ONE-DIMENSIONAL THERMOELASTIC CONTACT WITH A STRESS-DEPENDENT RADIATION CONDITION

A one-dimensional quasistatic thermoelastic contact problem with a stress-dependent boundary condition is considered. The problem models the evolution of the temperature and the displacement of a long thin elastic rod that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. The local existence of a strong solution to the problem and local dependence on the initial-boundary data are proved. In addition, the uniqueness of the solution is established. The proof rests on an abstract result dealing with perturbations of monotone operators, as well as some a priori estimates which permit an application of Schauder's fixed point theorem.