# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 4 Improved Unloading/Reloading Relationship
### Comparison  
In this section, comparisons of the three models are drawn using the example in [39], where the following set of parameters is used: \(\phi_{n}=100\,\mathrm{N/m}\), \(\phi_{t}=300\,\mathrm{N/m}\), \(\sigma_{max}=2\) MPa, \(\tau_{max}=4\) MPa, \(\alpha=3\), \(\beta=5\), \(\lambda_{n}=0.20\), and \(\lambda_{t}=0.25\). The loading process consists of three steps. First, a proportional mixed-mode loading where \(\Delta_{n}=\Delta_{t}\) is applied up to a predefined value \(\Delta\). Then, a proportional mixed-mode unloading where \(\Delta_{n}=\Delta_{t}\) is carried out down to \(0\). Finally, a mode I reloading (keeping \(\Delta_{t}=0\)) is conducted. Unlike the work in [39], the simplified PPR traction-separation law proposed in this paper is used here instead of the PPR model. Therefore, \(\overline{\delta}_{t}\) is not used, and the fracture occurs for either \(\Delta_{n}=\delta_{n}\) or for \(\Delta_{t}=\delta_{t}\). Based on the parameters above, \(\delta_{n}=0.099\) mm, \(\delta_{t}=0.171\) mm, \(\delta_{t}^{peak}=0.020\) mm, and \(\delta_{t}^{peak}=0.043\) mm. Figure 12: Variations of the normal traction component (**a**) and tangential component (**b**) under mode I (mode II) loading, mixed-mode \(\Delta_{n}=\Delta_{t}\), and mixed-mode \(\Delta_{n}=0.5\Delta_{t}\). Figure 13 shows how the dissipated energy for the three models change with the increase in the proportional loading amplitude \(\Delta\). The three models give the same energy values at the beginning and end. The variation of the energy value for model (i) is quite different from model (ii) and model (iii), while model (ii) and model (iii) are almost the same. The dissipated energy given by model (iii) is identical to model (i) when \(\Delta<0.017\) mm, which is a constant equal to the mode I fracture energy of \(\phi_{n}=100N/m\). When \(\Delta\geq 0.017\) mm, the change law of the energy value for model (iii) is exactly the same as model (ii). There are three discontinuities for model (i), at \(\Delta=\delta_{n}^{peak}=0.020\) mm, \(\Delta=\delta_{t}^{peak}=0.043\)\(mm\), and \(\Delta=\delta_{n}=0.099\) mm, whereas model (ii) gives a smooth continuous curve, and model (iii) only has one discontinuity at \(\Delta=0.017\) mm. These differences are explained below by analyzing the changes of the traction components during the loading process. First, consider the proportional loading amplitudes \(\Delta\) around 0.017 mm. Figure 14 presents the variations of the traction components during the loading process for a proportional loading amplitude of \(\Delta=0.016\) mm. The computation results for model (i) and model (iii) are identical; thus, both are presented using Figure 14a. As can be observed in Figure 14a, during unloading, both traction values back up along the same curves that they followed during loading, when the peak of tractions was not reached. Consequently, the final energy is only dissipated in pure mode I reloading, which is equal to \(\phi_{n}=100\) N/m. In contrast, both unloading curves follow straight lines with the model (ii), as shown in Figure 14b. This is due to no peak value being needed to start using the linear response. That is say, damage is assumed to occur at the beginning, and the initial elastic region is ignored. Due to the assumed damage, the energy dissipated in pure mode I reloading for model (ii) is smaller than for model (i) and model (iii). As a result, the total dissipated energy (\(97.0\) J/m\({}^{2}\)) for model (ii) is lower than \(\phi_{n}\). When \(\Delta=0.017\) mm, the peak of normal traction is reached under the mixed-mode loading (Figure 15a,b), and therefore, the linear unloading response of model (iii) is activated. As a consequence, the variations of the traction components during the whole loading process for model (iii) become the same as for model (ii), both presented in Figure 15b. They still apply for larger proportional loading amplitudes \(\Delta>0.017\) mm. Thus, in the following analysis, the results for model (ii) and model (iii) are all displayed with the same diagrams.