# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 2 Original Models
### Original PPR Model  
The fundamental issue in cohesive zone modeling is the definition of traction-separation law, which gives the constitutive behavior of the fracture. The original PPR model defines the traction-separation law by taking the derivative of the cohesive fracture potential. The potential consists of polynomials formulated in terms of a normal separation (\(\Delta_{n}\)) and a tangential separation (\(\Delta_{t}\)), and it is expressed as [32]:  
\[ \begin{array}{l}\Psi(\Delta_{n},\Delta_{t})=\min(\phi_{n},\phi_{t})+\left[T_ {n}\Big{(}1-\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{\alpha}\Big{(}\frac{m}{ \alpha}+\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{m}+\langle\phi_{n}-\phi_{t} \rangle\right]\\ \left[T_{t}\Big{(}1-\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta}\Big{(} \frac{m}{\beta}+\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{n}+\langle\phi_{t}- \phi_{n}\rangle\right]\end{array}  \tag{1}\]  
Therefore, the traction-separation law is calculated  
\[ \begin{array}{l}T_{n}(\Delta_{n},\Delta_{t})=\frac{\partial\Psi}{\partial \Delta_{n}}=\frac{T_{t}}{\delta_{n}}\Bigg{[}m\Big{(}1-\frac{\Delta_{n}}{ \delta_{n}}\Big{)}^{\alpha}\Big{(}\frac{m}{\alpha}+\frac{\Delta_{n}}{\delta_ {n}}\Big{)}^{m-1}-\alpha\Big{(}1-\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{\alpha -1}\Big{(}\frac{m}{\alpha}+\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{m}\Bigg{]}\\ \left[T_{t}\Big{(}1-\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta}\Big{(} \frac{m}{\beta}+\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{n}+\langle\phi_{t}- \phi_{n}\rangle\right]\end{array}  \tag{2}\]  
\[ \begin{array}{l}T_{t}(\Delta_{n},\Delta_{t})=\frac{\partial\Psi}{\partial \Delta_{t}}=\frac{T_{t}}{\delta_{t}}\Bigg{[}n\Big{(}1-\frac{|\Delta_{t}|}{ \delta_{t}}\Big{)}^{\beta}\Big{(}\frac{m}{\beta}+\frac{|\Delta_{t}|}{\delta_{t }}\Big{)}^{n-1}-\beta\Big{(}1-\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta-1 }\Big{(}\frac{m}{\beta}+\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{n}\Bigg{]}\\ \Big{[}\Gamma_{n}\Big{(}1-\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{\alpha}\Big{(} \frac{m}{\alpha}+\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{m}+\langle\phi_{n}-\phi _{t}\rangle\Big{]}\frac{\Delta_{t}}{|\Delta_{t}|}\end{array}  \tag{3}\]  
where \(\langle\cdot\rangle\) is the Macaulay bracket, i.e., if \(x\leq 0\), then \(\langle x\rangle=0\), otherwise \(\langle x\rangle=x\). There are eight basic parameters (\(\phi_{n}\), \(\phi_{t}\), \(\sigma_{max}\), \(\tau_{max}\), \(\alpha\), \(\beta\), \(\lambda_{n}\), and \(\lambda_{t}\)) involved in the PPR model [32]. The PPR model considers different normal and tangential fracture energies (\(\phi_{n}\) and \(\phi_{t}\)), different cohesive strengths (\(\sigma_{max}\) and \(\tau_{max}\)), and controls the shape of the traction-separation law using the parameters \(\alpha\), and \(\beta\) and the initial slope indicators \(\lambda_{n}\), and \(\lambda_{t}\). The influence of \(\alpha\), \(\beta\), \(\lambda_{n}\), and \(\lambda_{t}\) on the material softening response were introduced in detail in [32]. These eight parameters could be obtained by fitting the interface stress--displacement relation measured in the splitting and shearing model test of concrete and mortar bonded composite specimens [40].