Fractal geometry modeling with applications in surface characterisation and wear prediction

Pergamon

Int. J. Mach. Tools Manufact. Vol. 35, No.2, pp. 2 0 3 - 2 0 9 , 1995 Elsevier Science Ltd. Printed in Great Britain 0 8 9 0 - 6 9 5 5 / 9 5 5 7 . 0 0 + .00

FRACTAL GEOMETRY MODELING WITH APPLICATIONS IN SURFACE CHARACTERISATION AND WEAR PREDICTION

G. ZHOU,'~ M. LEU* AND D. BLACKMORE~t -DEPARTMENT OF MECHANICAL ENGINEERING, TEMPLE UNIVERSITY, PHILADELPHIA, ~'qNSYLVAINA,U S A SDEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, DEPARTMENT OF MATHEMATICS, NEW JERSEY INS'ITI'UTE OF TECHNOLOGY, NEWARK, NEW JERSEY, U S A

Abstract Mat:ufactured surfaces such as those produced by electrical discharge machining, waterjet cutting and ion-nitridlng coating can be characterized by fractal geometry. A modified Gaussian random fractal model coupled with structure functions is used to relate surface topography with fractal geometry via fractal geometry via fractal dimension (D) and topothesy (L). Thls fractal characterization of surface topography complements and improves conventional statistical and random process methods of surface characterization, Our fractal model for surface topography is shown to predict a primary relationship between D and the bearing area curve, while L affects this curve to a smaller degree. A fractal geometry model for wear prediction is proposed, which predicts the wear rate in terms of t~ese two fractal parameters. U s i ~ g ~ h l s model we show that the wear rate V r and the true contact area A have the relationship V r~ ( A )m~uj, where m(D) is a function of D and has a value between 0.5 and I% We next study the optimum (~e the lowest wear rate) fractal dlemsnsion in a wear process. It is fou~id that the optimum fractal dimension is affected by the contact area, material properties, and scale amplitude. Experimental results of bearing area curves and wear testing show good agreement with the two models. they include various methods for computing fractal dimensiom. A I. Introduction good start has been made in applying fractal geometry to surface topography, but there is still much work to do in determining the In machines or mechanisms, parts are assembled to perform extent to which fractais can be used to describe and model surfaces. certain functions or tasks. The work usually is done by transmitting In this study we shall develop a modified Ganssian fractal model power, energy, supporting loads, and transferring motion through for surf~e topography. This model connects frnctal geometry with pairs of contacted surfaces of machine elements. Surface topography bearing area curve and provides a quantitative and more efficient is crucial to mechanical design and manufacturing. It has been shown representation of surface properties. We also shall develop a fractal that fractals provide useful insights into the nature of surfaces [1-7]. It geometry model for wear prediction. This model integrates surface is highly desirable to establish fractal-hased techniques to study topography information into wear theory and can predict wear rate in surface engineering and trihology, and to apply the techniques to terms of fractal parameters. engineering applications such as surface contacts, wear processes and friction. 2. Techniques for O b t a i n i n g F r a c t a l D i m e n s i o n s Characterization of surface topography has become increasingly important in many engineering fields. The methodology of analyzing Fractals have mmly dimensions such as Hansdorff dimension, surface topography has been developed for more than fifty y,ars. compass dimension, box dimension, mass dimension and areaOver thirty parametem are used to quandfy topography in the perimeter dimension, and there are several methods for computing National and ISO Standards of Surface Roughness. This rash of each of these dimensions. These dimensions can be most readily parameters causes endless disputes about the methods of ~ n t calculated for self-similar fractals. However, for self-affine fractals and evaluation of rough surfaces. Over the last dozen or so years it which are not self-similar, the dimension cannot be obtained so has been realized 18. 91 that much of the difficulty in using standard readily [17]. In this study, we are mainly concerned with the measures of surface roughness stems from the non-statiotmry and microtopogtaphy of surfaces. The observed magnifications in vertical multi-scale nature of surface topography as well as the dependenoe of and horizontal directions usually differ by a factor of 100 to 10.000. the measurements on instruments. The shortcomings of conventional Experiments indicate that the surface profiles tend to be only selfstatistical and random process methods have stimulated interest in new affin¢, not self-similar. We found that the smicaure function is wellapproaches to characterizing surface roughness. suited to computing fractal parameters. Suppose z(x) is a fractal Fractal geometry was proposed as a means of characterizing function; it is known [18] that the correlation (z(xl)z(x2)) (and hence surface topography by Mandelbrot [1, 10]. The application of fractals the variance (z2(x))) is infinite, and z(x) is not differemiable. Berry in surface related phenomena is very recent and shows great potential suggested that a structure function be used to characterize z(x). The in such fields as trihology, surface contact mechanics, thermal increment Iz(x+~.)-z(x)l is assumed to have a Gaussian distribution conductance, optical scattering on rough surfaces, chemical reactivity with the following second moment (called the structure function) of surf~es, currents in superconductors. The use of fractal geometry

