## 2 CZM Theory and Cohesive Element Embedding Methods
### CZM Damage
#### 2.2.2 Damage Evolution  
The term \"damage evolution\" refers to the state of the material after damage has appeared. This stage of the stiffness weakening process can be used to describe the material's subsequent mechanical properties during the degeneration process. A scalar damage variable, \(D\), represents the overall damage. It initially has a value of 0. If damage evolution is modeled, \(D\) monotonically evolves from 0 to 1 upon further loading after damage initiation. The nominal traction stress vector and stiffness are affected by the damage according to  
\[k_{ij}=k_{ij}^{0}(1-D) \tag{5}\]  
\[t_{n}=\left\{ \begin{array}{c}(1-D)t_{no},t_{no}\geq 0\\ t_{no},t_{no}<0(\text{no damage to compressive stiffness})\end{array} \right. \tag{6}\]  
\[t_{s}=(1-D)t_{so} \tag{7}\]  
\[t_{t}=(1-D)t_{to} \tag{8}\]  
where \(t_{no}\), \(t_{so}\), and \(t_{to}\) are the contact stress components predicted by the elastic traction-separation behavior for the current separations without damage. \(k_{ij}^{0}\) is the stiffness of the material without damage. It is evident from the preceding equation that the computation of \(D\) holds the key to explaining the damage evolution process of the material. Several intrinsic models are typically used to compute the damage variable \(D\). Many different intrinsic models can often be used to calculate the damage variable \(D\). [PERSON] et al. [43] compared them and came to the conclusion that the bilinear model could strike a balance between the needs of computing efficiency and accuracy. Thus, in this paper, modeling is completed using the bilinear model. In the model, \(D\) is expressed as follows:  
\[D=\frac{\delta_{m}^{f}(\delta_{m}^{\text{max}}-\delta_{m}^{o})}{\delta_{m}^{ \text{max}}\left(\delta_{m}^{f}-\delta_{m}^{o}\right)} \tag{9}\]  
where \(\delta_{m}^{o}\) represents the element node's opening displacement at the beginning of damage (i.e., \(D>0\)), \(\delta_{m}^{f}\) represents the node's opening displacement at \(D=1\), and \(\delta_{m}^{\text{max}}\) represents the node's maximum opening displacement during the loading history. Rather than tensile damage or pure shear in one direction, materials and structures are usually destroyed by a combination of factors. The default CZM in this investigation is the same for II and III. According to the cohesive crack model, the crack surface has two tangential traction forces, \(t_{s}\) and \(t_{t}\), and one normal traction force, \(t_{n}\). These traction forces are assumed to decrease monotonically with the corresponding displacement of the crack surface. Figure 1 displays a schematic of the mixed-mode cohesive traction response. Where \(\delta_{n}^{o}\), \(\delta_{s}^{o}\), and \(\delta_{t}^{o}\) are the relative I, relative II, and relative III at the onset of damage, respectively. As the traction force decreases, \(\delta_{nf}\), \(\delta_{sf}\), \(\delta_{lf}\), and \(\delta_{mf}\) are the pure I displacement, pure II displacement, pure III displacement, and effective displacement, respectively. The increasing linear upward branch in each softening curve describes the initial intact material. The I, II, and III fracture energies are represented, respectively, by the areas \(G_{m}^{c}\), \(G_{s}^{c}\), \(G_{t}^{c}\) and bounded by the traction-detachment curves and the axes in Figure 1. The Benzeggagh-Kenane fracture criterion is applied to characterize the outcomes of various damage mechanisms efficiently. This criterion is most useful when the pure II and III fracture energies are equal and is expressed as follows:  
\[G_{n}^{c}+(G_{s}^{c}-G_{n}^{c})\bigg{\{}\frac{G_{S}}{G_{T}}\bigg{\}}^{\eta}=G ^{C} \tag{10}\]Denoting by \(G_{n}\), \(G_{s}\), and \(G_{t}\) the work completed by the tractions and their conjugate separations in the normal, first, and second shear directions, respectively, and defining \(G_{T}=G_{n}+G_{s}+G_{t}\). It is also useful to define the quantity \(G_{S}=G_{s}+G_{t}\) to denote the portion of the total work completed by the shear traction and the corresponding separation components.