A novel generic model at asperity level for dry friction force dynamics

Transcription

1 Tribology Letters, Vol. 16, No. 1, January 2004 (# 2004) 81 A novel generic model at asperity level for dry friction force dynamics F. Al-Bender, V. Lampaert and J. Swevers Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300B, 3001 Heverlee, Belgium Received 12 January 2003; accepted 8 June 2003 This paper presents a theoretical model for (dry, low-velocity, wear-less) friction force dynamics based on asperity interaction considerations subject to the phenomenological mechanisms of creep/relaxation, adhesion and (elasto-plastic) deformation in their most generalized forms. The model simulates the interaction of a large population of idealised, randomly distributed asperities with arbitrarily chosen geometrical and elastic properties. Creep and adhesion are simulated by an expedient local coefficient of friction that increases with time of contact, while deformation effects are accounted for by rate-independent hysteresis losses occurring in the bulk of the material of an asperity that is breaking loose. An energy method is adopted to calculate the instantaneous, local friction force leading to better insight into the problem as well as higher numerical efficiency. The results obtained by this model show both qualitative and quantitative agreement with the known types and facets of friction force dynamic behaviour; in particular, pre-sliding quasi time-independent frictional hysteresis in the displacement, velocity weakening, slider lift-up effect and frictional lag, in addition to the influence of the various process parameters, all in a single formulation, such as no extant friction model could show before. Moreover, the model is still open for and capable of further refinement and elaboration so as to incorporate local inertia and viscous effects and thus to be extended to include velocity strengthening and lubricated rough contacts. KEY WORDS: friction modelling, simulation, pre-sliding hysteresis, frictional lag, lift-up effect, velocity weakening, asperity dynamics 1. Introduction The pioneering work of Amontons, Coulomb, and Euler tried, albeit qualitatively, to explain the friction phenomenon in terms of the mechanics of relative movement of rough surfaces in contact with one another. Since then, only sporadic attention has been paid to this important question, i.e., friction as a dynamical process that evolves in the contact. Instead, the greatest part of the research effort has been concentrated on characterising and quantifying the complex mechanisms, such as adhesion and deformation, that give rise to frictional resistance; often overlooking the dynamical aspects of the problem. As a result, despite the fact that those mechanisms have been relatively well researched, characterised and understood, no effective and comprehensive model for the evolution of the friction force, as a function of the states of the system, namely time and displacement (or equivalently, the displacement and its time derivatives), in the totality of a rubbing contact has been advanced to date. The need for such a model is becoming now more urgent than ever, since an understanding of the frictionforce dynamics proves crucial for the knowledge and control of systems involving rubbing elements, from machines to earthquakes. Spikes [1] predicts that a To whom correspondence should be addressed. farid.albender@mech.kuleuven.ac.be major part of the tribological research activity in the coming decade and beyond will be devoted to modelling. Stick-slip motion, jittering windscreen wipers,...and earthquakes [2,3,4] are all phenomena involving a mechanical system with friction-force dynamics. By the latter, we mean the friction force evolution in the contact as a result of relative tangential movement. It is the interaction of this dynamics with the rest of the mechanical system that can give rise to the unstable motion that is characteristic of those phenomena. Therefore, if we are able to qualify and quantify this frictionforce dynamics, it would be a relatively simpler step to treat the dynamics of a whole system comprising friction. Realising this, researchers, towards the end of the first half of the last century, in their endeavour to explain the stick-slip phenomenon (the most commonly known frictional instability), first identified velocity weakening to be the main culprit. Velocity-weakening of kinetic friction, in dry contact, akin to the Stribeck effect in lubricated contacts, is the phenomenon of decreasing friction force with increasing sliding velocities, (or, in more simplified treatments, the assumption that kinetic friction be lower than static friction, both values being assumed constant). At a latter stage, it was proposed that another phenomenon, namely the rising static friction with dwell (or rest) time, be a friction dynamics that could by analysis be shown to give rise to stick-slip motion. (See, e.g. [19a,b,40 (chapter 7)]. It appeared, much later, that the two aforementioned phenomena were interconnected [23,31], i.e., they were /04/ /0 # 2004 Plenum Publishing Corporation

2 82 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics manifestations of one and the same process. But researchers until then just took them for granted, i.e., without attempting to explain why they were there in the first place. It is quite a remarkable fact that Coulomb, with the rather limited resources at his disposal, was able both to measure and to advance plausible explanations to both phenomena, while in the space of almost 200 years that followed, nobody seems to have paid the question a serious thought. In the mean time, friction force dynamics was beginning to gain in importance in a variety of fields; the main thrust coming from machine control [], geoand applied physics [6] and structural dynamics [7], with obvious and important motivations in each of those fields of application. So that, at a latter stage, the seminal work of Rice and Ruina [6], showed, at least for the cases they examined, that the friction force had a more complex dynamics (than the quasi-static velocity weakening), which depends on the rate of change of velocity and on a state of the contact; hence the ratestate model. That model, which has now become well established, has been able to give a more satisfactory explanation to observed (macro) dynamics of frictions in geo-physical systems. However, beside its heuristic nature, this model suffers at least from one drawback, namely that it can deal only with (gross) sliding friction dynamics: it contains a characteristic speed below which it cannot yield a valid result. As a matter of fact, there appears to be a pre-sliding phase of friction (see later) in which the friction force (if it could still be called that, since there is no gross sliding) is nearly independent of the rate of relative displacement, i.e., the