the use of the fractal description to characterize engineering surfaces and wear particles

8/8/2019 The Use of the Fractal Description to Characterize Engineering Surfaces and Wear Particles

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Wear 255 (2003) 315326

The use of the fractal description to characterize engineeringsurfaces and wear particles

C.Q. Yuan a,, J. Li b, X.P. Yan a, Z. Peng c

a Reliability Engineering Institute, Yujiatou Campus, Wuhan University of Technology, Wuhan 430063, PR Chinab Wuhan Research Institute of Materials Protection, Wuhan 430030, PR China

c School of Engineering, James Cook University, Townsville, Qld 4811, Australia

Abstract

Fractals can be extremely useful when applied to tribology. Obtaining fractal descriptions of engineering surfaces and wear particlesrequires surface topography information to be measured, digitized and processed. Such procedures can be rigorous. This article compares

various methods to calculate profile and surface fractal dimension. Profile fractal dimension is computed using three available methods,

corresponding to the yard-stick, the power spectrum and the structure function method. The precision of the three methods is analyzed

and compared in this paper. Surface fractal dimension is calculated using the slit island and the box counting method. Both profile fractal

dimension and surface fractal dimension are used to describe TiN coating surfaces and wear particles.

2003 Elsevier Science B.V. All rights reserved.

Keywords: Tribology; Fractal; Surface; Wear particle

1. Introduction

A surface is composed of a large number of length scalesmutually superimposed roughnesses that are generally char-

acterized by the standard deviation of surface peaks [1,2].

Due to the multiscale nature of the surface, the variances and

derivatives of surface peaks and other roughness parameters

strongly depend on the resolution and the filter processing

of measuring instruments. Hence, measuring instruments of-

ten provide different values on surface roughness. Ideally,

rough surfaces should be characterized in such a way that

the structural information of roughness at all scales is re-

tained. To do so, quantifying the multiscale nature of surface

roughness is essential.

A unique property of a rough surface can be obtained by

its scale-independent measurement. This requires the rough-

ness reflecting the real surface structure at all magnifications.

The similarity of a surface profile under different magnifi-

cations can be statistically characterized by fractal geom-

etry since its topography is statistically self-affinity [35].

A typical sketch of fractal surface topography is shown in

Fig. 1. The ability to characterize surface roughness using

scale-independent parameters is a specific feature of frac-

Corresponding author.

E-mail addresses: [email protected], [email protected]

(C.Q. Yuan).

tal approach. Fractal analysis provides information of the

roughness at all length scales that exhibit fractal behavior.

Normally, a large number of wear particles, which can be ex-amined using ferrography and microscopy, are generated in

wear processes. Wear types can usually be analyzed accord-

ing to the experience of analysts, which may lead to large

subjective deviation and quantitative divergence. However,

researches have proved that the fractal character of the edge

profile and texture surface of wear particles may be quan-

tified reliably by fractal dimension using available fractal

theory [69].

In recent years, many researchers have applied the fractal

theory to the field of tribology. The calculation of fractal

dimension is the foundation of studying tribology using

the fractal theory. Generally speaking, it is a very compli-

cated process to calculate fractal dimensions of engineering

surfaces and wear particles because it involves not only

a number of mathematical models but also surface topog-

raphy images to be digitized and processed. Therefore, it

is important to develop a computer program or devices to

calculate and analyze the fractal information. Up to date,

the calculation of surface fractal dimension calculations is

available in some commercial surface instruments, such as

atomic force microscope (AFM) [10] and Talysurf PGI [11].

However, these instruments are very expensive. Further-

more, the function of calculating surface fractal dimension

attached with these instruments cannot be used to calculate

0043-1648/03/$ see front matter 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0043-1648(03)00206-0


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316 C.Q. Yuan et al. / Wear 255 (2003) 315326

Fig. 1. An engineering surface with fractal character.

the surface fractal dimension of surface data by other equip-

ments. Podsiadlo and Stachowiak [6], Kirk et al. [7] and Ge

and Suo [8] have conducted research in the development

of new methods to calculate fractal dimension. Although

each fractal calculating method has individual advantage

and disadvantage on practical application, these researches

mainly focused on using a single method to calculate the

profile fractal dimension (note: in this project, the profile

of an engineering surface means its surface profile in 2D

while the profile of a wear particle means its boundary).

