JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 1, pp. 3-35, Warsaw 2005 WEAR DEBRIS: A REVIEW OF PROPERTIES AND CONSTITUTIVE MODELS Alfred Zmitrowicz Institute of Fluid-Flow Machinery, Polish Academy of Sciences e-mail: azmit@imp.gda.pl Wear appears as gradual removal of a material from contacting and rubbing surfaces of solids during their relative sliding. The mechanism of wear involves formation of debris particles. The particles have small sizes and different shapes. The wear debris can be ”rolled over” into cy- lindrical, spherical and needle-like particles. Particles are detached from rubbing surfaces and they form amore or less continuous interfacial lay- er. They transmit forces,moments and displacements (translational and rotational) at the contact interface. The presence ofwear debris between sliding surfaces affects frictional and wear behaviour significantly. Con- stitutive relations characterize quasi-solid, quasi-fluid and granular-like behaviour ofwear particles. In the paper, two constitutivemodels of we- ar debris are discussed: (a) material continuum, (b) granular medium. The continuum models are formulated for a micropolar thermoelastic material, micropolar fluid and thermo-viscous fluid. Key words: contact mechanics, wear, friction, constitutive equations 1. Introduction The friction process of elements operating in contact conditions always in- volves heat generation and wear of their surfaces. Wear and frictional heat, apart from fatigue, fracture and corrosion are main factors which restrict a life time of machines and mechanical devices. Numerous machine component parts must be taken out of service not due to failure caused by exceedance of the limit stress, but due to wear manifested in removal of a material. Theore- tically, wear ofmachine component parts should not occur if their surfaces are separatedwith the aid of a lubricant film,but frommany reasons (e.g. frequent 4 A.Zmitrowicz starts, stops and reversals ofmotion of amachine, operations in dusty air con- ditions, etc.) breakdown of a lubricant film occurs, andwear cannot be avoid. Many machine components operate in conditions of partial fluid-lubrication (mixed lubrication), where an average lubricant film thickness is of the same order of an average surface roughness.Then, an applied load is carried partial- ly by a hydrodynamic action of the lubricant film and partially by contacts of surface asperities. In general, ”dry” andmarginally-lubricated contacts are as common as ”wet” contacts and also govern operating conditions of machines. Understanding and controlling the friction and wear places is of considerable practical importance in engineering. Phenomena of friction andwear are inse- parable. Wear in sliding systems is usually a very slow process, but it is very steady and continuous. Thefirstwear experimentswere carried out at thebeginning of 19th centu- ry.With the aid of a special test device, Hatchett (1803) investigated abrasion of gold, silver and copper coins. Rennie (1829) investigated wear of solid sur- faces of metals, wood and other materials. Contemporary wear investigations belong to fundamental tests onmechanical properties of engineeringmaterials. Quantitative prediction of wear is difficult. At present, there are a few very simple calculation procedures of wear, seeArchard (1953), Rabinowicz (1995). In this study, wear appears as gradual removal of a material from con- tacting and rubbing surfaces of solids during their relative motion (sliding). Normal pressure and sliding action are necessary for wear as they depend, first of all, on the rubbing process. Themechanism of wear involves formation and circulation of debris particles in any sliding system. Particles detached from solids circulate in the contact region and form amore or less continuous interfacial layer. The presence of a thin layer of wear debris separating two sliding bodies is an extremely important topic of the present analysis. When there is a layer of particles in the interface between contacting solids instead of a liquid, then themodelling situation requires newmethods and theoretical tools. The present study is devoted to phenomenological models of properties and constitutive relations of wear debris. 2. An outline of wear physics Many mechanical, physical and chemical phenomena are responsible for wear of materials. Several types of wear have been recognized, e.g.: adhesive, abrasive, contact fatigue, fretting, oxidation, corrosion, erosion. Inmachinery, wear occurs most frequently as an abrasive or contact fatigue process. Each Wear debris: a review of properties and constitutive models 5 wear type has a differentmechanism, cause and effect. For example, fretting is awear process of contacting solids occurring during oscillatory sliding of small amplitudes.A low amplitude of displacements typical in fretting is usually less than 30µm, and a low velocity amplitude is expressed in µm/s. Oscillatory tangential relativemovements arise fromvibrations or cyclic stressing of one of the components. In the caseof fretting,weardebris is trappedandaccumulated in the area between the oscillating surfaces. Wear of solids is usually treated as a mechanical process. However, oxi- dation, corrosion and other chemical processes are exceptions of this general rule. In some contacts, chemical reactions can play an important role. Fur- thermore, it can be supposed that mechanical energymay provide the energy for chemical reactions directly or indirectly by means of thermal energy (i.e. it may accelerate chemical reactions in solids). During dry, boundary andmixed lubricated sliding, wear particles are ge- nerated by a number of different physicalmechanisms, depending onmaterials and sliding conditions. The wear particles are generally detached mechanical- ly bymicro-stresses resulting from the applied load and relative sliding.Wear particles are detached from sliding solids by the following microscopic me- chanisms: micro-cutting of adhesive junctions between surfaces, mechanical failure of contacting asperities (i.e. non-elastic deformations, cutting, abra- sion, break off, fracture, fatigue of surface asperities etc.), surface spalling, plastic deformations of surfaces in a form of grooves and scratches, nucleation and propagation of surface and subsurface cracks and voids, oxidation, cor- rosion, chemical reactions, see Fig.1. Wear particles are also generated as a consequence of ploughing. Surfaces can be plowed by the wear particles, hard particles entrapped from the environment and by hard asperities of the coun- terface. Certain forms of surface damage do not occur slowly and continuously but may suddenly cause a gross disruption of rubbing surfaces, after which large wear particles are removed. Thismechanism is called scuffing or galling. The wear process can be examined in three stages. The first stage is con- cerned with debris generation and deals with various mechanisms of material removal fromsliding surfaces.The secondone is devoted to the evolution of de- bris inside contact (i.e. mechanical physical and chemical changes). The third stage refers to own behaviour of debris, which can be either eliminated out of the contact, or accumulated between sliding surfaces. Taking into account an amount of the removed material from solids, me- asures of wear have been formulated with respect to changes of the following quantities: mass, volume and linear dimensions of sliding bodies (Rabinowicz, 1995). 6 A.Zmitrowicz Fig. 1.Wear micro-mechanisms and possible ways to produce wear debris Mostwear observations are carried out indirectly (post factum). It hasma- ny disadvantages, i.e. the rubbing process must be stopped, wearing elements must be disassembled, and after that the effects of the wear process can be observed. All what can be done is to measure or weight the rubbing bodies before experiments, repeat this procedure after the experiment, and note the differences.Weighting is the simplest way of detecting wear. It gives the total amount of wear in the form of a single number, but the distribution of the removed material in the sliding surface is unknown. Formation of loose wear particles is of crucial importance in the wear pro- cess. In pin-on-disc test machines, as in many other friction and wear test devices, wear particles are observed inside and outside of the wear track, see Fig.2. In every-day life, one can easily observe wear debris during abrasion, e.g.: pencil drawingmarks on paper, a piece of chalk writing on a blackboard, a rubber eraser rubbing out pencil marks on paper. There is a large number of modern devices which only operate in such a way so that the friction and wear can be closely controlled. Wear cannot be eliminated completely, but it can be reduced to a high degree. The simplest methods of reduction of friction and wear are as follows: lubrication, forma- tion of sufficiently smooth surfaces,modification of near surface layers ofmetal components (e.g. achieved through surface treatments and/or coatings which increase surface hardness andwear-resistance). It can be supposed that some- Wear debris: a review of properties and constitutive models 7 Fig. 2.Wear traces observed in typical friction and wear test devices: (a) pin-on-flat surface, (b) pin-on-disc times considerable advantages canbe achievedbyoptimization of friction itself or by an optimal choice of structural, kinematical andmaterial parameters of the rubbing system. In general, consequences of wear are negative, however one should remember the advantages of manymethods for producing surfaces on manufactured objects exploiting the abrasion phenomenon (e.g. finishing operations). 