# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 3 Simplified PPR Traction-Separation Law
### Path Dependence of Work-of-Separation
#### 3.2.2 Non-Proportional Separation  
The non-proportional separation path is shown in Figure 7. Path 1 is that the interface is loaded in the normal direction until \(\Delta_{n}=\Delta_{n,max}\); then, complete tangential separation occurs. Accordingly, path 2 is that the interface is first loaded in shear up to \(\Delta_{t,max}\), and then completely broken in the normal direction [41]. The expressions of the work-of-separation for the two paths were given by [32]:  
\[W_{sep}=\int_{0}^{\Delta_{n,max}}T_{n}(\Delta_{n},0)d\Delta_{n}+\int_{0}^{ \delta_{n}}T_{t}(\Delta_{n,max},\Delta_{t})d\Delta_{t} \tag{18}\]  
\[W_{sep}=\int_{0}^{\Delta_{n,max}}T_{t}(0,\Delta_{t})d\Delta_{t}+\int_{0}^{ \delta_{n}}T_{n}(\Delta_{n},\Delta_{t,max})d\Delta_{n} \tag{19}\]  
Figure 6: The work-of-separation \(W_{sep}\) (**a,b**); work conducted by the normal traction \(W_{n}\) (**c,d**); and work conducted by the tangential traction \(W_{t}\) (**e,f**); with respect to the change of the proportional angle \(\theta\). Park–Paulino–Roesler (PPR) model (**a,c,e**); simplified PPR traction–separation law (SPPR) model (**b,d,f**). For the first path (Figure 7a), \(\Delta_{n,max}=0\) represents the pure mode II fracture, while \(\Delta_{n,max}=\delta_{n}\) describes the pure mode I fracture. Similarly, for the second path (Figure 7b), when \(\Delta_{t,max}\) is zero, the separation path illustrates the pure mode I failure, while \(\Delta_{t,max}=\delta_{t}\) represents the pure mode II fracture. The change of \(\Delta_{t,max}\) from 0 to \(\delta_{t}\) (resp. \(\Delta_{n,max}\) from 0 to \(\delta_{n}\)) demonstrates the gradual change of the mode mixity from the mode I fracture to the mode II fracture (resp. from the mode II fracture to the mode I fracture). Based on Equations (18) and (19), the work-of-separation may change with the increasing of \(\Delta_{n,max}\) or \(\Delta_{t,max}\). If the work-of-separation has a monotonic variation from one fracture mode to the other fracture mode, this demonstrates the consistency of the cohesive constitutive model [32; 41]. Figure 8 shows the variation of \(W_{sep}\), \(W_{n}\), and \(W_{t}\) with respect to the two paths, under the condition of \(\phi_{n}<\phi_{t}\). The results for PPR model are on the left and for the SPPR model are on the right. \(W_{sep}\), \(W_{n}\), and \(W_{t}\) all change monotonically for both models. For path 1 (Figure 8a,b), the curves of \(W_{sep}\), \(W_{n}\), and \(W_{t}\) for the SPPR model are exactly the same as the PPR model. Figure 8a,b show that the work conducted by the tangential traction \(W_{t}\) gradually decreases from \(\phi_{t}\) to 0, while the work conducted by the normal traction \(W_{n}\) increases from 0 to \(\phi_{n}\). The work-of-separation \(W_{sep}\) is the sum of \(W_{n}\) and \(W_{t}\), and this monotonically varies from the value of \(\phi_{t}\) to the value of \(\phi_{n}\) by increasing \(\Delta_{n,max}\) from 0 to \(\delta_{n}\). For path 2 (Figure 8c,d), the change rules of \(W_{sep}\), \(W_{n}\), and \(W_{t}\) are the exact opposite of those in path 1. There is a kink point on the curves of \(W_{n}\) and \(W_{sep}\), as shown in Figure 8c, but not in Figure 8d. The separation at the kink point corresponds to the border \(\Delta_{t}=\overline{\delta}_{t}\) of the original PPR model, where \(\overline{\delta}_{t}\) is the tangential conjugate final crack opening width as previously described in Section 2.1. When \(\Delta_{t}\) is smaller than \(\overline{\delta}_{t}\), the normal cohesive interaction is obtained based on Equation (2). When \(\Delta_{t}\) is greater than \(\overline{\delta}_{t}\), the normal traction is set to zero. The normal cohesive interaction is then not smooth but piece-wise continuous at \(\Delta_{t}=\overline{\delta}_{t}\). As a result, the \(W_{n}\) and \(W_{sep}\) also have the kink point at the same location. In contrast, the curves of \(W_{sep}\), \(W_{n}\), and \(W_{t}\) for the SPPR model, as shown in Figure 8d, are continuous and smooth.