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An energy-based model of friction and its application to coated systems
195
Wear, 72 (1981) 195 - 217 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
AN ENERGY-BASED MODEL OF FRICTION AND ITS APPLICATION TO COATED SYSTEMS
P. HEILMANN and D. A. RIGNEY Department (U.S.A.)
of Metallurgical
Engineering,
The Ohio State
University,
Columbus,
OH 43210
(Received March 24,198l)
Summary
An energy-based friction model is used to develop expressions for the friction coefficient; these expressions depend on familiar mechanical parameters, stress-strain curves and microstructural features of the materials. The main assumption is that the frictional work performed is equal to the work of plastic deformation during steady state sliding. The results include as a special case the well-known expression for the friction coefficient derived from adhesion theories. However, the new model can also be applied to more complex systems such as those involving coatings. In particular, for soft coatings on hard substrates as well as for hard coatings on soft materials, the friction coefficient can be predicted as a function of layer thickness in accordance with reported observations. Furthermore, guidelines are presented for achieving low friction by using coatings. The new model can also be applied to systems involving reaction layers, transferred material and solid lubricants of varying thickness. In addition, the model offers some insight into the variation in friction coefficients with time and into the processes which occur during running-in.
1. Introduction
Examples of friction are commonly observed in everyday life but despite the familiarity of this subject frictional phenomena are not well understood. Friction seems to involve complex processes. Therefore, although a number of simple models have been proposed, they are inadequate for explaining or predicting most friction phenomena. Neither acceptable explanations of the mechanisms involved nor reliable predictions of friction coefficients are currently available for most materials. The most widely accepted models of friction are based on adhesion and/or the interlocking of asperities. Descriptions of these models can be found readily in other publications, e.g. refs. 1 - 5, and they are not presented in detail in the present paper.
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Friction is often regarded as purely a surfaee’process. However, it usually involves changes in structure and properties below the surfaces of interacting materials and it can be accompanied by loss of material from either or both of the sliding members. Such damage might be very small, in which case it would not be detected over a long period, or the damage might be large enough to be recognized as wear. In both cases, irreversible processes are involved and this is one of the factors which makes it difficult to predict what will happen. Tribologists do not yet agree on which properties of sliding materials and which additional physical parameters are essential and should be included in a model of friction. Apparently, many factors influence friction [ 2, 4 - 71. If more of these are incorporated in models of friction, the models will become increasingly complex. At this stage it seems more appropriate to select only the most impo~nt parameters; special care is taken to see that none of these is omitted. Most available models of friction use a very limited number of parameters. With the exception of a yield stress or a hardness parameter, basic materials characteristics tend to be neglected. In particular, microstructural information is generally omitted. Yet, details of microstructure profoundly influence the nature of the materials behavior, and they should be included in a realistic friction model. Except for the lowest loads, some plastic deformation occurs near the surfaces of sliding materials, even under lubricated conditions. In fact, a principal mechanism for the dissipation of frictional work is through plastic deformation [ 5 3 . Therefore, the ease of plastic deformation and associated parameters such as those of the microst~c~re should be important considerations in setting up a friction model. As in other physical processes, energy considerations can be very useful. This is true even for an irreversible process like friction. The usefulness of energy in a friction model is that it introduces a kind of average to the individual events which take place and which may be of a wide variety. The energy point of view of friction becomes even more valuable if one makes two simplifying assumptions. The first of these involves the generation of wear particles. These appear as discrete local events which should contribute to the “noise” of a friction versus time curve but which need not be included in the basic friction model. Secondly, the model described here assumes that steady state conditions are achieved. Together, these assumptions lead to a highly idealized model which may never correspond in detail to actual sliding situations. However, the assumptions lead to the following conclusions. Frictional work will result in plastic deformation near the interface between interacting materials. Heat is produced during deformation and during subsequent recovery processes. This frictional heat will tend to raise the temperature of the friction couple and this in turn will change some of the mechanical properties of the materials, e.g. the workhardening rate and recovery rate. These in turn change the ease of plastic deformation and influence the frictional work until a dynamic balance is
197
achieved. At this steady state condition the average local temperature of the friction couple remains constant and the recovery just compensates the additional work hardening caused by the plastic defo~a~on during sliding. A full analysis would require calculation of this equilibrium temperature for the steady state from basic assumptions and structural properties. This approach could result in a calculated friction coefficient. However, the result would contain too many uncertainties and the effort would not be justified. Given the present state of knowledge, perhaps a better way is to measure some of the steady state parameters and to use them to calculate the friction coefficient p. This method is used in the following sections since, despite the simplifications used, it gives valuable insight into the dependence of the friction coefficient on structural parameters. It also has practical implications, since the results suggest ways to modify ~1.
2. Basic friction model 2.1. Material properties used in the friction c~icu~t~on Defo~ation near the surface of at least one of the infecting materials commonly accompanies sliding. The ease of this deformation can be described with the aid of a stress-strain curve; this curve reflects implicitly many basic characteristics of a material, e.g. the grain size and hardness. Therefore, the appropriate stress-strain curve can be taken as a convenient way to describe certain properties of a material in a concise way. Furthermore, during sliding, local regions in a material are exposed to different stresses; this results in various strains which are governed by the stress-strain curve. Choosing a proper stress-strain curve for a friction calculation is not easy. First, it should be described by a relatively simple formula for ease of calculation. A numerical representation might also be used, but the results would then be more difficult to interpret. Second, since shear stresses are involved during friction, an appropriate shear stress-shear strain curve is needed. However, most stress-strain curves are measured in simple tensile tests and they must be converted to shear curves by considering resolved shear stresses. Third, even if a suitable curve is available, the measured range of shear strains y is usually limited to low values, i.e. y 5 1. During friction, much higher strains will often arise; these will require the extrapolation of available stress-strain curves to higher strain levels. For low strainsthere exist several more or less sophisticated empirical laws to describe stress-strain curves. Reviews can be found in refs. 8 and 9 and, more recently, in ref. 10. A very widely accepted correlation between the shear stress 7 and shear strain y is 7 = kylln or 7 = kr”‘, depending on the definition of n and n’ [ 111. An even simpler law is 7 = h71j2, which is a good approximation for some materials, e.g. polycrystalline f.c.c. metals at low strains. At higher strains, the stress commonly deviates from these simple power laws and eventually reaches a saturation level [lo] .
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An empirical relation, introduced by Vote [12]- for compression stressstrain curves, incorporates a saturation stress 7,,, : 7 = Ti + (7,,,
-TijlII.
-&XP(-~)j
For consistency with the present paper, the original mathematical representation [ 121 and the symbols have been changed. Here Ti and yi are the shear stress and shear strain respectively at the beginning of the test. A shear stress-shear strain relation very similar to that of Vote will be used for the present friction model. Since Ti is usually small compared with T,,, and yi is much smaller than the strains actually achieved during steady state sliding, 7: and ri are set equal to zero. Furthermore, a square root will be introduced; this can be justified by two arguments (this is discussed below). The resulting stress-strain relation to be used for friction calculations is then 7=7 max(1 - exp(--cy )P2
(1)
The stress-strain curve which results from plotting this function is shown in Fig. 1.
ShearStraqy Fig. 1. Plot of a simple stress-strain function 7 = T,, (1 - exp(-cy)} 1’2 which reduces to a power law(T - y*‘2)at low strains and approaches a constant value T,, at high strains.
For small y values, the exponential function can be expanded easily and eqn. (1) reduces to a simple power law: 7 - yff2. The second reason for using an exponent equal to l/2 in eqn. (1) arises from the following semiquantitative derivation. Dislocation densities during plastic deformation have been carefully studied and a theoretical expression for the dislocation density p during stages II and III is presented in ref. 13. This work was later extended to higher strains by introducing a further correction [ 141. As pointed out recently [ 151, this model predicts an upper limit for the dislocation density at high strains; this is consistent with available experimental data. The dislocation density in stage III is given by P = pC expj-a(7
-yCN + z 11 -expC-Q(r
-yc)l]
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The parameters $2 and U are as defined in ref. 14. For strains larger than about 0.5, #is expression can be simplified to p = (U/Q){1 - exp(-SZ2y)). We can also use a commonly accepted relation between p and T (desc~bed in ref. 13): r = cvGbp112,where a x 0.5, G is the shear modulus and b is the dislocation Burgers vector. Combining these expressions results in a stress-strain relation equivalent to eqn. (1). In an earlier model [7 ] , which also used a stress-strain curve, a simpler function was used. This was a combination of the power law 7” yljn for low strains and a constant value of 7 for high strains. This function has the same limits as eqn. (1) but it has an arbitrary discontinuity in its derivative, which complicates its use in the friction model. Nevertheless, the results are very similar to those described in the present paper. However, some of the limitations of the earlier model have been removed in the present work. Of course, for some materials, eqn. (1) will not be a good representation for the stress-strain curve. Examples would include certain single crystals oriented for easy glide and perhaps materials involving deformation-induced phase transformations, such as me&stable austenitic stainless steels. For such cases, a model like the one described here could still be used but different stress-strain relations, tailored to the particular interactions, would have to be used. Alternatively, a numerical representation could be used for complex stress-strain relations. 2.2. Contact between the surfaces during friction Real surfaces are never absolutely smooth. Therefore, contact between two surfaces will occur first at local “points” determined by asperity profiles. If there are N of these local contacts, each of which has an average area A, then the total real contact area will be NA. Johnson [ 161 has analysed the mechanics of sliding asperity contacts for two cases: in one case adhesion aids stress transfer and the other case depends simply on the ~~rlock~g of asperities. For the following calculations, as long as some means exist for stress transfer, the choice is not important; therefore we make no assumptions about how the stress is transmitted from one surface to the other. The differences which Johnson reported can arise in the present model through differences in the extent of plastic deformation. The model described in the present paper is based on a simple hard rider sliding on a sample material which deforms elastically and plastically. For simplicity at this stage, the rider is assumed not to deform. However, this assumption is not essential for the approach taken and the model can be generalized later to allow deformation and energy dissipation in both members of the sliding system. Each asperity on the surface of the sample material will deform when it interacts with the surface of the slider. This deformation will continue until the contact fractures, if the interaction is caused by adhesion, or until the sliding motion separates interlocking asperities. During each local asperity
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contact, the stress and strain at the surface will grow from some initial value (which will depend on its stress-strain history) to some final value. Relaxation or recovery can occur almost simult~eously (dynamic recovery) or during the intervals between asperity contacts. If the relaxed stress is equal to the initial value, this will define the steady state conditions. At any given time, different asperities in contact with the rider will be at different stages of deformation. However, we can define an average surface stress 7, to characterize this deformation; it will represent the average over all individual stress values at all contact points. When steady state conditions are achieved, 7, will be time independent. Then, given r5, it is easy to calculate the plastic work which is done when the asperities are deformed by an additional virtual displacement 6x,. The friction sample can be treated as if it were a simple shear sample of contact area NA pre-stressed to 7,. The corresponding shear strain would be ys, and an additional displacement 6x, would provide an additional shear strain Ays_ For this simple case, with T, and Ay, held constant over the deformed volume V, the work done is simply 7, &s V. 2.3. Depth dependence of the def~rrn~t~o~ The simple expression derived in Section 2.2 for the plastic work during sliding is not realistic and therefore it is not very useful. It would apply only for a simple shear test with constant deformation over the entire volume. For the case of sliding friction, deformation is known to decrease relatively rapidly with depth [ 1’7,18] . Therefore, material at a distance z below the surface will be less stressed than surface material, and a displacement of 6r, imposed at the surface (z = 0) will be smaller for z > 0. The depth dependence of the displacement field 6x = 6x(z) will be considered first. Experimental data are available in the form of marker profiles developed during sliding [ 17, 18 ] . A marker is embedded in the material in such a way that, at the start of testing, the projection of the marker, viewed in a lon~tudin~ section, is pe~en~cul~ to the sliding surface and parallel to the z axis..After sliding occurs, the marker is bent over in the direction of sliding, and its shape may be described roughly by an exponential curve. This observed profile is the result of many small displacements 6x, at the surface and smaller values 6x(z) below the surface. If the displacement profile can be described by a simple exponential function, e.g. - exp(-az), then the individu~ displacement 6 x(z) can be written 6x(z) = &r, exp(-az)
(2)
This expression is consistent with the interpretation of 6x = 8x, at z = 0. The shape of this displacement function is shown in Fig. 2. The constant a can be determined by fitting an exponential curve to the appropriate measured marker profile. Since the virtual displacements decrease with depth, the associated shear strain increments A-y(z) will also decrease. The strains and displacements are related by differentiation, so that
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Depth.2
Fig. 2. Assumed variation in virtual displacement 6x with depth z below the material surface using 6x = 6x, exp(-ilz). Fig. 3. Depth dependence of the shear stress T(Z) for two different choices of the surface stress 7, using eqn. (5) (T,(l) > rst2)).
