Dry Adhesive Friction of Elastomers: A study of the fundamental mechanical aspects

Dry Adhesive Friction of Elastomers

A study of the fundamental mechanical aspects

A.R. Savkoor

TR diss

1576

A study of the fundamental mechanical aspects

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft, op gezag van

de Rector Magnificus Prof.dr. J.M. Dirken,

in het openbaar te verdedigen ten overstaan van

een commissie door het College van Dekanen

daartoe aangewezen, op 15 oktober 1987 te 14.00 uur

door

Arvin Ramachandra Savkoor

geboren te Bombay

Master of Engineering

TR diss)

1576

Prof.ir. H. Blok

niet zonder.

The study reported in this work is concerned with the basic study of the phenomenon of friction between ordinary dry surfaces of solids. Under such conditions the main cause of friction is adhesion. In virtue of the special tribological features of friction between elastomers and hard solids, this combination of materials is ideally suited to carry out basic studies on friction. Besides this scientific aspect, its importance for engineering practice stems from applica­ tions for traction, especially for tyres of road vehicles. Keeping these dual areas of interest in mind the work described is broadly divided into two parts. The first part (chapters 2 and 3) concentrates on the basic aspects of adhesion and static friction of ideally smooth and elastic bodies, where the elastomer is seen merely as an ideal linear elastic material. In the second part (chapters 5 and 6) the em­ phasis is placed on the behaviour of extended rough surfaces in sliding friction. Here, the elastomer is treated as a thermo-rheologically simple material with linear viscoelastic properties, in order to investigate the relationship between these properties and kinetic friction under quasi-isothermal conditions of sliding. This relationship is of both theoretical and practical interest.

Following a general introduction the next two chapters (2 & 3) aim to probe into the basic nature of the interactions of bodies in adhesion and friction. Although the bodies are considered as being perfectly smooth and elastic, the influence of the real-world environ­ ment has not been ignored. The effects of adhesion on the mechanics of contact under the action of purely normal forces is considered in chapter 2. The most tangible of these effects are i) the contact area is larger than Hertzian, and ii) a finite tensile force is required to separate the bodies. The various formulations of contact mechanics for taking these effects into account are discussed. A concept of a "thermodynamical contact area" is introduced in order to treat problems of adhesive contact within the framework of thermodynamics. This formulation leads to the well-known energy balance approach of fracture mechanics. Some of the other formulations are illustrated with different models where the effects of adhesion are incorporated either through the introduction of non-contact forces or by using the critical stress intensity factor of fracture mechanics. The merits of these various formulations are discussed. For dissimilar materials it is shown, using the energy balance approach, that adhesion is reduced on account of shear tractions even though both bodies and the loading are rotationally symmetric.

The effect of a tangential force on the adhesive contact is considered in chapter 3- It is shown that the contact area shrinks in size by peeling as the tangential force increases until it attains a certain critical value. This effect is essentially similar to that of separating two bodies as discussed in ch.2. Above the critical value, the effect of increasing the tangential force is considered in the light of the stress intensity factors of fracture mechanics and the onset of slip is viewed as the shear mode of fracture. It is worth noting that the effect of the normal mode of separation (as in peeling), discussed in ch.2, is distinctly different from that produced by the shear mode discussed here. In contrast to the process

Certain mechanical features of contact between extended sur­ faces are introduced in chapter 4, using simple deterministic and statistical models of surface roughness. The objective here is to il­ lustrate the basic effects of surface roughness in adhesion and friction. It is assumed that an asperity of an extended rough surface, at least on a microscopic scale, behaves as though it were perfectly smooth. Consequently, the behaviour of any individual asperity can be calculated by applying the theory developed in chapters 2 and 3- The static fractional behaviour of spherical bodies is studied on this basis, i.e., using the fracture mechanics criteria for predicting slip instead of the conventional Cattaneo-Mindlin approach based on the Amontons-Coulomb law of friction. Next, considering kinetic friction, the relative motion between macroscopic bodies is seen as an oblique interaction of asperities of their surfaces, the contact kinematics being defined by Green's constraint. Preliminary results of experi­ ments performed using upscaled model asperities are presented to illustrate the concept of stagnation of the growth of the contact area as a result of strong adhesion. As a result of this stagnation shear tractions build up, leading to fracture of adsorbed layers at the interface. While awaiting an exact analysis of this complex contact-mechanical problem, only a rough calculation can be presented, using empirical results, for illustrating how the traction and consequently the tangential force build-up following the stagnation.

Chapter 5 deals with models of friction between extended sur­ faces of viscoelastic and rigid bodies. The presentation given here is divided into two parts mainly because of historical reasons. In part I, a highly simplified model, first developed by the author in 1965, is presented for studying the fractional problem of visco-elastic materials on hard solids. Although the model is rather crude it serves to illustrate the basic factors which govern the relationship between the viscoelastic properties and the speed dependent frictional properties of elastomers. The calculated curves show generally good qualitative agreement with the measured curves. In part II, a more detailed physical framework has been presented and certain significant improvements have been proposed towards a more refined theoretical analysis of the problem. Methods of fracture mechanics are used to predict adhesive fracture at the interface between asperities. Some new features introduced here are that the adsorbed material may recuperate, even after failing, when the surfaces are exposed anew to the healing action of the atmosphere. Preliminary results obtained with this theory already predict the expected bell-shape of the fric­ tion curve remarkably well.

The experimental work undertaken to correlate friction of elas­ tomers on hard solids with the different influential factors governing the process is reported in chapter 6 and the structure of a tentative relationship has been deduced on the basis of the theoretical trends and experimental results. Chapter 7 deals with the physico-chemical properties of dry surfaces of solids and leads to a particularly simple, empirical characterization of such properties by measuring contact angles of drops of liquids on these solids.

1 GENERAL INTRODUCTION 1

1.1 Some opening remarks 1 1.2 Background of dry adhesive friction 3

1.3 Remarks on the "adhesion anomaly" 5 1.4 Some remarks concerning the mechanics of dry friction 6

1.5 How is surface roughness involved in friction 7

1.6 Elastomer friction 9 1.7 Practical considerations and the need for empirical laws 10

1.7.1 Frictional forces and coefficients 10 1.7.2 Clean and dry or unlubricated state of surfaces 11

1.8 References 11

2 CONTACT MECHANICS OF ADHESION 13

2.1 Introductory survey 13 2.2 Outline of the chapter 16 2.3 Basic physical aspects of contact mechanics 17

2.3.1 The interaction between surfaces of solids 17

2.3.2 The interaction of rigid solids 21 2.3-3 The " thermodynamical " contact area of rigid solids 25

2.3.4 The thermodynamic area of deformable solids in mechanical

contact 27 2.4 The formulation of the adhesive contact problem of elastic

solids 35 2.4.1 Approaches for including the influence of adhesion 35

2.4.2 The Hertzian contact problem without adhesion 36 2.4.3 Interaction of deformable bodies based on the action of

"non-contact" surface forces 39 2.4.4 A hybrid formulation of the adhesive contact problem 4l

2.4.5 The thermodynamic area of the hybrid problem 46 2.5 The energy balance approach in contact mechanics 46 2.5.1 A general thermodynamical formulation of the contact problem 48

2.5.2 The axisymmetric contact problem with adhesion 51 2.5.3 The JKRS solution for adhesive contact of spherical bodies 54

2.5.4 Inherent limitations of the contact mechanics approach 55

2.5.5 The force of adhesion and the "pull-off" force 57 2.5.6 Validity of the JKRS results in the pull-off region 59 2.5.7 Surface forces across a mechanical point contact 62 2.6 The fracture mecht: tics approach to adhesion 64

