5 all the mass is involved equally in the deformation process and all deformations occur at constant volume. As the Poisson Coefficient decreases the deformation involves smaller fractions of the loaded mass, local1 the stress and strain increase and the defyormation is accompanied by volume increase. For practical purposes The Poisson Coefficient of plastics can be considered equal to 0.4 [2]. n the present context a more precise scale is necessary. Then, materials with a high coefficient will perform more volume work than those at the bottom of the scale so that using this criterion it should be possible to anticipate the friction behavior of plastics. Also worth considering is the quality of the glue and/or support. It is reasonable to expect that the effect of such a mechanical constraint on friction will be observed only when the hardness of the glue is higher than the hardness of the polymeric material.

220

4.2 The thermal constraint - The heat flow out of the system. The surface temperature is high due to friction heating [2] so that the heat produced by volume work must flow out of the plastic material through the aluminium base. The rate at which heat flows out, according to Fig. 7, controls the rate at which lhe strain follows the application of stress. The heat flux varies with the distance, therefore the thickness acts as a thermal constraint; increasing the thickness at constant thermal conductivity has the same effect as reducing the thermal conductivity at constant thickness. Again, there is no reason why this conclusion would not be valid for other polymeric materials. One can go a step farther and conceive another thermal constraint: Preventing the base to act as heat sink. This can be done in various ways according to the desired effect. - A thicker, thermally isolating, glue layer should decrease friction (in the present experiments the glue thickness was 2-3 pm). - In the same direction should act a poor thermally conducting base material or a small mass (thickness) heat sink. - Isolating or deliberately heating the support material should create the same thermal constraint. By reverse actions one could reduce the thermal constraint, increase volume work and energy dissipation, effects which would be of interest in damping problems. In this spirit it is the case to look once again at the effects of the friction variables on the torque. Fig. 3 displays the monotonous effect of load, in all cases as expected. Fig. 4 suggests that the thermal constraint operates in the case of both, thin and thick disks. The controlling variable is the heat flow rate. Observing that the main heat source is the volume work (Jq+) performed near the surface it follows that the temperature rise has two effects: it reduces the strain for a given stress and by this reduces the volume work; it rises the temperature of the heat sink reducing the temperature gradient and the heat flow rate (Jq-). Thermal steady

state can be expressed as Jq+/Jq-= 1 . Now the three thickness alternatives can be evaluated. Thin disks. As the temperature increases, Jq+ (the volume work) decreases. At the same time the temperature gradient decreases because it has less Jq+ to dissipate while the sink temperature rises (independent variable). A further rise of temperature and increase of polymer hardness ma follow. The overall result must be a smaier friction torque. 2-3 mm disks, In the absence of the mechanical constraint much more volume work is performed and a greater temperature gradient develops. Probably, up to '3 mm thickness the heat flow rate is sufficient to maintain Jq+ =Jq-. The externally controlled rise of temperature has only one effect: to rise equally the upper temperature of the gradient. Thick disks, Increasing the length of the path (disk thickness) for heat flow is the same as thermally isolating the material, more or less efficiently. A thermal constraint acts on the system; the temperature of the material rises at a rate which depends on the isolation efficiency and adiabatic conditions develop. Reference to Fig. 7 shows that this means less volume work and lower friction torque. The effect of velocity presented in Fig. 5 is significant in two respects; The amount of volume work (number of loading cycles per revolution) increases while friction heating increases the material strength to a certain extent so that the volume work per each cycle is reduced, independent of the cooling conditions. Beside, there is the surface contribution about which little is known. If hard plastic materials (in adiabatic conditions) behave (in this respect) like metals, the friction torque should decrease with increasing velocity. To conclude: Thin disks, The velocity effect should be small and go either way. At low temperature and small velocity increase the effect of more volume work cycles per unit time may exceed the effect of the diminished volume work per cycle. Then the friction torque will increase. As the velocity increases the magnitude of the effects may reverse and the friction torque will decrease, more so if

22 1

the surface contribution acts in the same direction. -2-3 mm disks, .Inthe absence of the thermal constraint the effect of volume work should prevail , more loading cycles per unit time increasing the friction torque accordingly. Thick disks, The thermal constraint is present. The stress/strain relationship shifts slowly to the adiabatic mode. Apparently there is more volume work per loading cycle than in the case of thin disks but another factor intervenes, the available time for stress relaxation between loading cycles. The time available for relaxation is -0.08 s at low velocity and only 0.02 s at high velocity. Assuming maximum material strength 1s reached and the maximum strain during each loading cycle is constant, the stress (relaxation) decreases as the velocity increases and so does the volume work per loading cycle. Therefore, the friction torque should not be too sensitive to the velocity. Although the paper discusses a very particular polymeric material, the reasoning refers to properties , thermodynamic interpretations, experimental conditions and assumptions similar to those adopted by other authors [2]. This justifies the hope that the conclusions reached for Akulon (Nylon 6.6) will prove valid for other alike materials.

7. SUMMARY AND CONCLUSIONS It was found that when metal balls slide on plates of Akulon (Nylon 6.6) glued on a etal support, the power dissipated by friction varies with the thickness of the plate: low for plates up to -1 mm or more than ‘3 mm thick and about twice as high in the intermediate thickness range. This behavior suggests that the effect is due to the volume work contribution to friction and if for some reasons the strength of the polymeric material is higher, less volume work is performed and the friction is lower. Therrniodynamic considerations show that there are two constraints controlling the volume work. For thin plates the constraint is mechanical, the hard glue preventing the polymer chains from moving freely under

the applied load. For the thick plates the constraint is thermal, it prevents rapid heat flow out of the system allowing the temperature to rise, the strength of the polymer to increase and the volume work to decrease. It is expected that the conclusions presented in this paper can be extended to other applications and to other polymeric materials.

REFERENCES 1. Lancaster J.K., Friction and Wear, Polymer Science, A.D. Jenkins (ed.) North-Holland Publishing Comp. ,( 1972) 959-1046 2. Tanaka K.,Kinetic Friction and Dynamic Elastic Contact Behaviour of Polymers, Wear, Elsevier, Lausanne, 100 (1984) 243-262 3 diBenedetto A.T., The Structure and Properties of Materials, McGrow-Hill Book Company, N.Y., International Students Edition, 1967 4 Jastrzebski Z.D., The Nature and Properties of Engineering Materials, J.Wiley & Sons, New York, N.Y.,1976

This Page Intentionally Left Blank

Dissipative Processes in l'ribology / I). Dowson et al. (Editors) 0 1994 Glsevier Science B.V. All rights reserved.

223

Interface friction and energy dissipation in soft solid processing operations

M. J. Adamsa, B. J. Briscoe and S. K. Sinha Department of Chemical Engineering and Chemical Technology Imperial College, London SW7 2BY, U.K. a Visiting Lecturer from Unilever Research, Port Sunlight, Bebington, L63 3JW, U.K.

The paper considers the interactive influences of the bulk rheology and interface friction boundary condition of plastic solids where they are deformed by rigid walls. Various experimental data and analyses are presented which indicates that the wall friction has a pronounced influence upon the nature of the flow induced in the soft solid and as a consequence greatly controls the extent of the energy dissipated during the deformation process. 1.1 INTRODUCTORY REMARKS available commercially and is marketed with the Soft solids constitute an important generic class of materials in so far as many manufacturing operations use materials of this type of rheological fom as part of routine manufacturing operations. These systems are diverse in types and include hot metals, ceramic pastes and food doughs as well as a variety of polymeric composites. Since these materials do not flow appreciably under gravitational forces they must be deformed by the action of solid walls which are nominally rigid. The commonalty is in the form of the rhological response with which these materials are endowed. They are soft, and thus readily deform, plastically deforming solids which have the intrinsic capacity to flow in a coherent way and retain their external form after processing. This paper is concerned with a description of the stresses which are at the plastic solid/rigid wall interfaces and the influence of these stresses upon the energy dissipation and the consequentflow of the deforming solid. 'Ibe main focus is upon the role of the interface friction (the boundary condition) upon bulk flow behaviour which will define the overall energy dissipation during the deformation. The studies to be described here used a model paste system which is constructed with a mixture of clay particles dispersed in a hydrocarbon liquid. The material is readily

trade name "Plasticine". It emulates all of the known bulk and interfacial rheological properties ' soft solid materials at relatively of low temperatures; these materials are often described as pastes. ' h e paper briefly reviews the accepted and aspired for methods of describing these two rheological characteristics. Several experimental methods for obtaining the interfacial and rheological characteristics of these paste systems are described. There are two generic groups. One in which the boundary friction arrests flow. Examples are provided for cylindrical upsetting or squeeze film deformation, ring compression and wedge indentation hardness. The other is where the interface traction is responsible for actually inducing the flow. The example chosen here is for the twin roll configuration. Data are provided for each example and a range of analyses, of various degrees of sophistication,are applied. These results provide first order estimates of the intrinsic rheology of the paste and the prevailing boundary conditions. These data serve to illustrate the pronounced influence of the interface friction, between the paste and the wall, upon the work required to institute flow in these systems. For example, a madest reduction in the wall fiiction leak to a matked reduction in energy dissipated during the deformation process.

224

1.2 Boundary Conditions The viscoplastic deformation or flow of solid materials invariably involves interactions at the interface or the boundary between the material and the processing equipment walls. 'he nature of the boundary is important in the paste processing context because of the fact that the material flow response is extremely dependent upon the interface frictional characteristics. Conversely, the interpretation of most experimental data is not generally viable, to any degree of accuracy, without some knowledge of the interface rheology. The interface characteristics determine the mechanical and thermal energy dissipations across the boundary between the paste and the equipment. It thus actually plays an important role in material processing operations in terms of the optimisation and the efficiency. These effects arise, in the main, because, during paste deformation the interfacial resistance induces inhomogeneity in the flow of the bulk. The flow inhomogeneity produces complex displacement, stress and strain rate fields in the bulk of the deforming paste material. Hence, the reaction or pmcming forces involved during deformationare also greatly influenced by the inhomogeneous flow conditions.

Figure 1: Grid distortion (quarter symmetry) at 6Wo compression of a cylinder with an initial aspect ratio (diameterheight) = 1 and boundary friction p = 0.4. The figure shows the displucement vectorfields of an initial square grid afer compression, and the inhomogeneitiessuch as rigidzone development, shear bandsfoMation geometric barrelling and 'Ifokiing which arise during compression. When the friction is zero a rectangular grid is developed I'

Wall Boundary Conditions 2 , = mz,:O 5 m 5 1

H 2 , 2,

=W =pv

Coulombic

W

Mooney

Biilklev

Table 1: Examples of wall boundary conditions : m, friction factor: p, coeflcient of friction; j, slip coeflcient; v, wall slip velocity; a, wall pressure coeficient; q, wall velocity index and 7 ~ 0wall , yield

stress.

Figure 1 illustrates the flow field inhomogeneity in the bulk of a cylinder specimen undergoing uniaxial compression between parallel platens when the wall friction is finite. The flow field shown by the grid distortion was obtained using a finite element analysis [l]. Similar data may be obtained by flow visualisation. An accurate knowledge of the boundary conditions provides, in principle, a means to obtain a m e estimation of the bulk rheology. A number of boundary conditions have been proposed by various authors [2]. In metalforming, generally, two types of slip boundary conditions are considered. They are referred to as the Coulombic and the Tresca boundary conditions. These boundary conditions define the wall tractional stress as fractions of the normal pressure (Coulombic) or the bulk shear yield stress of the material (Tresca). In the field of fluid rheology, the wall stress has been generally related to the slip velocity or the shear rate at the wall. Table 1 lists the commonly accepted relationships for the wall boundary conditions [2].

225

1.3 Intrinsic Bulk Ftheology

Table 2 lists the commonly adopted constitutive relationships for ideal and engineering materials. The viscoplastic constitutive relationships, which have been used for highly concentrated particulate dispersions, exclude the elastic components as well as the strain hardening process. A real paste material will show elastoplasto-viscous behaviour. To overcome this problem Lipscomb and Denn [5] proposed a biviscosity model which assumes that paste materials show a higher viscosity prior to yielding. The other limitation of these existing constitutive relationships is that they can be implemented only for a two dimensional flow condition. This restricts their applicability to real flow problems, where flow occurs in three dimensions. These points will be discussed in detail in the l a m part of this paper.

The rheological response of paste materials has been, in general, described by a combination of a yield criterion and a postyielding constitutive relationship. 'Ibis choice is based on the fact that the material shows a definite yield point when acted upon by normal or shear stresses. Like ductile metals, pastes exhibit a arguably clear plastic yield phenomena. The Herschel - Bulkley constitutive relationship assumes that the material behaves as a rigid body in the pre-yielding regime and then as a power law fluid after yielding. Similarly, the Bingham model assumes Newtonian flow after yielding

WSI. 3hMLZ Constitutive relationships ~ = G Y

2. ANALYSES

TYPe Hookean solid

2.1 Yield Criterion L

The idea of a yield criterion for ductile solids, such as metals, is well established. It defines a critical combination of stresses at which elastic deformation terminates and plastic deformation is initiated. Certain fluids, such as concentrated suspensions, also exhibit a quite distinct yield phenomena. The most commonly used yield criterion for plastic fluids is the von Mises yield criterion which is defined by the following equality [7];

where 1/2 r* :r* is is the second invariant of the deviatoric stress tensor. 70 is the shear yield stress and r* is the deviatoric stress tensor.

Table 2 I&alised shear constitutive relationships for engineering materials. The parametrs are as follows: .r, stress; strain; ;J, strain rate: G, elastic modulus: 70, yield stress: q, viscosity: Q, plastic viscosity; k, flow consistency: k p plastic flow consistency: n, flow index: z, strain exponent; a, material constant: p, pressure and a,pressure coeflcient.

2.2 Cylinder Upsetting (Squeezefilm Test) The axial compression of a cylindrical specimen bas been studied both in plasticity and in fluid mechanics. In plasticity, the equilibrium stress analysis for the upsetting of rigid-plastic solids is commonly used [8]. Here, the wall boundary condition at the interface is characterised by the Coulombic coefficient of friction, p. The method assumes homogeneous

226

following result for the mean pressure for a plastic fluid between parallel platens as [91;

- 27,R

(a)

I I I

Ro

J++(

* + 6 + -

t

+

I

'=

--

+

where '50 is the shear yield stress of the material; for a von Mises material, the shear yield stress is given by CJo/ff. Equation (3) is not valid when 2R/h tends to zero as it predicts a zero yield stress of the material which is not physically realistic. A full stick wall boundary condition is assumed in this fluid mechanics analytical treatment of cylinder upsetting. However, this condition is not applicable when a highly concentrated paste material is deformed at high shear rates [lo]. Presently, there is no analytical solution available for the slip wall boundary condtion with limited traction.

(b)

.

(c)

I

deformion behviour o f a ring specimen under axial compression between parallel platens. (a) Original ring before compression: (b) ring afrer compression with low wall friction; (c) ring after compression with high wallfrictwn. The arrows show the direction 5 ofjlow of the material. e FIGURE 2:

5 E

(3)

3h

75 5o m

fa deformation. However, in actual practice the a 0 deformation, for the majority of the cases with $ tractional interfaces, is highly nonhomogeneous; E note figure 1. This leads to an overestimation of 2 the yield stress of the material as the measured g, $ -25 mean compressive pressure is greater than that 5 predicted using the classical solution based upon 6 homogeneous deformation. The solution for the critical deformation pressure in this case, -75 assuming homogeneous deformation, is given 0 by (see ref. 8);

3

-

(2)

where

is fhe mean reaction pressure, 60is the

uniaxial yield Of the material* is the radius and h is the height of the cylindrical specimen. This solution predicts that the mean D ~ ~ S S U Rwhen . extrapolated to a zero aspect ratio i2R/h), is non-zeroand equal to the-uniaxial yield stress of the material. Using the lubrication approximation, which is employed in the fluid mechanics treatment, Scott obtained the

1

I

25

50

75

Compressive Slraln Ok

FIGURE 3: The calculated change in the inner diameter of 6:3:2 (Ro:Ri:h, figure 2) ring specimen as a function of compressive strain at dgerent magnitude of wall jiction for Upper Bound solution with a Tresca wall boundary condition. (Tk curves represent results for (jgerent values of m; where is the piction factorWafter Avitzur [Ill)

2.3 Ring The conventional plasticity analysis for ring compression utilises the Upper Bound

227

theorem (Avitzur [ l l ] ) . This analysis incorporates the friction factor (m) as the boundary condition ('Tresca). The friction factor is given by T~ = m ro,where rw is the wall shear stress and again rois the shear yield stress of the material. Hence, the maximum value of m is unity when the wall shear stress is equal to the shear strength of the material. It is possible to correlate the friction factor, m to the Coulombic friction coefficient, p using Kudo's approach. The relationship between p and m is then given as [111;

When a ring specimen is compressed between two parallel platens the direction in which the material flows is solely determined by the interfacial friction condition ( as shown in the figure 2). For each interface friction condition there is a neutral radius; the location where the deforming ring material flows in opposite directions. The Upper Bound theorem relates the neutral radius Rn, with the outer radius, Ro, the inner radius Ri, and m by the following tXpMi0n;

(4)

Equation (4). in combination with the continuity equation, can be used to obtain a plot of the fractional change in the inner radius of a ring specimen against the height reduction for different values of the friction factor, m. This plot is also known as the "calibration curve". Figure 3 shows a calibration curve for ring compression.

Slipline Theory of Wedge Indentation 2.4

In the wedge indentation hardness experiment, tbe material is indentated with rigid wedges and the indentation pressure is measured as a function of the depth of the indentation. Slipline analyses of wedge indentation are available for the cases, with and without the interfacial friction between the rigid wedge face

and the deforming material. Figure 4 shows the penetrarion of a smooth rigid wedge into a semiinfinite mass of a rigid - perfectly plastic material so that the bisector of the wedge angle 2y is perpendicular to the plane surface of the medium.

FIGURE 4: Indentation of a semi-infinite rigid perfectly plastic material by a rigid wedge. w is the semi wedge angle and h is the depth of indentation. a and b are the orthogonal sliplines. p is the indentation bod. Using the continuity equation and the force equilibrium analysis, the indentation pressure pm may be related to the coefficient of friction, p, between the wedge face and the material and the deformation geometry (semi-included wedge angle w) as WI;

pm = JL= p w (I+ p cosy) 2a

(5)

where p is the indentation load per unit width of the wedge, mh, is the pressure at the wedge face and is given by

where rois the shear yield stress of the material.

v and X are related by the following equation;

228

Using equations (3, (6) and (7) it is possible to plot pm/270 against semi wedge angle for different coefficientsof Hction [12].

I

Ap

' I / ' : I

where R2 is the radius of the rolls, h ( = ( ho + hi )/2 ) is the mean sample thickness in the flow field and, a, is the projected linear contact length of a specimen with the rolls ( see Fig.5); hi and ho are the initial and final thicknesses of the sample respectively. The geometric factors in equation (9) define the mean imposed strain and, together with the roll velocity, also define the mean s W n rate. Numerically evaluated integral mean strain and strain rates are employed in practice (see ref. 13 ).

3. EXPERIMENTAL METHODS I I

FIGURE 5: Geometry of hvo roll milling. R2 is the roll radius and hi and ho are the initial and j i ~ thicknesses l of the sample respectively. P is the roll separating force per unit width of the

specimen.

2.5 Rolling In rolling, a hot metal rolling theory may

be applied which assumes a stick wall boundary condition. In this analysis, elements of material are assumed to undergo homogeneous plane strain compression: ( see fig, 5). That is, the mean vertical and horizontal stresses are taken as the principal stresses and, hence, are

interrelated by the following yield criterion;

where 3 is the mean uniaxial flow stress. A mean value description is employed since the stress will be a function of the strain and snain rate and these quantities will vary along the flow field. This model leads to the following expression for the total roll separating force per unit width, P [12]

3.1 Material The experiments were carried out on a model paste material known as "Plasticine". This material is a dispersion of clay particle (Kaoline, A14Si4010(OH)g) in a liquid (hydrocarbon) medium (78% wt. solid particle). The "Plasticine" was homogenised in a z-blade mixer before making specimens. Cylindrical and ring specimens of specified dimensions were prepared by mould cutters from slabs of material. These slabs were prepared using an Instron universal testing machine (model 1122). In order to facilitate removal of the sheets from the die, waxed paper inserts were used. For the rolling and wedge indentation experiments, slabs of accurately defined geometries were used. The specimens were themally equilibrated for at least 2 hours at 21 OC prior to the measurements. The interface traction was changed by introducing lubricants between the model paste and the steel surface of the platens. Talcum powder (MggSig020(0H)4) and a proprietary silicone grease were used as lubricants. The unequivocal stick boundary condition was achieved by utilising emery paper as the interface.

3.2 Experimental procedure

3.2.1

Cylinder Upsetting

Cylinder specimens of 80 mm diameter and 15 m m height were prepared. The cylinders were compressed between two over-hanging parallel steel platens. Experiments were carried

229

out under lubricated and unlubricated interface conditions. The mean compressive stresses as functions of aspect ratio and the natural strain were recorded. In addition, pressure distributions using unlubricated platens were measured for a number of strains. The experimental procedure involved the use of a local pressure transducer mounted in the lower platen [2].

3.2.2

Ring Compression

Specimens with an initial ring geometry of 63:2 (external radius : inner radius : height) and height of 10 mm were prepared. These samples were compressed between parallel platens at nominal strain rates of 0.1 and 0.4 s-l and the inner diamem were measured fora range of imposed compression strains. This was achieved by containing the specimens between "acetate" (poly (ethyleneterephthalate)) sheets which could be peeled away after each increment of strain. Measurements were made for both the unlubricated and the lubricated interface conditions.

3.2.3

Rolling

"Plasticine" slabs of width 80 mm, length 70 mm and thickness 7 mm were used to feed a two roll mill which had 78 mm diameter rolls, a roll speed range of 0.5 - 150 rpm and a minimum gap of 7 mm. Transducers were fitted to the roll bearings in order to measure the roll separation force which is tenned subsequently the "rollforce".

3.2.4 Wedge Indentation Wedge indentation tests were performed on thick "Plasticine" slabs and the ratio of the width of the sample to the indentation depth was kept more than 15. This was done to ensure that the condition of almost plane strain deformation was present during the wedge indentation. Stainless steel wedges of included angles 30°, 60°, 90°, 120° and 150° and of widths 180 mm were used for the indentation experiments. The wedges were driven into the samples to a fixed indentation depth of 6 mm. The indentation loads were recorded during the indentation process. Finally, the samples were removed from the wedges and the actual area of the contact was measured by

observing the residue impressions of the "Plasticine" on wedge faces; "Plasticine" naturally transfers a film of oil and particulate material to contacting surfaces. The indentation pressure was recorded against wedge included angles. For the "hot wedge test", the wedges were heated using heating caruidges. The wedge indentation experiments were carried out on an Instron universal testing machine.

4 RESULTS AND PRELIMINARY DISCUSSION 4.1 Traction Arrested Flows 4.1.1

Cylinder Upsetting

Figure 6 shows the uniaxial compressive stress against the natural strain for lubricated and unlubricated interface conditions. Obviously, the flow stresses obtained under perfectly lubricated conditions should correspond to the yield stress of the material provided that the wall traction is zero. A stick condition at the interface (i.e. emery paper, p=0.577) gives the highest apparent flow stress whereas, the recorded flow stress is the lowest for the lubricated case. Data for the lubricated interface show only slight strain dependence of the flow stress.

0.3

02

0.1

0.0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

FIGURE 6: Mean compressive stress as a function of the natural strain in cylinder upsetting of "Plasticine The interfaces employed are; a, emery paper (400); b, polished steel surface; c, talcum powder; d, silicone grease. 'I.

230

Figure 7 shows the measured local n o d pressure distribution against the position from the centre for cylinder upsetting recorded using small (2 mm diameter pin) normal stress transducers. This figure shows the variations in the lccal normal pressure at the wall for different 0 strains. The edge pressure is equal to the yield lf3 stress which is about 0.13 MPa for this case CE c-r [14]. This result is consistent with those 5 obtained in the tensile measurements [15]. Figure 8 shows the deconvoluted coefficient of 2 friction from these upseaing data obtained using 8 equation (2); assuming homogeneous deformation. The cylinder upsetting experiment, which is a case of the traction arrested flow condition, shows that the interface boundary conditions influence the material flow characteristics. A lubricated or frictionless boundary induces a relatively homogeneous deformation which is basically an extensional flow. Finite traction at the interface produces inhomogeneity and complex flow field. Hence, a large amount of redundant work is done at the boundary as frictional work and also extra energy is needed for the generation of the inhomogeneousdeformation of the bulk.

1.o

Unlubricrted

.s " 0.0

I

0.0

0.2

.

I

0.4

.

I

0.6

.

'

0.8

.

1 -r/n

FIGURE 7: Wall pressure distribution as a function of position during unlubricated upsetting of '~Plasticine"at strain rate of0.1 s-1 for the following uniaxial strains; JC 0.6,, b 0.47, o 0.33, 0 0.22, 0 0.066. a is the current contact radius of the cylinder and r is the radial positionsfrom the centre of the cylider.

1 .o

0.4

0.6

0.8

1.0

Compressive Natural Strain FIGURE 8: Experimentally obtained coefficients offriction against the natural strain during upsetting of a "Plasticine" cylinder for lubricated and unlubricated interfaces. p is &convolutedfrom using equation (2).

4.1.2

0.0

0.2

Ring Compression

Figure 9 shows the measured and theoretical values of the fractional change in the internal radius against the fraction compressive strain. This figure gives a good initial estimate of the prevailing interface friction. For the silicone grease lubricated case, the interface friction factor, m is of the order of 0.04. For talcum powder interface the friction coefficient is around 0.6 and the unlubricated case gives the value m of around unity i.e. the stick condition. For the finite friction cases, it is seen that the friction coefficient starts to reduce at beyond around 30% of strain. This indicates a change in the boundary condition from Tresca to a more complex case. The ring compression test provides an excellent example of a friction retarded flow process. A change in the interface condition can induce an entirely different flow field. As is shown in the figure 9, the change in the internal radius is dependent only upon the friction conditions at the interface between the platen and the material.

23 1

condition upon the material flow characteristics and the overall response of the material.

\ I

1

0

.

1

.

20

1

40

'

1

60

'

1

'

80

100

1

% Height Reduction

frictionless

0.0

FIGURE 9: % decrease in internal radius as a function of % decrease in height during ring (initial geometry 6:3:2) compression for "plasticine",Interfaces are emerypaper, o talcum powder and x silicone grease.

4.1.3

Wedge Indentation

The "hot wedge indentation" experiments were carried out to investigate the effect of interface temperatures on the boundary condition and the bulk flow characteristics. Hot wedges also provide a suitable means of generating the desired amount of slip at the interface, which is, often, a problem in many cases using external lubricants. Many effective lubricants induce plasticisation in pastes of this type. Figure 10 shows the effect of interface temperatures on the pmnq, characteristicsof the material against the wedge semi included angle. The value of T~ was taken as 0.08 MPa. which was obtained from the cylinder upsetting results using the von Mises yield criterion. It is observed, from this figure, that the main influence of the interface temperature is to reduce the wall traction which leads to a lubrication effect. The contact times are relatively short in these experiments; ca 7 s. Although the heat transfer characteristics have not been studied in detail we may reasonably assume that a relatively thin "hot" lubricating region is produced at the paste surface. The "hot wedge" experiment provides a simple means of investigating the effect of the interface boundary

20 40 60 80 SEMIWEDGEANGLE degree FIGURE 10: P d 2 T~ us a function of the 0

semi wedge angle in hot wedge indentation of "Plasticine". The bulk "Plasticine temperature was kept at 21 0C.The intelface temperatures are in OC; Z 40, o 50, o 60, b 70, 80.

4.2 Traction Induced Flow 4.2.1

Rolling

The mean flow stresses calculated from the roll forces using equation (13) are plotted as a function of the mean natural strain in figure 11. Each data set was obtained at different mean strain rates [15]. The tensile data, at corresponding strain rates are also displayed, for strains upto 0.1. These data points clearly show dependence of the flow stress Dpon strain and strain rate. A common practice in metal plasticity is to use a relationship of the form;

a .B

G,=BE

E

where uo is the uniaxial yield stress, & and & are the mean extensional strain and strain rate respectively and, B, a and p are material parameters. The figure also shows data from uniaxial tension experiments. The best fit of

232

equation (10) to the rolling data produces B, a and p as 0.69 MPa, 0.26 and 0.21 respectively. It is interesting to note that the material is almost equally sensitive to the imposed strains and strain rates; in this respect it shows combined "solid" and "liquid" behaviom.

0.0

0.2

0.4

0.6

0.8

1.0

MEAN STRAIN

1.2

dissipated in other deformation geometries. Improvements in experimentation and analytical methods will progressively provide better approximations. The central difficulty is that the wall friciton is responsible for the creation of highly inhomogeneous flow fields within the soft solid and these conditions are not readily pred~cted,UIany degree of accuracy, by ftrst order analytical methods. Hence, the external reaction forces and energy dissipation can not be predicted even if the necessary bulk and interface rheological data are known to a high degree of precision. Naturally the converse situation also applies; knowing the energy dissipation in a given situation does not readily provide a means for an accurate description of the bulk and interface characteristics. These comments apart, the present data and the supplementary analyses serve to emphasise the critical influence of wall friction on the energy dissipation involved during the deformation of soft solid materials.

6. FINAL REMARKS FIGURE 11: ElongatioMl stress as ofunction of mean natural strain for "Plasticine"comparing the rolling and fensile results. (Solid lines show the tensile &a and the dotted lines are calculated using equation (10)for the roll mill data) The mean strain rates in rolling are; 1.0; x 0.5; o 0.1; A 0.05.

5. CONCLUDING DISCUSSIONS A variety of experimental data, derived from several types of experiments, have been described which demonstrate the way in which the interface friction, between a soft solid and a rigid wall, influences the energy dissipation when flow is induced in the soft solid by the action of forces imposed by a rigid wall. The scientific challenge is to uniquely abstract, from a given experiment, the constitutive relationships which accurately prescribe the boundary friction conditions as well as the intrinsic rheology of the soft solid. The methods for achieving this result, offered in this paper, must be regarded as only satisfactory approximations in so far as they provide useful data for the numerical simulation of the energy

Interface friction has a profound influence upon the flow behaviour of soft plastic solids when they are deformed by rigid counterfaces. Modest changes in the wall friction lead to large changes in the energy dissipated by the system. The behaviour of these systems is not accurately described by the available fvst order analytical treatments although a number of experimental configurations and the associated first order models do provide an indication of the extent of the influence of boundary friction upon the magnitude of energy dissipation during plastic flow.

ACKNOWLEDGEMENTS The authors wish to acknowledge the financial and organisational support provided by the MAFFDTI LINK Scheme for the project on "The optimisation of' soft solid processing operations".

List of notations TO = shear yield stress of

the deforming m a t e d -T* = deviotoric stress tensor p = mean compressive pressure in upsetting and ring compressions.

233

00 = uniaxial yield stress h = thickness of the cylinder specimen. R = radius of the cylinder. p = Coulombic coefficient of friction m = friction factor zw = wall shear stress Ri = inner radius of ring = outer radius of ring Rn = neutral radius pm = mean indentation pressure in wedge indentation. pw = wall pressure at the wedge face. 2y = included wedge angle P = roll separating force per unit width of the - specimen h = mean sample thickness in rolling.

REFERENCES

5. S.D.Holdsworth, Trans. IChemE, Part C, Food and Bio product processes, vol. 71, (1993)139,

6. G.G.Lipscomb and M.M.Denn, J . N o n Newtonian Fluid Mech. 14(1984)337. 7. M.J.Adams, B.J.Briscoe and M.Kamyab, Adv. Colloid lnte$ Sci., 44(1993), 141. 8 . G.W.Rowe, Principles of Metal Working Process. Edward Arnold, London, 1977.

9. J.R.Scott; Trans. Inst. Rubber Ind. , 7(1931)169. 10. U.Yilmazer and D.M.Kalyon, 33(8)( 1989)1197.

J Rheo.,

11 B.Avitzur in "Metal forming: Processes and Analysis", Mc Graw - Hill Book Co. NY,(1968) 12. K.L.Johnson, in "Contact Mechanics", Cambridge University Press, Cambridge, 1985.

1. M.J.Adams, S.K.Biswas, B.J.Briscoe and S. Shamasundar, proc. The 1993 IChemE Research Event' IChemE, Rugby (1993)61.

13.J.Chakrabarty in "Theory of Plasticity", Mc Graw Hill Book Co.,NY (1987) p.582.

2. M.J.Adams, S.K.Biswas, B.J.Briscoe and M.Kamyab, Powder Tech., 65(1991)381.

14. M.J.Adams, B.Edmondson, D.G.Caughy and R.Yahya, J.Non Newtonian Fluid Mech. in press.

3. M.J.Adams and B.J.Briscoe, Trans. IChemE, Part C, Food and Bio product processes, in press.

15. M.J.Adams, S.K.Biswas, B.J.Briscoe and S.K.Sinha, Mat. Res. SOC.Symp. Proc., Material Research Society, (eds: L.J.Stuble, C.F.Zukoski and G.C.Maitland, Flow and Microstructure of dense suspensions) USA, 289 (1993)245.

4. R.Byron-Bird, G.C.Dai and B.J.Yarusso;

Reviews in Chemical Engineering, Vol.1, No. 1, (1983)l.

This Page Intentionally Left Blank

Dissipative Proccsses in Tribology / D. Dowson et al. (Editors) 0 1994 Elsevier Science B.V. All rights rescrved.

