## 1 Introduction
[29; 30; 31; 32] proposed a multi-scale numerical method for the composite strands based on the asymptotic homogenization theory. Effective behaviors of various scales were generated by identifying the representative volume elements (RVE) simulated using the finite element method. Although the computational results were relatively ideal, the helical structure of the Nb\({}_{3}\)Sn wire and the fiber fracture damage were not considered. [PERSON] et al. [34; 35] developed a three-dimensional, multi-scale nonlinear model to investigate the mechanical behavior of a superconducting strand and its filaments, simulating the thermal stress due to changes from the heat treatment temperature to the operational temperature (4.2 K). The stress state of superconductors and their critical current-carrying capacity were calculated by the multiscale model. Taking the helical structure into consideration, [PERSON] et al. [36] developed a three-dimensional theoretical model based on the Mori-Tanaka approach to evaluate the internal strain and performance degradation of a Nb\({}_{3}\)Sn superconducting strand subjected to mechanical and thermal stress. The helical structure of the superconducting strands was shown to be effective in this investigation. [PERSON] et al. [37] investigated the elastic-plastic behavior of Nb\({}_{3}\)Sn superconducting strands using the finite element technique. The fluctuation of effective parameters with temperature was calculated using the Mori-Tanaka technique and the finite element numerical approach with consistent results. However, the Nb\({}_{3}\)Sn fiber is prone to brittle fracture, and the effects of the filaments breakage on anticipated mechanical properties were necessary as was the inaccuracy introduced when bronze was treated as a linear elastic material. The mechanical properties of Nb\({}_{3}\)Sn materials are similar to those of ceramics; therefore, brittle fracture occurs when they are subjected to tiny deformations. As such, many experts have started to concentrate on investigating Nb\({}_{3}\)Sn filament fracture [41; 42; 43; 44; 45; 46; 47; 48]. [PERSON] and [PERSON] [41; 42] hypothesized that filament breakage is a random process related to axial stress, and the filament fracture length could be described using the Weibull distribution function. Based on the Curtin-Zhou theory and the'shear-lag model', [PERSON] [43], and [PERSON] et al. [44] investigated the influence of filament fracture and damage progression on the mechanical characteristics of LMI-Nb\({}_{3}\)Sn strands. [PERSON] et al. [45] established a two-dimensional crack bridge model to investigate the fracture behavior of filaments within the strand and calculated the stress intensity factors of modes I, II, and III crack tips, as well as the relative opening displacements between the crack surfaces. [PERSON] et al. [46] investigated the influence of artificially generated surface defects on the microstructure and the electrical parameters of two kinds of Nb\({}_{3}\)Sn superconductors in 2021. Through SEM observations, it was found that the current-carrying capacity of the superconductor decreased proportionally with the increases in defect size and number. When the depth of the prefabricated crack reached 0.44 mm, the critical current was reduced by 10%. Consequently, the effect of internal fiber damage on the physical properties and mechanical behavior of the strand must be considered. To analyze the temperature stress caused by heat treatment and the influence of the twisting process on the fracture damage of strands, [PERSON] et al. [47; 48] used laser-scanning microscopy to study Nb\({}_{3}\)Sn strands in TF CICC cable. Comparing the micrographs of strands before and after heat treatment under artificial bending, it could clearly be observed that the internal fibers of strands after heat treatment and repeated bending were severely damaged. Previous studies on Nb\({}_{3}\)Sn strands have mostly focused on their mechanical properties, including the damage and fracture caused by mechanical and electromagnetic loads.