3. Morphology and kinematics of wear debris Morphology of wear debris deals with its structural properties and it is characterized by: forms (shapes), dimensions (sizes), concentration (number), distribution. Furthermore, particles generated by friction and wear can be described by: colour, texture, etc. (see Myshkin et al., 2001). In some cases, the transfer of a material between contacting surfaces is obvious to a unaided eye because of associated colour changes. Interfacial films are seen as layers of a different colour than that of parent bodies. Wear debris has been observed and studied for a very long time. There are many research studies in the field of tribology about size, shape, number and nature of wear particles, see e.g. Finkin (1964), Crone (1967), Larsen-Badse (1968), Wilson and Eyre (1969), Scott et al. (1974), Bowen and Westcott 8 A.Zmitrowicz (1976), Samuels et al. (1980), Anderson (1982), Don andRigney (1985), Swan- son and Vetter (1985), Georges and Meille (1985), Murti and Philip (1987), Hunt (1993), Dowson et al. (1992, 1996), Lin et al. (1992). Therefore, a large body of empirical data has been collected so far. Both optical and scanningmicroscope studies onwear debris revealed that the particles did not have one particularmorphology.Generatedwear debris is of different shapes (e.g. flakes, chips, thin platelets, slices, filings, powder-like particles, etc.) and different sizes. There are various classifications of shapes of wear particles, e.g.: • A rough division of shapes of wear debris into two categories, flake and non-flake, can be made. The flake-type debris refers to a particle that has a relatively uniform thickness whereas non-flake includes such debris types as powder, plates ribbons, cylinders, spheres, irregular chunks and loose clusters. • Debris can roughly be divided into two groups: long fibrous debris and very thin platelet debris. Both types of particles appeared e.g. in lubri- cant samples. • There are following types of wear debris: sheet-like, roll-like, long rope- like, aggregated and granular particles. • Wear particles can be classified into several types according to their origin (their generation process) for instance: fatigue chunk particles, severe sliding particles, laminar particles. Morphology of wear debris can be defined with the aid of the following dimensions: length,width and thickness. The flake debris can be characterized by their outline, e.g. straight edges/irregular edges orby curvature radii of arcs and by apex angles of corners, etc. Royalance et al. (1992, 1994, 2000) used differentmorphological attributes of the particles. Quantitative data, obtained byusing image analysis techniqu- es, providedmeasurements of shape, size, edge detail, surface roughness. The edge detail was based on a curvature pattern derived for a particle periphery and on statistical analysis performed to establish mean values of dispersion, skewness Rsk and kurtosis Rku (these parameters are often used in topogra- phical analysis of rough surfaces). Two additional parameters, an aspect ratio and a roundness factor, were also applied. Fractals have been used to describemorphology of wear debris byBerthier et al. (1988), Kirk et al. (1991), Zhang et al. (1997) andWrona (2003). There have been attempts to automate the analysis of wear particles, see Peng et al. (1997), Xu and Luxmoore (1997), Podsiadlo et al. (1997), Peng and Kirk Wear debris: a review of properties and constitutive models 9 (1997, 1998). Laser scanning microscopy has been applied to obtain three- dimensional images of wear particles and to surface roughness analysis of a single wear particle by Peng et al. (1997). Several numerical parameters to describe profiles of wear debris, based on automated computer image analysis systems, were developed by Kirk et al. (1995), Stachowiak (1998), Podsiadlo andStachowiak (1998, 1999a,b, 2000), Stachowiak andPodsiadlo (1999), Peng andKirk (1999a,b). Geometrical (morphological) characterization of wear particles were made with the aid of neural networks (Umeda et al., 1998), fuzzy systems (Cheng and Yang, 1998), and other techniques (Unchung and Tichy, 2000). The particle shape yields qualitative information while the concentration and size distribution provide quantitative data. It is necessary to know, or to estimate, the maximum size of different kinds of particles. Obviously, the shape and size of debris differ for different materials and sliding conditions. For example, when two metal surfaces slide against each other under a load, wear debris is produced in the form of both macroscopic (size from a few to several micrometers) and microscopic (size from sub-micrometer to a few micrometers) particles. It was found that the number of microscopic wear particles generated during sliding can be estimated as an exponential function of the particle size. The number of wear particles and their size distribution were studied in detail by Rabinowicz (1995), Xuan and Cheng (1992), Mizumoto and Kato (1992), Hou et al. (1997). In order to visualize the wear debris layer, high wear rate materials were used byKohen et al. (1980) andGodet et al. (1991). Godet recorded, on a video tape, a dynamic contact in which solids were separated by layers of chalk, graphite, etc., of a few tenth of micrometers in thickness. Godet andPlay (1975) andGodet (1988) conducted experiments onnume- rousmaterial combinationswith includedpolymers (PMMA,PC, etc.), elasto- mers, metals (various steels, aluminum alloys, titanium alloys, copper, etc.), ceramics (Al2O3, SiC, etc.), chalk, glass, sapphire, carbon etc., under both continuous and reciprocatingmotion, and all proved that wear debris are pro- ducedquasi-immediately, then trapped in the contact for some timeandfinally eliminated after the next time interval. A small particle (e.g. 10−6m) cannot be instantly eliminated from a comparatively wide contact (e.g. 10−2m), for purely geometrical reasons, see Fig.3. Hou et al. (1997) measured interfacial layers in rolling-sliding contacts consisting of wear debris and contaminants. The size of most particles was of 1µm in magnitude, with some larger particles measuring about 10µm. 10 A.Zmitrowicz Fig. 3. Schematic view of a small particle at a comparatively wide contact area and rolls-like wear debris observed in a wear track The thickness of the agglomerated and compacted layer was about 20µm. Rabinowicz (1995) investigated wear debris in a sliding pair of copper on a low-carbon steel (unlubricated). He concluded that small particles are seen to be much numerous than large ones. Xuan and Cheng (1992) conducted dry sliding wear tests on a three-pin- on disc machine using stainless steel pins and disc. Wear debris was collected in a test chamber, measured and examined. It has been found that the total wear debris comprisesmacroscopic andmicroscopic particles. Particles of sizes ranging from 0.2µmto 12µmandaboveweremeasured.The created particles have a size distribution to be described by an exponential function. Estimated numbers of generated particles were: 10 million of size 0.2µm, 1 million of size 1µm, and 500 of size larger than 12µm. Xuan and Cheng (1992) found that the complete wear process from running-in to steady-state involved four distinguishable periods when considering the source and the mechanism for creating wear debris. Mizumoto and Kato (1992) used a pin-on-disc frictional apparatus in a series of experiments to observe wear debris. Measured particles larger than 0.2µmwere classified into 11 rangesof diameter.Kato (1990) observed changes of wear modes and formation of powder-like and flake-like wear particles in reciprocating sliding of a diamond pin on a silicon carbide disc. Freshly generated wear debris escape to the interface and they are pro- cessed further e.g. crushed into finer particles, compacted and agglomerated Wear debris: a review of properties and constitutive models 11 into large debris. Therefore, the evolution of particle morphology should be investigated. Wear particles undergo the following processes: they are heavily deformed, theparticles are fragmented to a small enough size (by repeatedpla- stic deformations and fracture), the particles are oxidized, coagulated together or agglomerated (due to adhesion forces), reattached to either of the contac- ting surfaces (i.e. they can be transferred to the counterface by mechanical interlocking andby adhesion). Agglomerated particles are formed progressive- ly, by agglomeration of small particles. Some of thewear particles are removed from wear tracks to form loose particles. Ejection of particles from both the contact andwear track (sliding path) is predominant inmost situations. Some- times, wear particles are removed from the sliding interface by being brushed off (case of lubricated or openmechanical devices). It is often observed that wear debris of flake and chip forms. etc. can be ”rolled over” into cylindrical, spherical and needle-like particles, see Zanoria et al. (1995). It takes part especially, when contacting bodies realize relative oscillatory motions, see Fig.3. For instance, large rolls are formedwhen using a rubber eraser. An interaction between debris and bodies can govern the fact that the debris can roll. The roll-like wear debris are created by rubbing of two bodies. Each roll is subjected to opposing tangential forces at its top and bottom surfaces. The torque resulting from these forces causes the debris to roll. The axes of roll particles are aligned perpendicularly to the direction of sliding, indicating that they were actually rolling on the wear track (or that they were generated by rolling in the contact), and suggesting that the particle circulation is common in the contact, see Fig.3. In experiments with a silicon ball on a silicon flat reported by Zanoria et al. (1995), roll lengths varied from 0.1 to 30µm and a roll diameter ranged from 0.1 to 1.4µm. Observations revealed that those rollers aremore or less compact agglome- rates offinerwearparticles.Very small grains are compactedandagglomerated (through physical and chemical effects, contact pressure, temperature and hu- mid environments) inwear debris showing cylindricalmorphology. Aggregates are subsequently rolled into dense cylindrical particles. The rolls can grow as snow balls by collecting more and more particles. The rolls, formed from ele- ments of each contacting body, show a lamellar or composite structure. An increase in load can entail either destruction or fragmentation of the roll or its flattening without rapture. A spherically shapedwear particle is a particulary intriguing type of wear debris, often observedand subject tomuch interest, seeRabinowicz (1977), Sa- muels et al. (1980), Davies (1986), Wang et al. (1986), Jin and Wang (1989). 12 A.Zmitrowicz Formation of spherical wear particles during a dry grinding process of car- bon steels by aluminum oxide (Al2O3) abrasive wheel was shown by Lu et al. (1992). An approximate percentage of spherical particles in the grinding swarf was about 50%. Spherical particles ranging in size from 1 to 80µmwere observed. Small (less than 1µm) spherical particles were also seen by some investigators. The rolls and spheres were obtained under a very wide range of running conditions, and for very different materials (steels, ceramics, polymers, etc.). The rolls are found on surfaces of silicon nitride, silicon carbide, alumina, yttrium barium, copper oxide, ceramic matrix composites and single-crystal silicon. There is a number of reports in the subject literature that wear of non-metallic solids produces debris in the shape of cylindrical rolls. In rubber, wear debris apparently develops as the rolls of rubber. Numerous experimental investigations and every-day praxis indicate for- mation of awear debris layer at the sliding interface (i.e. an intermediate layer between the sliding surfaces) practically immediately after the rubbingprocess starts. The surfaces slide on each other separated by the wear debris layer. In the case of the wear debris layer, friction occurs partially between the surfaces of contacting bodies, partially between the surfaces of each body and wear particles as well as inside the layer between the wear particles. Some com- ponent of the friction force originates from the rolling of particles over each other. The resistance to rolling (i.e. rolling friction) follows from rotations of a single particle with respect to neighbouring particles. The interfacial layer can be also formed by hard particles and contami- nants entrapped into the sliding system from the operating environment (e.g. airborne debris of dust and sand, combustion products such as fly ash, con- struction dirt, contaminant particles from the lubricant, corrosion scale, weld beads) or particles specially introduced into the sliding interface with respect to theirbeneficial role during the frictionprocess (e.g. typical layer-lattice solid lubricants such as carbon graphite powder, molybdenum disulphide, PTFE, etc.) or abrasive debris (diamond, silicone carbide, etc.) in material-finishing operations. At normal operating conditions of machines, a lubricant generates a ve- ry thin film which separates sliding surfaces. The lubricant film thickness is as follows: in elastohydrodynamic systems 0.1µm–1.0µm, in hydrodynamic journal bearings 1.0µm–25.0µm, in hydrostatic journal bearings 0.5µm– 100µm, in gears 0.05µm–0.5µm, in roller bearings 0.1µm–1.0µm. It is well-known that contamination of oils is a serious problem. Any lubricant Wear debris: a review of properties and constitutive models 13 sample may have considerable quantities of contaminant particles of various sizes andmaterials, see Nikas et al. (1998). The size of contaminant particles found in a lubricant oil closes in a ran- ge from sub-micrometer to 1000 micrometers. The wear particles which are transported by the lubricant can pass easily through dynamic clearances in operating components. The largest particles will cross the contact by inducing plastic deformations, the smallest particles may only induce elastic deforma- tions. The contamination andwear particles of the size that can reach the dy- namic clearance and interact with contacting surfaces damage these surfaces, generate new particles and result in a catastrophic failure. Hard contaminant particles in an oil can induce additional elastic properties of the oil film. Some researchers, who studiedmorphology ofwear particles, assumed that the size, shape, concentration andcompositionofweardebris can reveal impor- tant information about friction andwear processes that generate the particles, see Reda et al. (1975), Barwell (1983), Santanam (1983), Hawthorne (1991), Hwang et al. (1999), Cho andTichy (2000), Sherrington andHayhurst (2001). Debrismorphology can relate towear processes. Analysis of wear debriswhich is transported by the lubricant and subsequently captured during filtration is often applied for machine condition monitoring and fault diagnosis. 4. Mechanical and physical properties of the wear particles Physical properties of wear particles include composition, microstructure, density, thermal expansion, thermal conductivity, melting temperature, etc. Depending on a type of parent materials, different kinds of wear particles are created; there are particles formed by metals, metal oxides, alloys, plastics, ceramics, etc. In the case of plastics wearing out in contact withmetals, wear debris forms a mixture of metal and plastic particles. The material which was removed from surfaces is not in fact the same, either structurally or chemically, as the base material. Instead, it is a very fine-grainedmaterial whichmay be derived from both parts of the contacting system andmay include components from the environment as well. It is clear that the worn surface and debris interact with the environment to yield reac- tion products. Chemical properties of the third body layer were investigated byWirth et al. (1994). Friction and wear imply the existence of a thin, uniform and almost con- tinuous intermediate layer between sliding solids called sometimes the third body.Acomplete separationwith full debris layer is not a rare occurrence.The 14 A.Zmitrowicz third body is characterized by trapped particles changing their size, composi- tion andmorphology, and influencing in turn the particle detachment process. These particles will be active in a rubbing systemuntil they are removed from the contact. The debris trapped in the interface will deform when subjected to high compressive and shear stresses. Therefore, the wear debris layer can undergo compression and tension.The stiffness of the thirdbody can influence deformation of the rubbing system. A significant amount of aggregated debris can be present on the sliding surface. Since aggregated debris is not a homogeneous solid, its strengthunder aparticular loading condition is lower than the theoretical strengthof the same material. The interfacial layer differs both in microstructure and composition from the base material. The layer have certain characteristic properties which de- pend upon physical andmechanical properties of wear particles and the layer as a whole. Some properties of wear debris layer have been investigated by Sheasby (1983), Godet (1989), Berthier et al. (1989), Jiang et al. (1998), De- scartes andBerthier (2002). Perhaps, hardness was themost widely discussed mechanical property of wear debris. Mechanical behaviour of the third body, as it transports stresses and stra- ins, is complicated enough. If the third bodies are to be prominent in theoreti- calmodels of contactmechanics, theirmechanical propertiesmust be properly represented. Identification and understanding of the third body behaviour is difficult since it requires direct observations of the interface during the wear process. In general, themost adequate is to carry outmeasurements of mechanical properties of materials in real working conditions. Since, it is often not possi- ble, then typical strength tests should be done, i.e. strength with respect to: tension, compression, bending, shear, torsion, creep, fatigue, hardness. Such fundamental tests have not been undertaken yet for the wear debris layer. 