AT(Z) = - ;
6x(z)
= a6x, exp(-uz)
(3) The shear stress will also decrease below the surface. If we assume steady state conditions, the average shear stress should remain constant for a chosen value of z. This average shear stress, which is the sum of external and internal stresses at any given time, can be correlated with the shear strain by using the stress-strain curve (eqn. (1) and Fig. 1). Because of this correlation, we can think of the material as being internally stressed to a level T(Z) or, equivalently, we can think of the material as being strained to a level y(z). However, this value of y is not the same as the strain which can be measured by marker displacements. Instead, it is simply the strain which corresponds to a stress T on the stress-strain curve. This distinction is critical and, since it is a possible source of confusion, it is emphasized here. If the function y(z) is known, we can readily calculate the depth dependence of the stress T(Z). If AT(Z) satisfies eqn. (3) then it is reasonable to assume that y(z) decreases exponentially in the same way: ~(2) = yS exp(--az)
(4)
With the aid of eqn. (l), the average surface strain ys may be expressed in terms of the average surface stress 7,: TS=--f
lnjl-(
c)‘/
Combining this expression with eqns. (1) and (4) gives an expression for the depth dependence of the shear stress: T(Z) = ~,a, [l_~~_(
foci’“‘“‘]“’
This function is shown in Fig. 3 for two different choices of the average surface stress 7,. Through eqn. (5), the parameter 7, also characterizes the
(5)
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stress beneath the surface. If a and 7,aX are known, 7, can be determined by measurements of T(Z). The curves in Fig. 3 exhibit three distinct regions, especially for high values of 7,. Near the surface, 7 is nearly constant and very close to its maximum value r,,, . In the next region, T declines more rapidly until it approaches a level characteristic of the undisturbed bulk material. In the present paper this value 7i has been assumed to be zero; larger values of Ti could be included in a way similar to that of Vote. The neglect of Ti does not generally have a large effect on the calculated value of P because the incremental displacements and strains are very small below the near-surface region. The stress ~(2) depends on the material and the local microstructure, in particular on the elastic modulus, crystal structure and dislocation density. These might be related through 7 = ~Gbp’/~ [ 131 or through some more complicated expression. In any case, T(Z) might be determined by using these measurable quantities. The different regions of the T(Z) profile probably correspond to regions with different dislocation substructures, such as those reported recently by several research groups [ 19 - 221. For example, the relatively flat portion of the curve is probably associated with a fine cell or subgrain structure with approximately uniform dimensions, while the steeply decreasing portion of the curve should be associated with substructure dimensions which increase with depth. If such stress-structure or stress-dislocation density correlations are possible then we can estimate the average stress T, at the surface. Also, if we can match the regions of the ~(2) curve with the thickness of different substructure layers then the depth dependence of 7 used in the present model would be justified.
2.4. Calculation of the friction coefficient p The basic concept used to calculate p at the steady state, as outlined earlier [ 5, 71, is to equate the frictional work and the plastic work. Thus it is assumed that at the steady state all frictional work is used to deform the material plastically and that the work is dissipated as heat. Other dissipative processes, reviewed in ref. 5, are assumed to be less important for most materials (the principal exceptions to this are elastomers [ 161). The incremental strain at steady state is Ay when a hard slider moves a virtual distance 6x, over the surface of a plastically deforming sample material. The plastic work done is then
vJ,l
=~WY
dV
(6)
with the integration over the deformed volume. Each asperity contact is assumed to have an average area A, over which the stress is constant, and the deformed material below each contact is represented by a cylinder with its
203
axis parallel to z. With this approach, the stress will depend only on the depth z below the surface and the volume integral becomes
WPl
=NA
$~(y)i\r
dz
(7)
0
where NA is the total area of asperity contacts. Combining eqns. (3), (5) and (7) yields
which gives
or WPl
=
NA SX,T,,,F
(9)
where F(u) =
1- 2
ln(1 + U) - u ln(l - u2)
(10)
F(u) is a monotonic function, defined for 0 6 u G 1, and it has the limits = 1. It is displayed in Fig. 4; a non-linear scale is used to show that the function approaches unity as 7, approaches rmax, i.e. as u + 1. At steady state and with reasonably heavy loads, 7, and r,,, should be very nearly equal for the highly deformed material near the surface. The frictional work which is done by moving the rider a distance 6x, is F(0) = 0 and F(l)
Wf = /.lLsx,
(11)
where L is the applied normal load. The basic assumption of this model is that W,, and WU~,. can be equated. Therefore, from eqns. (9) and (ll), we obtain pL Sx, = NA SX,T,,,F
or
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Fig. 4. Dependence of the function F(u) (defined in eqn. (10))on u = T,/T,,.
F(u)+ 1
=~,'~*,X.
NA /J = -TInax L
(12)
When F(T,/T,,,) = 1, p reaches its upper limit pmax = (NA /L)T~~~. Therefore, p may be written as (13) This means that if the plot shown in Fig. 4 is scaled by pmax, we can obtain the friction coefficient as a function of the ratio of the surface stress 7, to the ultimate shear stress 7mBx.
2.5. Discussion From eqns. (10) and (12) it can be seen that three parameters determine P ; these parameters are T,,, , T, and NA/L. The first of these, rmax, is the ultimate shear strength of the material which can be achieved during shear. This is a basic material property, and it can be measured by an independent test. The second parameter, rs, is the average shear stress actually achieved at the surface during sliding. It may depend on many experimental parameters, e.g. load, temperature and sliding velocity, and also on materials characteristics such as crystal structure, microstructure, work-hardening rate and recovery rate. Given our present state of knowledge it would be very difficult or even impossible to calculate 7, as a function of all these variables for a given material. Therefore, instead of attempting such a calculation from first principles, it is probably better to measure 7,. One must be careful, however, because T, should be an average value of T(Z) at the surface, and the usual methods for measuring stresses, e.g. with X-rays, result in an average value for material in an extended region below the surface.
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An alternative way to obtain 7, is to use transmission electron microscopy (TEM) to measure the dislocation density created during sliding. The stress 7, (and also r(z)) could then be calculated from an expression such as 7 = CuGbp1i2[ 131 or from some other function relating p and 7. This method would have two main sources of error. One would involve the appropriateness of the chosen correlation between r and p . The other would be related to the unknown amount of recovery which might take place between the time of sliding and the time of observation by TEM. Another way to estimate T, is to use the shape of the r(z) curve, as shown in Fig. 3. It seems very probable that the flat region of r(z) which exists for high r, values corresponds to a highly deformed layer at the surface. The thickness of a surface layer with a particular microstructure can be measured easily and compared with the extent of the flat portions of calculated 7(z) profiles. The best fit would yield 7, directly, The third parameter needed to calculate p is the ratio of the real contact area NA to the applied load L. Since hardness is a familiar and readily available material parameter, a comparison of L/NA with the hardness might be useful. The penetration hardness is defined as p = load/(contact area). Depending on which kind of area (projected or real contact area) and which kind of test indenter is used, p could be represented by the Brinell, Vickers or Knoop hardness. During sliding, asperities act as similar indenters. In this case, shear forces are present in addition to compressive forces, and the real contact area may be different from that which supports a purely compressive load [ 1, 2) ; therefore, L/NA need not be the same as p. If we bear this precaution in mind, it is interesting to examine the form of eqn. (12) when L/NA ;t: p or (14) This result is very similar to the well-known formula derived from an adhesion theory and first in~oduced by Bowden and Tabor [ 11, i.e. P=
shear strength penetration hardness
(1’5)
where the penetration hardness is defined as in eqn. (14). The shear strength to be used in eqn. (15) is not clearly specified. It might be chosen as the yield strength in shear of the bulk material or it could be as high as T,~~. It has commonly been chosen at the low end of this range. The factor P(~,/T,,,) of eqn. (14) has no equivalent in eqn. (15). When stresses at the surface are high, as with continuing sliding at high loads, 7, is nearly equal to 7maxand F(r,/r,,, ) = 1. Then the new model will have the same form as the older adhesion model but the friction coefficient for the new model will be higher because of the higher value used for T,,,. Thus the new model at least partially removes one of the earlier criticisms of the Bowden and Tabor model, i.e. that it yielded a calculated ~1that was lower than values found by experiment.
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The new model and the familiar adhesion model give similar results only for the special case just discussed. The new model is much more general and it can be applied to many different sliding situations. The factor F(T,/~,,,) becomes more important when the surface is only moderately stressed, as with light loads. Thus the factor F(T,/~~~~ ) introduces a dependence of P on load as is often observed. This load dependence may be weak, because the factor NA/L also depends on load, but it varies in the opposite direction. Although the present model is intended to describe steady state conditions, it provides some insight into behavior during running-in (break-in) when 7, values are less than those developed at steady state. If 7, increases slowly during running-in, the steady state model should be applicable, at least for qualitative results. With 7, increasing with time, a plot of the friction coefficient versus time will be similar to a plot of F(~,/T,,,) uersus T,/T,,, (Fig. 4). Such variations in friction coefficient are often observed during running-in. Equation (12) provides a formula for calculating the friction coefficient if data are available for three clearly defined parameters, i.e. T,,,, T, and NA/L. However, these data are not at present available from experiments. A qualitative estimate of the variables in eqn. (12) indicates that the use of this formula provides friction coefficients of the correct order of magnitude but a fair test of the model must await the results of additional experiments. Another difficulty with eqn. (12) is that NA and T, are defined as values observed during the sliding test and it is difficult, if not impossible, to measure these in situ except for very special situations. For example, if NA is measured with a hardness test after sliding, it may be in error because of the different mode of loading, because of recovery of the highly deformed material or because of the reduction in temperature. Finally, the parameters rmax, 7, and NA/L may not be independent variables. Therefore it is important to measure them under the same experimental conditions if they are to be used to calculate p. The above discussion indicates that the present model, like other available friction models, cannot yet provide good numerical values for p (again, elastomers seem to be exceptions [16]). However, the approach described here, based on plastic work, is able to predict trends in p and also the dependence of p on several different parameters. The model can easily be expanded to treat rather complex situations. As an example, in the next section of this present paper, the model will be applied to metals with coatings of different thicknesses.
