I was curios about Hausdorff dimensions. They seem to neatly describe rough surfaces. So I was wondering if there are common applications of Hausdorff dimensions in things like complicated friction research? I have a little bit of a desire to learn specifically about formation of cracks in surfaces, which seem as though friction and perhaps a little chaos might have an intuitive factor in their formation.
Fifteen years ago, there was much research interest in the applicability of fractals to the description of fractures in rocks. In particular, as a mathematical technique for modeling fracture populations, fracture networks, or fracture surfaces in systems exhibiting brittle-failure, fluid flow through porous rock, and sliding friction. I wouldn't say any of the applications became 'common' however, you probably still find most references to this in academic papers. Rather than searching for the term Hausdorff dimensions, I suggest searching for some combination of "fractal" and "fracture." At the time, much of this work was motivated (paid for) by interest in geothermal energy extraction and development.