This article is the third one in a series of papers by the authors on
vanishing-viscosity solutions to rate-independent damage systems. While in the
first two papers [KRZ13, KRZ15] the assumptions on the spatial domain $\Omega$
were kept as general as possible (i.e. nonsmooth domain with mixed boundary conditions), we
assume here that $\partial\Omega$ is smooth and that the type of boundary
conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields.
Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system.
Furthermore, in comparison to [KRZ13, KRZ15], in our vanishing-viscosity analysis we do not
switch to an artificial arc-length parameterization of the trajectories but we
stay with the true physical time. The resulting concept of Balanced
Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behavior at time discontinuities of the solution.