Frictional shakedown of a coupled continuous contact

The plane elastic problem of a circular disc shrink fitted into a hole in an elastically similar plane, with a frictional interface, is studied. The disc is subject to a radial force and torque both of which vary with time. In the steady-state the loads vary harmonically, and a relative phase shift is considered. Frictional slip is accounted for by edge dislocations distributed along the interface. It is shown that the steady-state response, which might, for example, be cyclic slip or full adhesion (shakedown) for a particular loading regime, depends on the loading history before the steady-state loading regime is entered. This contrasts with Melan's theorem for elastic-plastic shakedown, and is a consequence of the normal-tangential coupling in this frictional system.