A Fractal Theory of the Temperature Distribution at Elastic Contacts of Fast Sliding Surfaces

The statistical temperature distribution at fast sliding interfaces is studied by characterizing the surfaces as fractals and considering elastic deformation of the asperities. The fractions of the real contact area in the slow, transitional, and fast sliding regimes are determined based on the microcontact size distribution. For a smooth surface in contact with a rough surface, the temperature rises at the real contact area are determined under the assumption that most of the frictional heat is transferred to one of the surfaces. The interfacial temperature rises are bounded by the maximum temperature rise at the largest microcontact when the fractal dimension is 1.5 or less, and are unbounded when it is greater than 1.5. Higher temperature rises occur at larger microcontacts when the fractal dimension is less than 1.5, and at smaller microcontacts when it is greater than 1.5. For a fractal dimension of 1.5, the maximum temperature rise at a microcontact is independent of its size. The maximum temperature rise at the largest microcontact is expressed as a function of the friction coefficient, sliding speed, elastic and thermal properties, real and apparent contact areas, and fractal parameters. The closed-form solutions for the distribution density function of the temperature rise can be used to calculate the fraction of the real contact area of fast sliding surfaces subjected to temperature rises in any given range. The present theory is applied to boundary-lubricated and dry sliding contacts to determine the fractions of the real contact area where lubricant degradation and thermal surface failure may occur.