3. The friction
coefficient
of coated metals
It is well known that the application of a thin layer of a suitable coating material can reduce friction [ 23, 241. In such cases, the resulting friction depends on the materials chosen and on the thickness of the coating. In this Section, experimental observations on the friction of coated metals are
discussed in terms of the friction model presented above. One goal is to understand better the role of coatings. Another goal is to predict the friction behavior when a given coating/substrate system is used. This in turn would lead to the possibility of selecting the proper combinations of materials for desired friction characteristics. 3.1. Plastic work needed to deform a layer of thickness T The calculation of P for a coated material is based on the same model as that described for uncoated materials. The same concept of plastic work is used but the deformation in both the coating and the base material must be included. Since the work terms in the two materials are additive, the contribution of each region to friction can be calculated separately using appropriate boundary conditions. The total friction coefficient is then simply the sum of these separate con~butions. First, the plastic work needed to deform a layer of thickness T is calculated. The assumptions are the same as those in Section 2, except that here the deformed volume is restricted to a thickness T. A superscript L is used to distinguish the various parameters of the layer material from those used for the bulk material. By comparison with eqn. (8), we can write directly the corresponding expression for the plastic work of the layer of thickness T: wL,l
= (NA)L SqaLTLmaX SI1
_)I
-(~~~~~i'""-"Lz)]l'2~
0
X exp(-aLz)
d.2
(16) The integration is performed with the aid of the same substitution used in Section 2.4. The result is WL,t = (NA)L C~X,(T~,~~
- exp(-aLT)F
(17)
where rLr is the average stress‘at the interface (at z = T) between the deformed layer and the underlying material. For very thick layers (!I’ -+ -), eqn. (17) yields the same result as the initial friction calculation does, i.e. eqn. (12). 3.2. The friction coefficient for material I coated with material 2 For the case of material 1 coated with a different material 2, tbe boundary conditions become important. When the rider is displaced by 6x,, the displacement profile in the coating can be described by 6x(z) = 6x, exp(-aLz). At the coating-su~tra~ interface, the d~placement will be 6x, exp(-aLT). It is assumed that this displacement will be continuous across the boundary. The displacement should then decrease by another exponential factor, exp(-az), which is characteristic of the base material. The plastic work term from the substrate material is obtained directly from eqn. (12) if Sx, exp(-aLT) is substituted for 6x, and r1 for 7,. At this stage, it is assumed that the contact area is the same in the surface layer and in the bulk material and that it is determined by the
208
properties of the layer material. Therefore (NA)L is used for both materials. This assumption is reexamined in Section 3.3. When both contributions to the deformation work are combined and the sum is equated to the frictional work, we obtain the following expression for p as a function of the thickness of the coating material:
+7
max
exp(--aLT)F
i
~max )I
A-
(18)
In this expression, rI and rL1 have been retained as independent variables. This is not reasonable physically because the stress in the bulk is increased by the transfer of stress from the surface down through the coating and, for 7L1 > 71, across the boundary. Transfer of stress across the boundary would not occur for rL1 < rI unless the rate of recovery in the bulk material is appreciably greater than in the coating. For the following discussion, it is assumed that the stress is continuous across the boundary, so that 7I must be equal to Eli. The boundary stress is then determined by 7Ls, and it can be expressed from eqn. (5) as (19)
3.3. Correction for a thin layer In Section 3.2 it was assumed that the contact area is determined by the properties of the coating material, so that the contact area will be larger for a soft layer than for a hard layer. This can no longer be true when the layer is extremely thin, because the yield pressure of the substrate material will then influence the size of the contact area [ 1,2,23,25]. We could modify the friction model to allow for this effect for an ideal smooth surface. Or, more realistically, we can acknowledge the presence of surface roughness, so that with films of a small average thickness some asperities will be coated and some will contact the rider surface directly. This will be very likely when the average film thickness is of the same order as the surface roughness. For uncoated asperities, the contribution to ~1will have the same form as that of eqn. (12), in which all variables are properties of the base material. This term will disappear when the film thickness is large but it will dominate when the film is very thin. One way to incorporate these considerations in the friction model is to introduce a distribution function f(T) to describe the fraction of the asperities which is uncoated for a given value of the average film thickness. Since experimental data are not available to describe this function, a simple form which has appropriate limits will be used: f(T) = + {tanh(U-
UT) + 1)
(20)
209
-0.6 tz 04 -
0.2 -
Thickness,T
Fig. 5. The dependence of the distribution function f(T) = 0.5 {tanh( U - VT) + 1) on coating thickness 2 (U = 2 ; u = 4 pm-l ). This function represents the fraction of sample asperities in direct contact with the rider for a given value of the average coating thickness.
Figure 5 shows the form of this function for particular choices of U and u. The point of inflection is chosen by setting T = U/u. The parameter U is chosen so that f(0) is close to 1. The con~bution of the uncoated asperities can then be weighted by f(T) and the contribution of the coated asperities by 1 - f(T). The resulting form for p(T) is then
+7
max
exp(--aLT)F
-5-
+ f(T) N$
7s 7maxfit
-
(21)
( ?max )I ( rmax 1 When T approaches zero (no coating), this equation reduces to eqn. (12), and when T approaches m, this equation reduces to a similar expression for the bulk coating material. Halling [ 23 ] has used a similar approach to treat surface roughness for his calculation of ~1(T), but his model is based on the conventional adhesion model of friction. Halling’s distribution function is similar to the one used here, but he has not weighted the con~butions independency. Instead, he has used averages for all his parameters. The two models give similar results for T = 0 and for large T, but they differ for intermediate values of T. 3.4. Results for a soft layer on a hard base Equation (21) is a general result which may be used for different types of coatings and substrates. The first case to be discussed is that of a soft coating on a harder base material (7Lmax< T,,,). One example would be a soft metal film on a harder alloy substrate. The film material, treated by itself as bulk material, would have a friction coefficient of
Similarly, without any coating, the base material, denoted by B, would give
210 B _ cr
--?
NA max LJ
Using these expressions for pL and pB, together with eqn. (19) for 7r, eqn. (21) becomes p(T) = (1
F(WLmax)
1 -exp(--aLT)
-f(T)}pL
JY~LJ7Lnlax1
i --
Tmax
F(TIITrnax
rL max
F(~Ls/~Lmax)
-
1
(22)
+ fWPB
The shape of this function depends on the relative values of pL and pB. Several examples are shown in Fig. 6. These have been calculated as follows: f(T) as in Fig. 5; ~Lm,,/~maX= 0.2; 7Ls/~LmaX= 0.9999; uL = 0.05 pm-‘. The value of pB has been kept fixed, while pL has been varied. When pL < 1.3pB, there is a minimum in ~1(Figs. 6(b) - 6(d)). As pL decreases (Fig. 6(d)), the value of the minimum gets lower. However, it is shown below that the relative depth Prnin/P L of the minimum stays constant at about 0.75. Of course, for other choices of the parameters in eqns. (22) and (19), this number would vary. fn Fig. 6, the curve for ,uL < ,uBis similar to published experimental curves [ 1,23,24] . The reported minima in p are sometimes deeper than
0 (a)
(bf
(cl
(d)
0.1
IO
T
10
IOOpn
Fig. 6, Examples of the dependence of ~1on T for soft coatings on harder substrate materials(oL= 0.05fimP1 ;rLm,/~m, = 0.2;?Ls/~Lm, =0.9999;f(T) asin Fig.5): (a) /J~//J~ > 1.3; (b) 1.3 > /_tL&B > 1; (c) pL/pB = 1; (d) /.fL/pB < 1.
211
those predicted here but this discrepancy could be due to experimental conditions which do not correspond with assumed calculation parameters. It is instructive to examine eqn. (22) to see how the minimum will change as different parameters are varied. The influence of the distribution function NT) is the easiest to see. Changing U/u simply shifts the left part of the ~(7’) uersus T curve (T < Tmin)withrespect to the rest of the curve. This makes the minimum broader or narrower. Changing u in f(T)changes only the slope of p(T)at the left side of the minimum but the effect is small if this parameter is chosen in a reasonable way, as suggested in Section 3.3. The parameter uL affects the scale of T;uL canbe determined experimentally from marker experiments. The presence or absence of a minimum in I-((T) is certainly of interest for practical reasons. Because of the complexity of eqn. (22), a straightforward derivation of a condition for a minimum in /J is difficult. To obtain an approximate condition, the p(T)curve is divided into two parts. For small T the function f(T)accounts for the decrease in p(T) (Figs. 6(b) - 6(d) should be compared), whereas the function exp(--aLT) determines the increase in p(T)at large T.As soon as T 2 (1 + V)/u, f(T)is approximately zero and p(T) has reached its smallest value. It will increase again with increasing T assoon as T > O.l/aL because then exp(--aLT) < 1. To simplify the following discussion it& assumed that the surfaces are reasonably smooth. Then (1 + U)/V will be small, probably smaller than O.l/uL, and a minimum in p(T)might occur somewhere in the range (1 + U)/V < T < O.l/aL. To decide whether there is actually a minimum, a further investigation of eqn. (22) is necessary. The condition T > (1 + U)/u results in f(T)= 0 and the condition T < O.l/uL results in exp(--aLT) = 1; this simplifies eqn. (22) drastically. The same arguments applied to eqn. (19) yield pi = 7Ls; p for layers of the specified thickness will then be P
szp
L
~,a,
--
rL max
F(~Ls/cn,x) F(~~h~rnax 1
(23)
For a soft layer on a hard substrate (Tag_ < T,,,), P is smaller than yL, but for a minimum in p(T)it is also necessary that p < pB. This gives the following approximate condition for a minimum: fiL rLmax F(~~s/~~rnax) B<--(24) l-l rmax WLJ7,,,) The depth of the minimum is then given by eqn. (23). Figure 7 shows how the condition for a minimum (eqn. (24)) varies with rLJrLmaX for several values of Tag&,,,. When T~~_/T,,,, which expresses the dissimilarity of the coating and substrate materials, is less than about 0.5, it has little influence on the condition for a minimum. For a soft coating on a hard substrate, this condition will generally be satisfied. During sliding, the soft material will be highly stressed, close to its intrinsic limit, so that rLJrLmaX should be close to unity. Recognizing these conditions, one can see from Fig. 7 that pL/pB should be less than about 1.3
212
Fig. 7. Conditions for which a minimum will occur in p(T) us. 2’ (soft coatings on harder substrates). Fig. 8. Values of pL,in/pL for different choices of T~.JT~,~ coating on hard substrate).
