On the Fractal Dimension of Rough Surfaces

Most natural surfaces and surfaces of engineering interest, e.g., polished or sandblasted surfaces, are self-affine fractal over a wide range of length scales, with the fractal dimension Df=2.15±0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\mathrm{f}} = 2.15\pm 0.15$$\end{document}. We give several examples illustrating this and a simple argument, based on surface fragility, for why the fractal dimension usually is <2.3. A kinetic model of sandblasting is presented, which gives surface topographies and surface roughness power spectra in good agreement with experiments.