\phi \|^2 \right) {\,\mathrm{d}}\Omega$$ where $\phi$ is the phase-field that takes on values between 0 and 1, corresponding respectively to fully intact and broken states. On the other hand, $\kappa$ is a small positive constant meant to ensure positivity of the bulk energy as $\phi \rightarrow 1$. The two important features of the above expression are (a) the replacement of $\mathcal{H}^{n-1} \left( \Gamma \right)$ by an elliptic functional that calculates the combined length of all the cracks, and (b) the coefficient $\left( 1 - \phi \right)^2 + \kappa$ known as the *energy degradation function* that penalizes the material stiffness according to the value of $\phi$. is essentially a direct adaptation of the earlier functional of [@Ambrosio1992], for which a proof of $\Gamma$-convergence was subsequently given by [@Chambolle2004].

### Alternative variational problems

It was later suggested by [@Miehe2010_ijnme] that development of elliptic approximations to $\mathcal{H}^{n-1} \left( \Gamma \right)$ could also be motivated from a more physical standpoint by considering the 1-dimensional case of an infinitely long bar of uniform cross section. Assuming that the bar is aligned with the $x$-axis and that a single crack fully cuts the bar at $x = a$, the phase-field profile corresponding to this sharp crack 