235

The Effect of Interfacial Temperature on Friction and Wear of Thermoplastics in the Thermal Control &@me Francis E. Kennedy and Xuefeng Tian Thayer School of En 'neering Dartmouth Col ege Hanover, NH 03755, USA

P

ABSTRACT This paper describes an experimental study of the sliding behavior of two thermoplastic polymers, polymethylmethacrylate (PMMA) and ultra-high molecular weight polyethylene (UHMWPE),in oscillatory sliding contact. Sliding surface temperatures were measured with the aid of miniature thin film surface thermocouples, and the results were related t o friction and wear behavior. The temperature measurements were also correlated with predictions of a recently-developed surface temperature model for finite bodies in oscillatory sliding contact. The experimental results showed that the contact temperature rise is dominated by a steady state nominal temperature rise, upon which is superimposed a cyclic local temperature rise. When the sliding speed and normal load were high, the measured peak surface temperature reached a ceiling value for the thermoplastic materials. The critical temperatures for PMMA was the temperature at which it softened while under compressive stress, approximately 30-35°C below the melting temperature of the material, whereas UHMWPE's critical temperature was found t o be very close to its melting temperature. At the critical temperature, the wear rate drastically increased and there was a large increase in the real area of contact, but there was not a large change in friction coefficient. The increase in contact area caused a decrease in frictional heat flux, thus limiting the surface temperature rise. Owing t o the large changes in wear and contact area which occur at the critical surface temperature of thermoplastic polymers, conditions which would result in such temperatures must be avoided in sliding components using those materials. In-situ surface temperature sensors could help insure that the sliding temperatures don't reach the critical values. 1. INTRODUCTION

I t is well known that the dissipation of frictional energy results in an increase in the surface temperature of sliding bodies. In many cases those surface temperatures can, in turn, influence the friction that occurs a t the interface. This re 'me of tribological behavior has been ca led by Ettles the 'thermal control regime' 111. Thermal control of friction could be particularly important for materials that melt at a low temperature, such as ice and snow, and for materials such as thermoplastic polymers which soften at temperatures which can be attained on sliding surfaces. For those cases, it was postulated that the interfacial temperature

T

rise due to frictional heating could reach a limiting value which is related t o the melting or softenin temperature or the contacting materia s 121. Accordin t o Ettles' theory, since the temperature f m i t could not be exceeded, this resulted in a limit on the coefficient of friction in severe sliding conditions. As it was formulated, the thermal control theory had several limitations: 1)it was limited t o friction and said nothing about wear; 2) i t is difficult t o predict the actual contact temperature in sliding contacts, so the theory could not be used easily to predict when failure will occur. The objective of this work is t o extend the thermal control theory so that it can be used in predicting wear failure of thermoplastic components.

f

236

It has long been known that sliding surface temperatures have a strong influence on both friction and wear of thermoplastics [3]. The relationship between tem erature and friction or wear is not simple, owever but also depends on such parameters as siiding velocity, contact pressure, and the materials themselves [471. Lancaster and his co-workers found that the wear rate of many thermoplastics increases si nificantly when t h e temperature o f t e polymer surface exceeds a critical value 13, 81. They also found that p a r a m e t e r s which affect surface temperature, particularly sliding velocity, normal load, and ambient temperature, can all cause the surface temperature to reach the critical value [31. In fact, Lancaster showed that the 'PV limit', which is often used in the design of dry plastic bearings, is in reality a 'critical surface temperature limit'; the combination of pressure and velocity which causes severe wear of the polymer is that which causes the surface temperature to reach a critical temperature related t o the softening temperature of the material [81.

R

5

In most of the past research reports, the temperature of interest was either the ambient temperature or the local or "flash" temperature rise due to frictional heating. It has been found, however, t h a t the maximum surface temperature actually has three contributions: background temperature, local temperature rise and nominal surface temperature rise of the contacting materials [91. The background temperature is affected by ambient temperature and any temperature increases caused by the bearing support or housing, while the nominal temperature rise is a general change in the temperature of the entire contactin surface brought about by movement o the real area of contact. The local temperature rise occurs within the real area of contact and is due to frictional heat generated there.

P

Man models have been created to determine t e total contact temperature ratter rise in sliding contacts [8-101, but all such models require knowledge of friction force and the dimensions of the real contact spots. Since such information is not generally known a priori, the models cannot be easily used in a design situation

K

when polymer components a r e to be designed against tribological failure. An alternative approach is proposed in this paper. 2. METHODOLOGY

A combination of experimental and analytical methods was used in this work. 2.1 Experimental Study The objective of the experimental study was to measure the actual sliding contact temperature for several different thermoplastic materials in contact with ceramic and metallic counterfaces, and t o relate the peak contact temperature to the measured friction a n d wear of the polymers. Oscillatory sliding tests were carried out using a test apparatus which had been used in earlier studies of polymer wear [71. The tester has stationary polymer pins in contact with a n oscillatin flat specimen. In this work, the flat-ende pins had 2 mm x 4 mm cross-section (4 mm direction in the sliding direction). The oscillation amplitude was 4 mm and the fre uency ran ed from 0 to 20 Hz.Normal loa s ranging rom 9.8 to 52.4 N were used in the study. The specimen holders for both pin and flat s ecimens were fitted with thermoelectric eaters which enabled their background (or bulk) temperature to be controlled at temperatures ranging from 15OC below room temperature t o 50°C above room temperature. All tests were run in air.

8

a

7

R

To enable contact temperature to be measured, thin film thermocouples (TFTC) were fabricated on the contact surface of the flat specimens. TFTC are extremely small temperature sensors which are deposited on the surface of a sliding component a n d enable t h e accurate determination of actual sliding surface temperatures without requiring any further modification of the sliding components [ l l l . The structure and fabrication procedures of the thin film thermocouples are described elsewhere [lll. For this work, nickel-copper TFTC were used and they generally were about 0.5 pm thick with a measuring junction about 100 pm square. To protect the TFTC from damage, a thin (0.2 pm thick) layer of A1203 was deposited over

237

the thermocouples. The measuring junctions were located at the center of the nominal contact area between pin and flat when the test system was at rest.

developed model [91. The model assumes that total contact temperature is composed of three contributions, as in the following equation:

During a test, a chosen normal force was applied to the top of the polymer pin, background temperatures were set to their desired values, and the flat specimen was set in motion at the chosen oscillatory frequency. The friction force was measured by a piezolectric force transducer and the linear wear of the pin specimen w a s monitored using a displacement transducer (LVDT). Contact temperature, friction force, and linear wear were all monitored continuously with the aid of a computerized data acquisition system a n d a chart recorder.

TtOtal= ATlocal+ ATnominal+ Tbackground

2.2 Materials

Two quite different thermoplastic polymers were tested in the work reported here. One was polymethyl methacrylate (PMMA), an amorphous polymer with a glass transition temperature of about 105°C. A cast grade of PMMA without impact modifiers was used in this study. With the aid of a differential scanning calorimeter (DSC), the meltin point of the PMMA material was found to e 195-203OC. The second material was ultra-high molecular weight polyethylene, a linear, semi-crystalline polymer with a molecular weight ranging between 3 and 4 million. Its melting temperature range was measured to be 138-141OC.

%

For most tests the counterface was a flat glass slide, but stainless steel (304 SS) were used in some tests. The sliding surface of the stainless steel specimens was coated with a thin (approximately 2 - 5 pm thick) dielectric layer of Al2O3 to insulate the surface thermocouples from the stainless steel s u b s t r a t e . Thin film Ni-Cu thermocouples were deposited on the slidin surfaces of glass and stainless stee specimens for surface temperature measurement, as described above.

7

2.3 Analytical Study To accompany the experimental study, analytical predictions of surface temperature were made using a recently-

(1)

a

The back round temperature is assumed to be nown, while the nominal and local (or flash) temperature rises can be calculated as long as contact area, specimen eometry, velocity, and friction force are fnown, as are the thermal properties of the contacting materials. Following t h e procedures developed in reference [91, the nominal t e m p e r a t u r e increase was determined in this work by assuming that a percentage of the frictional h e a t is distributed uniformly over the area on the flat specimen which is swept out by the oscillating pin. The remaining frictional h e a t was assumed t o be uniformly distributed over the nominal contact area of the pin. Steady-state h e a t conduction solutions were used to find ATnominal for both flat and pin specimens. The maximum local temperature rise (ATl, ) was calculated using expressions for ?lash temperature described in a recent paper [121. Measured values of friction force and real contact area were used in the expressions, and the peak sliding velocity was used. The same heat partitioning coefficient was used in the expressions for both nominal and local temperature rise, and that coefficient was determined by setting the maximum total temperatures on the pin and flat s ecimens equal to each other (the Blok postu ate).

Y

3. RESULTS A typical plot of measured surface as a temperature rise ( ATl-1 + ATqominal) function of time during a test IS shown in Figure 1. The test in question had a UHMWPE pin in contact with an oscillating glass flat s ecimen. The measured friction coefficient uring the test was 0.14 and the background temperature was constant at 25OC. It can be seen that the temperature rise consists of a steady nominal temperature rise (ATnominal = 73.9'0 upon which a cyclic local temperature rise was added. The tem erature reached a peak twice during eac cycle, at the times when the absolute value of the sliding velocity

s

R

238

was at a maximum. The Deak measured surface temperature in this 'case was Tm 1 = 117.8"C (= 25" + 73.9" + 18.9'). T%is temperature is well below the melting temperature of UHMWPE a n d t h e measured wear rate in this case was rather low; the conclusion was that the sliding conditions were not severe enough to be in the thermal control regime.

[

;.

i

.

! .

:

:

!

:

C

i

;

;

['C

!

90.9 1 68.18

.

22.5

.

22.6

.

.

.

22.7

.

.

.

22.8

8

22.9

Seconds

Figure 1. Measured surface temperature in oscillatory slidin test of UHMWPE in against glass flat. scillation amplitu e = 4 mm, frequency = 10.8 Hz, normal load = 52.4 N.

B

8

To test the ability of the surface temperature model to determine the surface temperature, the measured value of friction coefficient was used, along with a post-test measurement of contact area. (The contact area was estimated by measuring the area of the wear scar on the pin surface.) Using those values, the following temperature rises were calculated:

= 71.4"C ATno,.,.,ina,

AT1i,

= 18.3"C

Using these values in equation (11, along with the known back ound temperature of = 114.7"C. 25"C, one gets a pre icted T, This value differs by less than 3% from the measured value. The heat artitionin factor was also calculated for t is case an it was found that over 94% of the generated frictional h e a t entered the flat glass specimen. This was due in large part to the low thermal conductivity of the polymer.

f

R

that during the first second of testing the surface temperature rose steadily owing to a steady increase i n t h e nominal was rather large temperature rise. during the first few oscillations, due to the large oscillations, due t o the large friction coefficient of this material combination (approximately 0.5). After about 1 second, however, the local temperature rise became smaller, even as the nominal temperature continued t o rise. This led t o a total temperature which remained approximately constant at about 1625°C 25°C). Continued testin of the (Tbac PMl!@?$i under the same con itions showed t h a t t h e peak temperature remained relatively constant at a value near or just below the 162°C level. There was a very high wear rate of the polymer pin d u r i n g t h i s t e s t , along with considerable evidence of softening and deformation of the pin (to be described later).

i

A test of PMMA under conditions even less severe than those used in Figure 1 led to quite different results. Figure 2 shows the measured surface temperature rise during the first 3.5 seconds of a test of PMMA pin vs. glass flat. It can be seen

.---____

-

I

lg3.j

.___-_ --!1 3 7 5

I

,

/

I

.?

0

1

i

4

I

2 1

'

I

2 8

/

,

3 5

Seconas

Figure 2. Measured surface temperature during first 3.5 seconds of oscillatory sliding test of PMMA pin against glass flat. Oscillation amplitude = 4 mm, frequency = 8.0 Hz, normal load = 52.4 N. Tests of both PMMA and UHMWPE materials were run at a large number of operating conditions leading t o a wide r a n e of surface temperatures. The resu ting wear rates are shown in Figures 3 and 4 as a function of the measured total surface tem erature. The wear coefficient used in the igures is defined as the volume lost per unit sliding distance per unit normal load, and it has units of m2/N. I t is evident from the figures that both materials experienced relatively low wear until the peak surface temperature reached a critical value. For the PMMA material, as noted above, the critical temperature was about 162"C, whereas for the UHMWPE material the critical temperature was about 137°C.

7

P

2 39

The critical temperature for UHMWPE is approximately equal t o the melting temperature of that material. For PMMA, however, the critical temperature is about 30-35°Cbelow its melting temperature and is determined by the temperature at which the PMMA material softens while under compressive stress. It might also be noted that the wear coefficient of UHMWPE was much lower than that for PMMA, even in the severe wear regime at Tdtical.

* + + ++

.-. ,

.

80

. , ". , 100

#.

120

.

,

I

140

.

PMMArestdata

X

0.6-

0.8

0.4

-

0.2

'I*

0.0,.

I 1

.

i

x I

.

.

8

.

.

x I

'

'

% 8

'

I

Figure 4. Wear data for PMMA pins in oscillatory sliding against glass flats at different contact temperatures. The wear coefficient is defined as volume lost per unit jliding distance per unit normal load.

+

0.060

1 .o

,

160

Ternprnture. "C

Figure 3.Wear data for UHMWPE pins in oscillatory sliding against glass flats at different contact temperatures. The wear coefficient is defined as volume lost er unit sliding distance per unit normal oad.

P

For many of the test points shown in Figures 3 and 4,the surface tem erature was varied by changin the bac ground temperature. It was Eund that small changes in background temperature had a great influence on wear rate when the total surface temperature was close t o the critical value. An example of this is shown in Fi re 5, which shows linear wear of a UH&PE pin a s the background temperature was varied and all other test conditions were held constant. In this case the total surface temperature reached Tcritical when the background temperature was about 51°C. If the temperature was just 3" below this value (the segment from a to b) the wear rate wasn't too severe. A 5°C rise in background temperature above that value, however, was sufficient to cause a transition t o very severe wear (segment from b to c). When the temperature was then dropped back down t o 50°C, the wear rate decreased back to the same value i t had at th a t tern erature at the beginning of the test. There ore, the increase in wear

R

f!

Time, hr

Figure 5. Wear rate as function of for UHMWPE pin in test against glass flat. Oscillation amplitude = 4 mm, fre uency = 11.3 Hz, normal load = 36.5 . = 50°C;ab, "background = 350 MPa)

Y

3

m

0.2 -

No. Ref. dp 0' -1 0

I

I

170

I

I

I

I

530

350

I

- (wrn)

I

710

oyy ( M W

Figure 20. Solid Fraction Venus Compression: MoS,, CeF,, NiO (Extrapolated for uyy> 350 MPa)

I

890 93581

325 3.0

2.4

TiO, c 0.4 (Anatose) 1.8

1.2

.6

No. Ref. d, 0 -10

62

278

- (urn) 350 93579

Figure 21. Bulk Density versus Compression: TiO,

326 Table 2. QIA of Particulate Ensemble of TiO, (Anatase Form)

Statistical Parameter

Compressed Face

Fractured Face

Sheared Face

0.36 0.008 0.092 0.128to 0.735

0.34 0.008 0.089 0.164to 0.59

Equivalent Diameter (pm)

Mean Value Variance Standard Deviation Limits

Mean Value Variance Standard Deviation Limits

I

0.33 0.010 0.101 0.167to 0.78

1.22 0.235 0.485 0.527to 3.85

I

I

1.33 0.103 0.322 0.575to 2.59

1.17 0.083 0.289 0.571 to 1.90

0.016 0.126

0.79 0.009 0.096 0.528to 0.987

Shape Factor

Mean Value Variance 0.027 Standard Deviation 0.164 Limits

P447

327

wn

I

t

-U

89579-1

Figure 22. Transverse and Longitudinal Variation in Density with Powder Lubrication

5c

5

10

h h

x

‘K

3

0.6

C

.-c. 0 0

e

un 0

0.4

a

0.2 U

Dh = 0.5 I

-0

0.2

I

I

0.4

I

I

I

0.6

I

0.8

Normalized Film Thickness (y/h)

Figure 23. Proposed Mathematical Model for Variation of Solid Fraction Across Powder Film

1.o 93582

328

Table 3.

Fuzzy Subset Values: Triboparticulate Shape Memberships

I

BN

MoS,

0.4

0.2

0.3

:nMoOzSz

%

cz3 6)@

a /

&

I

Triboparticulates Co losition

0.7

0.6

0.3

0.3

0.5

O3

0.8

0.8

0.5

0.3

0.05

0.1

0.3

0.9

0.8

0.05

0.2

0.4

0.05

0.4

0.05

0.2

0.3

0.05

0.05

0.1

0.6

0.6

935790

Dissipative Processes in l’ribology / D. Dowson et al. (Editors) 1994 Elsevier Science R.V.

329

Surface Chemistry Effects on Friction of Ni-P/PTFE Composite Coatings E.A. Rosset, S. Mischler, D. Landolt Materials Department, Ecole Polytechnique F6dCrale de Lausanne, 1015 Lausanne, Switzlerland Electroless and galvanic deposition are inexpensive and flexible methods for the fabrication of functional metal films on a wide range of substrates. These techniques are also suitable to produce metal matrix composite coatings such as self-lubricating coatingscontainingPTFEspheres in a metallic matrix. In the present work the tribological behaviour of electroless Ni-P/P”FE metal matrix composite coatings has been studied in air and in pure nitrogen by using a reciprocating motion test rig. The chemical state of the surface was characterised by Auger Electron Spectroscopy. Results are interpreted by considering the third body concept proposed by Godet and co-workers.

1. INTRODUCTION Despite the wide application range of self lubricating Ni-P/lTFEcompositecoatings[1,2,3], the friction and wear mechanisms of this class of materials is only partially understood [4,5]. In the literatureit is generally assumed that PTFEalone is responsible for the self lubricating properties of the composite even if surface oxides should, in principle, play a role in the tribological phenomena. The aim of the present paper is to investigate the influence of the surface composition on the dry sliding behaviour of an electrolessNi-P/PTFE Composite coating. For this friction and wear tests are performed in air and, in order to avoid surfaceoxidation, in a pure nitrogen atmosphere using a reciprocating motion wear test rig. Frictional coefficients and wear rates are measured and the surface composition and morphology of the wear scar are determined by Auger Electron Spectroscopy (AES)and ScanningElectronMicroscopy(SEM), respectively. For comparison the tribological behaviour of a Ni-P coating was investigated in air under the same conditions.

2. EXPERIMENTAL

Sliding friction conditions were established by rubbing a 1 0 0 0 6 steel ball (lOmm diameter) against a Ni-P/PTFE or a Ni-P coating plated on aluminium. The Ni-P coating was 10 pm thick and smooth (Ra0.05, cut off length 0.8 mm) The composite coating exhibited a rough surface (Ra 0.8 1, cut off length 0.8 mm) and the average film thickness was 9 pm. The PTFE content of the compositecoating was 30% PTFEby volume and the averagediameter of the PTFE spheres was 0.4 pm. Thephosphorus content of the matrix was 9% for both coatings. No heat treatment was performed on the coated sample. The hardness of the Nip coating was about 500HV corresponding to an approximate yield strength of 1200 MPa. Frictional tests were carried out in the reciprocating ball-on-plate rig described in more details elsewhere [6]. During the test the frictional and the normal force as well as the horizontal and vertical displacement of the rubbing interface were monitored using a Macintosh IIfx computer (Labview2 software from National Instruments). The coefficient of friction was determined by dividing the frictional and thenormal forcesmeas-

330

ured when the pin was in the middle of the wear scar. The ball was oscillating at a frequency of 5 Hz. The stroke length of 4 mm corresponded to an averagesliding velocity of 0.040 m/s. The applied normal load was of 0.6 N. Prior to the test the samples were cleaned in an ultrasonic ethanol bath, rinsed with fresh ethanol and dried using an argon jet. After the friction test the samples were transferred to the AES analysis system for chemical characterisation. Subsequently their morphology was studied using SEM and stylus profilometry. Some tests were carried out in a glove box allowing for working in a nitrogen atmosphere (measured oxygen level below 5 ppm). The transfer of the samples after the wear test to the AES system was carried out in the laboratory atmosphere. Each experiment was repeated three times in order to check for reproducibility. The surface compositionwas determined in a Perkin Elmer 660 Scanning Auger Microscope using a lOkeV (50nA)electron beam. Surface analysis was carried out by focusing the electron beam (beam diameter of 1pm) on selected points of the sample surface.

Depth profile acquisition was performed by rasteringa2keVAr-tbeamoveranareaof 1.5x1.5 mm. Under these conditions the sputter rate of Ta205 was 0.5 nm/min. The surface morphology was observed with a Cambridge Stereoscan 650 SEM or a JEOL 6300 F SEM. Wear track profiles were measured using a Taylor-Hobson Talysurf 10 profilometer.

3 RESULTS Ni-P in air: Generally the coefficients of friction were found not to depend significantlyon sliding distance and therefore the average p values listed in Tab.1 for each experiment can be used to characterisethe frictional behaviour of the systems studied here. The values of Tab. 1 were determined by calculating the average and the standard deviation of the coefficients of friction measured every 20s during sliding. The frictional behaviour of the Ni-P coating is characterisedby a coefficient of friction of 0.36 (average of three experiments) and a standard deviation of about 30% due to important oscillations in the p values during sliding. Such oscillations are usually observed on this rig when a significant amount of wear particles is present in the contact. ~~

Table 1 Averaged coeilkients &friction Sample Atmosphere

p: average

p:standarddeviation

Ni-P

air

0.38 0.34 0.35

0.12 0.09 0.09

Ni-PPTFE

air

0.12 0.11 0.14

0.03 0.03

0.17 0.18

0.03 0.04

0.16

0.04

Ni-P/PTFE

N2

0.05

33 1

----

The wear behadour of Ni-P was characterised by relative littledamage to the coated surface as shownbytheweartrackprofile(Fig. 1).Important wear of the steel ball and some material transfer from the coating to the steel was observed N P In Alr. 18'0ooStmb8 N P +TFE in Ah, 18'Ooo slrobs

induces shadowing effects of the ion beam used for sputtering. If locally the oxide film is not

N M InAlr, 11QOOO Irokcls

Figure 1. Wear track profiles (Talysurf). by SEM (Fig.2~). Wear particles were found aggregating at the end of the wear track (Fig. 2a). The size distribution was not uniform ranging from a maximum of 1 pm down to less than 0.1 pm with an aspect ratio near to the unity. Ploughinginducedridges and alarge number of cracks were observed using SEM on the Ni-P wear track (Fig.2b). This indicates that delamination by subsurface crack propagation was the dominant wear mechanism. The thickness of the delaminated sheets was about 1 p. Cracks were also observed on the steel ball. A typical Auger profile measured on the NiP wear track is shown in Fig.3a. It indicates the presence on the metal surface of a thin oxide film (about 2 nm thick). Carbon is present as a surface contamination only. ?his is suggested by the fact the carbon signal goes to zero at a depth of 0.3 nm. No iron was found on the wear track surfaceon the coating. Theoxygenprofiledoes not show asharp interface possibly because the surface roughness

Figure 2. NiP in air, 18'000 strokes. a) wear track b) detail of wear track c) steel ball

332

a

substrate contact as consequence of wear of the coating. The SEM investigation and the wear track profiles recorded at different sliding times indi-

I NIP In Air

-------.------------0

2

4

6

8

lo

Approx. Depth [nm] Figure 3. Auger depth profiles measured on wear tracks. removed by sputteringit contributesto the overall Auger signal even at large sputter times. Because of this effect no quantification of the profile of Fig. 3a was carried out. Surface analysis could not be performed on the wear debris and on the worn surface of the ball because of charging due probably to an important surface oxidation. Ni-PPTFE in air:The frictional behaviour (Tab. 1) of the composite coating sliding against a 100Cr6 ball was characterised by a relatively constant coefficient of friction of 0.12 (average value of 3 experiments, standard deviation 8 96). Oscillations and an average increase in p were observed only after about 18oooO strokes and were attributed to the beginning of the ball-

Figure 4. Ni-P / PTFE in air a) wear track, 1'800 strokes b) wear track (endon theright), 11O'OOOstrokes c) steel ball, 11O'OOOstrokes. Light grey zone correspondsto the material transferredfrom the coating

333

cates that asperity deformation (Fig. 4a) was followed after about 18'000 strokes by general plastic flow of the compositecoating (Fig. 4b). Some laminated sheet separating from the substrate are also observed by SEM (Fig.4b) on the wear track . Their thickness is about 0.1 pm, significantly lower than in the case of Ni-P coatings (Fig.2b).Important material transfer was observed on the ball (Fig.&) while wear debris generation was negligible. An Auger depth profile measured on a deformed surface similar to that pictured in Fig. 4a is reported in Fig. 3b. Again no iron was found here. Similar profile were obtained at different slidingdistances (1800,18000and 1loo00 strokes) on deformed zones of the coating and on the transferred layer on the ball. The simultaneouspresence of Carbon,Fluorine and Oxygen together with Nickel and Phosphorous observed by AES indicates the presence of thin surface films of P E E and a NickelPhosphorous oxide. The thickness of the oxide layer correspond to about 2 nm, i.e. in the same order of magnitude as the value observed on the Ni-P coating. The apparent thickness of thePTFE determined in Fig. 4b is about 1 nm, but is believed to be higher since the sputter rate of polymers is about 2-3 time higher than that for oxides. From the measured profile it is not possible to determine if the oxide and the PTFE layers coexist on the surface as islands or if the one layer is covered by the other. Scanning Auger mapping results obtained with a lateral resolution of 50 nm and not presented here show however a homogeneous surface distribution of the elements. This and the well known trend of PTFE to form thin interfacial films under tribological conditions [7] suggest the presence of a PTFE film covering the oxidised surface of deformed asperities. Further experimental evidence is needed to confirm this hypothesis.

N i - P m F E in N,: Rubbing the steel ball against the composite coating in absence of oxygen lead to an average coefficient of friction of 0.17 as shown in Tab.1. The wear rate in nitrogen is much higher than in air as seen in Fig. 1 a n d 5 The profilometry shows that the wear scar after 1800 strokes in nitrogen is nearly six microns deep. In comparison more than 1loo00 strokes are needed to wear off 5 pm in air. The important difference in wear

Figure 5. Ni-P / PTFE, N,, wear track, 1'800 strokes rate is evident when comparing the surface morphology after the same number of strokes: after 1800 strokes in air only some asperities have interactedwith the ball (Fig. 4a) whilst in nitrogen generalisedplasticdeformationhas occurzed (Fig. 5). The wear morphology in nitrogen does not differ significantly from the one observed in air flow on the wear track. This indicates that the presence of oxygen in the atmosphere influences the wear rate but not the wear mechanism. Surface analysis indicates that the Ni-PI PTFE wear track surface is covered with a few nm thick PTFE film (Fig. 3c). Because of its high surface reactivity, the metallic nickel is expected to oxidise during the transfer of the sample from the glove box to the Auger analysis system. However no oxygen was found by AES. This indicates

334

that PTFE film forms an adherent and compact layer on the metal surface acting as barrier against oxidation. Very thin PTFE films formed during sliding of PTFE containing polymer composites have been observed recently by Fletcher et a1 using imaging XPS [S]. DISCUSSION The obtained results show that the tribological behaviour of the analysed coatings depends critically on the presence of PTFE in the material and on the test atmosphere. In order to understand the mechanismsinvolvedit is convenient to make use of the concept of third body presented i.e. by Godet et al. [9] The simple three body contact model proposed by these authors includes two first bodies (in our case the coated sample and the steel ball), the third body and the two third-body/ first body interfaces also called screens. By separating the two first bodies the third body transmits the load from one first body to the other and accommodates by dissipation most of the velocity difference between the first bodies. A list of the nature of third bodies and screens possibly present under the present test conditionsisproposedinTab.2. Itis assumedthat the transfer layer observed on the ball as well as the plastic deformed zone on the coatings are part of the first bodies. The experimental results obtained here can be explained by considering the effect of the presence of screens and different

third bodies in the tribological systems investigated. Third bodies: During rubbing of the Ni-P coating against the steel ball particles are generated by delamination of the two first bodies. Once detached and before being ejected from the wear scar the particles form a powder bed (the third body) separating the two first bodies. Due to the inhomogeneous nature of the powder bed the load is not distributed uniformly in the contact so that locally the contact pressure may reach the critical stress required for subsurface crack nucleation and propagation in the first bodies. Under the assumption that the stress required for crack nucleation in delamination corresponds approximately to four times the shear yield strength of the deposit [ 101 one obtains a critical stress of about 2400 MPa (the shear yield strength is assumed to correspond to half the yield strength). The fact that this value is well above the nominal contact stress calculated according to the Hertzian formalism (400 m a ) support the hypothesis of the inhomogeneous load carrying behaviour of the particles bed. The third body formed during rubbing of the compositecoating consists in a low shear strength PTFE film formed by plastic deformation during the contact of the PTFE spheres present at the coating surface.This PTFE film can be assimilated to athin lubricant film and as such it reduces friction and distributes the load homogeneously in the contact so that the probability to reach the critical stress for subsurface cracking is much lower than in the case of Ni-P. According to the

Table 2

Summary of results and proposed mechanisms Sample Atmosphere p 1 Wear Wear mechanism NiP

Air

NiP-PTFE Air NiP-PTFE N2

0.36 0.12 0.17

1) Average of three experiments

low low severe

Delamination Asperity Deformation Asperity Deformation

Thirdbody

Screen

Weardebris P"FEfilm PTFEfilm

Oxide Oxide None

335

delamination theory of wear [ 101the thickness of the delaminatedsheets correspondsto the depth of crack nucleation which is proportional to the friction force and the local normal stress. This explains the formation of thinner sheets on the composite coating than on the Ni-P coating. Due to the different nature of the third body only little wear by delamination occurs during sliding of the composite coating, however. The damage is limited to the surface where asperity deform and a adhesive wear take place. Screens: The thin lubricating PTFE film can break down locally when first body asperities interact strongly or as a consequence of scuffing phenomena (thermally induced film breaking or insufficient film-substrate adhesion). In such an event metal-metal contact and adhesion may occur leading to the observed transfer of material to the ball and to the wear of the coating. The oxide screen can prevent metal-metal contact thus limiting the extent of adhesive wear. During rubbing in nitrogen no oxide screen is formed. This may explain the higher wear rate. Another application of this is that the presence of the oxide may improve the adhesion of the PTFE film, thus reducing the possibility of scuffing phenomena. Therefore, altough the wear mechanism of Ni-P / PTFE in air and in nitrogen is conditioned by the nature of the third body, the wear rate is governed by the presence or not of oxide screens. The oxide screen also plays a similar antiadhesive function during rubbing of the Ni-P coating in air. This explains why no significant material transfer was observed in air. The third body / screen approach allows one to explain the mechanismsleading to the observed sliding behaviour of Ni-P and Ni-P/PTFE coatings. However further experimentalwork is needed in order to better understand the role of intrinsic material properties, surfacecomposition and third

bodies for the tribological behaviour of these materials. CONCLUSION Under the present experimental conditions the wear and frictional behaviour of self lubricating Ni -P/PTFE compositecoatings aredetermined by the formation of very thin surface layers of PTFE or Ni-P oxide. The observed differences in tribological behaviour of Ni-P/PTFE coatings in air and nitrogen stress the important role of surface oxidation for friction and wear.

Acknowledgement This work was financially supported by the CERS (Bern). The coatings were manufactured by Steiger SA (Vevey). The authors thank M. P. Mettraux for the SEM analysis. REFERENCES 1. R.N. Duncan Hardness and wear resistance of electroless Nickel-Teflon composite coatings Metal Finishing, 9 (1987), 33-34 2. P.R. Ebdon Composite electroless NickeVPTFE coatings Surface Engineering, 3 (1987), 114-116 3. E. Steiger Nickelage chimique avec incorporation de FTFE Oberflkhe-Surface, 10 (1990), 19-22 4. K.Matsukawa,A. IchikawaandM. Kobayashi Effects of Heat treatment on tribological properties of electroless Nickel- P h o s p h o r o u s coated steel Proc. Japan International Tribology Conference, Nagoya (1990), 7-12 5. T. Sakamoto, M. Abo, 0. Takano and M. Nishira Friction and wear of electroless Nickel-PTFE composite coating

336

Roc. Japan International Tribology Conference, Nagoya (1990), 31-36 6. E.A. Rosset, S. Mischler and D. Landolt Wear and Friction Behaviour of Ni-Sic Composite Coatings in Thin Films in Tribology (Leeds-Lyon 19) D. Dowson et al (Editors), Elsevier Science Publishers B.V .(1993), 101- 108 7. S.K. Biswas and K.Vijayan Friction and wear of PTFE - a review Wear, 158 (1992), 193-211 8. I.W. Fletcher, M.Davies and D. Briggs Surface modifications introduced to a Pol ytetrafluorethylene-filled Polycarbonate compound by dry sliding against steel as revealed by imaging X P S Surface and Interface Analysis, 18 (1992), 303-305 9. M.Godet, Y.Berthier and J. Lancaster Wear modelling: using fundamental understanding or practical experience Wear, 149 (1991), 325-340 1O.N.P. Suh Tribophysics Prentice-Hall Inc., Englewood Cliffs, New Jersey (1986)

Dissipative Processes in 'I'ribology / 1994 Elsevier Science B.V.

D.Dowson et a]. (Editors)

337

Transfer layers in tribological contacts with diamond-like coatings J. Vihersaloa,H. Ronkainena,S . Varjusa,J. Likonenband J. Koskinenc "Technical Research Centre of Finland (VlT) , Laboratory of Production Engineering, P.O.Box 1 1 1, FIN-0215 1 Espoo, Finland bTechnical Research Centre of Finland (VTT), Reactor Laboratory, P.O.Box 200, FIN-02151 Espoo, Finland "Technical Research Centre of Finland (V'IT), Metallurgy Laboratory, P.O.Box 113, FIN-02151 Espoo, Finland

The tribological properties of amorphous hydrogenated carbon (a-C:H) films deposited on steel AISI 440 B in a radio frequency (RF) assisted plasma were studied. The films were studied using a pin-on-disc machine with steel AISI 52100 and alumina as the counterface materials. The sliding velocities varied from 0.1 to 3.0 m / s and the normal forces from 5 to 40 N. The tests were carried out unlubricated in room air at 22k2 "Ctemperature and with 50*5 % relative humidity. Tribofilms formed on the pin and coating wear surfaces were studied by secondary ion mass spectrometry (SIMS). The coefficient of friction strongly depended on both the sliding velocity and the load. For the steel pin sliding against coating the coefficient of friction varied from p=0.42 to p=O.l and for the alumina pin sliding against coating the coefficient of friction varied from p 4 . 1 3 to p=0.02. The formation of a tribofilm on the pin had a significant effect on both the friction and the wear properties of the coating.

1. INTRODUCTION

2. EXPERIMENTAL PROCEDURE

Diamond-like carbon films have favourable tribological properties, which make them good candidates for many tribological applications. The tribological performance of diamond-like carbon coatings can be affected by e.g. the deposition method [ 11. The environment has proved to have a considerable effect on the friction and wear properties of diamond-like carbon coatings [2-41. The tribological tests have typically been carried out with low sliding velocities and moderate normal loads. Only in a few studies have higher loads and velocities been applied.

