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Characterization of fractal surfaces
Wear 239 Ž2000. 36–47 www.elsevier.comrlocaterwear
Characterization of fractal surfaces Jiunn-Jong Wu ) Department of Mechanical Engineering, Chinese Cultural UniÕersity, F4, No. 270, Sec. 2, Li-Nong Street, Bei-Tou District, Taipei, Taiwan Received 5 August 1999; received in revised form 15 November 1999; accepted 6 December 1999
Abstract Fractal profiles are generated and analyzed. It is found that the root mean square Žrms. slope and curvature can be obtained from the structure function. Also, it is found that rms curvature is a good estimate of asperity curvature. Finally, a bifractal surface is analyzed. It is found that the critical wave number of the spectral density does not correspond to the critical length of the structure function. Again, the rms curvature is a good estimate for the asperity curvature of bifractal surfaces. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Fractal surface; Root mean square; Asperity
1. Introduction Surface roughness plays an important role in contact, friction and wear. Therefore, characterization of roughness is necessary in these studies. Conventionally, a rough surface is assumed to be a random process. Statistical parameters, such as the standard deviations of the surface height s , slope sm and curvature sk , are used for characterizing surface roughness. These statistical parameters are used by the famous contact model established by Greenwood and Williamson w1x and other revised models w2,3x. However, these statistical parameters depend on the resolution and the scan length of the roughness-measuring instrument. They are not properties of the surface alone w4x. In spite of that, they are still valuable. Greenwood and Williamson’s contact model, which uses instrument-dependent statistical parameters, is surprisingly good at predicting elastic contact phenomena w5x. Since statistical parameters are instrument-dependent, Majumdar and Bhushan w6x developed a fractal model for elastic–plastic contact between rough surfaces, based on fractal geometry. In their model, fractal surfaces can be characterized by two parameters: a non-integer fractal dimension D and a scale constant G. Further, Bhushan and Majumdar w7x discovered that some rough surfaces were bifractal, and proposed a bifractal contact model. )
Tel.: q886-2-2823-9548. E-mail addresses: [email protected], [email protected] ŽJ.-J. Wu..
However, there are still some problems in the fractal contact model. The asperity curvature predicted by fractal contact model has never been verified yet. Also, the characteristics of bifractal surface have not been investigated in detail, either. All these problems will be analyzed in this paper. The goal of this paper is to provide a further understanding of fractal surfaces. This paper discusses the statistical parameters and asperity curvature of fractal surfaces, and also discusses the characterization of bifractal surfaces. With the findings in this paper, new contact models for fractal and bifractal surfaces can easily be developed in the future.
2. Fractal analysis 2.1. Fractal properties Consider a fractal surface, for which the vertical cut is a fractal profile of dimension D. Its spectral density Žor called power spectrum. P Ž v . and structure function SŽt . are, respectively: PŽ v. s
G 2 Dy2
v 5y 2 D
,
S Ž t . s E z Ž x . y z Ž x q t .
0043-1648r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 Ž 9 9 . 0 0 3 6 2 - 2
s Kt Ž4y2 D. ,
Ž 1. 2
4 s2
R Ž 0. y R Ž t .
Ž 2.
J.-J. Wu r Wear 239 (2000) 36–47
where RŽt . is the auto-correlation function ŽACF. and Ks
p G 2 Dy2
1
.
2 G Ž 5 y 2 D . sin Ž 2 y D . p
37
higher wave number limit. They are related to length of sample l and sampling interval s, respectively:
p vls
The ACF and the structure function can be obtained from the spectral density by the following equations:
vhs
l p
, .
s
`
RŽt . s
H0 P Ž v . cos Ž vt . d v , `
SŽt . s2
H0 P Ž v .
1 y cos Ž vt . d v .
In Eq. 1, G is a scale constant. The dimension of G is length. D is the fractal dimension, where 1 - D - 2. With G and D, we can characterize a fractal profile or surface. Ganti and Bhushan w8x thought that the G in Eq. 1 is instrument-dependent. They claimed that the structure function should be expressed as: S Ž t . s Ch 2 Dy3t 4y 2 D ,
Ž 3.
where h is the resolution of the instrument and C is a constant related to the amplitude of all frequencies. Ganti and Bhushan thought that C is instrument-independent. Since h is the resolution of the instrument, it is not the same if different instruments are used to measure this fractal surface. Therefore, constant C in Eq. 3 is not independent of instrument as Ganti and Bhushan claimed. On the contrary, C depends on the instrument, and G is independent of instrument. Because K in Eq. 2 can be obtained from G, and vice versa, there are two sets of parameters that can characterize a fractal surface: Ž D and K . or Ž D and G .. Both of them are independent of scale. 2.2. Properties of fractal surfaces Some statistical parameters, s , sm and sk , are used to characterize a surface in some contact models. Majumdar and Tien w9x found that, for fractal surfaces, they are: ² z2:s dz
G 2Ž Dy1.
Ž4 y2 D.
2
¦ž / ;
ž
1
v lŽ 4y2 D .
G 2Ž Dy1.
s
dx
d2 z
2
¦ž / ; d x2
Ž 2 D y 2. G 2Ž Dy1. s 2D
1 y
v hŽ 4y2 D .
/
,
Ž v h2 Dy2 y v l2 Dy2 . ,
Ž 4.
Ž v h2 D y v l2 D . ,
Ž 5.
where v l is the lower wave number limit and v h is the
1
Unlike some other equations, this makes both gamma and sine functions positive for the practical range 1- D- 2.
Ganti and Bhushan gave different expressions for - z 2 ) ,- Žd z 2rd x . 2 ) and - Žd 2 zrd x 2 . 2 ) . But their result 2 w8x is simply the discrete form of the above three equations. Sayles and Thomas w10x found that the mean slope and curvature should be expressed by structure function in the discrete analysis. They are: ² z 2 : s s 2 s R Ž 0. , 2
dz
¦ž / ;
s sm2 Ž x . s
dx
s d2 z
dx
x2
x2
s sk2 Ž x . s
s
R Ž 0. y R Ž x .
s Kx 2y 2 D ,
2
¦ž / ; 2
SŽ x .
2
2 x4
3 R Ž 0. y 4 R Ž x . q R Ž 2 x .
4S Ž x . y S Ž 2 x . x4
Ž 6.
s Ž 4 y 2 4y 2 D . Kxy2 D .
Ž 7. Wu w11x found that Eqs. 6 and 7 are more suitable for the calculation of root mean square Žrms. slope and curvature at a finite sampling interval than Eqs. 4 and 5. Therefore, we should use Eqs. 6 and 7 for fractal surfaces. All these statistical parameters will be analyzed in Section 3. 2.3. Asperity curÕature The asperity curvature is important in fractal contact models. Therefore, we will investigate it in detail. It is hard to define the ‘‘asperity curvature’’, since usually no asperity is a perfect sine wave or parabola. However, asperities are usually assumed to be parabolic
2
For example, Ganti and Bhushan thought that 1 1 1 ² z 2 : sG 2Ž Dy1. 5y 2 D q 5y2 D q PPP q . 2 3 Ž Nr2. 5y2 D In fact, it is simply a discrete form of: 2Ž Dy1. vh G ² z 2 :s dv. 5y 2 D vl v So are ²Žd z rd x . 2 : and ²Žd 2 z rd x 2 . 2 :.
