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Nano metrology of cylinder bore wear
Pergamon PII:
lnt, J. Mach. Tools Manufact. Vol. 38, Nos 5~, pp. 519-527, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0890-6955(97)00097-7 0890-6955/98 $19.00 + 0.00
NANO M E T R O L O G Y OF CYLINDER BORE WEAR B.-G. Ros6n, R Ohlsson, T. R. Thomas Department of Production Engineering, Chalmers University, S-41296 Gothenburg
ABSTRACT Cylinder bores are multi-process surfaces whose roughness is difficult to characterise for tribological purposes by conventional methods. Statistical approaches may be used to compute asperity densities, summit curvatures and so on, but suffer from the usual disadvantage of tending to infinite values in the absence of a short-wavelength cutoff. A useful advance in tribological roughness assessment would be to find a means of establishing an appropriate scale of measurement. Using a form of the plasticity index corrected for anisotropy, a short-wavelength limit Ep is derived below which asperities will not take part in long-term tribological interactions. A general relationship is obtained between three dimensionless numbers, the short-wavelength limit )~p normalised by the topothesy A, the fractal dimension D and the material ratio (the ratio of the Hertzian elastic modulus E' to the hardness H). From this relationship, presented as a carpet plot, the appropriate scale of roughness measurement for any tribological investigation of a fractal surface may be determined. With a stylus instrument and an atomic force microscope, a number of cylinder bores were measured at locations of both high and medium wear before and after running in. By inspection of an ensemble of structure functions, it is shown that cylinder bore surfaces are multifractal, with a transition point (the so-called "comer frequency") at about 20 [tm, corresponding to the average size of a honing grit. Below this length the surfaces are selfsimilar fractals down to the limits of AFM resolution. The short wavelength limit using the above formulation appears to be about 40 nm, weIl below the range of instruments usually employed to measure tribological surface roughness. ~.5, 1998 ElsevierScienceLtd
1. I N T R O D U C T I O N
a much coarser one, the different properties of each being critical for the functional performance of the whole component.
In the characterization of cylinder bore roughness there are two separate, but related, problems. The first is how to describe the freshly manufactured surface for the purposes of, for instance, inspection and quality control. The second is how to describe, and if possible predict from contact mechanics, the changes that occur to the surface topography during runningin. These problems are complicated by the nature of cylinder bores as "multiprocess surfaces" ( Malburg and Raja 1993 ) which typically have a fine finish superimposed on
What features of topography are changed during running-in and how are we able to describe these changes numerically? An initial hypothesis might be that at least some of the higher regions of the surface are removed to begin with. One obvious way of describing this would be to investigate changes in the height distribution. King et al (1977) reported systematic changes in the higher moments of the height distribution, specifically in the
519
520
B.-G. Rost3n et cal.
third and fourth central moments (skewness and kurtosis). However, other changes can occur, such as the disappearance of a band of surface wavelengths (Whitehouse & Archard 1971), or both processes may be combined. Thomas (1978) proposed a typology for characterizing running-in which would require a description of changes both to the height distribution and to the power spectrum of the surface. In either case, some statistical description of the surface will be necessary. When Greenwood & Williamson (1966) formulated their original statistical description of elastic rough contact in terms of asperity density and peak radius of curvature it was rightly hailed as a major breakthrough in rough surface tribology. Problems began to arise when Whitehouse & Archard (1971) pointed out that asperity densities and curvatures were not intrinsic properties of surfaces but varied in their numerical value with the scale of measurement. These problems became clearer when Nayak (1971) applied statistical theories of random rough surfaces to engineering surfaces and workers faced the practical difficulties of quantifying his predictions. The central difficulty was easily defined: asprerity densities and curvatures are expressed in terms of the second and fourth moments m n of the power spectral density function G(m), where: mn =~2
conG(m) dm
(1)
and m 1, 0 2 are respectively the long and short-wavelength cutoffs of the spectrum. The long-wavelength cutoff in any tribological application must be set by the physical dimensions of the contact. It is not so easy, however, to see a physically obvious reason for truncating the spectrum at the short wavelength end. Given a
power spectrum of the form: G(m)
=
Bco~
(2)
where [3 > 1, m 2 and m 4 are therefore undefined. This is merely to say that any approach which attempts to include the smallest features on a random surface will predict that there are an infinitely large number of infinitely sharp asperities. An ingenious attempt has been made to avoid this difficulty by constructing arguments for contact behaviour based purely on contact size (Majumdar & Bhushan 1991), but these appear to yield predictions that are counter to intuitive; in any case a theory which does not take account of the known effect of asperity geometry on contact mechanism cannot be regarded as wholly satisfactory.
2. R O U G H C O N T A C T
2.1 Fractal surfaces More recently the random-process approach of Whitehouse and Archard and Nayak has been modified by a fractal view of surface behaviour. It has been known for some time that a number of machined surfaces exhibit fractal behaviour over at least a part of their range of surface wavelengths (Saytes & Thomas 1978, Thomas & Thomas 1988). The possible applications of fractals to tribology have been reviewed by Ling (1990). Fractal descriptions of engineering surfaces have been given by a number of workers (Majumdar & Tien 1990, Stupak et al. 1991, Vandenberg & Osborne 1992, Yordanov & Ivanova 1995), and fractal theories of contact mechanics (Majumdar & Tien 1990, Majumdar & Bhushan 1990,1991, Wang & Komvopoulos 1995, Warren & Krajcinovic 1995) and wear (Zhou et al. 1993) have been developed.