speed. In a quite separate development, this pre-sliding phase, sometimes termed micro-slip, has been treated by many researchers [8 11]. The models, reached by assuming a constant local coefficient of friction between the normal and shear tractions and solving the elastic contact problem, all boil down to the Maxwell-slip model structure [12]. Between all this, and spurred by control engineering needs, all sorts of models, mostly heuristic, were formulated with varying degrees of success, to fill an empty gap in this important field of application [13 17]. In summary, the last two decades or so have witnessed an intensified activity in friction modelling, which is expected further to increase in the coming decade. Research in this domain falls now roughly into two main categories: the search for (i) physically motivated friction models and (ii) empirically motivated models. The first category is becoming more popular in the domain of nano/micro tribology [18] since the invention of the Atomic Force and the Friction Force Microscopes. There, one looks for a physical explanation of the behaviour of a tip on a surface. Although this is providing valuable insight about friction mechanisms and behaviour at micro-scale, it is not an evident task to extend the results to the case of macroscopic contact. The second category, on the other hand, is popular among geomechanic physicists and control engineers. They try heuristically to fit a dynamic model to experimentally observed results. The resulting models are then, in most cases, only valid for the specific behaviour from which the model was drawn, and cannot provide a physical explanation. In this paper, we try to fill the gap between the two approaches by constructing a physically based model at the asperity level (a relative level between the macro and micro tribology) and show that it is satisfactorily able to explain experimentally observed macroscopic friction force behaviour. Here we present a novel generic model for wear-less, dry friction, which effectively simulates the dynamics of interaction of a large population of idealised surface asperities during relative motion, under normal loading, subject to the phenomenological mechanisms of creep, adhesion and deformation in their most general(ised) forms. Despite its utmost simplicity, the model is able to simulate all experimentally observed properties and facets of low-velocity friction force dynamics (that we are aware of), such as no extant friction model could do in a single formulation. The model considers the tangential force that evolves in a rubbing contact, in response to an inexorably imposed tangential, timevarying relative movement, in isolation from the rest of the rubbing system s dynamics. Adhesion models [19a,b], deformation models [20], and atomic (hysteresis response) models [21] have been able to explain satisfactorily the possible mechanisms responsible for friction. In our model formulation, we treat them as friction mechanisms that are a priori at our disposal and use them, together with other mechanisms and dynamical considerations, in a very simplified way, to formulate an integrated generic model for the macro friction force dynamics, and to show its effectiveness in revealing the complex behaviour that is observed in practice. This model, which is largely theoretical in its present formulation, corresponds to wear-less friction: asperities are assumed to remain intact after interaction. This might not always be the case in practice. Gross plastic deformation, ploughing and junction growth [19b] lead to irreversible changes in the surface and material characteristics. Likewise, wedge/prow formation [22] and temperature effects can lead to instability of the process that might render our model invalid to explain ensuing frictional behaviour, unless special measures are taken to account for such phenomena in the formulation. Our immediate concern here is to give a basis that reveals the essential dynamics from which one can proceed to incorporate other elements that affect the (otherwise) constant physical and geometrical parameters of the system. Such details are outside the scope of this communication. The next section sketches first the different friction regimes and types of behaviour that are observed in practice. The ingredients of the model are then presented

3 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics 83 and discussed, leading to the actual formulation of the generic model and initial analysis of its properties. This is followed by a simplified, yet effective, application of the model formulation to the contact of two idealised rough surfaces under constant compression, with an indication about how the model is normalised and basic calculations are performed. Results pertaining to the basic friction force dynamics are then presented and discussed. Finally, general remarks about the model together with further possible refinements are indicated and appropriate conclusions are drawn. 2. A sketch of generally observed dry friction force dynamics: basic notions In order to introduce the reader to the force dynamics we are looking for, a word about the generally observed friction behaviour may be in order first. If we consider an object in rubbing contact with another (see figure 2A), then an applied tangential force will always result in a displacement, unless the contact is infinitely stiff. Below a certain tangential force threshold, if that force is held constant, the displacement will likewise remain constant (except possibly for creeping motion). When the force is decreased to zero, not all displacement will be recovered, i.e., there will in general be a residual displacement. This is the pre-sliding regime, in which, although there is relative movement, there are still points of unbroken contact, and points of micro-slip, on the two surfaces of the objects resulting in hysteresis of the force in the displacement that marks the friction behaviour in this regime [9,]. Above that force threshold, the system will be critically stable: the object will suddenly accelerate; all connections are broken, and we have true sliding [23]. The term friction force is usually taken to mean the resistance to the motion during true (or gross) sliding ; it usually has its maximum value at the commencement of motion ð¼ static frictionþ and usually decreases with increasing relative velocity ð¼ dynamic or kinetic frictionþ: It has been shown that the force is predominantly a function of the displacement in the pre-sliding regime showing quasi rate-independent hysteresis with nonlocal memory [9], and predominantly a function of the velocity (and its derivatives) in the true sliding regime, showing lag behaviour [6,23]. The borderline between the two regimes is obviously the pre-sliding distance and/or the pre-sliding force ð¼ static friction forceþ threshold, which are not evident to determine (at least exactly) owing to many factors that will become apparent in the rest of the paper. 