5. Contribution of wear debris to load and displacement transmission Analysis of the contact problem of solid bodies generally requires deter- mination of stresses and strains within individual bodies, together with in- formation regarding distribution of displacements and stresses at the contact region. Figure 4 illustrates optical interference patterns representing stresses in a photoelastic model of rolling contact, Wuttkowski and Ioannides (1992). Wear debris: a review of properties and constitutive models 15 Three different contact conditions were investigated: (a) surfaces separated by lubricant film, (b) solid contaminant particles contained in the interface, (c) a single contaminant particle being rolled over. Experiments show that contaminant particles considerably change stress distributions near the con- tact surfaces. Therefore, examinations of wear debris are important because thewear debris gives rise to aunique stress pattern anddeformationbehaviour in sliding solids. Fig. 4. Optical interference patterns representing stresses in a photoelastic model of the rolling contact, (a) clean surfaces completely separated by the lubricant film, (b) solid contaminant particles contained in the lubricant film, (c) dent in the raceway caused by a contamination particle being rolled over, seeWuttkowski and Ioannides (1992) Godet suggested to model wear debris accumulated between sliding surfa- ces and separating the surfaces as a mediumwith a load carrying mechanism as occurs in fluid lubrication. In Godet’s opinion load carrying phenomena are not limited to fluid flow mechanics but that the arguments identical to those used in fluid analysis can be put forward in cases of rubbing surfaces, see Godet (1984), Colombé et al. (1984), Berthier et al. (1989, 1992), Denape et al. (2001). In a general case, the load carrying mechanism depends on fluid flow. By the effective load carrying capacity, we mean that the total normal load is entirely carried by transfer particles. This, for instance, implies that a pin (in pin-on-disc test) is able to withstand normal pressurewhich is then developed on the top of particles present within the apparent area of contact. If sliding surfaces are separated by an interfacial layer, then the sliding velocity difference between two sliding surfaces can be accounted for in a number ofways. The simplest case – a velocity distributionalong the thickness of the interfacial layer is a linear function. The contribution of the third body to velocity accommodation is very important. Godet suggested that the gradient of contact tangential velocity with respect to layer thickness can existwithin the third body.The thirdbody 16 A.Zmitrowicz accommodates the difference in sliding velocity of parent bodies with the aid of the two following dominant modes: (a) by interfacial sliding between the layer and parent bodies, (b) by shearing in the bulk of the third body. 6. Role of wear debris in modifying friction and wear The friction phenomenon is very sensitive to changes of sliding conditions. Experimental studies demonstrate that wear particles entrapped between sli- ding surfaces can affect frictional and wear behaviour very significantly. The presence of the debris implies modifications of the friction coefficient and the wear rate, see Kuwahara andMasumoto (1980). Circulation of wear particles is reflected by the friction coefficient, which increases when the particles are accumulated anddecreases when the particles are removed from the sliding interface. According to Suh and Sin (1980), the kinetic coefficient of friction for metals is in the neighbourhood of 0.1 to 0.2 (butmostly in the range of 0.12 to 0.17) regardless ofmaterials tested, i.e. gold on gold, steel on steel, brass on steel, etc. Wear particles entrapped between sliding surfaces affect frictional behaviour increasing the friction coefficient to 0.5-0.7. Variations in the friction coefficient show evolution of the amount of debris in the contact; a stability of the friction coefficient indicates a constant amount of debris in the contact interface. In some experiment study, a steady value of the friction coefficient was reachedwhen the number of newly formed wearparticleswas equal to thenumberofparticles leaving the contact interface (µ =0.28 initial value, µ =0.63 steadyvalue).The steady-statewear ratemay be larger for some sliding conditions where wear particles cannot be removed from the contact and act as abrasive particles. There is a number of reports in the literature that spherical and cylindrical wear particles roll over each other with resulting low friction. It has also been reported that, as a result of the roll formation, the coefficient of friction under- goes transition to lower values by a factor of three, and the wear decreases by several orders of magnitude. The coefficient of friction in the presence of rolls usually ranges from0.1 to 0.4 (Zanoria et al., 1995). It has been suggested that cylindrical debris can act asminiature roller bearings or ”solid lubricants”, so that sliding friction can be reduced. Wear debris: a review of properties and constitutive models 17 7. Wear debris between two rough surfaces Real engineering surfaces are rough.What happens when we put two sur- faces together to form an interface? Under normal and tangential tractions, two processes occur in the interface between two sliding rough surfaces, see Hou et al. (1997). First, asperities are elastically and plastically deformed and flattened, resulting in closure of voids between them. Secondly, interfacial wear particles are rearranged and plastically deformed, closing many of the voids present between wear debris. Although some cavities will always remain, it is reasonable to assume that these cavities do not significantly change homoge- neity of the interfacial layer. Therefore, one can assume that the interfacial layer consists of deformed asperities, wear and other debris (e.g. contaminant particles) which, under normal and tangential tractions, form a uniform layer. Themost frequently used statistical models of the surface roughness assu- me a symmetric Gaussian distribution for heights of surface asperities (taken with respect to a reference line or plane). The contact of two rough surfaces (that follow theGaussiandistributions) andwearparticlesmaybemodelled as contact of two smooth reference planes and an interfacial layer. Reference bo- undaries of bodies can be located a little inside the bodies (in comparisonwith real boundaries of the bodies) so that the Gaussian distribution of asperity heights of both boundary surfaces can be included (Hou et al., 1997). Usually, mechanical properties of two rough surfaces in contact are characterized by compliance. Microgeometry of sliding surfaces is not a given property but changes du- ring thewearprocess.After the running-inphase, someof asperities of surfaces are deformed and other ruptured. To take into account the effect of surface fi- nish, thewear rate and the distribution of asperity heights shouldbemodified. Wear particles produce geometrical changes in wearing surfaces. After some number of repeated slidings, the wearing surfaces may become rougher com- pared with the initial surfaces. A correlation between wear debris formation and surface topography was observed by Xuan and Cheng (1992). 8. Constitutive models of wear debris Some effort hasbeendevoted to characterizemechanical properties of third bodies and the way they deform in rubbing contact. For these purposes, one needs some equivalent but inscrutable properties to define quasi-solid, quasi- fluid and granular-like behaviour of wear particles. However, there is a lack of constitutive laws for wear debris. 18 A.Zmitrowicz In the papers by Szefer et al. (1994), Szefer (1998), the interfacial layer (described with the aid of a singular surface) was assumed to be attached to contacting bodies. Itmight be some promise in to be able to treat wear debris as a single continuum, see Zmitrowicz (1987, 1989, 2000, 2001, 2002, 2003). 