H
J.-J. Wu r Wear 239 (2000) 36–47
38
Fig. 1. Definition of asperity curvature.
Že.g., in the contact model of Greenwood and Williamson w1x, and in fractal contact model w6x.. Therefore, we use a parabolic line to fit the shape of asperity and define the ‘‘asperity curvature’’. Aramaki et al. w12x defined an asperity as ‘‘one which makes one contact spot’’. If an asperity is parabolic, its curvature is:
ks
12 A l3
,
where l is the contact length and A is the area above the contact line. So, we define the equivalent asperity curvature of any shape by the same equation Žsee Fig. 1.. When measuring a profile, we obtain data at grid points. So, when successive heights at grid points are higher than the contact level, an asperity is detected. Given the contact level, by interpolating between the neighboring grid points, we can obtain the asperity profile and its curvature. In Majumdar and Bhushan’s fractal contact model, they claimed that, according to the spectral density, the roughness amplitude is:
p 2 G Dy 1 lD
.
Ž 8.
However, the asperity curvature is not only caused by a single wave number mode. The amplitudes of all wave
Fig. 2. W–M function for Ds1.5, Gs1.5 and gs1.5.
number modes contribute to the asperity curvature. So, whether Eq. 8 is true or not needs to be investigated. In the unified theory of Greenwood w13x, the asperity curvature can be estimated by rms curvature sk . Although, it is the three-point curvature and is different from Aramanki et al.’s definition, it is still worth investigating. In Section 3, simulated fractal profiles will be used to check the asperity curvature.
3. Numerical experiment 3.1. Fractal simulation
d s G Dy 1 l 2y D . Therefore, the asperity curvature is: ks
Fig. 3. Spectral density of W–M function, Ds1.5, Gs1.5 and gs1.5.
There are several schemes that can generate fractal profiles or surfaces. All of them have their advantages, and also limitations. We adopt three schemes to simulate fractal profiles. The first one is called Weierstrass–Mandelbrot
Fig. 4. Structure function of W–M function, Ds1.5, Gs1.5 and gs1.5.
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 5. Profile generated by Ganti and Bhushan’s method, Ds1.5.
ŽW–M. function, which is widely used in fractal contact model. The second one is proposed by Ganti and Bhushan w8x. This method generates a fractal profile from its spectral density. The third method is called successive random addition method. The profile generated by this method has a power law structure function. They are described in the following sections. We compare all characteristics for the profiles generated by these three schemes. 3.1.1. W–M function W–M function w14x is the most widely used function for fractal contact model. It can be expressed as: 3 cos Ž g n x q fn .
`
zŽ x. sG
Ž Dy1.
Ý
g Ž2yD. n
nsn 1
,
Ž 9.
where fn is random phase. Its spectral density is: PŽ v. s
G 2Ž Dy1.
d Ž vyg n.
`
Ý
2
nsn 1
v Ž4y2 D.
.
39
Fig. 6. Spectral density of profile generated by Ganti and Bhushan’s method, Ds1.5.
Its structure function is: S Ž t . s E z Ž x . y z Ž x q t .
2
4 s Kt Ž4y2 D. ,
Ž 12 .
where K s wp G 2 D -2 xrw4 ln g G Ž5 y 2 D .sinwŽ2 y D .xp x. A profile generated by the W–M function is shown in Fig. 2. Its spectral density is shown in Fig. 3. From Fig. 3, we find that there are some spikes in the spectral density. Therefore, the spectral density follows Eq. 10, not Eq. 11. We can also find that the distance between each spike depends on g . The structure function is shown in Fig. 4. Although the spectral density does not follow Eq. 11, the structure function does follow Eq. 12. We compare spectral density in Fig. 3 with that of real surfaces. From the spectral densities obtained from real surface w4,8x, it is found that there are no such spikes in
Ž 10 .
Its equivalent continuous spectral density is w14x: PŽ v. s
G 2Ž Dy1. 2 ln g
1
v
Ž5y2 D.
.
Ž 11 .
3
Several papers w4–8x write the W–M function as: ` cos Ž 2pg n x q fn . z Ž x . sG Ž Dy1. . g Ž2yD . n ns n
Ý
1
With this convention, its spectral density is: PŽ f .s
PŽv.s
G 2Ž Dy1.
`
Ý
2
ns n 1
Ž 2p G . 2
Ž D y1 .
2
where f is frequency.
d Ž f yg n . f Ž4y2 D .
,
`
d Ž v y2pg n .
Ý
v Ž4y2 D .
ns n 1
, Fig. 7. Structure function of profile generated by Ganti and Bhushan’s method, Ds1.5.
J.-J. Wu r Wear 239 (2000) 36–47
40
Fig. 8. Profile generated by successive random addition method for Ds1.5.
spectral density as those in Fig. 3. So, there is no such parameter g existing. Some authors thought that g is 1.5 w8x. It is baseless. There is another problem with the W–M function. It cannot be extended into two dimensions. Ausloos and Berman w15x believed that the W–M function could be extended into two dimensions by the following function: zŽ x, y. sL
G
Ž Dy2 .
1r2
ln g
Ý Ý sg Ž Dy2. n
ž / ž / L
M
° =~cos f ¢
m , n y cos
=cos tany1
ms1 ns1
2pg n Ž x 2 q y 2 .
x
y M
/
q fm , n
hold for all Ž v x , v y .; i.e., the surface is not isotropic. So, a vertical cut of the surface is not necessarily a W–M function. Thus, this is not a really two-dimensional W–M function. From the above analysis, we know that: firstly, no real profile follows a W–M function; secondly, W–M function cannot be extended into two dimensions. But W–M function is still generated and investigated in the following sections.
1r2
L
pm
y
ž ž /
n max
M
Fig. 10. Structure function of profile generated by successive random addition method for Ds1.5.
¶• ß.
But from their paper, for the surface obtained from this equation, it is shown that P Ž v . s P Ž v x2 q v y2 . does not
(
Fig. 9. Spectral density of profile generated by successive random addition method for Ds1.5.
3.1.2. Profile generated from spectral density Ganti and Bhushan’s method is also used to generating the fractal profiles. For a fractal profile, the spectral density is of the form of Eq. 1. From Ganti and Bhushan’s method, a profile with fractal spectral density can be generated. The profiles with fractal spectral density have the property of self-similarity. The structure function of this generated profile is expected to have the form of Eq. 2.
Fig. 11. sm2 of W–M function for Ds1.5, Gs1.5 and gs1.5.
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 12. sk2 of W–M function for Ds1.5, Gs1.5 and gs1.5.
A profile generated by Ganti and Bhushan’s method is shown in Fig. 5. Its spectral density is shown in Fig. 6. The structure function is shown in Fig. 7. It is found that its spectral density follows the power law perfectly. There is only a small error in its structure function. Both the spectral density and structure function do follow the power laws.