2.6.1 Relationship between G and K 65 2.6.2 Adhesive contact of long cylinders 67 2.7 Adhesion of elastically dissimilar solids as affected

by shear traction 69 2.7.1 Contact of elastically dissimilar bodies 69

2.7.2 Axisymmetric bodies having arbitrary elastic constants 71 2.7.3 The conical punch (n=l) indenting an elastic half-space 74 2.7.4 The adhesion between a spherical punch and an elastic

half space 76 2.8 Adhesion between viscoelastic bodies 78

3 TANGENTIAL INTERACTION OF ELASTIC BODIES 83

3.1 Outline of the chapter 83 3.2 Effect of a small tangential force on the adhesive contact 85

3-3 Experimental verification 89

3.3.1 Experiment 90 3.3.2 Results 91 3.3-3 Discussion 91 3.4 The transition from peeling to sliding 94

3.4.1 The shear stress intensity factor Kx ( « K ^ ) 95

3.4.2 A physically motivated assumption about the transition 96 3;5 A simple model of interfacial breakdown in shear 99

3.6 Analysis of the mechanics of slippage 102 3.6.1 The sub-problem 1, that for the locked region rSc 103

3.6.2 The sub-problem 2, that for the slip region c<rSa„ 103

3.6.3 The determination of c. 104 3-7 The compliance characteristics of bodies during transition 105

Appendix 1 The sub problem 2 108 Appendix 2 Initial breakdown zone and the "size"effect" 112

Appendix 3 The angle between the vectors of shear traction

and slip 116 Appendix 4 The interaction between modes I,II and III under the

combined action of normal and tangential forces at

pull-off. 119

3.8 References 121

4 SURFACE TOPOGRAPHICAL EFFECTS IN ADHESION AND FRICTION

123

4.1 Outline of the chapter 123 4.2 Asperity models of disperse contact 124

4.2.1 Deterministic models 125 4.2.2 Statistical models 126 4.2.3 Plane contact of solid surfaces with Greenwood-Williamson

roughness model 128 4.2.4 Contact of a smooth elastic sphere with a nominally flat

rough surface of a rigid solid 130 4.3 Surface topographical effects in adhesion 135

4.3.1 The attenuating effects of surface roughness 135 4.3.2 The first basic effect of surface roughness 135 4.3.3 The second effect of surface roughness 138 4.3.4 The third effect of surface roughness 143 4.4 Surface topographical effects in static friction 145

4.4.1 The role of surface roughness in the transmission of

normal and tangential forces between spherical bodies 145

4.4.2 The asperity loads in a multi-junction contact 146 4.4.3 A simplified model of•slip-growth of a multi-junction contact 150

4.4.4 Discussion 154 4.5 Mode of asperity interaction in kinetic friction 156

4.6.1 Experimental procedure 160

4.6.2 Results 161 4.6.3 Empirical approximations suggested by the experimental results 161

4.7 An approximate calculation of the shear traction within

the adhering contact area 162 4.7-1 An approximate relation for the growth of the contact area

during oblique penetration 163 4.7-2 Distribution of shear tractions and the stress intensity

factor K 164

4.8 References 170

5 THE THEORY OF ISOTHERMAL FRICTION OF ELASTOMERS 173

CH.5 Pt.I

5.1 Outline of the chapter 173 5.2 The theoretical background of elastomeric friction 174

5.2.1 Importance of the study of elastomer friction to tribology 175

5.2.2 A brief survey of previous investigations 176 5.3 An elementary theory of friction between elastomers

and hard solids l8l 5-3-1 The frictional interaction between asperities 182

5-3-2 An outline of the simple rudimentary theory 184 Appendix 5/A Calculation of viscoelastic spectra from isochronous

dynamical data obtained at different temperatures 204

CH-5 Pt.II

5.4 A more detailed study of the mechanics of sliding

friction of elastomers 209 5.4.1 A brief summary of this section 209

5.4.2 Introductory remarks 209 5.4.3 An idealised model of sliding friction 210

5.4.4 The influence of the speed of sliding on the normal contact 214 5.4.5 Tangential interaction during the prestagnation period 217 5.4.6 Tangential interaction during the stagnation phase

and the initiation of fracture 220 5-4.7 Discussion of fracture criteria for inelastic materials 220

5.4.8 Implementation of the model of fracture initiation 222 5-4-9 Crack propagation along the interface through the

adsorbed layer 225 5-4.10 Results and discussion 230

5.4.11 Concluding remarks 236

5.5 References 237

6 EXPERIMENTAL STUDY OF ISOTHERMAL FRICTION OF ELASTOMERS 241

6.1 Outline 24l 6.2 Introduction 242

6.3-2 Material specifications 248

6.3-2.1 Rubber compounds 248 6.3-2.2 Track materials 250 6.4 Experimental work 251

6.4.1 Methods 251 6.5 The experimental test programme 253

6.6 Experimental results and discussion 253

6.6.1 Discussion of results 254 6.7 Summary of the experimental trends from sets a to g 268

6.8 The structure of the constitutive relation for f 270

6.8.1 General considerations 270 6.8.2 The influential factors 271 6.8.3 Friction during steady state sliding 273

6.8.4 Static friction as a limiting case of sliding friction 273

6.8.5 Displacement dependence in initial friction 274

6.8.6 Non-uniform motion 275 6.8.7 Measurements of friction in the falling region of the

friction curve 276 6.9 A tentative constitutive relation for f 280

6.10 Some thoughts concerning the frictional behaviour

under non-isothermal conditions of sliding 282 6.10.1 Higher speeds and the interrelated speed-temperature effect 282

6.11 References 282

7 THE INFLUENCE OF THE PHYSICO-CHEMICAL PROPERTIES OF THE COUNTERFACE 285

7.1 Outline of the chapter 285

7.2 Introduction 286 7-3 Surface physico-chemical properties in adhesion and friction 287

7-4 Thermodynamical background 288 7.5 The characterization of surface chemical properties

of solids in terms of the wetting behaviour of liquids 292 7.6 The relation of the contact angle to the available

surface energy 294 7-7 The relation of the contact angle to friction 296

7.8 Experimental correlation of contact angle and friction 298

7-9 Interpretation of the contact angle data 300 7.10 The influence of surface geometrical factors on the

correlation of friction with the contact angle. 302

7.11 Remarks and discussion 303

7.12 References 304

LIST OF SYMBOLS (Chapterwise) 307

SAMENVATTING 316 ACKNOWLEDGEMENTS & Curriculum vitae 319

GENERAL INTRODUCTION

1.1 Some opening remarks:

The work described here is partly based on some of the earlier publications of the author. At the outset it is fair to remark that because the subject of dry friction is essentially multidisciplinary in character (belonging formally to what is known as tribology) , the researcher is placed in a rather unique position as compared with scientists working in many other areas. This has certain obvious ad­ vantages as well as certain equally obvious disadvantages. However one may look at these aspects, it is usually true that the researcher finds himself being an expert and a layman at the same time! In many respects the search for newer results goes hand in hand with a reexamination of the old results since the foundations on which one stands are often fluids rather than solids. There are many theories of friction, some of these have been proposed in the distant past, some in the near past and this activity continues and will probably con­ tinue until some sort of consolidation has been attained.

A detailed review of the historical development of the various ideas has been meticulously compiled by Dowson in his famous book [1]. It is unnecessary and even futile to review the many theories of friction. However, we shall mention a few of these for bringing out some of the features of friction relevant to our discussion. One of the main aims of this work is to understand the basic mechanism of ad­ hesive friction between two bodies when their surfaces are in the dry state of contact. In order to appreciate the complexity of the fric-tional phenomenon, excluding its many-sided relationship with other equally (or more) complex phenomena like wear, heat dissipation, fric-tional welding, machining or cutting process, triboelectrification etc., we shall consider the frictional phenomenon in its purest and simplified form. One of the main requirements for obtaining friction in this form is that wear is not an intrinsic part of the process of friction. However, this restriction does not reduce the value of such a study merely to an academic excercise because there are many practi­ cal applications where it is both desirable and feasible to attain friction in this form. Especially in this respect, the friction of elastomers on nominally smooth and dry surfaces of hard solids provides us with an interesting example to carry out basic studies of friction relevant to practice.