3. Basic ingredients of the model Curtly stated, these ingredients fall into two categories: friction mechanisms and contact scenarios. They can be viewed as the premises or axioms of our model. The first category comprises the well-established phenomenological mechanisms governing friction: (i) Creep of the contacting asperities in the normal direction, which is fairly well known and understood for most materials. In the developed model, however, the creep mechanism is assumed to arise not (necessarily) only from material creep/relaxation upon loading, i.e., microcreep, but also (generally) from the gradual interlocking of mating asperities, in time, owing to various causes (e.g., stochastic variations in the surface forces) or owing to the oozing of third bodies (e.g., a viscous lubricant) out of the contacting surfaces, i.e., macrocreep [24]. This is intuitively obvious since the two objects will keep seeking a state of maximum stability (or minimum energy). (ii) Adhesion by which we shall generally mean tangential surface forces (resisting tangential movement) arising from a variety of sources, when the two surfaces are at a certain proximity of each other. Thus, not only metallic adhesion (asperity junction shear strength) is considered, but any other, short or long range forces, such as van der Waals, can lie at source of adhesion [24,2], (iii) Deformation will refer to elasto-plastic, hysteresis losses, of rate-independent nature [,26] in the bulk materials as a consequence of geometrical deformation of asperities (also called internal friction). In this formulation, only tangential deformation is considered, which, owing to adhesion, may persist long after the bases of the asperity have passed one another (cf. pull-off work). The effect of pure interlocking of asperities may likewise be treated. It should be obvious, from the above, that these three mechanisms are interconnected with each other to the degree that it be difficult to draw a clear borderline between the spheres of influence of each. For example, when we say that the adhesion friction increases with dwell time (of contact) owing to creep, then this effect may also be interpreted as arising from increased interlocking of asperities leading to increased deformation losses upon sliding. In our proposed model, we assume a priori the existence of these mechanisms, consider them abstractly, and represent the first two by a time increasing local friction coefficient, 1 while the last is accounted for by loss of stored (elastic and inertial) energy of the deforming asperities. Thus, by varying the parameter responsible for each mechanism, we can gain insight into its effect on the global friction behaviour. 1 This is common practice in the theory of elastic contacts (see e.g. reference [11].) The motivation therefore is that a friction law, such as Amontons law, is assumed to be valid on the micro-scale, so that at the point of micro-slip, the shear traction is some function of the normal traction (or contact pressure). In our formulation, the constant of proportionality, i.e., the local coefficient of friction is assumed to increase with contact time owing to creep.

4 84 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics It is the dynamic interaction of these three mechanisms with the asperity mass and stiffness that is responsible for the richly varied and rather complex behaviour of the friction force. This interaction is mainly induced by the surface topography. The second category of ingredients therefore involves the asperity interaction scenario itself, which will be seen to be equally significant in determining the friction force dynamics. This scenario only considers the vertical distance between asperities of both surfaces as a function of the relative horizontal translation between them. In the literature dealing with the contact modelling of two rough elastic surfaces the model is often reduced to that of the contact of one elastic surface, having the equivalent (or composite) roughness and elasticity of both surfaces, with a rigid perfectly flat second surface [,27]. While such a reduction is acceptable when considering stationary (i.e., not sliding) contact, it falls totally short of revealing the dynamics of the non-stationary case (i.e. sliding), since it masks the interlocking of asperities and gives a false picture of the continuously changing relative topography and hence of the resulting tangential forces. The contact scenario discussed in this paper, as presented schematically in figure 1, considers these topographical changes correctly. Figure 1A represents two rough objects in contact with each other, at their surfaces, while the dashed line shows the upper object translated horizontally to the left over a certain distance. As a result, some asperity contacts will persist (a and d), some will disappear (b and e) and new ones will occur (c). The distance between the two contacting surfaces can also be transformed to one flat surface and one rough surface (figure 1B). The shape of this rough surface corresponds to the distance between the two surfaces and will hence change with the horizontal translation (see the dashed line). Figure 1C shows the equivalent asperity, of contact point a; for four different time instances of its lifecycle. An overlap between the two surfaces corresponds to a contact between the two asperities. The contact results in a deformation of the asperity (not shown in the figure) where the size of the overlap is a measure for the size of deformation. The deformation process may in practice be quite complex, (elastic, plastic, mixed, prow formation, material transfer, asperity fatigue and wear particle formation, etc.). In this paper, we deal primarily with wear-less friction in which only elastic deformation and elasto-plastic hysteresis losses are considered, since the incorporation of other effects will only confuse the picture unduly. Finally, figure 1D transforms the contact scenario of figure 1C into the contact scenario used in this paper: A a b c d e B a b c d e C D Figure 1. The upper figure shows two surfaces in sliding contact with each other. The dotted line corresponds to the upper surface shifted to the right over a certain distance. The middle figure shows the transformation of the upper figure where the lower surface becomes a flat surface (note the different shape for the shifted surface). The left lower figure shows the transformed surfaces of point a for four different shift values. The first and the second one correspond to the full and dotted lines in the middle figure. The right lower figure gives the equivalent representation of the evolution of the contact point a. Note that this is only a schematic representation, the scales differ for each figure in x and y direction.