41
Fig. 14. sk2 of profile generated by Ganti and Bhushan’s method, Ds1.5.
that its structure function follows the power law perfectly. There is only small error in its spectral density. Both the spectral density and structure function do follow power laws, too. 3.2. The rms slope and curÕature
3.1.3. SuccessiÕe random addition method The third method is called successive random addition method w16x. It is a modification of the mid-point displacement method. This method can generate a profile whose structure function is in the form of Eq. 2. The spectral density of this profile is expected to have the form of Eq. 1. Fig. 8 shows the profile generated by successive random addition method. Its spectral density is shown in Fig. 9. The structure function is shown in Fig. 10. It is shown
The rms slope and curvature are analyzed. They are obtained by three different ways. Firstly, they are calculated directly from the profiles. Also, they can be obtained from the spectral density by Eqs. 4 and 5. Finally, they can be obtained from the structure function by Eqs. 6 and 7. The rms slope and curvature for profiles generated by W–M function are shown in Figs. 11 and 12, those for Ganti and Bhushan’s method are shown in Figs. 13 and 14; and those for successive random addition method are shown in Figs. 15 and 16.
Fig. 13. sm2 of profile generated by Ganti and Bhushan’s method, Ds1.5.
Fig. 15. sm2 of profile generated by successive random addition method for Ds1.5.
42
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 16. sk2 of profile generated by successive random addition method for Ds1.5.
It is found that rms slopes obtained from Eqs. 4 and 6 are both good estimates of that obtained from the profile. But, the rms curvature from Eq. 7 is always better than that from Eq. 5. Since Eq. 7 is based on the structure function and Eq. 5 is based on spectral density, we can assert that the structure function is more suitable for the calculation of rms curvature at a finite sampling interval. 3.3. Asperity curÕature In order to find the asperity curvature, we try different dimensions of fractal profiles and fractal profiles generated by different schemes. We also set the contact line at the mean line of the fractal profile. For W–M function and Ganti and Bhushan’s method, D s 1.2, 1.4, 1.6, 1.9 is used. For D s 1.1, the profile is
Fig. 17. Asperity curvature of W–M function, Gs1.5 and gs1.5; contact level: mean line; dot — real asperity curvature, dash line — sk , dotted line — k from Eq. 8.
Fig. 18. Asperity curvature, Ganti and Bhushan’s method; contact level: mean line; dot — real asperity curvature, dash line — sk , dotted line — k from Eq. 8, mean line.
too smooth, so there are too few asperities such that it cannot be used. For successive random addition method, D s 1.3, 1.4, 1.5, 1.7 is used, since this method cannot generate a fractal profile with good spectral density and structure function for too large or too small D. We compare asperity curvature with sk and k . sk is the rms curvature obtained from Eq. 7. k is the asperity curvature used by Majumdar and Bhushan’s fractal contact model, as obtained from Eq. 8. From Figs. 17–19, we find that both sk and k have the same trend. But, k overestimates the asperity curvatures for small D, and underestimates them for large D. We also found that there is always the same ratio between asperity curvature and sk . Obviously, sk is much better than k in predicting the asperity curvature.
Fig. 19. Asperity curvature, successive random addition method; contact level: mean line; dot — real asperity curvature, dash line — sk , dotted line — k from Eq. 8.
J.-J. Wu r Wear 239 (2000) 36–47
Recall that Eq. 8 predicted the asperity curvature only by a single wave number mode. In fact, the asperity curvature is affected by all wave number modes. When the fractal dimension is small, the amplitude of large wave number is smaller and makes less contribution. For large fractal dimension, the amplitude of large wave number is larger and makes more contribution. But k always makes the same prediction. Therefore, k will overestimate the asperity curvature for small D and underestimate it for large D. On the other hand, sk is the rms curvature based on three-point curvature. There could be some relation between asperity curvature and three-point peak curvature. There is also possible some relation between three-point peak curvature and rms curvature. Thus, it is reasonable to use s in predicting asperity curvature. The real relation between sk and asperity curvature is not discussed in the present paper. It may need much more investigation in the future.
4. Bifractal profile Bhushan and Majumdar w7x found that some surfaces have bifractal characteristics. In order to investigate the characteristics of bifractal surface, we start by forming a fractal surface with a limited band spectral density. A bifractal surface may be considered as a surface with two such fractal spectral density bands. 4.1. Fractal surface with band-limited spectral density
43
Fig. 21. Structure function for bandwidth spectral density.
a sampling interval equal to the smallest fractal length, and the specimen length equal to the longest fractal length: p vls , l p vhs . s The structure function of a fractal profile with bandlimited spectral density will be SŽt . s2
vh
Hv
P Ž v . 1 y cos Ž vt . d v ,
l
s 2G 2 Dy2
vh
Hv
l
1 y cos Ž vt .
v 5y 2 D
dv.
Consider a fractal profile whose wave number is restricted by two extreme wave numbers, one for the largest wave number and the other for the smallest one. v h is the maximum wave number and is related to the smallest fractal length, while v l is the minimum wave number and related to the longest fractal length. In our analysis, we use
We arbitrarily set v l s 10y4 and v h s 1. That means l s 10 4 p and s s p : 1 y cos Ž vt . 1 S Ž t . s 2G 2 Dy2 dv. v 5y 2 D 0.0001 Fig. 20 shows the structure function of a fractal profile with D s 1.3, and Fig. 21 shows the structure function of
Fig. 20. Structure function for bandwidth spectral density.
Fig. 22. Spectral density of a bifractal profile. D1 s1.2, D 2 s1.8.
H
J.-J. Wu r Wear 239 (2000) 36–47
44
Is Eq. 14 the structure function of the profile defined by Eq. 13? It needs investigation. The structure function of Eq. 13 is:
SŽ x . s2
½H
vc
v
0 `
Hv
q
Fig. 23. Structure function of a bifractal profile.
a fractal profile with D s 1.7. In both cases, the spectral densities are limited to wave numbers between 0.0001 and 1; i.e., l s 10 4 p and s s p . It is shown that the structure functions follow a power law in the range from around t s p to around 7 = 10 3. The range does not correspond to the maximum and minimum wave numbers.
4.2. Bifractal surface
4.2.1. Critical waÕe number and critical length A bifractal profile can be defined by spectral density or structure function. If a bifractal surface is defined by spectral density, its spectral density is:
°G v P Ž v . s~
2 Dy2 1 5y2 D
G 22 Dy2
¢v
5y2 D
c
v
G1 5y 2 D 1
G2 5y 2 D 2
1 y cos Ž v x . d v
5
1 y cos Ž v x . d v .
Ž 15 .
Fig. 22 shows a bifractal spectral density. The critical wave number is vc s 0.1. Fig. 23 shows its structure function obtained from Eq. 15. It is found that there are two asymptotic lines in Fig. 23. But there is no exact critical length existing for the structure function. If we regard the intersection of two asymptotic lines as the critical length, then it is tc s 14.9. It is different from the corresponding length of the critical wave number:
p vc /
tc
.
It can be explained from the fractal profile with bandlimited spectral density. The range of structure function that follows the power law does not correspond to the maximum and minimum wave numbers of the limit band spectral density. Thus, the critical length does not correspond to the critical wave number, either. Normally, there are two asymptotic lines for the spectral density of bifractal surfaces. The critical wave number
v G vc
Ž 13 . v F vc .
