is none other than the discontinuous scalar function $$\label{eq:sharpCrack}
    \phi \left( x \right) =
    \begin{cases}
        1, & x = a \\
        0, & \text{otherwise}.
    \end{cases}$$ A regularized approximation of the above can then be made via a function $\phi \in \Phi$, where $$\label{eq:setPhi}
    \Phi = \Set{ \phi \in H^1 \left( \mathbb{R}; \left[ 0,1 \right] \right) | 
        \begin{array}{l} 
            \phi \left( a \right) = 1 \\
            \phi^\prime \left( x \right) > 0 \text{ if } x < a  \\
            \phi^\prime \left( x \right) < 0 \text{ if } x > a  \\
            \phi \left( x \right) \rightarrow 0 \text{ as } x \rightarrow \pm\infty \end{array}}.$$ Some candidate functions are $$\begin{aligned}
    \phi \left( x \right) &= \exp \left( -\frac{\left| x-a \right|}{\ell} \right) \label{eq:approxExp} \\
    \phi \left( x \right) &= 
    \begin{cases}
    \left( 1 - \dfrac{\left| x-a \right|}{\sqrt{2}\ell} \right)^2, & x \in \left[ a - \sqrt{2}\ell, a + \sqrt{2}\ell \right] \\
    0 & \text{otherwise}
    \end{cases} \label{eq:approxVarIneq} \\
    \phi \left( x \right) &= \left( 1 + \frac{\left| x-a \right|}{\ell} \right) \exp \left( -\frac{\left| x-a \right|}{\ell} \right). \label{eq:approxHighOrder}\end{aligned}$$ We note that the function utilized by [@Miehe2010_ijnme] is in fact the solution obtained by minimizing the functional of [@Bourdin2000] in one dimension. On the other hand, is a 