in surface topography analysis is still in a beginning stase. Ling [3, 11 ] proposed an exponential law based on experimental results, and discussed several potential advantages of using fractals for the study of boundary lubrication. Majtmadar [4, 121 proposed a fractai surface model based on Weiet~'trass-Mamlelbrot functions to characterize rough surfaces, and gave two major applications in surface contact conductance and electromngnedc wave scattering by rough surfaces. Some other approaches using ftnctals in engineering analysis of surfaces were also reported [12-161. These studies represent attempts to introduce fractals as means of characterizing rough surfaces and


) sinl~2D-3)ll-'(2D-3) I ~-I 4zD (l)

where L is any displacement along the X direction, g is a constant, D is the fi.actal dimension of the z(x) function and 1"( ) is the Gamma function. It can be seen that the chords joining z-values separated by a distance ~, do have a finite mean square slope. If there exists a displacement k = L such that the chord has an r.m.s, slope of unity. then a concise fomanla can be written:

203

G. ZHOU et al.

204

ilz(x + L ) - z(x)l 2) L2

I

0(z) = 1+ 81zeZ2/2~2

(2)

(7)

SO that the height of the bearing area curve at z is where L is a characteristic paranleter of file ffactal function called its topothesy. By comparing equations (1) and (2), an equation relating the structure function with fractal geometry paranzeters is derived: ( I z ( x + L ) - z ( x ) l 2)

L zl)2

Z142D,

I < D < 2

z

It.._

Based on equations (1) and (3) the structure function can be used fi)r experimentally computing D and L of fractal surfaces 121. It can be used to identify whether a surface is fractal or non-fi'actal. Nonfractal surfaces do not possess the above nlentioned properties; in other words, their doubly logarithmic plots either do not have a umque slope, or the value of D lies outside the interval (1, 2). 3.

,.

(3)

Modified Gaussian Fractal Modeling of Surface Topography

0 Profi 1¢

Probaotht) Ocn$1t~

Figure 1 Constriction of bearing area cu~'e (M-M Reference Line) ,

We shall now describe the mathentadcal model which we used to characterize the topography of the surfaces. This model is used to analyze features of surfaces in engineering applications: in particular. it leads to some predictions about the qualitative relationship between the model parameters and bearing area curves. 3.1 The M~I,I We start with a mean zero Ganssian model for the ragdom variables A(x, ~) = Z(x + ~. *) - Z(x. *), where Z(x, y) represents the surface height. Then we perturb the Gaussian distribution slightly to add a bias (characteristic of self affine fractal graphs) as follows: For almost every .~. the random variable A(x, .~) = Z(x * ~, .) Z(x, *) defined on -b ~_ y _< b is normally distributed with nlean zero 3 -~x

~'%

l_~c

-2

13(z) = [2nL "m ') v-'" o,] 2 f O(z) exP[21.2(D ~ 2(2 - D ) d';'

(8)

Z

Let us fix z. and view B(z) as a function of D and L. We compute that OB(z) 0D

= (2~)':,~'" "~ ' tog~,Z )F(z) L L

,9)

where

F(z~: .~ ~<) I l- 4 I~.# ) " / I exp[ 2L2CD-~ _.:e2(2-D) ]d~

(10)

Z

Similarly, we find that

c~- I

0B(Z)0L= "(2r0 1,2 (D-I)(ZL)D ,v2 F(z)

and standard deviation L 2 I ~ I 2 : i.e.. P(A(x, ~)< z ) = ( 2 n L 3 a i ~ "'t'l'½ ~ ' e x - [ .~2 ]d r , J ~t2L3-'*i~,l= I J "~

(4)

(l I )

Observe that F(-OO) ~- 0 and

-oO

where ct and L are real paranleters such that 1 < ct < 3 and 0 < L. P denotes the usual probability measure and Z(x. -) denotes the function of y obtained by fixing the first argument of Z at x. The parameter ¢t is related to the fractal dimension of the profiles of the surface topography and has the expression ,:x = 5-2D 119]. It is shown in Berry 1181 that the structure functions of the profiles satisfy (1). We shall refer to the equation (4) as the Ga~,sianfractal model. Using fundamental characterizations of surfaces, we can rigorously show 1201 that the actual height distribution is a normal distribution multiplied by a power series. Based on this. we postulate a modified Gaussian fractal model for the height distribution @x) = Z(x, Yo) of the surface profile; namely