If a bifractal profile is defined by structure function, then, its structure function follows: SŽt . s
½
K 1t Ž 4y 2 D 1 .
t F tc
K 2t Ž 4y 2 D 2 .
t G tc
,
Ž 14 .
where: K1 s
K2 s
p G 12 D 1y2 2 G Ž 5 y 2 D 1 . sin Ž 2 y D 1 . p
p G 22 D 2y2 2 G Ž 5 y 2 D 2 . sin Ž 2 y D 2 . p
,
.
Fig. 24. Spectral density of bifractal profile generated by Ganti–Bhushan’s method.
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 25. Structure function of bifractal profile generated by Ganti– Bhushan’s profile.
45
Fig. 26. sk2 of bifractal profile generated by Ganti and Bhushan’s method.
i.e., exists at the intersection of these two asymptotic lines. From Eq. 13, we find that, for the critical wave number: G 12 D 1y2
vc5y 2 D 1
s
G 22 D 2y2
vc5y2 D 2
.
Therefore: G 1D 1y1 G 2D 2y1
s v cD 2yD 1 .
Ž 16 .
For the structure function, there are also two asymptotic lines existing for a bifractal surface, too. The critical length is again the intersection of two asymptotic lines. From Eq. 14, we obtain:
K1 K2
4 y 2 4y 2 D 2 s
4 y 2 4y2 D 1
x c2Ž D 1yD 2 . .
We find that all the critical wave numbers and critical lengths are different. In fact, for spectral density, structure function or rms curvature of bifractal surfaces, there is usually a transition region between two asymptotic lines. Therefore, the critical wave number and critical length do not exist strictly. We generate a bifractal profile by Ganti and Bhushan’s method. The profile is generated from a bifractal spectral density. The spectral density of the generated profile is shown in Fig. 24. Fig. 25 shows the structure function. Fig. 26 shows the rms curvature. All these figures show that there are two asymptotic lines for spectral density, structure function and rms curva-
K 1tc4y 2 D 1 s K 2t 24y 2 D 2 . So, K1 K2
s
G 12Ž D 1y1. G Ž 5 y 2 D 2 . sin Ž 2 y D 2 . p G 22Ž D 2y1. G Ž 5 y 2 D 1 . sin Ž 2 y D1 . p
s tc2Ž D 1yD 2 . . Similarly, for a bifractal surface, the rms curvature is:
sk2 Ž x . s
½
Ž 4 y 2 4y 2 D 1 . K 1 xy2 D 1 Ž 4 y 2 4y2 D 2 . K 2 xy2 D 2
x G xc x F xc .
The critical length is the intersection of two rms curvature lines. So, D2 D2 s Ž 4 y 2 4y 2 D 2 . K 2 xy2 , Ž 4 y 2 4y 2 D 1 . K 1 xy2 c c
Fig. 27. Asperity curvature of bifractal profile generated by Ganti and Bhushan’s method Ds1.5, at central line.
J.-J. Wu r Wear 239 (2000) 36–47
46
If W–M function is used, the result is exactly the same. There is also a discontinuity existing. In the bifractal contact model of Majumdar and Bhushan w6x, they analyzed the structure function, but applied Eq. 17 to obtain the asperity curvature, which is obtained from spectral density. In their model, they used the critical wave number corresponding to the critical length. There must be a discontinuity for asperity curvature. It is a mistake. 5. Conclusion
Fig. 28. Asperity curvature of bifractal profile generated by Ganti and Bhushan’s method Ds1.5, at hs s .
ture. All the asymptotic lines and the critical point are the same as we expect. 4.2.2. Asperity curÕature of bifractal surfaces By using the spectral density, Majumdar claimed that the asperity curvature is: 2
ks
p G
A C
s p 2yD G Dy1v D .
lD
½
p 2y D 1 G 1D 1y1v D 1 p
6. Nomenclature
Dy 1
If their formula is correct, the asperity curvature of a bifractal surface should be:
ks
In this paper, fractal profiles are analyzed. It is found that rms curvature can be related to the structure function. Also, it is found that rms curvature can be used to estimate the curvature of an asperity. Bifractal profiles are analyzed, too. It is found that the critical wave number in spectral density does not correspond to the critical length for the structure function. The rms curvature obtained from the profile is a good estimate for asperity curvature. The finding in this paper can be used to establish a fractal or bifractal contact model in the future.
2y D 2
G 2D 2y1v D 2
v G vc v F vc .
Ž 17 .
From Eq. 16, we find that, at critical wave number: G 1D 1y1 v cD 2 G 2D 2y1 vcD 1
.
Thus, from the first part of Eq. 17, the asperity curvature for the critical wave number is:
k s p 2y D 1 G 1D 1y1vcD 1 . But, from the second part of Eq. 17, it is different:
k s p 2y D 2 G 1D 2y1vcD 2 s p 2y D 2 G1D 1y1vcD 1 . So, there is a discontinuity at the critical wave number for the asperity curvature. We use the surface generated by Ganti and Bhushan’s method, and compare the real asperity curvature with k and sk , where k is obtained from Eq. 17 and sk is the rms curvature obtained from profiles. The results are shown in Figs. 27 and 28. In Fig. 27, the contact line is set at central line. In Fig. 28, the contact line is set at height s above central line. In both figures, it is found that there is a discontinuity for k . We also find that rms curvature is also a good estimate for asperity curvature.
the area above the contact line a constant related to the amplitude of all frequencies D fractal dimension G a scale constant for a fractal surface K a constant in structure function l sampling length, contact length P Ž v . spectral density SŽt . structure function s sampling interval x horizontal coordinate d roughness amplitude h the resolution of the instrument g scaling ratio fn random phase k asperity curvature, used by Majumdar and Bhushan s standard deviation of the surface height sm standard deviation of the surface slope sk standard deviation of the surface curvature t distance v wave number vh higher wave number limit vl lower wave number limit and References w1x J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proceeding of the Royal Society, London A 295 Ž1966. 300–319. w2x P.R. Nayak, Random process model of rough surfaces, Journal of
J.-J. Wu r Wear 239 (2000) 36–47
w3x
w4x
w5x
w6x
w7x w8x w9x
Lubrication Technology, Transaction of the ASME 97 Ž1971. 398– 407. D.J. Whitehouse, J.F. Archard, The properties of random surfaces of significance in their contact, Proceeding of the Royal Society, London A 316 Ž1970. 97–121. A. Majumdar, B. Bhushan, Role of fractal geometry in roughness characterization and contact mechanics of surfaces, Journal of Tribology, Transactions of the ASME 112 Ž1990. 205–216. Z. Handzel-Powierza, T. Klimczak, A. Polijaniuk, On the experimental verification of the Greenwood–Williamson model for the contact of rough surfaces, Wear 154 Ž1992. 115–124. A. Majumdar, B. Bhushan, Fractal model of elastic–plastic contact between rough surfaces, Journal of Tribology, Transactions of the ASME 113 Ž1991. 1–11. B. Bhushan, A. Majumdar, Elastic–plastic contact model for bifractal surfaces, Wear 153 Ž1992. 53–64. S. Ganti, G. Bhushan, Generalized fractal analysis and its applications to engineering surfaces, Wear 180 Ž1995. 17–34. A. Majumdar, C.L. Tien, Fractal characterization and simulation of rough surfaces, Wear 136 Ž1990. 313–327.
