## 2 Phase-Field-Regularized Cohesive Zone Model  
### Phase Field Representation of Crack Surface  
Considering an infinitely long circular rod with a cross-sectional area \(A\) arranged along the \(x\)-axis, assume that the rod is fully cracked at \(x=0\), as shown in Figure 1. An auxiliary field variable \(\phi(x)\in[0,\ 1]\) can be introduced to describe the sharp crack topology; then, we can get Equation (2) based on the variational principle. The crack phase field \(\phi(x)\) located at \(x=0\) can be derived by (3); see more details in [18; 1]. \[A=\int_{\Gamma}\mathrm{d}\Gamma=\int_{\Gamma}\int_{-\infty}^{\infty}\mathcal{ Y}(\phi)\mathrm{d}x\mathrm{d}\Gamma=\int_{\Omega}\mathcal{Y}(\phi)\mathrm{d} \Omega=A(\phi) \tag{2}\]  
\[x(\phi)=l_{0}\int_{\phi}^{1}\alpha^{-\frac{1}{2}}(\phi)\mathrm{d}\phi \tag{3}\]  
where \(A\) denotes the fracture surface area, and the crack surface density function is \(\mathcal{Y}(\phi)\):=\(\frac{1}{\ell_{0}}\left[\ell_{0}(\phi^{\prime})^{2}+\frac{1}{\ell_{0}} \alpha(\phi)\right]\); \(\alpha(\phi)=\xi\phi+(1-\xi)\phi^{2}\) is denoted as the crack geometry function, \(c_{0}=4\int_{0}^{1}\sqrt{\alpha(\phi)}\mathrm{d}\phi\) is the scaling parameter and \(\ell_{0}\) is a length scale parameter. As demonstrated by [PERSON] [1; 25], some conventional models can be obtained with different \(\xi\in[0,\ 2]\) of the crack geometry function \(\alpha(\phi)\)  
* AT2 model [16; 18] \[\xi=0,\ \phi(x)=exp\bigg{(}-\frac{|x|}{\ell_{0}}\bigg{)}\] (4)
* AT1 model [45] \[\xi=1,\ \phi(x)=\bigg{(}1-\frac{|x|}{2\ell_{0}}\bigg{)}^{2}\] (5)
* WN model [1; 46] \[\xi=2,\ \phi(x)=1-sin\bigg{(}\frac{|x|}{\ell_{0}}\bigg{)}\] (6)  
In this paper, we intend to use the WN model, i.e., \(\xi=2\); thus, we have \(\alpha(\phi)=2\phi-\phi^{2}\), \(c_{0}=\pi,\gamma(\phi)=\frac{1}{\pi}\Big{[}\ell_{0}(\phi^{\prime})^{2}+\frac{ 1}{\ell_{0}}\big{(}2\phi-\phi^{2}\big{)}\Big{]}\). Extending the 1D crack surface density function \(\gamma(\phi)\) into a multidimension gives Equation (7). \[\gamma(\phi,\
abla\phi)=\frac{1}{\ell_{0}}\Big{[}\ell_{0}|\
abla\phi|^{2}+ \frac{1}{\ell_{0}}\alpha(\phi)\Big{]}=\frac{1}{\pi}\bigg{(}\ell_{0}|\
abla \phi|^{2}+\frac{1}{\ell_{0}}(2\phi-\phi^{2})\bigg{)} \tag{7}\]  
Two-dimensional sharp and diffusive crack topology is shown in Figure 2. The phase field model diffuses sharp crack \(\Gamma\) into crack band \(\mathcal{B}\subseteq\Omega\) with finite scale \(\ell_{0}>0\). \(\partial\mathcal{B}\) is the external boundary of the crack band \(\mathcal{B}\), The outward unit normal vector of \(\partial\mathcal{B}\) is denoted  
Figure 1: (**a**) A one-dimensional (1D) rod with a crack located at \(x=0\) and a cross-section \(A\). (**b**) Damage representation with a shape crack at \(x=0\). (**c**) Diffusive phase field approximation with WN (black), AT1 (blue) and AT2 (red) model scaling by the length scale parameter \(\ell_{0}\). as \(n_{\mathcal{B}}\). The body force is denoted as \(\mathbf{b}^{*}\). The regularized surface energy can be obtained by Equation (8). \[\Psi^{\Gamma}=G_{c}\int_{\Gamma}\mathrm{d}\Gamma=G_{c}\int_{\Omega}\gamma(\phi, \
abla\phi)\mathrm{d}\Omega=G_{c}\int_{\mathcal{B}}\gamma(\phi,\
abla\phi) \mathrm{d}\Omega \tag{8}\]  
where \(G_{c}\) is the critical energy release rate. Correspondingly, the bulk energy is  
\[A=\int_{\Gamma}\mathrm{d}\Gamma=\int_{\Gamma}\int_{-\infty}^{\infty}\mathcal{ Y}(\phi)\mathrm{d}x\mathrm{d}\Gamma=\int_{\Omega}\mathcal{Y}(\phi)\mathrm{d} \Omega=A(\phi) \tag{2}\]0  
in which \(\psi(\mathbf{\varepsilon},\phi)=\omega(\phi)\psi_{0}(\mathbf{\varepsilon})\) is the bulk energy density, \(\psi_{0}(\mathbf{\varepsilon})=\frac{1}{2}\mathbf{\varepsilon}:\mathbb{E}_{0}:\mathbf{ \varepsilon}=\frac{1}{2}\mathbf{\sigma}:\mathbb{C}_{0}:\mathbf{\sigma}\) is the elastic strain energy, \(\mathbf{\sigma}\) and \(\mathbf{\varepsilon}\) are the stress tensor and the strain tensor, respectively. \(\mathbb{E}_{0}\) and \(\mathbb{C}_{0}\) are the elastic stiffness tensor and the flexibility tensor, respectively.