Wear, 53 (1979) 345 - 370 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
D. A. RIGNEY and J. P. HIRTH Department of Metallurgical Engineering, Avenue, Columbus, Ohio 43210 (U.S.A.)
The Ohio State
116 West 19th
(Received September 26,1978)
Summary Most theories of sliding friction have emphasized surface roughness or adhesion. In some cases plastic deformation has been invoked to account for energy dissipation. After a brief review of published explanations of friction, a new model is described for the source of friction during the steady state sliding of metals. It focuses on the plastic work done in the near-surface region, described in terms of work hardening, recovery and the microstructure existing during steady state sliding. The model is discussed with respect to several alternate ways in which plastic deformation has been incorporated in recent theories of friction. Reasonable results are found when the new model is used to estimate friction coefficients for metals. Also, the model appears to be consistent with a number of published observations on the relation of friction to load, sliding distance, surface temperature and microstructure, and with a model for sliding wear which has been presented earlier. -.--
1. Introduction The quantitative study of the friction effects present when two solids rub against each other is believed to have begun in the fifteenth century with the work of Leonardo da Vinci [l] . He was probably the first to report that the frictional force is approximately proportional to the normal load and independent of area. These two empirical “laws” of friction were rediscovered by Amontons in 1699 [ 21, and they are commonly associated with his name. During the eighteenth and nineteenth centuries, many of the attempts to explain frictional behavior involved surface roughness and the interlocking of surface asperities [ 3 - 1 l] . The various models included asperities as simple ramps [ 2 - 41, depressed obstacles [ 21, bent-over springs  and hard spheres [ 51. A second major approach to friction has emphasized the role of adhesion and direct interatomic forces [12 - 291. The most widely accepted
theory of friction today is an adhesion theory based on the impressive work of Bowden and Tabor at Cambridge (reviewed in refs. 21,22, 30 and 31). It incorporates adhesion of surface asperities to form junctions, growth of the junction area and shear at or near the junctions. It has been used to estimate typical values of coefficients of friction and to explain the dependence of friction coefficient on load, hardness, apparent contact area, sliding speed, temperature and surface films of various kinds. It is interesting to note that deformation, both elastic and plastic, has been invoked by those who have emphasized surface roughness (e.g. refs. 2 and 3) and adhesion (e.g. refs. 12,18, 21, 22, 26, 27, 30 and 31). Bowden and Tabor have proposed two main contributions to sliding friction, an adhesion term and a plowing term [ 21, 311, and both terms include plastic deformation. Buckley [ 321 has demonstrated that if adhesion occurs when two metals touch in a clean environment, then plastic deformation is also observed. Also, metallographic results from numerous investigators have demonstrated clearly that plastic deformation is common near the surface of sliding materials. Adhesion theories have typically concentrated on deformation of asperities [24, 251 and on the junctions which they may form, with little consideration given to the structure of the underlying material and its response. Microstructure effects have certainly been included in tribological studies, particularly for wear, but they have not been incorporated in theories of friction based on adhesion, except through such properties as average hardness or yield strength. Leslie  was one of the first investigators to be concerned about the energy dissipated during sliding friction *. He thought that neither roughness nor adhesion could adequately explain dissipation, and he suggested, in effect, that one must consider deformation losses. Bowden and Tabor [ 21, 221 have also recognized that the energy expended in deformation is important for friction, but they have discussed this energy primarily with respect to their plowing term and not their adhesion term. Although the Cambridge approach to explaining friction is widely accepted, some criticism has been offered. Rabinowicz [ 341 has summarized some of the more obvious questions raised concerning the role of adhesion in friction, and Bikerman  and Kimura  have discussed challenges to the adhesion approach in some detail. Others have offered alternatives to adhesion theories. For example, Moore  and Moore and Douthwaite [ 381 have concentrated on the large plastic strains observed at considerable distances from the wear surface, and they suggested that plastic deformation could account for most of the work observed. Liu , Walton [40,41], Giimbel  , Rowe [ 431, Kragelskii , Glaeser  , Wanheim et al. [ 461, Suh and Sridharan [ 471, Argon [ 481, Rigney [ 491 and Rigney and
is given in
Hirth [ 501 have also discussed the importance of plastic deformation for friction and wear. Some of these theories are briefly described in Section 2.
2. Plastic deformation
Liu  has suggested a way of estimating the contribution of plastic deformation to friction by considering the plastic work involved in the deformation of a model asperity. His approach yields the relation 2E,,ZVi i where F is the friction force, Ax the sliding distance, E, the total work expended in plastic deformation per unit volume and Vi the total volume of the asperity which is plastically deformed. The factor of 2 was included for the case of two identical materials in contact with matched asperities. Plastic deformation below the asperities was not considered in the model. The total contact area supporting the load was then given as ZiA; = LIP, where P is the bulk flow stress of the material. For asperities shaped like the corners of cubes, the friction coefficient could be estimated as ~1= 20 E; [P. To test this result, the energy term E, was estimated for copper by referring to data on plastic work and strain from Gordon  and Clarebrough et al. [ 521. However, the strain was estimated by using an expression for elastic dilatational strain de = dV/V, or FAX=
where Ui is the displaced volume at the tip of each asperity. This yielded a very small (elastic) strain of about 0.04, which was then used to estimate E,. Since dV = 0 for the plastic deformation actually assumed, this procedure is not appropriate, and the estimated value of the friction coefficient has little relation to the assumed physical situation. The derivation would be applicable to anelastic dissipation wherein the release of elastically stored strain energy was irreversible. Walton [40, 411 has also considered the relative contributions of adhesion and plowing to friction. He examined copper on copper, coppercoated indium pairs and gold-coated copper pairs, and from the results he concluded that “... a class of frictional phenomena exists which cannot be explained by the adhesion theory”. For the copper samples a friction coefficient of about 0.5 was observed at light loads without any measurable adhesion. Released elastic stresses could not account for the apparent absence of adhesion. Walton suggested that plastic deformation was the source of the frictional energy dissipated. He assumed that this could be described by a geometric plowing term similar to that proposed by Bowden and Tabor [ 20 - 221, but with the’emphasis now on plowing instead of adhesion. Plowing was assumed to be localized in the surface oxide and not in the underlying metal. Walton did not report
metallographic evidence which might have confirmed or disproved his conclusions. Suh and Sridharan  have used ideas from their delamination theory. of wear  to develop an approximate expression relating the friction coefficient and the wear factor, which is defined as the wear volume divided by the product of the normal load and the sliding distance. In their model they focused on the work done in creating a wear particle, thus associating friction and the wear process directly. The following parameters are involved: the dimensions of a typical wear particle, a strength term, a strain per cycle and an approximate strain distribution. Despite the many simplifying assumptions used, the model leads to values that are close to experimental observations for the friction coefficient and the wear factor. In a recent summary report  Tsuya and her colleagues have described the results of numerous friction measurements on copper (both single crystals and polycrystals), on films of soft metals and of various solid lubricants and on composites of metals and solid lubricants. Tsuya’s papers are a good source of experimental details on friction, and they contain microstructural information which is not commonly available in this area. Moreover, Tsuya has recognized the importance of plastic deformation, especially that which is concentrated in a well-defined region near the surface, which she calls the “micronized” layer. Tsuya has also proposed a model of friction which is baaed on the work done during plastic deformation. This model is described briefly here and discussed further in Section 5. For deformation over a width w and a depth t for a sliding distance S, Tsuya expresses the deformation energy as W s $
where E, is the deformation energy per unit mass and p is the density. This expression can be equated to the product of the friction force F and the distance slid, so that F=2wp
is the simple result. The factor of 2 is appropriate for the case of a metal sliding on itself, since there are then two deformation zones to consider. To proceed further, one needs to know the functional dependence of deformation energy on distance from the surface. Tsuya uses data on microhardness profiles to estimate E, . One can then readily calculate a friction coefficient of 1.6 for copper on copper in air under a 0.40 kgf load. This result is in very close agreement with Tsuya’s experimental value for friction under these same conditions. However, as discussed in Section 5, there are problems with the details of her calculation,
Tsuya has used a similar model to analyse situations involving surface films and coatings. Each region which deforms plastically is described by an integral term similar to the one described above, and characterized by appropriate values for p, t and E, . For cases in which plastic deformation is largely confined to a thin surface region of thickness t, Tsuya offers a simplified expression for the frictional force: F=2pwtE, where &, is “micronized” Tsuya’s experimental related model
the mean value of the energy per unit mass within this region. ideas on friction are interesting because they are closely tied to observations and because they involve materials properties. A is presented in Section 3.