2.1. Coating deposition

In previous studies we have evaluated the performance of hydrogen free carbon coatings[5,6]. In the present study the authors have evaluated the hydrogenated carbon coatings deposited in a RF plasma. In order to study the effect of the frictional power input and the tribofilm formation on the tribological performance of these diamond-like carbon coatings, the pin-on-disc tests were carried out in a wide range of sliding velocities and normal loads. The wear surfaces were characterized and analysed by SIMS.

The amorphous hydrogenated carbon films (a-C:H) were deposited in a RF-plasma using methane (CH,) as the process gas. In the deposition chamber the substrates were placed directly on a water-cooled cathode. The deposition parameters were kept constant during the a-C:H film deposition. The pressure during the deposition was 5.5 Pa and the bias voltage was -550 V. The deposition temperature was close to 100 "C. Prior to deposition all substrates were cleaned by ultrasonic washing in 1,1,2-trichloro-1,2,2-trifluoroethane followed by rinsing in ethanol and drying in hot air. Finally the substrates were sputter cleaned in argon atmosphere for 5 minutes at a bias voltage of -300 V and a pressure of 0.7 Pa. For improving the adhesion of the coating to the substrate, a titanium carbide (Tic) layer was applied as an intermediate layer.The Tic layer was deposited

338

discharge using methane as the process gas. During the Tic deposition the bias voltage was -680 V and the pressure was 0.8 Pa.

d -

E

@b

30-

'u

it P

-&

d

20-

10-

d d

d

b

Sliding velocity v [mh]

Figure 2. Experimental parameters in pin-on-disc tests.

Figure 1. The principle of the deposition chamber Stainless steel AISI 440B (DIN X 90 CrMoV 18) was applied as the substrate. The hardness of the substrate material was 620 HV,,, and the surface roughness (R,) after the deposition was 0.03 pm. The total thickness of the coating was about 1.O pm. The Tic-layer was up to 0.5 m thick and the a-C:H-film on the top was about 0.5 pm thick. 2.2. Tribological experiments

The tribological experiments were carried out as pinon-disc tests in order to evaluate the effect of the load and the sliding velocity on the tribological behaviour of the a-C:H coatings. Polished 10 mm balls manufactured from steel AISI 52100 (100Cr6) and alumina (-A1203)were used as pins. The tests were performed unlubricated in room air at 2222 "C temperature and 50*5 % relative humidity. The sliding distance in the tests was 2000 m. Both the normal force and the sliding velocity were vaned within a wide range. The parameters were chosen for evaluating the effect of the normal force and the sliding velocity separately and as a combination of both. The normal force varied in the range 5 to 40 N and the sliding velocity in the range 0.1 to 3.0 m/s.

The friction force was measured during the tests. The pin wear rates were calculated from ball wear scar diameters, which were determined by optical microscopy. The disc wear rates were calculated as an average value from four profilograms taken across the disc wear track. All the presented values are mean values of two tests at a specific test parameter combination. The film composition was determined by Rutherford backscattering spectroscopy (RBS). The hydrogen content of the sample was measured using standard forward recoil spectroscopy (FRES). The wear surfaces of the pins and the coated discs were analyzed by optical microscopy and SIMS. The SIMSanalyses were carried out using a VG IX 70s doublefocusing magnetic sector instrument. Ga ions with energies of 10 keV in depth profiling and 22 keV in elemental imaging were used as primary ions.The primary ion current was typically 10 - 15 nA during depth profiling and 1 nA in the elemental imaging. To achieve uniform bombardment, the focused ion beam was raster-scanned over an area of 140 x 175 pm2in depth profiling. The alumina pins were analyzed with optical microscopy only.

339

3. RESULTS 3.1. Tribological evaluation The coefficient of friction showed a strong dependence on both the normal force and sliding velocity. The highest values of the coefficient of friction were measured when a low load and a low sliding velocity were applied. An increase in load and/or sliding velocity caused a decrease in the coefficient of friction. In Fig. 3 the coefficient of friction for a steel pin sliding against the coating is presented. The highest value of the coefficient of friction (p=0.42) was achieved at a normal load of 5 N and a sliding velocity of 0.1 m/s. The lowest value of the coefficient of friction (p = 0.10) was achieved when a high load (FN=35N)and a high sliding velocity (v=2.6m/s)were applied.

Figure 4. The coefficient of friction at the end of the pin-on-disc tests with alumina pin against a-C:Hcoatings. The specific wear rate (later wear rate) i.e. the wear volume divided by the load and the sliding distance decreased slightly when the load and sliding velocity were increased separately. An increase in both the sliding velocity and the normal load led to a strong reduction in the wear rate of the pin (Fig. 5). The wear volume of the pin remained almost constant when both the load and the sliding velocity were increased simultaneously, but increased when the load was increased at a low sliding velocity.

Figure 3. The coefficient of friction at the end of the pin-on-disc tests with steel pins against a-C:H - coatings. Fig. 4 presents the coefficient of friction when the coatings were tested against aluminapins. The highest value, p = 0.13, was achieved when the normal load was 5 Nand sliding velocity 0.1 m/s. The lowest value, p = 0.02, was achieved when a load 22 N and a sliding velocity of 1.5 m / s were applied.

Figure 5. The Wear rate Of pins in the pin-on-disc tests with steel pins against a-C:H coating.

340

The wear rate of the coated disc when sliding against a steel pin exhibited a systematic behaviour when either the sliding velocity or the load or both were increased, being highest when a low sliding velocity and a low normal force were applied and lowest when a high sliding velocity and a high load were applied (Fig. 6).

Figure 6. The wear rate of the disc in the pin-on-disc tests with steel pins against a-C:H coatings. For aluminapins sliding against a-C:H coateddiscs no systematic wear behaviour was observed in terms of the wear rate. An increase in the normal load from 5 Nto22 Nledtoalowerwearrateofthepin, butwhen a 40 N normal load was applied the wear rate was increased but was still lower than at the 5 N normal load (Fig. 7).

The disc wear rate when sliding against alumina pins also behaved systematically when both the sliding velocity or the load or both were increased. The highest wear rate was observed when a low sliding velocity and a low normal force were applied, and the lowest wear rate was observed when a high sliding velocity and a high normal force were applied (Fig. 8).

Figure 8. The wear rate of the disc in pin-on-disc tests with alumina pins against a-C:H coatings.

3.2. Coating characterization According to the FRES analysis the coatings contained about 26 at.-% hydrogen. The RBS analysis showed that the intermediate layer beneath the a-C:Hfilm consisted of a Titanium layer and a Tio,z8Co,,z layer, approximately 100 nm and 400 nm thick, respectively.

3.3. Characterization of the wear surfaces The SIMS elemental imaging revealed that elements (Fe,Cr) from the steel pin were transferred onto the aC:H coated disc. A higher wear volume of the steel pin correlated with increased concentrations of iron and chromium in the wear track. The thickness of the layer, which contained material from the pin, was typically about 10 nm and it seemed to be independent of the sliding velocity and the load used in the pinon-disc tests (see Fig. 9). Figure 7. The wear rate of the pin in pin-on-disc tests with alumina pins against a-C:H

34 1

The formation of a tribofilm on the steel pins was observed with all test parameter combinations. At higher sliding velocities the appearance of the tribofilm was more evident. The thickness of the film was not constant across the contact area (Fig. 1 1..14).

10

-

10

u)

n 0

Y

T, l o 3 E

m .u)

10

10'

t

i

10 O

0

150

100

50

200

Depth [nm]

Figure 9. A S M S depth profile of a disc wear track from a test with steel against an a-C:H coating at 5 N load and 0.1 m / s sliding velocity.

Figure 11. A micrograph showing a steel pin surface (5N,0.1 ds).

0

10

20

30

Time [min]

Figure 10. A SIMS depth profile of a pin wear track from a test with steel against and a-C:H coating at 35 N load and 2.6 m / s sliding velocity. Accordingto~SIMS~p~p-ofilesthetin~ pin weartracks showedchromiumdepletionandiron enrichment.Furthertothis,thetribofilmcontainedhydrogenandoxygen.Thetriboslmspmbablyconsistedof amixture of iron oxide and chromium oxide. SIMS resultsindicatedthatthemtion layercontainedasurFigure 12. A micrograph showing a steel pin surface prisinglylowamountofcarbon.(Fig. 10). (40 N, 0.1 d s ) .

342

tracks after tests against steel. Different features of formed layers on the alumina pin surfaces are presented in figures15 ...18.

Figure 13. A micrograph showing a steel pin surface (5 N. 3.0 d s ) .

Figure 15. A micrograph showing an alumina pin surface ( 5 N, 0.1 d s ) .

Figure 14. A micrograph showing a steel pin surface (35 N, 2.6 d s ) .

When the alumina pin was sliding against the hydrogenated a-C:H coating, aluminium was transferred onto the disc wear track. The tribofilm formed on the pin wear track was transparent. As the worn volume of the alumina pin was lower the amount of aluminium in the disc wear track was smaller than the amount of iron and chromium in the a-C:H wear

Figure 16. A micrograph showing an alumina pin surface (40 N, 0.1 d s ) .

343

When investigating the pin wear surfaces, the formation of tribofilm seems to be more evident at higher loads and sliding velocities. The tribofilm covered more when the frictional power input in the contact was higher and when a higher contact pressure was applied.

The tribofilm has a strong influence on the tribological behaviour of the a-C:H-film. The thickness of the film on the pin was not constant across the contact area. When the tribofilm was thin the adhesive contacts between the transferred material in the disc wear track and pin possibly caused a high coefficient of friction. A thicker tribofilm on the pin acted as an protective layer decreasing the pin wear. Figure 17. A micrograph showing an alumina pin surface ( 5 N, 3.0 d s ) .

Based on the friction results and pin wear surface morphology the properties of the tribofilm governed the frictional behaviour. According to the SIMS analyses the tribofilmon the steel pin contains mainly iron, chromium, oxygen and hydrogen, but carbon content was surprisinglylow. The oxygen was probably bonded to the iron and chromium. A formation of a graphite layer on the steel pin has been reported as an explanation for the low coefficient of friction with a-C:H-films [ 5 ] . However, in the present study the formed layer contaned mainly iron, chromium, oxygen and hydrogen, which indicates that the role of the graphite leading to a low coefficient of friction is not so obvious.

CONCLUSIONS

Figure 18. A micrograph showing an alumina pin surface (35 N, 2.6 d s ) .

DISCUSSION The coefficient of friction showed a significant decrease when the sliding velocity and the load were increased. For the steel pin sliding against the coating the coefficient of friction varied from p=0.42to p=O. 1 and for the alumina pin sliding against coating the coefficient of friction varied from p=O. 13 to p=0.02.

Tribofilm formation governed the tribological properties of a-C:H films. The formation of protective tribofilm reduced the pin wear and the coefficient of friction with steel pins. The coating wear volume generally increased, whereas coating wear rate decreased with higher loads. When higher sliding velocities were used the coating wear volume slightly decreased. The tribofilms formed on the steel pin wear surfaces consisted of iron and chromium oxides, carbon and hydrogen.

344

REFERENCES 1. A. Grill and V. Patel, Diamond and related mate rials, 2 (1993) 597.

2 K.Enke, H. Dimigen and H. Hubsch, Appl. Phys. Lett. 36 (4), 1980 29 1. 3. K. Miyoshi, P. Pouch and S.A. Alterovitz, NASA TM 102379 (1989) 4. D.S. Kim, T. E. Fischer and B. Gallois, Surface

and Coatings Technology, 49 (1991) 537. 5 . H. Ronkainen, J. Koskinen, A. Anttila, K. Holmberg and J.-P. Hirvonen, Diamond and related Materials, 1 (1992) 639.

6. H. Ronkainen, J. Likonen, J. Koskinen, Surface and Coatings technology, 54/55 (1992) 570.

Dissipative Processes in Tribology / D. Dowson et al. (Editors) 0 1994 Elsevier Science B.V. All rights reserved.

345

SURFACE BREAKING CRACK INFLUENCE ON CONTACT CONDITIONS. ROLE OF INTERFACIAL CRACK FRICTION. THEORETICAL AND EXPERIMENTAL ANALYSIS M.C. DUBOURG" ,T. ZEGHLOUL"" ,B. VILLECHAISE""

* Laboratoire de MCcanique des Contacts, UR4 CNRS 856, INSA, 20 Av. A. Einstein, Bdt. 113, 69621 Villeurbanne Cedex, France. ** Laboratoire de Mecanique des Solides, URA CNRS 861, UniversitC de Poitiers, 861 Av. du Recteur Pineau, 86022 Poitiers Ctdex, France Abstract Numerous studies are devoted to the determination of two body contact conditions, i.e the contact area, the stick and slip zone repartition, the normal and tangential pressure distributions. Interfacial roughness, friction, worn profiles for instance are taking into account as they disturb the hertzian stress field, but no specific attention is paid to the influence of surface breaking cracks. The mutual influence of surface breaking cracks on two body rolling contact conditions was studied theoreticalyy in a previous paper by Dubourg and Kalker [l]. Significative overpressure relatively to the classical maximum hertzian pressure and split up of the contact area were obtained numerically. An original experimental simulation is undertaken to validate these results. The theoretical model is the combination of a two body rolling contact model and a fatigue crack model. The steady rolling contact between the wheel and the rail is solved as a unilateral contact problem with friction. Displacement and stress expressions derive from Boussinesq and Cermti potentials. The fatigue crack model is based on distributions of dislocations for crack modelling and unilateral contact analysis with friction for the contact solution between crack faces. These two problems are solved in turn as displacements generated by cracks modify the two body surface geometry. This process goes on until convergence is reached, i.e when the two body contact coqditions are stabilised from one iteration to the next. The experimental work is based on photoelastic technique. Birefringent slabs for both wheel and the cracked rail are employed. Isochromatic fields, normal and tangential loads and global displacements are recorded continuously during the loading. Visualization and calculation of pressure peaks in the wheel and extent of the contact area are performed. 1. INTRODUCTION

In a previous theoretical study [I] the mutual influence of surface breaking cracks on the wheel-rail contact conditions was investigated. Modifications of the traction distributions with and without split up of the contact area were determined, depending on the relative position of the wheel with respect to the cracks. This paper is concerned with the experimental validation of these results. The experimental part includes the simulation of the contacting movement of the wheel relatively to the cracked rail with vizualisation through photoelasticity technique of the

isochromatic fields in both contacting bodies. Theoretical simulation of this experiment will be conducted simultaneously. 2. THEORETICAL APPROACH A steady contact model and a fatigue crack model including frictional locking at crack interface are combined. Connection between the two problems is introduced through surface geometry modification caused by displacements generated by cracks. Both models are half analytical and numerical that imply inexpensive computer time and great accuracy.

346

d

2.1 Two-body contact

Steady state normal contact of a cylinder (the wheel) over an elastic half-plane (the rail) is considered. Perfectly elastic conditions are considered, the solids are homogeneous and isotropic. A theory of contact is required to predict the shape of the area of contact, the slip and stick zone repartition, the magnitude and distributions of surface tractions, normal and possibly tangential, transmitted accross the interface (cf. figure 1).

way that satisfies the boundary conditions along the faces of the presumed cracks.

-P

J

p ------0

/-i Figure 2 : Fatigue crack model I

h

slip zone

stick zone

Figure 1 : WheeYrail contact model This contact problem is solved as a unilateral contact problem following the method developed by Carneiro Esteves at al. [2]. As semi-infinite bodies are considered, Boussinesq and Cemti potentials are used. Relations between displacements and stresses are obtained. The potential area of contact is discretized into segments on wich stresses are assumed constant. 2.2 Crack model

These boundary conditions (cf. figure 3) are expressed as in a : contact zone

-

k=0,

4,( 0

(1)

d = O ,

&,)O

(2)

- open zone:

- backward slip zone: - 4=f.4, ~ , * a J t ) 0 - forward slip zone d=-f*d 4*dt)0 9

I

(d

(4)

-stick zone &t

A theoretical two-dimensional linear elastic model of multiple fatigue cracks was developed [3,4] to determine the stress and displacement fields in cracked solids and the stress intensity factors (SIFs) at crack tips. Multiple interactive cracks, straight or kinked, surface breaking or not, taking into account frictional locking and situated in an isotropic medium can be modelled (cf. figure 2). The model rests on the continuous dislocation theory, pionnered by Keer and Bryant [5,6] and on the unilateral contact theory developed by Kalker [7]. Resultant stress and displacement fields ,FU , FV ) are given by superposing the uncracked solid ( $"" ) and the crack ( d,6u, Fv) responses to the load in such a

(3 )

=0

I

ldtl(f

openzone I

stick zone backwardslipzone

----

Figure 3 : Boundary conditions at crack interface

(5)

347

The continuum stress f l C in the uncracked solid may be obtained numerically (finite element analysis for instance) or analytically in the case of a halfplane. The crack response corresponds to displacement discontinuities along its faces, opening and slip, that generate stresses. These displacement zones are modelled with continuous distributions of dislocations bx and by. Single distributions of dislocations bx and by are considered along each crack. It is assumed that by and bx are square root singular at crack tips, and at crack mouths for embedded cracks, bounded elsewhere. The correct behaviour of the stress field along cracks is thus guarantee. The strength of these singularities is then driven numerically to zero in the case of a contact zone or a stick zone at crack tip [3]. Consistent equations come from corresponding boundary conditions (6u, = 0, 6u, = 0). This method gives single stress and displacement expressions for the whole crack, independent from the final contact division:

g([ w w ~ . Y " . a

&+

or=9 &)

jb,(S)K,:(x.yq.Rd5 rl

i j = x,y

I

(6)

r;

hn= jbIJOd5

(8)

r;

where p is the shear modulus, k=3-4v for plane strain or (3-v)/(1+v) for plane stress, v the Poisson's ratio, Kvx,K; the stress kernels expressed in [3], crack 1 profile, m the number of cracks. Stress expressions are singular integral equations, solved following Erdogan et a1 [S]. Discretized stress and displacement expressions are obtained. The 2NI unknown are the bx and by values at the discretisation points, where NI is defined by

r,

rn

NI = c p i , p, the number of discretisation points i=I

for crack i. The contact problem solution between crack faces as a unilateral contact problem with friction gives automatically the contact area division, slip, stick and open zones, and the suitable distributions of dislocations. Load cycles are

described with an incremental description which takes into account the load history as hysteresis is generated by friction at crack interface. This model was used to determine the stress intensity factors experienced at crack tips under various loading conditions, sliding or rolling contact conditions, bulk tractions ... 2.3 Crack influence on two-body conditions

contact

Conditions at interface between the wheel and the rail influence significantly contact stresses. Surface roughness, interfacial friction, worn profiles are taken into account. No specific attention is paid to crack influence. For convenience of data treatement, the global problem is split into two parts, the two-body contact and the crack problems, which are solved independantly in turn. Connection between the two problems is introduced through surface geometly modification caused by displacements generated by cracks. Note that the displacement field generated by cracks is continuous except along the crack where displacement discontinuities correspond to slip or opening. The normal displacement at the smooth half-plane surface Vsurf is calculated at the discretization points of the contact area 2a from bx and by. Iterations on crack influence are organized in the following manner (cf. figure 4): First iteration: the surface geometry considered H(y) corresponds to a cylinder over a smooth half-plane. The contact problem is solved as exposed in the section 11.1. The contact area 2a and the normal traction p(y) are determined. The crack behaviour is then determined. Surface displacement V,,rf is calculated. H(y) is modified, h'(y) = H(y) + relax* V_ surf. Next iteration: h'(y) is considered as the new profile. The two-body contact is solved again. Continuuum stress field at iteration j is the resultant stress field calculated at the (i-1) iteration. Variations in normal tractions 6p cause variations in distributions dislocations 6bx and 6by. Slip and opening along crack faces are modified and SIFs too. Surface displacement Vsurf induces by theses variations is determined agam and added to the geometry. This

348

process goes on until1 convergence is reached, i.e normal tractions exerted on the half-plane surface are stabilized from one iteration to the next. For the next load step, the geometry is reinitiated to Hb) GIVEN LOAD STEP

iterafion n"l :

hertzian analysis fatigue crack model

-

2a, P,9 Vsud

iferation ny :

2a', p', q'

-

6p = p' - p 6q = q* - 4

bodies are presented on figure 5 . The loading frame is placed in an optical system of photoelasticity. Photoelastic picture acquisition is realised by the use of a CCD camera for numerical treatement. Both wheel and rail scale models are made in polyurethane PSMl of 9.6 mm width, a birefringent material sold by the Vishay-Micromesures society. The loading frame was realised to ensure the plane stress assumption. The apparatus is made of two parts where the two contacting bodies are fixed. These two parts can move in two perpendicular directions corresponding to normal and tangential loadings. The horizontal guide is realised through two ball columns. The vertical displacement compresses the force sensor N providing the wheel normal loading. The wheelrail sytem is based on a gas slider. The cracked rail scale model is realised from two trapezoi'd reamed parts. These two parts are partially sticked together along their inclined face, forming thus a parallepiped. The unsticked remaining surface corresponds to the crack. The stiffness of the glue, once dry, is identical to those of the polyurethane. This technique guarantees a good geometric definition of the crack.

--

E-2400MPa

Figure 4: Crack influence on wheelhail contact conditions.

v

3. EXPEFUMENTAL APPROACH

t-10rnrn b - 15 mm

C

R

The contact formation and behaviour and particularly in the case of frictional contact is still not yet understood in some cases. A formal definition of the problem particularly in terms of boundary conditions for theoretical modelling is difficult. Understanding frictional phenomena and interactive mechanisms between two body contact and surface breaking cracks on one hand and determing the key parameters on the other hand require an experimental analysis of the contact evolution. The work presented here is concerned with normal contact of a wheel over a cracked rail, and more precisely with the study of the influence of cracks on the contact conditions. 3.1 Loading frame

The experimental set-up and both the geometrical and mechanical characteristics of the two contacting

038

3.5 N/mmiFna~c 1000 mm

rl

h

Figure 5 : Experimental set-up 3.2 Tests

The main goal of the tests realised during this study is to show the influence of a crack on the contact conditions at the wheelhail interface, i.e changes in the contact conditions during the passage of the wheel over the cracked rail. In this first approach the rotating movement of the wheel is not realised

349

experimentally. Only the normal loading is performed. A load step consists in normal loading, unloading and wheel displacement. The running conditions are a normal load of 290 N, a wheel radius of 1 m that correspond to a contact area 2a of 10,5 mm. The crack length b is equal to 15 mm and its inclination is 8 = -40'. The position of the wheel is defined with respect of the trailing edge of the loading zone yt. yt varies from -27 mm to 11 mm with a 2 mm displacement increment. 19 load steps are thus realised. The isochromatic field is numerized for each wheel position.

model describes in that case both the overpressure and the split up effects. These phenomena are due to the location of the crack mouth inside the contact area.

Step 4: yt 2 5 m m The wheel moves away from the cracked region which therefore influences no more the wheel-rail contact. The behaviour observed is similar to those described in step 1: 4. NUMERICAL SIMULATION

3.3 Analysis of the isochromatic field evolution A phenomenological analysis of the tests was conducted. At yt = -10mm the leading edge of the contact area moves over the crack mouth and at yt = 0 mm the trailing edge moves away from crack mouth. Depending on the wheel position several steps were observed:

Step 1: yt I

- 25 m m

The normal loading is imposed by a vertical displacement of the wheel. The isochromatic field changes continuously and is slightly disymmetric. This disymmetry is due to a torque associated to macrogeometric difects created at the manufacturing stage of the wheel scale model. During this step, the the wheel is far from the crack and no interacting effect between them is observed.

Step 2:

- 23 m m

I yt I -11 m m

The wheel rolls nearer the crack; from yt = 9.64mm the wheels is above the crack tip. Crack faces are pressed together and sheared. Further stress concentration at crack tip is observed, corresponding to the slip of the left part of the rail. This stress concentration increases when the wheel comes nearer.

Step 3: - 9 m m I yt I 3 m m The surface breaking crack modifies the contact conditions at the rail surface. The pressure distribution along the contact area is modified locally with split up of the contact area in two parts without modification of its extent Accordingly the numerical

Concerning the two-body contact simulation, a potential contact area of 12 mm discretized with 121 points is considered. The maximum hertzian pressure Po is equal to 3.67 MPa and the contact area is 10,47 mm width. The origin of the reference axis is placed at the crack mouth. 30 discretization points are distributed along the crack. Different friction coefficient values were tested, ranging from 0.1 to 0.3. The best match between theoretical and experimental stress field is obtained for a friction coefficient equal to 0.1.The relaxation coefficient relax is equal to 0.3. For all the load steps the crack behaviour is determined (slip, stick and open zone distribution), the stress intensity factors are calculated (figure ) and the stress fields computed over an area of width 45 mm along oy and of length 17 mm over ox. The width is centered with respect to y = -7.5 mm. The mutual influence of crack on the contact conditions generates: stress intensity factor variations, pressure distribution modifications with local overpressures, contact area width changes, - split up of the contact area.

-

Step 1: No significative influence: yt I - 11 mm

No significative mutual influence between crack and the two-body contact is noticed as long as the wheel is not situated over the crack mouth, i.e yt less than -1 1 mm. At the load step 1, the the crack is partially open at its mouth, then a forward slip zone holds and the crack tip is sticking. KII is very small 55 Padm. The state of the crack changes at load step 2: from crack

350

Figure 6 a) : l o a d s t e p 6

yT =

-

1 7 mm

35 1

F i g u r e 6 b) : load step 9

yT =

-

11 mm

352

-2.0

-7.0

-12.0

-17.0

F i g u r e 6 c : l o a d s t e p 10

YT -

5 t h coupling i t e r a t i o n

-

9mm

353

Figure 6 d) : load s t e p 12 yT 4th coupling i t e r a t i o n

-

-

5

m

354

-R.S -10.2 -11.9

-13.6

-15.3 -17.C

-16.5

yT 4 t h coupling i t e r a t i o n

F i g u r e 6 e ) : load s t e p 14

-

1mm

355

-1.7

-

-3.4

-

-5.1

-

-6.8

-

-8.5

-

0.0

-1u.2-11.9-

F i g u r e 6 f) : l o a d s t e p 16

yT = 3 mm

356

tip to crack mouth a bakward slip zone, a stick zone, a forward slip zone and finally an open zone are distributed. This state doesn't change up to load step 9 included.

A qualitative comparison is performed between the experimental and the theoretical results. It is based on

- mainly on informations concerning the two-body

Step 2 : Influence: -9 mm I yt I-1 mm 4 to 5 iterations are needed to converge. Different

mechanisms are observed: -load step 10: yt = -9 mm: the leading edge of the contact area is situated over the crack mouth. A global backward slip zone holds along the crack. Coupling leads firstly to split up of the contact area in two parts situated on each side of the crack faces with local overpressures up to 6.8 MPa and increase in KII from 11250 to 113 10 Padm. But at the next iteration the contact area is again in one part, the pressure distribution is very similar to the initial one, and the stress intensity factors too. -load steps 11 and 12: yt = -7 mm and yt = 5mm: A particular distribution of displacement zones is observed: the crack is open at crack tip, and a backward slip zone spreads from it up to crack mouth. The compressive action of the two-body loading closes obviously the crack faces at crack mouth. Further the crack tip is situated outside of the compressive zone and therefore an open zone holds there. Coupling leads for load step 11 to overpressure up to 4.49 MPa, i.e a 22% increasing without modification of the contact area width. For the load step 12, overpressure up to 4.7 MPa, i.e a 28 % increasing is noted. For both load steps, MI variations are very small, less than 1%. -load step 13: coupling is negligible -load step 14: yt = -1 mm A global forward slip zone holds along the crack. A sligth increasing in the maximum pressure is noted, from 3,67 to 3,69 MPa. But split up of the contact area in two parts is obtained: from -1 to to 0.05 mm and from 0.15 to 9.7 mm.

-

Step 3: No more influence: yt 2 1 mm An open zone spreads out from the crack mouth, then a forward slip zone at crack tip. 5. COMPARISON

-

contact conditions and their modifications due to the crack influence. Split up of the contact area is observed for yt varying from -9 to 3 mm. This behaviour is numerically obtained for yt = lmm. experimental and theoretical isochromatic fields, presented for load steps 6, 9, 10, 12, 14, 16 (cf. figure 6). They are very much alike.

CONCLUSION

The influence of surface breaking cracks on two-body rolling contact conditions was previously studied theoretically by one of the authors. Significative overpressure relatively to the maximum hertzian pressure and slip up of the contact area were obtained numerically. An original experimental simulation was undertaken to validate these results. This experimental work is based on the photoelastic technique. Comparison between isochromatic fields and computed stress fields, experimental and theretical two-body contact behaviour (slip up of the contact area) confirms the previous results. REFERENCES

M.C. Dubourg, J.J. Kalker. Crack behaviour under rolling contact fatigue. In "Rail quality and maintenance for modern railway operation". Proceedings of the International Conference on "Rail quality and maintenance for modem railway operation", Delft (NL), June 1992, Ed. by J.J Kalker, D.F. Cannon, 0. Orringer, Kluwer Academic Publishers, p. 373-384, 1993. A. Carneiro Esteves, J. Seabra, D. Berthe. Roughness frequency analysis and particle depth. In Interface dynamics, Proceedings of the 14eme Leeds-Lyon Symposium, 8-1 1 Sept. 1987. Ed. by D. Downson, C.M. Taylor, M. Godet, D. Berthe. Amsterdam: Elsevier, p 209213, 1988.

M.C. Dubourg, B. Villechaise. Analysis of multiple fatigue cracks Part I: theory. ASME,

-

357

Journal of Tribology, Vol 114, pp. 455-461, 1992. 4

M.C. Dubourg, B. Villechaise, M. Godet. Analysis of multiple fatigue cracks - Part 11: results. ASME, Journal of Tribology, Vol 114, pp. 462-468, 1992.

5

L.M. Keer, M.D. Bryant and Hiratos G.K. Subsurface and surface cracking due to hertzian contact. ASME, Journal of lubrication technology, Vol. 104,, pp. 347-351, 1982.

6

L.M. Keer and M.D. Bryant. A pitting model for rolling contact fatigue. ASME, Journal of lubrication technology, Vol. 105, , pp. 198-205, 1983.

7

J.J. Kalker. Three-dimensional elastic bodies in rolling contact. Kluwer Academic Publishers, 1990,3 14 p.

8

F. Erdogan, G.D. Gupta and T.S. Cook. Numerical solution of a singular integral equation. In Method analysis and solution of crack problems. Ed. by Sih, Leyden, Nordhoff International Publishing, p. 368-425, 1973.

This Page Intentionally Left Blank

SESSION VIII MACROSCOPIC ASPECTS, FRICTION MECHANISMS Chairman:

Professor K L Johnson

Paper Vlll (i)

The Generation by Friction and Deformation of the Restraining Characteristics of Drawbeads in Sheet Metal Forming Theoretical and Experimental Approach.

-

Paper Vlll (ii)

A Model for the Estimation of Damping in Helical Strand Under Bending Vibration.

Paper Vlll (iii)

Energy Dissipation and Crack Initiation in Fretting Fatigue.

This Page Intentionally Left Blank

Dissipative Processes in Tribology / D. Dowson ct al. (Editors) 0 1994 Elsevier Science B.V. AU rights reserved.

36 1

The generation by friction and plastic deformation of the restraining characteristics of drawbeads in sheet metal forming - Theoretical and experimental approach E. Felder and V. Samper Groupe Surfaces et Tribologie - Ecole des Mines de Paris CEMEF - URA CNRS 1374 BP 207 , F 06904 Sophia-Antipolis Cedex, France During its drawing through the drawbead, the sheet undergoes successive bending cycles and friction along the surface of the shoulders and the bead. Starting from the work of Stoughton (1988) and Marciniak & Duncan (1989), we develop a simple theoretical model for describing the behaviour of an anisotropic sheet material whose flow stress is a power function of strain 00 = K en; an energy balance provides the contribution of plastic deformation to drawing load; then the holding force is deduced from a stress balance; the analysis of stress and strain increment distributions accross the sheet thickness provides the thickness reduction. The reliability of the rheological assumptions for steel and aluminium sheets is discussed by comparison with experiments performed between rollers by Nine (negligible friction) at the maximal depth of bead penetration. Then we study theoretically the influence of the depth of bead penetration and friction on the loads for strips holded by rollers in the original plane of entry (case 1) or sliding on blankholder (case 2) Reliability of the model is discussed by comparison with experiments published by Stoughton (case 1) or performed on a super formative sheet SPC 3C and a Zn-Ni coated sheet HPC 35 (case 2). Endly we study experimentally the restraining characteristics on the sheet HPC 35 of single and multiple drawbeads; they can be described by a single apparent friction law; the geometry of the drawbeads defines the range of accessible holding forces, but an increase in the number of drawbeads produces a significant decrease in the efficiency of the system and a significant increase in the thickness reduction of the strip. which is a power function of the drawing load. 1 .INTRODUCTION

Sheet metal forming for manufacturing autobody parts involves various plastic deformation modes, particularly drawing or simple sliding between die and blankholder; the stress balance induces at the boundary between these both modes marked shear and ultimately rupture of the blank. So drawbeads are implemented along the sliding parts of the workpiece in order to homogenize the metal flow towards the punch; but a good design of the operation, especially by numerical simulation (1-4) requires the knowledge of their restraining characteristics, namely the evolution with the depth of bead penetration 6 of the holding H and drawing F forces. Despite the previous work performed by Nine (8,9), Stoughton (11) and Yellup & Coll. (12,13) among others, the influence of the sheet rheology, friction, the sheet clamping system or number of drawbeads cannot be predicted easily. We aim to provide such information by theoretical and/or experimental work. Before any calculation, we can remark that

drawbead induces plane strain of the sheet; so forces increase in direct relation with the strip width w and are characterized by their value by unit width and the apparent friction coefficient pa :

For theoretical calculations, we assume that the strip is rigid- plastic and yields according to the Hill's quadratic plastic criterion; so its behaviour is characterized by the normal anisotropy factor r and the evolution of the effective stress 00 versus the effective strain E; we assume monotonic strain hardening Hollomon law:

Hill's plasticity criterion implies that any plane principal strain increment dep induces an effective strain increment de (11) :

362

work required for performing such strain increment for an unit length of strip; so 2. Forces generated by deformation

In this part, we assume that the friction of the strip on tools is negliaible; so forces depend only on the rheologicarpropertiesof the strip. 2.1 Theoretical model We assume that in the drawbead the shape of the mid-surface of the strip consists of three circular arcs with the radius Rb (angle 244 in the central part and R, in the lateral parts (angle @) (figure la). According to the elementary theory of the bending of thin strips (7),the effective strain increases in direct relation with the distance to the mid-surface y (figure 1c) and has the value at the limit between the two first circle arcs (with t the strip thickness):

t/2 AEl(Y) f d l ) = 2 jdy

End&

0 Kt (a)ltfl fdl)= (l+n)(2+n) R,

where the variation of the strip thickness for the moment is assumed negligibleSimilarly, we obtain (cf figure 1b) Kt

f ~ ( 2 ) = (1+n)(2+n)

1

1

( a t (-+-))'+" R, Rb

2 1 Kt (at(-+-))l+n fo = fo(3) = (l+n)(2+n) Rs Rb

(6)

(7)

The force balance of the central part of the strip and the bead provides the related holding load The related drawing force f o ( l ) is the plastic

ho(2)=( fo(l)+-fo(2)) sin (I

(8)

Figure 1: Analysis of the strip deformation in the drawbead a) Assumed geometry of the mid-surface b) Force balance on the bead c) stress and strain increment across the strip thickness

363

As demonstrated by previous calculations, the bending cycles are induced by increasing axial stress ox = Wt;. by an analysis of the stress and strain increment distribution across the strip thickness (cf figure l c ) Duncan & Marciniak (7) have demonstrated that any increase in the tensile bending strain (d&M)/a induces the thickness variation:

0.2. Kt

Steel sheets

0.18.

