Determination of the Fractal Scaling Parameter From Simulated Fractal-Regular Surface Profiles Based on the Weierstrass-Mandelbrot Function

Abstract

A fractal dimension and a fractal roughness parameter are usually used to characterize a fractal surface. For a fractal-regular surface, a fractal domain length is also included. Such a formulation is based on an approximation using a constant value of the fractal scaling parameter that represents the ratio of the spatial frequencies of adjacent harmonic components in the Weierstrass–Mandelbrot (W-M) function. Although there were some reasons for assuming a constant value of 1.5 for the fractal scaling parameter, it is still left more or less arbitrary to adopt this assumption in fractal modeling of solid contact. In the present study, the fractal scaling parameter was treated as a variable rather than a constant by using a form of the W-M function with randomized phases based on a random walk formulation. A simple numerical scheme with clear graphical interpretation was developed to determine the value of the fractal scaling parameter. The fractal dimension, fractal roughness parameter, and fractal scaling parameter were all recovered with reasonable accuracy from numerically generated surface profiles.