3. A simple model for steady state friction Rigney and Glaeser  have described a wear model in which attention is focused on steady state wear. They emphasized plastic deformation near the surface, particularly in the highly deformed region which has a very fine microstructure and a high degree of preferred orientation (Fig. 1). In metals and in some ceramic materials, this near-surface microstructure consists of dislocation cells developed during an initial break-in period. Under steady state conditions the average cell structure at a given distance from the surface remains constant, and the average thickness t of the cell region is a constant that depends on material properties and on details of the sliding wear test. In this section the ideas applied earlier to wear are extended to apply to ‘friction as well. In each case we are suggesting an alternative to traditional theories, such as those which emphasize the local shear of adhesive junctions. With the assumptions that the frictional force arises from plastic deformation and that most of the deformation work is confined to the cell region, one can develop an expression for the friction coefficient using a model that is consistent with the wear model of ref. 55 and with the steady state microstructures described in refs. 56 - 60.
t Undeformed Fig. 1. Longitudinal section of a wear specimen: the curved lines indicate strain and the arrow indicates the sliding direction.
Since steady state friction is assumed to depend on a steady state microstructure, all of the plastic work is dissipated as heat and none is added to the energy already stored in the microstructure. The simplest case is that in which one component of a sliding pair is much harder than the other, so that plastic deformation is essentially confined to the near-surface region of the softer material. The further development of a theoretical expression then requires definition both of the volume undergoing plastic deformation when a slider moves and of the plastic work performed per unit volume. Several limiting cases are treated in Appendix 2. We present the derivation of a theoretical expression for the friction coefficient for the case of a slider moving into a recovered region and causing plastic deformation without appreciable dynamic recovery. However, we show in Appendix 2 that the same result obtains when dynamic recovery occurs. For the case where there is no dynamic recovery the plastic work per unit volume is given by the product of the shear stress 7 in the cellular region and in the sliding direction and the strain per cycle E in that region*. Both 7 and E will depend on the depth below the sliding surface, and a more detailed calculation would include this variation. However, for this work, average values will be used for r and E as well as for w and t. The total work is then given by the product of the plastically deformed volume V and 7~. For a virtual displacement 8~ the frictional work is just F&x, where F is the frictional force, and these two expressions for frictional work must be equal: Vr.e = F6x
The volume is equal to z&&x, where w is the width of the highly deformed region (approximately equal to the wear track width for a pin-ondisk geometry). Since the friction coefficient is defined by p = F/L, the following expression for the friction coefficient is obtained: wtre P= --L
All four terms in the numerator are material properties or are closely related to material properties. Three of the terms (w, t, L) in eqn. (2) are directly measurable and readily available for a given sliding situation. The shear stress 7 is not the bulk value that might be available from simple stress-strain curves. However, it could be estimated from appropriate tests on severely cold-rolled material with similarly textured cell microstructure,
*In a more accurate model, work done in producing other (redundant) plastic strain components would need to be included in a work balance. Indeed in a crystal plasticity model the relevant work would be the integral work produced by the resolved shear stress on all active slip systems. However, the simple work term used here is expected to be the dominant term. The difficulty in defining E is such that a more detailed analysis is not warranted at present.
The average net strain per cycle E is somewhat more difficult to measure. A chemical marker technique could be used together with Auger analysis to measure the variation of gross strain with depth in the near-surface region. Another experiment would involve the use of a marked split sample. Finally, eqn. (2) indicates a proportionality of P to specimen size at large loads as discussed in Appendix 2, which is amenable to experimental test by using sliders of rectangular cross section moving in different directions. At first sight eqn. (2) seems to give the result that P depends on the load. However, both w and t should increase with the load. A model for such an increase is presented in Appendix 3. If the product wt is roughly proportional to load, then Amontons’ familiar first law is obeyed; i.e. the friction coefficient is independent of the load. If the relation between wt and L is not quite linear, then there would be some dependence of the friction coefficient on the load. Since stress falls off rapidly near the surface, one would expect that the thickness of the highly deformed (cell) layer would vary with load by some power less than unity, i.e. it would increase with load but in such a way that strain is concentrated near the surface and t does not reach large values. Limited data for copper [ 541 indicate that t - L”, where n = k. It is reasonable to expect that the width w would also be proportional to Lm with m < 1. In fact, data on the Vickers hardness number (VHN)  , which is roughly proportional to load/area, would indicate that m = 4, with perhaps a somewhat higher value at very light loads. However, one can do better than to use this plausibility argument, since Tsuya  has also measured track width uersus load for copper, and she finds m = $. Withbothnzi andm=i, eqn. (2) describes a friction coefficient that is approximately independent of load. When (n + m) # 1, there is some dependence of P on load, as is sometimes observed experimentally [ 22,621. If the conditions are such that the true area of contact is much less than the apparent area of contact (e.g. low load) 7 and E should be essentially independent of load L. This is so because the true area of contact adjusts with L in such a way that local stress conditions are the same where contact occurs, including both elastic and plastic components of stress. This situation should prevail until the intense elastic field of a given asperity overlaps that of a neighboring asperity or roughly until the fractional area of contact is about one-half. Strictly, the postulated invariance would hold only if all asperities were the same shape so that the distribution of elastic and plastic contact areas remained invariant with load, but effects of this type are expected to be of second order in importance. Once the elastic zones begin to overlap, the local stress would begin to increase with increasing load. Eventually, the entire area would be in contact with plastically deformed regions and increased load could only be sustained by work hardening, resulting in increased local stress with increased load. With increasing stress the local yield condition can become dependent on stress state, so that 7 and e would become load dependent. The dependence of the local critical resolved shear stress on stress components other than the