and rLmax/~max (soft
if a minimum in ,u(2’) is desired. Yet softer materials often have higher friction coefficients than hard materials. Therefore it might be difficult to find a soft material which fulfills this condition. The value of the minimum in p(T) (eqn. (23)) is also of practical interest. Figure 8 shows how this expression varies with ~~~~~~~~~and rL max17max. Again, the curves are insensitive to Tag&-,,, for values of this parameter which are less than about 0.5. When the surface stress rLSis high, the value of I-(~~ will typically be about 75% of pL. This discussion has been based on the assumption that (1 + U)/u < O.l/uL, which may not always be true. However, because of the complexity of eqn. (22), the preceding discussion has concentrated only on the development of some simple guidelines for selecting a coated system which will exhibit low friction. To summarize, the following approximate criteria emerge. (1) rL,ax for the coating material should be less than about half the 7maxvalue for the substrate material (2) The friction coefficient for the coating material alone (thick coating) should be as low as possible, at least satisfying pL 2 1.3~~. (3) The surface stress for the coating should reach a value as close to its maximum value as possible. (4) The minimum value of y(T) will be roughly 75% of pL. (5) The thickness of the layer at which p( 2’) reaches its minimum will be in the range (1 + U)/V 5 T 2 O.l/uL for smooth surfaces. The conditions summarized here are not necessarily independent. In general, an optimum coating will depend on a compromise among possibly conflicting criteria. 3.5. Results for a hard layer on a soft base Equation (22) can be used again for this case, but now 7L,aX will be larger than r,,, since the coating material is stronger than the substrate material This fact introduces an additional constraint on r which was not
213
important for the case of 7L,aX < T,,.,~~ : 7I at the coating-substrate interface cannot exceed T,,, of the bulk. With ~~~~~ and uL fixed for a given sliding system, 7I will be a function of 7Ls. Therefore this new constraint determines an upper limit for 7Ls. The physical reason for the upper limit of rLs can be understood easily. When the coating is thick, so that all stresses and strains diminish inside the layer (owing to the exp(--aLz) factor), the surface can support a stress as high as its intrinsic limit 7L,aX. However, if the layer is thin, shear stresses will be present at the interface and these must transfer across the interface into the underlying base material. Since that material is softer, it will deform more easily than the adjacent coating material The net result will be a shear fracture at or near the coating-substrate interface and the coating will skate across the surface of the substrate. If this is not to happen, 7Ls will be limited by 7i, which in turn will be limited by T,,, rather than by T~,,,~~.An extreme example may illustrate the problem more clearly. The soft bulk might be gelatin and the hard layer a thin metal plate. With this model system, it would be impossible to deform the metal plastically (i.e. 7LS would be less than the plastic yield strength of the metal) and the maximum shear stress which could be applied (assuming good bonding) would be that of the gelatin. For the calculation of p(T) curves with this model it is assumed that 7Ls is smaller than T,,, of the bulk. This ensures that the condition 71 < 7max is always satisfied. Figure 9 shows some typical curves. The following parameters are used: rLmax/r,,, = 4; rLs/rLmaX= 0.24; aL = 0.05 pm-‘. Again, pL is varied with respect to pB and it is assumed that both pL and pB are determined under the same conditions as the final p(T) curve. A comparison of this model with experimental data is difficult because only limited data are available. Also, the p(T) uersus T curves predicted here are based on very severe constraints, the continuity of 7 and 6x across the coating-substrate interface, and on the achievement of steady state conditions. The continuity conditions are particularly difficult to guarantee for the case of a hard coating on a soft substrate, especially for a very thin coating. One preliminary experimental curve for p(T) (hard layer on soft base) has been presented [ 251. The curve p(T) shows a clear maximum which agrees at least qualitatively with the result of the present model. For many applications of coatings it might be undesirable to have a maximum in P at a certain thickness of coating material, and therefore the conditions for a maximum are worth discussing. The same assumptions as in Section 3.4 are used and ~1for the region (1 + U)/u < T < O.l/uL can be approximated as in eqn. (23). Since 7LmaX> T,,,, p is now larger than pL; if ,u is also larger than (Us, this will result in a maximum in p (2’). The conditions for the existence of a maximum are therefore L !!-_> -rL max fT~Ls/~Lrnax)and
PB 7max JVLs/?nax) The value of the maximum is
rL s =GTmax
(25)
214
tL
P"
P 0:
0.1
IO
T
IO
loo pm
po01
1.0
T
IO
100 ptn
(a)
rl---: r-”
LLB
P 0
01
IO
IO
0.1
100 urn
T
(cl
1.0
IO
IOOpr
T
fd)
Fig. 9. Examples of the dependence of g on T for hard coatings on soft substrate materials (aL = 0.05 pm-l ; 7Lmax/7,,, = 4; TLs/TLm, = 0.24;f(T) as in Fig. 5): (a)pL/pB = 0.8; (b)pL/pB= O.S;(c)pL/pB= l.O;(d)pL/pB= 1.3.
Figure 10 shows the height of the maximum as a function of 7LS/~LmaX for several values of 7LmaX/rmax . A maximum will only occur when the conditions (25) are also satisfied. This maximum will be of noticeable size if 7LS is very close to
Fig. 10. Values of p,,/pL for different choices of rLs$rLmax and ?Lmaxfrmax (hard coating on soft substrate). The upper limit for prnax is given by the boundary condition 71<7,,.
215
T,,,. In this case the maximum might be as large as l.5r.tL. However, according to eqn. (25), pL has to be larger than about 0.7gB to get a maximum at all. This condition might be somewhat difficult to fulfill since hard materials tend to have lower friction coefficients than soft materials. With the present model no prediction can be made for the case of TI > rrnax (large Tag). Therefore, the following very approximate guidelines apply only for the case of rLs < T,,,. A maximum in y(T) for a hard coating will occur if 7Ls is very close to 7,,, and pL is larger than about 0.7~~. Then T,,, might reach a value of up to 1.5~~ and the thickness of the layer at the maximum value of P will be in the same range as in the case of a soft coating.
4. Conclusions Tbe friction model described in the present paper is not intended to give precise numerical results for the friction coefficient but to determine the dependence of the friction coefficient on familiar mechanical parameters as well as on microstructural features of a material. It is based on the idea of the equivalence of the frictional work done and the work of plastic deformation during steady state sliding. The plastic behavior of a material is described by parameters of its shear stress-strain curve and the average shear stress at the surface. The model includes as a special case the well-known model of Bowden and Tabor [ 1,2 ] but it does not require adhesion to be present. The new model is an extension of simpler energy-based models described recently by the present authors [5, 71. The model has been improved by using a more realistic stress-strain curve, by introducing a surface stress rather than a surface strain and by adding the effect of direct rider-substrate contact in the case of coated materials. The friction model represents a broad general basis from which work can be extended in a number of directions. All the parameters can be defined more precisely to include more variables and fewer assumptions. The process of dividing a sample into several layers and adding their contributions to the friction has proved to be straightforward with the present model. For example, in the case of coated materials the model has been used to predict the general form of the friction coefficient as a function of coating thickness and materials properties. In addition, some approximate guidelines for selecting coated systems have been established. The idea of using different layers to describe a friction system is not limited to well-bonded applied coatings. It can also be applied to layers of reaction products (such as oxides), contamination layers, transferred material, layers of solid lubricants (such as Moss or graphite) and even to lame&r structures. Fu~ermo~, the initial convenience of assuming a hard rider material is not really necessary. The model can readily be extended to apply to any combination of materials in which plastic deformation is the dominant dissipative mechanism.
216
Although the present model has been developed for steady state conditions, it offers some insight into the processes which occur during running in. It takes time for TV,7 and f(T) and the ~mperature to reach their steady state values, so the model contains an implicit time dependence for the friction coefficient. Thus, changes during the early stages of sliding, as well as transitions which might occur later, can be accommodated. The main difficulty in using this model is the lack of appropriate experimental data. Even for pure metals, it is difficult to find suitable data for the m~imum shear stress T,,, and for alloys and oxides the situation is worse. Data are even more limited for the average surface stress 7, and the real contact area NA during steady state sliding. Until these data are available, this friction model, and probably others as well, will be restricted to predicting trends rather than reliable quantitative results.
Acknowledgments The authors would like to thank Professor K. L. Johnson, University of Cambridge, and Professor J. P. Hirth, The Ohio State University, for their helpful comments. This material is based on work supported by the National Science Foundation under Grant DMR 7805719.
References 1 F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Part II, Clarendon, Oxford, 1964. 2 F. P. Bowden and D. Tabor, Friction: An Introduction to Tribology, Anchor, New York, 1973. 3 D. Dowson, History of Tribology, Longman, London, 1978. 4 D. G. Teer and R. D. Arnell, in J. Halling (ed.), Principles of Tribology, Macmillan, London, 1975, pp. 72 - 93. 5 D. A. Rigney and J. P. Hirth, Wear, 53 (1979) 345 - 370. 6 W. A. Glaeser, An engineer’s guide to friction, DMIC Memo. 246, 1970 (Defense Materials Information Center, Battelle Memorial Institute). 7 P. Heilmann and D. A, Rigney, Sliding friction of metals. In D. Dowson, C. M. Taylor, M. Godet and D. Berthe (eds.),Proc. Leeds-LyonSymp. on Tribology, Leeds, September 9 - 1.2, 1980, Westbury House, Guildford, 1981. 8 J. P. Hirth and J. Weertman (eds.), Work Hardening, Gordon and Breach, New York, 1968. 9 A. W. Thompson, Work Hardening in Tension and Fatigue, Metallurgical Society of AIME, New York, 1977. 10 U. F. Kocks, Strain hardening and strain-rate hardening, American Society for Metals Symp. on Mechanical Testing for Deform&ion Model Development, Bal Harbor, FL, November 1980. 11 J. H. Holioman, Trans. Metall. Sot. AIM.&‘, 162 (1945) 268 - 290. 12 E. Vote, J. Inst. Met., 74 (1947 - 1948) 537 - 562. 13 D. Kuhlmann-Wilsdorf, in J. P. Hirth and J. Weertman (eds.), Work Hardening, Gordon and Breach, New York, 1968, pp. 97 - 132.
217 14 A. Van den Beukel, Ser. hfetall., 13 (1979) 83 - 86. 15 D. A. Rigney, Ser. Metall., 13 (1979) 353 - 354. 16 K. L. Johnson, Aspects of friction. In D. Dowson, C. M. Taylor, M. Godet and D. Berthe (eds.), Proc. Leeds-Lyon Symposium on ~iboiogy, Leeds, September 9 - 12, 1980, Westbury House, Guildford, 1981. 17 J. H. Dautzenberg, wear, 60 (1980) 401 - 412. 18 M. A. Moore and R. M. Douthwaite, Metall. Trans., A, 7 (1976) 1833 - 1839. 19 N. Ohmae, in N. Saka and N. P. Suh (eds.), Fundamentals of Tribology, Massachusetts Institute of Technology Press, Cambridge, MA, 1980; Philos. Mag. A, 40 (1979) 803 - 810. 20 L. K. Ives, in K. C. Ludema, W. A. Glaeser and S. K. Rhee (eds.), Weur of Materials, American Society of Mechanical Engineers, New York, 1979, pp. 246 - 256. 21 A. W. Ruff, L. K. Ives and W. A. Glaeser, Characterization of worn surfaces and wear debris, American Society for Metals Materials Science Semin. on Fundamentals of Friction and Wear of Materials, Pittsburgh, October 4 - 5, 1980, American Society for Metals, Metals Park, OH, 1981. 22 0. Vingsbo and S. Hogmark, Wear of steels, American Society for Metals Mater~~ Science Semin. on Fundamentals of Friction and Wear of Materials, Pittsburgh, October 4 - 5, 1980, American Society for Metals, Metals Park, OH, 1981. 23 J. Halling and M. G. D. Sherbiney, The role of surface topography in the friction of soft metallic films, Conf. on Materiah Performance and Conservation, Swansea, April 3 - 4. 1978, Institution of Mechanical Engineers, London. 24 N. P. Suh, N. Saka and S. Jahanmir, Wear, 44 (1977) 127 - 134. 25 J. Haliing, USME Rep. T/44/75, 1975 (Department of Aeronautical and Mechanical Engineering, University of Salford).