Cylinder bore wear Self-similar fractals of the kind originally described by Mandelbrot are described completely by a single parameter, the fractal dimension D, which is an intrinsic property of the surface and does not change with the scale of measurement. Real surfaces of the kind measured by a stylus instrument have the additional restriction that they must be everywhere single-valued, that is they cannot contain re-entrants (as an electron microscope, for instance, might reveal). Surfaces like this are not self-similar; as the scale is reduced, steeper and steeper slopes are revealed. Some additional scaling factor is needed to compensate for this change in appearance. This scaling factor is called the topothesy A (Sayles & Thomas 1978) and has units of length. The topothesy is defined as the separation of heights corresponding to an average slope of one radian (Berry 1978). Such surfaces are called self-affine.
521
It can be shown that for a fractal profile (Russ 1994): S(x)
= A(2D-2)x2(2°D)
In other words, the structure function of a fractal profile obeys a power law, so it plots as a straight fine on a log-log scale. This is an easy way of establishing fractal behaviour, and from the slope and intercept of this straight line both the fractal dimension D and the topothesy A can easily be calculated. The corresponding power spectrum obeys a similar power law and will also plot on a log-log scale as a straight line, this time with slope [3 and intercept B. The respective slopes and intercepts are related by (Russ 1994): [3 = 2 D - 4
(4)
B = (2r~)13 2 F ( - ~ )
cos(-[3r~/2)
• A(3+13) Most methods of calculating the fractal dimension were developed for self-similar fractals, like coastlines, cracks in rocks and particles of powder. They do not work very well for self-affine surfaces with gentle slopes. An effective way to calculate the fractal dimension of a profde z ( x ) is to compute the structure function (Sayles & Thomas 1977): S(x)
= <[z(x)
- z(x+x]2>
where x is the separation of pairs of points in the plane of the surface. The structure function is related closely to the autocorrelation function and hence by Fourier transform to the power spectrum. It contains no new information compared with these other functions but presents it in a more accessible way. Also, it is quicker to compute than an autocorrelation function and is much more stable than a power spectrum.
(3)
(5)
where F is a Gamma function. In practice no real surface can be fractal over an infinite range of wavelengths, because no natural or man-made process can operate over an infinite range of wavelengths. A real surface will be formed by several different processes each with its characteristic features. For instance, a mountain landscape may be formed by erosion on a scale from kilometres down to meters. Below this scale the landscape may be covered with vegetation, which may also be fractal down to a scale of minimetres, but with completely different values of fractal properties. Such surfaces are called multifractal (for an extensive discussion, see Russ 1994) and typically will present a structure function as two or more straight lines of different slope meeting at a more or less sharp discontinuity. On the contrary, such a structure function is evidence of multifractal behaviour and the wavelength corresponding to the discontinuity will
522
B.-G. Rosen eta/.
mark the transition from one mechanism of surface formation to another which may be completely different. This transition point has been termed a comer frequency (Majumdar & Tien 1990). Machined surfaces are likely to be multifractal because they are usually produced by several processes. The approximate shape of the surface is produced by casting or rough preliminary machining. This will produce the form and waviness of the surface, possibly depending on the dynamics of the machine tool (chatter, spindle runout and so on). One or more finishing processes will produce the final roughness of the surface. One would expect transitions in the structure function to depend on the dimensions of the actual cutting element, which may produce surface features far smaller than itself but cannot produce features any larger, Plateau honing should be a good example of a multifractal surface. We would expect to see a transition at a wavelength corresponding to the diameter of the abrasive particles, because although these can create a range of surface features smaller than themselves they can create none larger. Note that the depth of honing may affect the magnitude of the structure function at a particular separation (because this is a measure of the roughness) but should not affect the transition wavelength. 2.2 The limits of elastic behaviour
Many attempts have been made to calculate the elastic parameters of a rough contact (for a review see Johnson 1985). They all require a knowledge of the numbers of asperities per unit area of the surface and their curvatures or slopes. This causes a philosophical difficulty in that statistical theories of surface roughness all predict that as asperities become smaller they become more numerous and sharper The problem is explicit in Nayak's (1971)
approach, where he expresses the statistical rnicrogeometry of a rough surface in terms of the first three even moments m n of the power spectrum G ( m ) . As was pointed out above, slopes, curvatures and numbers of asperities depend on the second and fourth moments, and it is easy to see that unless the power spectrum falls away very sharply with increasing frequency, neither of the corresponding integrals will be finite. Thus to obtain finite values of slopes and curvatures requires a restriction of the domain of integration. In physical terms it requires the setting of some limiting short wavelength below which all surface features will be ignored. Are there any reasons for such a rejection? A possible argument may be constructed (Thomas 1982) by considering the mechanism of deformation. As asperities get smaller and sharper, a size will be reached below which they will deform plastically during the very first cycle of contact and so disappear. During the subsequent lifetime of the component, it will behave elastically as if the corresponding range of surface wavelengths did not exist. The critical wavelength may be found from a relationship between the second moment and the plasticity' index ~ (Greenwood & Williamson 1966). Greenwood and Williamson looked for a criterion which would determine the mode of deformation of an array of asperities of varying heights. They found that the mode of deformation of the highest asperities was almost independent of load. Sharp asperities would deform plastically under even the lightest loads, while blunt asperities would deform elastically under even the heaviest loads. The criterion of deformation was the so-called plasticity index, which depended only on material properties and roughness parameters. Their roughness parameters are rather inconvenient to use, and we have shown
Cylinder bore wear elsewhere (Rosen et al 1997) that a more convenient formulation for machined surfaces is: m
E'
/
where E" is the Hertzian elastic modulus and H is the hardness. The second moment may be found directly from measurements of surface slope. The validity of this approach has been confirmed independently by Pawlus & Chetwynd (1996). The difficulty again is that the numerical value of the slope, and hence the plasticity index, depends on the scale of measurement. When measurements were made on the same areas of worn and unworn cylinder liners, using a stylus instrument and an atomic force microscope
523
(Rosen et al 1996), the smaller sampling interval of the AFM yielded higher slopes and hence higher values of the plasticity index (Figure 1), and the AFM values correlated better with the observed wear behaviour of the surfaces. If Up is the critical value of the plasticity index above which deformation will be plastic at any load, then the critical second moment m2p = {r~ ( 2 - 7r,/2 ) }-Y2 ~p2 ( H / E" ) 2
(6) Integrating Equations 1 and 2 yields: m2=B(3+13)-l(2=)-13(oY23+13 _to13+13)
(7)
Plasticity index Elastic and Plastic regions m e
lO
v= .s N t t I
I
J
i1
m
N+
I o
0
t
Plastic
Elastic
[ d o 0
i 1--
10
1 ! i l
100
E'/H • •
stylus TDC UW • stylus Mid UW &
afro TDC UW afro Mid UW
x
stylus TDC W stylus Mid W
<~ X
afro TDC W afm Mid W
Figure 1. Different plasticity indices measured on the same surface by AFM and stylus (Rosen et al. 1996).