The various reasons why elastomers are particularly suited to the basic study of dry friction shall be considered more fully in the text. At this point it is sufficient to mention these very briefly: low modulus and highly elastic behaviour, easily observable effects of adhesion , Poisson ratio of (nearly) 1/2 which considerably simplifies solutions of contact problems, the strong rate-dependence and the

close relationship between the visco-elastic properties and their frictional behaviour, the extremely high wear resistance which permits to dissociation of wear from friction and finally resistance to con­ tamination because of their low surface energy. All these special features prove useful at various stages of our theoretical as well as experimental work. However, we must mention some of the negative points, like complications arising from geometrical and physical non-linearities (even ignoring additional difficulties on account of their viscoelastic behaviour), material and surface stability (both in the physical sense of non-linear effects at large deformations and in the chemical sense of the influence of exposure to the environment for long periods of time or at high temperatures). Fortunately, elastomers used in engineering which contain anti-oxidants and fillers are suffi­ ciently stiff and well cross-linked, so most of these difficulties do not appear to be serious.

Properties of rubbers such as viscoelasticity and their low elastic modulus make the study of friction of this material not only rewarding for its own sake but also to understand the complex mechanism of friction. It is well known that hysteresis losses during cyclic deformation of rubber are mainly responsible for the rolling resistance of tyres. However, in this work we shall be concerned mainly with the mechanism of sliding friction and in particular with the way in which the viscoelastic properties affect the sliding fric­ tion between elastomers and hard solids. The linear viscoelastic behaviour of rubber at small strains is typically characterized by the effect of time or frequency of deformation on the elastic modulus. The molecular structure of such materials also implies a relationship be­ tween temperature and the rate of deformation. The relationship has been established by a number of material-scientists and in particular a simple time-temperature transformation formula of William, Landel and Ferry (W.L.F.) found a wide following. The effect of varying the rate or frequency of deformation at a given temperature may be simu­ lated merely by a suitable change of temperature without varying the rate. Experimental results on sliding friction of rubber show that the aforementioned transformation also applies to the coefficient of slid­ ing friction of rubber. The variation of p with speed and temperature is related to the viscoelastic properties of rubber.

In developing a phenomenological theory of friction one has to recognize the paramount importance of the following elements.

1) The existence of adhesion, however small, between any two bodies and the way it affects the contact between these bodies.

2) The existence of boundary layers of adsorbed material on the sur­ faces of bodies and their role in adhesional and frictional contact of bodies (However we shall not treat these layers as separate elastic bodies but we do assign a distinct strength property to these layers). 3) The existence of surface roughness leads to the formation of dis­ perse areas of real contact between the two bodies. If the normal load transmitted through the contact is relatively small this leads to a wide spacing between any two contacts and hence such micro-contacts may be treated as though they were isolated from one another. This tacit assumption is made in order to simplify the analysis of the friction of extended surfaces. Furthermore, we assume that the

asperities of such surfaces have smooth tips and that, essentially, the surface geometry may be modelled in some deterministically or statistically simple manner.

4) The deformational properties of the elastomer are thermo-rheologically simple and can be represented in terms of linear theories of elasticity and viscoelasticity.

For illustrating the basic problems of the contact mechanics of adhesion we shall restrict the analysis to be presented in chapters 2 and 3 to well-known problems of Hertz (see e.g. the recent book of. Johnson [2]), Cattaneo [3] and Mindlin [4], for perfectly elastic and smooth spherical bodies. These problems may be regarded fundamentally as two of the most significant problems of contact mechanics of solids. The first one highlights the mixed boundary conditions of nor­ mally loaded elastic bodies, where the contact region is not entirely fixed by the geometry of the bodies but depends in a non-linear manner on the applied load.. The second problem is the most relevant extension of the Hertz problem where a tangential force appears in addition to the normal load. It serves as a benchmark for developing and testing various models of the frictional phenomenon.

We shall mention here two other contact problems of interest to the tribological studies. The problem of indentation of a semi-infinite elastic body by a rigid, flat-ended punch of circular cross-section with its solution by Boussinesq (see e.g.[2]) will be used in this work. In this work we shall be dealing only with kinematically simple problems of static or steady-sliding contact. In many applica­ tions in transportation technology the problem of rolling contact becomes important. The exact solution of this kinematically complex problem of contact through which forces may be transmitted by friction has been given by Kalker [ 5 ] . This problem has since become the benchmark in the case of rolling contact of bodies. Although the popular solution of this theory makes use of the Coulomb friction law, his formulation is sufficiently general to accomodate more complex laws of friction.

1.2 Background of dry adhesive friction

Amongst the many diverse concepts encountered in the field of of mechanical engineering and particularly in solid mechanics, fric­ tion and adhesion are probably two of the most elusive ones. The subject of this work is adhesive friction, which emphasizes our view that although we will be dealing mainly with the process of friction, adhesion has an important part to play in that process. Intuitively we know that when two solids are brought into contact it becomes possible to transmit forces and moments from one body to the other according to certain definite rules. The questions how such rules may be defined and applied remain as yet unanswered to the dissatisfaction of scien­ tists and engineers.

An important issue of scientific interest is how friction comes into being. As Suh and Sin [6] aptly put it " One of the fundamental issues of tribology is the genesis of friction between sliding

surfaces". This issue has a bearing upon the nature and the relative importance of the basic disciplines which must be consulted in order to understand and describe the processes involved. Our position con­ cerning the origin of friction is obvious from the choice of the title of this work. Unlike the views expressed e.g. in [6], we regard adhe­ sion between bodies as one of the most important factors governing ordinary friction.

For some time in the past it was customary to assume that the presence of surface roughness was the main cause of friction between bodies. The asperities on surfaces were seen as minor obstacles to the sliding motion. In order to slide one body over another it is neces­ sary either to drag and lift the bodies over such obstacles or to break the obstacles in the process of sliding. Clearly, these models depict rigid and not deformable bodies. Although this view is often credited to (or blamed upon) Coulomb, a study of his works (1781), reveals that this was just one out of the many alternative ideas that he conceived. In the historical perspective it is interesting to know that the notion of friction as a distinct process appears in the famous notebooks of Leonardo da Vinci, some hundreds of years earlier. Amontons (1699) was probably the first to make a systematic experimen­ tal study of the frictional behaviour of bodies. For this reason one of the most viable of the so called "laws of friction" is known as the Amontons-Coulomb law. It states that the force of friction required to slide bodies is nearly proportional to the normal force transmitted through the contact. Parker and Hatch [7] note that Coulomb discounted one of his alternative models based on the attraction between the sur­ face molecules of the two bodies mainly on the ground that it was not consistent with his experimental observation. That observation being the insensitivity of friction to the nominal size of the rubbing sur­ faces provided the normal load remains constant.

The modern era in tribology based on the use of macroscopic concepts and properties started round about 19^*0 with the recognition of the importance of adhesion to friction. The strong influence of tiny amounts of surface active material on friction has been known since the work of Desaguliers as early as in 1725- It becomes clear from his work that while the sparse application of a surface active substance cannot affect the roughness of a surface to any significant extent its influence on friction can be explained only in terms of its effect on surface adhesion. In this connection we should also mention the works of Hardy [8] and Holm [9]- For a complete historical ac­ count, the reader may consult the work by Dowson [1]. Since 19^0, various different interpretations of the theory of adhesive friction have been proposed in the literature. Notably, it was a great service of Bowden and Tabor [10] to raise the level of acceptance of their basic idea that adhesion between bodies is the main source of friction and that surface roughness plays, by itself, only a secondary role. In fact, surface roughness will cause a reduction of the "real area of contact", thereby reducing the adhesion and consequently the friction between bodies.