Wear, 72 (1981) 195 - 217 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
AN ENERGY-BASED MODEL OF FRICTION AND ITS APPLICATION TO COATED SYSTEMS
P. HEILMANN and D. A. RIGNEY Department (U.S.A.)
of Metallurgical
Engineering,
The Ohio State
University,
Columbus,
OH 43210
(Received March 24,198l)
Summary
An energy-based friction model is used to develop expressions for the friction coefficient; these expressions depend on familiar mechanical parameters, stress-strain curves and microstructural features of the materials. The main assumption is that the frictional work performed is equal to the work of plastic deformation during steady state sliding. The results include as a special case the well-known expression for the friction coefficient derived from adhesion theories. However, the new model can also be applied to more complex systems such as those involving coatings. In particular, for soft coatings on hard substrates as well as for hard coatings on soft materials, the friction coefficient can be predicted as a function of layer thickness in accordance with reported observations. Furthermore, guidelines are presented for achieving low friction by using coatings. The new model can also be applied to systems involving reaction layers, transferred material and solid lubricants of varying thickness. In addition, the model offers some insight into the variation in friction coefficients with time and into the processes which occur during running-in.
1. Introduction
Examples of friction are commonly observed in everyday life but despite the familiarity of this subject frictional phenomena are not well understood. Friction seems to involve complex processes. Therefore, although a number of simple models have been proposed, they are inadequate for explaining or predicting most friction phenomena. Neither acceptable explanations of the mechanisms involved nor reliable predictions of friction coefficients are currently available for most materials. The most widely accepted models of friction are based on adhesion and/or the interlocking of asperities. Descriptions of these models can be found readily in other publications, e.g. refs. 1 - 5, and they are not presented in detail in the present paper.
196
Friction is often regarded as purely a surfaee’process. However, it usually involves changes in structure and properties below the surfaces of interacting materials and it can be accompanied by loss of material from either or both of the sliding members. Such damage might be very small, in which case it would not be detected over a long period, or the damage might be large enough to be recognized as wear. In both cases, irreversible processes are involved and this is one of the factors which makes it difficult to predict what will happen. Tribologists do not yet agree on which properties of sliding materials and which additional physical parameters are essential and should be included in a model of friction. Apparently, many factors influence friction [ 2, 4 - 71. If more of these are incorporated in models of friction, the models will become increasingly complex. At this stage it seems more appropriate to select only the most impo~nt parameters; special care is taken to see that none of these is omitted. Most available models of friction use a very limited number of parameters. With the exception of a yield stress or a hardness parameter, basic materials characteristics tend to be neglected. In particular, microstructural information is generally omitted. Yet, details of microstructure profoundly influence the nature of the materials behavior, and they should be included in a realistic friction model. Except for the lowest loads, some plastic deformation occurs near the surfaces of sliding materials, even under lubricated conditions. In fact, a principal mechanism for the dissipation of frictional work is through plastic deformation [ 5 3 . Therefore, the ease of plastic deformation and associated parameters such as those of the microst~c~re should be important considerations in setting up a friction model. As in other physical processes, energy considerations can be very useful. This is true even for an irreversible process like friction. The usefulness of energy in a friction model is that it introduces a kind of average to the individual events which take place and which may be of a wide variety. The energy point of view of friction becomes even more valuable if one makes two simplifying assumptions. The first of these involves the generation of wear particles. These appear as discrete local events which should contribute to the “noise” of a friction versus time curve but which need not be included in the basic friction model. Secondly, the model described here assumes that steady state conditions are achieved. Together, these assumptions lead to a highly idealized model which may never correspond in detail to actual sliding situations. However, the assumptions lead to the following conclusions. Frictional work will result in plastic deformation near the interface between interacting materials. Heat is produced during deformation and during subsequent recovery processes. This frictional heat will tend to raise the temperature of the friction couple and this in turn will change some of the mechanical properties of the materials, e.g. the workhardening rate and recovery rate. These in turn change the ease of plastic deformation and influence the frictional work until a dynamic balance is
197
achieved. At this steady state condition the average local temperature of the friction couple remains constant and the recovery just compensates the additional work hardening caused by the plastic defo~a~on during sliding. A full analysis would require calculation of this equilibrium temperature for the steady state from basic assumptions and structural properties. This approach could result in a calculated friction coefficient. However, the result would contain too many uncertainties and the effort would not be justified. Given the present state of knowledge, perhaps a better way is to measure some of the steady state parameters and to use them to calculate the friction coefficient p. This method is used in the following sections since, despite the simplifications used, it gives valuable insight into the dependence of the friction coefficient on structural parameters. It also has practical implications, since the results suggest ways to modify ~1.
2. Basic friction model 2.1. Material properties used in the friction c~icu~t~on Defo~ation near the surface of at least one of the infecting materials commonly accompanies sliding. The ease of this deformation can be described with the aid of a stress-strain curve; this curve reflects implicitly many basic characteristics of a material, e.g. the grain size and hardness. Therefore, the appropriate stress-strain curve can be taken as a convenient way to describe certain properties of a material in a concise way. Furthermore, during sliding, local regions in a material are exposed to different stresses; this results in various strains which are governed by the stress-strain curve. Choosing a proper stress-strain curve for a friction calculation is not easy. First, it should be described by a relatively simple formula for ease of calculation. A numerical representation might also be used, but the results would then be more difficult to interpret. Second, since shear stresses are involved during friction, an appropriate shear stress-shear strain curve is needed. However, most stress-strain curves are measured in simple tensile tests and they must be converted to shear curves by considering resolved shear stresses. Third, even if a suitable curve is available, the measured range of shear strains y is usually limited to low values, i.e. y 5 1. During friction, much higher strains will often arise; these will require the extrapolation of available stress-strain curves to higher strain levels. For low strainsthere exist several more or less sophisticated empirical laws to describe stress-strain curves. Reviews can be found in refs. 8 and 9 and, more recently, in ref. 10. A very widely accepted correlation between the shear stress 7 and shear strain y is 7 = kylln or 7 = kr”‘, depending on the definition of n and n’ [ 111. An even simpler law is 7 = h71j2, which is a good approximation for some materials, e.g. polycrystalline f.c.c. metals at low strains. At higher strains, the stress commonly deviates from these simple power laws and eventually reaches a saturation level [lo] .
198
An empirical relation, introduced by Vote [12]- for compression stressstrain curves, incorporates a saturation stress 7,,, : 7 = Ti + (7,,,
-TijlII.
-&XP(-~)j
For consistency with the present paper, the original mathematical representation [ 121 and the symbols have been changed. Here Ti and yi are the shear stress and shear strain respectively at the beginning of the test. A shear stress-shear strain relation very similar to that of Vote will be used for the present friction model. Since Ti is usually small compared with T,,, and yi is much smaller than the strains actually achieved during steady state sliding, 7: and ri are set equal to zero. Furthermore, a square root will be introduced; this can be justified by two arguments (this is discussed below). The resulting stress-strain relation to be used for friction calculations is then 7=7 max(1 - exp(--cy )P2
(1)
The stress-strain curve which results from plotting this function is shown in Fig. 1.
ShearStraqy Fig. 1. Plot of a simple stress-strain function 7 = T,, (1 - exp(-cy)} 1’2 which reduces to a power law(T - y*‘2)at low strains and approaches a constant value T,, at high strains.
For small y values, the exponential function can be expanded easily and eqn. (1) reduces to a simple power law: 7 - yff2. The second reason for using an exponent equal to l/2 in eqn. (1) arises from the following semiquantitative derivation. Dislocation densities during plastic deformation have been carefully studied and a theoretical expression for the dislocation density p during stages II and III is presented in ref. 13. This work was later extended to higher strains by introducing a further correction [ 141. As pointed out recently [ 151, this model predicts an upper limit for the dislocation density at high strains; this is consistent with available experimental data. The dislocation density in stage III is given by P = pC expj-a(7
-yCN + z 11 -expC-Q(r
-yc)l]
199
The parameters $2 and U are as defined in ref. 14. For strains larger than about 0.5, #is expression can be simplified to p = (U/Q){1 - exp(-SZ2y)). We can also use a commonly accepted relation between p and T (desc~bed in ref. 13): r = cvGbp112,where a x 0.5, G is the shear modulus and b is the dislocation Burgers vector. Combining these expressions results in a stress-strain relation equivalent to eqn. (1). In an earlier model [7 ] , which also used a stress-strain curve, a simpler function was used. This was a combination of the power law 7” yljn for low strains and a constant value of 7 for high strains. This function has the same limits as eqn. (1) but it has an arbitrary discontinuity in its derivative, which complicates its use in the friction model. Nevertheless, the results are very similar to those described in the present paper. However, some of the limitations of the earlier model have been removed in the present work. Of course, for some materials, eqn. (1) will not be a good representation for the stress-strain curve. Examples would include certain single crystals oriented for easy glide and perhaps materials involving deformation-induced phase transformations, such as me&stable austenitic stainless steels. For such cases, a model like the one described here could still be used but different stress-strain relations, tailored to the particular interactions, would have to be used. Alternatively, a numerical representation could be used for complex stress-strain relations. 2.2. Contact between the surfaces during friction Real surfaces are never absolutely smooth. Therefore, contact between two surfaces will occur first at local “points” determined by asperity profiles. If there are N of these local contacts, each of which has an average area A, then the total real contact area will be NA. Johnson [ 161 has analysed the mechanics of sliding asperity contacts for two cases: in one case adhesion aids stress transfer and the other case depends simply on the ~~rlock~g of asperities. For the following calculations, as long as some means exist for stress transfer, the choice is not important; therefore we make no assumptions about how the stress is transmitted from one surface to the other. The differences which Johnson reported can arise in the present model through differences in the extent of plastic deformation. The model described in the present paper is based on a simple hard rider sliding on a sample material which deforms elastically and plastically. For simplicity at this stage, the rider is assumed not to deform. However, this assumption is not essential for the approach taken and the model can be generalized later to allow deformation and energy dissipation in both members of the sliding system. Each asperity on the surface of the sample material will deform when it interacts with the surface of the slider. This deformation will continue until the contact fractures, if the interaction is caused by adhesion, or until the sliding motion separates interlocking asperities. During each local asperity
200
contact, the stress and strain at the surface will grow from some initial value (which will depend on its stress-strain history) to some final value. Relaxation or recovery can occur almost simult~eously (dynamic recovery) or during the intervals between asperity contacts. If the relaxed stress is equal to the initial value, this will define the steady state conditions. At any given time, different asperities in contact with the rider will be at different stages of deformation. However, we can define an average surface stress 7, to characterize this deformation; it will represent the average over all individual stress values at all contact points. When steady state conditions are achieved, 7, will be time independent. Then, given r5, it is easy to calculate the plastic work which is done when the asperities are deformed by an additional virtual displacement 6x,. The friction sample can be treated as if it were a simple shear sample of contact area NA pre-stressed to 7,. The corresponding shear strain would be ys, and an additional displacement 6x, would provide an additional shear strain Ays_ For this simple case, with T, and Ay, held constant over the deformed volume V, the work done is simply 7, &s V. 2.3. Depth dependence of the def~rrn~t~o~ The simple expression derived in Section 2.2 for the plastic work during sliding is not realistic and therefore it is not very useful. It would apply only for a simple shear test with constant deformation over the entire volume. For the case of sliding friction, deformation is known to decrease relatively rapidly with depth [ 1’7,18] . Therefore, material at a distance z below the surface will be less stressed than surface material, and a displacement of 6r, imposed at the surface (z = 0) will be smaller for z > 0. The depth dependence of the displacement field 6x = 6x(z) will be considered first. Experimental data are available in the form of marker profiles developed during sliding [ 17, 18 ] . A marker is embedded in the material in such a way that, at the start of testing, the projection of the marker, viewed in a lon~tudin~ section, is pe~en~cul~ to the sliding surface and parallel to the z axis..After sliding occurs, the marker is bent over in the direction of sliding, and its shape may be described roughly by an exponential curve. This observed profile is the result of many small displacements 6x, at the surface and smaller values 6x(z) below the surface. If the displacement profile can be described by a simple exponential function, e.g. - exp(-az), then the individu~ displacement 6 x(z) can be written 6x(z) = &r, exp(-az)
(2)
This expression is consistent with the interpretation of 6x = 8x, at z = 0. The shape of this displacement function is shown in Fig. 2. The constant a can be determined by fitting an exponential curve to the appropriate measured marker profile. Since the virtual displacements decrease with depth, the associated shear strain increments A-y(z) will also decrease. The strains and displacements are related by differentiation, so that
201
Depth.2
Fig. 2. Assumed variation in virtual displacement 6x with depth z below the material surface using 6x = 6x, exp(-ilz). Fig. 3. Depth dependence of the shear stress T(Z) for two different choices of the surface stress 7, using eqn. (5) (T,(l) > rst2)).
