Tribology in Machine Design

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You already recently rated this item. Your rating has been recorded. Write a review Rate this item: Preview this item Preview this item. Tribology in machine design Author: T A Stolarski Publisher: Oxford [England] ; Boston: English View all editions and formats Summary:.

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Add a review and share your thoughts with other readers. Similar Items Related Subjects: Linked Data More info about Linked Data. Tribology in Machine Design. It shows how algorithms developed from the basic principles of tribology can be used in a range of practical applications within mechanical devices and systems. The computer offers today's designer the possibility of greater stringency of design analysis.

Dr Stolarski explains the procedures and techniques that allow this to be exploited to the full. This is a particularly practical and comprehensive reference source book for the practising design engineer and researcher. It will also find an essential place in libraries catering for engineering students on degree courses in universities and polytechnics. Important surface topography parameters are: Generally speaking, the behaviour of metals in contact is determined by: Depending on the deformation mode within the contact, its real area can be estimated from: The introduction of an additional tangential load produces a phenomenon called junction growth which is responsible for a significant increase in the asperity contact areas.

The magnitude of the junction growth of metallic contact can be estimated from the expression where CY z 9 for metals. In the case of organic polymers, additional factors, such as viscoelastic and viscoplastic effects and relaxation phenomena, must be taken into account when analysing contact problems. Friction due to adhesion One of the most important components of friction originates from the formation and rupture of interfacial adhesive bonds. Extensive theoretical and experimental studies have been undertaken to explain the nature of adhesive interaction, especially in the case of clean metallic surfaces.

The main emphasis was on the electronic structure of the bodies in frictional contact. From a theoretical point of view, attractive forces within the contact zone include all those forces which contribute to the cohesive strength of a solid, such as the metallic, covalent and ionic short-range forces as well as the secondary van der Waals bonds which are classified as long-range forces.

An illustration of a short-range force in action provides two pieces of clean gold in contact and forming metallic bonds over the regions of intimate contact. The interface will have the strength of a bulk gold.

In contacts formed by organic polymers and elastomers, long-range van der Waals forces operate. It is justifiable to say that interfacial adhesion is as natural as the cohesion which determines the bulk strength of materials. The adhesion component of friction is usually given as: In the case ofclean metals, where the junction growth is most likely to take place, the adhesion component of friction may increase to about The presence of any type of lubricant disrupting the formation of the adhesive junction can dramatically reduce the magnitude of the adhesion component of friction.

This simple model can be supplemented by the surface energy of the contacting bodies. Then, the friction coefficient is given by see Fig. Recent progress in fracture mechanics allows us to consider the fracture of an adhesive junction as a mode of failure due to crack propagation where o , , is the interfacial tensile strength, 6, is the critical crack opening displacement, n is the work-hardening factor and H is the hardness.

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It is important to remember that such parameters as the interfacial shear strength or the surface energy characterize a given pair of materials in contact rather than the single components involved. Friction due to ploughing bl Figure 2. The asperities on the harder surface may penetrate into the softer surface and produce grooves on it, if there is relative motion.

Because of ploughing a certain force is required to maintain motion. In certain circumstances this force may constitute a major component of the overall frictional force observed. There are two basic reasons for ploughing, namely, ploughing by surface asperities and ploughing by hard wear particles present in the contact zone Fig. The case ofploughing by the hard conical asperity is shown in Fig. Asperities on engineering surfaces seldom have an effective slope, given by O, exceeding 5 to 6; it follows, therefore, that the friction coefficient, according to eqn 2.

This is, of course, too low a value, mainly because the piling up of the material ahead of the moving asperity is neglected. Ploughing of a brittle material is inevitably associated with micro-cracking and, therefore, a model of the ploughing process based on fracture mechanics is in place. Material properties such as fracture toughness, elastic modulus and hardness are used to estimate the Basic principles of tribology 17 coefficient of friction, which is given by where K t is the fracture toughness, E is the elastic modulus and H is the hardness.

The ploughing due to the presence of hard wear particles in the contact zone has received quite a lot of attention because of its practical importance. It was found that the frictional force produced by ploughing is very sensitive to the ratio of the radius of curvature of the particle to the depth of penetration. The formula for estimating the coefficient of friction in this case has the following form: The usual technique in analysing the deformation of the single surface asperity is the slip-line field theory for a rigid, perfectly plastic material.

