\phi \right) \psi \left( \bm{\varepsilon}\right) \text{ on } \Omega \label{eq:evolution} \\
    \nabla\phi \cdot \bm{n} &= 0 \text{ on } \partial\Omega. \label{eq:evolution_bc}
    \end{aligned}$$

comprise the linear momentum equation and its corresponding boundary conditions, while is the phase-field evolution equation with the associated boundary condition given by . As mentioned earlier, $\phi$ should go to zero away from the crack. Thus it is tacitly assumed that the domain is of sufficient size to provide adequate separation between the regularized crack and the boundary, allowing the phase-field to decay to values that are small enough to approximate this condition.

Extension of the irreversibility condition to the regularized case is not immediately obvious, since the intermediate states $0 < \phi < 1$ do not have a straightforward physical interpretation. The natural course from the perspective of damage mechanics is to enforce the condition $$\label{eq:phaseField_growth}
    \phi \left( \bm{x} \right)_{t + \Delta t} \geq \phi \left( \bm{x} \right)_t \quad \forall \bm{x} \in \Omega.$$ This can be imposed via an additional penalty term in the phase-field evolution equation [@Miehe2010_ijnme], or alternatively through a history variable $\mathcal{H}$ that replaces the quantity $\psi \left( \bm{\epsilon} \right)$ in , defined as [@Miehe2010_cmame] $$\label{eq:histField}
    \mathcal{H} \left( \bm{x},t \right) = \max\limits_{s \in 