A General Fractal Distribution Function for Rough Surface Profiles

Starting with a very general functional description, involving fractal parameters, of the height along a given line on a rough surface, a distribution function for the corresponding surface profile is derived. This distribution is found to differ from Gaussian form by a convergent power series and to be directly dependent on two fractal parameters: the fractal dimension and topothesy. It is shown how the distribution function can be used to determine the effects of varying the fractal parameters on the height of the bearing-area curve (a standard measure of surface roughness). By truncating the series representation for the distribution function for the surface profiles, two approximate models for the height distribution are obtained. These models are shown to compare favorably with experimentally obtained distributions.

[1]  Ming-Chuan Leu,et al.  Fractal geometry model for wear prediction , 1993 .

[2]  J. Bosse,et al.  Caractérisation fractale de la rugosité tridimensionnelle d'une surface , 1994 .

[3]  M. Berry,et al.  On the Weierstrass-Mandelbrot fractal function , 1980, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[4]  Emanuel Parzen,et al.  Modern Probability Theory And Its Applications , 1962 .

[5]  F. F. Ling.  Fractals, engineering surfaces and tribology , 1990 .

[6]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[7]  P. Nayak,et al.  Random Process Model of Rough Surfaces , 1971 .

[8]  Emanuel Parzen,et al.  Modern Probability Theory And Its Applications , 1960 .

[9]  B. Bhushan,et al.  Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces , 1990 .

[10]  R. A. Fox,et al.  Introduction to Mathematical Statistics , 1947 .

[11]  B. Mandelbrot.  Fractal Geometry of Nature , 1984 .

[12]  A. Majumdar,et al.  Fractal characterization and simulation of rough surfaces , 1990 .

[13]  Ming-Chuan Leu,et al.  Fractal geometry modeling with applications in surface characterisation and wear prediction , 1995 .

[14]  M. Longuet-Higgins.  The statistical analysis of a random, moving surface , 1957, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[15]  F. F. Ling.  Scaling law for contoured length of engineering surfaces , 1987 .

[16]  J. Peklenik,et al.  Paper 24: New Developments in Surface Characterization and Measurements by Means of Random Process Analysis: , 1967 .

[17]  B. Mandelbrot,et al.  Fractal character of fracture surfaces of metals , 1984, Nature.

[18]  S. James Taylor,et al.  Mathematical Proceedings of the Cambridge Philosophical Society The measure theory of random fractals , 2022 .

[19]  R. Sayles,et al.  Surface topography as a nonstationary random process , 1978, Nature.

[20]  C. Roques-carmes,et al.  Fractal approach to two-dimensional and three-dimensional surface roughness , 1986 .