5 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics 8 the lower rigid profile represents a measure for the overlap whereas the horizontally moving spring is a representation of the equivalent asperity which will deform during contact. This representation provides, in a simplified but effective way, for asperities to be present on both surfaces independently and includes the dynamics of making and breaking the individual contacts, which is decisive for friction force dynamics. With these ingredients at our disposal, it turns out to be a relatively simple matter to assemble all the foregoing aspects into a single model whereby the three mechanisms are applied to a large population of individual (idealized) asperities that vary randomly in the time of sliding. 4. Formulation of the model-contact scenario with the friction mechanisms The contact surfaces of two blocks rubbing against each other (see figure 2A) can be represented by a flexible surface containing all the possible equivalent asperity contacts, each with its own equivalent stiffness, mass and shape depending on the characteristics of the two corresponding interlocking asperities. Each possible equivalent asperity contact has its own individual rigid, shaped lower surface. This representation allows determining whether or not an equivalent asperity will be active. Since we shall be considering a large number of such random asperity contacts for any simulation run, we find it justifiable to simplify the behaviour of each (generic) single asperity without jeopardising the global (emergent) behaviour. Figure 2B depicts the life cycle of one such equivalent asperity, where it is assumed that the upper surface is moving from left to right with respect to the fixed lower surface. Topographical characteristics are assigned to both surfaces. The equivalent characteristics of the two interacting asperities (namely stiffness, mass, compression and adhesion) are lumped into one point (), for simplicity of treatment. This point (figure 2B) is initially moving freely (i), until it touches the lower rigid surface (ii), after sticking to and then slipping over the lower profile it breaks completely loose from the lower profile (iii). In case (ii) the asperity is said to be in an active state, for the other cases the asperity is said to be inactive. (This may be reminiscent of the Tomlinson Prandtl atomic model, except that it accounts for creep, adhesion and load-carrying, which prove essential in revealing friction force dynamics). In this analysis, we have ignored the possible vibrations of a contacting asperity, deeming it for the time being to be of secondary importance. From the moment the asperity becomes active, it will begin to follow the profile of the lower surface, by deforming normally & and tangentially ; resulting in a normal and tangential force. The normal force, F n ðtþ; is given by k n &ðtþfðþ; where k n is the normal stiffness and f 1 is a weighting function; and the tangential force is given by F t ðtþ ¼k t ðtþ; where k t is the tangential stiffness. The maximum tangential force an asperity can sustain, before slipping, equals the adhesion force: F ðtþ ¼ðtÞk n &ðtþ: Here, the expedient local friction coefficient ðtþ is function of the contact time, owing to normal creep. This behaviour can be deduced from the static friction versus dwell-time relation. Coulomb in 178 [28] was probably the first to explore this time dependence; he reported a power-law increase of static friction with time of contact. As a rule, static friction F s grows with dwell time. An overview of the normal load force (A) (i) (ii) (iii) spring force (iia) (iib) x z k n k t k t 2 2 spring extension (iii) (C) ζ ξ w λ w αw (B) Figure 2. Figure A shows a general contact between two objects. Figure B shows the life cycle of one asperity contact: (i) no contact, (ii) contact, (iii) loss of contact. Figure C shows the spring force behaviour as a function of the spring extension during a life cycle of an asperity contact: (ii a) during stick, (ii b) x during slip, and (iii) when loosing contact.