?(L/y)tnl~ , it is not difficult to show that when 81 is not equal m zero and L is sufficiently snudl (as is the case for many surface tound in engineering practice) then F(z) > 0 for all finite values of z. Consequently, it follows from (9) and (1 I) that B(z) is an increasing ~nction of D and a decreasing function uf L fi)r each z, whenever the bias term is not zero and L is sufficiently sntall. The relative rates of change of B(z) with D and L are given by

p(¢_
,?B(Z)~DI ~ l

z ~2 y [1 +E t~e-~2/2Wz] exp( 2 L2(D - l)y2(2_D))dg

(5)

~'dF :. -O(z)ll" ZZ(L) 2d)'l' y 21 exP[L2, D.~f?2(2-z- D ) '

2

v 2(I).11

As 0 is nonnegtive and 11- z (L)

aud

7 'l is nonpositive when Izl >

" = .v. (~)" Iog(~ L )(D-I)",

(13)

tile quantity y(~)qog(~V v ) tends to be tairly large ti)r most

engineering surface profiles when tile refit o l length is expressed in

-cO

where 8 | is a real parameter, ~ is a very small positive number, "/is a size factor relating with measuring area and they are all determined by sampling conditions. Observe that this modified distribution is simply obtained by adding a Weibull distribution to (4). 3.2 Normalized Bearirm Curve and Fractal Paramet¢~ The concept of the bearing area curve of a surface topography was introduced by Abbott [211. Figure I illustrates the bearing area curve which is obtained from the bearing ratio cO

th = P(z ~ h) = f p(t)dt h

(6)

where z is the profile height, p is the probability density of the height, and h is the height from the reference line M-M. By plotting th according to (6) over a range of dis.crete heights h for a profile, one obtains the bearing area curve as shown in Figure 1. For purposes of comparison, we introduce the normalized bearing area curve. In practice, the exact nature of the height distribution of a profile is usually not known, so the normalized beating area curve can be obtained experimentally. We shall now investigate how the be, ring area curve depends on the fractal paranleters D and L for our model. Define

microns. Hence assuming a bias in the distribution (131 is not equal to zero), our model predicts that B(z) is an increasing thncdon of D and a decreasing function of L, where the rate of increase with D is larger than the rate of decrease with L, for most engineering surfaces. In order m relate the fractal parameters to the bearing area curve for our model, we assume that only D and L are allowed to vary fi,r the surface sample under consideration. Suppose that we measure profiles for two samples and find that the fractal parameters are D], LI and D 2. L2. respectively. The height distribution data is used to obtain the corresponding normalized bearing area curves Bt(z} and B2(z), where z is the normalized height 0 -< z _< I. We scale each data set and associate modified Gaussian model as shown in Figure 2 in order to obtain the normalized height distributions. Let PI and P2 be the respective probability density functions for the normalized height distribution. Then the normalized bearing area ratios are 1

Bl(z) = f p l ( X ) d t

I

and B2(z) = f p 2 ( x ) d r

Z

t14)

Z

rile dependence on the fractal parameters is defined by Bi(z ) = Bl(z : Di, Li)

(i = I, 2)

; 00"~

B ¢ ~ R -Xtea Car~e

(15)

Fractal geometry modeling

It follows from our analysis of (9) and (11) that

205

2o

A

B(z ; DI. 1,) < B(z " D2, L)

(16)

and B(z : D, L1) < B(z : D, L2)

(17)

for each 0 < z < I if DI < D2 and LI > L2. Thus our conclusions about the relationship between the fractal parameters and the normalized bearing area curve is as follows: Prediction: The modified Gaussian fractal model for surface topography defined by~ (5) implies that the height of the normalized bearing area curve B(z, D, L) increases with D and decreases with L for all normalized heights 0 ~z < I. Furthermore, for raojt surfaces the height depends more .~trongly on D than L, so that B I ~ ) < B2(z) if DI < D2for all 0 -~ z < 1, except possibly for some z very close to 0 or 1.

15

~o "~ s ~

o

~

.s .~o .is .2o 10OO

1SO0

2500

2000

3000

3500

LENGTH (gin) (a) 2s

15 10

r

le

Ity Dens4ty p l

Profile 2

Probability DenstW pZ

-10

5OO

100o

ISO0

2Ooo

~

3OOO

35oo

L E N G T H (u,m) (b) 1

s2(I)

0%

20%

40% 5 0 ~ eO% V ~ J N 6 gATIO

~%

Figure 3 Two profiles having nearly equal values of conventional roughness parameters but differem features: (a) profile BR1 generated by waterjet cutting, Ra =4.26~tm, Rq=5.351am, and Rmax=31.6!am, and (b) profile BR2 generated by electrical discharge machining, Ra =4.251am, Rq =5.68~m. and Rrmx=33. lrtm.