47
w10x R.S. Sayles, T.R. Thomas, Measurement of the statistical microgeometry of engineering surfaces, Journal of Lubrication Technology, Transaction of the ASME 101 Ž1979. 409–418. w11x J.-J. Wu, Spectral analysis for the effect of stylus tip curvature on measuring rough profiles, Wear 230 Ž1990. 194–200. w12x H. Aramaki, H.S. Cheng, Y.W. Chung, The contact between roughness surfaces with longitudinal texture: I. Average contact pressure and real contact area, Journal of Tribology, Transactions of the ASME 115 Ž1993. 419–424. w13x J.A. Greenwood, A unified theory of surface roughness, Proceeding of the Royal Society, London A 393 Ž1984. 3133–3157. w14x M.V. Berry, Z.V. Lewis, On the Weierstrass–Mandelbrot fractal function, Proceedings of the Royal Society, London A 370 Ž1980. 459–484. w15x M. Ausloos, D.H. Berman, A multivariate Weierstrass–Mandelbrot function, Proceedings of the Royal Society, London A 400 Ž1985. 331–350. w16x D. Saupe, Algorithms for random fractals, in: H.O. Peitgen, D. Saupe ŽEds.., The Science of Fractal Images, Springer, 1988, pp. 71–136.
Characterization of fractal surfaces Jiunn-Jong Wu ) Department of Mechanical Engineering, Chinese Cultural UniÕersity, F4, No. 270, Sec. 2, Li-Nong Street, Bei-Tou District, Taipei, Taiwan Received 5 August 1999; received in revised form 15 November 1999; accepted 6 December 1999
Abstract Fractal profiles are generated and analyzed. It is found that the root mean square Žrms. slope and curvature can be obtained from the structure function. Also, it is found that rms curvature is a good estimate of asperity curvature. Finally, a bifractal surface is analyzed. It is found that the critical wave number of the spectral density does not correspond to the critical length of the structure function. Again, the rms curvature is a good estimate for the asperity curvature of bifractal surfaces. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Fractal surface; Root mean square; Asperity
1. Introduction Surface roughness plays an important role in contact, friction and wear. Therefore, characterization of roughness is necessary in these studies. Conventionally, a rough surface is assumed to be a random process. Statistical parameters, such as the standard deviations of the surface height s , slope sm and curvature sk , are used for characterizing surface roughness. These statistical parameters are used by the famous contact model established by Greenwood and Williamson w1x and other revised models w2,3x. However, these statistical parameters depend on the resolution and the scan length of the roughness-measuring instrument. They are not properties of the surface alone w4x. In spite of that, they are still valuable. Greenwood and Williamson’s contact model, which uses instrument-dependent statistical parameters, is surprisingly good at predicting elastic contact phenomena w5x. Since statistical parameters are instrument-dependent, Majumdar and Bhushan w6x developed a fractal model for elastic–plastic contact between rough surfaces, based on fractal geometry. In their model, fractal surfaces can be characterized by two parameters: a non-integer fractal dimension D and a scale constant G. Further, Bhushan and Majumdar w7x discovered that some rough surfaces were bifractal, and proposed a bifractal contact model. )
Tel.: q886-2-2823-9548. E-mail addresses: [email protected], [email protected] ŽJ.-J. Wu..
However, there are still some problems in the fractal contact model. The asperity curvature predicted by fractal contact model has never been verified yet. Also, the characteristics of bifractal surface have not been investigated in detail, either. All these problems will be analyzed in this paper. The goal of this paper is to provide a further understanding of fractal surfaces. This paper discusses the statistical parameters and asperity curvature of fractal surfaces, and also discusses the characterization of bifractal surfaces. With the findings in this paper, new contact models for fractal and bifractal surfaces can easily be developed in the future.
2. Fractal analysis 2.1. Fractal properties Consider a fractal surface, for which the vertical cut is a fractal profile of dimension D. Its spectral density Žor called power spectrum. P Ž v . and structure function SŽt . are, respectively: PŽ v. s
G 2 Dy2
v 5y 2 D
,
S Ž t . s E z Ž x . y z Ž x q t .
0043-1648r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 Ž 9 9 . 0 0 3 6 2 - 2
s Kt Ž4y2 D. ,
Ž 1. 2
4 s2
R Ž 0. y R Ž t .
Ž 2.
J.-J. Wu r Wear 239 (2000) 36–47
where RŽt . is the auto-correlation function ŽACF. and Ks
p G 2 Dy2
1
.
2 G Ž 5 y 2 D . sin Ž 2 y D . p
37
higher wave number limit. They are related to length of sample l and sampling interval s, respectively:
p vls
The ACF and the structure function can be obtained from the spectral density by the following equations:
vhs
l p
, .
s
`
RŽt . s
H0 P Ž v . cos Ž vt . d v , `
SŽt . s2
H0 P Ž v .
1 y cos Ž vt . d v .
In Eq. 1, G is a scale constant. The dimension of G is length. D is the fractal dimension, where 1 - D - 2. With G and D, we can characterize a fractal profile or surface. Ganti and Bhushan w8x thought that the G in Eq. 1 is instrument-dependent. They claimed that the structure function should be expressed as: S Ž t . s Ch 2 Dy3t 4y 2 D ,
Ž 3.
where h is the resolution of the instrument and C is a constant related to the amplitude of all frequencies. Ganti and Bhushan thought that C is instrument-independent. Since h is the resolution of the instrument, it is not the same if different instruments are used to measure this fractal surface. Therefore, constant C in Eq. 3 is not independent of instrument as Ganti and Bhushan claimed. On the contrary, C depends on the instrument, and G is independent of instrument. Because K in Eq. 2 can be obtained from G, and vice versa, there are two sets of parameters that can characterize a fractal surface: Ž D and K . or Ž D and G .. Both of them are independent of scale. 2.2. Properties of fractal surfaces Some statistical parameters, s , sm and sk , are used to characterize a surface in some contact models. Majumdar and Tien w9x found that, for fractal surfaces, they are: ² z2:s dz
G 2Ž Dy1.
Ž4 y2 D.
2
¦ž / ;
ž
1
v lŽ 4y2 D .
G 2Ž Dy1.
s
dx
d2 z
2
¦ž / ; d x2
Ž 2 D y 2. G 2Ž Dy1. s 2D
1 y
v hŽ 4y2 D .
/
,
Ž v h2 Dy2 y v l2 Dy2 . ,
Ž 4.
Ž v h2 D y v l2 D . ,
Ž 5.
where v l is the lower wave number limit and v h is the
1
Unlike some other equations, this makes both gamma and sine functions positive for the practical range 1- D- 2.
Ganti and Bhushan gave different expressions for - z 2 ) ,- Žd z 2rd x . 2 ) and - Žd 2 zrd x 2 . 2 ) . But their result 2 w8x is simply the discrete form of the above three equations. Sayles and Thomas w10x found that the mean slope and curvature should be expressed by structure function in the discrete analysis. They are: ² z 2 : s s 2 s R Ž 0. , 2
dz
¦ž / ;
s sm2 Ž x . s
dx
s d2 z
dx
x2
x2
s sk2 Ž x . s
s
R Ž 0. y R Ž x .
s Kx 2y 2 D ,
2
¦ž / ; 2
SŽ x .