6 86 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics various formulae for such dependence, as found or postulated by various authors, is given by Gitis and Volpe [24] (see also a list thereof in Appendix I). While any appropriate relation may be used according to the situation at hand, we have chosen exponential saturation for simplicity of interpretation: ðtþ ¼ 0 þð 1 0 Þ 1 exp t t 0 ð1þ with t 0 the time instance when contact occurs, 0 and 1 the local friction coefficients at initial and infinite contact time, and a characteristic creep time constant. (Note that not all postulated formulae lead to saturation of the friction force in time, but they all lead to saturation in the rate of increase.) Thus, depending on the relative values of ; k n ; k t and the relative topography, the asperity tip () will initially stick to the lower profile then slip on the profile and, finally, break completely loose from the profile. Assuming, without loss of generality, linear tangential (and normal) elasticity of the top asperity, the force increases initially linearly with the displacement until slip occurs (in the mean time keeps increasing with contact time). When the asperity tip has fully traversed the bottom surface, it will break loose, vibrate (tangentially and normally) and thereby dissipate (part of) its elastic and inertial energy, by internal hysteresis, until it comes to rest or comes in contact with the next bottom asperity. In the version of the model that will follow, we assume that all of the asperity energy is lost. This assumption is evidently valid for low sliding speeds and relatively large separation of consecutive asperities. (With some additional programming effort, the general case can also be considered, however at the expense of unduly complicating the picture). If the motion reverses during interlocking of the asperity (case (ii) of figure 2B), hysteresis loops will be formed due to alternating stick and slip phases as shown in figure 3 (the local friction coefficient being assumed here constant for clarity). Let us here make the following important remark. It should be clear from the above that the friction mechanisms and the contact scenario are valid for a very broad range of length and time scales. If we add to that fact that such a process can, in fact, take place anywhere on the surfaces or in the bulk materials, the self-similar, fractal nature of friction is revealed.. Simplified application to a complete surface Referring to figure 2 the contact is modelled as that taking place between one nominally flat object, covered by a population of mutually independent elastic point asperities having randomly chosen constant heights and masses, z and m; and a given distribution of stiffness values, k; (in the normal and tangential directions), and a rigid surface having a square wave profile (dashed line), for each asperity, with a constant height but randomly chosen wavelengths, w (that are fixed only during the life time of a single asperity contact, when contact is lost a new randomly chosen wavelength is valid). The mean ratio between the characteristic length of non-activity and activity is represented by a topography parameter : The surface profiles of the two objects will therefore be described by the parameter together with the distribution functions used for z ( z with a mean value of z 0 ) and w ( w with a mean value of w 0 ). The choice of a square wave asperity profile has been made in order to simplify calculation while retaining the essential feature of the problem: since the point asperities have random height distributions and are assumed to be independent of one another, the local form of the bottom asperities is reckoned to have only a negligible effect on the global behaviour especially when a large population is considered. In other words, we consider an averaging of the behaviour of each asperity. Without loss of generality & can be chosen equal to z if there is contact, otherwise & equals 0 (Bjo rklund showed that neither the type of height distribution is critical for a qualitative characterization of presliding behaviour). The ðtþ curve is chosen to be the same for each asperity. As may be apparent in figure 2B, the asperity will tilt so that its load-carrying capacity will also vary in function of the relative position of its tip. As a last simplification, the normal force ( ¼ asperity load capacity) is assumed to be equal to k n z if the base of an active asperity is within the region ½0; ð1 þ ÞwŠ of the beginning of the active square wave profile, where 0: Otherwise the normal force equals 0. (In a more refined treatment, this can be made a continuous function of the relative asperity position.) 6. Basic calculations The proposed model uses an energy method to determine the friction force at each time instance, viz. by calculating for each asperity the work done between two consecutive time steps. Dividing the calculated work by the corresponding relative displacement between the two objects results in the instantaneous tangential force. This implementation not only overcomes the problem of excessively small time steps but gives also better insight into the friction phenomenon. Figure 4 explains this way of calculation. The two full lines show the force characteristics: (i) a linear spring characteristic as a function of the tangential deformation of the asperity F k ðxþ (stick) and (ii) the maximum adhesion force which is a function of the contact time F ðtþ (slip). At time instants k and k þ 1 the relative displacement between the two objects equals x k and x kþ1 ; respectively. At time step k the tangential force has not yet reached the maximal adhesion force and the asperity remains in the

7 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics 87 displacement a b c d e f g h i time force h i b c e d force a displacement g f time Figure 3. The left figure show the displacement and tangential force as a function of time for one asperity contact. The right figure shows the tangential force as a function of the displacement. During the time instances a b; c f; and g h the asperity tip sticks, during the other time intervals the tip slips. The hatched area represents the hysteresis losses. stick phase until the displacement equals x b (diagonal hatching equals the work done). From then on the asperity slips and consumes plastic energy (horizontal hatching). At a displacement x c the asperity tip loses contact and the elastic energy (vertical hatching) as well as the inertial energy ð¼ mž 2 =2Þ are dissipated due to the vibration of the asperity. Between the displacement x c and x kþ1 there is no contact thus no work is consumed. The total work needed between time steps k and k þ 1is thus the diagonal and horizontal hatched area in figure 4 plus the inertial energy. When the micro-friction coefficient ðtþ ¼ 0 ¼ const; the mean friction force will be given by: F t ¼ 1 þ 1 0 k n z 0 mž 2 0 k n z 0 þ 2 k t w k n z 0 w 0 1 þ where the first term, in the bracketed expression, is due to slip, the second to stick and the last to inertia of the asperity. This gives us an idea about the relative importance of each mechanism, which will depend on the used materials, surface properties and surface texture. F F k (x) F µ (t) tangential force displacement x x b x a x k c x k+1 x Figure 4. Basic asperity contact scenario. The friction force as a function of the relative displacement.