)w%

Figure 2 Relation between fr~tal dimension and bearing curve OSIal oStq2

3 3 Expenment,Td Resadts ~pndDiscussion We have performed measurements on several surface profiles in order to test the validity of our modified Gaussian model of surface topography. The widely used parameters of surface roughness are Average Height Ra, Root-Mean-Square Height Rq, and Maximum Peak-toValley Height Rmax. Some surfaces have nearly equal values of Ra, Rq, and Rmax even though their surface features are different. For example, BRI and BR2 in Figure 3 are two surface profiles generated by different machining methods. They have nearly equal values of Ra, Rq. or Rmax. The bearing area ratios of these surfaces are calculated by the tbllowing procedure. First the sampled profile height data are normalized between 0 and 1, with the lowest point heing 0 and highest point being I. Then the normalized height range is divided into 20 equal divisions. For each division the probability of the height data is computed by dividing the number of data points in this range by the total number of data points. The cumulative density function forms the bearing area curve. It can be seen from Figure 4 that the bearing area curve of BRI is above that of BR2. This implies that when the surfaces are in contact BRI has more contact area, i.e. the surface support ability of BRI is greater than that of BR2. Another example is given by the two profiles in Figure 5. The parameter values of Ra, Rq, and Rmax for BR3 and BR4 are nearly identical. This time, however, the two profiles are not as differem as in Figure 3. Figure 6 shows their bearing area cutwes, which imply that BR4 has weaker support ability in surface contact than does BR3. Fractal dimension therefore appears to give a good qualitative measure of surface contact ability. It is interesting that not only D6RI > DKq.2 and DBR3 > DBR4 in agreement with the sizes of the reslx~tive bearing area curves and ~ r t abilities, but that the agreement applies to all of the four surfaces, even thoagh the Ra, Rq, and Rmax values of BRI and BR2 are very different from those of BR3 and BR4 (refer to Figure 7).

0%

2o%

,*o%

6o%

to%

)oo%

8EARING RATIO

Figure 4 Bearing area curves of BRI and BR2. Their fractals are DBr~= 1.47, L6r, = 0.53faro and D0r: = 1.34, L,m =0.177~m.

4. Fractal

41W

Geometry " "

M

and

Wear

Theory

I

We shall develop a fractal model for surface wear processes. This model reve.als an interesting relationships between wear characteristics and fractal parameters. The construction of our model is based on the fractal property of islands, Weierstrass-Mandelbrot functions for surface profiles, adhesiye wear theory, and the work of At'chard 1221, Mandelbrot II, 101 and Majumdar & Bhushan 1231. We find that the total area of all islands, Ar. can be expressed in terms of the fractal dimension and wear volume can be derived as (see details in (Zhou, Leu and Blackmore, 1993b)) V=(I+~H'2)I/2Ar[Ke'(Ke-Kp)((2 D~A (Q~ '?E) 21I'J'l)(zD)/2]d -

r

y'

(18) where V is the wear volume, Ke and Kp are the elastic and plastic wear coefficients, respectively, Ar is the true contact area, 13 and Q are constants, ~ is frictioncoefficient, o3, is yield stress, E is Young's modulus, G is a scale amplitude relat~f to t o ~ y L, and d is the sliding distance. W e next normalize the variables in (18) as follows:

G. ZHOU et al.

206

75 S, 25

~

.25

r,+ ,+,+

0 B-~3

-'0

.15

~00

t000

1,500

20C0

250¢

3CO0

BEARING RATIO

35 ~'~0

LENGTH (~m) (a)

Figure 6 Bearing area curves of BR3 and BR4. Their fractals are D~+~3-: 1.50, L ~ = 0 . 3 1 8 ~ m and Drm~: " 1.42, Lm~.~=0.189tam.

OB,q~

++ +'+ tO

r'l B,,~

+' + I +'+++" 500

)000

1500

2000

LENGTH

2500

3~0

4 fl~ 0 80,,1

0%

(lain)

2o%

~%

+0%

no%

~ooc~

BEARtNG RATIO

(b)

Figure 5 Another two profiles having nearly equal values of conventional roughness parameters but different features: (a) profile BR3 generated by waterjet cutting, Ra =2.52p.m, Rq =3. 151Jnh and Rttmx = 21.Opm, and (b) profile BR4 generated by electrical discharge machining. Ra = 2.551am, Rq = 3.158~m, and Rmax = 22.0p+m.

v

V*= ~ a

A__+_, G . _ (Aa)h, G 2 ,andqJ =

" A r * = Aa

(19)

where Aa is the appareut coutact area; V* is rite uomhalized wear rate; Ar" is tile nomlalized true contact area; G* is tile nomm]ized scale amplitude; and qJ is a material property cottstant. With the nomlalized variables, equation (18) can now be rewritten as

