  The aim of this paper is to prove existence of weak solutions of
hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with
mixed boundary conditions describing, for instance, damage processes and
elasticity with inertia terms. To this end, a suitable weak formulation to deal
with such evolution inclusions in a non-smooth setting is presented. Then,
existence of weak solutions is proven by utilizing time-discretization,
$H^2$-regularization of the displacement variable and variational techniques
from [C. Heinemann, C. Kraus: Existence results of weak solutions for
Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl.
21 (2011), 321--359] to recover the subgradients after the limit passages.