2
2 x4
3 R Ž 0. y 4 R Ž x . q R Ž 2 x .
4S Ž x . y S Ž 2 x . x4
Ž 6.
s Ž 4 y 2 4y 2 D . Kxy2 D .
Ž 7. Wu w11x found that Eqs. 6 and 7 are more suitable for the calculation of root mean square Žrms. slope and curvature at a finite sampling interval than Eqs. 4 and 5. Therefore, we should use Eqs. 6 and 7 for fractal surfaces. All these statistical parameters will be analyzed in Section 3. 2.3. Asperity curÕature The asperity curvature is important in fractal contact models. Therefore, we will investigate it in detail. It is hard to define the ‘‘asperity curvature’’, since usually no asperity is a perfect sine wave or parabola. However, asperities are usually assumed to be parabolic
2
For example, Ganti and Bhushan thought that 1 1 1 ² z 2 : sG 2Ž Dy1. 5y 2 D q 5y2 D q PPP q . 2 3 Ž Nr2. 5y2 D In fact, it is simply a discrete form of: 2Ž Dy1. vh G ² z 2 :s dv. 5y 2 D vl v So are ²Žd z rd x . 2 : and ²Žd 2 z rd x 2 . 2 :.
H
J.-J. Wu r Wear 239 (2000) 36–47
38
Fig. 1. Definition of asperity curvature.
Že.g., in the contact model of Greenwood and Williamson w1x, and in fractal contact model w6x.. Therefore, we use a parabolic line to fit the shape of asperity and define the ‘‘asperity curvature’’. Aramaki et al. w12x defined an asperity as ‘‘one which makes one contact spot’’. If an asperity is parabolic, its curvature is:
ks
12 A l3
,
where l is the contact length and A is the area above the contact line. So, we define the equivalent asperity curvature of any shape by the same equation Žsee Fig. 1.. When measuring a profile, we obtain data at grid points. So, when successive heights at grid points are higher than the contact level, an asperity is detected. Given the contact level, by interpolating between the neighboring grid points, we can obtain the asperity profile and its curvature. In Majumdar and Bhushan’s fractal contact model, they claimed that, according to the spectral density, the roughness amplitude is:
p 2 G Dy 1 lD
.
Ž 8.
However, the asperity curvature is not only caused by a single wave number mode. The amplitudes of all wave
Fig. 2. W–M function for Ds1.5, Gs1.5 and gs1.5.
number modes contribute to the asperity curvature. So, whether Eq. 8 is true or not needs to be investigated. In the unified theory of Greenwood w13x, the asperity curvature can be estimated by rms curvature sk . Although, it is the three-point curvature and is different from Aramanki et al.’s definition, it is still worth investigating. In Section 3, simulated fractal profiles will be used to check the asperity curvature.
3. Numerical experiment 3.1. Fractal simulation
d s G Dy 1 l 2y D . Therefore, the asperity curvature is: ks
Fig. 3. Spectral density of W–M function, Ds1.5, Gs1.5 and gs1.5.
There are several schemes that can generate fractal profiles or surfaces. All of them have their advantages, and also limitations. We adopt three schemes to simulate fractal profiles. The first one is called Weierstrass–Mandelbrot
Fig. 4. Structure function of W–M function, Ds1.5, Gs1.5 and gs1.5.
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 5. Profile generated by Ganti and Bhushan’s method, Ds1.5.
ŽW–M. function, which is widely used in fractal contact model. The second one is proposed by Ganti and Bhushan w8x. This method generates a fractal profile from its spectral density. The third method is called successive random addition method. The profile generated by this method has a power law structure function. They are described in the following sections. We compare all characteristics for the profiles generated by these three schemes. 3.1.1. W–M function W–M function w14x is the most widely used function for fractal contact model. It can be expressed as: 3 cos Ž g n x q fn .
`
zŽ x. sG
Ž Dy1.
Ý
g Ž2yD. n
nsn 1
,
Ž 9.
where fn is random phase. Its spectral density is: PŽ v. s
G 2Ž Dy1.
d Ž vyg n.
`
Ý
2
nsn 1
v Ž4y2 D.
.
39
Fig. 6. Spectral density of profile generated by Ganti and Bhushan’s method, Ds1.5.
Its structure function is: S Ž t . s E z Ž x . y z Ž x q t .
2
4 s Kt Ž4y2 D. ,
Ž 12 .
where K s wp G 2 D -2 xrw4 ln g G Ž5 y 2 D .sinwŽ2 y D .xp x. A profile generated by the W–M function is shown in Fig. 2. Its spectral density is shown in Fig. 3. From Fig. 3, we find that there are some spikes in the spectral density. Therefore, the spectral density follows Eq. 10, not Eq. 11. We can also find that the distance between each spike depends on g . The structure function is shown in Fig. 4. Although the spectral density does not follow Eq. 11, the structure function does follow Eq. 12. We compare spectral density in Fig. 3 with that of real surfaces. From the spectral densities obtained from real surface w4,8x, it is found that there are no such spikes in
Ž 10 .
Its equivalent continuous spectral density is w14x: PŽ v. s
G 2Ž Dy1. 2 ln g
1
v
Ž5y2 D.
.
Ž 11 .
3
Several papers w4–8x write the W–M function as: ` cos Ž 2pg n x q fn . z Ž x . sG Ž Dy1. . g Ž2yD . n ns n
Ý
1
With this convention, its spectral density is: PŽ f .s
PŽv.s
G 2Ž Dy1.
`
Ý
2
ns n 1
Ž 2p G . 2
Ž D y1 .
2
where f is frequency.
d Ž f yg n . f Ž4y2 D .
,
`
d Ž v y2pg n .
Ý
v Ž4y2 D .
ns n 1
, Fig. 7. Structure function of profile generated by Ganti and Bhushan’s method, Ds1.5.
J.-J. Wu r Wear 239 (2000) 36–47
40
Fig. 8. Profile generated by successive random addition method for Ds1.5.
spectral density as those in Fig. 3. So, there is no such parameter g existing. Some authors thought that g is 1.5 w8x. It is baseless. There is another problem with the W–M function. It cannot be extended into two dimensions. Ausloos and Berman w15x believed that the W–M function could be extended into two dimensions by the following function: zŽ x, y. sL
G
Ž Dy2 .
1r2
ln g
Ý Ý sg Ž Dy2. n
ž / ž / L
M
° =~cos f ¢
m , n y cos
=cos tany1
ms1 ns1
2pg n Ž x 2 q y 2 .
x
y M
/
q fm , n
hold for all Ž v x , v y .; i.e., the surface is not isotropic. So, a vertical cut of the surface is not necessarily a W–M function. Thus, this is not a really two-dimensional W–M function. From the above analysis, we know that: firstly, no real profile follows a W–M function; secondly, W–M function cannot be extended into two dimensions. But W–M function is still generated and investigated in the following sections.
1r2
L
pm
y
ž ž /
n max
M
Fig. 10. Structure function of profile generated by successive random addition method for Ds1.5.
¶• ß.