shear stress itself has been observed in simple mechanical tests of a variety of materials and is denoted as the strength differential or SD effect [63 - 701. The most commonly tested SD effect is the dependence of the critical resolved shear stress on the hydrostatic stress component. The effect can involve persistent local strain events or transient ones; e.g. the hydrostatic stress can interact with the volume expansion associated with vacancy emission (a persistent effect) or could involve an activation volume for dislocation motion (a transient effect). The model presented in this section has been based on the simple case of a hard material sliding on a softer material in which plastic work is largely confined to the cell layers in the softer material. However, since contributions to friction from more than one region are additive, one can extend eqn. (2) to other cases of practical significance. In fact, whenever plastic deformation is largely confined to a well-defined region of a material in a sliding system, an expression like that of eqn. (2) applies for that region. Some examples are given below. (1) If metal A slides against a second piece of the same metal A, each develops a cell region of thickness t, and the friction coefficient is given by P = ~(2t)~e/L. This is the case treated by Tsuya [ 541. (2) If a thick metal film is applied, and the substrate does not deform at all, then /1 = wtre/L applies, but t, T and E characterize the cell region of the metal film. (3) If the metal film is fully cellular, and the substrate also develops a cell layer, then a two-term expression is needed for friction:
wtsT,E, Y = ----
WtfTfff + _ ~~--
where s refers to the substrate and f to the film. Solid lubricants could be treated by a similar two-term expression, one term representing the lubricant layer and the other term representing the substrate. (4) A two-term expression also results if two dissimilar materials slide against each other: y = WtlTlEl ..--.-_-_ L
+ wt272e2 __I_L
(5) For a composite with a lamellar structure, if the lamellae become aligned parallel to the sliding surface, then a separate term can be written for each lamella in which plastic shear is appreciable. A multi-term expression is then appropriate for the friction coefficient: WhTlE1 p=_-L_+.L-_
In all these cases, the contributions to friction from deformation below the “easy” shear region are neglected. The work of Tsuya [ 541 shows that this approximation is reasonable, since most of the deformation is localized near the surface. Therefore the error involved in using this simplified model is expected to be small.
One can use a typical stress-strain curve together with a series of successive strain profiles (Fig. 2) to demonstrate further that our approximation is reasonable. Region A would be associated with Stage I of the stress-strain curve, and region B with Stage II (rapid work hardening). Region C would correspond to Stage III, in which dynamic recovery is important; work hardening is then balanced by recovery and cell structures can form. Evidently, only region C would contribute to plastic dissipation at steady state, as hypothesized in the above model. If a steady state were reached without recovery, as in Fig. 2(b), then the work-hardened material could only deform elastically and friction would be low. Such cases might exist at very low temperatures for selected materials. Of course, during the break-in period, these contributions would be more important, but the model described here and in refs. 50 and 55 is not meant to apply before steady state conditions are established. N
v SlrOln B
Fig. 2. Strain profiles
times during sliding.
The quantities w, t and L can be measured directly for a given test system. The quantities 7 and e are not directly available. However, it is possible to estimate these quantities from data published in the literature. Therefore it is possible to estimate friction coefficients. As an example, we have selected the case of an OFHC copper sample tested in a LFW machine with a 440 C alloy ring in an argon atmosphere. The load was 20.4 kgf (45 lbf) and the sliding speed was 0.05 m s-l. Since the 440 C alloy is much harder than the copper, most of the deformation should be in the copper and the simple one-term expression can be used for the friction coefficient, i.e. ~1= (wt~e)/L. The resulting track width was the width of the sample block, i.e. 6.3(5) mm. The value of t was determined from a transverse section of the copper sample, so that a proper average could be obtained. The approximate value for t,, was 12 pm. Since the textured microstructures generated by large reductions during cold rolling [ 591 and those generated during sliding [60, 711 are similar, one can use tabulated strength data on cold-rolled material to estimate T [ 721. With the assumption that the preferred orientation of the near-surface material is that which allows easiest shear, then 7 should be approximately half the ultimate tensile strength (UTS). For copper, the appropriate UTS is
about 416 MPa (60 000 lbf in2); therefore 7 = 208 MPa (30 000 lbf in2). Since the dislocation cell size is known to vary with distance from the sliding surface [60, 711, T probably varies also. Therefore the value used here is an estimate for an average value of r. Suh and coworkers [ 731 have used an intercept method described by Dautzenberg and Zaat [ 741 to obtain approximate strain profiles for copper samples. While it is not clear that this method can be used directly when the reference region has a qualitatively different microstructure from the deformed region, Suh’s value of about 10 for E will be used here*. The resulting calculated value of the friction coefficient is p = 0.78. For comparison, the measured value was 0.7(5) + 0.1. The agreement is much better than might be expected from the use of so many approximate values. It is interesting to estimate the range of friction coefficients that might be expected for a wide variety of metal systems. If one excludes the low melting point metals which recrystallize readily near room temperature, then a reasonable range for 7 is 70 - 1000 MPa (10 000 - 150 000 lbf in2). Then, using values of 5 - 20 for e and 5 - 50 pm for t, one can predict a range of friction coefficients from roughly 0.1 to 30 for unlubricated metal-to-metal sliding. Smaller values of p could probably be traced to surface films of reaction products or lubricant. The higher values observed in a high vacuum environment have been explained by citing the absence of such films . Surface films in our model affect the way in which stress is transferred from one surface to another in a sliding system. 5, Discussion The model for friction proposed by Tsuya [ 541 and the one presented in this paper appear to be closely related. Both involve an energy argument baaed on plastic deformation near the surface, and both result in an expression for the friction force which is the product of the width of the deformed region, the thickness of the region in which the deformation is concentrated and a deformation energy per unit volume (pE, or TE). The first two terms, w and t, are readily available by examination of friction or wear test specimens which have been operated under steady state conditions. However, there are differences in the energy terms in the two models. For copper specimens in air, with L = 0.40 kgf, Tsuya  has used E,(~)=3XlO-~z-~/~calg-~ (1.3~-~/~ Jkg-l)forz>lO~mandE,(z)= 3 x 10-4(1o-s)-s’z cal g-l (1.3 (10-3)-3/2 J kg-‘) for 0 Q z < 10 pm. The boundary of the two regions was selected by examination of microhardness profile data, which indicated approximately constant values near the surface for a range of applied loads. Tsuya then correlated microhardness, determined as a function of z, with strain and stored energy data from the work of Clarebrough et al. [76, 771 and McLean , and identified &, with the stored energy. This procedure is not applicable for steady state conditions because the net stored energy per cycle is then close to zero and most *Note that for the present theory the strain should be the net value per cycle of contact of a rider and not the total strain in a given test.
of the deformation work per cycle is dissipated. Earlier in the sliding process, during break-in, a larger fraction of the deformation energy is stored, because that is when the substructure is being produced which eventually evolves to the steady state microstructure. Even in the early stages of deformation, however, the fraction of work stored is small compared with the total deformation work done, as shown in Figs. 3 and 4 [ 76, 781. Thus, only during break-in would the term determined by Tsuya contribute to the friction coefficient. Even then a calibration curve like that shown in Fig. 3 would be needed to determine the total energy dissipated and stored, which would completely describe the friction
Fig. 3. Variation of stored (adapted from ref. 76). Fig. 4. Variation from ref. 78).