/

/'

/'

I

aoo

,/*

oo is some mean effective stress across the strip thicknes and is equal to a first approximation i I

IKEndde= K(AEM)" l+n

%=AEM

so as X'

0,

Aluminium sheets n 0.25-0.29

I+r

v

AEM

n=

-

0.14.

1F x -_d _ - -dEM t

Theory

fo

A

1

0.08

b

I

0.09

t

G%G ., 1

0.1

.

0.11

0

--_

K(A&M)~+~d AEM =(l+n)(2+n) t - a(2+n) 'EM I

and by integration, we obtain the final strip thickness: (AE3M)2

1

In(-)= t 2a(2+n) = 2a(2+n) (

2

1

~ ( G + )*G(9))

2.2 Comparison with experiments Nine (8,9) has performed measuments of drawing and holding loads for drawing two aluminium and steel sheets (0,761110.99mm) between rollers of radius p = 5.5 and 4.75 mm at the maximal depth of bead penetration 6 ~ ; for his experimental conditions where the clearance is very small (LdP+t&M) (ci fig lb)

Conditions of strip clamping imply that ho(2) is equal to the the total holding force (cf 9 3), so formula (7) and (8) imply that

In order to compare the behaviour of the various alloys, we have reported with logarithmic scale on fig. 2 the reduced drawing force fo/Kt versus the increase in effective strain induced by one bending Ae3~/6.

Figure 2: Comparison between the theoretical drawing force and the experimental one (8,9) for negligible friction and maximal depth of bead penetration (drawing speed V=85 mmls) In such representation, the theoretical values are located on a straight line which depends only on the value of the strain-hardening exponent n. Notice that the rheological data are related to classical uniaxial tensile testing performed at very low strain rate 10-3 s-l in the range of strain ~ ~ 0whereas . 2 drawing through the drawbead at the velocity V induces cyclic hardening at high strain rate = V/R 520 s-' and an effective strain EI 0.6. Nevertheless we observe a good agreement between theory and experiments for the steel sheets, but theory overestimates the drawing forces of aluminium sheets from about 20 %. Such conclusions have been previously drawn by Nine (8,9) ; the probable explanation is that the hardening effect of high strain rate present in steel (cf 94) is cancelled by the lesser hardening produced by cyclic straining, whereas the absence of strain rate effects in aluminium sheets produces the gap between theory and experiments. Theory predicts that the apparent friction coefficient depends only on n and increases slightly with it (formula (10);

364

we observe some scatter in the experimental values, but the mean values increase with n and generally are slightly higher than the theoretical ones (figure 2), again the gap is maximal for aluminium sheets but is lesser than 6 Oh. The drawing of the 0.97 mm A.K. steel between rollers p= 5.5 mm, produces a 8.7 O h reduction in thickness (8); formula (9) predicts a good order of magnitude: 6.5 'lo. So the theory can be considered as providing easily satisfactory results when the strip shape in the drawbead is known. 3. Influence of the bead depth and friction for strip clamped by rollers

3.1 Theoretical model For such clamping, the maximal strip deflection is equal to the bead depth 6 ; so the figure 3 summarizes the main assumptions on the strip shape: it is tangent to the plane part of the shoulders and the bead at its deeper point: elementary geometrical analysis provides the basic relations:

the shoulders radius p s ; for small bead depths where R according to the relation (13) is greater than pb+t/2, we assume that Rb=Rs=R; for 6>SC we assume: t L R (14) 2 S-sin$ Rb For analysing the effect of friction, we adopt the same assumptions as Nine (8,9): Coulomb friction between tool and strip and the contact traction uniformlv distributed along the circular parts of the strip: So stress and energy balance in the first part provide the relations:

&,=Po+-

h(1) = f ( l ) sin@=p i Rs sin$ (1 + p tan($/2)) f(1) = fo(1) + @ Rs PP1

so

f(1) = f o ( l ) +

f(1) 1+ptan($/2)

Similarly, we obtain : h = h(2) = ( f ( l ) + f(2) ) sin

= 2 sin4

Rb p2

',.

Two cases are currently met: -The radius of the shoulders and the bead are equal; we assume that the radius of the strip mid-surface is uniform R = Rb = R s and according to the relations (11) has the value

L R= 2sind1 -the bead radius pb for example is greater than

rollers

I

''

f(2)=fo(2)+ cl4 ( 1+pt;($,2)

\

+f(l)+f(2)) (16)

h(3) = f(3) sin@=p3 Rs sin$ (1 - p tan($/2) ) f = f(3) = fO(3) + $p(Rs p1 + 2 Rb P2 + Rs P3)

Elimination of the mean contact pressures p i , p2 and p3 in the last equation by using the previous stress balance equations provides

sheet

Figure 3: Assumed geometry of the strip and distribution of contact traction for a strip clamped on the shoulders by rollers

365

h 100-

Theory 11 = 0

- - - 0.163 d r aw be ad,

\


- K

sin a

de

dT< 0

if

d0

wire-wire normal contact force is obtained from Eqs.

whcre

10 and 11. Depending on the sign of

de'

Eq. 23

becomes (18)

tan cp = p sin a

E=JL[TSLCL de sin a r

1

2

and

aui .-

G(8, +_ 9)= h, sin hc8 cos (8 + 00) t

a= -L [ de sin a

- 2pcosp ifdT>O

(24)

a o .

1-2v 4xG

Finally, by using Eqs. (5) and ( 6 ) we derive the analog of Reynolds’ equation, h/2

( V , L1 dz 1 FE(g+tVp)dt)+

’

-h/2

(u,,Vh)-0

2

-h/2

for h(x,y)> O , (8)

,

sine (T;,t2,,) R

1 &&+

404

sine-- Y-9 R t

cose--

non-compressible material of the bodies for which ~ 0 . 5 and

X-2 R '

where fl is the contact area, G and v are the shear elastic modulus and Poisson's ratio of the bodies material , 7+= (4 and 7 - = ( 7xz',T ~ ~ are The tangential stresses acting on the surfaces of the upper and lower bodies which for dry contact regions satisfy conditions ( 2 ) and for lubricated regions are determined by the equations (see Eqs. (11, ( 6 1 , and ( 7 ) ) +,

T*'rg--Vpr h 2

(11)

Using well-known considerations applied to contact problems of elasticity and the expression for difference in vertical displacements AW we get the relation for the gap h between the contacting bodies 2Rx

2R,

2xG,

2R,

2R,

4xG,

R

- )

4.

BOUNDARY FRICTION

The solution of the problem for boundary lubrication involves determination of the pressure P(XtY) and gap h(X1Y) distributions (normal problem) and the frictional stress 7 and sliding velocity s distributions (tangential problem). Consider the normal problem. In the dry regions the gap between the contacting bodies is zero

R

for h ( x , y )- 0 . (14)

where h, is an unknown constant, R, and are the effective radii surface curvature for the contacting bodies in the directions of the Xand yaxes respectively. Eq. (12) can be simplified for the case of a

,"r

I

Under the conditions of a slow stationary motion and small slippage the equation for the sliding velocity s can be obtained by applying the differential operator 0.5(u,+u2,V) to both sides of the first two equations in ( 1 2 ) (It means differentiating with respect

40 5

to time.) This equation can be expressed in the form s--B(r)

+v, V = ~ - U 1 ,

behavior of the conjunction in a partial lubrication regime is governed by the combination of boundary friction and fluid film effects. Everywhere within the partially lubricated conjunction the pressure p(x,y) is assumed to be nonnegative, P ( X 1 Y ) 20.

case (3~sin~0-1) Dx11--

Dx22-

-

Several additional conditions must be imposed on the derived equations. One of them is the static condition

I

XGR~

(16)

co s0 ( v - 1.- 3 v sin28 I

XGR~

where P is the external force sin0 ( ~ - 1 - 3 ~ ~ 0 ~ ~ 0 ) applied to the bodies. Dy11Everywhere at the contact XGR~ boundary r vcos0 (1-3sin20) Dy12 -Dy21- pl r-0 I XGR~ I

I

Dy22-

-

s i n 0 ( 3vcos20-1) XGR~ (15)

Here 7 is the sliding frictional stress, which coincides with the frictional stress in the dry regions and is equal to g in the lubricated regions. 5. PARTIAL LUBRICATION

When some contact takes place between the asperities, partial lubrication (sometimes referred to as “mixed lubricationw1)will occur. The

r-

unknown if d r-0.

(Is)

Note that boundary condition (19) is the local one. Therefore, some pieces of the contact boundary r are known and some are unknown. Also, the problem solution must satisfy the boundary conditions for p, different at different parts of the contact region boundary, whether belonging to dry or lubricated regions, to the inlet or exit zones of the contact. Suppose that ri and re are the inlet and exit parts of the contact area boundary, which are given and unknown in advance,

406

respectively, r=riUre. The boundary condition for the dry contact region boundary is given in (18). Let us consider the lubricated contact regions. This case requires different boundary conditions. Here is the precise meaning of this assumption: r i - g i v e n if h lrl>O A ( 9 , d 1,/0;

(11)). The boundary conditions will be based on the requirement of the frictional stress continuity on the internal boundaries. Suppose li and ni are the i-th internal boundary and a unit normal vector to it, respectively. Therefore, the continuity condition can be expressed in the form: lime+ [ T ( x-enx,y-en,) -

eu

T (x+en,,y+eny)1 -0

f o r ( x , y )Eli.

The indicated boundary conditions (18) and (19) must be combined with conditions (16) and (17). Besides, the frictional stress 7 must vanish at the contact region boundary

TI ,-Om

(20)

The formulation of the boundary conditions imposed on the external boundaries of the contact is completed. In the case of mixed friction conditions the contact area may contain some internal boundaries between dry and lubricated regions that, apparently coincide with flow lines of the lubricant. Now, it is necessary to formulate the boundary conditions that must be held on the possible internal boundaries. A s we know, the frictional stresses for lubricated and dry conditions are described by Eqs. (1) and (2) (see also

Let us consider the physical nature of the transition from liquid to dry friction. A fundamental experimental study [ 3 ] shows that a continuous transition of the frictional stress from liquid friction to dry friction occurs in a small number of molecular liquid layers on a solid surface (Fig. 2). The specific features of this transition have been studied to a small extent and depend on the adsorption properties of the lubricantsolid surface interface. Let us consider the internal boundary between lubricated and dry regions on which slippage occurs i.e., la1>0. Then by using Eq. (21) together with (2), and (11), we obtain limh,os(p,h , p I I d

-

407

Owing to adsorption effects, the lubricant boundary layers show the properties of structural anisotropic fluids, and p=p(p,h, Is[). For a particular case of Newtonian fluid with F(x)=x, Eq. (22) becomes more transparent,

for Id>0.

':Boundary : rPartial rElastohydrodynmic

II 11 1

I

I I

1

1

I--

-

a :

c-c

.$ .0 p \

--

0

Hydrodynamic

I

5

10

15

I

20

Figure 2. Variation of friction coefficient with film parameter c43

(8),(7),(11) for the fluid film lubrication regions, and by equations (2),(14), and (15) for the dry regions together with additional conditions (16)-(21). In addition to these equations and inequalities the relations for functions F(@), p , f, and ri must be given. The problem solution consists of the functions p(x,y), h(x,y), r(x,y), and s(x,y), the exit boundary re, some parts of the inlet boundary ri that represent the boundaries of the dry regions, the internal boundaries li between the lubricated and dry contact regions, and the constant h,. In the case of contacting bodies made of the same material and small influence of elastic deformations on slippage the normal and tangential problems can be solved separately. It means that in this case s=uzu,+o ( I uz-u, I ) and the normal problem becomes independent of the tangential problem. Thus, first the normal problem must be solved for p(x,y) and h(x,y), and after that the tangential problem must be solved for r(x,y) and a(x,y).

A=PISI/P. 6.

It can be shown that (22) can be condition satisfied if the fluid viscosity approaches zero as the pressure approaches zero (see Eq. (36)). It is obvious from Eq. (23). Finally, the considered problem is completely described by equations

STATISTICAL APPROACH

A s already mentioned the shape of the inlet meniscus depends on many different factors that currently cannot be registered. Therefore, it is appropriate to treat the shape of the inlet meniscus ri as a random function depending on several

408

p a r a m e t e r s e.g. 7 . NARROW CONTACT UNDER PARTIAL LUBRICATION ri ( a1 / a2/ / an) Then the problem solution depends on the values of this set of Let us consider the case parameters [i.e. , p= p(x,y, of an incompressible elastic (-0.5) and a al/az, tan) / h=h(~/~/al/a2/material long in the direction *.*/an) ~ = ~ ( ~ , ~ / a l / a 2 , * * . / acontact n) .]. of the y-axis and narrow in s=s(x,y,al,a2,. ,an), In most practical cases it the direction of the x-axis. is important to know the This condition is equivalent average values of such to the inequality R,/%O). It can be shown that sliding velocity s, is unbounded at x=a and x=cy if the sliding frickonal stress g, does not vanishes at these points. The only way to make s, bounded everywhere is to accept the fact that limpop(p,h,ld)-0. (37) This condition is in a perfect agreement with the conclusion obtained from Eq. (22) (see also (23)) and the frictional stress continuity condition Eq. (21). Under this condition the regular solution of the problem Eqs. (32)-(34) can be found numerically or asymptotically for the case r l < < l . Suppose that the line y=y is not lubricated (i.e.,* H,=O) In this case the analytical solution of the problem is well known [ Z ] . It depends on the value of /78f

.

412

and may contain segments of relative slippage and adhesion. In the particular case of a Newtonian lubricant the normal problem [Eqs. ( 2 5 ) (31)1 can be solved separately from the tangential problem [Eqs. (32)-(34)]. The reason is that all integrations in Eqs. (7)- (9) can be performed analytically and

This is true for the original (non-simplified) problem as well. under equal Usually , conditions dry frictional stress is greater than liquid frictional stress. Therefore, the friction force is higher for mixed friction than for a purely liquid regime. The increase in friction force depends on the relative portion of the contact area occupied by dry friction regions. Figure 3 illustrates the type of pressure, film shape and boundary that can be expected in a partially lubricated conjunction. The top view of the conjunction [Fig. 3 (a)] shows the dry and lubricated regions as well as the Hertzian contact zone. Figure 3(b) shows a threedimensional view of the pressure within the conjunction. Note that in the dry region the pressure profiles are Hertzian but in the lubricated regions a

pressure spike exists. Figure 3(c) gives the film shape and boundary in a two-dimensional view along the y-axis. CONCLUSIONS

The work presented in this paper is only a beginning in understanding the major mechanisms governing mixed friction. A mathematical formulation of the contact problem with mixed friction is given. The problem is reduced to a system of mechanisms governing mixed friction A mathematical formulation of the contact prdblem with mixed friction is given. The problem is reduced to a system of alternating nonlinear

.

Figure 3 (a). Two-dimensional view of conjunction.

41 3

.

Figure 3 (b) Three-dimensional view of pressure profile.

equations and inequalities with some unknown external and internal boundaries. A statistical approach to the problem of mixed friction is proposed. The necessity of taking into account structural non-Newtonian lubricant behavior in thin films and the approach of the lubricant viscosity to zero as contact pressure vanishes is shown. The asymptotical analysis of the problem is conducted for the case of a contact stretched in the direction perpendicular to the lubricant flow. It is shown that for starved (limited) lubrication and mixed friction conditions the behavior of the lubricant film thickness resembles the behavior of the distance between the corresponding points of the input oil meniscus and the input Hertzian boundary of a dry contact. It is also shown that the friction force is higher under mixed friction conditions than for purely liquid friction and becomes greater as the dry friction region in the contact area enlarges. ACKNOWLEDGMENTS

.

Figure 3 (c)

Two-dimensional view of film thickness and boundary along the y-axis while setting X=O.

The authors wish to thank SKF Engineering and Research Centre (The Netherlands) for its support of this effort. The first author wishes also to thank the College of Arts and Sciences of the University of Scranton for supporting research and financial assistance.

414

REFERENCE8 1. 2.

3.

4.

5.

H. Schlichting, Boundary Layer Theory. McGrawHill, New York, 1960. L.A. Galin, Contact Problems of Elasticity a n d V i s c o elasticity. Nauka, Moscow, 1980 (in Russian). A.S. Akhmatov, Molecular Physics of Boundary Friction. Fizmatgiz, 1963 (in Moscow, Russian). B.J. Hamrock, B.J., and D. Dowson, Ball Bearing The Lubrication Elastohydrodynamics of Elliptical Contacts. Wily-Interscience, New York, 1981. J.J. Kalker, On Elastic Line Contact. ASME Trans., Ser. E, J. Appl. Mech., Vol. 39, No. 2,

9.

10.

11.

1972. 6.

7.

8.

Dowson and G.R. Higginson, Elastohydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication. Pergamon, Oxford, 1966. B.J. Hamrock and B.O. Jacobson, Elastohydrodynamic Lubrication of Line Contacts. ASLE Trans., Vol. 24, No. 4, D.

1984. 1.1. Kudish, Asymptotic

Methods of Investigating

12.

Plane Problems of Elastohydrodynamic Theory of Lubrication Under Heavy Loading Conditions. Part 1. Isothermal Problem. Izvestija Akademii Nauk Armjanskoj SSR, Mekhanika, Vol. 35, N0.5, 1982 (in Russian). L.G. Houpert and B.J. Hamrock, Fast Approach for Calculating Film Thickness and Pressures in Elastohydrodynamically Lubricated Contacts at High Loads. J. Tribology, Vol. 108, No. 3, 1986. V.M. Aleksandrov and 1.1. Kudish, Problem of Contact-Hydrodynamic Theory of Lubrication for a Viscous Fluid with Complex Rheology. Mechanics of Solids, Vol. 15, No. 4, 1980. 1.1. Kudish, Some Problems of Elastohydrodynamic Theory of Lubrication for a Lightly Loaded Contact. Mechanics of Solids, Vol. 16, No. 3, 1981. 1.1. Kudish,

Extremely Heavy Loaded Lubricated Contact and Critical Analysis of Some Approximate Analytical Theories. Proceedings of the 6th Intern. Congr. o n T r i b o l o g y EUROTRIB I 9 3 'I, Budapest, Hungary, 1993.

Dissipative Processes in Tribology / D. Dowson et al. (Editors) 0 1994 Elsevier Science B.V. All rights reserved.

41 5

Third body theoretical and numerical behavior by asymptotic method G.Bayadaab , M.Chambatb , K.Lhalouanib and C.Licht“

“Mecanique de Contacts CNRS URA 856 Institut National des sciences Appliquees 113, 69621 Villeurbanne Cedex France bLab.Analyse Numerique, CNRS URA 740,Universite Claude Bernard, 69622 Villeurbanne Cedex France ‘L.M.G.C, CNRS, Universite Sciences Techniques du Languedoc, 34095 Montpellier Cedex France We consider the behavior of a device with a thin soft layer between a rigid support and an elastic first body. We show how the expression of the interface law depends on the relative asymptotic behaviors of the parameters of thickness, stiffness and dissipation.

1. INTRODUCTION A trend in the study of friction between two bodies involves introducing a very thin third body between them. Sticking two elastic bodies also implicates a third thin material with a rather different mechanical behavior. It is then of interest to study the asymptotic behavior of a thin layer between a rigid body and an elastic one. A major assumption in this paper is that the third body is softer than the first body. A lot of papers have been devoted to such study with various geometries, behavior laws and more or less rigorous approach. In [l] closed analytical form solution with Bessel series are gained. In [2] the asymptotic method has already been used for a particular value of Lami coefficients with respect of the thin height E of the joint. A lot of related references are discussed in this paper. Preliminary results concerning our approach can be found in [3],[4]and [5]. The thin layer is stuck to the rigid body while various boundary conditions are considered between the thin body and the elastic one

as well as various relationship between stress and displacement for the thin body. The second section is devoted to a basic linear problem with perfect adhesion boundary condition. The problem is similar to an adhesively bonded joint. A rescaling of the coordinate through the joint is performed. An identification of the leading terms for displacement and stresses after asymptotic expansion allows us to obtain new boundary conditions at the interface between first body and the rigid support. In the third section, numerical results using finite element codes are given and compared with the theoritical preceeding results. To gain more realistic model of a third body, we replace in the fourth section the perfect adhesion condition by a Tresca’s boundary condition, acting on the tangential part of displacement and stresses vector. The asymptotic study leads to a similar behavior as in the second section, with the exeption of a new boundary condition occuring for a peculiar value of the stiffness Lame coefficient of the joint. The last section involves an evolution equation with dissipative generalized stan-

416

dard material whose caracteristics relie on E . Whether the ratio between the dissipative part and the purely elastic one is big or not, the range of the assymptotic boundary conditions goes from a sticking behavior to a slipping one.

strain.

The interface is located at x3 = 0 (see fig 1) while the joint lies between 23 = 0 and x3

This section is concerned with a model of soft joint of thickness e bounded by an elastic solid with perfect adhesion condition and a rigid body with zero-displacement condition (see figl).

2.1. Equations

So that equilibrium equations are

+ fi = o

aj.5'

+ fi

a+' 'I

= Ae,+Jij

i

= 1,2,3

x3 = --E

,4

I

in the joint

(1)

in the first body

(2)

Fig 1 : the Basic device

=0

= Xsae;k6ij

=0

:

2.2.

With the stress-displacement relation : 0:' IJ

and adhesion at the bottom implies U;

Various conditions cau be introduced (imposed displacement, external forces). Being applied to the first body, they don't modify the present asymptotic study. Let (A, p ) be the Lame coefficients of the first body and A', p c those of the joint. As the joint is assumed to be softer than the first body, we are led to introduce a and 4 such that :

a .Ju - c11

= --b.

At the interface, perfect adhesion boundary condition are :

BASIC PROBLEM

2.

= 1 if i = j and zero if i # j .

6ij

+ P&

e;

(3)

+2pet

(4)

Where ut (resp. u') and u+ (resp.)'u are stress and displacement in fitst body (resp. in the joint) and = i(6'i.j'- 6'ju;-) the

et-

+

Rescaling and Asymptotic Expansion

As a first step towards obtaining an approximate solution of problem (1). . . (7), we define an equivalent problem posed over a domain that does not depend on E by way of the rescaling :

417

Tentatively, we assume that a solution ( u , u) can be written in the following form :

By inserting (9) in (10) and identifying the leading terms, we obtain :

By inserting (9)-(10) in (2), we have to examine various situations.

we use equation (11) for i = id we have to discuss the leading terms witu respect of cr and p for the equation :

The same arguments as for the tangential problem can be used and we obtain the following table 1 which summarizes the various boundary counditions for z = 0 so obtained. u;=o u;=o

u; = x u ,0 u+o

=0 '1. i,j = 1,2 UP.

If ,f3 > 1,leading term is uGO= 0 and taking (6) into account we gain : uTfo = O

z =O

for

(12)

If ,f3 < 1, leading term is &(ti;') = 0 and taking (5) and (7) into account, we deduce Table 1 : Asymptotic boundary condition at the bottom of the first body (pure adhesion condition)

If p = 1, equation (2) reduces to /&(u;O)

= us;0

a2 aZu-0

and from (14) and ( l l ) , we get that all = 0 in the joint so that the following relation holds for L = 0 UT-0

By inserting (9) (10) in (2)-(4) we find that the classical linear elasticity equations are fulfilled by (u+',uf0) as E infers neither in the change of variable in the first body nor in the coefficients, so :

=jlu;o

and condition (5)-(6) allows us to carry on these condition from the joint to the first body:

To get information about the normal displacement and normal stress on the interface

Equations (15) (16) together with boundary condition of table 1 at the bottom 1 3 = 0 give a closed system which can be solved without any reference to the third body. So the asymptotic behavior of the device is well known by solving this system.

418

3

COMPARISON BETWEEN NUMERICAL AND THEORICAL RESULTS

a = p = 1 and UT and un for 0 < LY < 1 and p = 1. The results in the table 2 shows that the convergence of the characteristic ratio q - / q and Un/Un is very fast for the inner points of the interface. For E = 0.01, the convergence towards the asymptotic value is obtained within 1%. However, strange results appear a t the corner points (tl = 0.33 and 1 1 = 9.33) for which no convergence seems to occur. This is the consequence of boundary layer phenornena near the corner (see [3]). The asyrnptotic method gives average convergence and not a point- wise convergence. In other words, only the overall elastic energy of the system is proved to be converging. In the table 3, overall view of the convergence is shown for various values of a and p . The recorded values have been computed at an inner point of the interface to avoid boundary layer problem. The good convergence of the process is confirmed for all values of Q and p. To be noticed also is that results appearing in the columns E = 0.0005 and E = 0.0001 are comd e t e h different. This is caracteristic of an other difficulty for computing directly such

The previous asymptotic analysis is valid for

‘‘ small values of E. ”

To gain quantitative informations on the precise values of E for which the theoritical study is applicable, we perform some numerical computation for the initial problem with two different codes : Systus and Modulef. Both give the same results. The device is a square first body of length L = 10. A tangential displacement of L/lo is imposed at the upper surface while lateral sides are free. The corresponding Lame coefficients are A = 1,15 10l1 and p = 0 , 7 l o l l . At the bottom, there is an horizontal thin layer of thickness stucked to a rigid support, Regular finite elements discretization has needed 3000 triangles in the first body and 700 in the joint. We compute for various value of E = Q and /3 the displacement and stress in the whole device and retain the value related to the asymptotic theoritical behavior for five points on the interface . For example, we retain uT/uT and Un/Un for

4,

= 0.05 I 0.585E + 11 I 0.781E + 11 0.779E + 11 0.781E + 11 0.5856 + 11 0.2416 + 12 0.2736 + 12 0.2716 + 12 0.271E + 12 0.241E + 12

E

= 0.01 1 0.541E+ 11 I 0.775E + 11 0.773E + 11 0.776E + 11 0.541E + 11 0.1936 + 12 0.271E + 12 0.270E + 12 0.270E + 12 0.193E + 12

e

e=0.005

I ~=0.001 I 0.342E+ 11 I

0.491E+ 11 I 0.771E + 11 0.770E + 11 0.771E + 11 0.491E + 11 0.174E + 12 0.270E + 12 0.269E + 12 0.270E + 12 0.174E 12

+

+ +

0.7686 11 0.7696 11 0.7686 + 11 0.3426 + 11 0.135E + 12 0.2696 + 12 0.269E + 12 0.2686 + 12 0.135E 12

theoretical values

0.769E

+ 11

0.2696 + 12

+

Table 2 : numerical values obtained at the interface using MODULEF code for various x and

Q=p=l

E

for

419

E

(Y = 1 /3= 1 (Y = 2 P=1

I

= 0.2

I 0.9456 + 12 % -

1

I

1

.. I

0.3216+12 0 . 4 3 3 6 11 0.8796 12

+ +

E

= 0.01

+ 11

0.7736 0.2706+ 0.7656 0.155E

+ +

12 11 12

E

= 0.005

E

+

0 . 7 7 0 6 11 0.2696+12 0.767E 11 0 . 1 5 4 6 12

+ I + I

= 0.001

+

0 . 7 6 9 6 11 0.2696+12 0.769B 11 0 . 1 5 4 6 12

theorie

e = 0.0001

I 0 . 5 7 4 6 + 08 I 0 . 7 6 9 6 + 11

1

0.119E+09

I

0.2696+12

+ I 0 . 4 3 5 6 + 08 I 0 . 7 6 9 6 + 11 + 1 0.155E + 09 I 0 . 1 5 3 6 + 12

4 = 0.5 I U T I 0 . 5 6 8 6 - 05 I 0.181E - 10 I 0 . 1 4 6 6 - 10 1 0 . 1 3 2 6 - 10 I 0 . 1 1 2 6 - 06 I Table 3 : comparison between theory and numerical results for various a and /3