8 88 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics 7. Results and discussion For better analysis and presentation, we normalise the variables with respect to the parameters as follows. F n ðtþ ¼f 1 ðn; k n ; z ;;;xðtþþ! F nðþ ¼ ð1 þ ÞF n k n z 0 ð1 þ ÞN F t ðtþ!f t ðþ ¼ ð1 þ ÞF t 0 k n z 0 N ¼ f 2ðk t ; 1; z ; w ;;;x ðþ; m Þ with, N being the number of asperities and 1 ¼ 1 = 0 ; ¼ t= ; x ðþ ¼xðtÞ=w 0 ; ž ðþ ¼žðtÞ =w 0 ; k t ¼ k t w 0 = 0 k n z 0 ; m ¼ mw 0 = 0 k n z 0 2 : The simulation results are all expressed as a function of the normalized variables. Here, we look at the behaviour of the generic model for the different experimentally well-established characteristics of dry friction: (i) the pre-sliding hysteresis (friction force versus position), (ii) velocity weakening (friction force versus steady state velocity) highlighting another, less known, but very significant sliding characteristic, viz. variation of normal separation (lift-up of slider), and (iii) the dynamical behaviour of the friction force in the velocity during true sliding. Transition between presliding and gross sliding is also discussed. The parameters used in the simulations are: k t ¼ 200; 1 ¼ ;¼ 1;¼ 0:; m 0 (except in figure 8), N ¼ 2000 ;000: The distribution functions z and w are chosen as a uniform distributions between the values 0 and 2z 0, and 2w 0, respectively. (Other distributions will yield qualitatively similar results but not quantitatively.) We have chosen, on purpose, rather unrealistic values for the stiffness and adhesion parameters in order to exaggerate their effects for better visualisation. 8. Pre-sliding hysteresis At very small displacements, i.e., in the pre-sliding regime, different researchers found a displacement dependent friction force [9,29,30]. The displacement dependency is a hysteresis function with nonlocal memory. Rogers and Boothroyd [29], and Prajogo [9] showed that this relationship is quasi time (i.e. rate) independent. Figure shows the results obtained by the generic model. The position signal is chosen such that there are two inner loops inside the outer hysteresis loop. The resulting hysteresis curve is quasi independent of the time scale of the applied position signal. When an inner loop is closed, the curve of the outer loop is followed again, showing the nonlocal memory character of the hysteresis. The shape of the hysteresis function is determined by the distribution of the asperity heights z and the distribution of the square wave profile w : The results are also qualitatively in correspondence with the results of Bjo rklunds paper [] or with the Maxwell slip model [17] indicating that the latter are special cases of our model. The resulting hysteresis function is a weighted sum of the different individual hysteresis characteristics shown in figure Gross sliding When the tangential force has reached a certain threshold, gross sliding will ensue. This threshold will generally depend on the rate of applying the force and, therefore, on the mechanical system incorporating the rubbing contact. To circumvent this difficulty, we assume the input to the rubbing system to be an inexorable displacement, and consider the resulting friction force resulting thereof. The gross sliding regime is rich with diverse behavioural facets. We shall examine them one by one and conclude with the behaviour of a complete cycle incorporating pre-sliding as well as gross sliding. The first property is velocity weakening, namely decreasing friction force for increasing steady state velocity, is a similar type of behaviour as the Stribeck curve for lubricated friction (and is often confused with it). Figure 6 gives the typical behaviour predicted by our model. An obvious explanation for the velocity weakening is that for a higher sliding velocity, the slip phase will be reached earlier, thus resulting in a lower adhesion force. This effect is, however, magnified by another less obvious one, namely the lift-up effect; about which more is given in a separate paragraph below. The actual form of the friction-velocity curve is determined by the process parameters, namely, the tangential stiffness, the surface topography and the normal creep (i.e., the time evolution of adhesion), which are discussed in the next two paragraphs. A higher tangential asperity stiffness results in shorter stick time and lower elastic energy stored in the asperity, for the same constant velocity and, consequently, in lower adhesion and deformation work. The mean tangential force will therefore be generally higher for lower stiffness values, as is clearly seen in figure 6. For very low velocities the friction force is almost independent of the velocity, i.e., the curve posses a flat part. This phenomenon, which has a bearing on the uniformity or otherwise of kinetic friction, will occur if the saturation of the micro-friction coefficient is reached before the asperity tip begins to slip, for the majority of asperities. Using the exponential curve (equation (1)) and assuming that the saturation value 1 is reached at 3 ; the velocity curve will possess a flat part for

9 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics A b c c b e e B c f b x * [ ] 0.1 F * t [ ] d f d a a a τ [ ] a e x * [ ] C D F * t [ ] 0 F * t [ ] x * [ ] x * [ ] Figure. Simulation results for the generic model in pre-sliding. Figure A shows the reference position signal. Figures B, C, and D show the hysteresis curves for a position signal respectively 1,, 20 times faster than the reference position signal. velocities lower than ž ¼ 1=3k t : The corresponding value of the friction force can be shown to be F t 1ð1 þ 1=2k t Þ; having the value of 12. for our set of parameters. A lower tangential stiffness also means that the relative availability of an asperity for load carrying is less (lift-up effect again), which enhances the aforementioned effect (see further below). The time evolution of adhesion, i.e. the ðtþ curve will evidently influence the form of the friction-velocity curve. The parameters 0 and 1 obviously determine the lower and upper bounds of the friction force. Restricting our attention to the exponential ðtþ form of equation (1), then the only parameter left is : Normalisation shows, however, that this parameter only scales the velocity. That is to say, a slowly increasing ðtþ leads to a slowly decreasing friction force in the velocity. This result has been confirmed experimentally by Rabinowicz [31] and, more recently, by Baumberger [23], in the form: D 0 static ¼ dynamic ðžþ ž where the (process characteristic) number D 0 is termed the creep-length.. The lift-up effect In the foregoing, we have alluded to the fact that the apparent load carrying capacity of an asperity increases with the sliding speed (figure 6). This is so because, owing to decreasing adhesion influence with speed, a single asperity will be more active per unit displacement, or more available for load carrying. This situation may be likened to trying to walk on the surface of a sticky quicksand: if you cannot free you feet in time to take the next step you will just sink down. Consequently, when the normal load is constant, for a given slider, then the number of contacting asperities will decrease with increasing sliding speed, i.e., the slider will lift up, just as in hydrodynamic lubrication. This behaviour, which has been observed by several researchers [7,32], emerged rather unexpectedly as a result of our problem formulation. Several explanations have been offered in the past, notably (i) Amontons classical, but F * t [ ] v * [ ] F * n [ ] v * [ ] F * t /F* n [ ] v * [ ] Figure 6. The influence of the tangential stiffness. The figures show from left to right the normalized tangential force, the normalized normal force, and the ratio of the two forces as a function of the normalized velocity. The dotted, full, and dashed line correspond to a normalized tangential stiffness equal to 00, 200, and 20, respectively.