A slip-line deformation model of friction, shown in Fig. Three distinct regions of plastically deformed material may develop and, in Fig. The flow shear stress of the material defines the maximum shear stress which can be developed in these regions. The proportion of load supported by the plastically deformed regions and related, in a complicated way, to the ratio of the hardness to the elastic modulus is an important parameter in this model.

For completely plastic asperity contact and an asperity slope of45", the coefficient of friction is 1. It decreases to 0. Another approach to this problem is to assume that the frictional work performed is equal to the work of the plastic deformation during steadystate sliding.

This energy-based plastic deformation model of friction gives the following expression for the coefficient of friction: Energy dissipation during friction In a practical engineering situation all the friction mechanisms. In general, frictional work is dissipated at two different locations within the contact zone.

The first location is the interfacial region characterized by high rates of energy dissipation and usually associated with an adhesion model of friction. The other one involves the bulk of the body and the larger volume of the material subjected to deformations. Because of that, the rates of energy dissipation are much lower.

Energy dissipation during ploughing and asperity deformations takes place in this second location. It should be pointed out, however, that the distinction of two locations being completely independent of one another is artificial and serves the purpose of simplification of a very complex problem. Friction under complex motion conditions I I 1- - - - - A - I chemical degradation 7 T y p e of wear and their mechanisms 19 is not only a function of the usual variables, such as load, contact area diameter and sliding velocity, but also of the angular velocity.

Furthermore, there is an additional force orthogonal to the direction of linear motion. Assuming that the slip at the point within the circular area of contact is opposed by simple Coulomb friction, the plate will exert a force 7 dA in the direction of the velocity of the plate relative to the pin at the point under consideration. To find the components of the total frictional force in the x and y directions it is necessary to sum the frictional force vectors, 7 dA, over the entire contact area A.

The integrals for the components of the total frictional force are elliptical and must be evaluated numerically or converted into tabulated form. Friction and wear share one common feature, that is, complexity. It is customary to divide wear occurring in engineering practice into four broad general classes, namely: Wear is usually associated with the loss ofmaterial from contracting bodies in relative motion.

It is controlled by the properties of the material, the environmental and operating conditions and the geometry of the contacting bodies. As an additional factor influencing the wear of some materials, especially certain organic polymers, the kinematic of relative motion within the contact zone should also be mentioned. Two groups of wear mechanism can be identified; the first comprising those dominated by the mechanical behaviour of materials, and the second comprising those defined by the chemical nature of the materials. In almost every situation it is possible to identify the leading wear mechanism, which is usually determined by the mechanical properties and chemical stability of the material, temperature within the contact zone, and operating conditions.

For an adhesive junction to be formed, the interacting surfaces must be in intimate contact. A number of well-defined steps leading to the formation of adhesive-wear particles can be identified: The volume of material removed by the adhesive-wear process can be 20 Tribology in machine design estimated from the expression proposed by Archard where k is the wear coefficient, L is the sliding distance and His the hardness of the softer material in contact.

The wear coefficient is a function of various properties of the materials in contact. Its numerical value can be found in textbooks devoted entirely to tribology fundamentals. In the case of lubricated contacts, where wear is a real possibility, certain modifications to Archard's equation are necessary. The wear of lubricated contacts is discussed elsewhere in this chapter. While the formation of the adhesive junction is the result of interfacial adhesion taking place at the points of intimate contact between surface asperities, the failure mechanism of these junctions is not well defined.

There are reasons for thinking that fracture mechanics plays an important role in the adhesive junction failure mechanism. It is known that both adhesion and fracture are very sensitive to surface contamination and the environment, therefore, it is extremely difficult to find a relationship between the adhesive wear and bulk properties of a material. It is known, however, that the adhesive wear is influenced by the following parameters characterizing the bodies in contact: For example, hexagonal metals, in general, are more resistant to adhesive wear than either body-centred cubic or face-centred cubic metals.

Abrasive wear Abrasive wear is a very common and, at the same time, very serious type of wear. It arises when two interacting surfaces are in direct physical contact, and one of them is significantly harder than the other. Under the action of a normal load, the asperities on the harder surface penetrate the softer surface thus producing plastic deformations.

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When a tangential motion is introduced, the material is removed from the softer surface by the combined action of micro-ploughing and micro-cutting. In the situation depicted in Fig. The amount of material removed in this process can be estimated from the expression Figure 2. The simplified model takes only hardness into account as a material property. Its more advanced version includes toughness as recognition of the fact that fracture mechanics principles play an important role in the abrasion process. The rationale behind the refined model is to compare the strain that occurs during the asperity interaction with the critical strain at which crack propagation begins.