,,.=(l +

>('--0>':I (20)

We call this the fractal geometry ,uulel of wear prediction. With (20) the wear rate V Call be evaluated as a t:ulK~tlon (')f A r . D, G , alKl q~. If the area of tile largest contact slx)t S L is less than the critical area of plastic detormation, i.e.. S L < Sc. then only plastic detomtation will take place, hi that case Ke = 0 iu (20). A few remarks are in order contenting (20). Our equation treats Ix)th the plastic and elastic aspects of wear together. This is not to suggest that both plastic aJ~l elastic wear are of equal magnitude at each iztstant of the wear processl We emphasize that (20) is a dynamic equation in which D, G, and the other parameters vary during tile wear cycle. At the start of the entire process, the plastic wear is dominant. As the pn~ess continues with D and G changing, the dominance of the plastic mode subsides and tile elastic component becomes a more significant contributor to tile total meclmnism of wear. In our wear experiments (described in section 6) the fractal parameters are measured at various times during the wear process in order to test for a correlation between the changes in the fractal geometry and changes in the stages of wear. We note that ergodicity of the surface was not assumed in the derivation of (20). In fact. we showed in 15, 61 that a self-affi~ fractal model for the surfaze topography leads to a non-stationary random process which is not ergodic.

Figure 7 Bearing area curves of BRI. BR2. BR3, and BR4. Note that D~m~ > Dtml > DnR4 > Du~ (refer to Figures 3 alvJ 5), 4.2 Effect of Fract~l Dimel~i0n on Wear Rate The effect of fractal dimcl~ion on wear rate i~ of colLsidcrablc interest. To nunlerically hlvestigate how V* is affected by D using equation (201. tllc values of other parameters ueed to be ~ h o s e l l . Based on the literature 12. 4, 51, for ordinary cases the paranlctcr valuesnlay be chosen as G * = 1 0 .9 , qs- 0.01, la=0.2. 13 9, Ke :10 4 • * . . . . . and Kp =0. I. Log(V*) is ph)tted against Iog(A r ) in h g u r e 8 Ior various o v a l u c s . It can be seen that there are two regions uf D that have significantly different wear rate behavior. In the first region, tot D between 1.15 aald 1.5, V* decreases with increasing D. Ill the second region, for D betweeu 1.6 and 1.9, V* increases slightly with increasing D. To show this more clearly, the relations in these two regions are plotted separately in Figures 9 and 10. Figure 9 shows that when D increases V* decreases and that this relationship is notthnear. When D increases from 1.15 to 1.2, an increa.,,e of D b~, old y 0.05, V* decreases by au average of 3.2 decades fi)r the range of At-considered. When D increases ft+om 1.2 to 1.3. V* decreases b~, 2.5 decades. As D iw.:reases from 1.3 to 1.4 and front 1.4 to 1.5, the decreases ill V* are 1.2 and 0.5 decades, respectively. Figure 10 shows thal V* increa,,+es with iucreasing D h)r (lie range of D between 1,6 and 1.9. This wear rate behavior in the two fractal dimension regions call be explained as fi)llows. When D increases from I. 15 to Dnl, which is approxinlatcly 1.5 lbr chosen parameter values and later in Section 5 will be called optimtml F?actal dimen.sion, there is a corresponding upward shift ot the beariug area curve which signals au increase in tile surface couta,.t area. Consequently, the nomlal conlact pressure between tile surfaces under the ,marne load decreases with increasing D, thus tile wear rate decreasc,s. However, the rate of upward shift in the beariug cur~,e decreases as D approaches D m from the left, which is in accord with the prediction of the fractal surface model developed in 15. 61. One Ix)ssible explanation of file subsequent increase in the wear rate as D a~unles values ill excess of Dm is as t~)llows: Tile rate of upward shift ill the bearing area curve ix small when D > Dm. Over the wear process, an accumulation of worit material ix:curs between the surta~:e and the number of asperities per unit surface area increa,ses with the tips of the asperities beconling sharper alvJ weaker. The net effect is a rather substautial increase ill the tolx>thesy which tends to lower the bearing area curve. It may I~ that the increase ill L becomes su|'ficielll to overcome the effect on the contact support ability due to int:reasiug D, so that there tv,:curs a very small downward shift of tile bearing area curve in the interval (D m, 2) and a con.sequent increase in tile wear rate.

Fractal geometry modeling

The above observations can be summarized a~ follows: For fractal surface with 1 < D < Din, the contact s u p p ~ ability is dominated by the size of D; but wheaD > D m other b c t o n inch as the growth of L tend to exert mine inflnence on the wenr ntte than D. It slmuld be noted that the value of Dm dmt ~ g t n s tbe two regions of wear rate I x ~ v i o r may vary for & f f e t ~ wear parameter values. For example, the Dm value is about 1.7 for the wear material and process de~ribed in Secdon 6.