524
B.-G. Roscn ~,t a /
~-~-'/
If [3 > ! and the bandwidth of surface wavelengths is reasonably wide, i.e. m2 ,~
-1175, -zo-4 . _ 2 D ) - 2 - b - - ~ (2n') F(4 *
o)1, then Equation 7 reduces to:
\
2 J\HJ (io)
m2 _= Bo)23+~,a / ( 3 + [ 3 ) ( 2 r t ) ~
(8)
This is a relationship between three dimensionless numbers: the critical wavelength normalized by the topothesy, the plateau fractal dimension and a material property ratio. The dimensionless wavelength is highly sensitive to the other parameters (Figure 2), and for a given fractal dimension and material property ratio the critical wavelength increases as the topothesy. Thus we can in principle now find a unique short-wavelength cutoff, depending only on material properties and intrinsic topography parameters, which we can use to determine the elastic behaviour of the contact.
The critical wavelength Lp = 2rt / o,'2p, so combining Equations 6 and 8,
B (_~12 8rc3/2(2-1r / 2)J/z
Zp3+~=
3+8 (9) From
Bush
et
al
1978,
~p
= 0.7.
Combining this with the numerical constant, replacing B and 13 by the corresponding fractal parameters from Equations 4 and 5 and rearranging, we have finally:
3 JIImllllmm m'mm Log ( Zp / A ) as-
2-
1,510,5 0
90 70
E'/H
D
~
""
Figure 2. Dimensionless critical wavelength as a function of material ratio and fractal dimension, from Equation 10.
Cylinder bore wear
525
3. E X P E R I M E N T A L
4. RESULTS AND DISCUSSION
The engine used in this study has been undergoing a so-called "dynamic test" simulating the everyday use of an ordinary car engine. The cylinder bore was made of grey cast iron, inserted into an injection moulded aluminum engine block and honed. In order to follow the surfaces from an unworn to a worn state without having to disassemble the test engine used, a replication and relocation method was employed. Roughness measurements were made with two instruments covering different but overlapping ranges of surface wavelengths. The first was a 3D scanning stylus system designed and constructed inhouse around a Perthen stylus instrument, with vertical and horizontal resolutions of 20 nm and 5 ~tm respectively. The second was a Rasterscope atomic force microscope (AFM) with vertical and horizontal resolutions of 3 pm and 125 pm respectively. Measurements were made in the unworn and worn states at the midpoint of the bore and at TDC. The measurement techniques, test surfaces and results have been discussed more fully elsewhere (Rosen et al. 1996).
The structure functions of all the unworn surfaces and the worn mid-cylinder surfaces, whether measured by stylus or AFM, combine into a remarkably consistent multifractal (Figure 3) There is a clear transition at about x = 20 l.tm, corresponding to the dimensions of the largest honing grits. Features smaller than this are evidently produced by a single continuous and almost ideally fractal process, which must be associated with the fracture of the grits.
topothesy is 1.3 x 10 -4 lxm. The numerical value of the topothesy is consistent with those reported in the literature (Russ 1994). Turning to the structure function of the AFM measurements on the worn TDC,
Structure Function
S(t) I
0,1
It is of course not necessary that the surface of the honing stone itself be fractal in order to create a fractal surface on the workpiece, but it is interesting to note in this connection that the surface of a diamond grinding wheel has recently been recognized as fractal (Liao 1995). From Eqn. 3, the fractal dimension associated with this process is about 1.2 and the
÷ TDC, unworn ¢" TDC, worn • MID, unworn A MID, worn
_
.