0

~~~

device : the locking phenomena, well known for plates, which prevents the right calculation for devices with thin layer using classical codes. This emphasis the interest of the theoritical study.

4 . THIRD BODY LAYER As a tentative to model the thin layer as a third body, we replace the boundary conditions (5) (6) by the Tresca condition, allowing a possible jump of the tangential displacement uf and u7 i = 1 , 2 at the interface. Let g a given upper bound for the tangentical stress modulus, the Tresca conditions are

[ut] denoting the jump of the tangential displacement at the interface. Clearly, this condition is nothing else than a Coulomb like condition with a fixed upper bound for bt in (18). Equations (1)-(4) and (17)-(20) together with loading conditions define a well posed problem which is a non linear one due to (18)-( 19), so that the formal identification procedure as in section 2 is no longer valid. However, mathematical analysis approach [6] allows us to obtain the asymptotic behavior of the device. New asymptotic boundary conditions associated to the equations (15)-( 16) are shown in table 4. By comparing table 1 and table 4, differences only appear for ,f? = 1. In that case the boundary condition .to = p u t o is replaced by a Tresca’s like condition. An important feature is that in the case ,f3 = 1, it is not possible to solve the asymptotic problem without taking the joint into account and we must add to (15)-(16) the asymptotic elasticity equations in the joint :

420

P>1

u;

=0 =0

uo

=0

UT 0

u: = xu; u; = 0

UQ.

i,j

=0 = 1,2

uo = 0

uo

=0

J.

p=1

05p TEST TEMPERATURE ("C) Fig.3. Curves of wear rate vs. operating temperature (Test conditions: Loading P = 2520N, Number of impacts 6 x I @ . Impactfrequencyf = 20h)

3.1.2 Effect of temperature and material strength on wear The effect of temperature on wear is shown in Fig.3, in which the end-point wear after 6 x lo5 impacts is recorded from 100°C to 800OC. Data is shown for the tool steel 2 1-4N material (curve B) as well as Stellite 12 (curve A). It can be seen that Stellite 12 wears less. Curves C and D are respectively the 0.2% yield strength curves of 214N and Stellite 12 [6]. Observing figure 3, it is evident that there is an inverse relation between wear rate and yield stress. 3.1.3 Dissipativeprocess of the valve-seat wear

Scanning electron microscope metallographic studies have been made of the Stellite 12 valve-seat

Fig.4 Micrograph cross-slip deformation of subsurface of contact zone Material: Stellite 12.) (Test conditions: Number of impacts 6 x 10'. Loading P=201 ON,

Impactjrequency f = 20h)

448

On the other hand, high temperature also changes the stability of the micro-structure. In particular, during its manufacture, the exhaust valve is heattreated, to enhance wear and corrosion resistance, to create uniform and fine carbides (e.g. Cr2C3 and Cr& ) in micro-structure. Under high temperature and high contact stress, these microparticles can diffuse again toward these high energy interfaces. Figure 5 shows the distribution of carbon along a line-scan through a slip-band region. Peaks in figure 5 are a result of dissolution of fine carbides in the microstructure and their precipitation again to form larger carbides. Figure 6 is a magnified view of part of figure 5 , focusing on a region of carbon peaks. It shows examples of such precipitated carbides. Obviously this change of microstructure will reduce wear resistance.

crystal boundaries. This would speed up the fatigue process of deformed material. Because the corrosion process needs a relatively longer period than wear, it is not easy to assess its effect in a simulation test in the laboratory. However, observation of the micro-structure of a worn section of a valve-seat from an actual engine test shows the effects of exhaust-gas are great. Fig.7 is a section through the subsurface of a valve-seat. It can be observed that grains in the micro-structure have plastically deformed and the deformed degree gradually become weaker from the surface toward substrate, and that microcracks exist along boundaries of these grains, and that some fatigue wear particles exist in the path of the cracks.

Fig.6. Carbides separated out in slipdeformation process (Test conditions are the same as in Fig..?) Fig.5. Carbon wavelength spectrum analysis of the deformed section (Test conditions are the same as in Fig.4)

3.2 Real engine test results 3.2.1 End-point wear depth After lOO,OOOkm, Stellitel2 valves were removed from an engine. Their wear depth was measured to be between 0.5 and 0.65mm. 3.2.2 Metallographic studies While an automotive engine is operating, exhaust gas passes through the gap between the valve-seat and its seat-insert. Some unburned matter in the exhaust gas will deposit on the contact surface of the valve-seat, elements such as Pb and C1 have a strong activity. They would penetrate any surface film, and migrate to the surfaces of deformed

Fig.7 Micrograph of fatigue cracks of worn subsurface of exhaust valve-seat in an actual engine (Material: Stellite 12. Opemting distance 100,000 hl).

449

Fig.8 Energy spectrum elements of wear micro-particle of exhaust valve-seat in an actual engine (Material: Stellite 12. Opemting distance 100,000 km.)

Obviously, the topography reflects both dissipative processes of corrosion and fatigue wear of the exhaust valve-seat. If we analyse these wear products (as shown Fig 8), it is found that in these products there are high concentrations of C1 and Pb which originally exist in petrol , as well as Cr and Co which originally exist in grain structure of the material. It is evident that entering of these corrosive materials into the fatigue cracks accelerates the wear process of the valve-seat material. Table2. Elemental compositions of a fatigue

Abrasion of the frictional surface between the valve-seat and its insert may also result. It could occur in two ways: from unburned matters in petrolldiesel and fatigue wear debris of material of the valve-seat or its insert. The latter plays a very important part in acceleratingthe valve-seat wear.

Generally, the fatigue wear debris from alloy materials contains some alloy elements such as Co and Cr. Table 2 is the elemental composition of micro-wear particles in Fig.8a. We can see that in these particles the amount of elements Co , Cr ,Pb and C1 is higher, and that the shape of the particle is a ball. This shows that forming of the abrasion is a result of a combination action of fatigue and corrosion. It is possible that, after these fatigue wear particles enter the frictional surface and being further oxidised at high temperature, the particles will become hard and abrasive and contribute to wear of the valve-seat. This also shows that the abrasion accelerating the valve-seat wear partly comes from fatigue wear debris of the material of the valve-seat.

4. DISCUSSION 4.1 Wear rate and contact stresses The wear depth from the engine test after 100,OOkm

450

was from 0.5 to 0.65mm, that from the simulation test at a load of 20210N was 0.085mm after 6 x lo5 impacts. Extrapolation of the linear wear rate in the simulation test to lo8impacts(the estimated number in the engine test) gives a wear depth of 5 to 6 nun, some ten times greater than observed. The simulation test, at two loads, however, shows the wear rate to be sensitive to load raised to a power greater than one. Part of the greater wear depth in the simulation than the engine test is a result of greater loading in the simulation test.

have produced a much lower wear rate, and further work is planned at lower loads. In terms of its geometry of contact between the valve-seat and its seat-inserts, the force P can be resolved into two components: normal N and tangential T (as shown in Fig.9). Because the seat-insert is fixed into the top of the cylinder, it has no normal displacement. If elastic deformation of the valve stem is ignored, the effect of the T force on wear of the valve-seat can be ignored. Therefore, only the normal component N of the P force is considered as causing a contact pressure stress resulting in wear of the valve-seat. If we consider the contact region and ignore the change of the contact width B during wear, the normal contact stress a,,canbe written: K d .P c o s a a, = (2) An

Fig.9 Contact state between valve-seat and seatinsert. While the exhaust-valve is closing, the valve-seat impacts its seat-insert with an impulse seat force P, The force theoretically consists of a static spring force F, (=kq)and a dynamic spring force Po . However, P changes with the increase of the valve clearance 6 which will increase with wear between the valve stem-top and tappet-top. Therefore, a dynamic loading factor I& should be introduced: p~~vol = Kd * (1) A test result shows that at a running speed of 1200 r.p.m, when the valve clearance 6 is 0.4 mm, a dynamic load set is P = 2010N. However, when 6 is adjusted to 0.8mm, the dynamic load set is increased between 1.8 and 2.2 times (up to between 3620N and 4400N, i.e. I(d = 1.8 2.2, Pac&d= (1.8 2.2) x 2010N). The values of I(d in simulation test, in fact, is approximately consistent with measurement value in actual engine[2]. At the same time, variation of 6 through the life of the engine would have caused to vary between 800 lOOON(6 = 0.4mm) and 2000 - 2500N(6 =0.8mm). The load in the simulation test may have been up to 2 times as high as that in actual engine. Inspection of figure 2 suggests a lower load would

-

-

-

where An = Id)B--- normal contact area, D is the diameter of the valve head and a is the taper angle of the valve seat. For the present test, D = 38mm, B = 1.5mm and a = 45". Taking P = 2010N and Kd = 2 gives a,,= 16I"a. The yield stress of Stellite 12 at 800°C is 280MPa (figure 3). It is therefore clear that the plasticity observed in figure 5 is associated with real contact area, asperity, stresses and not with bulk overloading. This is consistent with the depth of the plastic zone, 30pm, in figure 2. It is therefore strange that wear rates in the simulation tests (figure 2) are loaddependent to a power greater than one, as contact statistics[7] would suggest that wear be proportional to load. The results indicate that the running-in process is important in conditioning the surface state of the valve seat, and that is another reason for further work at lower loads.

4.2 Microstructural observations However, the simulation tests do compare with the real engine tests in showing precipitation of carbides in slip-bands .They focus attention on the interaction of high contact stress and temperature in causing element solution and precipitation to change the microstructure, and hence to influence wear resistance through reduced yield strength and changed fatigue wear resistance . The engine test in addition focuses attention on the role of exhaust gases in causing corrosion and further changing the composition of the micro-structure.

45 1

Fig. 10 Dissipative diagram in automotive engine valve-seat

4.3 A dissipative process diagram On basis of the discussion above, it is evident that high temperature reduces yield strength of the valve-seat materials so that plastic deformation occurs at the surface or subsurface of the contact zone under high contact stress, and that corrosion accelerates the fatigue process of the deformed material and that hard particles of the fatigue wear debris provide abrasion of the frictional surface. Thus, we can deduce a dissipative process diagram of the valve-seat wear. By the diagram(as shown in Figlo), a mutual connection among various mechanisms in exhaust valve-seat wear can be seen to exist. Their relative importance and interactions are the subject of continuing study.

5. CONCLUSIONS 1. A dissipative process diagram in automotive engine valve-seat wear has been developed on the basis of simulation wear tests and analysis. The diagram effectively describes the effect of contact stress, elevated temperature and exhaust gas on dissipative processes of the valve-seat wear. 2.High temperature decreases yield strength of valve-seat materials so that the wear rate of the valve-seat decreases with increase of the temperature. 3.During the dissipative processes in the valve-seat wear, deformation fatigue is a main factor. High

temperature and corrosion of the exhaust gas accelerate the fatigue process of materials. The fatigue wear debris provides abrasion for the contact surface between the valve-seat and its insert.

REFERENCES 1. C.F.Taylor, Internal combustion design" 1977 V01.2 $521-575. 2. K. Akiba and T. Kakiuchi "A dynamic study of engine valve mechanisms: Determination of the impulse force acting on the valve" SAE 88 0389. 3. T. Kurisu, K. Hatamura and H. Omoti "A study of jump and bound in a valve train" SAE 91 0426. 4. C.B. Allen. J.L. Sullivan and T.F.J.Quinn,"The wear of valve-seat materials at elevated temperatures" Tribology of reciprocating engines 1982 ~279-284 5. Y. Hagiwara, M. Ishida and T. Oh, "Development of Nickel-base super alloy for exhaust valves" SAE 91 0429. 6. American Society of Metals, 'I Metals Handbook" 8th edition 1961 Vol. 1 p666 7. S.P. Bhat and C. Laird, I' Cyclic stress-strain response and damage mechanism at high temperature" ASTM Stp 675 1979 p592-623 8. J. F. Archard, "Theory and mechanism of wear" Wear control handbook 1980 p35.

This Page Intentionally Left Blank

Dissipative Processes in 'l'ribology / D.Ilowson et al. (Editors) 1994 Elsevier Science I3.V.

453

Thermal Matching of Tribological Systems by A. V. Olver Department of Mechanical Engineering, Imperial College

When there is si@cant sliding or other dissipative processes present, simplistic application of the elastohydrodynamic film thickness regression equations can be highly misleading. This paper describes a simple model which seeks to predict and account for the consequent temperature rises. The case of a twin-disc machine and that of a high-speed spur gearbox are examined and the results compared with earlier measurements. 1. Introduction: Macrotribology

It is rare for the mechanical conditions in a rubbing contact to be independent of its environment. More commonly there is a dynamic interaction such that the conditions within the contact affect those in its surroundings and vice versa. Examples of this interaction may be found in such diverse phenomena as lubricant supply, debris transport and heat flow and often have the capability to alter radically the practical outcome. This can lead, for example. to a poor agreement between simple test machines and real equipment. In the present paper. the thermal behaviour of some elastohydronamic systems is modelled and the results compared with earlier experiments. 2. Thermal Model

It is, of course widely recognised that elastohydronamic lubrication is highly sensitive to temperature. Film thickness, for example, depends on inlet viscosity which depends on surface temperature. However, film thickness itself affects friction which affects heat generation, bulk temperature and hence inlet viscosity. The contact cannot be considered in isolation; a thermal model of the whole system is necessary. Such a model is presented here and has the following components:

- a conventional ehl model based on disc or gear kinematics and an appropriate

film thickness regression equation, - a traction model based on measurements carried out in a controlled test. - a partition theorem which allows the calculation of heat input to each component in the set notwithstanding its (different) bulk temperature, a conduction and convection model of heat transfer, - a simple treatment of windage and friction - an iterative solution method which allows compatible values of traction, temperature and film thickness to be found.

-

This model is identical to that presented in (1) except that here we introduce the traction model of Bair and Winer (2) in order to predict the coefficient of sliding friction. Details of this and some other aspects of the model are given in appendix 1. The key concept is that the heat partition, determined from Jaeger's theory of moving heat sources is applied, not just to the determination of the flash temperature, as by Blok (3), but also to the steady state heating of the body. T h s enables the calculation of the surface temperature of the disc or gear tooth under the (realistic) circumstances of sigrufcant frictional heating. In order to apply the method to gear pairs, some additional steps are necessary. Firstly the excess of the ambient temperature, T,, over the oil supply temperature, To,,. is calculated from the theories of windage (4) and churning (5,6) and knowledge of the oil flow rate and heat capacity.

454

Figure 1. Comparison of the predictions of bulk temperature TB of rubbing discs with the measurements of reference 7.

Heat generated by other sources, for example bearings is neglected; this is considered justifiable on the grounds that tooth temperature is not greatly affected by heat from the bearings (or vice versa) but again some caution may be advisable if, for example, hot oil from a bearing installation can impinge upon the gearset. Next, the parWion of heat between the two gears is found using the modified Jaeger method. Because the contact between the gears involves a cyclic meshing action, it is necessary first to find an average heat partition coefficient,a,, for the whole cycle. This is used to find a mean tooth temperature, TB, and contact temperature, Tc, the latter varying throughout the mesh. At each stage, the necessary iteration to find the local friction coefficient is carried out, noting that this is itself a function of the mean contact temperature. Details of the method are given in appendix 2. 3. Results: Disc Machines

Some predictions of the model are now given and compared with temperatures measured in earlier experiments. Because the novel part of the calculation is that associated with prediction of the inlet (or 'skin') temperature,TB, we shall first concentrate on measurements of this quantity. Paliwal and Snidle (7)have made extensive measurements on their disc machine using embedded thermocouples.

Figure 1 shows an example of their results together with the predictions of the present model. The operating parameters and the assumed material and lubricant parameters are given in the tables in appendix 3. It is noted that the faster disc of the two is the hotter because it absorbs a larger proportion of the frictional heat, this effect being greater than that of its higher convective heat transfer coefficient. Overall, the measured temperature rise is within about 10Y0 of the predicted value, the main error being in the prediction of the friction coefficient, which nevertheless shows a similar qualitative trend with applied load. Predictions of the behaviour of a smaller, higher speed, disc machine are shown in figure2 which shows the contact, and skin temperatures, film thickness and friction coefficient as a function of applied load. The discs are rotating at a speed of 10000 and 20000rev/min respectively and the other parameters are given in appendix 3. Increasing the load beyond a critical value c a w s the friction coefficient and temperatures to rise sharply, while the film thickness declines to a low value. This occurs because the declining film thickness causes an increase in the friction which in turn increases the inlet temperature causing the film to become yet thinner. An instability results which in its main features (temperature and friction rise and loss of film)bears a striking resemblence to the phenomenon of scuffing.

455

1800 IWJ

14m 1ZUI lm,

800 Bm Qo

am 0 0

91)

lm,

l r n a m a a s x a Mill

Figure 2. Predictions of maximum contact (T(C)) and skin (T(B)) temperature, Jilm thickness (expressed as a lambda value), and friction coeflcient (Mul for a small (19 mm diameter) disc machine For details see appendix 3.

Figure 3. Predictions of temperature, Jilm thickness (expressed as a lambda value), and friction coeflcient for the disc machine ofJigure 2, but running at higher speeds (see tex?).

Figure 3 shows the same prediction with the speeds increased to 15 0oO and 30 OOO rev/min. In this case the instability is still present but of lower severity. This is because the the rate of change of viscosity with temperature is lower at the higher temperature and the higher speed makes continuing elastohydrodynarmcsupport easier. Photographs of discs run under these conditions are shown in figures 4 and 5.

The higher speed test does indeed result in a less severe failure; because of the absence of a marked friction increase, the test was continued resulting in wear rather than d n g as was the case at the lower speed. The predictions of failure load are quite inaccurate but this is not unexpected as the model in its present form does not recognise the possibility of boundary lubrication.

456

Figure 4. Photograph of discs run under the conditions offlgure 2.

Figure 5. Photograph of discs run under the conditions ofjigure 3. 4. Results: Gears

Next, the temperature predictions of the model are compared with measurements taken with thermocouples embedded in high speed spur gear teeth. The results of Mizutani, Isikawa and Townsend (8) were used for the comparison and are distinguished for being unusually thorough in their provision of the background data needed for the modelling. Figures 6(a) to (d) show the effects of operating (pinion) speed and (wheel) torque on temperatures, mean friction coefficient and minimum lubricant film thickness (expressed as a lambda value) in the high speed spur gearbox of Mizutani & al. Throughout the useM operating range of the gearbox the film

thickness is highly sensitive to torque but relatively little affected by speed. This again is because of the thermal effect on the contact inlet. The temperature increase diminishes the beneficial effect of speed and greatly adds to the detrimental effect of torque. The tooth temperatures are compared with the experimental results in figures 7 and 8. The closeness of the agreement is almost certainly fortuitous in view of the uncertainties in the analysis. The predicted friction coefficients (figure 6d) differ somewhat from those given by Mizutani & al. but these are believed to be unrealistically low, having been derived by an indirect method.

457

Figures 6a and 6b. Predictions of maximum contact temperature (top) and pinion tooth temperature in the gearbox of reference (8).

458

Figures 6c & 6d. Predictions ofpiction coeflcient (top) and lambda value in the gearbox of reference (8).

459

55

-

50

Pinion

45

---- wheel

0

P

U \

2 40 a E 35

Oil

1.1.1.-

CI

L! f

Pinion

*

30

I-

25

I

20 4 2000

55 6o

d,

U \

x a, L

Wheel

Oil ,

4000

I

I

6000 8000 Speed I rpm

10000

12000

T

9.66

---- 19.32

501

-.--..28.98

45

- - - - -38.64

(D L

E

48.3

,!! 30 20 25

48.3

i 35

A

0

2000

4Ooo

6oOo

8Ooo

loo00

12000

O

2

Speed I rpm Figures 7 & 8. Comparison of the predicted temperatures in a test gearbox with the measurements of reference (8). figure 7 (above)shows the pinion, wheel and oil temperatures as a function of speed at a wheel torque of 196 Nm and an oilflow of 35 ml/s. Figure 8 (below)shows the eflect of varying the oilflow rate. The line plots are the theoreticalpredictions and the discrete points the measurements; the units of oil flow rate are ml/s.

2

460

5. Discussion and Conclusion

Useful application of ehl in situations such as those encountered in gearboxes. where significant dissipation occurs requires a unified thermal model of the type presented.

of Concentrated Contacts'. National Aeronautics and Space Administration. Wa~hington.D. C.. U. S. A.. 1970. 153-248. 4. P. H. Dawson. Windage Loss in Larger.

High Speed Cears. Proc. lnstn. Mech. Engineers London, 198 (1984) part A No.1

Results show that a simplistic application of the ehl regression equation can be misleading; for example the film thickness is seen to be highly dependent upon load (torque in the gearbox model). In addition a wide range of Merent thermal characteristics exist, some situations involving a rapid increase of skin temperature with load (e.g. the discs of figures 2-6) whde others show high contact severity (high T(:) with little bulk heating. The presence of the singularity, which is a consequence of rapid bulk heating and specific traction behaviour, suggests that some smaller test machines could fail in ways that larger equipment might not. Some of the well known discrepancies between different lubricant test methods - many of which are dependent on small and hence rapidly heating test components - might be related to this. The analysis presented here enables tests to be thermally as well as kinetically matched to the application they are intended to simulate. RReferences. 1. A. V. Olver. Testing Transmission Lubricants - the Importance of Thermal Response. Proc. Instn. Mech. Engineers London, 205 part G (1991) 35-44.

2. S . Bair and W. 0. Winer, Regimes of Traction in Concentrated Contact Lubrication, Trans. Am. SOC. Mech. Engineers, J. Lub. Technol. 104 (1982) 382-391. 3. H Blok, The Constancy of Scoring Temperature. in P. M. Ku (ed.) 'Interdisciplinary Approach to the Lubrication

51-59. 5. R. I. Boness. Churning Losses of Discs and

Gears Running Partly Submerged in Oil. Proceedings of thc 1989 International Power Transmission and Gearing Conference. Chigago, U. S . A.. Am. Soc.Mech. Engineers (1989). 355-365. 6. N. E. Anderson and S. H. Lowenthal, Design of Spur Gears for Improved Efficiency, Am. Soc. Mech. Engineers. I. Mech. Design, 108 (1986) 767-774. 7. M. C. Paliwal and R. W. Snide. Runninginand Scuffing Failure of Marine Gears, University College, Cardiff, Report for Ministry of Defence, Contract No. D/ER1/912072/045. (1987). 8. H. Mizutani, Y. lsikawa and D. P. Townsend, Effects of Lubrication on the Performance of High Speed Spur Gears, Proceedings of the 1989 International Power Transmission and Gearing Conference. Chigago, U. S. A., Am. Soc.Mech. Engineers (1989), 327-334.

9. D. Dowson. G. R. Hiwnson, J. F. Archard. and A. W. Crook. Elastohydronamic Lubrication, Pergamon, 1966. 10. Anon.. ASTM Standard D341, Appenlx X1, American Society for Testing and Materials (1977). 11. R C. Gunther, Lubrication, Bailey Brothers and S W e n (1971) 141-142.

46 1

Appendix 1

friction and u the s p e d of the surface mlative to the contact.

(a)Heat Transfer The contact temperature is given (1) as:

Expressions used for M here are now given. For results in figure 1 the 'radial' approximation was used:

= TA+ A T , +AT,

1/ M = l/MWd+l/Mdiscwhere where, for body 1 of a pair:

(AT,), =aQ(l.GOB,) and

M,, = (27dUh)-'and MhC= (8.88(hk)112 I 112R x I, (nR)/ I,,(nR)]-'

where the frictional heat generated, Q, is given by:

a is the proportion of heat going into body 1 and is given by:

where R is the radius. f the track width, I . and I, are the appropriate Bessel functions and n = (2h/ kf)'" For the results relating to figures 2-6, where the geometry is shaft-like as opposed to discl i e , the 'axial' approximation (1) is used:

a=(l.06B2+ M 2 )

+ {1.06(B,+ B , ) + ( M , +M2)) Here, the quantitiesB and M are the transient and steady state thermal compliances of the two bodies where:

Mhc = (4.44(hk)"2R3I2 x (tanhmL, + tanhmL,))-' where m = (2h/ kR)'".

B = (11 A k ) ( X b l U )

h, the convective heat transfer coefficient, was

and M, defined for the two bodies by:

b) Film thickness

must be calculated for each body from a suitable steady-state heat transfer model. x is the diffusivity, k the conductivity and b and A the contact area and width respective1y.W is the total load, and p the coefficient of slidmg

found by the method given in (1).

The Dowson-Higginson film thickness equation (9) was used together with the ASTM kinematic viscoSity relation (10) and an empirical relation (11) for the variation of density with temperature. c) Traction The fiiction coefficient, following Bair and Winer (2) is given by

462

where ,fqis the friction at 0 OC,A is the ratio of film thickness to composite (rms)roughness and Tea"is thc mean contact temperature. The constant C1 is taken, using the data of (2) for esters as unity and is given in appendix

c2

3.

d) Solution An iterative solution method was used to obtain consistent values of friction coefficient, temperatures and film thickness. The iteration was terminated when successive values of p Wered by less then .0005.

Appendix 2 Application of the model to gear pairs. The average heat parbtion coefficient, Orav is first found by stepping through the entire mesh, and applying the method of appendix 1. The local heat partition is now found by the following which accounts for the mean tooth temperature being held constant:

a=I%f4-am(4+M2) + 1.06B,} / 1.06(B,+ B, ) where &is the ratio of average, to instantaneous, heat generation. The radial approximation (appendix 1) was used to find the M's, the value of hfpd being modified for the increased peripheral area due to the teeth. In order to estimate the local ambient temperature relevant to the gears, the assumption is made that half the windage and churning energy and all that due to friction, is trwsfqqed to the oil prior to lubricating the gqq. Jp practice, it is accepted that th~smay yqy pn$jderably.

463

Appendix 3. Data for disc machines; Figure 1:top;Figures 2-6: bottom

Speed of Slow disc , rev/min, ratio of disc speeds Oil supply temperature, deg C Radii of slow, Pst, disc,m Shaft lengths, m Track width, m Plane Strain Elastic Constants JY(1 4 ) : fast, slow disc Pa Thermal Conductivity of fast, slow disc, W m-1 K-1 Thermal Diffisivity of fast, slow disc, m2 s-l ASTM viscosity constants of oil: A, 8,Density at 60 deg F, g cm-3 Alpha value at 0, 100 deg C, GPa-1 Composite RMS Roughness, m. Friction coefficient at 0 deg C, rate of change with film temperature, deg C-1 Specific heat capacity of lubricant, J K-1 kg-1 Thermal conductivity of lubricant, W m-1 K-1 Speed of Slow disc, revhin, ratio of disc speeds Oil supply temperature, deg C Radii of slow, fast, disc,m Shaft lengths, m Track width, m Plane Strain Elastic Constants E/(l-vz): fast, slow disc Pa Thermal Conductivity of fast, slow disc, W m-1 K-1 Thermal Difisivity of fast, slow disc, m2 s-1 ASTM viscosity constants of oil: A, B, Density at 60 deg F, g cm-3 Alpha value at 0, 100 deg C, GPa-1 Composite R M S Roughness, m. Friction coefficient at 0 deg C, rate of change with film temperature, deg C-1 Specific heat capacity of lubricant, J K-1 kg-1, Thermal conductivity of lubricant, W m-I KI

485.1. 3.0 45 0.038125,0.038125

(radial approximation used) 0.0047625 2.2X10~~,2.2XlO~~ 35.4, 35.4 8.56x104, 8.56~1 0" 24.661, -4.0313, 0.895 20, 14.6 0.925~10~ 0.06, 0.00010 2040, 0.138 10 000,2.0 75 0.0095, 0.0095 0.046, 0.046, 0.042, 0.042 0.00I 87s 2.2X10~~,2.2XlO" 35.4, 35.4 8 . 5 6 lOd.8.56x10-6 ~ 22.525, -3.6425, 0.995 14.9, 10.9 0.925~10" 0.025,0.00020 2040 0.138

Appendix 3 continued: Data for gearbox

Tooth numbers: Pinion, wheel Oil supply temperature, deg C Centre distance (C. D.), m, Tip Radi2C.D.: pinion,wheel. Root Radi2C.D: pinion, whee Nominal Pitch Radius/C.D., Effective Facewidth/C.D., Nominal Pressure angle (deg) Plane Strain Elastic Constants E/( I-v2): pinion, wheel, Pa Thermal Conductivity of pinion, wheel, W m-I K-1 Thermal Diffisivity of pinion, wheel, tn2 s-1 ASTM viscosity constants of oil: A, B, Density at 60 deg F, g cm-3 Alpha value at 0, 100 deg C, GPa-1 Composite R M S Roughness, m. Friction coefficient at 0 deg C, rate of change with film temperature, deg C-1 Specific heat capacity of lubricant, J K-I kg-1, Oil jet flow rate, m3 s-1 Thermal conductivity of lubricant, W m-1 KI Wheel Torque ,Nm, Pinion Speed,rev/min

40,77 28 0.1753, 0.36337. 0.67992 10.3171I , 0.65476 0.3423,O.14261, 20 2.2x1011,2.2x1011 35.4, 35.4 856x10". 8.56~10" 26.3883, -4.3756, 0.895 20, 14.6 5x 10" 0.06, 0.00010 2040,0.0000483 0.138 500, 12000

This Page Intentionally Left Blank

Dissipative Processes in Tribology / D. Dowson et al. (Editors) 0 1994 Elsevier Science B.V. AU rights rescrved.

465

POWER LOSS PREDICTION IN HIGH-SPEED ROLLER BEARINGS D. NELIAS*, J. SEABRA**, L. FL,AMAND* and G. DALMAZ*

*

**

-

INSA de Lyon - Labratoire de M M q u e des Contacts CNRS URA 856 - BAt.113, 20 Av. A. Einstein, 6962 1 Villeurbanne Cedex, France CETRIB Departamento de Engenharia Mecanica e Gestao Industrial - Faculdade de Engenharia da Universidade do Porto, 4099 Porto Codex, Portugal

-

ABSTRACT

Friction between the various elements in a rolling bearing results in power loss and heat generation. Therefore, an estimation of the rolling bearing power losses is necessary to refine lubrication techniques and to optimize machine component design. A new model which predicts and locates power losses in a high-speed cylindrical roller bearing, operating under purely radial load, is presented. Its new features come from the consideration of both cage action and the effect of lubricant film thickness in the computation of bearing kinematics at equilibrium. Lubricant rheological properties are used in order to calculate hydrodynamic and elastohydrodynamic forces in each lubricated contact. This model considers cage and roller kinematics to be unknown. These are obtained by solving the equations of motion for each bearing element. The computation and the location of power losses are given by the friction forces and the sliding speeds among the various bearing elements, i.e., contact between roller and inner or outer ring, roller edge-raceflange, roller-cage pocket and cage-ring pilot surface. The authors compare first the values of the calculated power loss with experimental data to assess the program's predictive capability. AJenvards the model is used to estimate and locate power losses in a welllubricated high-speed roller bearing. Results show that the total power loss varies strongly with the rotational speed, the lubricant inlet temperature and the oil flow through the bearing. Nevertheless, it is less sensitive to radial load Power loss results are also given as a function of the bearing internal geometry. NOMENCLATURE CC Friction torque between roller ends and cage pocket CE Friction torque between roller ends and race flanges DM Bearing pitch diameter DR Roller diameter Ei.2 Elastic modulus F Traction force between roller and raceway FC Centrifugal force acting on the rollers X I ,Traction Z force at the roller-cage pocket contact E?? Traction force between roller ends and race flanges FOL Oleo dynamic drag force acting on the rollers FR Applied radial load h Elastohydrodynamiclubricant film thickness at the roller-ring raceway contact H J , ~Roller position in the cage pocket JCC Roller to cage pocket circumferential clearance JD Bearing diametral clearance I Contact length Q Normal load between roller and raceway

QC Hydrodynamic normal load at the cage-ring contact QCJ,ZNO load ~ at the roller-cage pocket contact x Angular direction of the normal load at the cagering contact E Cageeccentricity S Roller-raceway contact deformation Sr Radial deflection of the bearing # Cage attitude angle (short journal bearing effect) v1.2 Poisson's ratio w Rotationalspeed wc Cage rotational speed wr Roller rotational speed 5 Angular location of the cage center (gq+@ ty Roller angular location

Subscripts Referstoherring Refers to outer ring Refers to roller number j

,-

,

466

1.

INTRODUCTION

It is commonly known that lubrication in rolling bearings prevents metallic contact, reduces friction and wear of interacting elements, and serves as a coolant by evacuating heat dissipated inside the bearing. Lubricant film thicknesses which separate friction surfaces range usually from micrometer up to tenth of millimeter. That means that the amount of oil addressed to a bearing mainly has a cooling function. Over recent years the operating environment in engines has become increasingly severe. The lubricant supply has therefore become increasingly important for its cooling function. Oil flow rates need to be high to achieve the desired temperatures and thus lead to excessive parasitic thrashing and churning of the oil before it is scavenged from the chamber. Furthermore, the oil flow to the bearing which was adequate when the emphasis was on provision of lubricant, is not at all appropriate when the cooling aspect becomes so important. Simple empirical expressions have been developed in order to give the overall bearing heat generation in terms of load, speed, lubricant supply, etc. However they do not identifj the distribution of heat generated among the various dissipative mechanisms as the parasitic churning and the contact between roller and raceway, roller-cage pocket and cage-ring pilot surface. Experience has also shown that such empirical rules cannot be applied to applications that require extrapolation sigmfkantly beyond the boundaries of the actual test data. The present, unsophisticated method of throwing more lubricant at the bearing to achieve a few degrees temperature reduction increases oil heating and leads to oversized and overpriced engine oil systems (oil mass, oil tank, exchangers, filters, pumps). It is clear from this analysis that it is advantageous to reduce the oil flow to the bearings, with associated benefits on oil system components. Most bearing users and manufacturers still estimate rolling bearing performance from 1960's and 1970's approaches, based on Harris' work [l], without taking into account kinematic effects of force balance in the bearing or recent progress in hydrodynamic or elastohydrodynamic lubrication.

These ball and roller bearing calculation methods remain helpful to establish general trends in a parametric design study, but do not accurately predict or locate the power losses of a welllubricated high-speed rolling element bearing. Since 1970, a few attempts have been carried out in order to evaluate the internal kinematics [24] or the heat generation (7-121 in oil lubricated roller bearings. In this field, the studies based on computer programs, such as Cybean [6], Shaberth [8] or Adore [ l l ] can be noted. They point out that the energy dissipated depends clearly on the lubricant drag forces and churning moments due to the substantial quantity of oil present in the bearing cavity. Shaberth software is capable of steady state and transient thermal analysis of shaft and rolling element bearing systems whereas Adore deals with the dynamics of rolling elements. However, the lack of knowledge about the equivalent density of the oilair mixture present in the bearing, implies that only a first approximation to the actual losses can be made. More recently, Chittenden [13] and Ndlias [14,15] have shown the internal bearing kinematic effects on the estimated power loss and its location for ball bearings. Furthermore, since the internal kinematics depends on the amount of lubricant in the bearing, it is not correct to separately study bearing cage slip and heat generation. We present here a new roller bearing model, including hydrodynamic and elastohydrodynamic lubrication analyses. It predicts and locates power losses in a high-speed cylindrical roller bearing, operating under purely radial load and oil lubricated. Rheological properties of the MIL-L23699 type lubricant are given by Houpert [16], Gupta et al. [17] and Ndlias et al. [IS]. An empirical model for the windage torque acting on the rotating element, constituted by both the cage and the rollers, is proposed. Other sources of energy dissipation are evaluated and located from the friction forces and the sliding speeds between the various bearing elements, including roller-raceway, roller-cage and cage-ring contacts. 2.