8.1. Continuum formulations In the published literature, various continuum mechanics-based models for the third body have been proposed: (a) solid-like models, (b) fluid-like models, (c) other models (e.g. mixtures), see Elrod (1988), Berker and Van Arsdale (1992), Heshmat andBrewe (1992), Hou et al. (1997), Iordanoff et al. (2002a,b). Advantages of the continuumapproaches are following: (a) themain stages of the wear process can be considered in this approach, i.e. the formation and circulation of wear debris, and the ejection from the contact area, (b) wear debris can undergo fragmentation and/or agglomeration, (c) the third body is a thin layer, and sliding occurs at the interface between the layer and the parent bodies. Disadvantages of the continuum models are as follows: (a) these models cannot be used of only a few wear particles are formed (a non-continuous interfacial medium), (b) difficulty to find quantitative values for physical con- stants, (c) discontinuity in the velocity profile, i.e. sliding velocity accommo- dation cannot be realized in the bulk of the third body. 8.1.1. Micropolar thermoelastic layers Amicropolar thermoelastic layer can describe solid-like behaviour of wear debris formedas rolls. Thermolasticmaterials with internal degrees of freedom in the form of micro-rotations were studied by Cosserat and Cosserat (1909), AeroandKuvshinskii (1960),Aero et al. (1965),GauthierandJahsman(1975), Eringen (1975/76), Pietraszkiewicz andBadur (1983), Alts andHutter (1988, 1989), Sansour and Bufler (1992), Blinowski (1994). Let us consider two contacting solids A and B and an interfacial layer S of wear debris taken as a two-dimensional continuum. Positions of particles in the layer S at time t are determined by vectors L and l in the reference and current configurations of the layer, respectively. The thermodynamical process in the interfacial layer is determined by position vectors of layer particles with respect to the reference and current configurations xS(L, t),XS(l, t), a tensor ofmicro-rotations χ(L, t) and a function of temperature ΘS(L, t). The tensor of micro-rotations χ describes systematic rotations of independent layer par- ticles, each on another, without a change of their positions. A representation Wear debris: a review of properties and constitutive models 19 of the rotation tensor in terms of an angle of rotation ψ and the unit vector of the rotation axis k is as follows χ=cosψ1S +(1− cosψ)k⊗k+sinψk×1S (8.1) k×1S =−�Sk where 1S is the identity tensor for the layer, �S is the Levi-Civita tensor for the layer S. Gradients of deformation for the two-dimensional layer are given by FS = GradSxS F −1 S = gradSXS (8.2) As measures of deformation of the micropolar layer, we use Cosserat’s de- formation tensors. They are functions of displacements and micro-rotations. Cosserat’s strains read: the Lagrangian surface deformation tensor Γ=χ>FS −1S (8.3) the Eulerian deformation tensor γS =1S −χ > F −1 S (8.4) the Lagrangian wryness tensor K=− 1 2 �S(χ >GradSχ) (8.5) the Eulerian wryness tensor κS =χKF −1 S (8.6) We can define positions of the layer at time t by the displacement vectors given with respect to the reference and current configurations US(L, t)=xS(L, t)−L (8.7) uS(l, t)= l−XS(l, t) Within the linear theory, small deformations are postulated, i.e. |GradSUS|� 1 |gradSuS|� 1 (8.8) Furthermore, small micro-rotations within the linear theory are considered cosψ ≈ 1 sinψ ≈ ψ (8.9) 20 A.Zmitrowicz We introduce a vector of small micro-rotations, i.e. Ψ = ψk (8.10) with the following properties ψ = |Ψ| k= Ψ ψ (8.11) For an isotropic and linearly elastic micropolar solid, the strain- displacement relations are given by γS = gradSuS −�SΨ κS = gradSΨ (8.12) The symmetric part of themicropolar strain tensor is equal to the linear elastic strain tensor γ sym S ≡ εS = 1 2 [gradSuS +(gradSuS) >] (8.13) The antisymmetric part is a relative rotation γ asym S = 1 2 [gradSuS − (gradSuS) >]−�SΨ (8.14) Within the linear theory it is assumed that deviations of temperature from the reference value T (0) S are small ΘS = T (0) S +TS |TS|� T (0) S (8.15) Two measures of stresses are assumed, i.e. a stress tensor σS and a co- uple stress tensor µS. The stress constitutive relations for the layer take the following forms σS = −βTS1S +λS tr(gradSuS)1S + + (µS +κ)(gradSuS −�SΨ)+µS(gradSuS −�SΨ) > (8.16) µS = αS tr(gradSΨ)1S +βS gradSΨ+γS(gradSΨ) > where, β is the thermal expansion, λS and µS are Lamé’s constants λS = ESνS (1+νS)(1+2νS) µS = ES 2(1+νS) (8.17) ES is theYoungmodulus, νS is thePoissonnumber, κ,αS,βS,γS arematerial constants of the micropolar continuum, see Eringen (1975/76). Wear debris: a review of properties and constitutive models 21 Two equations of motion define unknown displacements and micro- rotations in the layer, see Zmitrowicz (1987, 1989). The equations of motion of the micropolar thermoelastic layer with friction and wear effects included are as follows ρS ∂2uS ∂t2 +βgradSTS − (λS +2µS +κ)gradSdivSuS + +(µS +κ)curlS curlSuS +κcurlSΨ − tAS − tBS −mAVAS −mBVBS =0 (8.18) ρSj0 ∂2Ψ ∂t2 − (αS +βS +γS)gradSdivSΨ +γS curlS curlSΨ −κcurlSuS + +2κΨ −c+(mA+mB)j0 ∂Ψ ∂t =0 where tAS, tBS are friction forces between the layer S and the bodies A and B. VAS, VBS are sliding velocities between the layer S and the bodies A and B. mA, mB are masses supplied to the layer from the bodies A and B during the wear process, j0 is themicroinertia tensor of wear debris, c is the couple of friction forces in the layer. 8.1.2. Micropolar fluid films Micropolar fluid films can describe fluid-like behaviour of cylindrical or spherical wear debris. Micropolar fluids were studied by Więckowski (1955), Stokes (1966), Eringen (1975/76), Leslie (1979), Eringen (1980), Kirwan (1986), Lin (1997), Walicka (2000). The absolute velocity of a layer particle is defined by vS = DxS Dt ≡ vαaα+znn α =1,2 (8.19) where, {aα,n} are the unit vectors of the layer, v α, zn are tangential and normal components of the velocity. It is seen that the velocity of the layer particle is a sum of the superficial velocity vαaα and the velocity znn along a trajectory normal to the layer. The vector of the angular velocity about an instantaneous axis of rotation is defined by ω= ψ̇k+sinψk̇+(1− cosψ)k× k̇ (8.20) Two deformation rate measures are introduced using the vectors of trans- lational and rotational velocity. The following deformation rate measures are assumed in the case of the micropolar fluid layer A= gradSvS −�Sω B= gradSω (8.21) 22 A.Zmitrowicz Let us decompose the stress tensor as follows TS = π1S +T (d) S (8.22) where π is the thermodynamic pressure in the fluid, T (d) S is the dissipative part of the stress tensor. The following constitutive relations are used for the stress tensor T (d) S and the couple stress tensor M T (d) S = λv(trA)1S +(µv +κv)A+µvA > (8.23) M= α�SgradSΘS +αv(trB)1S +βvB+γvB > where λv, αv, µv, κv, α, αv, βv, γv are material constants, see Eringen (1975/76), Eringen (1980). Two governing equations define the unknown translational and rotational velocities in the layer, see Zmitrowicz (1987, 1989). The equations of motion for the micropolar fluid layer are as follows ρS DvS Dt + gradSπ− (λv +2µv +κv)gradSdivSvS + +(µv +κv)curlS curlSvS −κv curlSω− tSA− tSB − −mAVAS −mBVBS =0 (8.24) ρSj0 Dω Dt − (αv +βv +γv)gradSdivSω+γv curlS curlSω−κv curlSvS + +2κvω−c+(mA+mB)j0ω=0 In the equations of motion, the following effects are included: friction forces, friction couple and the supply of wear debris to the interfacial layer S. 8.1.3. Thermo-viscous fluid films As an example, we consider a non-polar fluid. To this end, it is assumed that the rotational velocity ω and the couple stress tensor M as well as the friction couple c are equal to zero. The surface gradient operator of the layer velocity is decomposed into symmetric and antisymmetric parts gradSvS =DS +WS (8.25) Wear debris: a review of properties and constitutive models 23 where DS = 1 2 [gradSvS +(gradSvS) >] (8.26) WS = 1 2 [gradSvS − (gradSvS) >] In classical fluidmechanics of viscous flow, the deformation rates are com- pletely measured by DS. The thermodynamic pressure can be introduced si- milar as in the case of the micropolar fluid. The dissipative part of the stress tensor is of the form T (d) S =T (d) S (DS) (8.27) The constitutive equations for the stress tensor take the form typical forNew- tonian fluids, i.e. TS = π1S +λ(trDS)1S +2µDS (8.28) where, λ and µ are viscosity constants. In the subject literature, exceptionally non-Newtonian models of lubricant fluids are proposed, e.g. viscoelastic of Rivlin-Ericksen, pseudoplastic of Reiner-Rivlin, etc. The governing equations of motion of the layer given in the component notation, α,β,ε =1,2 and n ≡ 3, are as follows ρS (∂vα ∂t +vα|βv β −2znbαβv β −znzn,α ) +π,α− (λ+µ)(v β|βα−2Kmzn,α)− −µvα| β β +µvεbεβ(b β α+2Kma β α)+3µzn,βb β α− t(SA)α− −t(SB)α−mAV(AS)α−mBV(BS)α =0 (8.29) ρS (∂zn ∂t +2zn,αv α+vαbαβv β ) −zn,αβa βα−2Km(λv α|α−π−2λKmzn)− −µbαβ(2vα|β +vβ|α−2znbαβ)− t(SA)n− t(SB)n =0 where, Km is themean curvature of the layer, a αβ is themetric tensor, bαβ is the curvature tensor, see Zmitrowicz (1987, 1989). 