AFM vs. Stylus
P°
_
0,01
f
0,001
0,0001
0,110001 0,001
0,01
0,1
AFM |r,ompllng nm) 41. IDC, unworn (500) 0 MID, unworn (240) © TDC, worn (500) • I"DC, worn (1201 • MID, unworn (500) <> MID, unworn (7.4) [] MID, worn {500)
I
L
I
I
1
10
100
1000
t(.ml 10000
Figure 3. Structure functions of regions of the same cylinder liner in worn and unworn conditions, measured with AFM and stylus instrument
526
B.-G. Rosdn eta/.
the fractal dimension of the shorter wavelengths is about 1.6. These figures may be compared with the prediction of Zhou et al. (1993) that D < 1.4 corresponds to a surface still experiencing wear, while 1.45 < D < 1.55 is a regime of minimum wear. From Eqns. ( 4 ) and ( 5 ), the constants of the power spectrum are 13=-1.6 and B = 2.86 x 10 -7 . I f E ' / H = 9.5, then from Eqn. ( 6 ), m2p= 0.0046, and finally from Eqn. ( 10 ), the point of
more convenient range of wavelengths. He went on to dismiss this as a practical possibility on the grounds that few real surfaces showed monofractal behaviour over a wide enough range of wavelengths. The present demonstration of unchanging fractal properties over three decades of wavelength leads us to believe that the geometry at, say, 20 nm may in fact be predicted with some confidence from measurements in the region of 1 ~tm.
transition to a plastic regime COp= 158 gin1, corresponding to a critical wavelength Zp of 0.04 gm. With this figure in mind, it may be significant to observe an apparent transition in the AFM-measured structure function of the unworn mid cylinder at about 0.03 gm. We would argue that this wavelength represents the lower limit of elastic behaviour; it follows that calculations of elastic contact based on measurements which do not extend to wavelengths as short as this, i.e. any measurements made with conventional stylus instruments alone, will be in error. Russ (1994) notes that it has not hitherto been possible to assign any physical significance to the topothesy, and that its typical numerical values represent implausibly small lengths. In this paper we believe that we have succeeded for the first time in linking the topothesy with some physically meaningful property of an engineering surface. Furthermore, the product of the topothesy with the large numerical constants predicted by Equation 7 yields critical wavelengths comfortably within the measurement range of modern instruments. Whitehouse (1994), in a striking phrase, suggested the possibility of using the fractal properties of a surface to"abseil" down the scale of surface wavelengths, that is to infer the microgeometry of a surface at the smaller scales from measurements made over a much higher and thus instrumentally
REFERENCES Berry, M. V., "Diffractals", J. Phys. A12, 781-797 (1979) Bush, A.W., R.D. Gibson and G.P. Keogh, "Strongly Anisotropic Surfaces", ASME Paper 78-Lub-16 (1978). Greenwood, J.A., and J.B.P Williamson, "Contact of Nominally Flat Surfaces", Proc. R. Soc., A295, 300 (1966). Johnson, K. L. Contact Mechanics, Cambridge University Press, London (1985). King, T.G., D.J. Whitehouse and K.J. Stout, "Some Topographic Features of the Wear Process -Theory and Experiment", Annals of the CIRP, 25 351-356 (1977). Liao, T. W., "Fractal and DDS characterisation of diamond wheel profiles", J Mat. Proc. Tech. 53, 567-581 (1995). Ling, F. F., "Fractals, engineering surfaces and tribology", Wear 136, 141-154 (1990) Majumdar, A. and B. Bhushan, "Fractal model of elastic-plastic contact between rough surfaces", J Trib. 113, 1-11 (1991) Majumdar, A., and B. Bhushan, "Role of fractal geometry in roughness characterisation and contact mechanics of surfaces", J. Trib. 112, 205-216 (1990).
Cylinder bore wear
Majumdar, A., and C. L. Tien, "Fractal characterisation and simulation of rough surfaces", Wear 136, 313-327 (1990) Malburg, M.C. and J. Raja, "Characterization of Surface Texture Generated by Plateau Honing process", Annals of the CIRP, 42/1,637-639 (1993) Nayak, P.R., "Random Process Model of Rough Surfaces", Trans. ASME: J. Lub. Tech., 93F, 398-407 (1971). Pawlus, P. and D. G. Chetwynd, "3D surface texture characterisation of cylinder bores", Proc. 9th. Int. Surface Colloquium, 473-484, TU Chemnitz-Zwickau (1996). Rosen, B.-G., R. Ohlsson and T. R. Thomas, "The plasticity index and its application to the tribology of machined surfaces", JSME Int. J. (in press). Rosen, B.-G., R. Ohlsson and T. R. Thomas, "Wear of cylinder bore microtopography", Wear 198, 271-279 (1996). Russ, J. C., Fractal Surfaces, Plenum Press, New York (1994). Sayles, R. S. and T. R. Thomas, "The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation", Wear 42, 263-276 (1977). Sayles, R. S. and T. R. Thomas, "Surface topography as a non-stationary random process", Nature 271, 431-434 (1978) Stupak, P. R., C. Y. Syu and J. A. Donovan, "The effect of filtering profflometer data on fractal parameters", Wear 154, 109-114 (1992) Thomas, T. R. and A. P. Thomas, "Digital analysis of very small scale surface
527
roughness", J Wave-Material Interaction 3, 341-350 (1988) Thomas, T. R., Rough Surfaces, Longman, London (1982) Thomas, T.R., "The Characterization of Changes in Surface Topography in runningin", Proc. 4th. Leeds-Lyon Symp., 99-108, MEP, London, (1979). Vandenberg, S., and C. F. Osborne, "Digital image processing techniques, fractal dimensionality and scale-space applied to surface roughness", Wear 159, 17-30 (1992). Wang, S., and K. Komvopoulos, "A fractal theory of the temperature distribution at elastic contacts of fast sliding surfaces", J. Trib. 117, 203-215 (1995). Warren, T. L. and D. Krajcinovic, "Fractal models of elastic-perfectly plastic contact of rough surfaces based on the Cantor set", Int. J. Solids Structures 32, 2907-2922 (1995). Whitehouse, D. J., Handbook of Surface Metrology, Institute of Physics, London (1994) Whitehouse, D.J. and J.F. Archard, "The Properties of Random Surfaces of Significance in their Contact", Proc. R. Soc., A316, 97-121 (1970). Yordanov, O. I., and K. Ivanova, "Description of surface roughness as an approximate self-affme random structure", Surface Science 331-333, 1043-1049 (1995). Zhou, G. Y., M. C. Leu and D. Blackmore, "Fractal geometry model for wear prediction ", Wear 170, 1-14 (1993).
