# Comparative Theoretical Analysis of Halden Reactor Creep Tests on M5 Cladding Material
## 5 Proposed M5 Creep Model  
The above results have been used for comparing the creep rate obtained here for M5 with the one expected for Zr-4 run with the same conditions as M5 in IFA-617 and IFA-663. The expected creep for Zr-4 has been calculated with the Franklin correlation mentioned earlier, while the secondary creep rate values obtained in this work have been used for M5. The results are shown in Figures 4 and 5. Based on the above arguments, an attempt at deriving an overall M5 creep model has been made by combining the Franklin model for the primary creep, which seems to also reproduce the primary creep phase in IFA-617 reasonably well, with the slope of the secondary creep obtained here, which appears to linearly evolve with time. The proposed model is as follows:  
* At the start, the primary creep evolves in accordance with the Franklin equation. * At some point, the secondary creep will start, evolving linearly with a slope provided by the following equation: \[\frac{\mathrm{d}\mathrm{D}}{\mathrm{d}\tau}=4.48\cdot\left(\frac{\Phi}{2.8 \times 10^{13}}\right)^{0.55}\cdot\left(\frac{\sigma}{64}\right)^{0.579}\cdot \mathrm{e}^{-\frac{\mathrm{Q}}{K}\cdot\left(\frac{1}{4}-\frac{1}{642}\right)}\]  
where \(4.48\) is in \(\upmu\)m/1000 h and provides the slope of the IFA-663 secondary creep (see Figure 4 with \(K=1.3\)), which is a realistic value considering the temperature difference in IFA-663 and IFA-617. The values \(2.8\times 10^{13}\), \(64\), and \(632\) are the fast neutron flux in \(\mathrm{n}/\mathrm{cm}^{2}\cdot\mathrm{s}\), the cladding hoop stress in MPa, and the cladding mean temperature in \(K\) in IFA-663; \(\uptau\) represents the time variable in this equation. The creep will then start and proceed according to (a) as long as the creep rate (i.e., the time derivative of the Franklin equation) remains higher than the creep rate provided by Equation (4). Calculations show that this transition occurs at about 500 h. After that, the creep will proceed linearly with time, with a derivative provided by Equation (4). Figure 4: Comparison of the M5 IFA-617 secondary creep results discussed earlier in this paper, with the Franklin model for Zr-4 cladding operated at same conditions as in IFA-617. The M5 secondary creep rates obtained in Figure 3 with both \(K=1\) and \(K=1.3\) are shown. The M5 secondary creep is linear and starts at about 500 h after loading. At this point in time, the secondary creep rate obtained here equals the slope of the Franklin model. The findings from this study align with recent research on creep behavior in zirconium-based cladding materials, particularly those investigating the role of stress and environmental factors such as corrosion and neutron flux. For instance, [PERSON] et al. [4] demonstrated that elevated neutron flux accelerates creep deformation in zirconium alloys, particularly under tensile stress. This is consistent with the higher creep rates observed in the IFA-663 test, where tensile stress played a significant role in enhancing deformation. Similarly, [PERSON] et al. [5] highlighted the exacerbating effects of localized corrosion and crud deposition on cladding materials, suggesting that surface oxidation and corrosion contribute to weakening the structural integrity of fuel rods. Our study's inclusion of corrosion and crud effects in the mathematical model corroborates their findings, emphasizing the necessity of accounting for these phenomena when evaluating long-term cladding performance. However, while previous studies have generally focused on the primary effects of mechanical stress and neutron flux, the present study goes further by integrating the impact of corrosion and crud deposition into a comprehensive model. This model provides a more accurate prediction of M5 cladding behavior under complex operational conditions.