# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 3 Simplified PPR Traction-Separation Law
### Modification  
From Figure 2, we concluded that the cohesive interaction regions for the normal and tangential fractions were the same only if \(\phi_{n}=\phi_{t}\). Substituting \(\phi_{n}=\phi_{t}\) into Equations (2) and (3), we obtain the traction-separation law as follows.  
\[\begin{array}{l}T_{n}(\Delta_{n},\Delta_{t})=\frac{\phi_{n}}{\delta_{n}} \left(\frac{\alpha}{m}\right)^{m}\left(\frac{\beta}{n}\right)^{n}\left(1-\frac {|\Delta_{t}|}{\delta_{t}}\right)^{\beta}\left(\frac{n}{\beta}+\frac{|\Delta_ {t}|}{\delta_{t}}\right)^{n}\\ \left[\alpha\left(1-\frac{\Delta_{n}}{\delta_{n}}\right)^{\alpha-1}\left(\frac {m}{\alpha}+\frac{\Delta_{n}}{\delta_{n}}\right)^{m}-m\left(1-\frac{\Delta_{ n}}{\delta_{n}}\right)^{\alpha}\left(\frac{m}{\alpha}+\frac{\Delta_{n}}{\delta_{n}} \right)^{m-1}\right]\end{array} \tag{13}\]\[T_{t}(\Delta_{n},\Delta_{t})=\frac{\phi_{n}}{\phi_{t}}\left(\frac{\alpha}{m}\right)^ {m}\left(\frac{\beta}{n}\right)^{n}\left(1-\frac{\Delta_{n}}{\phi_{n}}\right)^{ \alpha}\left(\frac{m}{\alpha}+\frac{\Delta_{n}}{\phi_{n}}\right)^{m}\] \[\left[\beta\Big{(}1-\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta-1 }\Big{(}\frac{n}{\beta}+\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{n}-n\Big{(}1- \frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta}\Big{(}\frac{n}{\beta}+\frac{| \Delta_{t}|}{\delta_{t}}\Big{)}^{n-1}\right]\frac{\Delta_{t}}{|\Delta_{t}|}. \tag{14}\]  
The traction-separation law only depends on mode I fracture energy \(\phi_{n}\). To account for different values of \(\phi_{n}\) and \(\phi_{t}\), the mode II fracture energy \(\phi_{t}\) is substituted for \(\phi_{n}\) in the equation for the tangential traction. Therefore, the final form of simplified PPR traction-separation law is given by  
\[T_{n}(\Delta_{n},\Delta_{t})=\frac{\phi_{n}}{\delta_{n}}\left( \frac{\alpha}{m}\right)^{m}\left(\frac{\beta}{n}\right)^{n}\Big{(}1-\frac{| \Delta_{t}|}{\delta_{t}}\Big{)}^{\beta}\Big{(}\frac{n}{\beta}+\frac{|\Delta_{t }|}{\delta_{t}}\Big{)}^{n}\] \[\left[\alpha\Big{(}1-\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{\alpha -1}\Big{(}\frac{m}{\alpha}+\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{m}-m\Big{(}1 -\frac{\Delta_{n}}{\delta_{n}}\Big{)}^{\alpha}\Big{(}\frac{m}{\alpha}+\frac{ \Delta_{n}}{\delta_{n}}\Big{)}^{m-1}\right] \tag{15}\]  
\[T_{t}(\Delta_{n},\Delta_{t})=\frac{\phi_{t}}{\delta_{t}}\left( \frac{\alpha}{m}\right)^{m}\Big{(}\frac{\beta}{n}\Big{)}^{n}\Big{(}1-\frac{ \Delta_{n}}{\delta_{n}}\Big{)}^{\alpha}\Big{(}\frac{m}{\alpha}+\frac{\Delta_{ n}}{\delta_{n}}\Big{)}^{m}\] \[\left[\beta\Big{(}1-\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta -1}\Big{(}\frac{n}{\beta}+\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{n}-n\Big{(}1 -\frac{|\Delta_{t}|}{\delta_{t}}\Big{)}^{\beta}\Big{(}\frac{n}{\beta}+\frac{| \Delta_{t}|}{\delta_{t}}\Big{)}^{n-1}\right]\frac{\Delta_{t}}{|\Delta_{t}|} \tag{16}\]  
The simplified PPR traction-separation law is similar to the original PPR model and can also consider different fracture energies, cohesive strengths, and various material softening behaviors. The noteworthy merits of the simplified model are that the energy constants \(\Gamma_{n}\) and \(\Gamma_{t}\) are omitted (other parameters are the same as the original model), and the formulas are unified regardless of what the fracture energies are. Taking \(\phi_{n}=100\) N/m and \(\phi_{t}=200\) N/m as an example, the normal and tangential cohesive tractions of the simplified model are plotted in Figure 4. Figure 4 shows that the normal and tangential tractions are both properly defined in the same regions as expected. In the following section, the applicability of the simplified model is demonstrated using multiple cases.