lnt, J. Mach. Tools Manufact. Vol. 38, Nos 5~, pp. 519-527, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0890-6955(97)00097-7 0890-6955/98 $19.00 + 0.00
NANO M E T R O L O G Y OF CYLINDER BORE WEAR B.-G. Ros6n, R Ohlsson, T. R. Thomas Department of Production Engineering, Chalmers University, S-41296 Gothenburg
ABSTRACT Cylinder bores are multi-process surfaces whose roughness is difficult to characterise for tribological purposes by conventional methods. Statistical approaches may be used to compute asperity densities, summit curvatures and so on, but suffer from the usual disadvantage of tending to infinite values in the absence of a short-wavelength cutoff. A useful advance in tribological roughness assessment would be to find a means of establishing an appropriate scale of measurement. Using a form of the plasticity index corrected for anisotropy, a short-wavelength limit Ep is derived below which asperities will not take part in long-term tribological interactions. A general relationship is obtained between three dimensionless numbers, the short-wavelength limit )~p normalised by the topothesy A, the fractal dimension D and the material ratio (the ratio of the Hertzian elastic modulus E' to the hardness H). From this relationship, presented as a carpet plot, the appropriate scale of roughness measurement for any tribological investigation of a fractal surface may be determined. With a stylus instrument and an atomic force microscope, a number of cylinder bores were measured at locations of both high and medium wear before and after running in. By inspection of an ensemble of structure functions, it is shown that cylinder bore surfaces are multifractal, with a transition point (the so-called "comer frequency") at about 20 [tm, corresponding to the average size of a honing grit. Below this length the surfaces are selfsimilar fractals down to the limits of AFM resolution. The short wavelength limit using the above formulation appears to be about 40 nm, weIl below the range of instruments usually employed to measure tribological surface roughness. ~.5, 1998 ElsevierScienceLtd
1. I N T R O D U C T I O N
a much coarser one, the different properties of each being critical for the functional performance of the whole component.
In the characterization of cylinder bore roughness there are two separate, but related, problems. The first is how to describe the freshly manufactured surface for the purposes of, for instance, inspection and quality control. The second is how to describe, and if possible predict from contact mechanics, the changes that occur to the surface topography during runningin. These problems are complicated by the nature of cylinder bores as "multiprocess surfaces" ( Malburg and Raja 1993 ) which typically have a fine finish superimposed on
What features of topography are changed during running-in and how are we able to describe these changes numerically? An initial hypothesis might be that at least some of the higher regions of the surface are removed to begin with. One obvious way of describing this would be to investigate changes in the height distribution. King et al (1977) reported systematic changes in the higher moments of the height distribution, specifically in the
519
520
B.-G. Rost3n et cal.
third and fourth central moments (skewness and kurtosis). However, other changes can occur, such as the disappearance of a band of surface wavelengths (Whitehouse & Archard 1971), or both processes may be combined. Thomas (1978) proposed a typology for characterizing running-in which would require a description of changes both to the height distribution and to the power spectrum of the surface. In either case, some statistical description of the surface will be necessary. When Greenwood & Williamson (1966) formulated their original statistical description of elastic rough contact in terms of asperity density and peak radius of curvature it was rightly hailed as a major breakthrough in rough surface tribology. Problems began to arise when Whitehouse & Archard (1971) pointed out that asperity densities and curvatures were not intrinsic properties of surfaces but varied in their numerical value with the scale of measurement. These problems became clearer when Nayak (1971) applied statistical theories of random rough surfaces to engineering surfaces and workers faced the practical difficulties of quantifying his predictions. The central difficulty was easily defined: asprerity densities and curvatures are expressed in terms of the second and fourth moments m n of the power spectral density function G(m), where: mn =~2
conG(m) dm
(1)
and m 1, 0 2 are respectively the long and short-wavelength cutoffs of the spectrum. The long-wavelength cutoff in any tribological application must be set by the physical dimensions of the contact. It is not so easy, however, to see a physically obvious reason for truncating the spectrum at the short wavelength end. Given a
power spectrum of the form: G(m)
=
Bco~
(2)
where [3 > 1, m 2 and m 4 are therefore undefined. This is merely to say that any approach which attempts to include the smallest features on a random surface will predict that there are an infinitely large number of infinitely sharp asperities. An ingenious attempt has been made to avoid this difficulty by constructing arguments for contact behaviour based purely on contact size (Majumdar & Bhushan 1991), but these appear to yield predictions that are counter to intuitive; in any case a theory which does not take account of the known effect of asperity geometry on contact mechanism cannot be regarded as wholly satisfactory.
