Analytic solution for the limiting shape of profiles due to fretting wear

We consider fretting wear due to tangential oscillations of two contacting bodies. For small oscillation amplitudes, the wear occurs only in a circular slip zone at the border of the contact area. With increasing number of cycles, the wear profile tends to a limiting form, in which no further wear occurs. Under assumption of a constant coefficient of friction, the limiting form of the wear profile does not depend on the particular wear criterion and can be calculated analytically. An explicit analytic solution is presented for arbitrary initial shape and illustrated for the cases of parabolic and conical shapes.