8.1.4. Mixtures Solid-liquid and solid-gas mixtures as constitutive models of wear debris were proposed byBerker andVanArsdale (1992). Heshmat andBrewe (1992) investigated a solid powder lubricant film between sliding surfaces. They po- stulated that a powder lubricant exhibits some of the basic features of hydro- dynamic lubrication, and they proposed a semi-empirical rheological model of the powder film. 24 A.Zmitrowicz Hou et al. (1997) investigated rheological behaviour of the interfacial layer which forms the third body in rolling-sliding contacts and consists of wear debris and contaminants. It was assumed that the wear and other debris, under traction and extreme pressure, form a uniform, solid, incompressible layer. Hou et al. (1997) suggested an elastic-plastic rheological model for the compressed layer. 8.2. Discrete formulations Some authors assumed that equations of continuum mechanics cannot be used in the case of the third body. They postulated that the third body is discontinuous, heterogeneous and anisotropic. For example, wear of steel can lead to a situation in which there is a layer of a granular material separating sliding bodies. In that case, mechanics of granularmedia should be applicable to define mechanical properties of the third body. That is why the discrete approach has been applied to the study of the third body. Advantages of the granular material model are following: (a) it is possible to simulate a non-continuous interfacial medium, (b) it is well known that the granular media can exhibit both solid-like and fluid-like behaviour, (c) the velocity accommodation takes place in the bulk of the third bodyby shearing. Disadvantages of the granularmodel: (a) a number of particles in the layer is constant; formation of new particles and ejection of the particles from the contact region cannot be considered, (b) a choice of inter-particle forces is arbitrary, (c) the layer has a finite thickness, and the size and shape of the particles have to be assumed. 8.2.1. Granular material models Granular material models have been used to describe quasi-granular be- haviour of wear debris, see Ikramov (1983), Berker and Van Arsdale (1992), Iordanoff et al. (2002a,b), Iordanoff andKhonsari (2004). Almost all classical theories of the granular flow involve an assumption that particles are sphe- rical in shape, similar in size and non-cohesive. An incompressible granular medium is commonly supposed. Motion in a granular medium is attributed to boundary conditions and to interactions between particles at points of con- tact. Interaction forces between the particles can bedefined in the frame of the Hertzian contact theory (see e.g. Kantani, 1981). They can include friction, adhesion and impact (with the restitution coefficient). Elrod (1988) assumed that the interactions between the particles transport normal and tangential Wear debris: a review of properties and constitutive models 25 contact forces which depend on the shape and roughness of a particle. The granular medium theory has been intensively studied by Hutter et al. (1996). Some generalizations of the rigid bodywere done byWięckowski (1973), who took into account polar effects. In order to extend and specialize mechanics of granularmedia to the third body, it was assumed that the interfacial layer occurs as a discontinuous me- dium consisting of isolated discrete particles. For a given type of particles, it is reasonable to assume that all particles have the same shape. Therefo- re, particles of the third body are modelled by assemblies of elastic spheres (or cylinders). The particles are of micrometer size, and the assembly has a finite thickness. Motion of the particles through the contact interface takes place. Then, equations ofmotion are applied to each particle considered as an isolated solid body. Let us consider a thick interfacial layer in the reference system Oxz. The particles candisplace along the axes Oxand Oz, and they can rotate along the axis Oy normal to the plane Oxz. Equations of translational and rotational motions are as follows miüi = k ∑ j=1 F (ij) Iiψ̈i = k ∑ j=1 My(ij) uti = [x t i,z t i] > F (ij) = [Fx(ij),Fz(ij)] > (8.30) where, xti, z t i,ψ t i define theposition of theparticle i at time t, Fx(ij),Fz(ij) are the inter-particle forces on the particle i due to particle j, My(ij) is the inter- particle moment on the particle i due to particle j, mi is the mass and Ii is the inertia moment of the particle i. Notice that displacements and rotations of the particles are finite in this formulation. Algorithms of calculations are based on approaches used inMolecular Dy- namics. The algorithm of discrete elements was formulated at first byCundall (1971) and Cundall and Strack (1979). Steps of calculation are as follows: — Accelerations of particles are computed using equations of motion (8.30)1,2, i.e. ü t+∆t i = k ∑ j=1 F (ij) mi ψ̈t+∆ti = k ∑ j=1 My(ij) Ii (8.31) where, ∆t is a small time step. 26 A.Zmitrowicz — New velocities of the particles are calculated by a numerical integration procedure u̇ t+∆t i = u̇ t i+ üti+ ü t+∆t i 2 ∆t (8.32) ψ̇ t+∆t i = ψ̇ t i + ψ̈ti + ψ̈ t+∆t i 2 ∆t — New positions of the particles are computed as follows u t+∆t i =u t i+ u̇ t+∆t i ∆t+ ü t+∆t i (∆t)2 2 (8.33) ψt+∆ti = ψ t i + ψ̇ t+∆t i ∆t+ ψ̈ t+∆t i (∆t)2 2 The contact force F ij can be decomopsed into normal Fn and tangential F t components. The normal component can have an elastic term (a penetration function) and a damping term (e.g. viscous damping). The tangential compo- nent is friction. Boundary conditions for the upper and lower walls must be defined, e.g. they canmove or befixedhorizontally or vertically. Theaccommodation of the sliding velocity between the upper and lower walls can be realized by shearing in the layer of granular particles. 9. Conclusions Ourmain results are as follows: • Wear particles are of small sizes and different shapes. • Wear debris can be ”rolled over” into cylindrical, spherical and needle- like particles. • Wear particles transmit translational and rotational displacements at the contact interface. • Wear particles transmit forces andmoments at the interface. • The presence of wear debris implies modifications of the friction coeffi- cient, wear rate and surface roughness. Wear debris: a review of properties and constitutive models 27 • Various constitutive models can characterize quasi-solid, quasi-fluid and granular-like behaviour of wear particles. Both continuumbasedmodels and granular material models are discussed in this study. • The practical importance of wear particles depends on a sliding system. It is extremely important in contacts where formation of loose particles is permanent, since they control the friction process. For example, in bearings, in fretting processes and in prostheses of human joints, we- ar particles are always present inside the contact region. Wear debris in the prostheses of human joints may be responsible serious illnesses (Podsiadlo et al., 1997, 1998; Tipper et al., 2001). • Analysis of wear debris is an important indicator in fault diagnosis of machines andmechanisms. Acknowledgements The financial support was provided by the State Committee for Scientific Rese- arch, grant No. 8T07A03420. References 1. AeroE.L.,BulganinA.N.,Kuvshinskii E.V., 1965,Nonsymmetric hydro- dynamics,Prikl. Mat. Mekh., 29, 297, in Russian 2. AeroE.L., Kuvshinskii E.V., 1960, Fundamental equations of the theory of elastic media with rotationally interacting particles, Fizika Tverdogo Tela, II, 1399-1409, in Russian; Trans. Soviet Physics Solid State, II, 1272-1281, 1961, in English 3. Alts T., Hutter K., 1988, 1989, Continuum description of the dynamics and thermodynamics of phase boundaries between ice andwater, Part I-IV, J. Non-Equilib. Thermodyn., 13, 3, 1988, 221-257, 259-280, 13, 4, 1988, 301-329, 14, 1, 1989, 1-22 4. AndersonD.P., 1982,Wear Particle Atlas, Telus Inc., Naval Air Engineering Center, Technical Report NAEC-92-163, revised edn. 5. Archard J.F., 1953,Contact and rubbing of flat surfaces, J. Appl. Phys., 24, 8, 981-988 6. BarwellF.T., 1983,The contribution of particle analysis to the studyofwear of metals,Wear, 90, 1, 167-181 28 A.Zmitrowicz 7. Berker A., Van Arsdale W.E., 1992, Phenomenological models of third body rheology, [in:] Wear Particles: From the Cradle to the Grave, D.Dowson et al. (Edit.), Elsevier, Amsterdam, 337-346 8. Berthier Y., GodetM., BrendleM., 1989,Velocity accomodation in fric- tion,Tribology Transactions, 32, 4, 490-496 9. Berthier Y., Vincent L., Godet M., 1992, Velocity accommodation sites andmodes in tribology,Eur. J. Mech., A/Solids, 11, 1, 35-47 10. BerthierY.,WehbeiD.,WackJ.,Roques-CarmesC.,GodetM., 