to the category of continuum approaches for fracture propagation, utilizing a diffuse representation of cracks in place of actual discontinuities. The amount of crack regularization is controlled via a prescribed length scale $\ell$, which constitutes an additional parameter of the model.

The aim of the present work is to address two long standing issues associated with the phase-field formulation that arise in conjunction with use of the now-classical quadratic degradation function. The first has to do with premature stiffness degradation stemming from the evolution of damage around regions of stress concentration. The second and more serious issue deals with the observed dependence of simulated failure loads on the phase-field regularization parameter, a phenomenon that has largely gone unexplored in the literature until very recently. The problem is most noticeable in cases of crack growth emanating from a notch and undermines the usefulness of phase-field approaches in solving problems that involve crack initiation at a priori unknown locations [@Klinsmann2015]. The parameter $\ell$ was initially introduced by [@Bourdin2000] as a purely mathematical construct allows for the Griffith energy corresponding to a discrete crack to be recovered in the limit as $\ell$ goes to zero, in the sense of $\Gamma$-convergence [@Braides2006]. Of the 