Optimal temperature paths for thermorheologically simple viscoelastic materials with constant Poisson’s ratio are canonical

In this note we discuss the thermal stress problem for a thermorheologically-simple linearly-viscoelastic body, subjected to a spatially-uniform temperature field and homogeneous boundary conditions, assuming that Poisson’s ratio is constant and inertia negligible. In particular, we consider the following optimization problem: of all temperature paths $\theta (t), 0 \le t \le {t_f}$, which belong to a given function class, is there one which renders a given stress measure a minimum at time ${t_f}$. We show that a resulting optimal path $\theta \left ( t \right )$ (if it exists) is canonical: $\theta \left ( t \right )$ is independent of the shape of the body and of the particular homogeneous boundary conditions.