There is very limited study on calculating fractal dimension

of engineering surfaces and wear particles. Furthermore,

fewer studies on applying both profile fractal dimension

and surface fractal dimension to characterize engineering

surfaces and wear particles have been conducted. When us-

ing profile fractal dimension and surface fractal dimension

separately, due to calculation errors, the profile or surface

fractal dimensions are sometimes the same for different

engineering surfaces or wear particles. It is helpful to avoidor decrease this phenomenon if profile fractal dimension

and surface fractal dimension are both used.

Based on above reasons, this project is to investigate dif-

ferent fractal calculating methods to seek a better method

for calculating profile fractal dimension and surface fractal

dimension. The software for calculating surface dimension,

which will be suitable for both direct 2D/3D data stored in

a data file and digital images, is developed in this project.

Then, both of profile fractal dimension and surface fractal

dimension are used to characterize TiN coating surfaces and

distinguish different types of wear particles.

2. Fractal calculation

Fractal calculation mainly includes the calculation of pro-

file fractal dimension (1 < D < 2) and the calculation of

surface fractal dimension (2 < D < 3) in tribological frac-

tal research. Fractal calculation is generally involved with

computer assisted image analysis of topography images in

2D or 3D of a surface obtained in analog or digital signals

using profilometer or microscopy, etc. An effective method

to convert these signals into the required data for calculating

fractal dimensions must therefore be sought.

2.1. Profile fractal calculation

Profile instruments can be used to obtain data in 2D, which

are then directly used to calculate fractal dimension. How-

ever, for the generic surface topography obtained using fer-

rography, SEM, and optical microscopy, etc. the information

need be preprocessed before it can be used to work out thefractal dimension of surface profiles and/or particle shapes.

In calculation of the fractal dimension of an image, surface

topography must be converted into a digital colorized image,

then into a gray image that may be thresholded optimally

into a binary image. Finally, profiles are obtained through

edge detection. Through tracking boundary, coordinates of

all points on the profile are gained and the fractal dimen-

sion of profile with programming can then be worked out

by using proper methods.

2.1.1. Principle

The methods for calculating profile fractal dimension

mainly include the yard-stick, the box counting, the varia-tion, the structure function and the power spectrum method

[8]. Generally, the fractal calculation is involved with

many mathematical models. Workload by manually calcu-

lating of fractal dimension is thus very heavy and results

obtained may not be accurate. This article provides a com-

puter program for auto-calculation of fractal dimension

using the yard-stick, the power spectrum and the struc-

ture function method, which incorporates with the data

gained from surface roughness measuring instruments,

and also from the digitized surface topography images

acquired using microscopes. The schematic approach of

calculating the fractal dimension of profile is illustrated inFig. 2.

2.1.1.1. Yard-stick method. The yard-stick method em-

ploys the technique of walking around a profile curve in

a step length, r. A point on the profile curve is chosen as

a starting point of divider, whilst another point at a dis-

tance r from the starting point is taken as its end point.

Repetitively, find the point-pair of dividers in the same

way until the profile curve is entirely measured. Then, the

summing up of the step lengths enables the curve length

to be determined. The repetition of this calculation pro-

cess at various step lengths allows all the curve length tobe evaluated. Further, plotting of the curve lengths verses

the step lengths on a loglog scale gives the slope m of

a fitting line to be related to the fractal dimension D as

[6,8]:

D = 1 m (1)

It is possible that this method has abandoned some pivotal

points, resulting in calculation error.

2.1.1.2. Power spectrum method. The modified Weier-

strassMandelbort (WM) function [9] for a rough surface


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C.Q. Yuan et al. / Wear 255 (2003) 315326 317

Fig. 2. Fractal calculation of profiles.

can be described as

z(x) = GD1

n=n1

cos2nx

(2D)n; 1 < D < 2, > 1 (2)

where D is the fractal dimension; n the discrete frequency

spectrum of the surface roughness; and nl the low cut-off

frequency of the profile and G the characteristic length scale

of the surface.