In the case of abrasive wear there is a close relationship between the material properties and the wear resistance, and in particular: The ability of the material to resist abrasive wear is influenced by the extent of work-hardening it can undergo, its ductility, strain distribution, crystal anisotropy and mechanical stability. The relative motion of the surfaces in contact is composed of varying degrees of pure rolling and sliding.

When the loads are not negligible, continued load cycling eventually leads to failure of the material at the contacting surfaces. The failure is attributed to multiple reversals of the contact stress field, and is therefore classified as a fatigue failure. Fatigue wear is especially associated with rolling contacts because of the cycling nature of the load. In sliding contacts, however, the asperities are also subjected to cyclic stressing, which leads to stress concentration effects and the generation and propagation of cracks.

This is schematically shown in Fig. A number ofsteps leading to the generation of wear particles can be identified. A number of possible mechanisms describing crack initiation and propagation can be proposed using postulates of the dislocation theory. Analytical 22 Tribology in machine design models of fatigue wear usually include the concept of fatigue failure and also of simple plastic deformation failure, which could be regarded as low-cycle fatigue or fatigue in one loading cycle.

Theories for the fatigue-life prediction of rolling metallic contacts are of long standing. In their classical form, they attribute fatigue failure to subsurface imperfections in the material and they predict life as a function of the Hertz stress field, disregarding traction. In order to interpret the effects of metal variables in contact and to include surface topography and appreciable sliding effects, the classical rolling contact fatigue models have been expanded and modified. It should be mentioned that, taking into account the plastic-elastic stress fields in the subsurface regions of the sliding asperity contacts and the possibility of dislocation interactions, wear by delamination could be envisaged.

Wear due to chemical reactions It is now accepted that the friction process itself can initiate a chemical reaction within the contact zone. Unlike surface fatigue and abrasion, which are mainly controlled by stress interactions and deformation properties, wear resulting from chemical reactions induced by friction is influenced mainly by the environment and its active interaction with the materials in contact.

There is a well-defined sequence of events leading to the creation of wear particles Fig. At the beginning, the surfaces in contact react with the environment, creating reaction products which are deposited on the surfaces. The second step involves the removal of the reaction products due to crack formation and abrasion.

In this way, a parent material is again exposed to environmental attack. The friction process itself can lead to thermal and mechanical activation of the surface layers inducing the following changes: As a result of that the formation of the reaction product is substantially accelerated; ii increased brittleness resulting from heavy work-hardening.

The model, given by eqn 2. Sliding contact between surface asperities The problem of relating friction to surface topography in most cases reduces to the determination of the real area of contact and studying the mechanism of mating micro-contacts. The relationship of the frictional force to the normal load and the contact area is a classical problem in tribology. The adhesion theory of friction explains friction in terms of the formation of adhesive junctions by interacting asperities and their subsequent shearing.

This argument leads to the conclusion that the friction coefficient, given by the ratio of the shear strength of the interface to the normal pressure, is a constant of an approximate value of 0. This is because, for perfect adhesion, the mean pressure is approximately equal to the hardness and the shear strength is usually taken as of the hardness. This value is rather low compared with those observed in practical situations. The controlling factor of this apparent discrepancy seems to be the type or class of an adhesive junction formed by the contacting surface asperities.

Any attempt to estimate the normal and frictional forces, carried by a pair of rough surfaces in sliding contact, is primarily dependent on the behaviour of the individual junctions. Knowing the statistical properties of a rough surface and the failure mechanism operating at any junction, an estimate of the forces in question may be made. The case of sliding asperity contact is a rather different one. The practical way of approaching the required solution is to consider the contact to be of a quasi-static nature.

In the case of exceptionally smooth surfaces the deformation of contacting asperities may be purely elastic, but for most engineering surfaces the contacts are plastically deformed. Depending on whether there is some adhesion in the contact or not, it is possible to introduce the concept of two further types of junctions, namely, welded junctions and non-welded junctions. These two types of junctions can be defined in terms of a stress ratio, P, which is given by the ratio of, s, the shear strength of the junction to, k, the shear strength of the weaker material in contact 24 Tribology in machine design For welded junctions, the stress ratio is i.

For non-welded junctions, the stress ratio is A welded junction will have adhesion, i. On the other hand, in the case of a non-welded junction, adhesive forces will be less important. For any case, if the actual contact area is A, then the total shear force is where 0 and the area of contact is given by Here w is the geometrical interference between the two spheres, and E' is given by the relation where El, E2 and v,, v2 are the Young moduli and the Poisson ratios for the two materials.