ROLLER BEARING MODEL

A comprehensive quasi-static model which provides a simulation of the heat generation in a

467

roller bearing is reported here. More details are given in reference [19]. The quasi-static model basically consists of the formulation of the geometrical relations and the equilibrium equations for the various roller bearing elements. The different assumptions are presented and the contact loads are identified. The force and moment equilibrium equations for each bearing element and the geometrical relations are presented. Finally, the algorithm developed to simultaneously solve the equilibrium and geometrical equations is discussed. 2.1. Rheological Behaviour of Non-Newtonian Fluids Traction studies in lubricated Hertzian contacts (E.H.D.) have shown that lubricant behaviour cannot be described by a simple Newtonian model. More complex rheological models have been suggested and have relied on transient viscosity and viscoelasticity to explain discrepancies between classical theory and experiments. Gupta and Forster et al. [20,2l]have shown that a simple non-linear viscoelastic model can be used to predict with good accuracy traction results obtained on a twodisc machine. The lubricant behaviour is represented by three parameters, which are considered to vary with pressure and temperature:

- the dynamic viscosity - the shear modulus - and a critical shear stress

p@,0,

G@*7')s roe,7').

The linear viscous law is written as:

where y is the shear strain rate, r the shear stress and p the viscosity. According to the Ree-Eyring theory [22],nonlinear viscous behaviour can be expected at high shear stress. That behaviour is described by a hyperbolic sine function. To take into account the transient effect, a transient viscosity concept can be introduced.

where ro is a critical shear stress which must be defined. A viscouselasto-plastic behaviour of the fluid is also considered. The model used is similar to that presented by Johnson and Tevaarwerk [23], and combines an elastic and a viscous shear rate. It reads: 1 dr y=--+F(r) G dt

I dr ro =--+-&G dt

p

r ro

(3)

where G is the shear modulus and where F(7) takes the Ree-Eyring form. 2.2. Roller Bearing Element Interactions The existing hydrodynamic and elastohydrodynamic models are used to determine the normal and traction forces corresponding to the geometrical interactions between the different elements of a roller bearing. This part of the paper is limited to the discussion of the models and formulations used in the computer program presented here. Solving bearing equilibrium requires a few simplifying assumptions. As it is a quasi-static analysis, inertial effects except centrifugal forces are neglected in this model. The geometry is assumed to be perfect, i.e., rings, cage and rollers are cylindrical. As it is a plane model, ring misalignment and roller skewing are not considered. The outer ring is stationary. Though, it is possible to take into account; an outer or innerring guided cage, an outer or inner-ring guided roller, and race flanges, which are guiding rollers, with or without flange angle. The basic geometry of a roller bearing is presented in figure 1. The different formulations relating cage-ring pilot surface forces and moments to the cage rotational speed and eccentricity; tangential and normal cage forces to the roller position in the cage pocket; and those defining traction forces at the roller-raceway interactions are summarized in table 1. From most of the available literature, it can be established that contact forces resulting from rollercage pocket and cage-ring pilot surface contacts are small in comparison to the forces at the roller-raceway contacts. Furthermore, due to a lubricant film at the interfaces, hydmdymmic models can be considered, and elastic

468

deformations are neglected in steady-state operating conditions. The computer program shows that rollers moving through the loaded zone, are located in the front of the cage pocket and consequently are driving the cage. While the opposite rollers, in the unloaded zone, are located at the rear of the cage pocket and act on the cage as a braking system. The interaction between cage and outer or inner ring pilot surface is assumed to be purely hydrodynamic, and no elastic deformations resulting from the hydrodynamic forces are considered. In a rolling bearing, the ratio of pilot surface width to cage diameter is always lower than

CONTACTS

0 WINDAGE

a CAGE I RING

ASSUMPTIONS

1/6. So the hydrodynamic of the cagehace contact is simply modelled by the well-known "short journal bearing" solution [24] in laminar or in turbulent flow. In this model, the attitude angle of the cage denotes the angle between the center line direction and the external load direction. Resulting from a lubricant film at the interface, low load and conform surfaces, the interaction between roller and cage pocket is considered being purely hydrodynamic. This contact is very simply simulated by the well-known "long journal bearing" solution [24], using the analytical solution of Martin for isoviscous fluid.

1

MODELS

RESULTS

Bearing Tests

Air-oil mixture generates oleodynamic drag

- light load

- experimental results

- windage torque

Hyddpamic

- load - torque - attitude angle

- "shortjournal bearing"

- rigid surfaces - laminar or turbulent flow

as h c t i o n of the eccentricity Elastohydrodyamic

- lubricant film thickness - Cheng theory - Gupta, Cheng et al. factor - thermal reduction factor

- from none up to heavy load ROLLER I RING

( h e r ring) I

Elastohydr+amic

RACEWAY

- elastic deformations

- rolling and sliding speeds @ ROLLER / CAGE POCKET

0 ROLLER EDGESlPOCKET EDGES

GI ROLLER EDGES/RING RIDING

- light load - rigid surfaces - no load - no skewing

- no load - no skewing - with race flange angle

- Johnson and Tevaarwerk theory Hydrodynamic

- Martin theory

- hction force - normal load - friction force as h c t i o n of the roller position

- couette flow

- hction force

Hydrodynamic

- couette flow

Table 1 - Roller bearing element interactions

- friction force - hction torque

469

Where c c c c

N*

I' 2'

S P @i @C

Qh

and

d,,,

are appropriate coefficients, is the number of rollers, is the roller frontal area, is the lubricant viscosity, is the inner ring speed, is the cage speed, is the lubricant flow, is the bearing pitch diameter.

With g=O for a lubrication by an external jet and g=l for a lubrication through the inner ring. The last term in equation (4) quantifies the lubricant drive effect when oil is provided through the inner ring, assuming that all the rotational energy of the lubricant is absorbed by the cage. Fig. 1- Roller bearing element interactions

2.3. Basic Eauations Concerning the roller endcage pocket edge interaction, assuming there is no roller skewing in the cage pocket and no normal load between roller ends and cage pocket edges, the model is similar to the hydrodynamic solution of couette flow [191. The roller-ring raceway contact is described by Cheng 1251 for the lubricant film thickness in an E.H.D. line contact, including a thermal corrector factor given by Gupta, Cheng et al. [20], and by Johnson and Tevaanverk for the traction force [23]. We assume that, at the roller end-race flange interaction, which can include a race flange angle, there is no normal load and that the roller is rolling centred between the two guiding shoulders without skew angle. Then we can use the well-known couette flow model [191. The empirical model for the windage torque acting on cage and roller elements, which is proposed, comes from several experimental studies of power loss in high-speed roller bearings, for different bearing geometries (35 to 142 mm pitch diameter) and different lubrication types (by jet and through the inner ring). Experimental results have been reduced to a simple model, with appropriate coefficients derived by curve-fitting the drive torque measured to the operating conditions and bearing geometries. From reference [26], the windage torque acting on the cage is predicted as follows:

Equilibrium equations In this model, the kinematics of the cage and of each roller are unknown. They are determined by the force and moment balance of the different bearing elements. Figure 2 shows the interactions around a roller, with an outer ring guided roller. These interactions may be represented by traction forces and moments, as shown in figure 3. So, equilibrium equations for the rollers are defined as follows:

-

0 = Qij Qoj - FClj + FC2j + FC

(5)

Fij-FOJ.-QClj+QC2j + FEij + FEoj - FOL

(6)

O=

0 = (-Fij - Foj + FClj + FC2j).DW2

+ CEij + CEOJ. + CCj

(7)

The cage equilibrium equations (Fig.4), along the horizontal and the vertical axes are written as: N O = Cl(eC2j-eClj).cos~jl+(eCi + eCo).sin(~I j=l (s)

470

N 0 = / y [ - ( F c 2 j -Fclj).COSylj J + ( Q c i + Q c , ) . C O . ( ~ ) j =I

(9) and the moment equilibrium equation is given by:

The outer ring is stati0~1-y and mounted in its housing. The mounting forces ensure the outer ring equilibrium without influencing the bearing behaviour.

N 0 = z [ ( Q c l j -Qc2j).DM / 2 - ( F C l j + F c 2 j ) . D R / 2 j=l -CCj J + CCi CCo

-

(1 0)

If

rotating inner nn9

Fig. 4- Cage balance, forces and moments

Fig. 2- Roller balance, geometrical interactions

Fig. 3- Roller balance, forces and moments The quasi-static equilibrium of the inner ring (Fig.5) can be reduced to one equation defined as follows:

0

=

N z[Q,ij.co~vj J - FR j =I

(11)

if the hydrodynamic load QCi at the cage-inner ring contact is neglected compared to the normal loads Qg at the roller-raceway interface and the applied radial load FR.

Fig. 5- Inner ring balance, forces and moments Geometrical relations When the roller moves through the loaded zone, its circumferential position in the pocket is modified. In this model this position is an unknown. From figure 2 a geometrical relation is given by:

0

=

Hl.+H2j-JCC J

(12)

47 1

For a line contact, according to Palmgren [27], the normal load Q is related to the normal contact deformation 6 as follows:

S = 0.39 [4(1-~12)/E1+4(1~22)/E2]0.9 .(Q.9 / Cds)

(13)

where vl. E l , v2 and E2 are the Poisson's ratio and the elastic modulus of the two interacting bodies, and 4 is the contact length. From figure 6, the knowledge of the load deflections and lubricant film thicknesses at the inner and outer ring contacts, for one roller, allows to determine the total elastic deflection for other rollers. Contact exists only for a positive value of S, and the contact load is determined by equation (13). As shown in figure 6, the inner race deflection is given by Cforj =I ) : 4 . = J D / 2 + 8'I. . + 8OJ. - h '.I. - hOJ.

An algorithm has been developed to solve simultaneously the non-linear equilibrium and geometrical equations set out above. This set of (4Nf3) equations, related to (4N+3) unknowns, is solved using the Newton-Raphson iterative method. The (4N+3) unknowns are the cage eccentricity, E, and its angular location, 6, the rotational speed of the cage, a+, and for each roller; its location in the cage pocket, HI, the inner and outer normal load, Qi and Qo, and the rotational speed of the roller about its own axis w,.. One of the most important advantages of this procedure is the small number of iterations required to obtain convergence. Convergence is reached within a maximum relative error of 10-6 after 20 to 80 steps, depending on the number of rollers, the initial conditions and the relaxation coefficient.

(14)

Then, for a prescribed roller azimuth rvj, i.e. for j = 2 to N, the geometrical relation is written as: O=(JD/2+ 6..+6 ~ - h . . - h o , ) - 8 r C O S ~ 'I OJ 'I

2.4. Numerical Procedure

(15)

3.

RESULTS

3.1. Data

The specifications of the bearing example are listed in table 2. Both cage and rollers are guided by the inner-ring. The lubricant is a tetra ester, qualified to the MIL-L-23699 specification. Lubrication is provided under the inner race, therefore, the whole input flow rate goes through the bearing. The range of operating conditions is described in table 3. 3.2. Test-Model Correlation

Fig. 6- Inner ring radial deflection The set of equations (5)-(1 I) and (15) describes the roller bearing balance.

The authors first compared the values of calculated drive power with experimental results. It must be noted that the drive power is equal to the total power loss. The experimental work used to validate the computer program has been performed on several high-speed roller bearing test rigs. The test-model correlation was originally reported in [26]. The predicted bearing heat generation agreed very well with the experimental data obtained from different sizes of roller bearings (35 to 142 mm pitch diameter), for a lubricant provided by jet or through the inner ring and over a speed range from 0.3 up to 3 million DN. The appropriate coefficients derived to estimate the windage torque appear to be valid over the range of shaft speeds, lubricant flow

412

rates, lubricant inlet temperatures and radial loads for the four bearing geometries investigated. ROLLER BEARING SPECIFICATIONS External geometry (ID,OD,width),mm Pitch diameter, mm Number of rollers Roller diameter, nun Roller length, mm Bearing diametral clearance, pm

3.3. Total Power Loss Prediction Vs. Operating

19x164~40 142 30 12 14 30

Cage guidance type

inner ring

Cage diametral guiding clearance, pn

480

Roller guidance type

Roller axial guiding clearance, pn Race flange angle, degree

inner ring 30 0.375

Race and ball material

AISI M-50

Table 2 - Roller bearing specifications

I

RANGE OF OPERATING CONDlTlONS ShaA speed, rpm Radial load, daN Lubricant flow, l/h Lubricant inlet temperature, "C Lubricant speciJication

Typical results of power dissipated versus lubricant flow rate are shown in figure 7, for the 142 m m pitch diameter bearing described in table 2.

I

0 to 20000 0 to 5 000 0 to 300 60 to 200 MIL-L-23699

Conditions Figure 8 shows the total power loss prediction versus the shaft speed for several radial loads varying between 500 and 4000 daN, a lubricant flow of 150 Uh and a lubricant inlet temperature of 100 "C. Heat generation increases greatly with the shafl speed, whereas it seems independent of radial load (all plots seem as one). In fact, a decrease in power loss can be observed for radial loads lower than 200 daN, as shown in figure 12. It must be emphasized that, as earlier noted for ball bearings [15], power loss prediction and location depend strongly on the roller bearing internal geometry, and more specifically on cage and roller guidance type, i.e. guided by inner- or outer-ring. For example with this specific internal geometry, where both cage and rollers are guided by the inner ring, a small cage slip occurs for radial load lower than 200 daN. This phenomenon is important because changes in cage speed produce a strong variation in windage loss, according to equation (4).

Table 3 - Range of operating conditions

-

0

5

3000-

8

2000-

,c

e

=

500 daN

N I = 2500 rpm 4

0

4000

8000

12000

I6000

20000

INNER RING SPEED (rpm)

/

0

0

FR

50

100

I50

200

1 250 MO

LUBRICAN1 FLOW (I/h)

Fig. 7- Predicted and experimental total power loss vs. lubricant flow and inner ring speed. Lubricant inlet temperature, 100°C; radial load, 2500 daN. (Symbols for tests; and line for model)

Fig. 8- Predicted total power loss vs. inner ring speed and radial load. Lubricant flow, 150 lh;lubricant inlet temperature, 100°C.

Figure 9 shows the total power loss prediction versus the lubricant inlet temperature varying from 6OoC up to 2OO0C, over a speed range from 2500 to 14000 rpm, with a lubricant flow of 150 Uh and a radial load of 2500 daN. It can be noted that the

473

decrease in total power loss with increased lubricant inlet temperature is very marked. The predicted trends of increased heat generation with increased shaft speed, increased lubricant flow rate and decreased lubricant inlet temperature were verified by the experimental data. 10000

+ 8000

g

6000

\+

+

N i = 14000 rpm

A

N i = 10500 rpm

7000 rpm N i = 4500 rpm N i = 2500 rprn

m Ni =

\

v 0

3.5. Power Loss Location Vs. Internal Geometry

v

[L W

3 a

0

4000

2000

0

300

350

400

4 50

500

LUBRICANT INLET TEMPERATURE ( K )

Fig. 9- Predicted total power loss vs. lubricant inlet temperature and inner ring speed. Radial load 2500 daN; lubricant flow, 150 Vh.

Loss Location Vs. Operating Conditions Figures 10, 1 1 and 12 present the estimated distribution of the heat dissipated in the bearing, versus the lubricant flow rate (Fig. lo), the inner ring speed (Fig. 11) and the radial load (Fig. 12). The cage contribution to the total power loss is important, mainly due to the fluid windage, and also due to the short journal dissipation at the cageinner ring pilot surfaces and to the power dissipated at the rollercage pocket interfaces. The decrease of the total power loss with decreased lubricant flow rate is mainly due to the windage loss reduction, as shown in figure 10. Predicted power loss location shows that cagering pilot surface contribution to the total energy dissipated can be more important than the windage contribution at very high speeds (Fig. 11). A shaft speed of 20000 rpm for a 142 mm pitch diameter bearing corresponds to 2.84million DN. As it was mentioned above, figure 12 presents some interesting results on power loss distribution 3.4. Power

under light radial load, typically for radial loads lower than 200 daN. The cage slip which produces a decrease in windage and total power losses also strongly modifies the power loss distribution. Then, dissipation can reach 300 W at the roller-race flange interface and 150 W by sliding at the rollerinner ring raceway contact. Note that both cage slip phenomenon and power loss distribution are strongly dependent on the roller bearing internal geometries, and then results for other geometries can be completely Werent. A parametric study was conducted in order to evaluate the effects of bearing internal clearances on power loss prediction and location. Power loss versus the bearing diametral clearance, the cage diametral guiding clearance, the roller axial guiding clearance and the race flange angle value are presented respectively in figure 13, 14, 15 and 16. The basic roller bearing geometry is presented in table 2. Operating conditions are the following; an inner ring speed of 10500 rpm, a radial load of 2500 daN, a lubricant flow of 150 l/h and a lubricant inlet temperature of 100OC. Figure 13 shows that no important change can be observed in the distribution of heat generated in the bearing versus the diametral clearance, as far as this clearance remains positive. Increase of the total power loss with decreased cage diametral guiding clearance is mostly due to the cage-ring pilot surface contribution, as shown in figure 14. Local effects of roller-race flange geometry are presented in figures 15 and 16. Decreasing both roller axial guiding clearance and race flange angle value increases power loss generated at this interface, without significant effects on the total power loss. 4.

CONCLUSION

The authors have presented a new roller bearing model based on cage and roller kinematics. This model includes both hydrodynamic and elastohydrodynamic analyses to describe interactions among the various rolling bearing elements, and an empirical windage torque model.

474

Experimental results were compared to computer predictions.

permission to publish this work, and to Gerard Paty of Turbomdca and Guy Dusserre-Telmon of SNECMA for their assistance.

The following major results were obtained:

REFERENCES 1. Although it is a quasi-static model, a good correlation between theory and experiment was obtained on power loss. 2. The contribution to the total energy dissipated of each lubricated contact is established, i.e., interface between roller and raceway, roller and race flange, roller and cage pocket, cage and ring pilot surface, etc. 3. Results show that parameters af€ecting the power loss may be classified in descending order as follows: - for operating conditions; the rotational speed, the lubricant inlet temperature and the lubricant flow, whereas the radial load effect is considered to be less important. - for internal geometry; the cage guiding clearance and its location on inner- or outerring, the roller guiding clearance, its location and the value of its race flange angle. The influence of the bearing diametral clearance can be considered as negligible. Finally, the authors point out that the knowledge of internal kinematics and contact loads in high-speed roller bearings is of great interest on heat generation, since power losses are related to sliding velocities and friction forces.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support given under a BRITE-EURAM contract number AER04007-C(A) (proposal number PL1103), by the Commission of the European Communities (CEC-DGXII). This work was also jointly sponsored by Turbomdca, SNECMA, Hispano-Suiza, GLCS and MSA in France, Rolls-Royce and Leeds University in England, BMW-RR and MTU in Germany and FEW at Port0 University in Portugal. The authors wish to express their thanks to the SNECMA and Turbomica Companies for performing the roller bearing tests and their

Harris T.A., Rolling Bearing Analysis, Wiley Intersciences, New York (1966). Boness R.J., #*TheEffect of Oil Supply on Cage and Roller Motion in a Lubricated Roller Bearing," A W E Jour. of Lubr. Tech., Paper 69-LUB 8-73, pp.39-53 (1970). Poplawski J.V., "Slip and Cage Forces in a High-speed Roller Bearing," A W E Jour. of Lub. Tech., Paper 71-LUB-17, pp.143-152 (1972). Rumbarger J.H., Filetti E.G. and Gubernick D., "Gas Turbine Engine Mainshaft Roller Bearing-System Analysis," A W E Jour. of Lub. Tech., 95, pp.401-416 (1973). Berthe D. and Flamand L., "Skidding in Roller Bearings, Effect of Lubricant," Proc. AGARD Symp., 323, Problems in Bearing Lubrication, Ottawa (1982). Kleckner R.J., Pirvics J. and Castelli V., "High-Speed Cylindrical Rolling Element Bearing Analysis "CYBEAN" - Analyt~c Formulation," A W E Jour. of Lubr. Tech., 102, 3, pp.380-388, discussion, pp.388-390 (1980). Coe H.H. and Schuller F.T., "Comparison of Predicted and Experimental Performance of Large-Bore Roller Bearings Operating to 3.0 Million DN," NASA, Washington, D.C. NASA Technical Paper 1599, 18p. (1980). Hadden G.B., Kleckner R.J., Regan M.A. and Sheynin, L., "Research Report - User's Manual for Computer Program AT81YOO3 Shaberth. Steady State and Transient Thermal Analysis of a Shaft Bearing System Including Ball, Cylindrical and Tapered Roller Bearings," NASA, Washington, D.C. NASA CR 165365 (1981). Coe H.H., "Predicted and Experimental Performance of Large-Bore High-speed Ball and Roller Bearings," NASA Conference Publication 2210, Advanced Power Transmission Technol., pp.203-220 (1983).

475

i 400

- Total Power Loss 0

-3

m

3000 2500

350

Windage

300

Cage-Ring

2000

T

Roller-Outer Ring

A

Roller-inner Ring

+

Roller-Ring Riding

*

Roller-Cage

I

Roller Edges-Pocket

m

2 50

0

1500

2

:: 1000

200 150 100

we-------*

1

500

LUBRICANT FLOW (I/h)

LUBRICANT FLOW (I/h)

Fig. 10-Power loss location vs. lubricant flow. h e r ringspeed, 10500 rpm; radial load, 2500 daN,lubricant inlet temperature, 100°C. 1000

8000

- Total

Power Loss

m

Windage Cage-Ring

v

3

v

Roller-Outer Ring

m

A

Roller-lnner Ring

9

0

6000

5

,-, 8

'z

800

600

-1

4000

Roller-Ring Riding

+

[L W

Roller-Cage a

400

a

Roller Edpes-Pocket

2000

0

-

0

5000

15000

10000

20000 INNER RLNG SPEED (rprn)

INNER RING SPEED ( r p m )

Fig. 11- Power loss location vs. inner ring speed. Radial load, 2500 daN, lubricant flow, 150 vh, lubricant inlet temperature, 100°C.

,

3000

2500

z-

- Total

-

1

Power Loss

-

-

-

-

a

*

-

-

--.---.-

,

Cage-Ring

Roller-Inner Ring

1500

+

I

Roller-Outer Ring

,------.

m

3oo

a Windage

-

2000

u,

9

1

s

2oo

W (L

3

2

1000

500

0

0' 0

10000 20000 30000 40000 50000 RADIAL LOAD (N)

0

10000 20000

30000

40000

50000

RADIAL LOAD (N)

Fig. 12- Power loss location vs. radial load. Inner ring speed, 10500 rpm; lubricant flow, 150 vh, lubricant inlet temperature, 100°C.

476

100

- Total Power Loss

E v

t1

2500 2000

Windage h

cn cn

5

80

Cage-Ring

1500

v

Roller-Outer Ring

A

Roller-Inner Ring

+

Roller-Ring Riding

*

Roller-Cage

1

Roller Edges-Pocket

5 R 53 LL

W L11

?

g

1000

5n

6o 40

20

n 50

0

100

200

150

BEARING DIAMETRAL CLEARANCE ( 1

0

.e-6*m)

50

100

150

200

BEARING DIAMETRAL CLEARANCE ( 1 e-6.m)

(regular view on the left; zoom 011 the right) Fig. 13- Power loss location vs. bearing diametral clearance.. Inner ring speed, 10500 rpm; radial load, 2500 daN; lubricant flow, 150 fi;lubricant inlet temperature, 100°C. - Total Power Loss

--

0

100

200

300

400

500

Roller-Outer Ring Roller-Inner Ring

+

Roller-Ring Riding

.

t

Cage-Ring

A

-

0

loo

Windage

Roller-Cage Roller Edges-Pocket

600

CAGE DlAMElRAL GUIDING CLEARANCE (1.e-6.m)

LL

E

40

0

0

100

200

300

400

500

600

CAGE DIAMETRAL GUIDING CLEARANCE ( 1 .e- 6.m)

(regular view on the lett; zoom on the right) Fig. 14- Power loss location vs. cage diametral guiding clearance. Inner ring speed, 10500 rpm; radial load, 2500 daN; lubricant flow, 150 yh; lubricant inlet temperature, 100°C.

[lo] Pirvics J. and Kleckner R.J., "Prediction of Ball and Roller Bearing Thermal and Kinematic Performance by Computer Analysis," NASA Conference Publication 2210, Advanced Power Transmission Technol., pp. 185-202 (1983). [ l l ] Gupta P.K., Advanced Dynamics of Rolling Elements, Springer-Verlag, New-York, 295p. (1 984). [I21 Schrader, S.M., "Performance of a Hybrid Cylindrical Roller Bearing," Lubr. Eng., 48, 8, pp.665-672 (1992).

[I31 Chittenden R.J., Dowson D. and Taylor C.M., "Power Loss Prediction in Ball Bearing," Proc. of the 15th Leeds-Lyon Symp. on Trib., pp.277-286 (1989). [14] NClias, D., "Skidding in High-speed Aircraft Turbine Engine Ball Bearings: Effects of Lubricant Contamination," Doctoral Thesis, INSA Lyon, 292p. (1989). I151 Nelias D., Sainsot P. and Flamand L., "Power Loss of Gearbox Ball Bearing Under Axial and Radial Loads," Presented at the 48th STLE Annual Meeting in Calgary, Alberta, Canada, May 17-20, 1993, Preprint 93-AM-4C-3.

477

100.

3000

- Total 2500

1

I

I

10

20

30

I

I

50

60

Power Loss

Windage

80 :

Cage-Ring

y

2000

w

Roller-Outer Ring

A

Roller-Inner Ring

v

ul

9

c

5 VI

SOL

i

VI

4

1500

1

*'

W Lz

b

a

1000

500

0

10

0

20

30

40

50

0

60

40

ROLLER AXIAL GUlOlNG CLEARANCE (1 .e-6+rn)

ROLLER AXIAL GUIDING CLEARANCE (1 .e-6-m)

Fig. 15- Power loss location vs. roller axial guiding clearance. (regular view on the left; zoom on the right) Inner ring speed, 10500 rpm; radial load, 2500 daN; lubricant flow, 150 yh; lubricant inlet temperature, 100OC. 3000

- Total

Power Loss

Windage

80 :

Cage-Ring

ul

9

1500

w

a 1000

E I 500

0

1 I- -*.I

i 1

3

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 RACE FLANGE ANGLE (degres)

Fig. 16- Power loss location vs. race flange angle.

.

.. . . .

^--^

. .-. . .

[I61 Houpert L., "A Contribuhon to the Study ot

Friction in E.H.D. Contacts," Doctoral Thesis, INSA Lyon, 265 p. (1980). [ 171 Gupta P.K., Flamand L., Berthe D. and Godet M., "On the Traction Behaviour of Several Lubricants," ASME Jour. of Lubr. Tech., 103, 1, pp.55-64 (1981). [18] Nelias D., Dalmaz G. and Flamand L., "Roller Bearings, Part 111: Traction Behaviour of a MIL-L-23699 Type Lubricant," Brite-Euram Contract, Aero-0007-C(A), Bearing with Minimum Lubrication, LMC INSA Lyon, 3 lp. (1992).

.

-

(regular view on the left; zoom on the right)

---.". . .

L l Y ] Nelias D.,

.

.

...

.

.^^^^

Dalmaz ti. and Flamand L., "Roller Bearings, Part I: Quasi Static Analysis of Lubricated Roller Bearings," Brite-Euram Contract, Aero-0007-C(A), Bearing with Minimum Lubrication, LMC INSA Lyon, 7 lp. (1993). [20] Gupta P.K., Cheng H.S., Zhu D., Forster N.H. and Schrand J.B., "Viscoelastic Effects in I: MIL-L-7808-Type Lubricant, Part Analytical Formulation," Trib. Truns., 35, 2, pp.269-274 (1992). [21] Forster N.H., Schrand J.B. and Gupta P.K., "Viscoelastic Effects in MIL-L-7808-Type

478 Lubricant, Part 11: Experimental Data Correlations," Trib. Trans., 35, 2, pp.275-280 (1992). [22] Ree T. and Eyring H., "Theory of NonNewtonian Flow. Part I- Solid Plastic System, and Part II- Solution System of High Polymers," Jour. of Appl. Phys., 26, pp.793800 (Part I), pp.800-809 (Part 11) (1955). [23] Johnson K.L. and Tevaarwerk J.L., "Shear Behaviour of Elastohydrodynamic Oil Films," Proc. Roy. SOC. Lond., A356, pp.2 15-236 (1977). [24] F r h e J., Nicolas D., Degueurce B., Berthe D. and W e t M., Hydrodynamic Lubrication, Journal and Thrust Bearings, Eyrolles, Paris, 488p. (1990). [25] Cheng H.S.,"A Numerical Solution of the Elastohydrodynamic Film Thickness in an Elliptical Contact," ASME Jour. of Lubr. Tech., 92, F, pp. 155-162 (1970). [26] Nelias D., Dalmaz G.and Flamand L., "Roller Bearings, Part V: Experimental Validation and Typical Results of the Computer Program QUASAR," Brite-Euram Contract, Aero-0007C(A), Bearing with Minimum Lubrication, LMC INSA Lyon, (1993). [27] Palmgren A., Ball and Roller Bearing Engineering, 3rd Edition, SKF Industries Inc., Burbank, (1959).

DissipativeProcesses in Tribology / D. Dowson et al. (Editors)

479

Q 1994 Elsevier Science B.V. All rights resewed.

POWER DISSIPATION IN ELASTOHYDRODYNAMIC TRACTION DRIVES I.M.CIORNEI", E.N.DIACONESCU", V.N.CONSTANTINESCUb and G.DALMA2'. "University of Suceava, 58OO,Suceava, Romania bPolytehnic Institute of Bucharest, Romania "Laboratoire de Mecanique des Contacts, INSA Lyon,69621, Villeurbanne, France SUMMARY The paper presents an analysis of friction power losses in elastohydrodynamic traction drives and underlines the possibilities of optimisation of these transmissions. Finally, several high performance traction drives are described. INTRODUCTION Elastohydrodynamic traction drives represent one of the few fields in which the fluid friction is an useful phenomenon. Mechanical power is transmitted between the active elements of these drives by shear of elastohydrodynamic oil films. A part of the input power is dissipated in the film by parasitic shears. The remaining part represents the useful, transmitted power. In order to optimise the efficiency of elastohydrodynamic traction drives, the parasitic shears must be minimised. Traction drives conceived to this end are finally described in the paper. FRICTION SOURCES IN ELASTOHYDRODYNAMIC TRACTION DRIVES The fluid friction in EHD traction drives depends on many physical factors, of which the most important are macro and microgeometry of contacting elements, contact deformations as functions of material properties, lubricant nature and lubrication procedure, [l]. Nowadays it is convenient to measure global power losses in an EHD traction drive and to assess theoretically the components of friction. Consequently, the possibilities of optimisation of traction drives are severe limited,

[1,17,19,20]. As a general view, the power losses in a traction drive are composed of: - friction in kinematic pairs; - friction between moving elements and lubricating medium; - friction in sealing elements; - air ventilation for cooling the drive. As shown in figure 1, each of these components is a sum of several sources and can be identified finally as a part of global friction torque of the drive. Losses by elastic hysteresis. During rolling, the inlet zone of the contact is subjected to loading, whereas the output is unloaded. The loading and unloading loaddeformation curves do not coincide and the area limited by them is a measure of the energy loss by elastic hysteresis per cycle. This l o s s depends, [1,2,6,18], on: - elastic properties of contacting materials: as the material behaves more elastic, the energy losses decrease whereas a viscous behaviour increases the power dissipation; - contact stresses: at low levels of contact stresses the material behaves more linearly and the hysteresis decreases; - stressed volume. Hysteresis losses are small in comparison to other losses. For instance, Drutowski, [6], attributes

P W

0

I

DISTRIBUTION OVER CONTACT AREA

FRICTION IN EHD TRACTION DRIVES

FUNCTIONAL

PARASITIC (LACK OF SYNCHRONISATION)

I

WITH FLUID ENVIRONMENT

WITH POSITIONING AND SUPPORTING ELEMENTS

1

LIMITING FRICTION WITH HOUSING MATERIAL

Figure 1. Power disipation in EHD t r a c t i o n d r i v e s

48 1

a friction coefficient of to hysteresis losses. Although small, these losses increase as the transverse reduced radius of curvature of the raceways increases, as a result of increasing stressed volume. Losses by nonuniform pressure distribution over the contact area. The hydrodynamic pressure generated in an EHD oil film yields a resultant force which is displaced towards the entry zone into the contact. It is assumed that this effect is responsible for the major part of rolling friction, [3]. Losses by microslip inside the contact. Two loaded active elements of a traction drive make contact over a finite area. Only a small fraction of the points placed in this area belong to the axoydes of motion. In the remaining points microslip occurs. This determines a local microshear of the oil film and consequently, shear stresses opposing the relative displacement. The resultant tangential force generates power dissipation, which increases with normal load nad contact ellipticity, [ E l . Losses by lonqitudinal slidinq. Longitudinal sliding occurs in an EHD contact either as an useful result of the operating process or due to parasitic shear produced by lack of synchronisation between parallel intermediate elements. This sliding is the result of shear behaviour of the oil film and is characterised by the traction [ 14 1. Specific coefficient, measures, such as the use of special lubricants, small rolling speeds or low temperatures are required to reduce the longitudinal sliding. Diminution of this sliding also requires a better synchonisation of multiple intermediate elements of the drive, [ 4 ] , figure 2 . Losses by spin. The spin motion is a result of contact kinematics. Experimental and theoretical investigations,[1,5,7,13,15,19],

indicate that spin greatly affects the traction curves.

Figure 2 . Lack of sincronisation due to Losses by side slip. Side slip occurs in a contact when the axes of the contacting elements are crossed. It reduces the traction capacity of the contact, as the spin does, [ 1 4 ] . Losses in main bearinqs. Usually, the rotating elements are supported in traction drives by rolling element bearings. These are heavily loaded by the normal load applied to the contact. As a result the friction in bearings is high and it dissipates an important part of the drive input power, [ll]. Losses in auxilliary bearinqs. Friction pairs are formed between active elements and their housing. The sliding friction in these pairs can be high, especially when lubrication is poor. Auxilliary, lightly loaded bearings are used to support and position the assembly of intermediate elements. The friction in these bearings is usually small. Losses by churninq become important when the surface of moving elements is large and the oil level is high. These can be reduced by using an incorporated pump to lubricate the drive and a low oil level. Losses in sealinq elements cannot be eliminated due to the design of the drives. Sealing elements having low power losses are therefore required.

482

Losses by air ventilation occur because the drive case must be cooled by means of a fan placed on the input shaft. A low temperature is required by a correct opperation of the drive.

v= 6,5 m/s

POSSIBILIES OF FRICTION DIMINUTION IN EHD TRACTION DRIVES The major part of power losses in a traction drive are caused by rolling, longitudinal sliding, spin, side slip, churning and ventilation, [ 91. The other power losses, shown in figure 1, are of secondary importance. The negative effect of spin upon power losses into an EHD contact is well known, [1,5,7,9,13,15,16]. It consist of a reduction of the slope of traction curves and of an increase of longitudinal sliding. This effect increases with rolling speed and it depends essentially on the rheological behaviour of the lubricant, [14]. The spin also reduces the maximum value of the traction coefficient, as shown in figure 3 . At small values of spinroll ratio this effect is unimportant, but it increases drastically above o limiting treshold of about (2-3)%, [ l ] . The influence of spin upon maximum traction coefficient, at various rolling speeds, is shown in figure 4.

Relations to estimate the spinroll ration in a point contact, as deduced by authors, are given in Table 1. Some of these are experimentally verified, as seen in figures 5 and 6 for, respectively, a ball on disc and a kopp B traction drives. Side slip occurs in a contact due to cross missalignment of the element axes. The negative effect of this parasitic shear is comparableto that of spin, [6]. High precision machining and mounting are required to reduce the side slip.

0.01

1

I I I

I

I I

I

1

I

0

2

1

3

L pivotement

Figure 3.Effect of spin upon maximum traction coefficient for oil T90EP2

Ci, = 3GPa

I

2

t

6

8

c

so vfdd

Figure 4.Effect of rolling speed upon maximum traction coefficient.

483 Table 1. Equations of spin calculation for point-contact traction drives Nr.crt.

1

Versiod

PERBURY

3

KOPP B

Ball on disc

Diagramma

I R

Input contact

x-

[R,+RcosaI s i n a - [ R , t g ( a - p ) + R s l n a ]c o s a - + R s i n a ]cos ( a -p) - [R,+Rcosa] s i n ( a -p ) Y= Rcosa [ s i n a - c o s a t g ( a - p ) I [R, t g ( a -P) * R s l n a ]cos ( a -p) - [R,+Rcosa] s i n ( a -p ) [ R ,t g ( a

output contact

T.= X ' s i n ( a + p )- Z ' c o s ( a + p ) X'=Z'=-

Observations

S-

nondirnensional longitudinal sliding

X'sina -2'cosa RI +Rcosa sina-cosatg(a+p) R s l n a +R,t g (a +p ) sina-cosa t g ( a t P )

Contact ball-retaining ring

o,*=o,xsinp-

w,R,S

(l+;)Rcosp

Ysinp

Figure 7 indicates the values of the cross angle as function of maximum Hertz pressure which reduce the transmitting capacity of the contact by 10% and 20%. It is advisable to keep this angle below 50'.

Figure 5.Variation of maximum traction coefficient and of efficiency with nondimensional spin for ball on disc traction drive.

Figure 6.Spin in Kopp B traction drive

Figure 7.Limiting values ior crossangle: a)for 10% diminuation of transmitting capacity; b)for 20% diminuation of transmitting capacity; When thermal conductivity is predominant, the energy balance equation yields relationships for the ratio of traction force under spin and side slip to the traction force when no parasitic shears act, [lo]. These can be written as:

where S'=o,,/o,under spin and S'=Av/v under side slip. In these relations p is the thermo-viscous coefficient, i l c the viscosity inside the contact, u the rolling speed and K, the thermal conductivity. Two possibilities exist to reduce the parasitic longitudinal sliding caused by lack of synchronisation between multiple intermediate elements of a traction drive. The first consists in rising the precision of the drive but it rises the price of the transmission. The second relies on supplimentary mechanisms to allow the selfpositioning of the intermediate elements. It seems to be more efficient than the former. As shown above, the friction in the main bearings of the drive can

485