The mutiscale nature of z(x) can be characterized by its

power spectrum, which gives the amplitude of the roughness

at all length scales. The parameters G and D can be foundfrom the power spectrum of the WM function by

S() =G2(D1)

2 ln

1

52D(3)

where S() is the power of the spectrum, and is the fre-

quency of the surface roughness profile. Usually, the power

law behavior would result in a straight line if S() is plot-

ted as a function of on a loglog graph. Using fast fourier

transform (FFT), the power spectrum of profile can be calcu-

lated and then be plotted verses the frequency on a loglog

scale. Thereafter, the fractal dimension, D, can be related to

the slope m of a fitting line on a loglog plot as:

D = 12 (5+ m) (4)

2.1.1.3. Structure function method. This method considers

all points on the surface profile curve as a time sequence

z(x) with fractal character. The structure function s() of

sampling data on the profile curve can be described as [8]

s() = [z(x+ ) z(x)]2 = c42D (5)

where [z(x + ) z(x)]2 expresses the arithmetic average

value of difference square, and is the random choice value

of data interval. Different and the corresponding s() can

be plotted verses the on a loglog scale. Then, the fractal

dimension D can be related to m of a fitting line on loglog

plot as:

D = 12 (4m) (6)

Fractal dimensions of wear particles are usually calculated

using the yard-stick method although the above-described

methods can be also used to calculate the fractal dimension

of wear particles. Reports on other methods are few.

2.1.2. Precision analysis

The WM function curve is a standard fractal function

curve and is used to conduct the precision analysis in this

study. The WM function may be constructed to a form as

below [12]:

f(t) =

+k=

(D2)k sin(kt); > 1, 1 < D < 2 (7)

Nine WM fractal function curves generated by computer

with = 1.55 are shown in Fig. 3. Table 1 tabulates the

difference between the ideal fractal dimension and computed

fractal dimension using the above methods.

The values in Table 1 have clearly shown that the

yard-stick method, the power spectrum method and the

structure function method have relatively high reliability in

calculating the fractal dimension. The three methods have

individual advantages and disadvantages on calculating the

fractal values. Abandoning some pivotal points when using

the yard-stick method would create large error. Although

the power spectrum method uses all points in fractal calcu-

lation, the conversion of those discrete peaks to frequency

seems to be somehow an approximate approach. Compared

to the power spectrum method, the approximation in the

structure function method is relatively small because its

calculation of fractal information is directly based on the


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Fig. 3. Standard WM fractal function curves generated by computer.


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Fig. 3. (Continued).


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320 C.Q. Yuan et al. / Wear 255 (2003) 315326

Table 1

Comparison of computed fractal dimension with ideal fractal dimension

Ideal

fractal

dimension

Computed fractal

dimension using

yard-stick method

Computed fractal

dimension using

power spectrum

method

Computed fractal

dimension using

structure function

method

Relative error

of yard-stick

method (%)

Relative error of

power spectrum

method (%)

Relative error of

structure function

method (%)

1.10 1.07 1.12 1.16 2.73 1.82 5.45

1.20 1.28 1.20 1.26 6.67 0.00 5.00

1.30 1.29 1.28 1.38 0.77 1.54 6.15

1.40 1.52 1.36 1.46 7.86 2.86 4.29

1.50 1.57 1.44 1.54 4.67 4.00 2.67

1.60 1.65 1.52 1.62 3.13 5.00 1.25

1.70 1.69 1.62 1.71 0.59 4.71 0.59

1.80 1.86 1.73 1.81 3.33 3.89 0.56

1.90 1.88 1.84 1.90 1.05 3.16 0.00

peak information. However, its subtracting approach for the

very close points does bring some computation error in the

calculation. According to the data in Table 1, the relative er-

ror of the power spectrum method is less than three percentwhen the fractal dimension of profile is lower than 1.5. The

relative error of the structure function method is less than

three percent when the fractal dimension of profile is greater

than or equal to 1.5. However, for the yard-stick method, the

computed relative error is waved all through the range of the

fractal dimension. This study has revealed, for the profile

of fractal dimension greater than or equal to 1.5, the struc-

ture function method should be preferred because it gives

calculations with a relatively higher precision than the oth-

ers. For the profile of fractal dimension lower than 1.5, the

power spectrum method provides the best precision among

the three. Therefore, when the calculated fractal dimension

is lower than 1.5, the fractal dimension using the power spec-

trum method will be adopted. When the calculated fractal

dimension is greater than or equal to 1.5, the fractal dimen-

sion by the structure function method will be adopted. All

fractal dimensions calculated using the yard-stick method

are only references due to the relative poor accuracy of the

method.

