is far from resolved. In view of this, we begin with a short review of theory pertaining to the phase-field formulation for brittle fracture along with a discussion of significant developments in the field in order to provide context for the present work.

Brittle fracture: from Griffith to Francfort-Marigo
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[@Griffith1921] can be credited as being the first to formally state the thermodynamic principles governing the propagation of fractures in brittle materials that has become the foundation of modern linear elastic fracture mechanics. According to Griffith’s theory, an existing crack will propagate when the rate of energy release $G$ associated with crack extension exceeds a critical value equal to the material fracture toughness, $\mathcal{G}_c$. This can be expressed via the following set of Kuhn-Tucker conditions [@Negri2008]:

\[eq:griffith\_principle\] $$\begin{aligned}
        G - \mathcal{G}_c &\leq 0 \label{eq:griffith_criterion} \\
        \dot{a} &\geq 0 \label{eq:griffith_irreversibility} \\
        \left( G - \mathcal{G}_c \right) \dot{a} &= 0 \label{eq:slacknessCondition}
    \end{aligned}$$

where $\dot{a}$ denotes the rate of crack length increase. The first inequality precludes the case of unstable cracking where $G > \mathcal{G}_c$, while the second is an irreversibility constraint that prevents unphysical healing of fractures. Finally, condition implies that $G$ must be equal to $\mathcal{G}_c$ when the crack is growing, 