## 2 Numerical Analysis of the Constitutive Relationship of EAS-Nb\({}_{3}\)Sn Strands
### Fiber Damage Caused by Thermal Stress  
With the increasing loads, the brittleness of the Nb\({}_{3}\)Sn material results in fiber cracks and fractures, impacting the overall stress-strain behavior of the strand. Therefore, this research employed the Curtin-Zhou model [41,42] to investigate the effect of the axial fracture of Nb\({}_{3}\)Sn fibers. The Curtin-Zhou model was extended to include the effect of early thermal damage caused by a temperature difference between the preparation and operating environments. [PERSON] et al. [41,42] assumed that the growing stress is equally distributed across all fibers in the unbroken region around the breaking fiber, and stress concentration is neglected, as shown in Figure 5b. The fiber stress away from the breakpoint is a constant, \(S\)  
Figure 4: 3D model of the homogenized EAS-Nb\({}_{3}\)Sn strands (length \(L\) = 20 mm). Figure 3: (**a**) Effective homogenization model of the EAS-Nb\({}_{3}\)Sn strand (yellow region represents the level 2 effective superconducting region, red region represents the tantalum isolation layer, and orange region represents copper); (**b**) the RVE of the level 2 (blue region represents the level 1 homogeneous fiber bundle, and cyan represents the matrix); (**c**) the RVE of the level 1 (green region represents the Nb\({}_{3}\)Sn fiber, and cyan region represents bronze). The Poisson distribution of fiber breakpoints may be used to characterize the composites' axial dimensionless stress [43]:  
\[\frac{\widetilde{\sigma}}{V_{F}}=\frac{1}{\widetilde{\rho}}\Big{[}1-e^{ \widetilde{\rho}\widetilde{S}}\Big{]}+\frac{V_{M}}{V_{F}}\widetilde{\sigma}_{M} \tag{4}\]  
Here, \(\widetilde{\sigma}=\sigma_{F}/\sigma_{c}\) represents the dimensionless stress of the composite material, which depends on the value of the average of the fiber stress, \(\sigma_{F}\), and characteristic stress, \(\sigma_{c}\).\(\widetilde{S}=S/\sigma_{c}=E_{F}\epsilon/\sigma_{c}\) represents the dimensionless stress of the fiber far from the breaking point, where \(\epsilon\) is the far-field homogeneous strain.\(\widetilde{\rho}=\rho\delta_{c}\) represents the dimensionless damage density of the fiber,\(\widetilde{\sigma}_{M}=\sigma_{M}/\sigma_{c}\) represents the dimensionless yield stress of the matrix, \(V_{F}\) and \(V_{M}\) represent the volume fractions of fiber and matrix, respectively. The characteristic stress, \(\sigma_{c}\), and feature length, \(\delta_{c}\), can be obtained through the material-stretching mirror experiment. The relationship between \(\widetilde{\rho}\) and \(\widetilde{S}\) is given as follows [43]:  
\[\text{d}\widetilde{\rho}=e^{-\widetilde{\rho}\widetilde{S}}m\widetilde{S}^{m- 1}d\widetilde{S} \tag{5}\]  
where \(m\) is the Weibull modulus. The influence of fiber fracture on the constitutive relationship of composites can be considered by reducing the average stiffness of the fibers in this model. With the Nb\({}_{3}\)Sn fiber fracture, the average stress in the fiber may be calculated by setting the fiber volume fraction to one and the matrix volume fraction to zero in Equation (4):  
\[\widetilde{\sigma}_{F}=\frac{1}{\widetilde{\rho}}\Big{[}1-e^{\widetilde{\rho} \widetilde{S}}\Big{]} \tag{6}\]  
Nb\({}_{3}\)Sn strands operate in liquid helium (4.2 K); therefore, the temperature differential between the preparation temperature (\(\sim\)1000 K) and the extremely low-temperature conditions may result in significant thermal stress to Nb\({}_{3}\)Sn fibers, which is referred to as _'initial thermal damage'_.