2.2. Surfae fractal calculation

Calculating the fractal dimension of profiles is compara-

tively easy. But the fractal dimension of profiles is usually

not sufficient to describe surface topology in 3D and the

fractal dimension of surfaces (2 < D < 3) must be used.

Calculation of surface fractal dimension is often involved

with a large amount of data. It is necessary to work out the

fractal dimension of surface using computer assisted analy-

sis techniques.

2.2.1. Principle

The methods for calculating surface fractal dimension

mainly include the box counting method, the slit island

method (SIM) and the project method, etc. This article fo-

cuses on the box counting method and the slit island method.

The slit island method is often applied to calculate surface

fractal dimension based on 3D surface data files while the

box counting method is used to calculate surface fractal di-

mension based on surface images. The diagram of calculat-ing surface fractal dimension in this study is illustrated in

Fig. 4.

2.2.1.1. Slit island method (SIM). SIM is widely applied

to calculate the fractal dimension of engineering surfaces.

Generally, for anomalistic topography with fractal character,

the relation of the perimeter P and area S is described below

[13]

P1/D S1/2 (8)

where D is fractal dimension. The logarithm converting of

formulation (8) islog P= 0.5D log S+ C (9)

where C is the constant parameter.

The surface fractal dimension is obtained by fitting the

slope of the line on the double logarithm chart of P and

S. To calculate the surface fractal dimension, firstly 3D

data is inputted into a computer and consequently the

three-dimension surface fitting equation is obtained by data

processing. According to the equation, the surface is inter-

cepted in the direction of paralleling the surface. Then, a

series of perimeter P and area S is obtained, and a series of

sectional fractal dimension d1

, d2

, . . . , d n

are obtained us-

ing the fractal calculation methods of profiles. The structure

function method is adopted in this study. Finally, the final

fractal dimension D is determined by the follow formula:

D = 1+

ni=1

di

n

n = 30 is usually sufficient.

2.2.1.2. Box counting method. The box counting method

is used to calculate surface fractal dimension based on

surface images. For fractal assemble ARn, the fractal


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C.Q. Yuan et al. / Wear 255 (2003) 315326 321

Fig. 4. Fractal calculation of surfaces.

dimension based on the box counting method defined by

Pentland [14] is as follows:

D = limr0

log Nr

log(1/r)(11)

Here, Nr is a minimum assemble number needed to cover

target object in r diametric assemble. The box counting

process suggested by Sarker [15] has been widely used.It is expressed as follows: suppose a three-dimensional

image has M M pixel and G gray level progression,

presenting in a form of (x, y, z). The first dimension

and the second dimension are the position of a pixel in

a two-dimensional image plane and the third dimension

shows its gray scale. When reducing its scale to S S

(M/2 S > 1, S is integer) using a proportion factor of

r (r = S/M) in the two-dimensional plane, the gray level

expressed by the third dimension also reduces in accord-

ing proportion. So, the volume of each box is S S S,

where the new gray level progression S meets the following

formula:G

S

=

M

S

(12)

[G/S] is the minimum integer greater than G/S. [M/S] is

the minimum integer greater than M/S. The space ofMM

is composed of a series boxes with the space of S S.

Suppose that in an i j area the minimum and maximum

gray level grade is, respectively, dropped in the area of no. K

and no. L box according to the new gray level progression,

so the box quantity cover the i j area is:

nr(i, j) = L K + 1 (13)

The box quantity needs to cover the whole target object is:

Nr =

Mi,j

nr(i, j) (14)

Thus, the fractal dimension of image can be estimated by

formula (11).

The effect of a number of selected rhas been studied [14],and the results have shown that the mean square slope value

of logNr and log(1/r) are suitable to be used to calculate the

fractal dimension. The studies have further suggested that

S= 2i should be used, where i is integer (2 S M/2).

