## 2 Improved Cohesive Model  
### Basic Principles  
The PPR cohesive model was proposed by [PERSON] (2009) [35] and [PERSON] et al. (2009) [36] and has a clear physical meaning for each parameter. The model ensures that both the cohesive bonding and the tangent stiffness matrix are consistently governed by a continuous potential function. The Mode I and Mode II cohesive tractions, denoted as \(T_{n}\) and \(T_{t}\), are derived by taking the first-order derivatives of the potential function with respect to the Mode I and Mode II crack separations. The tangent stiffness matrix is determined by solving the second-order differential equation of the potential energy function. Because of the continuous and smooth nature of the traction-crack separation curve, the model guarantees satisfactory convergence in the process of numerical iteration. The potential function of the PPR cohesive model is provided in Equation (1), as follows:  
\[\left\{ \begin{array}{l}\psi(\Delta_{n},\Delta_{t})=\min(\phi_{n},\phi_{t})+ F_{n}(\Delta_{n},\Delta_{t})\cdot F_{t}(\Delta_{n},\Delta_{t})\\ F(\Delta_{n},\Delta_{t})=\Gamma\Big{(}1-\frac{\Delta}{\sigma}\Big{)}^{A} \Big{(}\frac{\partial}{A}+\frac{\Delta}{\delta}\Big{)}^{Q}+O\\ O=\left\{\begin{array}{l}\left\langle\phi_{n}-\phi_{t}\right\rangle/\left( \phi_{n}-\phi_{t}\right),\text{Model}\\ \left\langle\phi_{t}-\phi_{n}\right\rangle/\left(\phi_{t}-\phi_{n}\right), \text{ModelII}\end{array}\right.\end{array} \right. \tag{1}\]  
where \(\Delta\) represents the Mode I (\(\Delta_{n}\)) and Mode II (\(\Delta_{t}\)) crack separations; \(\phi_{n}\) and \(\phi_{t}\) correspond to the Mode I and Mode II fracture energies, respectively; \(A\) is the parameter that governs the softening behavior of the normal and tangential tractions once tensile or shear strength is achieved; and \(A=\alpha\) in Mode I and \(A=\beta\) in Mode II; \(\Gamma\) and \(Q\) can be calculated using Equations (2) and (3). \[\left\{ \begin{array}{l}\Gamma=\left(-\phi\right)^{p}M\\ p=\left\{\begin{array}{l}\left\langle\phi_{n}-\phi_{t}\right\rangle/\left( \phi_{n}-\phi_{t}\right),\text{Model}\\ \left\langle\phi_{t}-\phi_{n}\right\rangle/\left(\phi_{t}-\phi_{n}\right), \text{ModelII}\end{array}\right.\\ M=\left(\frac{A}{Q}\right)^{Q}\end{array} \right. \tag{2}\]  
where \(p\) and \(M\) are process variables; \(\Gamma\) represents the normal (\(\Gamma_{n}\)) or tangential (\(\Gamma_{t}\)) fracture energy coefficients. \[Q=\left\{ \begin{array}{l}\frac{\alpha(\alpha-1)\lambda_{n}^{2}}{1-\alpha \lambda_{n}^{2}},\text{Model}\\ \frac{\beta(\beta-1)\lambda_{n}^{2}}{1-\beta\lambda_{n}^{2}},\text{ModelII} \end{array} \right. \tag{3}\]  
The potential function is limited in a square area called the cohesive region, which has boundaries from 0 to \(\delta_{n}\) and \(-\delta_{t}\) to \(\delta_{t}\), where \(\delta_{n}\) and \(\delta_{t}\) are the final crack separations for the Mode I and Mode II fractures, respectively, and can be calculated using Equation (4). \[\delta=\frac{\phi}{\sigma}A\lambda(1-\lambda)^{A-1}\bigg{(}\frac{A}{Q}\lambda+ 1\bigg{)}^{Q-1} \tag{4}\]  
In Equations (2)-(4), \(\phi\), \(\Gamma\), \(\delta\), and \(\lambda\) are directional variables, and when their subscripts are marked \"\(n\)\", they correspond to the behavior of Mode I; subscript \"\(t\)\" corresponds to the behavior of Mode II, where \(\sigma\) is the strength of the interface material, and \(\sigma=\sigma_{max}\) in Mode I fractures and \(\sigma=\tau_{max}\) in Mode II fractures. \(\sigma\) is also the maximum traction in the process of crack separation; \(\lambda_{n}\) and \(\lambda_{t}\) are the stiffness parameters in the elastic stage along the normal and tangential crack separations, which equal the ratio of the separation when the traction reaches the strength (\(K_{n}\) and \(K_{t}\)) and residual boundaries (\(\delta_{n}\) and \(\delta_{t}\)) in the scope of the potential energy function. The parameters have a relationship which can be expressed in Equation (5), and the cohesive region is shown in Figure 1.  
\[\left\{