followed by a discussion which aims to explain why the latter alone is not sufficient to ensure the proper behavior of the model. In Section 5, we propose a new parametric family of degradation functions that aims to increase the accuracy of phase-field simulations by addressing key issues discussed in the previous section. Superiority of the resulting formulation over the classical model employing quadratic degradation is demonstrated via several numerical examples in Section 6. Finally, concluding remarks and outlook are given in Section 7.

Theoretical aspects of phase-field modelling
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The phase-field framework was first introduced by [@Fix1983] and [@Langer1986] for modeling phase transitions in materials, and later extended to free discontinuity problems by [@Ambrosio1990] who worked on image segmentation. Its specific application to crack propagation in solids is much more recent, and is the result of independent work by researchers coming from the fields of physics [@Aranson2000; @Karma2001] and applied mechanics [@Bourdin2000]. We adapt the latter viewpoint in this study, and furthermore note that while the original formulation introduced by Bourdin et al. has remained virtually unchanged in current usage, the argument on what constitutes proper solutions to the resulting equations as well as the meaning of key quantities 