Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions; if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V(0) above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V>V(0) there are some initial conditions from which the contact pressure grows without limit in time, whereas for V<V(0) the system will always tend to one of the stable steady states
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