A New Method Developed for Fractal Dimension and Topothesy Varying With the Mean Separation of Two Contact Surfaces

Instead of a general consideration of the fractal dimension $(D)$ and the topothesy $(G*)$ as two invariants in the fractal analysis of surface asperities, these two roughness parameters in the present study are varied by changing the mean separation $(d*)$ of two contact surfaces. The relationship between the fractal dimension and the mean separation is found first. By equating the structure functions developed in two different ways, the relationship among the scaling coefficient in the power spectrum function, the fractal dimension, and topothesy of asperity heights can be established. The variation of topothesy can be determined when the fractal dimension and the scaling coefficient have been obtained from the experimental results of the number of contact spots and the power spectrum function at different mean separations. A numerical scheme is developed in this study to determine the convergent values of fractal dimension and topothesy corresponding to a given mean separation. The theoretical results of the contact spot number predicted by the present model show good agreement with the reported experimental results. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. Significant differences in the contact load or the total contact area are shown between the models of constant $D$ and $G*$ and variable $D$ and $G*$ as the mean separation is reduced to smaller values.