## 2 Research Methodology
### Theoretical Background  
#### 2.1.1 Cohesion Unit Model  
The cohesive model uses fracture criteria to determine the damage condition. Damage occurs on cohesive units, and when the damage criterion is met, the unit starts to deteriorate. According to the predefined damage evolution model, when the unit completely fails, the mid layer of the unit \"splits into two\", resulting in a geometric discontinuity and the formation of fractures. A complete damage process can be divided into the initiation stage and the damage stage. In the initiation stage, as the tensile displacement increases, the traction force also increases. When it reaches the damage initiation point, the material begins to deteriorate, and the traction force starts to decrease until it reaches zero, and the fracture fully opens. Before damage occurs, the stress and strain satisfy the following relationship:  
\[\sigma=\left\{ \begin{array}{c}\sigma_{n}\\ \sigma_{s}\\ \sigma_{t}\end{array} \right\}=\left[ \begin{array}{cc}K_{nn}&K_{ns}&K_{nt}\\ K_{st}&K_{ss}&K_{st}\\ K_{tn}&K_{bs}&K_{tt}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{n }\\ \varepsilon_{s}\\ \varepsilon_{t}\end{array} \right\}=K\varepsilon \tag{1}\]  
where \(\sigma\) is the stress vector, \(Pa\); \(\sigma_{n}\) is the normal stress of the element, \(Pa\); \(\sigma_{s}\) is the first shear stress, \(Pa\); \(\sigma_{t}\) is the second shear stress (which does not exist in the two-dimensional case),\(Pa\); \(K\) is the stiffness matrix of the element, dimensionless; \(\varepsilon\) is the strain, dimensionless; \(\varepsilon_{n}\) is the strain generated in the normal direction of the element, dimensionless; \(\varepsilon_{s}\) is the strain generated in the first shear direction, dimensionless; and \(\varepsilon_{t}\) is the strain generated in the second shear direction, dimensionless. The expression is as follows:  
\[\varepsilon_{n}=\frac{d_{n}}{T_{0}},\varepsilon_{s}=\frac{d_{s}}{T_{0}}, \varepsilon_{t}=\frac{d_{t}}{T_{0}} \tag{2}\]  
where \(d\) is displacement, \(m\), and \(T_{0}\) is the constitutive thickness, dimensionless. The damage model is used to describe the stiffness degradation and structural failure of a material after the traction force reaches its maximum critical value. The damage model consists of two parts: damage initiation criteria and damage evolution laws. Once the damage reaches the damage initiation criteria, the failure will occur according to the defined damage evolution laws. In this case, the damage initiation criteria used is the maximum nominal stress criterion. \[f=\max\left\{\frac{\{t_{n}\}}{t_{n}^{0}},\frac{t_{s}}{t_{s}^{0}},\frac{t_{t}}{ t_{t}^{0}}\right\} \tag{3}\]  
where \(t_{n}^{0}\) is the traction force perpendicular to the fracture surface at the beginning of damage, \(N\); \(t_{s}^{0}\), \(t_{t}^{0}\) is the traction force perpendicular to the fracture surface at the beginning of damage, \(N\); \(t_{n}\) is the traction force perpendicular to the fracture surface, \(N\); \(t_{s}\), \(t_{t}\) is the traction force perpendicular to each other on the fracture surface, \(N\); and \(f\) is the fracture criterion, and fracture will occur when \(1.0\leq f\leq 1.0+f_{tol}\); \(f_{tol}\) is the tolerance, with a default value of 0.05. The evolution law of damage describes the rate of the stiffness degradation of a material once it reaches the corresponding initiation criteria. The overall damage of the material is represented by \(D\), with an initial value of 0, indicating no damage. After the initiation of damage, \(D\) monotonically changes to 1, indicating complete failure. The stress components are represented as follows:  
\[t_{n}=\left\{ \begin{array}{cc}(1-D)\overline{t_{n}}&\overline{t_{n}}\geq 0 \\ \overline{t_{n}}&\overline{t_{n}}<0\end{array}