Y] •

0-t.*

4. 0-1,1 • G..I• • ~1.7 • at D-t •

0-11

207

4.3 D,',,,~'~'~" of We~r Rate on Co-__.~_f_~ Under Suui¢ Loadin~ From Figun: 8. it ~ be seen duu for every D. log(V*) increases linearly with log(A/), since each #or is a straight line on V log-tog graph. In Archard's equation, the wear rate (V r = ~ ) is proportional to the true contact area under stafc loading, i.e., V r Ar. As he pointed out in 1221 this relationship was based on the assumption that all deformations are plastic and that the asperities are isolated. On the other hand if all deformations are elastic and the asperities are isolated, the wear rate should satisfy the relation V r ~c (Ar)2/3 (see 1221). If deformations include elastic and plastic deformafiom and the aslg'rities are not isolated, the wear rate can be expected w have the relation V r oc (Ar)q. where q is a constant related to the surface topography. In the wear prediction model (18), we have included both elastic and plastic deformations. This equation can be used to predict the power q in the relation V r oc (Ar)q. Rewrite equation (18) as follows: Vr

[KeA r - j(K e Kp)(Ar)D/21

(21)

where

j =( D

c

)<=,),:

(2-D) (QOy/2E)2/(D- 1)

(22)

For a typical case encountered in practice, D=I.5, G= 10.7 m. •

.)

4

:s

-~ -~ LoI(A,*)

QO'¥/2E=0.001 (for steel material), Ke=10 "4, and Kp=0.1, then j =0.42 and equation (21) can be estimated as

:

.i

Vr x[ 104At - 0.42 10-4(Ar) D/2] +0.042(Ar) D/2 = (Ar)re(D)

Figure 8 Effect of fmctal dimension (D) and normalized conmca area (A,*) on normalized wear rate (V*)

V r oc (Ar)m(D)

•

O.l.ll • D-LI • O.t 3 •

.~

0-1.4 40*LS

4

•

.5

•

.i

.,i

4

-|

.i

Figure 9 The V*-A,* relation for the first range of fractal dimension D: 1.15-1.5 j.

i

,

i

,

•

--

(24)

where re(D) ~ D/2 (note that the other items of Ar in (23) arc extremely small) is an increasing function of D and has a value approximately between 0.5 (when D = I ) and I (when D=2). Once the fractal dimension of a surface is given, re(D) can b¢ determined. By checking the slope of log(V*) versus Iog(Ar*) for each D in Figure 8, one sees that these slopes are between 0.6 (when D= 1.15) and 0.94 (when D=I.9). These slopes are in good agreement with (24). Confirmation of this fractal power proportionality can also be found in the experimental results provided by An:hard [24]. In the adhesive were theory the tree contact area has the relation with the normal lead as Ar ~ W, these experimental results can also be used to find the relation between wear rate and tree contact area. These experimemal results show that for brass material the relation is Vr (Ar)°'9s, and for stellite V r oc (Ar)°'92. This is consistent with our predicted fractional power proportionality. 5. O ~

Log(Ar*)

Fractal Dimensions of Wear Processes

Low wear rate is the most important requirement for a surface in a wear process. The fractal dimension of the lowest wear can be found by differentiating (20) with respect to D and setting it to 0. i.e. dV" _dD

(I +[~g'2)I/2Ar'(Kp-Ke)[(2 D)D?;Zzqu.,, ](2-D)/2

1½ 0.I.7 A OQIJi

(23)

or

1 +

I

~

O~Ij

From our prior discussion there exist Ar%0, KpaK e. G**0 and 1 < D < 2. so equation (25) leads to *

2

2 Solutions of (26) for D are fm~tiom of thz~ v~rial)l~: D = F(Ar*, ~, G*). Thus we see that d~ optimum ~ ' ~ (~m~mion (k'~eds on the tree consul area. material property constant, and scale amplitude. We observe that solving (26) for D is equivalent to solving .~

•

4

4 I.,o~A r

4

-t

-1

*)

Figure 10 The V*-A~* relation for the second range of fracud dimension D: 1.6-1.9

D = 4)(D) = (R*)-I(2-D) e2/D ~ 2(2D-I)/(D't)2

(27)

where R* = G . 2 / A r * and e is the base of the natural logari0mL It is easy to see that ¢ is a strictly decreasing fimcdon of D on the interval