AT(Z) = - ;
6x(z)
= a6x, exp(-uz)
(3) The shear stress will also decrease below the surface. If we assume steady state conditions, the average shear stress should remain constant for a chosen value of z. This average shear stress, which is the sum of external and internal stresses at any given time, can be correlated with the shear strain by using the stress-strain curve (eqn. (1) and Fig. 1). Because of this correlation, we can think of the material as being internally stressed to a level T(Z) or, equivalently, we can think of the material as being strained to a level y(z). However, this value of y is not the same as the strain which can be measured by marker displacements. Instead, it is simply the strain which corresponds to a stress T on the stress-strain curve. This distinction is critical and, since it is a possible source of confusion, it is emphasized here. If the function y(z) is known, we can readily calculate the depth dependence of the stress T(Z). If AT(Z) satisfies eqn. (3) then it is reasonable to assume that y(z) decreases exponentially in the same way: ~(2) = yS exp(--az)
(4)
With the aid of eqn. (l), the average surface strain ys may be expressed in terms of the average surface stress 7,: TS=--f
lnjl-(
c)‘/
Combining this expression with eqns. (1) and (4) gives an expression for the depth dependence of the shear stress: T(Z) = ~,a, [l_~~_(
foci’“‘“‘]“’
This function is shown in Fig. 3 for two different choices of the average surface stress 7,. Through eqn. (5), the parameter 7, also characterizes the
(5)
202
stress beneath the surface. If a and 7,aX are known, 7, can be determined by measurements of T(Z). The curves in Fig. 3 exhibit three distinct regions, especially for high values of 7,. Near the surface, 7 is nearly constant and very close to its maximum value r,,, . In the next region, T declines more rapidly until it approaches a level characteristic of the undisturbed bulk material. In the present paper this value 7i has been assumed to be zero; larger values of Ti could be included in a way similar to that of Vote. The neglect of Ti does not generally have a large effect on the calculated value of P because the incremental displacements and strains are very small below the near-surface region. The stress ~(2) depends on the material and the local microstructure, in particular on the elastic modulus, crystal structure and dislocation density. These might be related through 7 = ~Gbp’/~ [ 131 or through some more complicated expression. In any case, T(Z) might be determined by using these measurable quantities. The different regions of the T(Z) profile probably correspond to regions with different dislocation substructures, such as those reported recently by several research groups [ 19 - 221. For example, the relatively flat portion of the curve is probably associated with a fine cell or subgrain structure with approximately uniform dimensions, while the steeply decreasing portion of the curve should be associated with substructure dimensions which increase with depth. If such stress-structure or stress-dislocation density correlations are possible then we can estimate the average stress T, at the surface. Also, if we can match the regions of the ~(2) curve with the thickness of different substructure layers then the depth dependence of 7 used in the present model would be justified.
2.4. Calculation of the friction coefficient p The basic concept used to calculate p at the steady state, as outlined earlier [ 5, 71, is to equate the frictional work and the plastic work. Thus it is assumed that at the steady state all frictional work is used to deform the material plastically and that the work is dissipated as heat. Other dissipative processes, reviewed in ref. 5, are assumed to be less important for most materials (the principal exceptions to this are elastomers [ 161). The incremental strain at steady state is Ay when a hard slider moves a virtual distance 6x, over the surface of a plastically deforming sample material. The plastic work done is then
vJ,l
=~WY
dV
(6)
with the integration over the deformed volume. Each asperity contact is assumed to have an average area A, over which the stress is constant, and the deformed material below each contact is represented by a cylinder with its
203
axis parallel to z. With this approach, the stress will depend only on the depth z below the surface and the volume integral becomes
WPl
=NA
$~(y)i\r
dz
(7)
0
where NA is the total area of asperity contacts. Combining eqns. (3), (5) and (7) yields
which gives
or WPl
=
NA SX,T,,,F
(9)
where F(u) =
1- 2
ln(1 + U) - u ln(l - u2)
(10)
F(u) is a monotonic function, defined for 0 6 u G 1, and it has the limits = 1. It is displayed in Fig. 4; a non-linear scale is used to show that the function approaches unity as 7, approaches rmax, i.e. as u + 1. At steady state and with reasonably heavy loads, 7, and r,,, should be very nearly equal for the highly deformed material near the surface. The frictional work which is done by moving the rider a distance 6x, is F(0) = 0 and F(l)
Wf = /.lLsx,
(11)
where L is the applied normal load. The basic assumption of this model is that W,, and WU~,. can be equated. Therefore, from eqns. (9) and (ll), we obtain pL Sx, = NA SX,T,,,F
or
204
Fig. 4. Dependence of the function F(u) (defined in eqn. (10))on u = T,/T,,.
F(u)+ 1
=~,'~*,X.
NA /J = -TInax L
(12)
When F(T,/T,,,) = 1, p reaches its upper limit pmax = (NA /L)T~~~. Therefore, p may be written as (13) This means that if the plot shown in Fig. 4 is scaled by pmax, we can obtain the friction coefficient as a function of the ratio of the surface stress 7, to the ultimate shear stress 7mBx.
2.5. Discussion From eqns. (10) and (12) it can be seen that three parameters determine P ; these parameters are T,,, , T, and NA/L. The first of these, rmax, is the ultimate shear strength of the material which can be achieved during shear. This is a basic material property, and it can be measured by an independent test. The second parameter, rs, is the average shear stress actually achieved at the surface during sliding. It may depend on many experimental parameters, e.g. load, temperature and sliding velocity, and also on materials characteristics such as crystal structure, microstructure, work-hardening rate and recovery rate. Given our present state of knowledge it would be very difficult or even impossible to calculate 7, as a function of all these variables for a given material. Therefore, instead of attempting such a calculation from first principles, it is probably better to measure 7,. One must be careful, however, because T, should be an average value of T(Z) at the surface, and the usual methods for measuring stresses, e.g. with X-rays, result in an average value for material in an extended region below the surface.
205
An alternative way to obtain 7, is to use transmission electron microscopy (TEM) to measure the dislocation density created during sliding. The stress 7, (and also r(z)) could then be calculated from an expression such as 7 = CuGbp1i2[ 131 or from some other function relating p and 7. This method would have two main sources of error. One would involve the appropriateness of the chosen correlation between r and p . The other would be related to the unknown amount of recovery which might take place between the time of sliding and the time of observation by TEM. Another way to estimate T, is to use the shape of the r(z) curve, as shown in Fig. 3. It seems very probable that the flat region of r(z) which exists for high r, values corresponds to a highly deformed layer at the surface. The thickness of a surface layer with a particular microstructure can be measured easily and compared with the extent of the flat portions of calculated 7(z) profiles. The best fit would yield 7, directly, The third parameter needed to calculate p is the ratio of the real contact area NA to the applied load L. Since hardness is a familiar and readily available material parameter, a comparison of L/NA with the hardness might be useful. The penetration hardness is defined as p = load/(contact area). Depending on which kind of area (projected or real contact area) and which kind of test indenter is used, p could be represented by the Brinell, Vickers or Knoop hardness. During sliding, asperities act as similar indenters. In this case, shear forces are present in addition to compressive forces, and the real contact area may be different from that which supports a purely compressive load [ 1, 2) ; therefore, L/NA need not be the same as p. If we bear this precaution in mind, it is interesting to examine the form of eqn. (12) when L/NA ;t: p or (14) This result is very similar to the well-known formula derived from an adhesion theory and first in~oduced by Bowden and Tabor [ 11, i.e. P=
shear strength penetration hardness
(1’5)
where the penetration hardness is defined as in eqn. (14). The shear strength to be used in eqn. (15) is not clearly specified. It might be chosen as the yield strength in shear of the bulk material or it could be as high as T,~~. It has commonly been chosen at the low end of this range. The factor P(~,/T,,,) of eqn. (14) has no equivalent in eqn. (15). When stresses at the surface are high, as with continuing sliding at high loads, 7, is nearly equal to 7maxand F(r,/r,,, ) = 1. Then the new model will have the same form as the older adhesion model but the friction coefficient for the new model will be higher because of the higher value used for T,,,. Thus the new model at least partially removes one of the earlier criticisms of the Bowden and Tabor model, i.e. that it yielded a calculated ~1that was lower than values found by experiment.
206
The new model and the familiar adhesion model give similar results only for the special case just discussed. The new model is much more general and it can be applied to many different sliding situations. The factor F(T,/~,,,) becomes more important when the surface is only moderately stressed, as with light loads. Thus the factor F(T,/~~~~ ) introduces a dependence of P on load as is often observed. This load dependence may be weak, because the factor NA/L also depends on load, but it varies in the opposite direction. Although the present model is intended to describe steady state conditions, it provides some insight into behavior during running-in (break-in) when 7, values are less than those developed at steady state. If 7, increases slowly during running-in, the steady state model should be applicable, at least for qualitative results. With 7, increasing with time, a plot of the friction coefficient versus time will be similar to a plot of F(~,/T,,,) uersus T,/T,,, (Fig. 4). Such variations in friction coefficient are often observed during running-in. Equation (12) provides a formula for calculating the friction coefficient if data are available for three clearly defined parameters, i.e. T,,,, T, and NA/L. However, these data are not at present available from experiments. A qualitative estimate of the variables in eqn. (12) indicates that the use of this formula provides friction coefficients of the correct order of magnitude but a fair test of the model must await the results of additional experiments. Another difficulty with eqn. (12) is that NA and T, are defined as values observed during the sliding test and it is difficult, if not impossible, to measure these in situ except for very special situations. For example, if NA is measured with a hardness test after sliding, it may be in error because of the different mode of loading, because of recovery of the highly deformed material or because of the reduction in temperature. Finally, the parameters rmax, 7, and NA/L may not be independent variables. Therefore it is important to measure them under the same experimental conditions if they are to be used to calculate p. The above discussion indicates that the present model, like other available friction models, cannot yet provide good numerical values for p (again, elastomers seem to be exceptions [16]). However, the approach described here, based on plastic work, is able to predict trends in p and also the dependence of p on several different parameters. The model can easily be expanded to treat rather complex situations. As an example, in the next section of this present paper, the model will be applied to metals with coatings of different thicknesses.
