## 2 Phase-Field Fracture Model for Brittle Fracture  
As can be seen in Figure 2, the phase-field fracture model approximates sharp cracks \(S\) as diffuse bands whose thickness is defined by the length scale parameter \(l_{c}\). As \(l_{c}\) approaches zero, the diffuse crack reverts to a sharp crack [48]. The phase-field variable \(\phi\in[0,1]\) represents the degree of damage, with \(\phi=0\) indicating no damage and \(\phi=1\) a fully damaged state. As indicated in Equation (1), it is imperative that the time derivative of the phase-field damage parameter remains non-negative, ensuring the irreversibility of the induced damage, i.e.,  
\[\dot{\phi}(x,t)\geq 0. \tag{1}\]  
Within the framework of the phase-field fracture method, the internal potential energy of the solid takes the form [49]  
\[\psi(\varepsilon,\phi,\
abla_{x}\phi)=\psi_{c}(\varepsilon,\phi)+\psi_{c}(\phi,\
abla_{x}\phi), \tag{2}\]  
where \(\
abla_{x}\bullet\) represents the spatial gradient of any scalar-valued field quantity \(\bullet\), \(\varepsilon\) denotes the strain tensor pertaining to small deformations, \(\psi_{c}\) denotes the stored strain energy, and \(\psi_{c}\) represents the fracture surface energy. The determination of the stored strain energy is achieved through the relation  
\[\psi_{c}(\varepsilon,\phi)=\omega(\phi)\,\psi_{0}(\varepsilon), \tag{3}\]  
Figure 2: A Solid undergoing fracture, regularized using a phase-field model: Left-hand side picture shows the sharp crack representation, and right-hand side picture shows the phase-field regularized representation. where \(\omega(\phi)\) represents the damage degradation function, responsible for reducing the stiffness of the undamaged material. The damage degradation function must adhere to the conditions  
\[\omega(\phi=0)=1,\qquad\omega(\phi=1)=0,\qquad\omega^{\prime}(\phi=1)=0. \tag{4}\]  
To mitigate the occurrence of cracking in regions subjected to compression, it is imperative to incorporate the damage degradation function in conjunction with the tension split of stored elastic energy such that  
\[\psi(\varepsilon,\phi)=\omega(\phi)\,\psi_{0}^{+}(\varepsilon)+\psi_{0}^{-}( \varepsilon). \tag{5}\]  
where \(\psi_{0}(\varepsilon)=\frac{1}{2}\,\varepsilon:\mathbb{E}_{0}:\varepsilon\) represents the strain energy density function of the undamaged solid and \(\mathbb{E}_{0}\) represents the fourth-order elastic stiffness tensor. The fracture surface energy \(\psi_{c}\) can be expressed as a volume integral via  
\[\psi_{c}(\phi,\
abla_{\mathrm{x}}\phi)=\int_{\Omega}G_{c}\gamma(\phi,\
abla_{ \mathrm{x}}\phi)\,\mathrm{dA}, \tag{6}\]  
where \(G_{c}\) is the critical energy release rate, and \(\gamma(\phi,\
abla_{\mathrm{x}}\phi)\) is the fracture surface density function. The calculation of \(\gamma(\phi,\
abla_{\mathrm{x}}\phi)\) is achieved using the relation  
\[\gamma(\phi,\
abla_{\mathrm{x}}\phi)=\frac{1}{c_{0}}\left[\frac{1}{l_{c}} \alpha(\phi)+l_{c}\
abla_{\mathrm{x}}\phi\cdot\
abla_{\mathrm{x}}\phi\right], \tag{7}\]  
where \(\alpha(\phi)\) is the crack geometric function, and is introduced to regularize \(\psi_{c}(\phi,\
abla_{\mathrm{x}}\phi)\). The total potential energy of the solid domain is expressed as  
\[P_{\mathrm{total}}=\int_{\Omega}\omega(\phi)\,\psi_{0}(\varepsilon)\,\mathrm{ dV}+\int_{\Omega}\frac{G_{c}}{c_{0}}\left[\frac{1}{l_{c}}\alpha(\phi)+l_{c} \,\
abla_{\mathrm{x}}\phi\cdot\
abla_{\mathrm{x}}\phi\right]\mathrm{dV}-\int_{ \Omega}\mathbf{b}^{\star}\cdot\mathbf{u}\,\mathrm{dV}-\int_{\Omega}\mathbf{t}^{\star} \cdot\mathbf{u}\,\mathrm{dA}. \tag{8}\]  
The total energy functional represented by Equation (8) is characterized as being quadratic and convex. Consequently, through the minimization of \(P_{\mathrm{total}}\), it is feasible to determine the displacement and phase-field fracture parameters denoted as \((\mathbf{u},\phi)\) in a staggered manner.