But from their paper, for the surface obtained from this equation, it is shown that P Ž v . s P Ž v x2 q v y2 . does not
(
Fig. 9. Spectral density of profile generated by successive random addition method for Ds1.5.
3.1.2. Profile generated from spectral density Ganti and Bhushan’s method is also used to generating the fractal profiles. For a fractal profile, the spectral density is of the form of Eq. 1. From Ganti and Bhushan’s method, a profile with fractal spectral density can be generated. The profiles with fractal spectral density have the property of self-similarity. The structure function of this generated profile is expected to have the form of Eq. 2.
Fig. 11. sm2 of W–M function for Ds1.5, Gs1.5 and gs1.5.
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 12. sk2 of W–M function for Ds1.5, Gs1.5 and gs1.5.
A profile generated by Ganti and Bhushan’s method is shown in Fig. 5. Its spectral density is shown in Fig. 6. The structure function is shown in Fig. 7. It is found that its spectral density follows the power law perfectly. There is only a small error in its structure function. Both the spectral density and structure function do follow the power laws.
41
Fig. 14. sk2 of profile generated by Ganti and Bhushan’s method, Ds1.5.
that its structure function follows the power law perfectly. There is only small error in its spectral density. Both the spectral density and structure function do follow power laws, too. 3.2. The rms slope and curÕature
3.1.3. SuccessiÕe random addition method The third method is called successive random addition method w16x. It is a modification of the mid-point displacement method. This method can generate a profile whose structure function is in the form of Eq. 2. The spectral density of this profile is expected to have the form of Eq. 1. Fig. 8 shows the profile generated by successive random addition method. Its spectral density is shown in Fig. 9. The structure function is shown in Fig. 10. It is shown
The rms slope and curvature are analyzed. They are obtained by three different ways. Firstly, they are calculated directly from the profiles. Also, they can be obtained from the spectral density by Eqs. 4 and 5. Finally, they can be obtained from the structure function by Eqs. 6 and 7. The rms slope and curvature for profiles generated by W–M function are shown in Figs. 11 and 12, those for Ganti and Bhushan’s method are shown in Figs. 13 and 14; and those for successive random addition method are shown in Figs. 15 and 16.
Fig. 13. sm2 of profile generated by Ganti and Bhushan’s method, Ds1.5.
Fig. 15. sm2 of profile generated by successive random addition method for Ds1.5.
42
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 16. sk2 of profile generated by successive random addition method for Ds1.5.
It is found that rms slopes obtained from Eqs. 4 and 6 are both good estimates of that obtained from the profile. But, the rms curvature from Eq. 7 is always better than that from Eq. 5. Since Eq. 7 is based on the structure function and Eq. 5 is based on spectral density, we can assert that the structure function is more suitable for the calculation of rms curvature at a finite sampling interval. 3.3. Asperity curÕature In order to find the asperity curvature, we try different dimensions of fractal profiles and fractal profiles generated by different schemes. We also set the contact line at the mean line of the fractal profile. For W–M function and Ganti and Bhushan’s method, D s 1.2, 1.4, 1.6, 1.9 is used. For D s 1.1, the profile is
Fig. 17. Asperity curvature of W–M function, Gs1.5 and gs1.5; contact level: mean line; dot — real asperity curvature, dash line — sk , dotted line — k from Eq. 8.
Fig. 18. Asperity curvature, Ganti and Bhushan’s method; contact level: mean line; dot — real asperity curvature, dash line — sk , dotted line — k from Eq. 8, mean line.
too smooth, so there are too few asperities such that it cannot be used. For successive random addition method, D s 1.3, 1.4, 1.5, 1.7 is used, since this method cannot generate a fractal profile with good spectral density and structure function for too large or too small D. We compare asperity curvature with sk and k . sk is the rms curvature obtained from Eq. 7. k is the asperity curvature used by Majumdar and Bhushan’s fractal contact model, as obtained from Eq. 8. From Figs. 17–19, we find that both sk and k have the same trend. But, k overestimates the asperity curvatures for small D, and underestimates them for large D. We also found that there is always the same ratio between asperity curvature and sk . Obviously, sk is much better than k in predicting the asperity curvature.
Fig. 19. Asperity curvature, successive random addition method; contact level: mean line; dot — real asperity curvature, dash line — sk , dotted line — k from Eq. 8.
J.-J. Wu r Wear 239 (2000) 36–47
Recall that Eq. 8 predicted the asperity curvature only by a single wave number mode. In fact, the asperity curvature is affected by all wave number modes. When the fractal dimension is small, the amplitude of large wave number is smaller and makes less contribution. For large fractal dimension, the amplitude of large wave number is larger and makes more contribution. But k always makes the same prediction. Therefore, k will overestimate the asperity curvature for small D and underestimate it for large D. On the other hand, sk is the rms curvature based on three-point curvature. There could be some relation between asperity curvature and three-point peak curvature. There is also possible some relation between three-point peak curvature and rms curvature. Thus, it is reasonable to use s in predicting asperity curvature. The real relation between sk and asperity curvature is not discussed in the present paper. It may need much more investigation in the future.
4. Bifractal profile Bhushan and Majumdar w7x found that some surfaces have bifractal characteristics. In order to investigate the characteristics of bifractal surface, we start by forming a fractal surface with a limited band spectral density. A bifractal surface may be considered as a surface with two such fractal spectral density bands. 4.1. Fractal surface with band-limited spectral density
43
Fig. 21. Structure function for bandwidth spectral density.
a sampling interval equal to the smallest fractal length, and the specimen length equal to the longest fractal length: p vls , l p vhs . s The structure function of a fractal profile with bandlimited spectral density will be SŽt . s2
vh
Hv
P Ž v . 1 y cos Ž vt . d v ,
l
s 2G 2 Dy2
vh
Hv
l
1 y cos Ž vt .
v 5y 2 D
dv.
Consider a fractal profile whose wave number is restricted by two extreme wave numbers, one for the largest wave number and the other for the smallest one. v h is the maximum wave number and is related to the smallest fractal length, while v l is the minimum wave number and related to the longest fractal length. In our analysis, we use
We arbitrarily set v l s 10y4 and v h s 1. That means l s 10 4 p and s s p : 1 y cos Ž vt . 1 S Ž t . s 2G 2 Dy2 dv. v 5y 2 D 0.0001 Fig. 20 shows the structure function of a fractal profile with D s 1.3, and Fig. 21 shows the structure function of
Fig. 20. Structure function for bandwidth spectral density.
Fig. 22. Spectral density of a bifractal profile. D1 s1.2, D 2 s1.8.
H
J.-J. Wu r Wear 239 (2000) 36–47
44
Is Eq. 14 the structure function of the profile defined by Eq. 13? It needs investigation. The structure function of Eq. 13 is:
SŽ x . s2
½H
vc
v
0 `
Hv
q
Fig. 23. Structure function of a bifractal profile.
a fractal profile with D s 1.7. In both cases, the spectral densities are limited to wave numbers between 0.0001 and 1; i.e., l s 10 4 p and s s p . It is shown that the structure functions follow a power law in the range from around t s p to around 7 = 10 3. The range does not correspond to the maximum and minimum wave numbers.