The geometrical interference, w, which equals the normal compression of the contacting hemispheres is given by Basic principles of tribology 25 where d is the distance between the centres of the two hemispheres in contact and x denotes the position of the moving hemisphere. By substitution of eqn 2.

A limiting value of the geometrical interference w can be estimated for the initiation of plastic flow. According t o the Hertz theory, the maximum contact pressure occurs at the centre of the contact spot and is given by The maximum shear stress occurs inside the material at a depth of approximately half the radius of the contact area and is equal t o about 0. Thus Substituting P and A from eqns 2. The foregoing equation gives the value of geometrical interference, w, for the initiation of plastic flow.

F o r a fully plastic junction o r a noticeable plastic flow, w will be rather greater than the value g v e n by the previous relation. An approximate solution for normal and shear stresses for the plastic contacts can be determined through slip-line theory, where the material is assumed to be rigid-plastic and nonstrain hardening. However, in order to make the analysis feasible, the Green's plane-strain solution for two wedge-shaped asperities in contact is usually used.

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Plastic deformation is allowed in the softer material, and the equivalent junction angle u is determined by geometry. Quasi-static sliding is assumed and the solution proposed by Green is used at any time of the junction life. Thus V and H may be determined as a function of the position of the moving asperity if all the necessary angles are determined by geometry.

In this section the probability of asperity contact for a given lubricant film of thickness h is examined. The starting point is the knowledge of asperity height distributions. It has been shown that most machined surfaces have nearly Gaussian distribution, which is quite important because it makes the mathematical characterization of the surfaces much more tenable. Thus if x is the variable of the height distribution of the surface contour, shown in Fig. Therefore, the probability density function f x may be expressed as The probability that the variable x i will not exceed a specific value X can be expressed as The mean or expected value x of a continuous surface variable xi may be expressed as The variance can be defined as where a is equal to the square root of the variance and can be defined as the standard deviation of x.

It is possible to establish the statistical relationship between the surface height contours and the peak heights for various surface finishes by comparison with the comulative Gaussian probability distributions for surfaces and for peaks. Thus, the mean of the peak distribution can be expressed approximately as and the standard deviation of peak heights can be represented as when such measurements are available, or it can be approximated by When surface contours are Gaussian, their standard deviations can be 28 Tribology in machine design represented as or approximated by where r.

T o determine the statistical parameters, B, and Bd, cumulative frequency distributions of both asperities and peaks are required or, alternatively, the values of X,, osand o,. This information is readily available from the standard surface topography measurements.

Thus, the probability of interference between any two asperities is Thus - Ah is a new random variable that has a Gaussian distribution with a probability density function Basic principles of tribology 29 so that is the probability that Ah is negative, i. In the foregoing, his the mean value of the separation see Fig. The probability P Ah , where R and R2 are the r. If 1, is known, then In the case of heavily loaded contacts, plastic deformation of interacting asperities is very likely.

Therefore, it is desirable to determine the probability of plastic asperity contact. Although understanding of the various mechanisms of wear, as discussed earlier, is improving, no reliable and simple quantitative law comparable with that for friction has been evolved.

An innovative and rational design of sliding contacts for wear prevention can, therefore, only be achieved if a basic theoretical description of the wear phenomenon exists. In lubricated contacts, wear can only take place when the lambda ratio is less than 1. The predominant wear mechanism depends strongly on the environmental and operating conditions.

Tribology in Machine Design

Usually, more than one mechanism may be operating simultaneously in a given situation, but often the wear rate is controlled by a single dominating process. It is reasonable to assume, therefore, that any analytical model of wear for partially lubricated contacts should contain adequate expressions for calculating the volume of worn material resulting from the various modes of wear. Furthermore, it is essential, in the case of lubricated contacts, to realize that both the contacting asperities and the lubricating film contribute to supporting the load.

Thus, only the component of the total load, on the contact supported directly by the contacting asperities, contributes to the wear on the interacting surfaces.

First, let us consider the wear of partially lubricated contacts as a complex process consisting of various wear mechanisms. This involves setting up a compound equation of the type where V denotes the volume of worn material and the subscripts f, a,c and d refer to fatigue, adhesion, corrosion and abrasion, respectively. This not only recognizes the prevalence of mixed modes but also permits compensation for their interactions.

Because all the available mathematical models for primary wear assume clean components and a clean lubricating medium, there will therefore be no abrasion until wear particles have accumulated in the contact zone. Thus Vd becomes a function of the total wear V of uncertain form, but is probably a step function. It appears that if Vd is dominant in the wear process, it must overshadow all other terms in eqn 2. When Vd does not dominate eqn 2.

