compactly supported function that leads to a variational inequality problem [@Pham2011], while is obtained by solving a 4th order governing equation and was introduced by [@Borden2012] with the aim of loosening the mesh size requirements with respect to $\ell$ at the same time taking advantage of numerical methods that can adequately model $C^1$-continuous solutions. shows a comparison of the three functions mentioned above. It can be observed that for a given value of $\ell$, the amount of crack diffusion is additionally dependent on the specific form of $\phi \left( x \right)$.

![image](phase_field_1d){width="40.00000%"} \[fig:phaseField\_1d\]

Higher order phase-field formulations for fracture have so far not achieved the same popularity as their lower order counterparts. This is mainly due to the higher order of continuity that they require in conjunction with numerical solutions, which is expensive to obtain with traditional frameworks such as finite elements. In addition, $\Gamma$-convergence is yet to be proven for these formulations. On the other hand, [@Li2015] point out that the incorporation of general anisotropic effects relating to the surface energy requires a formulation that is at least 4th order.

### Damage

The connection between phase-field approaches and nonlocal damage models was explored by [@Pham2011], who noted that elliptic 