If a calibration curve of the above type for stored energy uersus total energy input could be obtained for the very high strains characteristic of a steady state microstructure in a friction test, then the type of microhardness correlation used by Tsuya could be used for estimates of friction. Unfortunately, there are at present two barriers to this approach. The first is that microhardness-strain data are needed at strains much higher than presently available if the calibration curve is to be useful for the very high strains generated near sliding surfaces. The second is directly related to the shape of the calibration curve itself, the early stages of which are shown in Fig. 3. At higher input energy and strain, the curve will bend over and approach a limit which corresponds to the maximum stored energy for the deformed region. As this limit is approached, a small uncertainty in stored energy (or microhardness) would give a large uncertainty in total energy input, and at the limit which corresponds to a steady state the constant level of stored energy would give no information about further increases in energy input.
Tsuya has presented
two simple expressions
for the frictional
and F = 2utp t& Either of these is basically correct, and either could be used if E, were identified with the total energy stored and dissipated per contact event at the surface and if such energy data were available. However, such data are not yet published for either E,(z) or &+., and it appears difficult to design experiments which would provide them. In contrast, the formulation described in Section 3 involves terms which are measurable, ~though the experiments needed to provide good values for r and E have not yet been done. As in the friction analysis of Tsuya  and that of the present authors, the analysis of Suh and Sridharan [ 47 ] involves consideration of the energy dissipated during plastic deformation. However, there are some major differences in approach, p~icul~ly in the treatments of strain and in the ways in which friction and wear are interrelated. Suh and Sridharan define an equivalent plastic strain F as the permanent deformation which occurs during the N cycles of loading which precede a typical delamination event. The corresponding plastic work is then given as
The coefficient m is the ratio of the maximum equivalent strain t-t for a given cycle to the net plastic strain AT, C is an equivalent stress (assumed constant), b is the average width and i the average length of a typical wear particle, and .F* is a decreasing function of the distance .a from the wear surface. The strain profile -* E has been taken from the data of Agustsson  for an AISI 1020 steel. These data were baaed on the grain intercept method described by Dautzenberg and Zaat 1’741, which may not be directly applicable across boundaries between regions of different microstructural character. Thus, it may not be an appropriate method to use in the cellular region near the surface, where grain boundaries and other bulk features are totally obliterated [ 80, 811. In fact, in this highly strained region there appears to be no simple standard technique available for measuring plastic strain. Therefore, the values reported for E* in the cellular region are questionable. The major difficulty with the experimental strain profile used by Suh and Sridharan is that it is the steady state strain profile resulting from the
entire previous plastic deformation history; i.e. it is not the strain profile developed between wear particle delamination events. Closely related to this difficulty is the question of the proper strain term to use in the calculation of plastic work. Suh and Sridharan have recognized the potential usefulness of cyclic stress-strain curves for defining Fi and Aeip. Certainly, Aeip could be obtained from a set of cyclic stress-strain curves appropriate to the variable stress history of a wear surface exposed to a set of loading asperities. This would probably involve a compressive stress/zero stress cycle rather than the compressive/tensile cycle suggested by Suh and Sridharan and used more commonly in fatigue testing. However, they have not attempted to use a value for AeiP obtained from either type of cyclic stress-strain curve. Another problem with the approach used is in the curve fitting, in which the Ed data are approximated by Ed = ec - 1y.afor 0 < z < z, and 7 = er~-~ for z > z,. The value of z, is obtained by matching the values of these functions and their slopes. This procedure has no connection with microstructural features, such as the depth of the cellular layer, and the resulting value of z, is not simply related to any material property. In using eqn. (3) Suh and Sridharan appear to have fixed the strain profile to the original interface position, rather than allowing it to progress into the material as delamination wear occurs. However, if eqn. (3) were correct for one wear particle, it would be correct for the next one as well. Yet, after one delamination process the additional work needed to delaminate the next layer is given as
where 6 is the thickness of a typical debris particle. If one continued in this manner for later delamination events, one would subtract another term each time and A IV, would steadily decrease. This would be inconsistent with the existence of steady state friction and wear. The problem here may arise from the attempt to tie friction too directly to wear events. Suh and Sridharan consider the work in a selected volume which will become a debris particle after N cycles. However, their emphasis is on the delamination event rather than on the plastic work dissipated during each of the cycles between delamination events. Therefore they do not predict the friction coefficient directly. Instead, they derive an expression for p/K, where K is the wear factor which is defined as wear volume divided by the product of the normal load and the sliding distance. Hence their result involves two complex phenomena instead of one. Figure 5, from the work of Lancaster , shows a good example of a materials system for which the relation between changes in friction and wear rate is far from clear.
Fig. 5. Variation of wear rate and coefficient of friction with temperature for 60-40 pins on high speed tool steel rings (adapted from ref. 82).