1.3 Remarks on the "adhesion anomaly" :

It has been mentioned already that if adhesion between bodies is the main cause of friction we must bear the burden of "proving" that adhesion does exist, that it is a viable concept in the sense of physics and possibly demonstrating that its effects can be made tan­ gible in some way. But before we set out to do that it is worthwhile to examine the nature of the anomalous behaviour of adhesion.

A well-established methodology in solid mechanics for calculat­ ing the internal stresses and deformation of a "single body" subjected to external loads, is based on the concept of equilibrium of forces of action and reaction across fictitious mathematical surfaces which divide the body into its constituent parts. Physically, this approach reflects the action of internal cohesive forces due to the proximity of its atoms and molecules which hold the different particles of mat­ ter together. Hence there is no essential contradiction between the phenomenological and molecular interpretations of the phenomenon of cohesion. Macroscopic properties like elasticity, plasticity and frac­ ture of solid bodies may be translated, albeit in a qualitative form, into microscopic properties of matter from which such bodies are made.

For some reason this harmony between macroscopic and micro­ scopic views is disturbed if a single body is broken say, into two pieces and these pieces are subsequently pressed against one another so as to form a contact. Macroscopic experience indicates that or­ dinarily, it is not possible to recover the original body permanently by joining parts in this way. As soon as the joining pressure is released, contact is broken and the two pieces fall apart. Each piece must be simply considered as a separate body. Thus while it is pos­ sible to transmit a compressive force through their contact, it cannot sustain a tensile force! On the other hand, a certain amount of tan­ gential force may be transmitted readily through the contact by friction.

Mainly as a result of the extensive work of Bowden and Tabor [10], the adhesion theory of friction as it is sometimes called, has become one of the most widely accepted theories of friction. However, this acceptance has been given generally with some reservation. There are several reasons why this is so, the most important of them being the lack of credibility of the postulated action of adhesion. In every day life one does not experience any special forces while making or breaking contact between bodies, i.e. forces over and above those ex­ plicable simply in terms of gravity and inertia. Under ordinary circumstances, even if adhesion should exists at all, it would seem to be too weak to explain the substantial resistance against sliding. The explanation of this "adhesion anomaly" is given in Ch.2. The first part of our work (Chs.2 and 3) deals with the basic nature of adhesion and we shall consider how and why adhesion affects the contact defor­ mations and tractions both in situations of normal separation and friction.

In the broader macroscopic framework of thermodynamics of deformable solids the complex action of forces of attraction between the molecules of the solids may be more conveniently expressed in terms of energy per unit of the contact interface. This specific energy may be viewed as the work of adhesion required to separate the surfaces in a reversible manner. From physical chemistry of solid sur­ faces we know that it is given by the Dupré equation which will enter more explicitly in chapter 7 of this work. The explanation of this anomaly becomes transparent by studying the mechanics of normal separation in Hertzian contact. In this process the work of adhesion plays an important role. Using an energy balance approach it may be shown that the normal tensile force required to separate bodies may be small but it is most certainly finite.

1.4 Some remarks concerning the mechanics of dry friction

The transmission of force through contact between a pair of counterformal bodies has been the subject of intensive study since the classical work of Hertz. In many practical applications, in addition to the purely normal loading considered by Hertz the contact interface also transmits tangential forces by means of friction. A detailed study of the mechanism whereby slip may occur between surfaces is of interest both from the scientific viewpoint of understanding the phenomenon of friction and also to alleviate effects such as wear and fretting damage.

The basic problem of force transmission by friction was studied by Cattaneo and independently by Mindlin. Starting from the Hertzian normal contact they .extended the Hertzian analysis to include the ef­ fect of an additional tangential force on the slipping of bodies. They made use of the so called Coulomb's law of friction to determine the extent of the area of slip. As a consequence of their analysis they came to the conclusion that if slip is prevented completely the shear traction rises to infinity towards the boundary of the contact area, whereas the normal pressure there tends to zero. So when Coulomb's law is applied slip must ensue as soon as a tangential force is applied, no matter how small this force is. In the later part of their analysis they assume a finite coefficient of friction and show that the contact area is divided into two concentric regions. An inner circular region, where finite shear tractions are sustained without slipping is known as the "locked" region. The slip is confined only to an outer annular region.

Some of the experimental investigators have presented indirect experimental evidence which lends support to the theory. However, these experiments mainly concentrate on the overall behaviour of the bodies, e.g. the tangential force-displacement curves. Other indirect evidence has been that concerning the size of the annular region of no-slip estimated from experiments involving an oscillating tangential force. The results of these experiments, are often presented as "visually" observable pictures of areas of wear. However, these pic­ tures lack clarity in the crucial region of moderate to large slip.

I n t e r e s t i n g l y , i n t h e same r e g i o n e x p e r i m e n t a l d e t e r m i n a t i o n of t h e s h a p e of t h e t a n g e n t i a l f o r c e - d i s p l a c e m e n t c u r v e p r o v e s t o b e a d i f ­ f i c u l t t a s k . I t may be a r g u e d t h a t t h e s e e x p e r i m e n t a l r e s u l t s a r e , by t h e m s e l v e s , s t i l l i n c o n c l u s i v e e v i d e n c e o f t h e p h y s i c a l v a l i d i t y o f t h e C a t t a n e o - M i n d l i n t h e o r y . Moreover, t h e wear e x p e r i m e n t s a s c o r ­ r o b o r a t i v e e v i d e n c e a r e o p e n t o c r i t i c i s m s i n c e a l t e r n a t i v e i n t e r p r e t a t i o n s of t h e p r o c e s s may be g i v e n .

Of more fundamental s i g n i f i c a n c e i s t h a t t h e l o c k e d r e g i o n may b e s e e n a s t h e r e g i o n where t h e m a t e r i a l p o i n t s i n c o n t a c t p h y s i c a l l y a d h e r e t o one a n o t h e r . So s l i p can o c c u r o n l y i f t h e a d h e s i v e b o n d s a r e r u p t u r e d . Recent work on c o n t a c t of s p h e r e s made of e l a s t i c s o l i d s shows how s u c h b o n d s a r e r u p t u r e d i n t h e p r o c e s s of p u r e l y n o r m a l s e p a r a t i o n of b o d i e s . T h i s l e d t o t h e m o d i f i c a t i o n of t h e r e s u l t s f o r t h e H e r t z i a n c o n t a c t when a d h e s i o n i s p r e s e n t . C o n s e q u e n t l y , t h e t r e a t m e n t o f t h e e f f e c t of a t a n g e n t i a l f o r c e on t h e c o n t a c t between s o l i d s s h o u l d be r e - e x a m i n e d i n t h e l i g h t of t h e mechanics of f r a c t u r e o f a d h e s i v e b o n d s b e t w e e n s o l i d s . T h u s t h e n o n - s l i p s o l u t i o n s of C a t t a n e o and M i n d l i n can be i n t e r p r e t e d d i f f e r e n t l y t h a n they d i d . The s h e a r t r a c t i o n r i s e s t o i n f i n i t y t o w a r d s t h e boundary of t h e c o n t a c t a r e a w i t h t h e c h a r a c t e r i s t i c s q u a r e - r o o t s i n g u l a r i t y o f t h e l i n e a r f r a c t u r e m e c h a n i c s t h e o r y .