10 90 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics F t * [ ] v * [ ] Figure 7. The normalized friction force as a function of the normalized velocity for non-steady-state velocities. The dots represent the steadystate curve. The full, dotted, and dashed line show the time lag curves for different frequencies (0., 0.2, and 0.1, respectively). controversial hypothesis that the asperities on the mating surfaces have to surmount each other (the mean asperity slope representing then the coefficient of friction), and (ii) Tolstoi s questionable hypothesis that the effect be owing to induced normal vibrations coupled with the non-linear asperity stiffness behaviour. Our model presents yet another, and fresh view of the phenomenon, namely that it is a direct consequence of adhesion, deformation and creep, which in fact resolves the contradiction of the first view without excluding the second. Increasing load capacity with sliding means also that the inertia of the slider will play a role in the friction force dynamics; a view that confirms Baumberger s [23] experimental findings that friction is machine dependent. Rice and Ruina [6] developed a general heuristic rateand state-dependent friction law based on experiments where there is a suddenly imposed step change in the velocity. Such experiments suggest that the friction force response to this velocity step be a combination of an instantaneous increase with the velocity coupled with a first-order-like decay with the evolving state. The first factor, namely that the friction force increases simultaneously with the suddenly imposed increase of velocity, corresponds to the asperity inertia (or the viscous ) effect. 2 The second factor results in a non-linear lag response. Our model is able to simulate this behaviour (figure 8) and show that the time lag can be approximated as a first order system. However, the time constant depends not only on the velocity step, but also on the initial steady state velocity. 11. Sliding friction force dynamics Frictional lag, also called hysteresis in velocity or frictional memory, is a well-known effect in lubricated friction [14], where the physical process giving rise to it appears to relate to the time required to modify the lubricant film thickness. The same effect has been observed in dry friction experiments [23] where no form of lubrication is used. Figure 7 shows the friction force as a function of the velocity using the generic model. The applied velocity signal is sinusoidal, of different frequencies, plus a constant offset in order to ensure a positive velocity signal at all times. The simulations show that the dynamic friction curves lie above the steady-state curve for acceleration, and under it for deceleration. The area enclosed by the loop increases with the frequency. 12. Transition behaviour Another important aspect of friction is the behaviour at transition from pre-sliding regime to sliding regime and vice versa, which most continuum models, such as the rate-state model, are not able to deal with. This transition behaviour can be investigated by applying a 2 Note that the origin of viscosity, e.g., according to molecular gas theory, lies at the momentum transport losses between adjacent layers of the fluid that are moving relative to each other. In other words, it is the mass of molecules that leads to viscous effect. Our model shows that, in a dry contact, frictional resistance that increases with the velocity may be generated by asperity inertia effects. We could have also included visco-elastic losses in the bulk of the asperities, but these would in their turn be caused by the inertia effects of the molecules in the material.

