  The modeling of cracks has been an intensely researched topic for decades -
both from the mechanical as well as from the mathematics point of view. As far
as the modeling of sharp cracks/interfaces is concerned, the resulting free
boundary problem is numerically very challenging. For this reason, diffuse
approximations in the sense of phase-field theories have become very popular.
Within this paper, the focus is on rate-independent damage models. Since the
resulting phase-field energies in general are non-convex, we are faced with a
discontinuous evolution of the phase-field variable. Solution concepts have to
be carefully chosen in order to predict discontinuities that are physically
reasonable. We focus here on the concept of balanced viscosity solutions and
develop a convergence scheme that combines alternate minimization with a local
minimization ansatz due to Mielke/Efendiev, [EM06]. We proof the convergence of
the incremental solutions to balanced viscosity solutions. Moreover, the
discretization concept is implemented and several carefully selected examples
show the performance of this combined approach. Particularly, the effect of
different norms and arc-length parameters in the local minimization scheme is
investigated.