4.2. Bifractal surface
4.2.1. Critical waÕe number and critical length A bifractal profile can be defined by spectral density or structure function. If a bifractal surface is defined by spectral density, its spectral density is:
°G v P Ž v . s~
2 Dy2 1 5y2 D
G 22 Dy2
¢v
5y2 D
c
v
G1 5y 2 D 1
G2 5y 2 D 2
1 y cos Ž v x . d v
5
1 y cos Ž v x . d v .
Ž 15 .
Fig. 22 shows a bifractal spectral density. The critical wave number is vc s 0.1. Fig. 23 shows its structure function obtained from Eq. 15. It is found that there are two asymptotic lines in Fig. 23. But there is no exact critical length existing for the structure function. If we regard the intersection of two asymptotic lines as the critical length, then it is tc s 14.9. It is different from the corresponding length of the critical wave number:
p vc /
tc
.
It can be explained from the fractal profile with bandlimited spectral density. The range of structure function that follows the power law does not correspond to the maximum and minimum wave numbers of the limit band spectral density. Thus, the critical length does not correspond to the critical wave number, either. Normally, there are two asymptotic lines for the spectral density of bifractal surfaces. The critical wave number
v G vc
Ž 13 . v F vc .
If a bifractal profile is defined by structure function, then, its structure function follows: SŽt . s
½
K 1t Ž 4y 2 D 1 .
t F tc
K 2t Ž 4y 2 D 2 .
t G tc
,
Ž 14 .
where: K1 s
K2 s
p G 12 D 1y2 2 G Ž 5 y 2 D 1 . sin Ž 2 y D 1 . p
p G 22 D 2y2 2 G Ž 5 y 2 D 2 . sin Ž 2 y D 2 . p
,
.
Fig. 24. Spectral density of bifractal profile generated by Ganti–Bhushan’s method.
J.-J. Wu r Wear 239 (2000) 36–47
Fig. 25. Structure function of bifractal profile generated by Ganti– Bhushan’s profile.
45
Fig. 26. sk2 of bifractal profile generated by Ganti and Bhushan’s method.
i.e., exists at the intersection of these two asymptotic lines. From Eq. 13, we find that, for the critical wave number: G 12 D 1y2
vc5y 2 D 1
s
G 22 D 2y2
vc5y2 D 2
.
Therefore: G 1D 1y1 G 2D 2y1
s v cD 2yD 1 .
Ž 16 .
For the structure function, there are also two asymptotic lines existing for a bifractal surface, too. The critical length is again the intersection of two asymptotic lines. From Eq. 14, we obtain:
K1 K2
4 y 2 4y 2 D 2 s
4 y 2 4y2 D 1
x c2Ž D 1yD 2 . .
We find that all the critical wave numbers and critical lengths are different. In fact, for spectral density, structure function or rms curvature of bifractal surfaces, there is usually a transition region between two asymptotic lines. Therefore, the critical wave number and critical length do not exist strictly. We generate a bifractal profile by Ganti and Bhushan’s method. The profile is generated from a bifractal spectral density. The spectral density of the generated profile is shown in Fig. 24. Fig. 25 shows the structure function. Fig. 26 shows the rms curvature. All these figures show that there are two asymptotic lines for spectral density, structure function and rms curva-
K 1tc4y 2 D 1 s K 2t 24y 2 D 2 . So, K1 K2
s
G 12Ž D 1y1. G Ž 5 y 2 D 2 . sin Ž 2 y D 2 . p G 22Ž D 2y1. G Ž 5 y 2 D 1 . sin Ž 2 y D1 . p
s tc2Ž D 1yD 2 . . Similarly, for a bifractal surface, the rms curvature is:
sk2 Ž x . s
½
Ž 4 y 2 4y 2 D 1 . K 1 xy2 D 1 Ž 4 y 2 4y2 D 2 . K 2 xy2 D 2
x G xc x F xc .
The critical length is the intersection of two rms curvature lines. So, D2 D2 s Ž 4 y 2 4y 2 D 2 . K 2 xy2 , Ž 4 y 2 4y 2 D 1 . K 1 xy2 c c
Fig. 27. Asperity curvature of bifractal profile generated by Ganti and Bhushan’s method Ds1.5, at central line.
J.-J. Wu r Wear 239 (2000) 36–47
46
If W–M function is used, the result is exactly the same. There is also a discontinuity existing. In the bifractal contact model of Majumdar and Bhushan w6x, they analyzed the structure function, but applied Eq. 17 to obtain the asperity curvature, which is obtained from spectral density. In their model, they used the critical wave number corresponding to the critical length. There must be a discontinuity for asperity curvature. It is a mistake. 5. Conclusion
Fig. 28. Asperity curvature of bifractal profile generated by Ganti and Bhushan’s method Ds1.5, at hs s .
ture. All the asymptotic lines and the critical point are the same as we expect. 4.2.2. Asperity curÕature of bifractal surfaces By using the spectral density, Majumdar claimed that the asperity curvature is: 2
ks
p G
A C
s p 2yD G Dy1v D .
lD
½
p 2y D 1 G 1D 1y1v D 1 p
6. Nomenclature
Dy 1
If their formula is correct, the asperity curvature of a bifractal surface should be:
ks
In this paper, fractal profiles are analyzed. It is found that rms curvature can be related to the structure function. Also, it is found that rms curvature can be used to estimate the curvature of an asperity. Bifractal profiles are analyzed, too. It is found that the critical wave number in spectral density does not correspond to the critical length for the structure function. The rms curvature obtained from the profile is a good estimate for asperity curvature. The finding in this paper can be used to establish a fractal or bifractal contact model in the future.
2y D 2
G 2D 2y1v D 2
v G vc v F vc .
Ž 17 .
From Eq. 16, we find that, at critical wave number: G 1D 1y1 v cD 2 G 2D 2y1 vcD 1
.
Thus, from the first part of Eq. 17, the asperity curvature for the critical wave number is:
k s p 2y D 1 G 1D 1y1vcD 1 . But, from the second part of Eq. 17, it is different:
k s p 2y D 2 G 1D 2y1vcD 2 s p 2y D 2 G1D 1y1vcD 1 . So, there is a discontinuity at the critical wave number for the asperity curvature. We use the surface generated by Ganti and Bhushan’s method, and compare the real asperity curvature with k and sk , where k is obtained from Eq. 17 and sk is the rms curvature obtained from profiles. The results are shown in Figs. 27 and 28. In Fig. 27, the contact line is set at central line. In Fig. 28, the contact line is set at height s above central line. In both figures, it is found that there is a discontinuity for k . We also find that rms curvature is also a good estimate for asperity curvature.
the area above the contact line a constant related to the amplitude of all frequencies D fractal dimension G a scale constant for a fractal surface K a constant in structure function l sampling length, contact length P Ž v . spectral density SŽt . structure function s sampling interval x horizontal coordinate d roughness amplitude h the resolution of the instrument g scaling ratio fn random phase k asperity curvature, used by Majumdar and Bhushan s standard deviation of the surface height sm standard deviation of the surface slope sk standard deviation of the surface curvature t distance v wave number vh higher wave number limit vl lower wave number limit and References w1x J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proceeding of the Royal Society, London A 295 Ž1966. 300–319. w2x P.R. Nayak, Random process model of rough surfaces, Journal of
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