11 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics 91 7 v * = 0. v * = 1.2 v * = F t * [ ] τ [ ] Figure 8. The normalized friction force as a function of the normalized time for two different velocity step sizes ðm ¼ Þ: periodic velocity input to the system. Figure 9 shows the normalized tangential force as a function of the normalized sinusoidal velocity input. Not only frictional lag is observed, but more importantly, the crossover from positive to negative velocities and vice versa, passing through pre-sliding. When the velocity goes through zero, the direction of motion is reversed and all the active asperities will be relaxed and reloaded in the new direction of displacement. The dots on figure 9 represent the friction force for steady-state velocity. As in the case of frictional lag, the steady state curve lies between the curves of acceleration and deceleration. The results obtained by the generic model have the same qualitative behaviour as experimental results reported in [33]. Note that the great diversity observed in this type of behaviour depends not only on the process parameters but also on the velocity and its time evolution. This is yet another topic that is of great importance in friction modelling and on which other existing models have very little, if anything at all, to say. 13. And further The model we have presented is evidently open to further refinement and extension in many obvious, and F t * [ ] Figure 9. The normalized friction force as a function of the normalized time for a sinusoidal velocity signal. v * [ ]

12 92 F. Al-Bender et al./a novel generic model at asperity level for dry friction force dynamics some less obvious ways. In the first instance, the coupling of stiffness, mass and geometrical properties of real surfaces may be considered, based on stochastic/ fractal analysis; (likewise, the consideration of interaction of neighbouring asperities). This question is also related to how we can convert basic tribological and topographical data, of given real rubbing surfaces, into equivalent input to the model; we have made no attempt here to address this. In the second instance, extension to high velocity or asperity-inertia dominated regime should pose no serious problems. Here, preliminary analysis shows us that the model is capable of simulating all kinds of viscous effects so that it can be adapted to thin-film and non-newtonian rheology (where we think that there is a certain gap that can be filled); indeed, the model has shown that dry friction dynamics is similar in all respects to lubricated friction and that all extant continuum models of friction can be seen as special cases of this model. The model answers thus the basic questions about friction force dynamics and poses others. We expect the model to help us explore yet more frictional properties, e.g., the influence of the slider dynamics (inertia) on the friction force dynamics. So far, ad hoc exploratory tuning of the model parameters to quantitative experimental results (not reported here) has proven viable. One thing we are sure of, however; this model formulation may open a window on the understanding and quantification of friction and its dynamics. 14. Conclusions This paper discussed a theoretical model for dry friction force dynamics based on all well-known different aspects of friction mechanisms (deformation of asperities, adhesion theory, hysteresis losses due to internal friction, normal creep,...). The main contribution of this paper is threefold: (i) it unifies all the different theories into one generic model and (ii) this model can account, despite the gross simplifications it contains, very clearly and effectively for the known (low velocity, wear-less, dry) friction force dynamics observed and measured at macroscopic level by different researchers. (iii) Based on this model the relevance (namely, the relative contribution) of the different friction mechanisms can be investigated. Appendix I This appendix shows the different relations between the static friction and the contact time experimentally found by different researchers during the past decades. The shape and parameters of the functions are materialdependent. These functions can also be used in the analysis of the model described in this paper. However, the global results of the paper will remain valid, because all the functions have an increasing value as a function of the contact time. For more information about these functions see reference [24]. Exponential law: Power law: F ¼ F 1 ðf 1 F 0 Þ expð C 1 ðt stick Þ C 2 Þ F ¼ F 1 ½1 expð C 1 ðt stick Þ C 2 ÞŠ C 3 : Logarithmic law: t stick F ¼ F 1 ðf 1 F 0 Þ t stick þ C 1 F ¼ F k þ C 1 ðt stick Þ C 2 F 1 F ¼ F 1 =F 0 1 ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 t stick þ 1 þ 1Þ: F ¼ F 1 ðf 1 F 0 Þ logðc 1 t stick þ 1Þ: ½34Š ½3Š ½36Š ½37Š ½38Š ½39Š Acknowledgments The authors wish to thank Professors H. Van Brussel and J. De Schutter for their support and encouragement. The first author wishes to acknowledge the partial financial support of the Volkswagenstiftung, under Grant no. I/ The second author wishes to acknowledge the financial support of the K. U. Leuven s Concerted Research Action GOA/99/04. The scientific responsibility is assumed by its authors References [1] H. Spikes, Tribol. Int 34(12) (2001) 789. [2] B. Feeny, A. Guran, N. Hinrichs and K. Popp, Appl. Mech. Rev. 1() 321. [3] C.H. Holz, Nature 391 (1998) 37. [4] B.N.J. Persson, Sliding Friction: Physical Principles and Applications (Springer, Heidelberg, 1998). [] B. Armstrong-He` louvry, D. Dupont and C. Canudas de Wit, Automatica 30(7) (1994) 83. [6] J.R. Rice and A.L. Ruina, Trans. ASME, J. Appl. Mech. 0 (1983) 343. [7] J.T. Oden and J.A.C. Martins, Comp. Meth. Appl. Mech. Engng. 2 (198) 27. [8] P.R. Dahl, Proceedings of 6th Annual Symp. on Incremental Motion, Control Systems and Devices (1977), p. 49. [9] T. Prajogo, Experimental Study of Pre-rolling Friction for Motion-reversal Error Compensation on Machine Tool Drive Systems (Katholieke Universiteit Leuven, Leuven, 1999). [] S. Bjo rklund, Trans. ASME J. Tribol. 119 (1997) 726. [11] D.A. Hills, D. Nowell and A. Sackfield, Mechanics of Elastic Contacts (Butterworth Heinemann, 1993) [12] B.J. Lazan, Damping of Materials and Members in Structural Mechanics (Pergamon Press, London, 1968). [13] D.A. Haessig, Jr. and B. Friedland, Trans. ASME, J. Dyn. Syst. Meas. Cont. (113) (1991) 34. [14] D.P. Hess and A. Soom, J. Tribol. (112) (1990) 147.