# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 4 Improved Unloading/Reloading Relationship
### Comparison
Due to the variation of response, the dissipated energy for model (iii) changes from \(100\) J/m\({}^{2}\) (\(\phi_{n}\)) at \(\Delta=0.016\) mm to \(97.0\) J/m\({}^{2}\) at \(\Delta=0.017\) mm. Such an energy discontinuity is inherent to any CZM model that obeys a curved line in the reversible range and an unloading straight line when irreversibility has appeared, which was discussed in detail by [PERSON] et al. [39]. Therefore, the small energy jump is accepted. When \(\Delta=0.019\) mm, the peak of normal traction is exceeded, as shown in Figure 16a,b. However, because of \(\Delta<\delta_{n}^{peak}=0.020\) mm, the normal traction for model (i) still returns along the loading path during the unloading process, leading to a questionable response that the traction increases with the decrease in separations. When \(\Delta=0.020\) mm, the \(\delta_{n}^{peak}\) value is reached, and thus the model (i) is activated. Similar to that for model (iii) at \(\Delta=0.017\) mm mentioned above, there is a small energy jump for model (i) due to the change from an elastic region to a softening region. In contrast, for model (ii) and model (iii), the dissipated energy varies continuously. Consider now the proportional loading amplitudes \(\Delta\) around \(\delta_{t}^{peak}=0.043\) mm. Figure 17a, for \(\Delta=0.042\) mm, shows that tangential traction component \(T_{t}\) still returns along the loading path and increases significantly during the unloading process. When \(\Delta=0.043\) mm (Figure 17c), the \(\delta_{t}^{peak}\) value is reached, and therefore, the tangential unloading component in model (i) is activated as well. On account of the added energy that is dissipated by \(T_{t}\) during the proportional loading/unloading process, the total energy given by model (i) increases sharply, which induces a jump at \(\Delta=0.043\) mm in Figure 13.  
In contrast, model (ii) and model (iii) have a smooth evolution of dissipated energy, as can be observed in Figure 17b,d. Finally, consider the proportional loading amplitudes \(\Delta\) around \(\delta_{n}=0.099\) mm. When \(\Delta=0.098\) mm (Figure 18a,b), which is slightly below the critical value \(\delta_{n}\), both tractions are near 0 at the end of loading. For model (i), there is an increase in \(T_{t}\) during unloading, and, due to that, \(\Delta_{n}\) in the tangential term of model (i) varies during proportional unloading. When the proportional loading amplitude \(\Delta=0.099\) mm, the critical value \(\delta_{n}\) is reached, and hence fracture is complete (Figure 18a,b). As a result, the unloading and reloading phases no longer exist, and the dissipated energy for the three models becomes the same, equal to \(155.2\) J/m\({}^{2}\). From the above analysis, the original unloading/reloading relationship (referred to as model (i) here) may induce some questionable responses, such as increasing traction during unloading. The new unloading/reloading relationship proposed by [PERSON] et al. (2018) (referred to as model (ii) here) may bring damage at the beginning and ignore the initial elastic region. The improved unloading/reloading relationship (referred to as model (iii) here) proposed in this paper combines the merits of the above two models, which prevents the issues mentioned above, and defines an elastic region before a softening regime. Figure 14: Variations of the traction components \(T_{n}\) (solid lines) and \(T_{t}\) (dashed lines) during the process of proportional loading/unloading and mode I reloading, for a proportional loading amplitude of \(\Delta=0.016\) mm. Model (i) and model (iii) (**a**); model (ii) (**b**). Figure 13: Dissipated energy in the process of proportional loading/unloading and mode I reloading, with the increase in the amplitude of proportional loading. Model (i) (solid line), model (ii) (dashed line), and model (iii) (dotted line).