## 2 Improved Cohesive Model
### Basic Principles
\begin{array}{c}T_{n}(K_{n},0)=\sigma_{\max}\\ T_{t}(0,K_{t})=\tau_{\max}\end{array} \right. \tag{5}\]  
The normal and tangential cohesive tractions (\(T_{n}\) and \(T_{t}\)) according to the crack separations are controlled by Equations (6) and (7). \[\left\{ \begin{array}{c}T_{n}(\Delta_{n},\Delta_{t})=\frac{\partial\Psi}{ \partial\Delta_{t}}=\frac{\Gamma_{n}}{\delta_{n}}\frac{\partial F_{\mathrm{F} }(\Delta_{n},\Delta_{t})}{\partial\Delta_{t}}.F_{\mathrm{I}}(\Delta_{n}, \Delta_{t})\\ T_{t}(\Delta_{n},\Delta_{t})=\frac{\partial\Psi}{\partial\Delta_{t}}=\frac{ \Gamma_{t}}{\delta_{t}}\frac{\partial F_{\mathrm{I}}(\Delta_{n},\Delta_{t})}{ \partial\Delta_{t}}.F_{\mathrm{n}}(\Delta_{n},\Delta_{t})\\ \frac{\partial F(\Delta_{n},\Delta_{t})}{\partial\Delta_{t}}=Q\Big{(}1-\frac{ \Delta}{\delta}\Big{)}^{A}\Big{(}Q\frac{Q}{A}+\frac{\Delta}{\delta}\Big{)}^{Q -1}-A\Big{(}1-\frac{\Delta}{\delta}\Big{)}^{A-1}\Big{(}\frac{Q}{A}+\frac{ \Delta}{\delta}\Big{)}^{Q}\end{array} \right. \tag{6}\]  
\[\left\{\begin{array}{c}T_{n}(\Delta_{n},\Delta_{t})=\left\{ \begin{array}{ c}\frac{\Gamma_{n}}{\delta_{n}}\frac{\partial F_{\mathrm{F}}(\Delta_{n},\Delta_{t})}{ \partial\Delta_{n}}.F_{\mathrm{I}}(\Delta_{n},\Delta_{t}),\text{Tension}\\ \frac{\partial\Gamma_{n}}{\delta_{n}}\Delta_{n}=\frac{\Gamma_{n}}{\delta_{n}} \frac{\partial^{2}\Psi_{\mathrm{F}}(0,0)}{\partial\Delta_{t}^{2}}.F_{\mathrm{ I}}(0,0)\cdot\Delta_{n},\text{Compression}\\ T_{t}(\Delta_{n},\Delta_{t})=\frac{\partial\Psi}{\partial\Delta_{t}}=\frac{ \Gamma_{t}}{\delta_{t}}\cdot\frac{\partial F_{\mathrm{I}}(\Delta_{n},\Delta_{ t})}{\partial\Delta_{t}}\cdot F_{\mathrm{n}}(\Delta_{n},\Delta_{t})\\ \frac{\partial^{2}F_{\mathrm{F}}(0,0)}{\partial\Delta_{t}^{2}}=\big{(}Q^{2}-Q \big{)}\Big{(}\frac{Q}{A}\Big{)}^{Q-2}+\big{(}A^{2}-A\big{)}\Big{(}\frac{Q}{A }\Big{)}^{m}-2 AQ\Big{(}\frac{Q}{A}\Big{)}^{Q-1}\end{array} \right. \tag{7}\]  
It is important to highlight that Equations (6) and (7) delineate the method for calculating cohesive traction under a mix mode of tension and compression. For the tensile state, \(T_{n}\) is derived as a partial derivative of the potential function with respect to normal crack separation. In the compression state, drawing from studies by [PERSON] and [PERSON] (2018) and [PERSON] et al. (2018), \(T_{n}\) increases linearly with normal crack separations. The stiffness in the compression is the initial tensile stiffness of \(T_{n}\) and remains unaffected by a Mode II crack separation of 0. The PPR cohesive model requires eight input parameters, including fracture energy (\(\phi_{n}\) and \(\phi_{t}\)), strength (\(\sigma_{max}\) and \(\tau_{max}\)), initial stiffnesses (\(\lambda_{n}\) and \(\lambda_{t}\)), and softening parameters (\(\alpha\) and \(\beta\)). According to the above displacement traction function, the relationship between cohesion and displacement can be drawn as shown in Figure 2.  
Figure 1: Description of the cohesive region. It can be seen in Figure 2 that the normal and tangential cohesive tractions are affected by both normal and tangential crack separations. Simultaneously, the cohesive traction can vary continuously and smoothly with the normal or tangential crack separation growing.