2. R O U G H C O N T A C T
2.1 Fractal surfaces More recently the random-process approach of Whitehouse and Archard and Nayak has been modified by a fractal view of surface behaviour. It has been known for some time that a number of machined surfaces exhibit fractal behaviour over at least a part of their range of surface wavelengths (Saytes & Thomas 1978, Thomas & Thomas 1988). The possible applications of fractals to tribology have been reviewed by Ling (1990). Fractal descriptions of engineering surfaces have been given by a number of workers (Majumdar & Tien 1990, Stupak et al. 1991, Vandenberg & Osborne 1992, Yordanov & Ivanova 1995), and fractal theories of contact mechanics (Majumdar & Tien 1990, Majumdar & Bhushan 1990,1991, Wang & Komvopoulos 1995, Warren & Krajcinovic 1995) and wear (Zhou et al. 1993) have been developed.
Cylinder bore wear Self-similar fractals of the kind originally described by Mandelbrot are described completely by a single parameter, the fractal dimension D, which is an intrinsic property of the surface and does not change with the scale of measurement. Real surfaces of the kind measured by a stylus instrument have the additional restriction that they must be everywhere single-valued, that is they cannot contain re-entrants (as an electron microscope, for instance, might reveal). Surfaces like this are not self-similar; as the scale is reduced, steeper and steeper slopes are revealed. Some additional scaling factor is needed to compensate for this change in appearance. This scaling factor is called the topothesy A (Sayles & Thomas 1978) and has units of length. The topothesy is defined as the separation of heights corresponding to an average slope of one radian (Berry 1978). Such surfaces are called self-affine.
521
It can be shown that for a fractal profile (Russ 1994): S(x)
= A(2D-2)x2(2°D)
In other words, the structure function of a fractal profile obeys a power law, so it plots as a straight fine on a log-log scale. This is an easy way of establishing fractal behaviour, and from the slope and intercept of this straight line both the fractal dimension D and the topothesy A can easily be calculated. The corresponding power spectrum obeys a similar power law and will also plot on a log-log scale as a straight line, this time with slope [3 and intercept B. The respective slopes and intercepts are related by (Russ 1994): [3 = 2 D - 4
(4)
B = (2r~)13 2 F ( - ~ )
cos(-[3r~/2)
• A(3+13) Most methods of calculating the fractal dimension were developed for self-similar fractals, like coastlines, cracks in rocks and particles of powder. They do not work very well for self-affine surfaces with gentle slopes. An effective way to calculate the fractal dimension of a profde z ( x ) is to compute the structure function (Sayles & Thomas 1977): S(x)
= <[z(x)
- z(x+x]2>
where x is the separation of pairs of points in the plane of the surface. The structure function is related closely to the autocorrelation function and hence by Fourier transform to the power spectrum. It contains no new information compared with these other functions but presents it in a more accessible way. Also, it is quicker to compute than an autocorrelation function and is much more stable than a power spectrum.
(3)
(5)
where F is a Gamma function. In practice no real surface can be fractal over an infinite range of wavelengths, because no natural or man-made process can operate over an infinite range of wavelengths. A real surface will be formed by several different processes each with its characteristic features. For instance, a mountain landscape may be formed by erosion on a scale from kilometres down to meters. Below this scale the landscape may be covered with vegetation, which may also be fractal down to a scale of minimetres, but with completely different values of fractal properties. Such surfaces are called multifractal (for an extensive discussion, see Russ 1994) and typically will present a structure function as two or more straight lines of different slope meeting at a more or less sharp discontinuity. On the contrary, such a structure function is evidence of multifractal behaviour and the wavelength corresponding to the discontinuity will
522
B.-G. Rosen eta/.
mark the transition from one mechanism of surface formation to another which may be completely different. This transition point has been termed a comer frequency (Majumdar & Tien 1990). Machined surfaces are likely to be multifractal because they are usually produced by several processes. The approximate shape of the surface is produced by casting or rough preliminary machining. This will produce the form and waviness of the surface, possibly depending on the dynamics of the machine tool (chatter, spindle runout and so on). One or more finishing processes will produce the final roughness of the surface. One would expect transitions in the structure function to depend on the dimensions of the actual cutting element, which may produce surface features far smaller than itself but cannot produce features any larger, Plateau honing should be a good example of a multifractal surface. We would expect to see a transition at a wavelength corresponding to the diameter of the abrasive particles, because although these can create a range of surface features smaller than themselves they can create none larger. Note that the depth of honing may affect the magnitude of the structure function at a particular separation (because this is a measure of the roughness) but should not affect the transition wavelength. 2.2 The limits of elastic behaviour
Many attempts have been made to calculate the elastic parameters of a rough contact (for a review see Johnson 1985). They all require a knowledge of the numbers of asperities per unit area of the surface and their curvatures or slopes. This causes a philosophical difficulty in that statistical theories of surface roughness all predict that as asperities become smaller they become more numerous and sharper The problem is explicit in Nayak's (1971)