208

G. Z H O U

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1 < D _<2 such that (~(2) = 0 and ~(D)--+ do as D approaches 1. We conclude, therefore, that (27) has a uttique solution, say D = Din, in the interval I < D < 2 which corresponds to the intersection of the line y = D with the curve y = d~(D). Referring back to (25). it can be dV* dV* readily shownma - - ~ < 0 f o r I < D < D mand ~ > 0when D m < D < 2. Hence, the minimum value of V* on 1 < D < 2 is attained at D = Drn, so Dm is the optimum value of the fractal dimension. Since it is not possible to obtain a closed form solution of (27), it is necessary to use numerical methods to obtain values of Din. For finding the optimum dimension D for various At*. Figure 11 is obtained by replotting Figure 8. This figure shows that for different Ar* the optimum D's are different, but they all are roughly in the range I •45-1.55• The effect of the material constant ~ and scale • * . . . amphtude G on the optimum fractal dimension can also be found in ~e way.

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6. Implementation in Wear Testing We present our wear testing results to qualitatively support our fractal model for wear prediction. In our wear testing experiments the wear mechanism consisted of an alloy steel roller (rotating pan) and an ion-nitriding treated shoe (fixed part). When the testing was performed, a load was exerted on the mating surfaces and the wear rate was measured periodically. During the wear process the fractal dimensions of the roller and shoe were obtained using the method of surface topography measurement described in [251. The fractal dimensions of four pairs of rollers and shoes were calculated in the experiments and they had very similar values. The histogram of the wear rate from one of our experiments is shown in Figure 12. It can be seen from the figure that during the first 30 minutes of testing the wear rate was fairly high, and then decreased and stayed at a very low rate until 120 minutes had elapsed; after this time the wear rate increased dramatically. These observed changes of wear rate are consistent with the well-known three stages n f a wear process: run-in, mild wear, and severe wear. Changes of fractal dimensions associated with the above testing are given in Figure 13 (a) and (b) for the roller and shoe, respectively. During the first 30 minutes, the fractal dimensions increased from their initial values. This is referred to as the enhancement stage of fractal dimension in the figure. Apparently when the surfaces first came into contact, the fresh and sharp layers of surfaces were removed: the surfaces became smoother and the surface contact support ability increased. After this stage, from the 30th to the 120th minute, D increased slightly but maintained an overall balance. This is referred to as the balance stage of fractal dimension. After 120 minutes D decreased greatly a,s the process continued. This is called the descent stage of fractal dimension. This implies that clearly the contact support ability of the surfaces become poorer and poorer because the surfaces become rougher and rougher• Comparing Figure 12 with Figure 13 we see that the three ,,~tagesof wear rate correlate with the three stages of fractal dimension fairly well• This suggests that fractal dimension can be used to monitor the wear process. The same observations hold for the other three pairs of wear components.

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Using this wear testing process sonle parameter values for the roller were obtained as follows: la=0.06, qJ=(1.0034. G*-- 65x10 -t~ (average value of the whole process), Ke= 10-4, and Kp=O.l. For examining the validity of our mtxlel, these parameter val/Jes are used in equation (20) and the wear prediction graph is plotted in Figure 14. The figure shows that the optimum fractal dimension fnr this particular wear process is around 1.7. Hence D = 1.7 is a change point of wear rate, i.e. if D is less than 1.7, V* decreases with increasing D. and if D is larger than 1.7, V* increases with increasing D. Using the relation between V* and D in Figure 14, we see that the wear r.ate in Figure 12 can be correctly predicted by the measured fractal dimension in Figure 13.

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REFERENCES

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7. Concluslon~ Using our modified Gaussian fractal model, we have derived equations to relate the bearing area curve with tbe fractal dimension D and topothesy L. and we predict that the bearing area curve shifts higher as D increases or 1, decreases. Experimental results obtained from a number of waterjet cut. electric discharge machined and ionnitriding treated surfaces appear to verity this prediction. We may therefore conclude that the fractal dimensio, can serve as a replacement for the bearing area curve in the study of surface contact ~KI wear. Our fractal geometry model for wear prediction leads to results which are consistent with experimental observations. This model predicts that there are two regions of D which have different wear rate behavior. In one region wear rate decreases greatly with increasing D, and in the second region wear rate increases slightly with increasing D. These phenomena are function.s of surface contact .support ability and sharpness of asperities. The model alg) shows that there is a relationship between wear rate and true contact area under static loading ;Is Vr oc (Ar) re(D), with re(D) between 0.5 and 1. This estimation is consistent with Archard's proposed relations of Vr oc A r for purely plastic deformation and Vr oc (At) 2/3 for purely elastic defon'oation. Our model gives useful expressions for Vr and Ar in terms of surface fractai properties. The optimum fractal dimension, con'esponcling to the minimum wear rate in a wear process, is derived by using the wear prediction model. This optimum fractal dimension is deternfined by Ar, ~s. and G. Most engineering surfaces have an optimum fractal dimension of about 1.5, and this value shifts with changes in the three key parameters. Our model provides useful information about how to prepare surfaces for wear reduction. Results of wear testing tend to support our wear prediction model.