Thus it is known that corrosion 32 Tribology in machine design greatly accelerates fatigue, for example, by hydrogen embrittlement of iron, so that V f cwill tend to be large and positive. On the other hand, adhesion and fatigue rarely, if ever, coexist, and this is presumably because adhesive wear destroys the microcracks from which fatigue propagates. Hence, the wear volume V f adue to the interaction between fatigue and adhesion will always be zero.

Since adhesion and corrosion are dimensionally similar, it may be hoped that V a cand V f a cwill prove to be negligible. If this is so, only V f cneeds to be evaluated. By assuming that the lubricant is not corrosive and that the environment is not excessively humid, it is possible to simplify eqn 2. Repeated stressing through the thin adsorbed lubricant film existing on these micro-areas of contact would be expected to produce fatigue wear.

The block diagram of the model for evaluating the wear in lubricated contacts is shown in Fig. It is provided in order to give a graphical decision tree as to the steps that must be taken to establish the functional lubrication regimes within which the sliding contact is operating. This block diagram can be used as a basis for developing a computer program facilitating the evaluation of the wear.

Rheological lubrication regime As a first step in a calculating procedure the operating rheological lubrication regime must be determined. It can be examined by evaluating the viscosity parameter g, and the elasticity parameter g, where w is the normal load per unit width of the contact, R is the relative radius of curvature of the contacting surfaces, E ' is the effective elastic modulus, po is the lubricant viscosity at inlet conditions and V is the relative surface velocity. The range of hydrodynamic lubrication is expressed by eqns 2.

The range of the speed parameter g, and the load parameter g, for practical elastohydrodynamic lubrication must be limited to within the following range of inequalities: Functional lubrication regime In the hydrodynamic lubrication regime, the minimum film thickness for smooth surfaces can be calculated from the following formula: If the lambda ratio is larger than 3 it is usual to assume that the probability of the metal-metal asperity contact is insignificant and therefore no adhesive wear is possible.

Similarly, the lubricating film is thick enough to prevent fatigue failure of the rubbing surfaces. However, if A is less than 1. Thus, the change in the operating conditions of the contact should be seriously considered. However, in the mixed lubrication regime in which A is in the range 1.

When the range is between 0. This is particularly true in the case of the adhesive wear resulting from direct metal -metal asperity contacts. If lubricant molecules remain attached to Basic principles of tribology 35 the load-bearing surfaces, then the probability of forming an adhesive wear particle is reduced. At slow rate of approach the adsorbed molecules will have ample time to desorb, thus permitting direct metal- metal contact case b in Fig. At high rates of approach the time will be insufficient for desorption and metal -metal contact will be prevented case c in Fig. In physical terms, the fractional film defect, B, can be defined as a ratio of the number of sites on the friction surface unoccupied by lubricant molecules to the total number of sites on the friction surface, i.

The relationship between the fractional film defect and the ratio of the time for the asperity to travel a distance equivalent to the diameter of the adsorbed molecule, t,, and the average time that a molecule remains at a given surface site, t,, has the form Time t, is given by Values of Z - the diameter of a molecule in an adsorbed state - are not generally available, but some rough estimate of Z can be gven using the following expression: Here, to can be considered to a first approximation as the period of vibration of the adsorbed molecule. Again, to can be estimated using the following formula: Values of T, are readily available for pure compounds but for mixtures such as commercial oils they simply do not exist.

Taking into account the expressions discussed above, the final formula for the fractional film defect, P, has the form Equation 2. For a lubricant containing two components, additive a and base fluid b , the area Am arises from the spots originally occupied by both a and b. Thus, The fractional film defect for both a and b can be defined as where A, and A b represent the original areas covered by a and b , respectively. According to eqn 2. Basic principles of tribology 37 Thus, eqn 2.

Tribology in machine design

Load sharing in lubricated contacts The adhesive wear of lubricated contacts, and in particular lubricated concentrated contacts, is now considered. The solution of the problem is based on partial elastohydrodynamic lubrication theory. In this theory, both the contacting asperities and the lubricating film contribute to supporting the load. Thus where W, is the total load, W, is the load supported by the lubricating film and W is the load supported by the contacting asperities. Only part of the total load; namely W, can contribute to the adhesive wear.

In view of the experimental results this assumption seems to be justified. The ratio of lubricant pressure to total pressure is given by where iis the specific film thickness defined previously, h is the mean thickness of the film between two actual rough surfaces and ho is the film thickness with smooth surfaces. It should be remembered however that eqn 2.

