1. Multi-scale Tribology Laboratory
An Analysis of Generated Fractal and Measured Rough Surfaces
Xiaohan Zhang
Yang Xu
Robert L. Jackson
Multi-scale Tribology Laboratory
Auburn University
Auburn, AL

2. Motivation
Fractal methods have been developed to characterize the complex structures of rough surfaces. This simplifies complex surfaces to a fractal geometry that depends heavily on the fractal dimension, D. The aim of this work is to analyze the effectiveness of the fractal dimension methods in characterizing first generated and then measured real rough surfaces. 1

3. Introduction
Rough surfaces are multi-scale in nature, which means surfaces usually have scales of roughness that span from the macro scale to the atomic scale .
Multi-scale nature of surface profile 2

4. Introduction Fractal defines how surface changes with the scale.
Self-similarity
Fractal
Self-affinity π π 2π 2π
Fractal self-similar
Fractal self-affine 3

5. Fractal characterization
Four different methods are implemented in calculating the fractal dimension value:
a) Box-counting method:
b) PSD method: 4

6. Generated methods
c) Roughness-length method: d) Variogram method or structure function method:
The relationship between Rq and w is plotted on a log-log scale, the slope of the plot can be used to calculate the fractal dimension.
A large number of pairs of points separated by different lateral distance (τ) along the profile are chosen, the differences in height between two positions are calculated (z);
Calculating the sum of squares of all the height differences and then computing the average value of the sum;
The fractal dimension can be derived from the log-log plot of the average values s(τ) versus lateral distances (τ). 5

7. Surface Profile Generation Methods
1. W-M function
Surface profile generated by the W-M function 6

8. Surface Profile Generation Methods
Fractal dimension of the correct values and values obtained from four methods
Roughness-length method can result in a relatively precise prediction;
The variogram method can only make a good prediction when the fractal dimension is larger than 1.5;
The box-counting method cannot predict the value of the fractal dimension very well when it is too large;
The results for the PSD method are very poor, and differ greatly from the other methods. 7

9. Surface Profile Generation Methods
2. PSD-iFFT (inverse Fourier transform based on a prescribed Power Spectral Density)
Surface profile generated by PSD-iFFT method 8

10. Surface Profile Generation Methods
Note: The relationship between the Hurst exponent and fractal dimension is D = 2-H; and since the profiles are generated with the PSD, the fractal dimension values calculated by the PSD method are exactly the same as the correct values
Fractal dimension of the correct values and values obtained from four methods
Differences between the correct values and values calculated by roughness-length method are relatively small;
The variogram method also appears successful at high values of D, as it also did for the W-M function;
The box-counting method only works for low values of D. 9

11. + Experimental setup Profilometer stylus tip radius: 2 micrometer
Vertical resolution: 1 nm + =
Representative surface profile 10

12. Results 1. Fractal dimension values for measured surface profiles
Fractal dimension values for real rough surface profiles
According to the roughness-length method, the fractal dimension values appear to be less than 1.5 and often close to 1;
The variogram method cannot be considered accurate because D<1.5;
The box-counting method and the roughness-length method appear to be in reasonable agreement for most of surface profiles;
The PSD method differs greatly for all the other methods. 11

13. Results 2.Fractal dimension value for self-similar surface profiles
B-asperity aspect ratio
Δ- amplitude
λ - wavelength
Relationship between B and λ
For a perfectly self-similar rough surface profile, when B is plotted versus λ, it results in a straight horizontal line, which means no matter how the scale changes, B is a nominally constant over the scales and the dB/dλ value is zero. 12

14. Inconsistent with D a) W-M function (D = 1) b) PSD-iFFT (D = 1.5)
Only when D = 1 is B a constant and the surface profile is self-similar
b) PSD-iFFT (D = 1.5)
Only when D = 1.5 is B a constant and the surface profile is self-similar 13

15. Results 3. Parameters for real rough surface profiles
Averaged parameters calculated for the measured rough surface profiles
All the averaged dB/dλ values for these six rough surfaces tend to be close to zero, so all the measured rough surface profiles are nominally self-similar. 14

16. Conclusions
1. The fractal dimension, D, for a self-similar surface profile generated through the PSD-iFFT method is shown to be 1.5, but for a W-M function generated self-similar profile it is 1. These two values contradict with each other, which means the fractal dimensions in these two methods are not defined the same. Its usage to characterize surfaces should be considered carefully and critically. 2. The roughness-length method is the most effective method in calculating the fractal dimension values by comparing with the other three methods used in this work. In addition, the PSD method always predicts an erroneously large value of fractal dimension for measured surfaces. 3. All the measured rough surfaces appear to be nominally self-similar, and not clearly self-affine as many of the popular fractal models assumed. 4. The measured surfaces appear to be self-similar when D=1. 15

17. This project has already published, which is: X. Zhang, Y. Xu, R. L
This project has already published, which is: X. Zhang, Y. Xu, R. L. Jackson, “ An analysis of generated fractal and measured rough surfaces in regards to their multi-scale structure and fractal dimension”, Tribology International, 105, , 2017. 16