1988, Fractals: amethod of characterisation of third bodymorphology, [in:] Interface Dynamics, Proceedings of the 14th Leeds-Lyon Symposium on Tribology Held at the INSA,Lyon1987,D.Dowson et al. (Edit.), Elsevier,Amsterdam,105-108 11. Blinowski A., 1994, Revolutions of deformable bodies. Part 1 – Geometry and kinematics,Prace IPPT, 7, Warszawa, 1-112, in Polish 12. Bowen E.R.,Westcott V.C., 1976,Wear Particle Atlas, FoxboroAnalyti- cal/Tranonics Inc., BurlingtonMass. USA 13. Cheng J., Yang Z.Y., 1998, Fuzzy synthetic decision-making system in fer- rographic analysis,Wear, 222, 1, 1-20 14. ChoU., Tichy J.A., 2000,A study of two-dimensional binary images of wear debris as an indicator of distinctwear conditions, J. of Tribology, 44, 1, 132-136 15. Colombié CH.,BerthierY., FloquetA.,VincentL.,GodetM., 1984, Fretting: load carrying capacity of wear debris, J. of Tribology, 106, 2, 194-201 16. CosseratE.,CosseratF., 1909,Théorie desCorpsDéformable,A.Hermann et Fils, Paris 17. Crone W.C., 1967,The Particle Atlas, Ann Arbor Science Publishers 18. Cundall P.A., 1971,A computermodel for simulating progressive large scale movements of blocky rock systems, [in:] Proc. of the Symposium of the Int. Society of Rock Mechanics, Nancy, 1, 132-150 19. Cundall P.A., Strack O.D.L., 1979, A discrete numerical method for gra- nular assemblies,Géotechnique, 2, 47-65 20. Davies J.E., 1986, Spherical debris particles in engine oil,Wear, 107, 2, 189 21. Denape J., Berthier Y., Vincent L., 2001,Wear particle life in a sliding contact dry conditions: third body approach, [in:] Fundamentals of Tribology and Bridging the Gap Between the Macro- and Micro/Nanoscales, Proceedings of the NATO Advances Study Institute, B.Bushan (Edit.), Kluwer Academic Publishers, Dordrecht, 393-411 22. Descartes S., Berthier Y., 2002, Rheology and flows of solid third bodies: background and application to anMoS1.6 coating,Wear, 252, 7-8, 546-556 Wear debris: a review of properties and constitutive models 29 23. Don J., Rigney D.A., 1985, Prediction of debris flake thickness,Wear, 105, 1, 63-72 24. DowsonD. et al. (Edit.), 1992,Wearparticles: fromthe cradle to the grave, Proceedings of the 18th Leeds-Lyon Symposium on Tribology held at the INSA, Lyon 1991, Elsevier, Amsterdam 25. Dowson D. et al. (Edit.), 1996, The third body concept: interpretation of tribological phenomena, Proceedings of the 22nd Leeds-Lyon Symposium on Tribology, Lyon 1995, Elsevier, Amsterdam 26. Elrod H.G., 1988, Granular flow as a tribological mechanism – a first look, [in:] Interface Dynamics, D.Dowson et al. (Edit.), Elsevier, Amsterdam, 75-88 27. Eringen C., 1975/76,Continuum Physics, 1-4, Academic Press, NewYork 28. EringenC., 1980,Theory of anisotropicmicropolarfluids, Int. J. Engng. Sci., 18, 5-17 29. Finkin E.F., 1964, The wear of copper, aluminum, mild steel and zinc, and their particle shape factors,ASLE Transactions, 7, 377-382 30. Gauthier R.D., Jahsman W.E., 1975, A quest for micropolar elastic con- stants, J. Appl. Mech., 42, 369-374 31. Georges J.-M., Meille G., 1985, Microscopical behaviour of the third bo- dy in dry friction or boundary lubrication, Proc. of the JSLE Int. Tribology Conference, Tokyo 1985, Part I, 885-890 32. Godet M., 1984, The third-body approach: amechanical view of wear,Wear, 100, 437-452 33. Godet M., 1988, Modeling of friction and wear phenomena, [in:] Approaches to Modeling Wear, F.F. Ling and C.H.T. Pian (Edit.), Springer, Berlin, 12-36 34. Godet M., 1989, Third bodies in tribology,Proc. 5th Int. Congress on Tribo- logy, Eurotrib 89, Helisinki 1989, I, 1-15 35. Godet M., Berthier Y., Lancaster J., Vincent L., 1991, Wear model- ling: using fundamental understandingorpractical experience?,Wear,149, 1-2, 325-340 36. GodetM., PlayD., 1975, Third-body formation and elimination on carbon- fibre/epoxy composite, [in:] Space Tribology, H.M. Briscoe (Edit.), European Space Agency, Neuilly, 165-174 37. Hatchett C., 1803, Experiments and observations on the various alloys, on specific gravity, and on comparative wear of gold. Being the substance of a re- portmade to theRightHonourable theLordof theCommitteeofPrivyCouncil, appointed to take into consideration the state of the coins of theKingdom, and the present, establishment andConstitution of hisMajesty’sMint,Phil. Trans. R. Soc. London, for the yearMDCCIII, Part I, 43-194 30 A.Zmitrowicz 38. Hawthorne H.M., 1991, On the role of interfacial debris morphology in con- forming contact tribosystem,Wear, 149, 1-2, 169-185 39. Heshmat H., Brewe D.E., 1992, On some rheological aspects of tribo par- ticulates, [in:] Wear Particles: From the Cradle to the Grave, D.Dowson et al. (Edit.), Elsevier, Amsterdam, 356-367 40. Hou K., Kalousek J., Magel E., 1997, Rheological model of solid layer in rolling contact,Wear, 211, 1, 134-140 41. Hunt T.M., 1993,Handbook of Wear Debris Analysis and Particle Detection in Liquids, Elsevier Applied Science 42. Hutter K., Svendsen B., Rickenmann D., 1996, Debris flowmodeling: A review,Continuum Mechanics and Thermodynamics, 8, 1, 1-35 43. HwangD.H.,KimD.E.,LeeS.J., 1999, Influence ofwearparticle interaction in the sliding interface on friction of metals,Wear, 225-229, Part I, 427-439 44. Ikramov U., 1983, Kinematics of abrasive particle interaction with friction surfaces, Zagadnienia Eksploatacji Maszyn, 55, 3, 271-282 45. Iordanoff I., Berthier Y., Descartes S., Heshmat H., 2002a, A review of recent approaches for modelling solid third bodies, J. of Tribology, 124, 4, 725-735 46. Iordanoff I., Khonsari M.M., 2004, Granular lubrication: toward an un- derstanding of the transition between kinetic and quasi-fluid regime, J. of Tri- bology, 126, 1, 137-145 47. Iordanoff I., SeveB., BerthierY., 2002b, Solid third body analysis using a discrete approach: influence of adhesion and particle size on macroscopic properties, J. of Tribology, 124, 3, 530-538 48. Jiang J., Stott F.H., Stack, M.M., 1998, The role of triboparticulates in dry sliding friction,Tribology International, 31, 5, 245-256 49. Jin Y., Wang C., 1989, Spherical particles generated during the running-in period of a diesel engine,Wear, 131, 2, 315-328 50. KantaniK., 1981,A theoryof contact force distribution in granularmaterials, Powder Technology, 28, 2, 167-172 51. Kato K., 1990, Tribology of ceramics,Wear, 136, 1, 117-133 52. KirkT.B., PanzeraD., AnamalayR.V.,Xu Z.L., 1995,Computer image analysis of wear debris for machine condition monitoring and fault diagnosis, Wear, 181-183, II, 717-722 53. KirkT.B.,StachowiakG.W.,BatchelorA.W., 1991,Fractalparameters and computer image analysis applied to wear particles isolated by ferrography, Wear, 145, 2, 347-365 Wear debris: a review of properties and constitutive models 31 54. KirwanA.D., 1986, Boundary conditions formicropolar fluids, Int. J. Engng. Sci., 24, 7, 1237-1242 55. Kohen I., Play D., Godet M., 1980, Effect of machine rigidity or degrees of freedom on the load-carrying capacity of wear debris,Wear, 61, 2, 381-384 56. Kuwahara K., Masumoto H., 1980, Influence of wear particles on the fric- tionandwearbetweencopperdiskandpinofvariouskindsofmetal,Lubrication Engineering, 36, 6, 362 57. Larsen-Badse J., 1968, Influence of girt diameter and specimen size on wear during sliding abrasion,Wear, 12, 35-53 58. Leslie F.M., 1979,On the thermodynamics of polar fluids,Archive for Ratio- nal Mechanics and Analysis, 70, 2, 189-202 59. Lin J.-J., Chen Y., Cheng Y.-Q., 1992, The generation of wear debris of different morphology in the running-in process of iron and steels, Wear, 154, 2, 259-267 60. Lin J.R., 1997, Effects od couple stresses on the lubrication of finite journal bearings,Wear, 206, 1-2, 171-178 61. Lu L., Farris T.N., Chandrasekar S., 1992, Slidingmicroidentation wear particles: spheres in grinding swarf, [in:] Wear Particles: From the Cradle to the Grave, D.Dowson et al. (Edit.), Elsevier, Amsterdam, 257-263 62. MizumotoM.,KatoK., 1992, Size distribution andnumber ofwear particles generated by the abrasive sliding of a model asperity in the SEM-Tribosytem, [in:] Wear Particles: From the Cradle to the Grave, D.Dowson et al. (Edit.), Elsevier, Amsterdam, 523-530 63. MurtiV.S.,PhilipP.K., 1987,Ananalysisof thedebris inultrasonic-assisted electrical dischargemachining,Wear, 117, 2, 241-250 64. Myshkin N.K., Kong H., Grigoriev A.Ya., Yoon E.