Fig. 6. Approximate variation of friction coefficient with load. This pattern is not necessarily typical, but it is one of those commonly encountered [ 22, 621,
Liu [ 391 has noted that changing the rider material in a pin-and-disk experiment does not change the friction coefficient if the rider is harder than the disk. This observation is consistent with the predictions of our model, since the plastic deformation would be restricted to the disk material in both cases. As the hardness and strength of the rider approached that of the disk, the rider would contribute to the friction by plastic deformation until both rider and disk contributed comparable amounts to the friction, differing partly because of geometrical factors [ 831. For a rider material much softer than the disk material friction would be largely determined by plastic deformation in the rider, and changing to another hard disk material should have little effect on friction. The load dependence of friction has been studied by many investigators. Their results include evidence of friction coefficients that are independent of load as well as examples of gradual changes, sudden changes and increases and decreases with increasing load. One of the commonly reported patterns is like that shown in Fig. 6 (e.g. refs. 22 and 62). The coefficient of friction is high for light loads and falls to a constant level at higher loads. This pattern has been found for a wide variety of bulk metals and thin metal films. One explanation that has been proposed is that the surface region recrystallizes at higher loads  . Another explanation is based on an increasing fraction of surface asperities deforming elastically rather than plastically
[221. Behavior like that represented in Fig. 6 can also be discussed in terms of the friction model presented in Section 3. At the lightest loads it would be difficult to form a textured cell layer. One could even imagine a critical
thickness of such a layer as being of the order of one cell thick, or less than 1 pm. A simple experiment could help to distinguish between this general mechanism and the one proposed in ref. 22. After a series of steady state measurements were made with increasing load, the load could be returned to a low value. The mechanism described in ref. 22 should yield data that would fall on the initial curve when plotted, whereas the mechanism suggested here would yield friction coefficients that would remain low for long times. This would be a pseudo steady state situation, because the cell layer formed at a high load would remain largely unchanged at the lower load. It should be emphasized here that our model is intended to apply only to steady state conditions. Transitions induced by changes in input conditions can only be incorporated if sufficient time is given to re-establish the steady state. Many of the friction experiments described in the literature were not run for times long enough to achieve steady state conditions. In fact, much of the published friction data have been for single-pass experiments. These are difficult to analyse because of the complicated and variable conditions which exist during break-in. We picture the break-in period as the time needed to deform surface asperities and to create a constant (in a statistical sense) surface topography and near-surface microstructure. At steady state the situation is much simpler. From our model one can predict that, if one waits long enough, measurements will be independent of initial surface finish. Of course, in practical devices, especially those with close tolerances, surface finish can be critical. However, for simple friction tests, if one waits for steady state conditions, the initial surface finish will not be important. Therefore one should find the same results at steady state for surfaces that have been prepared by machining, abrasion, mechanical polishing, chemical polishing, etching or electropolishing. Since most of the strain appears to be localized in the textured region created during break-in, the initial microstructure in single phase materials should also have very little influence on the steady state friction results. Therefore, for a given composition single crystals of different orientations and polycrystals with various grain sizes should all give about the same values for the friction coefficient. It is even possible that certain multiphase materials will have constant values of friction coefficient even when the bulk composition and hardness are variable. These would include alloys in which a solute has low solubility in the matrix and forms a finely dispersed precipitate. An example would be copper-beryllium. In such a material, as suggested in ref. 55, the nearsurface microstructure created during sliding could accommodate a large amount of solute. Therefore the fine precipitates would not be present in the highly strained region. A similar situation might be expected for various series of steels in which the substitutional solutes were held constant and the carbon content in the bulk microstructure was the principal variable. In the absence of
differing thermal effects or complications due to environmental interactions, the steels with the highest carbon content and highest bulk hardness should have the lowest friction coefficient during break-in, but the friction coefficients for these steels should converge to nearly the same values at large sliding distances because the near-surface microstructures would be similar (Fig. 7). The carbon associated with various high-carbon components of the bulk microstructure migrates to the cell walls, which can accommodate a larger amount of carbon (or other interstitial solute) [ 84 - 861. The result is that martensite and carbides in this region disappear, the martensite by direct diffusion and the carbides by deformation-induced dissolution and migration. These processes leave cells of a-ferrite with a small amount of carbon in solution and cell walls with a higher carbon concentration in the form of carbon trapped in the field of dislocations comprising the cell walls, A similar situation has been reported for heavily drawn wires [84,85] /I
Fig. 7. Predicted steels.
/.I with sliding distance
S for a variety
Borgese and coworkers [ 87, 881 have shown that this is in fact what happens in bearings made from AISI 52100 steel. A clear dislocation cell structure was observed by electron microscopy. Electron diffraction showed that the lattice parameter of the material near the surface was essentially identical with that of a-ferrite and fine carbides were noted at cell walls after annealing. Since this example is for rolling friction rather than simple sliding, it should be emphasized that plastic deformation is important for both cases [22, 87 - 891 and the strain profiles for rolling friction are similar to those observed in sliding [ 901. It is interesting to note that 52100 steel, as used for bearings, is usually given a double heat treatment with a final low temperature temper at about 510 K (233 “C). This is consistent with a duplex carbide structure with small to medium spheroidal carbides plus very fine carbides (probably produced during the last temper) . Also, the stress-strain curve for 52100 subjected to this set of conditions is fairly flat at large strains  . This
would be consistent with dissolution of the fine carbides in the highly strained region. A similar mechanism may be responsible for the observation that gray cast iron can be hot tinned easily if it is first shot-blasted with fine grit. The surface layers are severely deformed and “the majority of the graphite flakes near the surface appear to be disintegrated and dispersed, leaving a ferrite structure” [ 921. This work raises the possibility that certain cast irons may exhibit only weak dependence of the coefficient of friction on carbon content and bulk microstructure. Our model of friction has been developed primarily for simple sliding situations. However, under abrasive conditions, when a significant fraction of the debris indicates that cutting action is present, there will still be extensive shear deformation from local sliding. Therefore, the measured friction should be at least partly explainable in terms of the model presented here. If grit size is very small and if grit particles are relatively smooth, it should be difficult to distinguish abrasion from simple sliding. The two terms are part of a continuous spectrum of conditions in which various combinations of microcutting and sliding may be present.
For the steady state model of friction described in this paper, most of the deformation energy is dissipated in a region of average thickness t below the wear surface. This means that the frictional heat source is not a twodimensional heat source localized at the surface. Therefore, it is incorrect to speak of frictional heat as being generated at the interface. Instead, the heat source is dispersed over a volume. This has important implications for estimating surface temperatures, which are often difficult to measure directly. It is well known that solutions of the heat flow equation with localized heat sources indicate higher maximum temperatures than solutions for more dispersed heat sources. Therefore it is not surprising that evidence for high surface temperatures cannot always be found in sliding systems . Of course, under appropriate conditions high temperatures can result from either type of heat source, but the dispersed heat source must always give a lower surface temperature than a localized heat source if other conditions are comparable [94,95].
The existence of a steady state regime in friction and wear tests indicates that the material near the surface has a low net work-hardening rate. It seems reasonable that an ideal material for low friction would be one which has a low work-hardening rate near the surface and a high work-
hardening rate in the interior. This would force deformation toward the surface, reduce friction, delay fracture and minimize the maximum size of wear particles. The highly deformed layer which develops near the surface appears to be the result of nature’s attempt to reach that set of conditions. The model presented in this paper involves friction as a continuous process. Wear, however, is more complicated than friction, because it involves plastic deformation plus localized fracture events. These discrete events contribute in a random manner to friction but they are not required for friction; i.e. one can have friction between wear events or even without wear. Thus, friction and wear are related, but not in a simple way, and accurate theoretical correlations of friction and wear have yet to be obtained.