be comparable to the traction transmitted through the active contacts. As a result, if traction oils are unavailable, it is of utmost importance to unload the main bearings, [ 1 ] The reduction of churning and ventilation losses requires lubrication of the drive by oil circulation. To this end a small incorporated oil pump can be used.

performances of this ball on disc traction drive are high. A different version of this design is shown in figure 13.

.

VARIABLE RATIO, HIGH PERFORMANCE TRACTION DRIVES As stated above, the reduction of spin-roll ratio in the active contacts is a very efficient solution to decrease the power losses in a traction drive. This ideea led to two new designs of traction drives, namely an improved Perbury drive, shown in figure 8, and a reduced spin torical drive, illustrated in figure 9, [ 11. These use double rollers that lead to a decreased spin-roll ratio, a higher contact fatigue life and improved conditions of lubrication of both contacts. The corresponding analytical relationship are given in [7] and shown in figure 10.

Figure 9.Traction drive with reduced spin.

'1

Figure 10.Spin-roll ratio in Perbury traction drives CI classical version; 0 improved Perbury drive; A reduced spin drive; Figure 8.Improved Perbury traction drive. Another efficient solution consists in an internal design of the drive based on unloaded main rolling bearings and on an incorporated oil pump for lubrication, [12]. Such a traction drive is shown in figure 11,[11). As indicated in figure 12, the

CONCLUSIONS The following conclusions can be drawn as a result of the analysis performed above: - improved performances of EHD traction drives require reduction of functional longitudinal sliding by use of special lubricants, of

486

adequate cooling of the drive and a low treshold rolling speed just able to form a fluid film;

Figure 11.Ball on disc traction drive with unloaded rolling bearings

"1

G:ZPGPa

-

higher performances of EHD traction drives are obtained if the spin-roll ratio in the active contacts of the drive is reduced by an adequate geometrical design of these components; - a high precision machining and mounting is necessary to produce a high performance traction drive; - when multiple parallel intermediate elements are used, these must be synchronised by carefully conceived mechanisms; - normal contact load must be applied directly to the active elements in such a way that the main rolling bearings remain unloaded; - an internal oil pump to assure a correct lubricant flow through the contacts is essential for a high efficiency transmission; - examples of improved versions of traction drives which incorporate these principles can be seen in this paper. REFERENCES l.I.M.Ciornei, Ph.D.Thesis, Polytechnic Institute of Bucharest (1986) (in Romania) 2.D.R.Adam and W.Mirst, Frictional Traction in EHD Lubrication, Proc. Roy.Soc.Lond.(l973),332.

3.B.I.Klem2,R.Gohar and A.Cameron, Photoelastic Studies of Lubrication Line Contact,Proc. Inst.Mech.Engs.(l971). 4.0.S.Crefu,Ph.D.ThesisIPolytechnic Institute of Iassy, (198l),(in Romania). 5.E.N.Diaconescu,Ph.D.Thesisl

University of London,(1975). G.R.C.Drutovsk1,Energy Bases of Balls Roeling an Plotes',ASME, Series D,Journal of Basic Engineering,81,(1959). 7.I.M.CiorneiIThe Optimisation of Perbury Traction Drive by Spin Reduction,Proc.of the 5-th Europeean Trib.Congress,Eurotrib '89,~O~.5,~.100-105,(1989).

Figure 12.Perforrnances of ball on disc traction drive with unloaded rolling bearings.

8.A.V.Sprisevschi,Rolling Bearings, Masinostroenia,Moskval(1969),(in Russian).

487

Figure 13.High efficiency ball on disc traction drive S.W.Wenitz,Friction at Hertzian Contact with Combined Roll and Twist. 10.V.N.Constantinescu and 1.M.Cionei On the Evaluation of Traction in Concentrated Contacts, Acta Tribologyca,vol.I,1,(1992),p.21-31 ll.E.N.Diaconescu,O.S.Crefu and

I.M.Ciornei,Traction Drives,Proc. of the Seminar on Present and Future Trends in Research of Rolling Contact,Suceava,(l985), p.42-701(in Romanian). 12.E.N.Diaconescu and A.G.Graur, Effect of Bath Oil Level Upon Performances of an EHD Traction Drive,VAREHD 2,SuceavaI(1982), p.249-257,(in Romanian).

l3.I.L.TevaarwerktA Simple Thermal Correction for Large Spin Traction Curves,ASME,vol.l02,(1989),p.440-

446. 14.I.L.Tevaarwerk and K.L.Johnson, The Influence of Fluid Rheology on the Performance of Traction Drive, ASME,Jour.of Lubrication Tech., ~01.101,(1979),~.266-274.

lS.I.M.Ciornei,The Implication of Spin of EHD Traction Drives, International Scientific Conference in Traction,Wear and Lubricants,Taskent,URSS,(1985),

p.22-28. 16.E.N.Diaconescu and I.M.Ciornei, Traction EHD,TCMMl,Editura Tehnica Bucharest,(l987),p.l58-165,(in

Romanian).

17.K.Okamura,Mechanisrn and Performance of Traction Drives, Japanese Journal of Tribology, vo1.35,no.1,(1990),p.23-33. 18.I.Koizmui and O.Kuroda,Analysis of Damped Vibration of a System with Rolling Friction,Japanese Journal of Tribology,vol.35,no.6, (1990),P.733-739.

19.M.PattersonITraction Drive Contact Optimisation,Proceedings of the 17-th Leeds,Lyon,Symposium in Tribology held at the Institute of Tribology,(l99l),p:295-300. 20.A.IshibashiIS.Hoyashita and H. Takedomi,Evolution of Efficiencies and Speed Ratios of CVT's with Planetary Cones,Proceedings of the 17-th Leeds,Lyon,Symposium on Tribology held at the Institute of TribologyI(l99l),p.277-294.

SESSION XI GENERAL ASPECTS OF FRICTION Chairman:

Professor F Kennedy

Paper XI (i)

Frictional Heating of Elliptic Contacts

Paper XI (ii)

Soil-Structure Interface Friction in Reinforced Soils

Paper XI (iii)

Diagrams for Estimation of the Solidified Film Thicknesses at High Pressure EHD Contacts

This Page Intentionally Left Blank

Dissipative Processes in Tribology I D. Dowson et al. (Editors) 1994 Elsevier Science B.V.

49 1

FRICTIONAL HEATING OF ELLIPTIC CONTACTS J. Bos and H. Moes University of Twente, Dept. of Mechanical Engineering, Tribology Group,

P.O. Box 217, 7500 AE Enschede, The Netherlands Wherever friction occurs mechanical energy is transformed into heat. The maximum surface temperature associated with this heat generation can have an important influence on the tribological behaviour of the mating components. For band contacts and circular contacts this temperature has already been studied extensively. However for elliptic contacts only approximate solutions exist. In this work a fast numerical algorithm is presented to calculate the steady state solution for the flash temperature for elliptic contacts with arbitrary entrainment angle. The heat generation may be due to either a uniform or a semi-ellipsoidal shaped heat source distribution, more or less representing EHL conditions and dry or boundary lubrication conditions, respectively. The asymptotic solutions for large and small PCclet numbers and numerical solutions will be presented. Function fits for the flash temperature will be proposed that are more reliable than the function fits in current use, even for circular contacts. Aspect ratios of the contact ellipse in the range of 0.20 - 5.0 are covered. Within this range the fits were found to be accurate within 5%.

1

Introduction

The publication by Blok (1937a) of a model for the surface temperature rise due to friction stimulated a series of studies about frictional heating and flash temperatures. For simplicity reasons most of the attention was directed to the contact temperature under steady state conditions between two semi-infinite solids, for either a band shaped contact or a circular contact. The work of Jaeger (1943) and of Carslaw and Jaeger (1959) [§10.7] extended Blok’s model to a band shaped contact with limited PCclet numbers. These were all band shaped contacts, though, whereas in practice elliptical contacts are much more common. Therefore Jaeger’s (1943) solution for rectangular contacts has generally been used as an approximation to the elliptic contact problem. For circular contact areas Archard (1959) introduced an approximate solution on the basis of Jaeger’s work together with a simple rule of thumb for the partitioning problem, a solution that meanwhile has proven to be quite useful. By introducing curve fit solutions Kuhlmann-

Wilsdorf (1986, 1987) introduced solutions for elliptic contacts that are quite generally applicable but differ slightly from Archard’s solutions. For a comprehensive review of the literature on flash temperatures see Kennedy (1984). Unfortunately most of these results are restricted to approximate solutions of the average contact temperature. Accurate flash temperature calculations, i.e. of maximum contact temperatures, that apply t o the general elliptic Hertzian contact are still lacking. The authors will try to fill up this omission as a first step in the implementation of a theory in which the tribological contact is part of a thermal network, as proposed by Blok (1989).

It has been a handicap that according to Carslaw and Jaeger’s theory heat exchange through a band shaped contact and a noninsulated stationary solid are incompatible. This incompatibility has been confirmed by calculations conducted by Allen (1962). In order to overcome the problems involved, Cameron, e.a. (1965) even assumed infinitely low temperatures distant from the heat source. This was dropped later (Cameron, 1966).

492

According to De Winter (1967) for a band shaped contact and a stationary solid a logarithmic singularity occurs. Therefore the introduction of an insulated stationary solid is essential when calculating the flash temperature. Fortunately, though, this singularity occurs only for band shaped contacts. Calculation of the heat exchange through an elliptical contact for a noninsulated, stationary solid is feasible. This singularity is the main reason why the solutions for band shaped contacts have not been incorporated in the present work. It would have led to unnecessary complications.

important role. Therefore keep in mind that it has been defined as

In order to present solutions that are generally applicable, function fits will be introduced, fitted to the numerical results obtained with the presented algorithm. The introduction of the function fits is based on the following three principles:

In the literature a uniform heat source has generally been assumed. This seems to be a fair approximation if full film lubrication conditions prevail, i.e. in EHL. For dry contact and boundary lubrication conditions, though, a semielliptic heat source distribution, due to an elastic (Hertzian) contact pressure distribution, seems more t o the point. Therefore in the present work both heat source distributions will be considered.

First, in the function fits a minimumnumber of similarity parameters should be figuring. Therefore the method of optimum similarity analysis introduced by Moes, (1992) has been applied. Second, the function fits should be based on simple algebraic relations only, i.e. additions, substractions, powers, exponential functions and logarithms. This facilitates reproduction and prevents self-deceit with erroneous data or inaccurate asymptotic solutions. Third, all the asymptotic solutions should be taken into account. Therefore they will be the building blocks of these function fits and may be distinguished at the first glance. Unfortunately, there is some confusion about the correct definition of the Pdclet number, i.e. the ratio between the bulk heat transfer and the conductive heat transfer. In the ensuing paper the characteristic length figuring in the PCclet number is the length of the heat source. However, this is four times the dimensionless group that has been applied by Carslaw and Jaeger (1959), Archard (1959) and Archard and Rowntree (1988) and eight times the dimensionless group that has been applied by Blok (1937b). Anyhow, in the derivations to follow, the Pkclet number of the solid in contact plays an

2aU K

where tc is the diffusivity and U is the velocity relative to the heat source. Whereas the characteristic source length a is half the maximum length of the heat source in the direction of the velocity.

Nomenclature a,, b, a b C

F Ir' P

Q U

2

Y

a

e 29

6J IC

P

40

4

- semi-axes of the heat source ellipse, . L velocity related heat source length, L velocity related heat source width, L - specific heat, L 2 / T 2 0 - rate of heat supply, FL/T - conductivity, F/TO - (2aU/n) Pdclet number - heat supplied per unit area, F/TL - velocity, L/T - coordinate in the direction of the velocity, L - coordinate perpendicular to the velocity, L - entrainment angle relative to a-axis - flash temperature number - temperature, 0 - flash temperature, 0 - ( K / p c ) diffusivity, L2/T - density, FT2/L4 - (b,/a,) aspect ratio of the heat source - (b/a) velocity related aspect ratio -

493

Asymptotic solutions

2 2.1

Y

Large P6clet numbers

An asymptotic solution for large PCclet numbers that applies to a band shaped heat source with a uniform distribution has been derived by Jaeger (1943). The authors will extend this solution to elliptic contacts with either a uniform or a semiellipsoidal heat source; the latter in correspondence with Hertz’ (1881) contact theory and a uniform coefficient of friction. For a uniform heat source distribution over an elliptic area the heat generation per unit area is given by

1F

Q ( ~ , Y= ) -ir ab

(1. - ~ Y I
5 O , p > 0.15) the flaky wear particles which might be expected from the cracks seen in the previous micrographs were observed. A typical sample is shown in figure 8. Rather surprisingly though, a completely hfferent shape of particle was observed at lower attack angles. An example is shown in figure 9. The particles are blocky and resemble irregular pyramids. This suggests that there must be a change in the stress field responsible for producing the wear debris at low attack angles.

3.2 Friction and wear The friction and wear results from these tests are presented in figure 10. Wear results from the lathe test are presented in non-dimensional form as Archard's wear coefficient defined in (2). The friction results are consistent with the predictions of the slip-line field of figure 1. For the lathe tests, which were lubricated, the ratio of interfacial shear strength to the shear strength of the aluminium is in the range 0.25-0.3, whereas for the unlubricated Bowden-Leben tests it is within the range 0.6-0.7. It would be possible to use these results as a test of some of the theories of friction and wear mentioned in the introduction. However, this is not

Figure 9. Wear particles from low angle test. a=2.5', p=0.13

our purpose here. It is rather to explain the way in which cracks propagate to form wear particles. Furthermore, to test properly these theories, a more comprehensive set of results would be needed which must await the completion of further experiments.

526

( 5

0.7 I

3,

I

11 4

0' 0

5

10

' 0 15

Attack angle

Figure 10. Friction (jt) and wear (k) coefficients. l:p, Bowden-Leben. 2: p, Lathe. 3: k, Lathe.

4. DISCUSSION.

Using the slip-line field of figure 1 alone, it is difficult to explain completely the crack directions and the form of the wear particles described in the previous section. As we have pointed out elsewhere (14), the most favourable place for cracks to propagate within the field is at the front of the wave, along AB, where the hydrostatic pressure is at a minimum. However, if the crack were to follow this slip-line, we would expect large, blocky particles to be produced in every case. Moreover, their thickness and length would be similar to those of the wave, whereas they are actually shorter and, in most cases, much thinner than this. Also, it would be very difficult for cracks to propagate within the region ECD where the hydrostatic pressure may be up to four times the shear yield strength. It is possible that easier crack propagation could take place immediately behind the wave, in region of contained plasticity or in the elastically stressed region. An approximate method of calculating the stresses here using a stress function has recently been developed (15) from an earlier indentation model of Olver and Johnson (13). A brief description is given in the appendix.

b) Q = 8O, p = 0.215, f = 0.3. Figure 11. Principal stresses behind wave. Scale of distance shown at top right. Scale of stress at bottom right.

This method was used to generate the diagrams shown in figure 11. The stress calculations have been restricted to a narrow region, 0 . 2 behind ~ the tip of the wedge and O.la beneath it, where 2a is the length of the wave (AD). Previous work (15) indicates that the magnitude of the stresses falls quite rapidly at greater distances from the wedge tip. The stresses at greater depths are probably not significant since it has already been shown that the cracks leading to wear occur within 0 . 1 ~of the surface. Restricting the stress calculation to this narrow region has the added advantage that it is here where it should be the most accurate, its basic assumption of the elastic region acting as a wedge loaded along one face with a

527

Figure 12. Hydrostatic pressure and shear stress at a depthofO.la. cx=2.5',p=0.109,f=0.3. 60

rIydrostatic pressurc

.

40

I

20 0

T i b 5 0

6

-20

Shear stress

-40

-60 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

xla

Figure 14. Hydrostatic pressure and shear stress at a depthofO.1u. a = 10',p=00.254,f=0.3. shear stress and a hydrostatic pressure being close to reality. The two loading conditions in figure 1 1 have been chosen to illustrate how the stresses change as the friction rises from a lower level, where blocky wear particles are produced (1 la) to a slightly higher value where flaky particles result ( 1 lb). Arrows point outwards for tensile stress and inwards for compressive stress. It can be seen at once from this figure that the stresses are almost everywhere compressive at p = 0.144, whereas at

Figure 13. Hydrostatic pressure and shear stress at a depth of O.la. a = 5 O , p = 0.144, f = 0.25. p = 0.215 there are significant tensile stresses close to the surface. Below the surface, the most compressive principal stress is more steeply inclined to the horizontal when the hction is higher. The magnitude and direction of the shear stresses which probably control most of the crack propagation are less obvious from this figure. It can be better seen in figures 12-14, which show plots of hydrostatic pressure, and shear stress magnitude and inclination against position at a depth of 0. la. Choosing a smaller depth would give the same form of plot, since the stresses in the elastic zone (HDI) are a function only of 8; but the stresses and inclination would change more rapidly with x. The origin of these plots is beneath the point D in fig. I . The positive direction of x is towards the right, i.e., behind the contact, and the range over which they have been made is indicated in fig. 1 by the mowed line. The inclination of the shear stress is taken as negative when it is against the direction of motion of the wedge. Thus the slip-line AB has a negative inclination whilst the slip-line CD has a positive inclination. Ease of shear crack propagation at any particular point can be judged relative to the conditions along AB (fig. l), where the hydrostatic pressure and the shear stress are both equal to K. A lower pressure or a higher shear stress will give easier crack propagation. Thus when a = 2.5' and

528

0.8

1.5

0.7

0.6

1

0.5

e

$

B

3.-

(c.

0

0.5

0.4

b

v)

0.3

0.2

0

3

‘E k u8

0.1

0

-0.5

5

0

15

10

20

25

Attack angle (degrees)

Figure 15. Maximum surface stress (upper curve) and coefficient of friction (lower curve) vs. a. when a = 5’, the hydrostatic pressure at the origin is around 1.6K, so crack propagation at this point will be more difficult than at the front of the wave even though the shear stress is the same. Further back from the origin, the hydrostatic pressure falls faster than the shear stress, and they both become equal to 0.8K at xla 0.05. In both cases, the inclination of the shear stress to the surface is positive here (10’ - 15”).Shear cracks will therefore propagate more easily at the front of the wave, against the motion of the wedge (negative inclination). An example is the right-hand crack in fig. 5 . At the rear, they may also propagate, but more slowly and with positive inclination. At no point will propagation parallel to the surface be favourable. The result of this would be to generate pairs of cracks like those shown in fig. 5 . If they join, a blocky wear fragment will leave the surface, but in the test of fig. 5, flakes form more quickly. At higher attack angles the hydrostatic pressure falls more rapidly behind the deformation region. Thus at a = 10’ both shear stress and hydrostatic pressure are equal to -K at the origin. However the inclination of the shear stress is now slightly negative (--8”), becoming parallel to the surface at x/a -0.1. Conditions will now be more favourable for crack propagation nearly parallel to the surface, which should favour the formation of flaky wear particles.

-

-

Another factor which may influence crack propagation paths and the shape of wear particles is the size of the surface stress. The way this changes with attack angle and friction coefficient is shown in figure 15. Below 01 = 4’ (p = 0.14) it is compressive, so there is no possibility of tensile cracks forming perpendicular to the surface to finish the process of detaching a flake. At higher values of OL and p the surface stress becomes steadily more tensile, whilst the depth of the shear zone becomes steadily smaller, increasing the tendency for flakes to form. Using these arguments, it is possible to explain the shapes of the wear particles observed and why there should be a change in particle morphology at low attack angles. The changeover between the two shapes, blocky and flaky, occurs when the maximum surface stress becomes tensile. This coincides with a reduced hydrostatic pressure behind the contact and a change in the orientation of the shear stress which together favour the formation of flakes. However, the exact inclination of the cracks to the surface is not easy to predict from the stress field. There are perhaps two reasons for this. Firstly, each crack, as it passes through the contact, is subject to a stress field which varies greatly in intensity and inclination. Thus the propagation direction may vary as the position of the crack changes. Secondly, the material passing through the high deformation zone (ABCDE in fig. 1) suffers considerable strain, especially at high friction, and this will be sufficient to rotate cracks out of their original orientations as Kopalinsky has already pointed out (10,12).

5. CONCLUSIONS. 1. The orientation of cracks formed in aluminium surfaces by the repeated passage of a blunt hard wedge has been found to be as described by previous workers. 2. Prolonged sliding of the wedge produces debris which is blocky at low friction and flaky at high fnction. 3. The morphology of the debris, and to some extent, the orientation of the cracks can be explained using stress fields calculated from an approximate elastoplastic analytical model.

529

REFERENCES. 1. R.C.D.Richardson, The wear of metals by hard abrasives, Wear, I0 (1 967) 29 1. 2. J.F.Archard and W.Hirst, The wear of materials under unlubricated conditions, Proc. Roy. SOC. London Ser. A , 236, (1956) 397. 3. N.P.Suh, The delamination theory of wear, Wear, 25 (1973) 11 1. 4. F.P.Bowden and D.Tabor, The Friction and Lubrication of Solids Vol 11, Oxford 1964. 5. J.M.Challen and P.L.B. Oxley, An explanation of the different regimes of friction and wear using asperity deformation models, Wear, 53 (1979) 229-243. 6. J.M.Challen, L.J.McLean and P.L.B.Oxley, Plastic deformation of a metal surface in sliding contact with a hard wedge, Proc. Roy. Soc. London Ser. A , 394 (1984) 161-181. 7. H.Moalic, J.A.Fitzpatrick and A.A.Torrance, The correlation of the characteristics of rough surfaces with their friction coefficients, Proc. I. Mech. E., 201 (1987) 321-329. 8. P.Lacey, A.A.Torrance and J.A.Fitzpatrick, The relation between the friction of lubricated surfaces and apparent normal pressure, Trans ASME, J. Trib., 1I I (1989) 260-264. 9. J.M.Challen, E.M.Kopalinsky and P.L.B.Oxley, An asperity deformation model for relating the coefficients of hction and wear in sliding metal friction, in Tribology - Friction, Lubrication and Wear fifty years on, Vol II. I.Mech E., London (1987) Paper C156/87. 10. B.S.Hockenhul1, E.M. Kopalinsky and P.L.B.Oxley, An investigation of the role of low cycle fatigue in producing surface damage in sliding metallic fnction, Wear 148 (1991) 135-46. 11. P.Lacey and A.A.Torrance, The calculation of wear coefficients for plastic contacts, Wear 145 (1991) 367-383. 12. E.M.Kopalinsky Investigation of surface deformation when a hard wedge slides over a soft surface. Trans ASME, J.Trib, 114 (1992) 100-105. 13. A.V.Olver, H.A.Spikes, A.F.Bower and K.L.Johnson, The residual stress distribution in a plastically deformed model asperity, Wear, 107 (1986) 151-174.

14. Zhou, F., S.Leavy and A.A.Torrance, The plastic deformation influence on the formation of wear particles under sliding conditions. Proc. Irish Advanced Manufacturing Technology Conference, Dublin City University (1993). 15. A.A.Torrance, The calculation of residual stresses in worn surfaces . Submitted to Proc I. Mech E. (Zond.) Ser. J, (1994).

NOMENCLATURE. ABCDEFGHI A’

a

f H J k

K P

PI 4 S

r

Points in stress field (see fig. 1). %W4+&-a-q. Half width of wave (a = %AD). Tresca’s factor (f= cos2~). Coefficient in stress function. Coefficient in stress function. Archard’s wear coefficient. Shear yield strength of Al. Hydrostatic pressure on CD. Hydrostatic pressure on DH. Pressure normal to a surface. Shear stress parallel to a surface. Radial distance from D.

c1

P E

Angles defined in figure 1.

11

e

0

P

Stress function Coefficient of friction.

6, 6 8

‘TI3

Q w

Elastic stress components. w4 + E-Ct-11. Angle between surface and boundary of plastic zone (HDI).

5 30

APPENDIX

Stresses hehind the wave. The starting point for the analysis is shown in figure A l A rigid wedge traverses an elastoplastic material pushing a plastic wave ahead of it, which is modelled by the slip-line field ABCDE. Below ABCD there is a region of contained plasticity, which is bounded near the surface at the rear of the field by the line AH. The passage of the wave leaves behind it a highly strained layer of depth FG below DI, as verified experimentally by Kopalinsky (12). Within the slip-line tield (ABCDE) the stresses may be found from Hencky’s equations as described in reference ( 5 ) . Outside this region, they may be found using the method of reference ( 13) as follows. In the region CDE, there is a hydrostatic pressure, p and a shear stress K where:

“461

,

Therefore, by Hencky’s theorem, there is a hydrostatic pressure p’ along DH where :

The stresses under load can now be calculated using the above equations [A3-A6] in the elastic zone (DHI) and Hencky’s equations in the plastic region, (CDH and ABCDE). Hard Wedge

Elastic/Plastic Boundary

Figure Al . Field used to calculate stresses. The region DHI may be treated as a semi-infinite wedge with a pressure y ‘ and a shear stress -k along DH. It can then be shown (13) that the following stress function satisfies the boundary conditions:

where K H = -2 sin(2yr)

and

.I

-Kco~wco~(~w)

0 is measured clockwise from DH, and r is the radial distance from D. The stresses can then be found from the usual relationships:

Dissipative Processes in Tribology / I). Dowson e l al. (Editors) 1994 Elsevier Science R.V.

531

The Influence of Lubricant Degradation on Friction in the Piston Ring Pack R.I. Taylor and J.C. Bell Shell Research Ltd., Thornton Research Centre, P.O. Box 1, Chester, CH13SH, UK

An improved understanding of lubrication within the piston ring pack is expected to contribute to

efforts aimed at improving the fuel economy of automotive engines. However, most studies of frictional losses within the piston ring-pack use viscometric data for 'fresh' lubricants. T h e changes that occur to the viscometric properties of the lubricant in a working engine are not generally taken into account. The lubricant in an engine is subjected to a harsh physical environment, and the aggressive chemistry t h a t takes place in the vicinity of the top piston ring causes lubricant degradation by both evaporation of the lower boiling point components and by oxidation. Top ring zone oil sampling studies show that this degradation changes the viscometric properties of t h e lubricant. Measured viscometric data have been used in a piston ring pack model to estimate t h e effects of oil degradation on frictional power losses within the ring pack. In a n extreme case, degradation increased these losses by up to 40%.

1. INTRODUCTION

The desire to improve the fuel economy of diesel and gasoline engines h a s focussed attention on frictional energy losses in the piston assembly. As the lubricant in an engine is subjected to a n increasingly harsh physical and chemical environment in the vicinity of the piston rings, due to changing engine design, the influence of oil degradation (either by evaporation of lower boiling point compounds or oxidation), and the resulting changes that occur in the frictional force between ring and cylinder, are of increasing importance. The changes in viscosity and composition of the lubricant in the region of the piston ring pack were monitored by the use of a top ring zone (TRZ) sampling technique'. Oil samples were obtained through an orifice in the top ring groove of a Caterpillar 1Y73 single cylinder diesel engine. A substantial increase in the viscosity of these samples was observed within the first ten hours of engine operation,

although the oil in the crankcase had shown no detectable change'. More recently, TRZ samples were collected at higher power outputs of 10 a n d 14 bar brake mean effective pressure (BMEP)~,using a very high quality SAE 10W/30 multigrade diesel engine lubricant. The viscosities of at these samples were measured temperatures between 40 a n d 150°C, using special viscometric techniques for very small oil samples. These data were input to a piston ring pack oil film thickness model to compare the frictional power losses with those of t h e fresh SAE 10W/30 multigrade oil and a n SAE 30 single grade oil. In this particular study, t h e degradation process resulted in substantially thickened oils. A simplification used here was t h a t t h e degraded lubricant lubricated t h e whole of the piston ring pack. In reality, although the oil in the vicinity of top dead centre would have the same viscometric properties as the TRZ samples, lubricant lower down the cylinder

532

would more closely resemble ‘fresh’ oil. In this sense, the study presented here represents a ‘worst-case’ scenario. Average power losses for the piston ring-pack were calculated for both ‘fresh’ and ‘used’ oils, and increases of up to 40% were found for the ring-pack power loss when ‘degraded’ oil viscometric properties were used in place of ‘fresh’ oil viscometric properties. For the top ring, the power losses were similar for both sets of oil, except that the proportion of hydrodynamic to boundary friction power loss decreased substantially for the ‘degraded’ lubricant. 2. PISTON RING-PACK LUBRICATION MODEL

The piston ring-pack lubrication model used a robust non-linear algebraic equation solver to solve for the minimum film thickness, and inlet and outlet positions on the piston ring for each ring in the ring-pack3. Reynolds’ equation was used, although the (very small) squeeze term was neglected, and a separation boundary condition was used a t the outlet position, whereby the lubricant pressure was constrained to equal the gas pressure on the outlet side of the ring, and the pressure gradient was taken to be zero4. The viscositytemperature variation of the lubricant was taken into account by using Vogel’s equation5.

In the above equation, K , h and 8 are Vogel parameters. T ( x ) is the liner temperature a t position x , assumed to be the same as the (x) is the lubricant temperature, and lubricant’s dynamic viscosity. Bottom dead centre and top dead centre cylinder temperatures were specified, and a linear variation of temperature was assumed inbetween. At any piston position, the lubricant temperature, and hence the viscosity, could be

determined. Changes in the viscometric properties of the lubricant were represented by changing the Vogel parameters. Vogel parameters for various ‘fresh’ and ‘degraded‘ lubricants a r e given in Table 1. Figure 1 shows the dynamic viscosities, of the ‘fresh’ and ‘used’ oils. The BDC temperature was taken to be 100°C, and the TDC temperature was 15OOC. T h e ‘degraded’ SAE 10W/30 oils h a d dynamic viscosities approximately 2-3 times greater than t h a t of the ‘fresh’ oils. The oil sampled at the higher power output was more heavily degraded and suffered a greater viscosity increase. In the model, the variation of viscosity with pressure was neglected, since t h e pressures were not expected to be high enough to cause more than a 10% increase in viscosity. The variation of viscosity with shear rate was also neglected. This was a more severe shortcoming for multigrade oils, since maximum shear rates of 3 x lo7 s-l were computed for the top ring a n d at these shear rates the multigrade lubricant would exhibit shear thinning. The ring pack h a s four piston rings ; a n oil control ring ; two identical tapered rings ; and one barrel shaped top compression ring. Talysurf profilometer traces of both “new” and “worn” piston rings for t h e Caterpillar 1Y73 single cylinder diesel engine were obtained and it was clear t h a t after t h e initial runningin period t h e “worn” ring profiles differed markedly from the “new” profiles. The profiles of t h e “worn” set of rings were used as inputs to t h e model. The oil control ring was assumed fully flooded during the downstroke, but the limited oil film left behind on the liner caused the higher rings to be starved of oil. On the upstroke, t h e rings ran on the oil film left on the liner from the downstroke. For each crank angle in the engine cycle t h e oil film thickness for each of the four rings was calculated. Also, the viscosity for each of t h e four ring positions a t each crank angle was known. T h e frictional

533

Table 1: Viscometric properties of ‘fresh’SAE 30, SAE 1OWl30. and ‘degraded’ SAE 10Wl30 lubricants

Fresh’ SAE3O ‘Fresh’ SAElOWl30

I I

2.275 x 13.64~ loT2

I I

1361.0

123.3

849.8

99.9

‘Degraded’ SAE 10W/30 (10 BMEP)

12.71 x

954.0

96.4

‘Degraded’ SAElOWl30 (14 BMEP)

6.361 x

1395.5

133.6

1000

100

10

1

0

50

100

150

200

Temperature (C) Figure 1 : The temperature dependence of the dynamic viscosity of the four different lubricants

534

force acting between ring and liner could thus be calculated. Multiplying the frictional force by the piston velocity gave the instantaneous power loss, and this was averaged over the engine cycle t o give the average power loss. If the film thickness fell below a certain value, taken t o be 0.25pm, typical of the r.m.s roughness of cylinder liner and piston surfaces, boundary lubrication w a s assumed to occur, and the frictional force was calculated using an appropriate friction coefficient (a value of 0.08 was used).

3. RESULTS The average power losses calculated for the oils described in Table 1are given in F i y r e 2. Viscous power losses were found to be higher when the ‘degraded’ viscometric properties were used, simply because the viscosity was higher. As a consequence, oil film thicknesses were found t o be larger so that boundary

friction for the top ring was reduced. The contribution of boundary friction to the power losses of the other rings was insignificant. The ring-pack average power loss was approximately 16% higher for the 10 bar BMEP ‘degraded’ oil, and approximately 40% higher when it was lubricated with oil having the 14 bar BMEP ‘degraded‘ viscometric lubricant properties compared with an oil having the ‘fresh’ viscometric properties, as shown in F i y r e 3. The results for average power losses mask large variations between the individual rings, as can be seen in Figure 2. For the top ring, in particular, the average power loss is roughly the same with both ‘fresh’ and ‘degraded’ oil properties, since the reduction in boundary lubricated power losses for the ‘degraded’ oils was compensated for by an increase in hydrodynamic power losses.

It must be stressed, however, that the calculation presented here represents a ‘worst-case’ scenario, in which the viscometric

Average power loss (Watts) Figure 2 : Ring pack power losses for each ring in the piston ring-pack for each of the four lubricants considered in this study

535

% cd

a en C

2 Figure 3 :

Lubricant type ring-pack power losses for each of the four oils considered here

properties of oil sampled from the top ring zone were assumed to hold for the whole ring pack. In addition to a gradation in temperature from BDC to TDC, a gradation in Vogel parameters should also be expected, since the lubricant near BDC is relatively ‘fresh’, whereas that near TDC would be expected t o exhibit the viscometric properties of the TRZ samples. Either way, an increase in the total average power loss is still expected, and since friction in the piston ring area is thought to contribute approximately 20-30O/o6 of total engine friction, the error incurred in using ‘fresh’ oil viscometric properties a s opposed to ‘degraded‘ oil viscometric properties is likely to be significant. In order to get information on the appropriate gradation in Vogel parameters, samples of oil from the lower rings need to be collected, and such a n exercise is now beginning. The results of these calculations show that short term lubricant degradation can significantly affect power losses in the piston ring assembly, and thereby overall engine fuel economy. I t follows t h a t to enable accurate estimates of engine power losses the viscometric properties of representative degraded oils must be used.

ACKNOWLEDGEMENTS The authors would like to thank the following colleagues at Thornton Reseach Centre, B. Bull, Dr. P.J. Burnett, Dr. R.C. Coy, D.J. Evans, Dr. L.E. Scales a n d R.J. Wetton together with M. Priest from Leeds University. The authors would also like to thank Shell Research Ltd. for permission to publish this work.

REFERENCES 1. M. Thompson, S.B. Saville, “The Use of Top Ring Zone Sampling and Analysis to Investigate Oil Consumption Mechanisms”, I. Mech. E. Conference on “Automotive Power Systems - Environment and Conservation”, Chester College, 10-12 Sept 1990 2. P.J. Burnett, B. Bull, R.J. Wetton, “Characterisation of the Ring Pack Lubricant and its Environment”, I. Mech. E. Conference on “Experimental and Predictive Methods in Engine Research and Environment”, Birmingham University, 17-18 Nov 1993

536

3. Y-R.Jeng, “Theoretical analysis of pistonring lubrication part I1 - starved lubrication and its application t o a complete ring pack”, Trib. Trans., 35, (19921, 707-714

4. D.E. Richardson and G.L. Barman, “Theoretical and experimental investigations of oil films for application to piston ring lubrication”, SAE92234 1

5. A. Cameron, “Basic lubrication theory”, Third Edition (published by Ellis Horwood Ltd., 1983)

6. P.K. Goenka, R.S. Paranjpe and Y-R. Jeng, “FLARE : An integrated software package f i r friction and lubrication analysis for automotive engines - part I : overview and applications”, SAE920487

Dissipative Proccsscs in 'I'ribology / D. Dowson ct al. (Editors) 1994 Elscvicr Science 13.V.