-S., 2001, The use of colour in wear debris analysis,Wear, 250-251, Part 2, 1218-1226 65. Nikas G.K., Sayles R.S., Ioannides E., 1998, Effects of debris particles in sliding/rolling elastohydrodynamic contacts, J. Eng. Trib., Proc. Inst. Mech. Eng., 212, J5, 333-343 66. PengZ.,KirkT.B., 1997,Two-dimensional fastFourier transformandpower spectrum for wear particle analysis,Tribology International, 30, 8, 583-590 67. Peng Z., Kirk T.B., 1998, Computer image analysis of wear particles in three-dimensions for machine condition monitoring,Wear, 223, 1-2, 157-166 68. Peng Z., Kirk T.B., 1999a, The study of three-dimensional analysis tech- niques and automatic classification systems for wear particles, J. of Tribology, 121, 1, 169-176 32 A.Zmitrowicz 69. PengZ.,KirkT.B., 1999b,Wearparticle classification ina fuzzygreysystem, Wear, 225-229, Part II, 1238-1247 70. Peng Z., Kirk T.B., Xu Z.L., 1997, The development of three-dimensional imaging techniques of wear particle analysis,Wear, 203-204, 418-424 71. Pietraszkiewicz W., Badur J., 1983, Finite rotations in the description of continuum deformation, Int. J. Engng. Sci., 21, 9, 1097-1115 72. PodsiadloP.,KusterM., StachowiakG.W., 1997,Numerical analysis of wear particles from non-arthritic and osteoarthritic human knee joints, Wear, 210, 1-2, 318-325 73. Podsiadlo P., Stachowiak G.W., 1998, Evaluation of boundary fractal methods for the characterization of wear particles,Wear, 217, 1, 24-34 74. PodsiadloP., StachowiakG.W., 1999a,3-D imagingof surface topography of wear particles found in synovial joints,Wear, 230, 2, 184-193 75. Podsiadlo P., Stachowiak G.W., 1999b, Applications of Hurst orientation transform to the characterizationof surface anisotropy,Tribology International, 32, 7, 387-392 76. PodsiadloP.,StachowiakG.W., 2000,Scale-invariantanalysisofwearpar- ticle surface morphology I: Theoretical background computer implementation and technique testing; II: Fractal dimension; III: Pattern recognition, Wear, 242, 1-2, 160-179, 180-188, 189-201 77. Rabinowicz E., 1977, The formation of spherical wear particles,Wear, 42, 1, 149-156 78. Rabinowicz E., 1995,Friction andWear of Materials, J.Wiley, N.Y., 2nd ed. 79. Reda A.A., Bowen R., Westcott V.C., 1975, Characteristics of particles generated at the interface between sliding steel surfaces,Wear, 34, 3, 261-273 80. Rennie G., 1829, Experiments on the friction and abrasion of the surface of solids,Phil. Trans. R. Soc. London, 34, Prt I, 143-170 81. Roylance B.J., Raadnui S., 1994, The morphological attributes of wear particles-their role in identifying wearmechanisms,Wear, 175, 1-2, 115-121 82. Roylance B.J., Wang G., Bovington C.H., 1992, The determination of particle morphological parameters to assist in elucidation of wear process, [in:] Wear Particles: From the Cradle to the Grave, D.Dowson et al. (Edit.), Else- vier, Amsterdam, 75-79 83. Roylance B.J., Williams J.A., Dwyer-Joyce R., 2000,Wear debris and associated wear phenomena-fundamental research and practice, J. Eng. Tribo- logy, Proc. Inst. Mech. Eng., 214, J1, 79-105 Wear debris: a review of properties and constitutive models 33 84. Samuels L.E., Doyle E.D., TurleyD.M., 1980, Slidingwearmechanisms, [in:] Fundamentals of Friction and Wear of Materials, D.A. Rigney (Edit.), American Society forMetals, Pittsburgh, 13-41 85. Sansour C., Bufler H., 1992, Exact finite rotation shell theory, its mixed variational formulation and its finite element implementation, Int. J. Num. Meth. Engrg., 34, 1, 73-115 86. Santanam N., 1983, Effect of wear debris on wear in rolling-sliding motion, Wear, 90, 2, 261-267 87. Scott D., Seifert W.W., Westcott V.C., 1974, The particles of wear, Scientific American, 230, 5, 88-97 88. Sheasby J.S., 1983, Direct observation of dry wear,Wear, 89, 2, 173-183 89. Sherrington I., HayhurstP., 2001, Simultaneous observation of the evolu- tion of debris density and friction coefficient in dry sliding steel contacts,Wear, 249, 3-4, 182-187 90. Stachowiak G.W., 1998, Numerical characterization of wear particles mor- phology and angularity of particles and surfaces, Tribology International, 31, 1-3, 139-157 91. Stachowiak G.W., Podsiadlo P., 1999, Surface characterization of wear particles,Wear, 225-229, Part II, 1171-1185 92. Stokes V.K., 1966, Couple stresses in fluids, Phys. Fluids, 9, 1709-1715 93. Suh N.P., Sin H.-C., 1980, On the genesis of friction and its effect on wear, [in:] Solid Contact and Lubrication, H.S. Cheng and L.M. Keer (Edit.), AMD- vol. 39, 167-183 94. Swanson P.A., Vetter A.F., 1985, The measurement of abrasive particle shape and its effect on wear,ASLE Transactions, 28, 2, 225-230 95. Szefer G., 1998, Contact problems in terms of large deformations,Z. Angew. Math. Mech., 78, 8, 523-533 96. SzeferG., JasińskaD., SalamonJ.W., 1994,Concept of a singular surface in contact mechanics,Arch. Mechanics, 46, 4, 581-603 97. Tipper J.L., Firkins P.J., Besong A.A., Barbour P.S.M., Nevelos J., Stone M.H., Ingham E., Fisher J., 2001, Characterization of wear debris from UHMWPE on zirconia ceramic, metal-on-metal and alumina ceramic- on-ceramic hip prostheses generated in a physiological anatomical hip joint simulator,Wear, 250-251, Part I, 120-128 98. Umeda A., Sugimura J., Yamamoto Y., 1998, Characterization of wear particles and their relations with sliding conditions,Wear, 216, 2, 220-228 34 A.Zmitrowicz 99. Unchung Cho, Tichy J.A., 2000, Quantitative correlation of wear de- bris morphology: grouping and classification, Tribology International, 33, 7, 461-467 100. Walicka A., 2000, Pressurized flows of amicropolar fluid between surfaces of revolution: The fixed one and the rotating one, Inż. Chem. Procesowa, 21, 1, 57-74 101. Wang Y., Cao C., Wang D., 1986, A study of the formative mechanism of spherical wear particles,Wear, 108, 3, 285-294 102. Więckowski J., 1955,Onmotion equations of real fluids,Arch. Hydrotechniki, 2, 3, 191-208, in Polish 103. Więckowski J., 1973, Elements of rigid bodymechanics in absolute notation and some generalizations,Biuletyn IMP PAN, Nr. 72/736/1973,Gdańsk, 1-46, in Polish 104. Wilson F., Eyre T.S., 1969, The generation of hard particles during sliding wear,Wear, 14, 285-288 105. Wirth A., EgglestonD.,Whitaker R., 1994,A fundamental tribochemi- cal study of the third body layer formed during automotive friction braking, Wear, 179, 1-2, 75-81 106. Wrona M., 2003, Applications of fractal methods in wear debris analysis, Tribologia, 189, 3, 103-117, in Polish 107. Wuttkowski J.G., Ioannides E., 1992, The effect of contaminants on be- aring life, SKF, Ball Bearing Journal, 239, 9-11 108. Xu K., Luxmoore A.R., 1997, An integrated system for automatic wear particle analysis,Wear, 208, 1-2, 184-193 109. XuanJ.L.,ChengH.S., 1992,Microscopicweardebris generationand surface topography, [in:] Wear Particles: From the Cradle to the Grave, D.Dowson et al. (Edit.), Elsevier, Amsterdam, 247-256 110. ZanoriaE.S.,DanylukS.,McNallanM.J., 1995,Formationof cylindrical sliding-wear debris on silicon in humid conditions and elevated temperatures, Tribology Transactions, 38, 3, 721-727 111. Zhang M.Q., Lu Z.P., Friedrich K., 1997, On the wear debris polyethere- therketone: fractal dimensions in relation to wear mechanism,Tribology Inter- national, 30, 2, 87-102 112. ZmitrowiczA., 1987,A thermodynamicalmodel of contact, frictionandwear: II constitutive equations for materials and linearized theories, Wear, 114, 2, 169-197 113. Zmitrowicz A., 1989, On the thermodynamics of contact, friction and wear, Zeszyty Naukowe IMP PAN, No. 287/1212/89,Gdańsk, 1-190 Wear debris: a review of properties and constitutive models 35 114. Zmitrowicz A., 2000,Variational approach to contact, friction andwear pro- blems, Z. Angew. Math. Mech., 80, S2, S463-S464 115. Zmitrowicz A., 2001, Variational descriptions of wearing out solids andwear particles in contact mechanics, J. Theoretical and Applied Mechanics, 39, 3, 791-808 116. Zmitrowicz A., 2002, Wear profiles and wear debris taken into account in variational approaches to contact mechanics, [in:] Proceedings of the Fi- fth World Congress on Computational Mechanics, H.A. Mang, F.G. Ram- merstorfer, J. Eberhardsteiner (Edit.), Vienna University of Technology, http://wccm.tuwien.ac.at, 1-10 117. Zmitrowicz A., 2003, Solution algorithms of contact problems with wear debris between sliding surfaces, [in:] CD-ROM Proceedings of the 15th Int. Conference on Computer Methods in Mechanics, T.Burczyński, P.Fedeliński, E.Majchrzak (Edit.), Silesian Technical University, Gliwice/Wisła, 1-10 Cząstki zużycia: przegląd własności i modeli konstytutywnych Streszczenie Zużycie jest to stopniowe usuwanie materiału ze stykających i trących się po- wierzchni ciał stałych podczas ich względnego poślizgu.Mechanizm zużycia powodu- je tworzenie cząstek zużycia. Cząstki są o małych wymiarach i różnych kształtach. Cząstki zużycia mogą być „zrolowane” w walce, kule lub igiełki. Cząstki oddzielone od trących się powierzchni tworzą prawie ciągłą warstwę pośrednią. Przekazują one siły, momenty i przemieszczenia (translacyjne i obrotowe) w obszarze styku. Obec- ność cząstek zużycia między ślizgającymi się powierzchniami oddziaływuje w sposób istotny na zjawiska tarcia i zużycia. Relacje konstytutywne charakteryzująwłasności cząstek zużycia typu quasi-ciało stałe, quasi-płyn i ośrodek granulowany. W pracy rozpatrzono dwamodele konstytutywne cząstek zużycia: (a) model ośrodka ciągłego, (b) model ośrodka granulowanego. Modele typu ośrodka ciągłego zostały sformuło- wane dlamikropolarnegomateriału termosprężystego, płynumikropolarnego i płynu termo-lepkiego. Manuscript received March 9, 2004; accepted for print July 14, 2004