Acknowledgments It is a pleasure to acknowledge project support under contracts administered by R. J. Reynik (National Science Foundation), G. Mayer (Army Research Office) and R. S. Miller (Office of Naval Research). We also thank Professor David Tabor for helpful comments on this work.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
D. Dowson, History of Tribology, Longman, London, 1978. G. Amontons, MBm. Acad. R., (1699) 206. P. de la Hire, Histoire de l’Acad6mie des Sciences, 1732, p. 104. Euler, Histoire de l’Acad8mie Royale des Sciences et Belle-Lettres, 1748, Berlin, 1750, p. 122. B. F. de Belidor, Architecture Hydraulique, ou 1’Art de Conduire, d’Elever et de Menager les Eaux, Jombart, Paris, 1737. C. A. Coulomb, MBm. Math. Phys. Acad. R., (1785) 161. S. K. Kotelnikov, Instruction on the Equilibrium and Movement of Bodies, St. Petersburg, 1774 (in Russian). G. Rennie, Philos. Trans. R. Sot. London, 34 (1829) 143. A. J. Morin, Nouvelles Experiences sur la Frottement faites 1 Metz en 1833, Paris, 1835. A. J. Morin, C.R. Acad. Sci., 7 (1838) 835; 10 (1840) 101; 12 (1841) 211; 13 (1841) 1022. G. A. Hirn, Bull. Sot. Ind. Mulhouse, 26 (1854) 188. J. T. Desaguliers, Philos. Trans. R. Sot. London, 33 (1724) 345. S. Vince, The motion of bodies affected by friction, Philos. Trans. R. Sot. London, (178E A. Ewing, Notes Proc. R. Inst., 13 (1892) 387. G. A. Tomlinson, Philos. Mag., 7 (1929) 905. W. B. Hardy, Collected Works, Cambridge Univ. Press, Cambridge, 1936. R. Holm, Wiss. Veriiff. Siemens-Werken, 17 (1938) 38. R. Holm, Electrical Contacts, Almquist and Wiksells, Uppsala 1946; Springer, Berlin, 1958. H. Ernst and M. E. Merchant, Proc. Special Summer Conf. on Friction and Surface Finish, M.I.T. Press, Cambridge, Mass., 1940, pp. 76 - 101. F. P. Bowden and D. Tabor, Bull. 145, Council for Scientific and Industrial Research, Commonwealth of Australia, 1942.
363 21 F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Oxford Univ. Press, Clarendon Press, Oxford, 1954. 22 F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Part 2, Oxford Univ. Press: Clarendon Press, Oxford, 1964. 23 A. P. Green, J. Mech. Phys. Solids, 2 (1954) 197. 24 A. P. Green, Proc. R. Sot. London, Ser. A, 228 (1955) 191. 25 T. Liu, Wear, 7 (1964) 163. 26 J. S. Courtney-Pratt and E. Eisner, Proc. R. Sot. London, Ser. A, 238 (1957) 529 550. 27 0. L. Anderson, Wear, 3 (1960) 253 - 273. 28 R. D. Mindlin, J. Appl. Mech., 71 (1949) 259. 29 P. K. Gupta and N. H. Cook, Wear, 20 (1972) 73 - 8’7. 30 D. Tabor, Surf. Colloid. Sci., 5 (1972) 245 - 312. 31 D. Tabor, in J. M. Blakely (ed.), Surface Physics of Materials, Vol. 2, Academic Press, New York, 1975. 32 D. H. Buckley, in Wear of Materials 1977, Proc. Int. Conf. on Wear of Materials, St. Louis, 1977, ASME, New York, pp. 12 - 29. 33 J. Leslie, An Experimental Inquiry into the Nature and Propagation of Heat, London, 1804. 34 E. Rabinowicz, Friction and Wear of Materials, Wiley, New York, 1965. 35 J. J. Bikerman, Wear, 39 (1976) 1 - 13. 36 Y. Kimura, in Proc. Int. Conf. on Fundamentals of Tribology, M.I.T., Cambridge, Mass., June 1978, M.I.T. Press, Cambridge, Mass., 1979. 37 M. Moore, Wear, 17 (1971) 51 - 58. 38 M. A. Moore and R. M. Douthwaite, Metall. Trans., 7A (1976) 1833 - 1839. 39 J. Liu, in P. J. Bryant, M. Lavik and G. Salomon (eds.), Mechanisms of Solid Friction, Elsevier, Amsterdam, 1964, pp. 163 - 174. 40 D. Walton, J. Appl. Phys., 33 (1962) 519. 41 D. Walton, Wear, 6 (1963) 257 - 261. 42 L. Giimbel, Reibung und Schmeirung im Maschinenbau, Krayn, Berlin, 1925, pp. 7 26. 43 G. W. Rowe, in P. J. Bryant, M. Lavik and G. Salomon (eds.), Mechanisms of Solid Friction, Elsevier, Amsterdam, 1964, p. 204. 44 I. V. Kragelskii, Friction and Wear, Butterworths, London, 1965. 45 W. A. Glaeser, An Engineer’s Guide to Friction, Battelle Rep. DMIC Memo 246,197O. 46 T. Wanheim, N. Bay and A. S. Petersen, Wear, 28 (1974) 251 - 258. 47 N. P. Suh and P. Sridharan, Wear, 34 (1975) 291 - 299. 48 A. S. Argon, Mechanical properties of near-surface material in friction and wear, Proc. Int. Conf. on Fundamentals of Tribology, M.I.T., Cambridge, Mass., June 1978, M.I.T. Press, Cambridge, Mass., 1979. 49 D. A. Rigney, Discussion of paper by A.S. Argon, Proc. Int. Conf. on Fundamentals of Tribology, M.I.T., Cambridge, Mass., June 1978, M.I.T. Press, Cambridge, Mass., 1979. 50 D. A. Rigney and J. P. Hirth, Plastic deformation and sliding friction of metals, Proc. Int. Conf. on Fundamentals of Tribology, M.I.T., Cambridge, Mass., June 1978, M.I.T. Press, Cambridge, Mass., 1979. 51 P. Gordon, Trans. Am. Inst. Mech. Eng., 203 (1955) 1043. 52 L. M. Clarebrough, M. E. Hargreaves and G. W. West, Proc. R. Sot. London, Ser. A, 232 (1955) 252. 53 N. P. Suh, Wear, 25 (1973) 111 - 124. 54 Y. Tsuya, Microstructures of wear, friction and solid lubrication, Tech. Rep. No. 81. Mechanical Engineering Laboratory, Igusa Suginami-ku, Tokyo, 1976. 55 D. A. Rigney and W. A. Glaeser, Wear, 46 (1978) 241 - 250;Proc. 1st Int. Conf. on the Wear of Materials, St. Louis, 1977, ASME, New York, p. 41. 56 D. R. Wheeler and D. H. Buckley, Wear, 33 (1975) 65. 57 R. C. Bill and D. Wisander, Wear, 41 (1977) 351. 58 W. M. Stobbs, J. S. Kallend and J. A. Williams, Acta Metall., 24 (1976) 1083 - 1093.
