\left[ 0,t \right]} \psi \left( \bm{\epsilon} \left( \bm{x},s \right) \right)$$ A closer look at the phase-field localization process however reveals that may not be the best extension of . For example in the 1-dimensional case [@Kuhn2015] demonstrate that localization of the phase-field at the center of a diffuse crack involves both the growth of $\phi$ near the crack tip, as well as a *decrease* of the same in adjacent regions, which enables the phase-field to correctly settle to the exponential profile given in . Based on this observed behavior, a strict imposition of may lead to an overestimate of the crack length. Consequently, we introduce a modified version of in which imposes irreversibility only when $\phi$ exceeds a certain threshold, i.e. $$\label{eq:histField_threshold}
    \mathcal{H} \left( \bm{x},t \right) =
    \begin{cases}
    \max\limits_{s \in \left[ 0,t \right]} \psi \left( \bm{\epsilon} \left( \bm{x},s \right) \right) & \text{if } \phi > \phi_c \\[1em]
    \psi \left( \bm{\epsilon} \left( \bm{x},t \right) \right) & \text{otherwise.} \end{cases}$$ The parameter $\phi_c$ represents the maximum value for damage that is allowed to heal during unloading. For a material point undergoing damage $\phi \leq \phi_c$, the resulting stress-strain curves will be nonlinear, with the amount of departure from linearity dependent on the specific 