I n c h a p t e r 3 . we s h a l l examine t h e mechanics of c o n t a c t between e l a s t i c b o d i e s , f i r s t u n d e r t h e a c t i o n o f a p u r e l y normal l o a d and l a t e r when a t a n g e n t i a l f o r c e i s a p p l i e d i n a d d i t i o n t o t h e n o r m a l l o a d . The i n f l u e n c e of a d h e s i o n on t h e c o n t a c t m e c h a n i c a l problem i s c o n s i d e r e d i n t e r m s of q u a n t i t i e s o f t h e r m o d y n a m i c s and f r a c t u r e m e c h a n i c s . 1.5 How i s s u r f a c e r o u g h n e s s i n v o l v e d i n a d h e s i o n and f r i c t i o n B e c a u s e a l m o s t a l l e x t e n d e d s u r f a c e s of m a c r o s c o p i c s o l i d s a r e rough on a m i c r o - s c a l e t h e r e a l a r e a of c o n t a c t i s o n l y a s m a l l f r a c ­ t i o n o f t h e n o m i n a l a r e a b e t w e e n t h e b o d i e s . I n i t i a l l y , t h e Bowden-T a b o r t h e o r y c o n c e n t r a t e d on m e t a l l i c c o n t a c t s a n d h e n c e t h e y c o n c l u d e d t h a t t h e l o c a l p r e s s u r e s a t t h e s u r f a c e a s p e r i t i e s w i l l be s u f f i c i e n t l y h i g h t o i n d u c e p l a s t i c d e f o r m a t i o n . Using v e r y e l e m e n t a r y p r o p e r t i e s o f m e t a l s n a m e l y , t h a t o f p l a s t i c f l o w a s i n an i n d e n t a t i o n - h a r d n e s s t e s t and t h e t e n d e n c y t o w e l d u n d e r p r e s s u r e e s p e c i a l l y a t h i g h t e m p e r a t u r e s , t h e t h e o r y o f f e r s an e x t r e m e l y s i m p l e and l u c i d e x p l a n a t i o n of t h e A m o n t o n s - C o u l o m b l a w of f r i c t i o n . I n s h o r t , t h e r e a l a r e a of c o n t a c t i s s i m p l y t h e r a t i o of t h e normal load and t h e h a r d n e s s of t h e s o f t e r of t h e two s o l i d s . The t h e o r y e x p l a i n s t h e p r o p o r t i o n a l i t y between t h e normal l o a d and t h e r e a l a r e a of con­ t a c t and a t t h e same time i t a l s o g i v e s a p l a u s i b l e e x p l a n a t i o n a s t o why t h e t a n g e n t i a l f o r c e i s r e l a t e d t o t h e r e a l a r e a , where t h e welded j u n c t i o n s must be b r o k e n i n t h e c o u r s e of s l i d i n g . The s i m p l e t h e o r y h a s u n d e r g o n e many r e f i n e m e n t s s i n c e i t s i n c e p t i o n .

Many s c i e n t i s t s have p u t i n a g r e a t d e a l o f e f f o r t t o e x p l a i n t h e A m o n t o n s - C o u l o m b law i n t h e hope t h a t i t may p r o v i d e t h e key t o

the mechanism of friction. But the main thrust of their effort has been to find the reason why the actual contact area between two bodies with rough surfaces increases in proportion to the normal force. Unlike metals where the postulate of plastic deformation immediately explains the latter linear relationship, this is not so obvious in the case of elastic materials. In the contact between asperities of elas­ tic bodies a simple model based on the Hertzian theory would predict a variation according to the 2/3 rd power of the normal load. However, if the statistical variation of the dimensions of asperities of a rough surface is taken into account, it turns out that the average size of the real area of contact varies almost linearly with the nor­ mal load. Thus we find that the linear relationship between the real area of contact and the normal load does not require per se that the contact deformation be plastic. Owing to surface roughness, the same linear relationship is obtained irrespective of whether the contact deformation is elastic or plastic.

However, such detailed statistical models are not strictly necessary for understanding the primary effects of roughness on adhe­ sion and friction. A number of workers do not seem to have realized that the more basic question is not how the real area varies with the load but how and why the frictional force varies with the contact area. Only this latter variation has a bearing on the mechanism of friction. Qualitatively, the two primary effects introduced on account of the surface roughness are: first, the total amount of the real area of where adhesive interactions are strong is much smaller than the nominal area and second because this implies a constriction and hence stress concentration which shall weaken the interface considerably.

A basic difficulty in tribological studies is that of choosing an appropriate model of the surface roughness. Clearly, it is highly unrealistic to depict asperities of surfaces as bristles of a brush or as microflats or as mating parts of a jig-saw puzzle. The modern view based on profilometric studies of practical surfaces after "running in" is that the microgeometrical contour of an asperity of is out­ wardly convex and rather smooth especially near the tip.

A rational analytical basis of the modern notions concerning the physical consequences of the disperse nature of contact of bodies with extended surfaces was laid by Blok [11-14]. Treating the bodies as half-spaces he studied both the mechanically as well as thermally significant parameters of such contacts. These fundamental investiga­ tions dealt with two basic questions, namely, the purely mechanical aspect of the transition from elastic to plastic deformation in the case of normally loaded metallic contacts [11,12] and the thermal aspect concerning the mechanism of the dissipation of frictional heat [13] and especially the "flash temperatures" arising in sliding con­ tacts [14]. In this work we shall not be concerned with this latter aspect of frictional heating. In chapter 4 of this thesis we shall study only the mechanical aspects of contact in greater detail. We shall describe the different models of rough surfaces and make use of a simple model developed by Greenwood and Williamson [14] to il­ lustrate the various physical effects associated with adhesion and friction of rough surfaces. Although the analytical treatment of the

problems formulated is still rather crude and elementary, the main ideas concerning the various different ways in which surface roughness modifies adhesion during the normal separation of bodies and friction between bodies has been illustrated quite clearly through simple examples.

1.6 Elastomer friction :

The subject matter concerning the theoretical models of fric­ tion of elastomers has been divided into two parts. In the first part is described the general background including a critical survey of previous literature. Since some of the past contributions of the author have been mainly in this area, the subject matter has been treated exhaustively in Ch.5. We give there the merits of choosing elastomers as test materials for basic tribological studies and also indicate in detail the various difficulties both of theoretical and experimental nature. A rudimentary theory of elastomer friction is also included in Part I of this chapter. Despite its crudeness in respect of the analysis of the contact kinematics and the ad-hoc na­ ture of the criteria to predict the fracture of the adhesive bonds, it remains useful as a guide in the selection of a rubber compound for a particular application. Sample calculations of frictional characteris­ tics are given making use of measured viscoelastic properties of different rubber compounds.

In Part II of the same chapter we describe a more detailed and physically realistic picture of the process of friction between asperities of the extended surfaces of solids. Here we shall consider the development of contact forces between two typical asperities of the opposite solids. It is assumed that the solids slide relative to one another at a uniform speed under a constant normal load. Furthermore assuming that there is no normal vibration of bodies during sliding it is possible to model the manner in which such a typical pair of asperities come into contact in terms of a kinematical constraint on their relative motion. As a result of such motion and the adhesion between the asperities we shall have to introduce new elements into the theory, namely the concept of stagnation of contact which leads to the initiation and ultimately to propagation of frac­ ture through adsorbed layers. Based on indirect experimental evidence, a preconceived path of fracture has been assumed. It is taken to traverse through the adsorbed layer on the hard solid. Calculations based on Griffith-type criteria of fracture have been made in order to obtain estimates of the frictional force as a function of the sliding speed of the elastomer. Although only tentative results can be given as yet, it is encouraging to note that the predicted trends of fric­ tion vs. speed curves as influenced by the different parameters are in qualitative agreement with experimental trends. Owing to the extremely complex set of contact problems involving non-stationary viscoelastic contact and large displacements it was not possible to pursue more detailed analysis at the present time. The application of finite ele­ ment techniques seems to be the only course open, but as yet we were

unable to implement successfully an existing programme. A more detailed study is contemplated in the near future.

