# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 2 Original Models
### Original PPR Model
From these eight parameters, the following quantities can be deduced, which are used in (1), (2), and (3):  
\[m=\frac{\alpha(\alpha-1)\lambda_{n}^{2}}{(1-\alpha\lambda_{n}^{2})},\ n=\frac{ \beta(\beta-1)\lambda_{t}^{2}}{\big{(}1-\beta\lambda_{t}^{2}\big{)}} \tag{4}\]  
\[\Gamma_{n}=(-\phi_{n})^{\phi_{n}-\phi_{1}/(\phi_{n}-\phi_{t})}\Big{(}\frac{ \alpha}{m}\Big{)}^{m},\ \Gamma_{t}=(-\phi_{t})^{\phi_{t}-\phi_{n}/(\phi_{t}-\phi_{n})}\bigg{(}\frac{ \beta}{n}\bigg{)}^{n}\ \text{for}\ (\phi_{n}\
eq\phi_{t}) \tag{5}\]  
\[\Gamma_{n}=-\phi_{n}\Big{(}\frac{\alpha}{m}\Big{)}^{m},\ \Gamma_{t}=\bigg{(}\frac{ \beta}{n}\bigg{)}^{n}\ \text{for}\ (\phi_{n}=\phi_{t}) \tag{6}\]  
\[\delta_{n}=\frac{\phi_{n}}{\sigma_{max}}\alpha\lambda_{n}(1-\lambda_{n})^{ \alpha-1}\Big{(}\frac{\alpha}{m}+1\Big{)}\Big{(}\frac{\alpha}{m}\lambda_{n}+ 1\Big{)}^{m-1} \tag{7}\]  
\[\delta_{t}=\frac{\phi_{t}}{\tau_{max}}\beta\lambda_{t}(1-\lambda_{t})^{\beta- 1}\bigg{(}\frac{\beta}{n}+1\bigg{)}\bigg{(}\frac{\beta}{n}\lambda_{t}+1\bigg{)} ^{n-1} \tag{8}\]  
where \(\delta_{n}\) and \(\delta_{t}\) are the normal final crack opening width and the tangential final crack opening width, respectively. If \(\Delta_{n}\geq\delta_{n}\) or \(\Delta_{t}\geq\delta_{t}\), the tractions \(T_{n}\) and \(T_{t}\) are set to zero. Therefore, the traction-separation law is only valid in a region. To keep things simple, the separations \((\Delta_{n},\Delta_{t})\) are assumed to be positive here. Then, the region can be expressed as \([\,(\Delta_{n},\Delta_{t})|0\leq\Delta_{n}\leq\delta_{n},0\leq\Delta_{t}\leq \delta_{t}]\). Considering the region, the normal and tangential cohesive tractions of the PPR model are plotted in Figure 2 with different fracture energies (e.g., \(\phi_{n}=100\) N/m, \(\phi_{t}=200\) N/m, and other cases), cohesive strengths (e.g., \(\sigma_{max}=40\) MPa, \(\tau_{max}=30\) MPa), shapes (e.g., \(\alpha=5\), \(\beta=1.3\)), and initial slope indicators (e.g., \(\lambda_{n}=0.1\), \(\lambda_{t}=0.2\)). The normal cohesive traction (on the left in Figure 2) illustrates the fracture behavior of a typical quasi-brittle material, while the tangential cohesive traction (on the right in Figure 2) describes a plateau-type behavior. If \(\phi_{n}<\phi_{t}\) (Figure 2a,b), the tangential cohesive traction was properly defined in the rectangular region corresponding to the final crack opening widths (\(\delta_{n}\), \(\delta_{t}\)) as mentioned above, while in the same region, the normal cohesive traction \(T_{n}(\Delta_{n},\Delta_{t})\) existed as negative (Figure 2a), which is contradictory to the nature of cohesive tractions. Similarly, if \(\phi_{n}>\phi_{t}\), the normal cohesive traction was properly defined in the rectangular region, while the tangential cohesive traction was negative in some areas, as illustrated in Figure 2c,d. If \(\phi_{n}=\phi_{t}\) (Figure 2e,f), the normal and tangential tractions were non-negative in the same region. To prevent the unphysical response, [PERSON] et al. [32] redefined the region by narrowing it to make the cohesive traction non-negative in new region, and the traction was set to zero if it was out of the new region. Taking \(\phi_{n}<\phi_{t}\) as an example, the change of region for the normal traction is demonstrated in Figure 3 (separations are assumed positive here). The parameter \(\overline{\delta_{t}}\) in Figure 3 is the tangential conjugate final crack opening width, and it is the single root of \(\Gamma_{t}\Big{(}1-\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta}\Big{(} \frac{n}{\beta}+\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{n}+\langle\phi_{t}- \phi_{n}\rangle=0\) between 0 and \(\delta_{t}\)[32]. For the new cohesive interaction region (on the right in Figure 3), one border of the new region is the normal final crack opening width \(\delta_{n}\). The other border is the tangential conjugate final crack opening width \(\overline{\delta_{t}}\).