364 59 Metals Handbook, Vol. 8, Am. Sot. Metals, Metals Park, Ohio, 1973, p. 220. 60 N. Ohmae, Transmission electron microscope study on inter-relationships between friction and deformation of copper single crystals, Proc. Int . Conf. on Fundamentals of Tribology, M.I.T., Cambridge, Mass., June 1978, M.I.T. Press, Cambridge, Mass., 1979. 61 Shimadzu Seisakusho Ltd., Kyoto, Japan, Microhardness tester literature, 1976. 62 D. H. Buckley, NASA Tech. Memo., TMX71781,1975. 63 W. C. Leslie and R. J. Sober, Trans. Am. Sot. Met., 60 (1967) 459. 64 D. Kalish and M. Cohen, Trans. Am. Sot. Met., 62 (1969) 353. 65 G. C. Rauch and W. C. Leslie, Metall. Trans., 3 (1972) 373. 66 R. Chait, Metall. Trans., 3 (1972) 365. 67 J. P. Hirth and M. Cohen, Metall. Trans., 1 (1970) 3. 68 D. C. Drucker, Metall. Trans., 4 (1973) 667. 69 F. B. Fletcher, M. Cohen and J. P. Hirth, Metall. Trans., 5 (1974) 905. 70 W. A. Spitzig, R. J. Sober and 0. Richmond, Acta Metall., 23 (1975) 885. 71 T. M. Ahn, Ohio State Univ., Columbus, Ohio, unpublished research. 72 Metals Handbook, Vol. 1, Am, Sot. Metals, Metals Park, Ohio, 1961. 73 N. P. Suh, S. Jahanmir and E. P. Abrahamson, II, The Delamination Theory of Wear, Report to ARPA, 1974. 74 J. H. Dautzenberg and J. H. Zaat, Wear, 23 (1973) 9 - 20. 75 D. H. Buckley, Friction, wear, and lubrication in vacuum, NASA Spec. Publ., SP-277, 1971. 76 L. M. Clarebrough, M. E. Hargreaves and M. H. Loretto, Acta Metall., 6 (1958) 725 735. 77 L. M. Clarebrough, M. E. Hargreaves and M. H. Loretto, Changes in internal energy associated with recovery and recrystallization. In L. Himmel (ed.), Recovery and Recrystallization of Metals, Interscience, New York, 1963, pp. 63 - 130. 78 D. McLean, Mechanical Properties of Metals, Wiley, New York, 1962, Chapter 5. 79 G. Agustsson, Strain field near the surface due to surface traction, M.S. Thesis, Mechanical Eng., M.I.T., Cambridge, Mass., 1974. 80 L. E. Samuels, Metallographic Polishing by Mechanical Methods, 2nd edn, Pitman, Melbourne and London, 1971. 81 Y. Tsuya, The behavior of the layer damaged by friction, Bull. Jpn Sot. Precis. Eng., 2 (1967) 214 - 220. 82 J. K. Lancaster, Proc. Phys. Sot., London, 70 (1957) 122. 83 M. Antler, IEEE Trans. Parts, Hybrids, Packag., PHPl (1973) 4 - 14. 84 V. K. Chandhok, A. Kasak and J. P. Hirth, Trans. Q. Am. Sot. Met., 59 (1966) 288 301. 85 J. D. Embury and R. M. Fisher, Acta Metall., 14 (1966) 147. 86 D. Langstaff, Ph.D. Thesis, Ohio State Univ., Columbus, Ohio, 1975. 87 J. A. Martin, S. F. Borgese and A. D. Eberhardt, J. Eng. Ind., (1965) paper no. 65WA/CF-4; J. Basic Eng., 88 (1966) 555. 88 S. Borgese, J. Lubr. Technol., (1970) 54 - 58. 89 K. F. Dufrane and W. A. Glaeser, Wear, 37 (1976) 21 - 32. 90 D. M. Fegredo and C. Pritchard, Wear, 49 (1978) 67 - 78. 91 C. A. Stickels, Metall. Trans., 8A (1977) 63 - 70. 92 C, J. Thwaites and J. J. Day, Metallurgia, 56 (1957) 263 - 270. 93 E. S. Sproles, Jr., and D. J. Duquette, Wear, 47 (1978) 387 - 396. 94 H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, London, 1959. 95 S. Malkin and A. Marmur, Wear, 42 (1977) 333 - 340.
Possible sources of energy dissipation in friction In general, a frictional force can be produced by storing or dissipating energy in a number of ways. Only a few of these have received much attention, and only a few provide the dominant contribution in typical cases. Nevertheless, for completeness, we enumerate in this appendix the various possible terms for crystalline solids. In most cases there is a parallel between these work terms and the corresponding contributions to internal friction in crystals, where the dissipation takes place on a microscale. First, there are a number of anelastic processes which in principle would be reversible for slow enough sliding rates, thus yielding no friction force, but which would give an irreversible contribution at higher rates. One example would be a geometric effect associated with surface irregularities (Fig. Al). For irregularities of the type shown in Fig. Al(a) the elastic work would be expected to be reversible (i.e. as much work is done by compressed asperities on the average as is done upon them). For those of Fig. Al(b), however, the elastic energy would be released abruptly without doing work and thus would be dissipated. (a)
Fig. Al. Effect of asperity shape on elastic energy dissipation. The arrows represent the relative motion.
Another category of anelastic contributions has an analogy in anelastic dissipation for single dislocation motion which is reviewed by Nabarrc [Al] . These contributions would include the thermoelastic effect, in which the elastic strain adiabatically heats or cools a region locally, leading to heat flow which would give an irreversible contribution if out of phase with the strain variation. There could be an analogous electron-elastic coupling term in which the elastic strains change the Fermi level locally, leading to energy dissipative electron flow. Other effects of this type include excitation of electrons, phonons or photons at the surface and electron and phonon scattering. A third category of dissipation includes stress-induced Rhase transformations such as martensite formation and stress-induced twinning [A2, A3], both of which can be anelastic (i.e. in principle reversible) in special cases [A4], but which are more likely to be completely irreversible or
plastic. Damping associated with stress-induced local atomic diffusion of atoms [A51 , such as the Snoek effect of single diffusive jumps of interstitial atoms, would give a contribution. Somewhat related would be structural rearrangements of atoms at the surface, whether of the solvent or of adsorbed solutes. Finally, in many cases the major contribution to friction is most likely to involve effects associated with dislocation motion [Al, A6]. These too can be anelastic, as for the bowing out of dislocations between pinning points, but they are more likely to be irreversible or plastic. Work done by dislocations moving against various constraints (phonon or electron emission or scattering, vacancy or interstitial generation at jogs, increases of dislocation line length, work against image or interaction forces, breakaway of dislocation loops from solute atoms or precipitates) would all contribute to work either stored as the energy of cold work or dissipated as heat, either of which would in turn contribute to the friction force. Even in the idealized case of adhesion, in which two crystals move over one another with only interface atomic bonds being broken, moving interface dislocations [A71 would provide the actual mechanism of bond breaking and energy dissipation. Without such dislocations, the resistance of the interface to shear would be given by a calculation analogous to that of Frenkel [AS] for shear of a perfect crystal. This would give a friction coefficient two or three orders of magnitude larger than those observed in practice. In this appendix we have tried to present a brief but fairly complete list of possible mechanisms for the energy dissipated when two crystalline solids slide against each other. The emphasis in this paper has been on plastic deformation arising from dislocation motion, which is widely recognized as an important energy dissipation process. Therefore, the model presented in Section 3 for steady state friction is directly related to energy dissipation. Before steady state conditions are achieved energy storage is accomplished by generation of defects which eventually rearrange to form a steady state microstructure. Thus the friction model described in this paper satisfies a basic requirement of any friction mechanism; the connection with energy storage and dissipation should be clear.