1.7 Practical considerations and the need for empirical laws

The basic research on the genesis of friction has still to go a long way before a sufficiently coherent picture can emerge. Meanwhile, a mechanical engineer cannot wait because he must get on with the job of constructing and running the "machine" with a large number of such "mechanical contacts". So hè requires a working set of rules which can be understood in his terminology and which are consistent and reliable. However easy as it may seem, a reliable quantitative expres­ sion for the force of friction in terms of the normal contact pressure, the relative motion, temperature, material properties and the most difficult of them all - the surface conditions and geometry does not exist as yet. The frictional behaviour in actual practice is not even amenable to simple empirical generalisation. Unfortunately there are no sets of rules in the form of "laws of friction" except the basic "Amontons-Coulomb law" which one can trust implicitly. Even this "first law" provides only a rough guideline; there is no guarantee that the machine will indeed behave as intended. Hence there is no escape from the requirement of a "sound engineering judgement" based on a "sufficiently deep insight" into the basic nature of fric­ tion if one intends to minimize the laborious "trial and error" procedures in engineering design and the rigourous "on site" testing. Considering the large variety of variables mentioned previously this insight must span a sufficiently wide range of topics in contact mechanics and tribology including both the disciplines of physical chemistry and materials science.

1.7-1 Frictional forces and coefficients:

The force of sliding friction is the quantitative measure of the level of friction. It is expressed usually in terms of the coeffi­ cient of friction (f or u ) , and the normal load acting on the area of contact. The coefficient is often regarded as an attribute of the friction pair which behaves according to the Amontons-Coulomb laws of friction. On dry surfaces, the first of these laws which describes the proportionality between the tangential force of friction and the nor­ mal load is usually obeyed in practice. It is worth remembering that the coefficient of friction may be regarded simply as a convenient non-dimensional representation of the force of friction obtained under a given condition, rather than as a basic physical property of the in­ teracting surfaces.

In Ch.6, we shall present results of our experimental work on the friction of elastomers. The trends are qualitatively similar to those which are predicted by the theory described in Ch.5- The rela­ tive influence of the different variables on the friction coefficient also shows a fairly good agreement. The theoretical results may be ap­ plied for predicting the various trends described in ch.6. In this chapter we shall also develop an empirical relationship for expressing

t h e c o e f f i c i e n t of f r i c t i o n a s a f u n c t i o n of t h e many d i f f e r e n t v a r i ­ a b l e s g o v e r n i n g t h e p r o c e s s . T h i s r e l a t i o n may b e v i e w e d a s a t e n t a t i v e form of t h e c o n s t i t u t i v e r e l a t i o n f o r f r i c t i o n . I t i s b a s e d p a r t l y on t h e r e s u l t s of t h e e x p e r i m e n t s and f o r t h e r e s t on t h e t h e o r e t i c a l r e s u l t s of Ch. 5-1 . 7 . 2 " C l e a n and d r y " o r " U n l u b r i c a t e d " s t a t e of s u r f a c e s : The p r a c t i c a l s i g n i f i c a n c e of t h e " c l e a n a n d d r y " s t a t e of a s u r f a c e s h o u l d n o t be u n d e r e s t i m a t e d , c o n s i d e r i n g t h a t an e n g i n e e r i s m o s t l y c o n c e r n e d w i t h t h e c o e f f i c i e n t o f f r i c t i o n a t t a i n e d on s u c h s u r f a c e s . A p r e c i s e c h a r a c t e r i z a t i o n of t h i s s t a t e of s u r f a c e s c a n n o t be made a s y e t b u t a p r a c t i c a l working d e f i n i t i o n may b e g i v e n b a s e d on t h e r e q u i r e m e n t t h a t no m a c r o s c o p i c c o n t a m i n a n t s e i t h e r i n s o l i d o r l i q u i d form a r e v i s i b l e t o t h e n a k e d e y e a n d t h a t t h e s u r f a c e s a r e f r e e from b o u n d a r y l u b r i c a n t s and o r g a n i c c o n t a m i n a t i o n . However, ad­ s o r b e d l a y e r s f o r m e d u n d e r t h e a c t i o n o f o r d i n a r y a t m o s p h e r e a r e t o l e r a t e d p r o v i d e d t h a t s u c h l a y e r s a r e n o t l o o s e l y bound ( t h i n r u s t ) . I t i s a l s o u s u a l t o t o l e r a t e an i n s i g n i f i c a n t a m o u n t o f p a r t i c u l a t e m a t t e r ( d u s t ) . I t i s a common p r a c t i c e t o r e f e r t o a p a r t i c u l a r s u r ­ f a c e b e l o n g i n g t o t h i s c l a s s by n a m i n g i t a f t e r t h e u n d e r l y i n g m a t e r i a l , q u a l i f i e d by t h e term " c l e a n and d r y " ( o r sometimes "dry") o r " u n l u b r i c a t e d " .

One of t h e most i m p o r t a n t p r o p e r t i e s of t h e s u r f a c e i s i t s s u r ­ f a c e e n e r g y . The h a r d s o l i d s u s e d a s c o u n t e r f a c e s o r " t r a c k s " a r e i n t r i n s i c a l l y h i g h e n e r g y s o l i d s . However, i n a g a s e o u s e n v i r o n m e n t , t h e s u r f a c e e n e r g y a v a i l a b l e f o r t h e p r o c e s s of a d h e s i o n i s o n l y a f r a c t i o n o f i t s i n t r i n s i c e n e r g y . On t h e o t h e r hand w i t h polymers the i n t r i n s i c e n e r g y i s q u i t e s m a l l a s c o m p a r e d t o s a y m e t a l s b u t s i n c e t h e r e i s p r a c t i c a l l y no a d s o r p t i o n t h e a v a i l a b l e e n e r g y i s almost e q u a l t o i t s i n t r i n s i c v a l u e . I n c h a p t e r 7 we s h a l l e x p l o r e t h i s s u b ­ j e c t f u r t h e r and d e s c r i b e a s i m p l e method u s i n g d a t a of c o n t a c t a n g l e s of l i q u i d s w h i l e w e t t i n g t h e t r a c k s u r f a c e s . The s u b j e c t of w e t t i n g a p p e a r s t o b e a h i g h l y s p e c u l a t i v e one and any c o n c l u s i o n s t h a t can be drawn must be viewed a s t e n t a t i v e o n e s .

1.8 R e f e r e n c e s :

[ 1 ] Dowson, D . , " H i s t o r y of t r i b o l o g y " , Longman, London & N . Y . , (1984) [ 2 ] J o h n s o n , K . L . , " C o n t a c t M e c h a n i c s " , C a m b r i d g e U n i v . P r e s s . , Cambr. , ( 1 9 8 5 ) ; (See t h e r e l e v a n t r e f e r e n c e s t o c l a s s i c a l w o r k s of H e r t z , H . , and B o u s s i n e s q , J . ) . [ 3 ] C a t t a n e o , C , "Sul c o n t t a t o d i due c o r p i e l a s t i c i " , R e n d . A c c a d . Naz. d e i L i n c e i , 2 7 , s e r . 6 , . p p . 3 4 2 , ( 1 9 3 8 ) . [ 4 ] M i n d l i n , R . D . , " C o m p l i a n c e o f e l a s t i c b o d i e s i n c o n t a c t " , Trans.ASME, S e r . E , 7 1 , J . A p p l . M e c h . , 16, p p . 2 5 9 , (19*19). [ 5 ] K a l k e r . J . J . , "On t h e r o l l i n g c o n t a c t of two e l a s t i c b o d i e s i n t h e p r e s e n c e of d r y f r i c t i o n " . D o c t o r a l D i s s e r t a t i o n , D e l f t U n i v . o f T e c h . , ( 1 9 6 7 ) .