The project uses this method, and the main procedure of

calculation is as follows:

(1) initialize two registers, Imax and Imin, to store the maxi-

mum and minimum gray level grade of the surface im-

age, respectively;

(2) set S= 2, r = S/M;

(3) calculate the S using formula (12);

(4) reduce the image map in a scale of 1/4. Use the formulas,Imaxi/2,j/2 = max(I

maxi,j , I

maxi+1,j, I

maxi,j+1, I

maxi+1,j+1), I

mini/2,j/2 =

max(Imini,j , Imini+1,j, I

mini,j+1, I

mini+1,j+1), to calculate the new

maximum and minimum gray level;

(5) calculate nr and Nr using formulas (13) and (14);

(6) double S, go to step 3 until S= M/2;

(7) calculate the fractal dimension using the mean square

slope value of logNr and log(1/r).

2.2.2. Precision analysis

Compared with the methods to calculate the profile frac-

tal dimension, it is more difficult to conduct the precision


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322 C.Q. Yuan et al. / Wear 255 (2003) 315326

Fig. 5. The 3D surface topographies from 3D surface data files.

analysis of calculating surface fractal dimension. This

project uses auto-correlation coefficient to evaluate the pre-

cision of the above two methods in calculating surface frac-

tal dimension. The auto-correlation coefficient [16] reflects

a statistical trend. Particularly in this study, it measures the

degree of the deviation between a straight fitting line and

the data points corresponding to the fractal dimensions.

The 3D data stored in data files and extracted from digital

images are used to carry out the precision analysis in this

project. The 3D surface topographies from 3D data files are

shown in Fig. 5 and the 3D surface topographies from opti-

cal images are shown in Fig. 6. Table 2 tabulates the results

of the surface fractal dimension calculated by the above

methods.

Table 2

Surface fractal dimensions calculated by the above methods on Figs. 5 and 6

Figures Slit island method Auto-correlation coefficient

of slit island method

Box counting

method

Auto-correlation coefficient

of box counting method

Fig. 5a 2.43 0.97

Fig. 5b 2.36 0.94

Fig. 5c 2.41 0.99

Fig. 6a 2.63 0.92

Fig. 6b 2.66 0.98

Fig. 6c 2.58 0.91

The results in Table 2 show that it is viable to use the

slit island method to calculate the surface fractal dimen-

sion when surface data files are available. The box counting

method can be used to calculate the surface fractal dimen-

sion based on surface images. More work need to be car-

ried out to develop a better method to evaluate the precision

analysis in calculating surface fractal dimension.

3. Fractal dimensions of TiN coating surface and wear

particles

The study of a traditional TiN coating surface in slid-

ing contacts of the rough surface was based on the surface


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C.Q. Yuan et al. / Wear 255 (2003) 315326 323

Fig. 6. The 3D surface topographies from optical surface images.

roughness in two dimensions. Since fractal theory provides

a new approach to describe the irregular performance [17],

researchers have shown that the fractal analysis is better

in representing the true contact and the variance in a tri-

bological surface pairs [1820]. The fractal methods have

successfully been employed to characterize non-coated sur-

faces. However, very limited quantitative study has yet been

performed to characterize coated surfaces, although study

[21] has indicated that TiN coated surfaces exhibit fractal

character.

Fig. 7. The surface images of the turned substrate before and after depositing TiN layer.

Surface images of substrate, respectively, machined by

fine turning, grinding and mirror polishing before and af-

ter TiN deposit process are shown in Figs. 79. These im-

ages, post of properly transformed, are used to work out the

fractal dimension using the above analyzing methods. Their

changes in fractal dimension D before and after depositing

TiN layer are clearly shown in Table 3.

The results of fractal dimension calculation have shown

that the surfaces of TiN coating can be characterized using

the fractal theory and the changes in fractal dimension for


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324 C.Q. Yuan et al. / Wear 255 (2003) 315326

Fig. 8. The surface images of the grinded substrate before and after depositing TiN layer.

Fig. 9. The surface images of the polished substrate before and after depositing TiN layer.

the various machined substrates are different. For the sub-

strate fine turned, both profile fractal dimension and surface

fractal dimension increase after depositing TiN coating. For

the substrate grinded, the profile fractal dimension increaseswhilst the surface fractal dimension decreases after deposit-

ing TiN coating. For the substrate polished, the profile frac-

tal dimension decreases whilst the surface fractal dimension

increases after depositing TiN coating.