537

HIGH SPEED DAMAGE UNDER TRANSIENT CONDITIONS 0.LESQUOIS~.J.J. SERRA~,P. KAPSAC. s. SEFU~OR~

aDGA/ETCA/CREA/PS40,Arcueil, France bGA/ETCA/CREA/CEOFont Romeu, France Qcole Centrale de Lyon, LTDS. URA CNRS 855 Ecully. France ABSTRACT:

The ETCA high speed tribometer is a facility able to achieve tests involving sliding speeds up to 350 m/s and loads up to 5000 N. The moving part is a 360 mm diameter stainless steel cylinder and the slider is a 5 mm side parallelepiped pin. The test duration does not exceed a few seconds. Several measurements have been carried out during the tests: - the evolution of the load and the friction forces by a piezoelectric triaxial sensor, - the evolution of the pin length loss by a capacitive sensor and of the pin weight loss during the test. - the evolution of the temperatures at different depths in the pin, using fast response intrinsic thermocouples. From this data, we can compute: the evolution of the friction coefficient, of the linear wear speed and of some thermal results. In addition. a high speed video camera (1000 picturedsecond) has been used to point out the different phases of the wear process. During the tests. competitive process between two types of physical phenomena appears: mechanical ones coming from the friction itself and thermal ones coming from the energy generated by the friction. When the pressure*velocity product (homogeneous to a heat flow) is low, mechanical phenomena are governing the processes. Thermal effects become more and more important when the PV product increases. and for very high values the pin frictional surface is partially molten. In such conditions, the incoming heat flux and surface temperature are important parameters for the understandng of pin wear process. These two parameters can be calculated from thermocouple measurements, using a space marching inverse conduction algorithm taking into account the free boundary. The obtained results show heat flux evolution curves versus time involving instability phases. These instabilities are due to metal particles transferred from the pin to the cylinder, which form rough asperities pushmg away the pin for a certain time. This aspect has been confirmed by the high speed film recorded during the tests. NOMENCLATURE -WR : wear rate, -f : friction coefficient, -V : slidingspeed. -P : contact pressure. -N : normal loa4 -Z : melted zone limit. -H heatflux -p : volumetric mass, -L : latent heat of fusion.

-T, : melting temperature, -To : room temperature, -T : temperature. -x : space, -t : time, -F : non Qmentional number, Fourier modulus type?

-C :speclfi~heat, -K q -Q,

: thermal conductivity, : interface heat flux, : heat flux used by melting.

538

1. LNTRODUCTION

High speed friction corresponds to situations where superficial melting of at least one of the two slidmg materials happens in the contact. This phenomenon generally occurs for high contact pressures (a few MPa or more) and for high slidmg speeds (several 10 m/s and more). High speed friction is. for example. observed for brakmg systems. rotating electrical contacts and friction between a projectile and a gun tube. The first important works on hgh speed friction are duc to Montgomery ( I ] during the fifties (and published in 1975). These stubcs have been made for the US Army aiming to reduce the wear of weapon's tubes. Several conclusions have been pointed out. the amount of heat which is received by a projectile dunng a shot is due to three phenomena: the friction, the deformation of the belt and the creation of heat by the propellant. Then the fomiation of a liquid metal film at the surface of the belt is observed earlier in real cases than in laboratory tests where we haven't the propellant effect. high friction coefficients and fluctuations have been recorded for low value of the PV product (pressure * speed). For values hgher than 6000 Mpa.m/s. the friction coefficient is low. stable and decreases slowly when the PV product increases. the wear rate is related to the speed of heat production. and then to the friction coefficient, the slidlng speed and the contact pressure:

where k is a constant. The product fPV is used because it is homogencous to a power per unit surface equal to a heat flux. It corresponds to the heat flux generated at the interface. the evolution of wear rate (weight loss divided by thc distance of sliding) versus the inverse of the melting temperature of sliding materials, on semi-log coordmates. is a straight line nith a positive slop indicating that high melting temperature materials are more resistant to wear. the results of these stuQes have shown that the w a r mechanism for high speed fnction is

superficial melting followed by an eliniination of a part of the melted layer. The surfaces are not in dlrect contact but are separated by a liquid layer. Then the wear speed is only a function of the melting temperature. The other material parameters. such as the thermal conductivity, the materials compatibility. the crystalline structure, the hardness. ... are of secondary importance or even negligble. However, if the slidmg duration is not sufficient to produce a total melting. these factors can become important. There is also important plastic defornmtions of the pin. Sternlicht and Apkarian 121 have modified the Montgomery friction machine to make experiments with a contact subjected to an electrical current. They have then observed a decrease of the friction force due to an increase of the energy dissipated in the contact by Joule's effect. Williams and G a e n 131 pointed out that the increase of the material temperature produces a decrease in the shearing force of the interface. The friction coefficient is then reduced. The friction coefficient is related to the sliding speed and the normal load (k is a constant):

High speed fnction situations are characterised by melting phenomena due to the energ?. dssipation in the slidmg interface. To have a better understanding of these situations. some authors have studed the propagation speed of the melted zone limit in the pin. One can notice in particular the works of Landau [7] used by Serra [S] who exhibited the important thermomechanical parameters. The dlsplacement speed of the melted zone limit can be evpressed by: (3)

The aim of the present work is Lo understand and modelize the phenomena governing the friction between two metals. at high speed and in a transient regme. The thermal exchanges in the contact have been considered in order to determine the thermal distribution coefficient

539

between the hvo slidlng W e s . This study have used a high speed friction machine developed at ETCA.

load is applied with an arm and dead load. The test duration is controlled by two pneumatic jacks which move the pin. T h s duration ranges from 0.2 to several minutes. Most of the tests were done for a rotational speed of 1500 rpm and a 1 s contact time.

2. EXPERIMENTATION 2.1. Friction machine

The whole pin support can be moved horizontally to change the wear track in the cylinder. Around 30 various tests can be perfornied on the same cylinder.

The high speed pin on cylinder tribometer is presented figure 1. The sliding speed ranges from 5 to 350 d s . correspondmg to a rotation speed up to 18 000 rpm. An orignality of this device is the use of active magnetic bearings (S2M) to support the cylinder which have a mass of around 3000 N. The axle and the cylinder are rotating around their inertia axis reducing $he vibrations. The stainless steel (Z6CN17 4 1 ) cylinder is of 360 nun diameter and was used for all the tests. The axle is horizontal and the pin, with a 5 x 5 mm square section and a 22 nun length. is underneath the cylinder. The cylinder is polished with abrasive papers to a 0.15 Ra value.

2.2. Measurements A triaxle piezo-electric transducer is used to measure the normal load and the friction force. A capacitive transducer is used to follow the evolution of the pin position during the test. indcating then the pin length worn. A tachometer gives the rotating speed of the cylinder.

The pins are weighed before and after the tests. The mass loss can be then compared to the indication of the position transducer. The data are collected during the test by a PC computer, then stored and processed after the tests.

For the pin. the tested materials yere iron, 100C6' (AISI 52100) steel. 35NCD16 , copper, aluminium. zinc, chromium, tungsten and lead and have been chosen for their known thermomechanical properties. All materials. except steel, are pure. For this paper, only the cases of pure iron and steels are considered.

CYLINDER

CYLINDER \MAGNETIC BEARINGS

n

I

I

L p

Figure 1: the friction machine.

For the set of test whch is studled here, the range of the normal load was 10 to 500 N. This

* AFNOR standard

I

Figure 2: Position of thermocouples on the pin. From some authors [1.2]. the pin surface is melted when the PV product (contact pressure x speed) is high. Then it has been interesting to perform thermal measurements of the surface temperature and the heat flux entering in the pin during the tests to validate this assumption. It seems diffkult to measure directly these values during the test. but it is possible to use thermocouples on the surface of the side of the pin at three different depths. As the pin is in iron or

540

steel. we can use an intrinsic thermocouple method using an iron wirc as a common reference and three aluniel wires which were welded by capacitive discharges at 1.5, 2.5, 3.5 mm from the contact surface (Figure 2). The main advantage of this intrinsic method, which is very important for such short tests. is to have a quicker response than with the classical two wires thermocouple method. 3. RESULTS

same wear, but the iron wear is much more important. At the beginning of the test, there is a competition between the wear and the linear expansion caused by the increase of the pin temperature. This competition causes low apparent wear which could be either positive or negative depending of the dominating phenomenon. If the contact time is long enough. a phase of rapid wear of a few mm/s occurs when the temperature of the surface of the pin is high enough.

3.1. Tribological results

Iron and steel pins create on the cylinder an irregular layer of transferred material composed of important rough deposits which can go up to a few tenth of millimetre high. The cylinder does not appear to be worn. However, an Electron Probe Mmoanalysis (EPMA) of the pins reveals sometimes some chronuum or nickel traces coming from the cylinder in the bulk. Figure 3 shows for three tests with same parameters (normal load 150 N. sliQng speed 28 d s . contact time 1 s) the evolution of the normal load and of the friction force. The n o d load decreases during the test when the linear wear is important because the mechanical system need to follow the wear. and its response is not instantaneous. The swingng that appear during the first 1/10 s come from the shock at the beginning of the test.

Figure 3: evolution during the test of normal load and friction force for iron and both steels. The curves in figure 4 are for the same tests as thc ones in figure 3. Both steels have about the

Figure 4:evolution during the lest of the linear wear. 3.1.1. Computation of tribological results

From the results of the measurements. we can compute the evolution of the friction force and the linear wear speed. Figure 5 shows these results for the same three tests.

Figure 5: evolution during the test of the friction coefficient and of the linear wear speed.

For iron. a friction coefficient of about 0.3 is

54 1

measured at the begmning of the test. Then. it increases up to a maximum and decrease to finish in the range of the initial values. The initial growth comes from the bad tribological behaviour of iron when the temperature increases. However when the temperature is hgh enough, iron in plastic flowing offers a lower resistance, and the contact is partially lubricated by melting iron. So the coefficient of friction decreases. The friction coefficients of both steels have the same behaviour of the iron ones. but the value obtained are lower. and may sometimes be less than 0.1 at the end of the test.

">,

1

I"

3l"

MEI\NNORMALLOADl NL

IRON

3.1.2. Comparaison criteria In order to compare the different tests. some criteria have been chosen to characterise friction and wear.

1OOC6

JSNC1)L

Figure 6: mean friction coefficient as a function of the mean normal load.

The three criteria which have been chosen for the friction coefficient were: - the initial fnction coefficient. - the mean value of the friction coefficient during the whole test, - the final fnction coefficient. For the wear, the criteria were: -the linear wear. - the bulk wear. - the linear wear speed at 0.5 nun and at 1 mm of wear. - the time to obtain 0.1 mm of linear wear which characterises the time to obtain the h g h wear phase.

"

.

IOU

.

li 0

20"

MEkrlNORMALLOADIM

RON

1OOC6

a JSNCDL

Figure 7: linear wear as a function of normal load.

c

I

I

1

All the three fnction coefficient criteria are proportional with the mean normal load; there is on some curves an evolution which may be either positive or negative. and some time no evolution. In figure 6. one can see that the mean value of the fnction coefficient as a function of the mean normal load decrease for 100C6 steel and is approximately constant for iron and 35NCD16.

The linear wear. the bulk wear and the linear wear speed increased proportionally with the normal load as it can be seen on figure 7 and 8. However. if the load is not hgh enough. there is a bend on the curve because the lines don't pass by 0. or with a load of 0 there must be a wear of 0.

Figure 8: linear wear speed as a function of normal load.

As it can be seen on figure 9. the time to obtain 0.1 mm of wear decreases exponentially with the mean normal load. One can observe that the beginning of the high wear began more

542

qwckly with iron at low load but at high load it is with the steels that the high wear began first. The explication is in a fnction coefficient effect, because if we plot the time to obtain 0.1 mm of wear against the mean friction force in spite of the mean normal load, both the steels began their hgh speed phase before iron for all the loads. We can assume that since the steel has a greater hardness than iron. the abrasion is more important in steel than in iron at the beginning of the test. II 8

,

I

Figure 10: contact between the cylinder and the pin.

Figure 9: time to obtain 0.1 mm of wear as a function of normal load. 3.2. Visualisation of the contact A high speed video system (60 to 1000 pictureds) was used to observe the behaviour of the pin during the test. There is a tranfer of the material of the pin on the cylinder and this transferred material temperature is quite low. because it has time to cool between two contact periods. Each time that this transfer pass in the contact, it plough the pin whch skip. Thus. we observe on the video film that from time to time. the pin is skipping, so the contact is periodically broken. With the increase of the temperature, the surface of the pin has a low hardness. and the transferred material produces sparks in relatively large area which are the real contact area. An hydrodynamic model has been developed for these important flash temperature areas. T h s model will be the subject of an other paper. The video film shows also the formation of the bulk. the occurrence of sparks and, at the back of the contact, the ejection of clouds of very small pamcles whch are droplets of molten metal or oxide dust.

Figure 11: break of the contact. Figure 10 and 11 are pictures coming from the video recording. The pin is at the bottom of the picture and the pad can be easily seen. Thc cylinder is at the top of the picture. and one can see the reflection of the pin in it. In t h s picture we have the contact at the left of the pin by a rough asperities stuck on the cylinder. This contact corresponds to the 1 cm length zone of the interface where there is a supplement of light. The figure 11 comes from the same test. but a few thousandth of second after. and the contact is broken because of the push caused by the passage of the tranferred material of figure 10. 3.3. Thermal results 3.3.1. Thermocouples results

Three temperatures are relavent to these

543

experiments: ambient temperature. global temperature of the contact surface and flash temperature whch are ephemeral and localised around the real contact zones, usually very small in dimension. The flash temperature can reach very high values even in low sliding speed ( [ 5 , 111). From the thermocouple data, we can compute to obtain thc temperature at the surface and the flux of heat entering into the pin with the method described below. Figure 12 a and b show the thermocouples results for two tests. one in iron and one in 35NCD16 steel respectively. In both cases, the linear wear is about 1 mm. The temperature increases more quickly on the iron pin than that on the steel, but it reaches a lower maximum temperature as compared to steel.

flux coming into the pin can be obtained in solving a one-dimensional inverse heat conduction problem with a moving boundary. This problem, called Stefan's problem. is not analytically solvable, and numerous authors have proposed numerical solutions. However, a method recently developedby Raynaud and Bransier can be used to take this conditions into account [9, lo]. So a code adapted to the high speed tribometer using this method has been developed. 3.3.2. Principle of thermograms interpretation Let us consider a one-dimensional heat conduction in the pin. initially at an uniform temperature To. At time t = 0. the contact surface of the sample, noted x = 0 in our problem, is subject to a frictional thermal flux. The objective is to estimate the interface heat flux q(0,t) at x = 0 (node I), given the inner temperatures at x = xj and x = Xk. The problem is solved using the space-marching finiteditTerence algorithm developed by Raynaud & Bransier [9, lo].

n+2 n+l n

Figure 12 a: evolution of wear and the 3 thermocouples measurementson an iron test (23 m / s , 75 N, 1 s).

n-1 n-2

1

2

3

i-1 abscissa

101

i

i+l

.

100

Figure 13 : principle of the space-marching finitedifference algorithm. Figure 12 b: evolution of wear and the 3 thermocouples measurementson a steel test (35NCD16.28 m / s . 230 N, 1 s). The temperature of contact and the thermal

The temperature field in the direct region x. < J: x I Xk can be calculated with a pure implicit scheme because the boundary conditions of the first kind in xj and Xk are known. Similarly, in the

544

inverse regon (0 I x 2 x.), J we note Tin the temperature at time nAt and at depth iAx. In this regon, the authors demonstrate that the temperature Ti-1". at point (i-1)Ax and time nAt, can be calculated as a function of the temperatures Tin. Tin-1 and T.+ln-l (past temperatures), Tin+1 and T ~ + I " +(future ~ temperatures). The relation obtained is:

last half (Ax1/2 width) control volume using an explicit formulation.

n+2 n+l n

n-I n-2

Where the quantities F1, F2 and F3 (non drmensional numbers. Fourier modulus type) are given by : 1+1

j-1

i-I

I

1+1

abscissa

Where C is the volumetric specific heat and k the thermal conductivity. The interface heat flux qln is calculated from an energv balance on the half (Ad2 width) control volume on the right of the node 1. q.AY q; =-[(2.F,4.At

l ) . z y +(2.F, +l)&-+l -2.4. - 2. F;.? + I ]

(6)

When the pin abrades (by melting, wear or any other way) the contact surface moves about the temperature sensors. The surface displacement is continuously measured using a capacitive sensor, so its abscissa s(t) is known at any time. The iterative calculation - equation (4) - giving the space forward temperature at a given time is repeated until the next node is over the ablation front location. Then, the calculation cell size is reduced in order to place the forward temperature on the contact surface and the backward temperatures symmetrically compared with the last calculated position (fig.14). These temperatures are interpolated between the known valucs Tinf1 and Ti+lM1, and the Ti,ln temperature is given by the relation (1) in with Ax is replaced by Ax' = iAx - s(t). The interface heat flux is calculated from an energy balance on the

Figure 14: arrival at the surface when using the space-marchingfiniteaerence algorithm. If the pin is worn by melting. the frictional heat flux involves an additional part which is used by the melting front displacement : As At

Q, = p.L.-

(7)

3.3.3. Thermal calculation results

In the present case, after the calculations (At = 0.004 s, Ax = 0.001 mm) we can see that the

global temperature of the surface reaches a maximum value of 120OOC (fig.15 a and b). The three curves of the figures correspond to the calculated values of the surface temperature using respectwely thermocouples 1 and 2, 1 and 3 and 2 and 3 as boundary conditions. TI-2 and Ti-3 results are in good agreement, but T2-3 is, mainly in the beginning of the curve, different. A numerical explanation is that the calculation by the inverse method begins at the thermocouple 1 in the case 1-2 and 1-3 and at thermocouple 2 in the case 2-3, and the distance from the surface to the thermocouple 2 is too large for the space marching algorithm. What is more, in the case 23, more errors appears mainly at the beginning of

545

the test where the thermal gradient is the highest and where the thermocouple 2 is farthest from the contact. We have a first phase corresponding to the low wear phase where the temperature increased very quickly. Then began the high wear phase where a part of the heat is used to melt the material which is going to leave the contact. In iron, the surface temperature increased more quickly than in steel, but iron has a temperature decrease during the high wear phase. Th~scan be explained by a vibrational problem on the tribometer. We saw with the hgh speed video camera that there is an adhesive transfer from the pin to the cylinder, and that the transfers are plouglung the pin at each rotation of the cylinder which cause the pin to skip. This phenomenon causes the real contact time to be much lower than the apparent one. The reduction in the real contact time leads to a reduction in the heat flux coming into the pin. This reduction in heat flux can cause the mean surface temperature to decrease.

The heat flux evolution curves versus time are presented in figure 16 a and b. In this curve, we observe instabilities in the form of abrupt falls. the biggest one which reaches zero a little bit before 0.9 s on the iron curve. This phenomenon can be explained by the break of the contact observed on the video recording. The important breaks of the curves comes from the great ruptures of the contact, but the video shows that there is also a lot of little breaks of a few thousandth of second. The existence of these little breaks can explain the little peaks in the incoming flux curves. They do not go down to zero because the acquisition frequency is actually only 250 Hertz, so it is difficult to see completely such fast phenomena. I

00

1

I,

0.

06

08

I 0

TQlD(S1

Figure 16 a: heat flux and heat dissipation coefficient on iron (23 m/s; 75 N, 1 s).

Figure 15 a: calculated surface temperatures on iron (23 m/s; 75 N, 1 s). 1500

1

1 I

"

00

01

6

o*

I 0

Tha (5)

OI

Figure 16 b: heat flux and heat dissipation coefficient on 35NCD16 (28 ds, 230 N, 1 s). The curve of the heat dissipation coefficient, which is calculated as the ratio of the heat flux coming into the pin versus the heat flux produced by the friction, follows approximately the variations of the heat flux. The values obtained are between 0.1 and 0.5.

Figure 15 b: calculated surface temperatures on steel (35NCD16, 28 m/s, 230 N, 1 s).

546

3.3.4. Metallographic observations

Figure 17: Metallographic cut, slidng direction from the left to the right (Magnification *20).

Figure 18: Metallographic cut, perpendicular to sliding direction from in front of to behmd (Magnification *20)

Some of the used pins have been cut, polished and observed with a microscope after etclung. Figure 17 and 18 show the pin of a steel test (100C6, 350 N, 28 m/s, 1 s). The cut was made in the direction of the sliding in figure 17. and in the other drection in figure 18. The contact surface is at the top of the figures. The contact surface is very irregular in figure 18 because of the rough transfers on the cylinder which plough the pin. One can see on the pin an important bulk coming from the plastic deformation at the end of

the contact, smaller ones on the sides, and a much smaller one at the beginning of the contact. There is also an important martensitic transformation area (in white) which take the all back bulk. But the most interesting comes from the fact that we can see the fibres of the pin. and these fibres are convolving on the bulks when they are near the end of the contact, but stop at the contact surface when they are too far away from the end of the contact. We can deduce from this observations that during the beginning of the test, the temperature of a part of the pin which is in the contact increases. When the temperature is lugh

541

enough. the material began to deform plastically and move to the end of the contact, and during this movement. the temperature continues to increase. If the distance to the end of the contact is small enough, the material is out of the contact before its temperature reaches the melting point. and the material form a bulk. If the distance to the end of the contact is large, the material is melted in the contact; it is then ejected in the form of droplet of molten material or in oxide dust. 4. CONCLUSIONS

A test machine was used to characterise tribological phenomena in high load and high speed conditions. Three thermocouples give thermal measurements in addition to tribological ones. The evolution of the normal load the friction force and the linear wear are computer recorded during the test. From this data, the evolution of the friction coefficient and of the linear wear speed are computed. Some criteria have been chosen to compare the various tests. The thermal measurements are treated by a code using the Raynaud and Bransier's space marching inverse conduction algorithm, which takes into account the free boundary. The results are accurate enough to show the skipping of the pin which is confirmed by the observation made on a high speed video recording. More tests with Merent materials and a better speed of acquisition are expected in the future.

REFERENCES 1 R.S.MONTGOMERY; Friction and wear at high sliding speeds; Wear, vol. 36, n03 pp 275278, Mars 1976. 2 B.STERNLICHT et HAPWAN: Investigation of melt lubrication; A.S.L.E. Trans. 2,248 (1956). 3 K.WLLLIAMS et E.GIFFEN; Friction between unlubricated steel surface at sliding speeds up to 750 feet per second; Roc. Inst. Mech. Engr 178,24 (1963/4). 4 F.P.BOWDEN, F.R.S. et E.H.FREITAG; The friction of solids at very high speed 1. metal on metal; 2. metal on diamond; Proc. Roy. Soc., Ser. A 248, 350 (1958). 5 S.C.LIM et M.F.ASHl3Y; Wear-mechsm maps; Acta metall., vol. 35. nO1, ppl-24 (1987). 6 H.S.CARSLAW & J.C.JAEGER; Conduction of heat in solids; 2nd ed.oxford: Clarendon Press (1959). 7 H.G. LANDAU; Heat conduction in a melting solid; Quart. Appl. Math. 8(1): 81-94 (1951). 8 J.J. SERRA et P RIBERTY; Analyse thermique de l'usure par ablation de metal fondu en frottement A grande vitesse.: Rapport ETCA NO89 R 139. 9 M.RAYNAUD et J.BRANSIER; A new finitedifference method for the nonlinear inverse heat conduction problem ? Numerical heat transfert, vol9, NO1,p27 a 42, 1986. 10 M.RAYNAUD et J.BRANSIER; Evaluation au moyen dune mdthode inverse. du partage du flux gentirk par frottement entre deux solides en mowement relatif; Soci6te Franpise des thermiciens,journke d'etude du 8 mars 1989. 11 T.F.J. QUINN and W.O. WINER; Wear 102, 67 (1985).

This Page Intentionally Left Blank

Dissipative Processes in Tribology / I). Dow'son et al. (Editors) 0 1994 Elsevicr Science R.V. AU rights reserved.

549

INCIPIENT SLIDING ANALYSIS BETWEEN TWO CONTACTING BODIES. CRITICAL ANALYSIS OF FRICTION LAW.

T. ZEGHLOUL", M.C.DUBOURG"", B. V I L L E C M E "

* UniversitC de Poitiers, Laboratoire de Mtcanique des Solides, URA CNRS 861, Av. Du Recteur Pineau, 86022 Poitiers Cedex, France. ** INSA, Laboratoire de Mecanique des Contacts CNRS URA 856, B3t. 113, 20 Av. A. Einstein, 69621 Villeurbanne CCdex. France. Two non-conforming bodies are brought into contact. They touch over a plane surface of extent size compared with the dimensions of the solids. Then they are subjected to a tangential force until gross sliding is observed at the contact interface. Under certain circumstances, this phenomemom is preceeded by local slidings over part of the contact area, named as "sliding waves". They are similar to the Schallamach waves. Relations between these instabilities and the stick limit and the friction conditions on one hand and with the body compliance on the other hand must be investigated as they generate vibratory phenomena like for instance judder. Progri et al. [l] have identified the experimental conditions for reproducing these phenomena. A two-body contact is established between a rectangular polyurethane slab (L = 80mm, 1 = lOmm, h = 40mm, Young modulus E = 7 MPa, Poissonk ratio v=0.48) and an araldite flat, considered as rigid comparatively. Phenomenological description of these phenomena was performed. Mouwakeh et al [2] performed a quantitative analysis and show that the energy dissipated during sliding was comparable with the energy associated with the propagation of a shearing mode interface crack. Numerical simulations based on variational formulations were proposed by numerous authors, and more particularly by Raous [3]. The friction law used is the Coulomb's law. Sliding waves are observed experimentally at one edge of the contact area, sweep through it first partially, then entirely and lead finally to gross sliding. Correct modelling of all stages of the sliding evolution from rest up to gross sliding is still not performed: a central slip zone is predicted, based on the assumption of a friction coefficient equal to the ratio of the tangential load corresponding to the gross sliding over the normal load. This study deals with these differences. Thus: - modelling of interface behaviours based on the Coulomb's law were listed. The numerical model is based on a combination of the finite element technique, the interfacial crack theory and the unilateral contact analysis with friction. Geometrical dimensions of the slab and the friction coefficient value have a great influence on the way the sliding stages progress. - tests were realised to give an accurate phenomenological description of the sliding waves. The influence of both the geometrical dimensions of the slab and the friction coefficient value have been verified. Further both load and displacement variations when the sliding waves sweep through the contact area, partially or totally, were performed. Particularly, informations on the influence of the loading speed on sliding were obtained. For the configuration studied, incipient slidings are highly dependent on the parameters mentionned above. These results can't be extrapolated directly to industrial configurations due to the difference in material, a very compliant one in tests, rigid ones in industry. But this device is of a great help for local frictional behaviour analysis. Comparisons between experimental and theoretical results and the behaviours noted from the parametric study lead to the conclusion that a parameter adjustement will allow to model sliding stages with a Coulomb law type.

550

tangential load leading to gross sliding over the normal load- defines the shear stress onti = fd

1. INTRODUCTION

A theoretical approach of friction and sliding phenomena between two frictional contacting bodies is presented in this paper. Progri et a1 [ l ] and Mouwakeh et a1 [2] have previously studied experimentally these phenomena and identified conditions leading to "sliding waves" resembling "Schallamach waves" [4] between two contacting bodies. A two-body contact is formed between a rectangular polyurethane slab and an araldite rigid flat (cf. figure 1). The slab is first pressed against the flat, then a tangential load is imposed until gross sliding is detected. Sliding is due to successive pertubations or waves located at the interface and wich travel through it. These pertubations are associated to the propagation of a shearing mode interface crack, as the energy dissipated during sliding and the energy dissipated during the propagation of an interface crack are comparable 121. Numerical simulations of these experiment based on variational formulations were proposed by Fbous [3] (finite element method) and Deshoullieres [ 51 (boundary integral equations). These simulations are based on the Coulomb's law with static and dynamic friction coefficients: a static coefficient of friction fs defines the "stick limit", ( < fs *

I

* Onn( ). The stick and slip zone evolutions in the contact are modelled. But the evolution of the state of the contact is still different numerically from the one observed experimentally [ 1,2]. Further a stress field singularity associated to the sliding zone extension is numerically obtained, but less marked than in the experimental case. The Coulomb's friction law and the experimental observations of the contact phenomena disagree. This study aims at : listing the different behaviours described by the Coulomb's law. The influence of the coefficient of friction and the slab dimensions will be studied with a parametric study. Location of initial slidings and the direction of extension of the sliding zones will be particularly observed. - an accurate experimental observation of "the sliding wave" phenomenom. This model combines numerical and analytical methods. The stress and displacement fields are calculated and the stick, slip and open zones at the contact interface are determined. Further the different behaviours at the contact interface depending on the Coulomb's law are identified and listed. This theoretical study is undertaken jointly with an experimental one. A loading frame was developed which allows us to perform accurate

-

1 anti

1

(I

annl ), a dynamic coefficient of friction fd equal to the ratio of the limiting value of the

t

L DT

-r3(

'

PolyurCthane E,= 7 MPn

h

H

Ve= 0.48

DD

V--T

L=80mm Araldite

EA2500MPa

A

A

'

5 niin

1

H

vA=0.38

A

A: comoressed air supply GasPad x Figure 1 : Experimental device. Mechanical Properties and geometrical dimensions of the contacting bodies

I

Y

55 1

analysis of the contact evolution during a load cycle. Both loads and displacements are measured continuously and their variations are studied in relation with sliding wave travelling. 2. MODELLING

The simplicity of the geometrical shapes of the solids, of their law of behaviour, of the cinematic conditions characterise the experimental configuration. Therefore the following simplifling assumptions are retained for the theoretical modelling:

displacements, and respects the interface equilibrium. The finite element method is used to calculate this field. The crack field (oF), corresponding to displacement discontinuities like slip and opening at the contact interface is a corrector field, added when the boundary conditions are no more respected. The resultant field (0) satisfies the boundary conditions listed underneath (7-10). Analytical expressions for the crack field exist in the literature [8]. The coordinate system (o,x,y) is introduced. x is the inner normal at the body 1 contacting interface, y is the tangential component (cf. fig. 1).

- steady-state sliding as -

the sliding speeds are small, elastic linearity : the deformable material of the slab behaves homogeneously and isotropically according to Hooke's law, two-dimensional and particularly plane strain: the specimens are assumed to be of infinite width compared with the contact dimensions, displacements are allowed accross the boundary of the two specimens.

The two-body sliding contact behaviour is particularly difficult to model. A gross sliding event is often preceeded by partial slips over part of the contact surface, causing: tangential displacement discontinuities within slip zones, - normal and tangential displacement discontinuities within open zones, - continuous normal and tangential displacements within stick zones. The slip and open zone distribution (number and location) is unknown at priori and changes during loading. This model uses the previous works of Comninou [6] and Dubourg [7]. It is based on the method of continuous distributions of dislocations for modelling the displacement discontinuities at the contact interface and the contact solution for determining the slip and open zone distribution as a unilateral contact problem with friction. The resultant stress and displacement fields are obtained by superposing the continuous and the crack responses to the load. The continuous field ( oc ) corresponds to sticking at the contact interface, i.e continuity of the normal and tangential

k, ( l - A ) + l - B 4 (k, + 1)

L

bx(q)dr

(3)

-

with A = - 1-r l+Tk,

B = k, - T k , k, + r

ki=3-4vi k.1 = (3 - vi) / (1 + vi)

r =P 2 PI

plane deformation plane stress

where the indices 1 and 2 correspond respectively to the flat and the slab, L is the contact width, bx and by are the Burgers vectors corresponding to open

552

and slip. Integration and observation points ri and s, are given by:

(::1:pj

ri =cos -

i=l,n

3. NUMERICAL RESULTS

j=l,n s, = cos- 2 j p 2n+l where n is the number of discretisation points. Stress and displacement expressions are determined from Airy stress functions, and satisfy therefore automatically to the compatibility and equilibrium equations. Further the stress field singularity at crack tip is correctly taken into account. The contact interface state is determined numerically by solving the unilateral contact analysis with friction. The boundary conditions for a model using the Coulomb's friction law are the following : Normal or opening problem (N) Contact zone onnI 0 Au = 0 AU = u 1412 Open zone an, = 0 A u > 0

reached, i.e when the distribution of open, stick and slip zones is stabilized from one iteration to the next.

(7) (8)

Tangential problem (T)

The starting point for studying the contact interface behaviour is to know the location of the first sliding, as this event is determinant for the interface behaviour. Location of the first sliding is predicted in previous studies [ I l l by the Coulomb's law depending on a stick threshold and slab dimensions WL. A normal displacement is first imposed. It corresponds to a normal load of about 76 N. Then a tangential displacement from the left to the right is applied. The terms "left" and the "right" corrcspond to the left and right of the reader. This convention is used in the remainder of the paper. The increment of the tangential load is constant and equal to 2N whatever the ratio WL is. The sliding zone evolutions were analysed for numerous contact configurations. The variation range of the parameters considered is : - friction coefficient: from 0.1 to 1.4, step 0.01 - ratio WL from 0.1 to 1, step 0.1 ratioH/L= 1.2 and 1.5

-

f

1.4

L: Location of the first opening at thc right edge of the contact

1.3

1.2

A v = v , -v, Slipzone lontl