[ 6 ] S u h , N . P . and S i n , H . C . , "On t h e g e n e s i s of f r i c t i o n and i t s ef­ f e c t on w e a r " , i n " S o l i d c o n t a c t and l u b r i c a t i o n " , AMD. V o l . 3 9 , ASME P u b l . , e d . H.S.Cheng and L.M.Keer, p p . 1 6 7 , Nov. 1 6 - 2 1 , ( 1 9 8 0 ) .

[ 7 ] P a r k e r , R.C. and H a t c h , D . , "The s t a t i c c o e f f i c i e n t o f f r i c t i o n and t h e a r e a of c o n t a c t " , P r o c . P h y s . S o c . , B 6 3 , p p . 1 8 5 - 1 9 7 , ( 1 9 5 0 ) . [ 8 ] Hardy,W.B., " C o l l e c t e d s c i e n t i f i c p a p e r s " , C a m b r i d g e U n i v . P r e s s , ( 1 9 3 6 ) .

[ 9 ] Holm, R., W i s s . V e r o e f f . , Siemens-Werk., 1 7 , p p . 3 8 , ( 1 9 3 8 ) ; a l s o s e e " E l e c t r i c c o n t a c t s " , Hugo Gebers F o r l a g , Stockholm, ( 1 9 4 6 ) .

[ 1 0 ] Bowden, F . P . a n d T a b o r , D . , "The f r i c t i o n a n d l u b r i c a t i o n o f s o l i d s " . V o l . I , I I , Oxford Univ. P r e s s , ( 1 9 6 4 ) . ; (For a more r e c e n t a n d c r i t i c a l r e v i e w o n e may c o n u l t t h e a r t i c l e o f T a b o r , D . , " F r i c t i o n - t h e p r e s e n t s t a t e o f o u r u n d e r s t a n d i n g " , ASME, J . L u b r . T e c h . , v o l . 1 0 3 , p p . 1 6 9 - 1 7 9 , A p r i l ( 1 9 8 1 ) .

[ 1 1 ] B l o k , H . , " F u n d a m e n t a l m e c h a n i c a l a s p e c t s o f b o u n d a r y l u b r i c a t i o n " , J . S A E . , ( 1 9 3 9 ) .

[ 1 2 ] B l o k , H. , "Comments on a p a p e r by R.Wilson " i n "A d i s c u s s i o n of f r i c t i o n " , P r o c . R o y . S o c . ( L o n d o n ) , A 212, p p . 4 8 0 - 4 8 2 , ( 1 9 5 2 ) . [ 1 3 ] B l o k , H . , "The d i s s i p a t i o n of f r i c t i o n a l h e a t " , Appl. S c i . R e s . , S e c t i o n A, V o l . 5 , PP- 1 5 1 - 1 8 1 , ( 1 9 5 6 ) . [ 1 4 ] B l o k , H. , " T h e o r e t i c a l s t u d y of t e m p e r a t u r e r i s e a t s u r f a c e s of a c t u a l c o n t a c t u n d e r o i l i n e s s l u b r i c a t i n g c o n d i t i o n s " , P r o c . G e n . D i s c u s s . L u b r i c a t i o n , 2 , I n s t . Mech. E n g r s . ( L o n d o n ) , p p . 2 2 2 - 2 3 5 . ( 1 9 3 7 ) .

[ 1 5 ] Greenwood, J . A . and W i l l i a m s o n , J . B . P . , "The c o n t a c t of n o m i n a l l y f l a t s u r f a c e s " , P r o c . R o y . S o c . , A 2 9 5 , P P - 3 0 0 - 3 1 9 .

CONTACT MECHANICS OF ADHESION

2.1 Introductory survey.

The theory of adhesive friction is based upon the assumption that adhesion is the primary cause of dry friction of solids. Despite the wide popularity of this concept, strong objections have been raised time and again, even questioning the credibility of the under­ lying hypothesis of adhesion itself! Indeed, everyday experience of separating two ordinary solid objects from one another hardly suggests the existence of adhesion, no matter how hard or for how long the ob­ jects were pressed together prior to the act of separating them. Naturally, one's prima facie reaction is to be sceptical of the postu­ lated role of adhesion in friction of bodies, because, unlike the former, friction is a very tangible force in practice. In defence of the adhesion theory Tabor [1] propounded the argument that these two seemingly different manifestations of adhesion need not be looked upon as incompatible entities. He went on to argue that it is in fact the imperceptible resistance to normal separation of bodies which ought to be regarded as an anomaly. Recognising that this anomalous behaviour 'cannot be explained away merely by appealing to the presence of sur­

face imperfections such as the roughness and contamination, he proposed that the elastic stresses released during the course of un­ loading which precedes separation were in some way responsible for the anomaly. Although it has recently become clear that Tabor's explana­ tion was essentially correct, the intuitive nature of his arguments failed to convince a number of critics. The controversy has per­ petuated for a long time and thereby has been a hindrance to the unreserved acceptance of the theory.

Recent progress in understanding the role of adhesion in the contact interactions of solids has been achieved through the applica­ tion of the methods of continuum mechanics to contact problems. The field of contact mechanics has progressed steadily since the beginning of the twentieth century when the significance of the pioneering works of Hertz and Boussinesq to engineering practice became clear. However, until very recently it was not realized that the classical approach of implementing problems of contact mechanics has to undergo some modification and reinterpretation with respect to the basic physical boundary conditions before it can be applied to the study of adhesion. The changes required are prompted by certain concepts evolved in another branch of continuum mechanics viz. fracture mechanics. This relatively new field has experienced an explosive rate of growth since its inception through the bold attempt of Griffith to explain the brittle fracturing of glass [2].

He put forward two new ideas. The first one was to suggest an expedient method of overcoming the dilemma which arises when the results of analysis based on the linear theory of elasticity assuming certain plausible boundary conditions lead to the prediction of stress singularities. Without being concerned unduly about the consequences of these unrealistically high albeit highly localized stresses he