Shape profiles of the images of wear particles can be fully

characterized by edge curves. Fractal dimension can not only

describes the boundary features but also the surface textures

of wear particles.

Figs. 10ad and 11 showthe typical shapes of four types of

wear particles: (a) a rubbing particle; (b) a triangle particle of

Table 3

Changes in fractal dimension D of various substrates after depositing TiN layer

Fractal dimension

using yard-stick

method

Fractal dimension

using power

spectrum method

Fractal dimension

using structure

function method

Fractal dimension

using slit island

method

Fractal dimension

using box counting

method

Fine turning Before TiN coating 1.65 1.63 1.62 2.27 2.19

After TiN coating 1.67 1.68 1.67 2.31 2.37

Grinding Before TiN coating 1.70 1.76 1.70 2.44 2.39


Polishing Before TiN coating 1.58 1.59 1.42 2.71 2.67


Note: The bolded figures in Table 3 are the best fractal dimension according to the precision analysis.

abrade wear; (c) a fatigue particle and (d) a cutting particle.

Their corresponding fractal dimensions calculated using the

above discussed methods are tabulated in Table 4.

The boundary fractal dimension combined with the sur-face fractal dimension shown in Table 4 can be regarded as

a shape and topological parameter of wear particles for dis-

tinguishing different types of wear particles and wear.

This project has investigated a large number of wear par-

ticles in different types. Statistically, five typical types of

wear particles have the following characteristics. For the

rubbing particle, both its boundary fractal dimension and

surface fractal dimension are small. The boundary fractal

dimension of the fatigue particle is medium whilst its sur-

face fractal dimension is high. The severe sliding particle


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C.Q. Yuan et al. / Wear 255 (2003) 315326 325

Fig. 10. Four different shapes of wear particles from Fig. 11.

Fig. 11. Ferrography images of wear particles.

Table 4

Fractal dimensions of wear particles calculated using different methods

Figures Fractal dimensionusing yard-stick

method

Fractal dimensionusing power

spectrum method

Fractal dimensionusing structure

function method

Fractal dimensionusing slit island

method

Fractal dimensionusing box

counting method

Fig. 10a 1.17 1.18 1.21 2.15 2.11

Fig. 10b 1.13 1.14 1.22 2.47 2.49

Fig. 10c 1.27 1.29 1.31 2.51 2.54

Fig. 10d 1.91 1.87 1.94 2.13 2.12

Note: The bolded figures in Table 4 are the best fractal dimension according to the precision analysis.

has a high value in both the boundary fractal dimension and

the surface fractal dimension. For the laminar particle, its

boundary fractal dimension is small and its surface fractal

dimension is medium. The cutting particle of abrasive wear

has a very high boundary fractal dimension, but its surface

fractal dimension is small.

4. Conclusion

Fractal dimension can be used to characterize the sur-

face topography of engineering surfaces and wear particles.

Results of this paper allow the following conclusions to be

drawn:

(1) Different fractal calculation methods have their

individual advantages and disadvantages. The routine

calculation of fractal dimension using fractal models

with image processing technology is feasible. However,

the improvement of the veracity of fractal dimension

evaluation is still increasingly attracting a great deal of

interest.

(2) TiN coating surfaces in sliding contacts give fractal

character. The surfaces of depositing TiN on various

processing substrates are characterized by fractal di-

mension. The fractal dimension, D, of TiN coating

is scale-independent and related to fractal calculation

methods. Fractal dimension is regarded as a parameter

to characterize the shape and surface features of wear

particles to distinguish different wear types.

(3) Fractal theory application on tribology is worth for

studying the intrinsic law of surface topography and

wear particles.


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326 C.Q. Yuan et al. / Wear 255 (2003) 315326

Acknowledgements

The authors would like to express their sincere gratitude to

National Natural Science Foundation of China for its fund-

ing this research project. A study on the Wear Mechanism

of the Friction and Creep Composite Wear and the Tribo-

logical Design for the Surface Topography of Thermoplas-tic Polymers (No.: 50175041) and Research on Intelligent

Interpretative Method of Condition Feature in Tribo-system

(No.: 50275111).

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