Models for the plastically deformed region in a friction experiment Once steady state is achieved in a slider-substrate friction experiment there are several limiting cases for the amount of plastic work associated with slider motion. The most probable situation at low temperatures and moderate to large sliding speed is illustrated in Fig. A2. One can imagine that this figure represents an entire slider in contact with the substrate at large loads or an asperity in contact with the substrate at low loads. As the slider advances, plastic work is performed in the vicinity of the leading edge of the slider. Work hardening ensues and most of the region under the slider
Fig. A2. A cross-sectional view of the slider moving at velocity u on a steady state wear track: (a) identification of hardened and recovered regions; (b) for an increase 6x in sliding distance the hardened region is extended by the volume indicated by crosshatching. is hardened to the point where deformation ceases and the load in this portion is supported elastically. Recovery occurs but only after the slider has moved past the region. This case is of the type describable by Tsuya’s deformation integral [ 541. As the slider advances by 6x new work per unit deforming volume 7~ is done, where 7 is the shear flow stress resolved in the sliding direction of the hardened region and E the resolved plastic shear strain. The deformed volume is wtsx where w is the width of the deforming region (the direction normal to the view of Fig. A2). This most probable case is the one discussed in detail in the text, and a work balance directly gives eqn. (2). At lower sliding velocities or higher temperatures, the situation shown in Fig. A3 can occur. Here deformation and work hardening occur at the leading edge of the slider (or asperity) but dynamic recovery occurs. Thus, the entire region in contact with the slider is undergoing plastic deformation. The volume deforming is now wtX and the rate of plastic work per unit volume is 76 where k is the plastic strain rate. The product of these two terms must equal the rate Fu of work done by the driving force F:
The average strain rate will equal the net strain E divided by the exposure time of a given infinitesimal volume element to the slider load, which is given by X/u. Substituting these relations into eqn. (Al), we find Fv = wtrcv
This yields the result
I I I J’Dynam’c“H,,rdened Recovered /I’ ,’ ReCOYery,’ I
Dynomx Recovery and New Deformation ’
Fig. A3. Cross-sectional view of the slider moving at velocity deforming and recovering dynamically ; (b) case intermediate and A3(a); (c) deformed regions for the case of Fig. A3(b).
F p =
u: (a) hatched region is that between those of Figs. A2( a)
L = I-
which is identical to eqn. (2). Similarly, the intermediate case shown in Fig. A3(b), with the deformation volume as shown in Fig. A3(c), is described by eqn. (2). Thus, eqn. (2) applies for all cases.
Model for the geometric
w and t
The parameter w need not correspond to the physical dimension of the slider, as illustrated in Fig. A4, where for generality a slider of rectangular cross section is depicted for six cases A - F. At low loads, the load will be sustained by a few asperity contacts. As a simple example, let us suppose that at a given load each asperity has the same circular area of contact with linear dimension d, . Each of these asperity contacts produces a plastic region of dimension d, . At low loads there is little statistical probability that two such zones overlap in their projections along X or Y. Hence, for either case A
‘\ \ :
I’ ‘\_________ ,’ _________*’ W‘ 4 (4 ‘1
Fig. A4. View normal to substrate of slider with rectangular cross section. The sliding directions for cases A to F are denoted by arrows. The solid circles represent regions of asperity contact and the regions of plastic deformation are within the broken lines: (a) small load; (b) intermediate load; (c) large load; (d) very large load.
or case B, w = nd, , where n is the number of asperity contacts. Moreover, if the shape of the asperity or the indentor is constant, then the size invariance of elastic-plastic solutions for constant geometrical shape problems* indicates that t is proportional to d,, or t = Cld,. Thus eqn. (2) becomes Clndp2rc I-(=_-_ L
However, the normal load L is sustained by a contact area proportional to ndc2 and this in turn is proportional to nd p2. The net coefficient of proportionality C2 should be constant for constant shape asperities or indentors for the reasons discussed above. Therefore the normal contact stress is given by u = L/C2ndp2. If this expression is combined with eqn. (A3), the result is P = C1re/Cso, which is similar in form to expressions resulting from simple adhesion theory [21,22,31]. For this case, p is independent of X, Y and L. In a more refined model (as in refinements relating true contact area to load in adhesion theories) a load dependence of p would appear, but it would remain independent of X and Y. For the intermediate load cases C and D, the projections along X and Y of the asperity contacts completely overlap. Thus, w = X in case D, w = Yin case C, but one still expects t = Cld,, giving
*If the shapes of the asperities are variable, then Cl would be replaced by a function of d,.
Cld, Y7e L
Since t or d, is proportional to L ‘I2 for this case one expects g both to decrease with increasing load and to’depend on the physical dimension of the slider. For the large load cases E and F, complete contact occurs. Again w = X or Y but now t is not related to asperity contact. Hence one expects tXTE p = .--L-
tYTE p =L
Here P still depends on the physical dimension of the slider but the dependence of ~1on load is not explicitly predicted. However, it is still likely to increase as L” with n < 1, so one expects P to decrease with increasing load. At still higher loads, the situation of Fig. A4(d) might obtain, with P again becoming independent of specimen geometry. At such large loads, however, the likelihood of a dominance of plowing in the work balance becomes greater as does the likelihood of seizing, which would terminate steady state behavior. Finally, it should be emphasized that many simplifying assumptions have been used in this appendix and elsewhere in the paper. In particular, if u is not a constant but depends on local microstructure and therefore on position, then it would be better to use u=-
1 s ACJ
with JdA, equal to the true contact area. Similarly, instead of using rough average values for 7 and E, a more detailed model would involve an exact average 1 ” 7E = TE dV J Vcl with V the volume refinements would tions which are not have been used for References Al A2 A3 A4 A5 A6 A7 A8
of the plastically deforming zone. Inclusion of these require experimental data on stress and strain distribureadily available. Therefore, constant or average values these parameters in this paper.
F. R. N. Nabarro, Theory of Crystal Dislocations, Oxford Univ. Press: Clarendon Press, Oxford, 1967, chap. 7. P. Van Houtte, Acta Metall., 26 (1978) 591. M. Kato, R. Monzen and T. Mori, Acta Metall., 26 (1978) 605. J. Perkins (ed.), Shape Memory Effects in Alloys, Plenum, New York, 1975. J. Snoek, Physica, 6 (1939) 591; 9 (1942) 862. J. P. Hirth and J. Lothe, Theory of Dislocations, McGraw-Hill, New York, 1968, chap. 15. W. Bollman, Crystal Defectsand Crystalline Interfaces, Springer Verlag, Berlin, 1970. J. Frenkel, Z. Phys., 37 (1926) 572.