Elasto-plastic contact problems with heat exchange and fatigue

We deal with a one-dimensional temperature dependent model for fatigue accumulation in a moving visco-elasto-plastic material in contact with an elasto-plastic obstacle. The problem for the unknown displacement and temperature is formulated using hysteresis operators as solution operators of the underlying variational inequalities. The existence result for this problem, consisting of the momentum and energy balance equations and an evolution equation for the fatigue, is obtained using a priori estimates established for the space discretized problem. The uniqueness result follows from the Lipschitz continuity of the nonlinearities.