strategy wherein the minimization is instead performed on an approximation of the Griffith functional having regularized jump sets so that $\bm{u}$ is continuous over the entire domain. This was inspired by the earlier work of @Ambrosio1990 [@Ambrosio1992] who solved a similar problem in image segmentation involving the functional of [@Mumford1989]. A scalar order parameter known as the *crack phase-field* is introduced to interpolate between fractured and intact regions, with the amount of regularization controlled by a characteristic length parameter denoted by $\ell$. The validity of such as strategy rests on whether the regularized approximation tends towards the original functional as $\ell$ goes to zero, in the sense of $\Gamma$-convergence [@Braides2006]. Although an additional equation governing the phase-field evolution must now be solved along with the linear momentum equation, the main advantage of this approach is that numerical solutions may be obtained via classical finite element algorithms as both $\bm{u}$ and the phase-field are continuous. [@Bourdin2000]’s regularization of took the form $$\label{eq:bourdinFunctional}
    \Psi \left( \bm{u}, \phi \right) = \int_\Omega \frac{1}{2} \left[ \left( 1 - \phi \right)^2 + \kappa \right] \bm{\varepsilon} \left( \bm{u} \right) : \mathbb{C}^e : \bm{\varepsilon} \left( \bm{u} \right) {\,\mathrm{d}}\Omega + \mathcal{G}_c \int_\Omega \left( \frac{1}{2\ell} \phi^2 + \frac{\ell}{2} \| \nabla 