## 2 Methods
### Traction-Separation Law  
In this work, cohesive crack is considered, which is consistent with the cohesive zone model. After the crack initiates, the traction across the two crack surfaces will drop from a prescribed value such as tensile strength to zero with the opening of crack [67], see Figure 1. The relationship between the traction and the crack opening is described by the so called traction-separation law.  
In this work, the exponential mixed-mode traction-separation law [68; 69; 70] is used in the CEM. Under 2D conditions, the equivalent crack opening is defined as  
\[\zeta_{eq}=\sqrt{\zeta_{n}^{2}+\zeta_{t}^{2}}, \tag{1}\]  
where \(\zeta_{n}\) and \(\zeta_{t}\) are the crack openings (as unknowns) along the normal and parallel directions, respectively, of the crack path and the corresponding unit vectors are denoted as \(\mathbf{n}=\left[n_{x},n_{y}\right]^{T}\) and \(\mathbf{t}=\left[t_{x},t_{y}\right]^{T}\). Clearly, \(n_{x}\)\(t_{x}+n_{y}\)\(t_{y}=0\). The traction components along \(\mathbf{n}\) and \(\mathbf{t}\), namely, \(T_{n}\) and \(T_{t}\), respectively, are obtained as  
\[T_{n}=T_{eq}\frac{\zeta_{n}}{\zeta_{eq}},\ T_{t}=T_{eq}\frac{ \zeta_{t}}{\zeta_{eq}}\] (2) with \[T_{eq}\left(\zeta_{eq}\right)=\left\{\begin{array}{ll}TL\left(\zeta_{eq} \right)=f_{t}\exp\left(-\frac{f_{t}}{G_{f}}\zeta_{eq}\right),&\text{loading}, \\ TU\left(\zeta_{eq}\right)=\frac{T_{mx}}{\zeta_{mx}}\zeta_{eq},&\text{ unloading/ reloading},\end{array}\right.\]  
where \(f_{t}\) is the uniaxial tensile strength, \(G_{f}\) is the fracture energy, \(\zeta_{mx}\) is the size of the maximum opening that the crack has ever experienced, and \(T_{mx}=TL\left(\zeta_{mx}\right)\) is the corresponding traction. The traction-separation law indicates that the CEM is consistent with the conventional cohesive zone model as a crack opening-based model but not a damage degree-based model.  
Correspondingly, the relationship between the traction differentials \(dT_{n}\) and \(dT_{t}\) and the crack openings \(\zeta_{n}\) and \(\zeta_{t}\) are described by  
\[\mathbf{D}\left[\begin{array}{c}d\zeta_{n}\\ d\zeta_{t}\end{array}\right]=\left[\begin{array}{c}dT_{n}\\ dT_{t}\end{array}\right], \tag{3}\]  
Figure 1: The basic of the cohesive zone model (redrawn based on the figure provided in [67]).  
with  
\[\begin{split}&\mathbf{D}=-\frac{T_{eq}}{\zeta_{eq}}\left[\begin{array}[ ]{cc}\frac{\zeta_{n}^{2}}{\zeta_{eq}^{2}}+\frac{f_{t}\,\zeta_{n}^{2}}{G_{f}\, \zeta_{eq}}-1&\frac{\zeta_{n}\,\zeta_{t}}{\zeta_{eq}^{2}}+\frac{f_{t}\,\zeta_ {n}\,\zeta_{t}}{G_{f}\,\zeta_{eq}}\\ \frac{\zeta_{n}\,\zeta_{t}}{\zeta_{eq}^{2}}+\frac{f_{t}\,\zeta_{n}\,\zeta_{t}}{ G_{f}\,\zeta_{eq}}&\frac{\zeta_{t}^{2}}{\zeta_{eq}^{2}}+\frac{f_{t}\,\zeta_{t}^{2}}{G_{f} \,\zeta_{eq}}-1\end{array}\right]\text{ for loading},\\ &\text{and}\\ &\mathbf{D}=\frac{T_{mx}}{\zeta_{mx}}\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right]\text{ for unloading/reloading},\end{split} \tag{4}\]  
in which \(\mathbf{D}\) obviously remains symmetric.  
On the other hand, some other types of traction-separation laws, such as linear, bilinear, and hyperbolic laws, can also be applied, but this has not yet been attempted. We prefer the exponential law because (i) it only needs two parameters, \(f_{t}\) and \(G_{f}\), both of which have strong physical meanings and can be obtained experimentally by standard tests; and (ii) it has \(C^{\infty}\) continuity, making \(\mathbf{D}\) very simple.