The modeling of brittle fracture has been a research topic of great interest for decades – both in the mechanical as well as in the mathematical community. In the past, the modeling of sharp cracks and the resulting free boundary problem posed significant numerical challenges. These difficulties have led to the rise of diffuse approximations in the sense of phase-field theories, which have become very popular, cf. [1]. This talk focuses on the numerical implementation of rate-independent phase-field damage models. In the case of rate-independence, phase-field models are characterized by (incrementally defined) non-convex energies. Such energies in turn lead to a time-discontinuous evolution of damage. Different mathematical solution concepts have been proposed for such an evolution, cf. [2], which are not equivalent in general. One mathematically and physically sound concept is the adaptive time-discrete scheme proposed by Efendiev & Mielke, cf. [3]. Within this talk, an efficient and robust finite element implementation of framework [3] is outlined. A detailed analysis of the physics induced by the Efendiev & Mielke scheme is given. REFERENCES [1] B. Bourdin, G.A. Francfort and J-J. Marigo, Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, Vol. 48, pp. 797-826, 2000. [2] D. Knees, Convergence analysis of time-discretisation schemes for rate-independent systems, ESAIM: Control Opt. Calc. Var., Vol. 25, 2019 [3] M. A. Efendiev, A. Mielke, On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity, J. Convex Anal., Vol. 13, pp. 151-167, 2006 [4] S. Boddin, F. Rörentrop, D. Knees and J. Mosler, Approximation of balanced vis- cosity solutions of a rate-independent damage model by combining alternate mini- mization with a local minimization algorithm, arXiv, 2022