s i m p l y c h o s e t o i g n o r e t h e s e and c o n c e n t r a t e i n s t e a d on a s p e c t s of t h e e n e r g y b a l a n c e of t h e p r o c e s s ! Thus he s t u m b l e d u p o n t h e s e c o n d i d e a w h i c h was t o p r o p o s e t h a t t h e c r e a t i o n of new f r a c t u r e s u r f a c e s must i n v o l v e an a d d i t i o n a l form of e n e r g y and t h a t s u c h a s e p a r a t e e n e r g y t e r m may b e i n c l u d e d i n t h e o t h e r w i s e c o n v e n t i o n a l e n e r g y b a l a n c e a p ­ p r o a c h by a s s i g n i n g a s p e c i f i c s u r f a c e e n e r g y t o a s u r f a c e . The l a t t e r p r o p e r t y of a s u r f a c e makes i t p o s s i b l e t o i n c o r p o r a t e v a r i o u s t e c h n i ­ c a l l y i m p o r t a n t phenomena r e l a t e d t o t h e p h y s i c o - c h e m i c a l a s p e c t s of s u r f a c e s w i t h i n t h e b r o a d e r t h e r m o d y n a m i c f r a m e w o r k o f c o n t i n u u m m e c h a n i c s . T h e s e i d e a s of G r i f f i t h h a v e shown how t h e c l a s s i c a l f o r ­ m u l a t i o n o f c o n t a c t m e c h a n i c s may be m o d i f i e d s o a s t o a p p l y t o t h e problem of a d h e s i o n . However, t h e i n c l u s i o n of s u r f a c e e n e r g y a s a p h e n o m e n o l o g i c a l q u a n t i t y o f t h e r m o d y n a m i c n a t u r e i n v o l v e s c e r t a i n c o n c e p t u a l d i f f i c u l t i e s . I n s e c t i o n 2 . 3 we w i l l c o n s i d e r t h e n a t u r e of t h e com­ p l i c a t i o n s which a r i s e i n t h a t f o r m u l a t i o n , i . e . i n t h e l i g h t o f an e l e m e n t a r y p h y s i c a l p i c t u r e o f i n t e r a c t i o n s b e t w e e n a t o m s a n d m o l e c u l e s of t h e s o l i d . But t h e p u r p o s e of t h i s e l e m e n t a r y p i c t u r e i s m e r e l y i l l u s t r a t i v e and a l t h o u g h i t h e l p s i n b r i d g i n g t h e gap b e t w e e n t h e m i c r o s c o p i c and m a c r o s c o p i c v i e w p o i n t s i t s p o t e n t i a l a s a t o o l f o r q u a n t i t a t i v e a n a l y s i s i s c l e a r l y v e r y l i m i t e d . Our i n t e n t i o n i s t o d e v e l o p and u t i l i z e t h e m a c r o s c o p i c a p p r o a c h of c o n t a c t m e c h a n i c s t o s t u d y t h e p r o b l e m s o f a d h e s i o n . The r e a s o n f o r t h i s p r e f e r e n c e i s t h a t t h e p o w e r f u l methods of c o n t i n u u m m e c h a n i c s u s e t r a c t a b l e m a t e r i a l p r o p e r t i e s a n d p e r m i t a c o m p r e h e n s i v e and c o n s i s t e n t d e s c r i p t i o n of t h e mechanics of c o n t a c t of s o l i d s , i n normal a s w e l l a s i n t a n g e n t i a l i n t e r a c t i o n s . The a n a l y s i s g i v e n i n s e c t i o n 2 . 4 i s b a s e d upon t h e c o n ­ t a c t mechanics a p p r o a c h ; i t e x p l a i n s t h e c h i e f c a u s e o f t h e a d h e s i o n anomaly i n n o r m a l s e p a r a t i o n . The v e r s a t i l i t y of t h e c o n t a c t m e c h a n i c s a p p r o a c h f a c i l i t a t e s t h e s t u d y of p r o b l e m s i n v o l v i n g t h e a d h e s i v e c o n ­ t a c t b e t w e e n d i s s i m i l a r m a t e r i a l s . I n t h i s c a s e s h e a r t r a c t i o n s a r i s i n g on a c c o u n t o f t h e s t a t i c f r i c t i o n a r e s e e n t o modify t h e n o r ­ mal a d h e s i o n o f s o l i d s ; t h e a n a l y s i s g i v e n i n s e c . 2 . 5 i s a c a s e i n p o i n t . B e f o r e g o i n g i n t o t h e s u b j e c t m a t t e r i t i s w o r t h c l a r i f y i n g some of t h e c o n f u s i o n a r i s i n g i n t h e open l i t e r a t u r e c o n c e r n i n g b o t h t h e t e r m i n o l o g y a n d n o t i o n s of a d h e s i o n . The t h r e e r e l a t e d p h y s i c a l p r o c e s s e s , m a k i n g c o n t a c t by b r i n g i n g two b o d i e s c l o s e t o g e t h e r , t r a n s m i t t i n g f o r c e and power t h r o u g h t h e c o n t a c t , and b r e a k i n g c o n t a c t i n o r d e r t o s e p a r a t e s o l i d s , i n v o l v e v a r i o u s m a n i f e s t a t i o n s o f t h e phenomenon o f a d h e s i o n . T h e s e a r e v i e w e d g e n e r a l l y from d i f f e r e n t a n g l e s by t h e d i f f e r e n t d i s c i p l i n e s of p u r e and a p p l i e d s c i e n c e s . The u s u a l m e c h a n i c a l v i e w p o i n t i s t o r e g a r d t h e f i r s t and t h e l a s t of t h e p r o c e s s e s a s t r i v i a l o n e s e x c e p t when s p e c i a l t e c h n i q u e s a r e e n ­ g i n e e r e d t o j o i n o r w e l d s u c h s u r f a c e s a r t i f i c i a l l y . Under o r d i n a r y c o n d i t i o n s i t i s assumed t h a t t h e normal t r a c t i o n s t r a n s m i t t e d a c r o s s t h e c o n t a c t a r e o n l y of c o m p r e s s i v e n a t u r e . I n t h e p r e s e n c e o f f r i c ­ t i o n t h e s e c o m p r e s s i v e t r a c t i o n s a r e a s s o c i a t e d w i t h s h e a r t r a c t i o n s and t o g e t h e r t h e y d e t e r m i n e w h e t h e r r e l a t i v e t a n g e n t i a l motion c a n o c ­ c u r b e t w e e n t h o s e m a t e r i a l p o i n t s w h i c h a r e i n c o n t a c t . T h e t e r m " a d h e s i o n " , u s e d t o d e s c r i b e t h e s t a t e of z e r o r e l a t i v e motion i n t h e t a n g e n t i a l p l a n e i s i n a p p r o p r i a t e b e c a u s e i n t h a t l i m i t e d s e n s e i t a c ­ t u a l l y i m p l i e s " s t a t i c f r i c t i o n " .

It

The two processes of making and breaking contact, which are usually ignored by mechanical engineers acquire special significance from the point of view of physicists and surface chemists. This is so because the attractive and repulsive forces between atoms and molecules which hold together particles of any solid in cohesion should interact just as naturally in adhesion across the plane of con­ tact between any two solids. The appreciation of these physical and physico-chemical interactions at the surfaces of contact has been at the heart of the rapid development of the present generation of com­ mercial adhesives used in engineering. Some of the surface chemists active in the evaluation of the strength of adhesive joints use the expressions "adhesion" or "force of adhesion" to refer to the applied force which causes the contact to be broken irrespective of where the joint fails.

Since this unfortunate usage has confounded fruitful dis­ cussions we will avoid it in this work. In our opinion, the notion of "adhesion" is aptly described by the physical and chemical interac­ tions which occur in the process of making contact. In this sense the terminology conforms to the ordinary meaning of the word in the non-specialist's vocabulary. The "force of adhesion" is the total force of attraction resulting from tensile tractions applied to the contact. The force required to separate bodies will be described by the some­ what less crisp but nevertheless lucid term "pull-off force". The distinction between the force of adhesion and the pull-off force is clearly illustrated in section 2.4. The unambiguous term "work of adhesion" is the reversible work of creating fresh surfaces by over­ coming the forces of intermolecular attraction (see section 2.3).

The complex process of making contact may be viewed as a com­ bination of two complementary sub-processes. The first refers to the macroscopic phenomenon of forcing the deformable bodies into conform­ ity while the second refers to the interfacial interactions which takes place on minute scales ranging from the sub-microscopic to the microstructural ones. In the present investigation we will be mainly concerned with the macroscopic aspects of the contact. However, this does not mean that the second part is unimportant, only that our in-t e n in-t i o n here is in-to simplify in-the picin-ture of in-the microscopic interactions to the utmost. Even amongst specialists there is little unanimity of view on the detailed picture of the various aspects of this microprocess. The various mechanisms proposed in the literature are the physical adsorption, chemisorption, diffusion, electrical, electronic interactions, and several variants of each of these. Among the reviews [3, 4, 5. 6, 7. 8] summarize the various theories and at­ tempt to unify the wide disarray of notions concerning the relative importance of one or more of these for adhesion between a particular pair of materials. In addition to these uncertainties, the presence of boundary layers resulting from adsorption and oxidation under the con­ taminating influence of the ambient gaseous environment implies that the contact interaction occurs between two solid-gas interfaces about which far less is known. It is neither within the scope of this work nor within the competence of the author to enter into a detailed dis­ cussion of the merits of the various types of interactions mentioned above.

For the present it suffices to pay attention only to those in­ teractions which occur most commonly between surfaces of elastomers