  We present a model for rate-independent, unidirectional, partial damage in
visco-elastic materials with inertia and thermal effects. The damage process is
modeled by means of an internal variable, governed by a rate-independent flow
rule. The heat equation and the momentum balance for the displacements are
coupled in a highly nonlinear way. Our assumptions on the corresponding energy
functional also comprise the case of the Ambrosio-Tortorelli phase-field model
(without passage to the brittle limit). We discuss a suitable weak formulation
and prove an existence theorem obtained with the aid of a (partially) decoupled
time-discrete scheme and variational convergence methods. We also carry out the
asymptotic analysis for vanishing viscosity and inertia and obtain a fully
rate-independent limit model for displacements and damage, which is independent
of temperature.