approach, where he expresses the statistical rnicrogeometry of a rough surface in terms of the first three even moments m n of the power spectrum G ( m ) . As was pointed out above, slopes, curvatures and numbers of asperities depend on the second and fourth moments, and it is easy to see that unless the power spectrum falls away very sharply with increasing frequency, neither of the corresponding integrals will be finite. Thus to obtain finite values of slopes and curvatures requires a restriction of the domain of integration. In physical terms it requires the setting of some limiting short wavelength below which all surface features will be ignored. Are there any reasons for such a rejection? A possible argument may be constructed (Thomas 1982) by considering the mechanism of deformation. As asperities get smaller and sharper, a size will be reached below which they will deform plastically during the very first cycle of contact and so disappear. During the subsequent lifetime of the component, it will behave elastically as if the corresponding range of surface wavelengths did not exist. The critical wavelength may be found from a relationship between the second moment and the plasticity' index ~ (Greenwood & Williamson 1966). Greenwood and Williamson looked for a criterion which would determine the mode of deformation of an array of asperities of varying heights. They found that the mode of deformation of the highest asperities was almost independent of load. Sharp asperities would deform plastically under even the lightest loads, while blunt asperities would deform elastically under even the heaviest loads. The criterion of deformation was the so-called plasticity index, which depended only on material properties and roughness parameters. Their roughness parameters are rather inconvenient to use, and we have shown
Cylinder bore wear elsewhere (Rosen et al 1997) that a more convenient formulation for machined surfaces is: m
E'
/
where E" is the Hertzian elastic modulus and H is the hardness. The second moment may be found directly from measurements of surface slope. The validity of this approach has been confirmed independently by Pawlus & Chetwynd (1996). The difficulty again is that the numerical value of the slope, and hence the plasticity index, depends on the scale of measurement. When measurements were made on the same areas of worn and unworn cylinder liners, using a stylus instrument and an atomic force microscope
523
(Rosen et al 1996), the smaller sampling interval of the AFM yielded higher slopes and hence higher values of the plasticity index (Figure 1), and the AFM values correlated better with the observed wear behaviour of the surfaces. If Up is the critical value of the plasticity index above which deformation will be plastic at any load, then the critical second moment m2p = {r~ ( 2 - 7r,/2 ) }-Y2 ~p2 ( H / E" ) 2
(6) Integrating Equations 1 and 2 yields: m2=B(3+13)-l(2=)-13(oY23+13 _to13+13)
(7)
Plasticity index Elastic and Plastic regions m e
lO
v= .s N t t I
I
J
i1
m
N+
I o
0
t
Plastic
Elastic
[ d o 0
i 1--
10
1 ! i l
100
E'/H • •
stylus TDC UW • stylus Mid UW &
afro TDC UW afro Mid UW
x
stylus TDC W stylus Mid W
<~ X
afro TDC W afm Mid W
Figure 1. Different plasticity indices measured on the same surface by AFM and stylus (Rosen et al. 1996).
524
B.-G. Roscn ~,t a /
~-~-'/
If [3 > ! and the bandwidth of surface wavelengths is reasonably wide, i.e. m2 ,~
-1175, -zo-4 . _ 2 D ) - 2 - b - - ~ (2n') F(4 *
o)1, then Equation 7 reduces to:
\
2 J\HJ (io)
m2 _= Bo)23+~,a / ( 3 + [ 3 ) ( 2 r t ) ~
(8)
This is a relationship between three dimensionless numbers: the critical wavelength normalized by the topothesy, the plateau fractal dimension and a material property ratio. The dimensionless wavelength is highly sensitive to the other parameters (Figure 2), and for a given fractal dimension and material property ratio the critical wavelength increases as the topothesy. Thus we can in principle now find a unique short-wavelength cutoff, depending only on material properties and intrinsic topography parameters, which we can use to determine the elastic behaviour of the contact.
The critical wavelength Lp = 2rt / o,'2p, so combining Equations 6 and 8,
B (_~12 8rc3/2(2-1r / 2)J/z
Zp3+~=
3+8 (9) From
Bush
et
al
1978,
~p
= 0.7.
Combining this with the numerical constant, replacing B and 13 by the corresponding fractal parameters from Equations 4 and 5 and rearranging, we have finally:
3 JIImllllmm m'mm Log ( Zp / A ) as-
2-
1,510,5 0
90 70
E'/H
D
~
""
Figure 2. Dimensionless critical wavelength as a function of material ratio and fractal dimension, from Equation 10.
Cylinder bore wear
525
3. E X P E R I M E N T A L
4. RESULTS AND DISCUSSION
The engine used in this study has been undergoing a so-called "dynamic test" simulating the everyday use of an ordinary car engine. The cylinder bore was made of grey cast iron, inserted into an injection moulded aluminum engine block and honed. In order to follow the surfaces from an unworn to a worn state without having to disassemble the test engine used, a replication and relocation method was employed. Roughness measurements were made with two instruments covering different but overlapping ranges of surface wavelengths. The first was a 3D scanning stylus system designed and constructed inhouse around a Perthen stylus instrument, with vertical and horizontal resolutions of 20 nm and 5 ~tm respectively. The second was a Rasterscope atomic force microscope (AFM) with vertical and horizontal resolutions of 3 pm and 125 pm respectively. Measurements were made in the unworn and worn states at the midpoint of the bore and at TDC. The measurement techniques, test surfaces and results have been discussed more fully elsewhere (Rosen et al. 1996).
The structure functions of all the unworn surfaces and the worn mid-cylinder surfaces, whether measured by stylus or AFM, combine into a remarkably consistent multifractal (Figure 3) There is a clear transition at about x = 20 l.tm, corresponding to the dimensions of the largest honing grits. Features smaller than this are evidently produced by a single continuous and almost ideally fractal process, which must be associated with the fracture of the grits.
topothesy is 1.3 x 10 -4 lxm. The numerical value of the topothesy is consistent with those reported in the literature (Russ 1994). Turning to the structure function of the AFM measurements on the worn TDC,
Structure Function
S(t) I
0,1
It is of course not necessary that the surface of the honing stone itself be fractal in order to create a fractal surface on the workpiece, but it is interesting to note in this connection that the surface of a diamond grinding wheel has recently been recognized as fractal (Liao 1995). From Eqn. 3, the fractal dimension associated with this process is about 1.2 and the
÷ TDC, unworn ¢" TDC, worn • MID, unworn A MID, worn
_
.