3. The friction
coefficient
of coated metals
It is well known that the application of a thin layer of a suitable coating material can reduce friction [ 23, 241. In such cases, the resulting friction depends on the materials chosen and on the thickness of the coating. In this Section, experimental observations on the friction of coated metals are
discussed in terms of the friction model presented above. One goal is to understand better the role of coatings. Another goal is to predict the friction behavior when a given coating/substrate system is used. This in turn would lead to the possibility of selecting the proper combinations of materials for desired friction characteristics. 3.1. Plastic work needed to deform a layer of thickness T The calculation of P for a coated material is based on the same model as that described for uncoated materials. The same concept of plastic work is used but the deformation in both the coating and the base material must be included. Since the work terms in the two materials are additive, the contribution of each region to friction can be calculated separately using appropriate boundary conditions. The total friction coefficient is then simply the sum of these separate con~butions. First, the plastic work needed to deform a layer of thickness T is calculated. The assumptions are the same as those in Section 2, except that here the deformed volume is restricted to a thickness T. A superscript L is used to distinguish the various parameters of the layer material from those used for the bulk material. By comparison with eqn. (8), we can write directly the corresponding expression for the plastic work of the layer of thickness T: wL,l
= (NA)L SqaLTLmaX SI1
_)I
-(~~~~~i'""-"Lz)]l'2~
0
X exp(-aLz)
d.2
(16) The integration is performed with the aid of the same substitution used in Section 2.4. The result is WL,t = (NA)L C~X,(T~,~~
- exp(-aLT)F
(17)
where rLr is the average stress‘at the interface (at z = T) between the deformed layer and the underlying material. For very thick layers (!I’ -+ -), eqn. (17) yields the same result as the initial friction calculation does, i.e. eqn. (12). 3.2. The friction coefficient for material I coated with material 2 For the case of material 1 coated with a different material 2, tbe boundary conditions become important. When the rider is displaced by 6x,, the displacement profile in the coating can be described by 6x(z) = 6x, exp(-aLz). At the coating-su~tra~ interface, the d~placement will be 6x, exp(-aLT). It is assumed that this displacement will be continuous across the boundary. The displacement should then decrease by another exponential factor, exp(-az), which is characteristic of the base material. The plastic work term from the substrate material is obtained directly from eqn. (12) if Sx, exp(-aLT) is substituted for 6x, and r1 for 7,. At this stage, it is assumed that the contact area is the same in the surface layer and in the bulk material and that it is determined by the
208
properties of the layer material. Therefore (NA)L is used for both materials. This assumption is reexamined in Section 3.3. When both contributions to the deformation work are combined and the sum is equated to the frictional work, we obtain the following expression for p as a function of the thickness of the coating material:
+7
max
exp(--aLT)F
i
~max )I
A-
(18)
In this expression, rI and rL1 have been retained as independent variables. This is not reasonable physically because the stress in the bulk is increased by the transfer of stress from the surface down through the coating and, for 7L1 > 71, across the boundary. Transfer of stress across the boundary would not occur for rL1 < rI unless the rate of recovery in the bulk material is appreciably greater than in the coating. For the following discussion, it is assumed that the stress is continuous across the boundary, so that 7I must be equal to Eli. The boundary stress is then determined by 7Ls, and it can be expressed from eqn. (5) as (19)
3.3. Correction for a thin layer In Section 3.2 it was assumed that the contact area is determined by the properties of the coating material, so that the contact area will be larger for a soft layer than for a hard layer. This can no longer be true when the layer is extremely thin, because the yield pressure of the substrate material will then influence the size of the contact area [ 1,2,23,25]. We could modify the friction model to allow for this effect for an ideal smooth surface. Or, more realistically, we can acknowledge the presence of surface roughness, so that with films of a small average thickness some asperities will be coated and some will contact the rider surface directly. This will be very likely when the average film thickness is of the same order as the surface roughness. For uncoated asperities, the contribution to ~1will have the same form as that of eqn. (12), in which all variables are properties of the base material. This term will disappear when the film thickness is large but it will dominate when the film is very thin. One way to incorporate these considerations in the friction model is to introduce a distribution function f(T) to describe the fraction of the asperities which is uncoated for a given value of the average film thickness. Since experimental data are not available to describe this function, a simple form which has appropriate limits will be used: f(T) = + {tanh(U-
UT) + 1)
(20)
209
-0.6 tz 04 -
0.2 -
Thickness,T
Fig. 5. The dependence of the distribution function f(T) = 0.5 {tanh( U - VT) + 1) on coating thickness 2 (U = 2 ; u = 4 pm-l ). This function represents the fraction of sample asperities in direct contact with the rider for a given value of the average coating thickness.
Figure 5 shows the form of this function for particular choices of U and u. The point of inflection is chosen by setting T = U/u. The parameter U is chosen so that f(0) is close to 1. The con~bution of the uncoated asperities can then be weighted by f(T) and the contribution of the coated asperities by 1 - f(T). The resulting form for p(T) is then
+7
max
exp(--aLT)F
-5-
+ f(T) N$
7s 7maxfit
-
(21)
( ?max )I ( rmax 1 When T approaches zero (no coating), this equation reduces to eqn. (12), and when T approaches m, this equation reduces to a similar expression for the bulk coating material. Halling [ 23 ] has used a similar approach to treat surface roughness for his calculation of ~1(T), but his model is based on the conventional adhesion model of friction. Halling’s distribution function is similar to the one used here, but he has not weighted the con~butions independency. Instead, he has used averages for all his parameters. The two models give similar results for T = 0 and for large T, but they differ for intermediate values of T. 3.4. Results for a soft layer on a hard base Equation (21) is a general result which may be used for different types of coatings and substrates. The first case to be discussed is that of a soft coating on a harder base material (7Lmax< T,,,). One example would be a soft metal film on a harder alloy substrate. The film material, treated by itself as bulk material, would have a friction coefficient of
Similarly, without any coating, the base material, denoted by B, would give
210 B _ cr
--?
NA max LJ
Using these expressions for pL and pB, together with eqn. (19) for 7r, eqn. (21) becomes p(T) = (1
F(WLmax)
1 -exp(--aLT)
-f(T)}pL
JY~LJ7Lnlax1
i --
Tmax
F(TIITrnax
rL max
F(~Ls/~Lmax)
-
1
(22)
+ fWPB
The shape of this function depends on the relative values of pL and pB. Several examples are shown in Fig. 6. These have been calculated as follows: f(T) as in Fig. 5; ~Lm,,/~maX= 0.2; 7Ls/~LmaX= 0.9999; uL = 0.05 pm-‘. The value of pB has been kept fixed, while pL has been varied. When pL < 1.3pB, there is a minimum in ~1(Figs. 6(b) - 6(d)). As pL decreases (Fig. 6(d)), the value of the minimum gets lower. However, it is shown below that the relative depth Prnin/P L of the minimum stays constant at about 0.75. Of course, for other choices of the parameters in eqns. (22) and (19), this number would vary. fn Fig. 6, the curve for ,uL < ,uBis similar to published experimental curves [ 1,23,24] . The reported minima in p are sometimes deeper than
0 (a)
(bf
(cl
(d)
0.1
IO
T
10
IOOpn
Fig. 6, Examples of the dependence of ~1on T for soft coatings on harder substrate materials(oL= 0.05fimP1 ;rLm,/~m, = 0.2;?Ls/~Lm, =0.9999;f(T) asin Fig.5): (a) /J~//J~ > 1.3; (b) 1.3 > /_tL&B > 1; (c) pL/pB = 1; (d) /.fL/pB < 1.
211
those predicted here but this discrepancy could be due to experimental conditions which do not correspond with assumed calculation parameters. It is instructive to examine eqn. (22) to see how the minimum will change as different parameters are varied. The influence of the distribution function NT) is the easiest to see. Changing U/u simply shifts the left part of the ~(7’) uersus T curve (T < Tmin)withrespect to the rest of the curve. This makes the minimum broader or narrower. Changing u in f(T)changes only the slope of p(T)at the left side of the minimum but the effect is small if this parameter is chosen in a reasonable way, as suggested in Section 3.3. The parameter uL affects the scale of T;uL canbe determined experimentally from marker experiments. The presence or absence of a minimum in I-((T) is certainly of interest for practical reasons. Because of the complexity of eqn. (22), a straightforward derivation of a condition for a minimum in /J is difficult. To obtain an approximate condition, the p(T)curve is divided into two parts. For small T the function f(T)accounts for the decrease in p(T) (Figs. 6(b) - 6(d) should be compared), whereas the function exp(--aLT) determines the increase in p(T)at large T.As soon as T 2 (1 + V)/u, f(T)is approximately zero and p(T) has reached its smallest value. It will increase again with increasing T assoon as T > O.l/aL because then exp(--aLT) < 1. To simplify the following discussion it& assumed that the surfaces are reasonably smooth. Then (1 + U)/V will be small, probably smaller than O.l/uL, and a minimum in p(T)might occur somewhere in the range (1 + U)/V < T < O.l/aL. To decide whether there is actually a minimum, a further investigation of eqn. (22) is necessary. The condition T > (1 + U)/u results in f(T)= 0 and the condition T < O.l/uL results in exp(--aLT) = 1; this simplifies eqn. (22) drastically. The same arguments applied to eqn. (19) yield pi = 7Ls; p for layers of the specified thickness will then be P
szp
L
~,a,
--
rL max
F(~Ls/cn,x) F(~~h~rnax 1
(23)
For a soft layer on a hard substrate (Tag_ < T,,,), P is smaller than yL, but for a minimum in p(T)it is also necessary that p < pB. This gives the following approximate condition for a minimum: fiL rLmax F(~~s/~~rnax) B<--(24) l-l rmax WLJ7,,,) The depth of the minimum is then given by eqn. (23). Figure 7 shows how the condition for a minimum (eqn. (24)) varies with rLJrLmaX for several values of Tag&,,,. When T~~_/T,,,, which expresses the dissimilarity of the coating and substrate materials, is less than about 0.5, it has little influence on the condition for a minimum. For a soft coating on a hard substrate, this condition will generally be satisfied. During sliding, the soft material will be highly stressed, close to its intrinsic limit, so that rLJrLmaX should be close to unity. Recognizing these conditions, one can see from Fig. 7 that pL/pB should be less than about 1.3
212
Fig. 7. Conditions for which a minimum will occur in p(T) us. 2’ (soft coatings on harder substrates). Fig. 8. Values of pL,in/pL for different choices of T~.JT~,~ coating on hard substrate).