HTM 35:2-F

1. Mandelbrot. B.B.. (1982). The Fractal Geometry of Nature. Freeman. New York. 2. Thomas, A., and T. R. Thomas. (19881. "Digital Analysis of very Small Scale Surface Roughness." Journal of Wave-Material Interaction, Vol. 3. No.4, pp. 341-350. 3. i,iug, F.F., (1990). "Fractals, Engineering Surt'aces and Tribology," Wear, 136. pp. 141-156. 4. Majumdar, A., and Bhushan, B., (1990). "Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces.," ASME J. of Tribology. Vol. II 2, pp. 205-216. 5. Zhou. G,. (19921, "Statistical. Random and Fractal Cltaracterizations of Surface Tol'x~graphy with Engineering Application,s," Ph.D. Dissertation, New Jersey Institute of Teclmology, NJ. 6. Zhou. G., M. C. Leu, and D. Blackmore, (1993a). "Fractal Characterization of Surface Topography and Implementation in Surface Contact," Submitted to ASME J. of Tribology. 7. Zhou, G.. M. C. Ix:u, and D. Blackmore, (1993b), "Fractal Geometry M(~el fl~r Wear Prediction." hnernational Journal of Wear, Vol. 170/1, pp. 1-14. 8. Sayles. R.S.. avKlThomas, T.R.. (19781, "Surface Topography as a Non-statiot~ary Random Process." Nature 271. pp. 431-434. 9. Jnrdan, D.L., Hollins, R.C.M., and Jakeman, E . (1986). "Measurement and Characterization of Multi-Scale Surfaces," Wear 109. pp. 127-134. 10. Mandelbrot. B.B., Passoja, D.E., and Paullay. A.J., (19841. "Fractal Character of Fracture Surfaces of Metals." Nature 308. pp. 1571-1572. II. Ling, F.F.. (19871. "Scaling Law tbr Contoured Length of Engineering Surfaces." J. Appl. Phys., Vol. 62(6). pp. 25702572. 12. Majuntdar. A., (19891. "Fractal Surta.ces and Their Applications to Surface Phenomena'. Ph.D. Thesis. University of California, Berkeley. 13. Gagnepain, J.J., alKI Rcvques-Carmes, C., (1986). "Fractal Approach to Two-Dimensional and Three-Dimensional Surface Roughness," Wear 109, pp. 119-126. 14. Kaye, B.H.. (19861, "The Description of Two-Dimensional Rugged Boundaries in Fine Particle Science by Means of Fractal Dimensions," Powder Technology. 46. pp. 245-254, 15. R(xtues-Carmes, C., Wehbi. D., Quiniou. J.F.. and Tricot, C.. (1988), "Modelling Engineering Surfaces and Evaluating Their Non-integer Dintension for Application in Material Science." Sur|ace Topography 1, pp. 435-443. 16. Dauw, D.F., Brown, C.A.. Griethuy~n. J., and Albert. J., (19901, "Surface Topography Investigations by Fractal Analysis of Spark-Eroded Electrically Conductive Ceramics." Annals of the CIRP, Vol. 39/I. 17. Mandelbrot, B.B., (19851, "Self-Affine Fractals and Fractal Dimen.sion." Physica Scripta 32, pp. 257-260. 18. Berry. M.V., (1979). "Dift~'actalx," Physics AI2, pp. 781-797. 19. Falconer. K.. (1990). Frat'tal Geometry:Mathematical Fundatio~ and Applications, John Wiley. 20. Blacknlore, D., and G. Zhou, (19941, "Derivation ot a General Fractal Distribution for Rough Surfaces." in preparation. 21. Abbott, E.J.. and Firestoue, F.A., (1933), "Specif3'ing Surface Quality." Mech. Engng.. 55. pp. 569-572. 22. Archard. J, F.. "Wear Theory and Mechanics." in Wear Control Handbook, Edited by Peterson, MB.. and Winter. W.O., ASME, New York, 1980. 23. Majumdar, A., and B. Bhushan, "Fractal Model of Elastic-Plastic Contact Between Rough Surfaces," ASME Journal of Tribology, Vol. 113, 1991. pp. 1-11. 24. Archard, J. F., "Contact and Rubbing of a Flat Surface." Applied Physics.. Vol. 24, 1953. pp. 981-988. 25. Zhou. G., M. C. Leu. and S. X. Dong. (1990). "Measurement and Assessment of Topography of Machined Surfaces," in Microstructural Evolution in Metal Proce~ing. Proceedings of ASME Winter Meeting. PED-Vol. 46. pp.89-100,