For rougher surfaces, a more advanced theory is clearly required. The point of intersection between the appropriate curves of asperity pressure and film pressure determines the division of total load between the contacting asperities and the lubricating film. Basic principles of tribology 39 2. Adhesive wear equation Theoretically, the volume of adhesive wear should strictly be a function of the metal-metal contact area, A,, and the sliding distance. This hypothesis is central to the model of adhesive wear.

Thus, it can be written as where k, is a dimensionless constant specific to the rubbing materials and independent of any surface contaminants or lubricants. Expressing the real area of contact, A,, in terms of W and P and taking into account the concept of fractional surface film defect, P, eqn 2. Although it has been customary to employ the yield pressure, P, which is obtained under static loading, the value under sliding will be less because of the tangential stress. According to the criterion of plastic flow for a twodimensional body under combined normal and tangential stresses, yielding of the friction junction will follow the expression where P is now the flow pressure under combined stresses, S is the shear strength, P, is the flow pressure under static load and u may be taken as 3.

An exact theoretical solution for a three-dimensional friction junction is not known. In these circumstances however, the best approach is to assume the two-dimensional junction. From friction theory where F is the total frictional force. Thus and eqn 2. The values of W and P can be calculated from the equations presented and discussed earlier. Fatigue wear equation It is known that conforming and nonconforming surfaces can be lubricated hydrodynamically and that if the surfaces are smooth enough they will not touch. Wear is not then expected unless the loads are large enough to bring about failure by fatigue.

For real surface contact the point of maximum shear stress lies beneath the surface. The size of the region where flow occurs increases with load, and reaches the surface at about twice the load at which flow begins, if yielding does not modify the stresses. Thus, for a friction coefficient of 0. The existence of tensile stresses is important with respect to the fatigue wear of metals. The fact, that there is a range of loads under which plastic flow can occur without extending to the surface, implies that under such conditions, protective films such as the lubricant boundary layers will remain intact.

Thus, the obvious question is, how can wear occur when asperities are always separated by intact lubricant layers. The answer to this question appears to lie in the fact that some wear processes can occur in the presence of surface films. Surface films protect the substrate materials from damage in depth but they do not prevent subsurface deformation caused by repeated asperity contact. Each asperity contact is associated with a wave of deformation. Each cross-section of the rubbing surfaces is therefore successively subjected to compressive and tensile stresses.

Assuming that adhesive wear takes place in the metal-metal contact area, A,, it is logical to conclude that fatigue wear takes place on the remaining part, that is A,- A, , of the real contact area. Repeated stresses through the thin adsorbed lubricant film existing on these micro-areas are expected to cause fatigue wear. To calculate the amount of fatigue wear in a lubricated contact, an engineering wear model, developed at IBM, can be adopted.

The basic assumptions of the non-zero wear model are consistent with the Palmgren function, since the coefficient of friction is assumed to be constant for any given combination of materials irrespective of load and geometry. Thus the model has the correct dimensional relationship for fatigue wear. Non-zero wear is a change in the contour which is more marked than the surface finish. The basic measure of wear is the cross-sectional area, Q, of a scar taken in a plane perpendicular to the direction of motion. The model for non-zero wear is formulated on the assumption that wearcan be related to a certain portion, U, of the energy expanded in sliding and to the number N of passes, by means of a differential equation of the type For fatigue wear an equation can be developed from eqn 2.

For non-zero wear it is assumed that a certain portion of the energy expanded in sliding and used to create wear debris is proportional to zma,S. Integration of eqn 2. The manner in which such an expression is obtained for the pin-on-disc configuration is illustrated by a numerical example. The procedure for calculating non-zero wear is somewhat complicated because there is no simple algebraic expression available for relating lifetime to design parameters for the general case.

The development of the necessary expressions for the determination of suitable combinations of design parameters is a step-like procedure. The first step involves integration of the particular form of the differential equation of which eqn 2. This step results in a relationship between Q and the allowable total number L of sliding passes and usually involves parameters which depend on load, geometry and material properties. The second step is the determination of the dependence of the parameters on these properties. From these steps, expressions are derived to determine whether a given set of design parameters is satisfactory, and the values that certain parameters must assume so that the wear will be acceptable.

The system under consideration is shown in Fig. The radius, r, of the wear track is 75 mm. The yield point in shear of the steel is The load W o n the system is 10 N. The system is lubricated with n-hexadecane.

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