AFM vs. Stylus
P°
_
0,01
f
0,001
0,0001
0,110001 0,001
0,01
0,1
AFM |r,ompllng nm) 41. IDC, unworn (500) 0 MID, unworn (240) © TDC, worn (500) • I"DC, worn (1201 • MID, unworn (500) <> MID, unworn (7.4) [] MID, worn {500)
I
L
I
I
1
10
100
1000
t(.ml 10000
Figure 3. Structure functions of regions of the same cylinder liner in worn and unworn conditions, measured with AFM and stylus instrument
526
B.-G. Rosdn eta/.
the fractal dimension of the shorter wavelengths is about 1.6. These figures may be compared with the prediction of Zhou et al. (1993) that D < 1.4 corresponds to a surface still experiencing wear, while 1.45 < D < 1.55 is a regime of minimum wear. From Eqns. ( 4 ) and ( 5 ), the constants of the power spectrum are 13=-1.6 and B = 2.86 x 10 -7 . I f E ' / H = 9.5, then from Eqn. ( 6 ), m2p= 0.0046, and finally from Eqn. ( 10 ), the point of
more convenient range of wavelengths. He went on to dismiss this as a practical possibility on the grounds that few real surfaces showed monofractal behaviour over a wide enough range of wavelengths. The present demonstration of unchanging fractal properties over three decades of wavelength leads us to believe that the geometry at, say, 20 nm may in fact be predicted with some confidence from measurements in the region of 1 ~tm.
transition to a plastic regime COp= 158 gin1, corresponding to a critical wavelength Zp of 0.04 gm. With this figure in mind, it may be significant to observe an apparent transition in the AFM-measured structure function of the unworn mid cylinder at about 0.03 gm. We would argue that this wavelength represents the lower limit of elastic behaviour; it follows that calculations of elastic contact based on measurements which do not extend to wavelengths as short as this, i.e. any measurements made with conventional stylus instruments alone, will be in error. Russ (1994) notes that it has not hitherto been possible to assign any physical significance to the topothesy, and that its typical numerical values represent implausibly small lengths. In this paper we believe that we have succeeded for the first time in linking the topothesy with some physically meaningful property of an engineering surface. Furthermore, the product of the topothesy with the large numerical constants predicted by Equation 7 yields critical wavelengths comfortably within the measurement range of modern instruments. Whitehouse (1994), in a striking phrase, suggested the possibility of using the fractal properties of a surface to"abseil" down the scale of surface wavelengths, that is to infer the microgeometry of a surface at the smaller scales from measurements made over a much higher and thus instrumentally
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Cylinder bore wear
Majumdar, A., and C. L. Tien, "Fractal characterisation and simulation of rough surfaces", Wear 136, 313-327 (1990) Malburg, M.C. and J. Raja, "Characterization of Surface Texture Generated by Plateau Honing process", Annals of the CIRP, 42/1,637-639 (1993) Nayak, P.R., "Random Process Model of Rough Surfaces", Trans. ASME: J. Lub. Tech., 93F, 398-407 (1971). Pawlus, P. and D. G. Chetwynd, "3D surface texture characterisation of cylinder bores", Proc. 9th. Int. Surface Colloquium, 473-484, TU Chemnitz-Zwickau (1996). Rosen, B.-G., R. Ohlsson and T. R. Thomas, "The plasticity index and its application to the tribology of machined surfaces", JSME Int. J. (in press). Rosen, B.-G., R. Ohlsson and T. R. Thomas, "Wear of cylinder bore microtopography", Wear 198, 271-279 (1996). Russ, J. C., Fractal Surfaces, Plenum Press, New York (1994). Sayles, R. S. and T. R. Thomas, "The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation", Wear 42, 263-276 (1977). Sayles, R. S. and T. R. Thomas, "Surface topography as a non-stationary random process", Nature 271, 431-434 (1978) Stupak, P. R., C. Y. Syu and J. A. Donovan, "The effect of filtering profflometer data on fractal parameters", Wear 154, 109-114 (1992) Thomas, T. R. and A. P. Thomas, "Digital analysis of very small scale surface
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roughness", J Wave-Material Interaction 3, 341-350 (1988) Thomas, T. R., Rough Surfaces, Longman, London (1982) Thomas, T.R., "The Characterization of Changes in Surface Topography in runningin", Proc. 4th. Leeds-Lyon Symp., 99-108, MEP, London, (1979). Vandenberg, S., and C. F. Osborne, "Digital image processing techniques, fractal dimensionality and scale-space applied to surface roughness", Wear 159, 17-30 (1992). Wang, S., and K. Komvopoulos, "A fractal theory of the temperature distribution at elastic contacts of fast sliding surfaces", J. Trib. 117, 203-215 (1995). Warren, T. L. and D. Krajcinovic, "Fractal models of elastic-perfectly plastic contact of rough surfaces based on the Cantor set", Int. J. Solids Structures 32, 2907-2922 (1995). Whitehouse, D. J., Handbook of Surface Metrology, Institute of Physics, London (1994) Whitehouse, D.J. and J.F. Archard, "The Properties of Random Surfaces of Significance in their Contact", Proc. R. Soc., A316, 97-121 (1970). Yordanov, O. I., and K. Ivanova, "Description of surface roughness as an approximate self-affme random structure", Surface Science 331-333, 1043-1049 (1995). Zhou, G. Y., M. C. Leu and D. Blackmore, "Fractal geometry model for wear prediction ", Wear 170, 1-14 (1993).