and rLmax/~max (soft
if a minimum in ,u(2’) is desired. Yet softer materials often have higher friction coefficients than hard materials. Therefore it might be difficult to find a soft material which fulfills this condition. The value of the minimum in p(T) (eqn. (23)) is also of practical interest. Figure 8 shows how this expression varies with ~~~~~~~~~and rL max17max. Again, the curves are insensitive to Tag&-,,, for values of this parameter which are less than about 0.5. When the surface stress rLSis high, the value of I-(~~ will typically be about 75% of pL. This discussion has been based on the assumption that (1 + U)/u < O.l/uL, which may not always be true. However, because of the complexity of eqn. (22), the preceding discussion has concentrated only on the development of some simple guidelines for selecting a coated system which will exhibit low friction. To summarize, the following approximate criteria emerge. (1) rL,ax for the coating material should be less than about half the 7maxvalue for the substrate material (2) The friction coefficient for the coating material alone (thick coating) should be as low as possible, at least satisfying pL 2 1.3~~. (3) The surface stress for the coating should reach a value as close to its maximum value as possible. (4) The minimum value of y(T) will be roughly 75% of pL. (5) The thickness of the layer at which p( 2’) reaches its minimum will be in the range (1 + U)/V 5 T 2 O.l/uL for smooth surfaces. The conditions summarized here are not necessarily independent. In general, an optimum coating will depend on a compromise among possibly conflicting criteria. 3.5. Results for a hard layer on a soft base Equation (22) can be used again for this case, but now 7L,aX will be larger than r,,, since the coating material is stronger than the substrate material This fact introduces an additional constraint on r which was not
213
important for the case of 7L,aX < T,,.,~~ : 7I at the coating-substrate interface cannot exceed T,,, of the bulk. With ~~~~~ and uL fixed for a given sliding system, 7I will be a function of 7Ls. Therefore this new constraint determines an upper limit for 7Ls. The physical reason for the upper limit of rLs can be understood easily. When the coating is thick, so that all stresses and strains diminish inside the layer (owing to the exp(--aLz) factor), the surface can support a stress as high as its intrinsic limit 7L,aX. However, if the layer is thin, shear stresses will be present at the interface and these must transfer across the interface into the underlying base material. Since that material is softer, it will deform more easily than the adjacent coating material The net result will be a shear fracture at or near the coating-substrate interface and the coating will skate across the surface of the substrate. If this is not to happen, 7Ls will be limited by 7i, which in turn will be limited by T,,, rather than by T~,,,~~.An extreme example may illustrate the problem more clearly. The soft bulk might be gelatin and the hard layer a thin metal plate. With this model system, it would be impossible to deform the metal plastically (i.e. 7LS would be less than the plastic yield strength of the metal) and the maximum shear stress which could be applied (assuming good bonding) would be that of the gelatin. For the calculation of p(T) curves with this model it is assumed that 7Ls is smaller than T,,, of the bulk. This ensures that the condition 71 < 7max is always satisfied. Figure 9 shows some typical curves. The following parameters are used: rLmax/r,,, = 4; rLs/rLmaX= 0.24; aL = 0.05 pm-‘. Again, pL is varied with respect to pB and it is assumed that both pL and pB are determined under the same conditions as the final p(T) curve. A comparison of this model with experimental data is difficult because only limited data are available. Also, the p(T) uersus T curves predicted here are based on very severe constraints, the continuity of 7 and 6x across the coating-substrate interface, and on the achievement of steady state conditions. The continuity conditions are particularly difficult to guarantee for the case of a hard coating on a soft substrate, especially for a very thin coating. One preliminary experimental curve for p(T) (hard layer on soft base) has been presented [ 251. The curve p(T) shows a clear maximum which agrees at least qualitatively with the result of the present model. For many applications of coatings it might be undesirable to have a maximum in P at a certain thickness of coating material, and therefore the conditions for a maximum are worth discussing. The same assumptions as in Section 3.4 are used and ~1for the region (1 + U)/u < T < O.l/uL can be approximated as in eqn. (23). Since 7LmaX> T,,,, p is now larger than pL; if ,u is also larger than (Us, this will result in a maximum in p (2’). The conditions for the existence of a maximum are therefore L !!-_> -rL max fT~Ls/~Lrnax)and
PB 7max JVLs/?nax) The value of the maximum is
rL s =GTmax
(25)
214
tL
P"
P 0:
0.1
IO
T
IO
loo pm
po01
1.0
T
IO
100 ptn
(a)
rl---: r-”
LLB
P 0
01
IO
IO
0.1
100 urn
T
(cl
1.0
IO
IOOpr
T
fd)
Fig. 9. Examples of the dependence of g on T for hard coatings on soft substrate materials (aL = 0.05 pm-l ; 7Lmax/7,,, = 4; TLs/TLm, = 0.24;f(T) as in Fig. 5): (a)pL/pB = 0.8; (b)pL/pB= O.S;(c)pL/pB= l.O;(d)pL/pB= 1.3.
Figure 10 shows the height of the maximum as a function of 7LS/~LmaX for several values of 7LmaX/rmax . A maximum will only occur when the conditions (25) are also satisfied. This maximum will be of noticeable size if 7LS is very close to
Fig. 10. Values of p,,/pL for different choices of rLs$rLmax and ?Lmaxfrmax (hard coating on soft substrate). The upper limit for prnax is given by the boundary condition 71<7,,.
215
T,,,. In this case the maximum might be as large as l.5r.tL. However, according to eqn. (25), pL has to be larger than about 0.7gB to get a maximum at all. This condition might be somewhat difficult to fulfill since hard materials tend to have lower friction coefficients than soft materials. With the present model no prediction can be made for the case of TI > rrnax (large Tag). Therefore, the following very approximate guidelines apply only for the case of rLs < T,,,. A maximum in y(T) for a hard coating will occur if 7Ls is very close to 7,,, and pL is larger than about 0.7~~. Then T,,, might reach a value of up to 1.5~~ and the thickness of the layer at the maximum value of P will be in the same range as in the case of a soft coating.
4. Conclusions Tbe friction model described in the present paper is not intended to give precise numerical results for the friction coefficient but to determine the dependence of the friction coefficient on familiar mechanical parameters as well as on microstructural features of a material. It is based on the idea of the equivalence of the frictional work done and the work of plastic deformation during steady state sliding. The plastic behavior of a material is described by parameters of its shear stress-strain curve and the average shear stress at the surface. The model includes as a special case the well-known model of Bowden and Tabor [ 1,2 ] but it does not require adhesion to be present. The new model is an extension of simpler energy-based models described recently by the present authors [5, 71. The model has been improved by using a more realistic stress-strain curve, by introducing a surface stress rather than a surface strain and by adding the effect of direct rider-substrate contact in the case of coated materials. The friction model represents a broad general basis from which work can be extended in a number of directions. All the parameters can be defined more precisely to include more variables and fewer assumptions. The process of dividing a sample into several layers and adding their contributions to the friction has proved to be straightforward with the present model. For example, in the case of coated materials the model has been used to predict the general form of the friction coefficient as a function of coating thickness and materials properties. In addition, some approximate guidelines for selecting coated systems have been established. The idea of using different layers to describe a friction system is not limited to well-bonded applied coatings. It can also be applied to layers of reaction products (such as oxides), contamination layers, transferred material, layers of solid lubricants (such as Moss or graphite) and even to lame&r structures. Fu~ermo~, the initial convenience of assuming a hard rider material is not really necessary. The model can readily be extended to apply to any combination of materials in which plastic deformation is the dominant dissipative mechanism.
216
Although the present model has been developed for steady state conditions, it offers some insight into the processes which occur during running in. It takes time for TV,7 and f(T) and the ~mperature to reach their steady state values, so the model contains an implicit time dependence for the friction coefficient. Thus, changes during the early stages of sliding, as well as transitions which might occur later, can be accommodated. The main difficulty in using this model is the lack of appropriate experimental data. Even for pure metals, it is difficult to find suitable data for the m~imum shear stress T,,, and for alloys and oxides the situation is worse. Data are even more limited for the average surface stress 7, and the real contact area NA during steady state sliding. Until these data are available, this friction model, and probably others as well, will be restricted to predicting trends rather than reliable quantitative results.
Acknowledgments The authors would like to thank Professor K. L. Johnson, University of Cambridge, and Professor J. P. Hirth, The Ohio State University, for their helpful comments. This material is based on work supported by the National Science Foundation under Grant DMR 7805719.
References 1 F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Part II, Clarendon, Oxford, 1964. 2 F. P. Bowden and D. Tabor, Friction: An Introduction to Tribology, Anchor, New York, 1973. 3 D. Dowson, History of Tribology, Longman, London, 1978. 4 D. G. Teer and R. D. Arnell, in J. Halling (ed.), Principles of Tribology, Macmillan, London, 1975, pp. 72 - 93. 5 D. A. Rigney and J. P. Hirth, Wear, 53 (1979) 345 - 370. 6 W. A. Glaeser, An engineer’s guide to friction, DMIC Memo. 246, 1970 (Defense Materials Information Center, Battelle Memorial Institute). 7 P. Heilmann and D. A, Rigney, Sliding friction of metals. In D. Dowson, C. M. Taylor, M. Godet and D. Berthe (eds.),Proc. Leeds-LyonSymp. on Tribology, Leeds, September 9 - 1.2, 1980, Westbury House, Guildford, 1981. 8 J. P. Hirth and J. Weertman (eds.), Work Hardening, Gordon and Breach, New York, 1968. 9 A. W. Thompson, Work Hardening in Tension and Fatigue, Metallurgical Society of AIME, New York, 1977. 10 U. F. Kocks, Strain hardening and strain-rate hardening, American Society for Metals Symp. on Mechanical Testing for Deform&ion Model Development, Bal Harbor, FL, November 1980. 11 J. H. Holioman, Trans. Metall. Sot. AIM.&‘, 162 (1945) 268 - 290. 12 E. Vote, J. Inst. Met., 74 (1947 - 1948) 537 - 562. 13 D. Kuhlmann-Wilsdorf, in J. P. Hirth and J. Weertman (eds.), Work Hardening, Gordon and Breach, New York, 1968, pp. 97 - 132.
217 14 A. Van den Beukel, Ser. hfetall., 13 (1979) 83 - 86. 15 D. A. Rigney, Ser. Metall., 13 (1979) 353 - 354. 16 K. L. Johnson, Aspects of friction. In D. Dowson, C. M. Taylor, M. Godet and D. Berthe (eds.), Proc. Leeds-Lyon Symposium on ~iboiogy, Leeds, September 9 - 12, 1980, Westbury House, Guildford, 1981. 17 J. H. Dautzenberg, wear, 60 (1980) 401 - 412. 18 M. A. Moore and R. M. Douthwaite, Metall. Trans., A, 7 (1976) 1833 - 1839. 19 N. Ohmae, in N. Saka and N. P. Suh (eds.), Fundamentals of Tribology, Massachusetts Institute of Technology Press, Cambridge, MA, 1980; Philos. Mag. A, 40 (1979) 803 - 810. 20 L. K. Ives, in K. C. Ludema, W. A. Glaeser and S. K. Rhee (eds.), Weur of Materials, American Society of Mechanical Engineers, New York, 1979, pp. 246 - 256. 21 A. W. Ruff, L. K. Ives and W. A. Glaeser, Characterization of worn surfaces and wear debris, American Society for Metals Materials Science Semin. on Fundamentals of Friction and Wear of Materials, Pittsburgh, October 4 - 5, 1980, American Society for Metals, Metals Park, OH, 1981. 22 0. Vingsbo and S. Hogmark, Wear of steels, American Society for Metals Mater~~ Science Semin. on Fundamentals of Friction and Wear of Materials, Pittsburgh, October 4 - 5, 1980, American Society for Metals, Metals Park, OH, 1981. 23 J. Halling and M. G. D. Sherbiney, The role of surface topography in the friction of soft metallic films, Conf. on Materiah Performance and Conservation, Swansea, April 3 - 4. 1978, Institution of Mechanical Engineers, London. 24 N. P. Suh, N. Saka and S. Jahanmir, Wear, 44 (1977) 127 - 134. 25 J. Haliing, USME Rep. T/44/75, 1975 (Department of Aeronautical and Mechanical Engineering, University of Salford).