## 1 Introduction  
Since the discovery of superconductivity more than 100 years ago, many diverse classes of superconducting materials have been found and subsequently studied in detail. Based on the different superconducting transition temperatures, the superconding materials can be classified into the low-temperature superconductors (LTS), which are represented by metals, alloys, and metal alloy materials, such as Nb\({}_{3}\)Sn, NbTi, et al., and the high-temperature superconductors (HTS). To date, more than 5000 superconducting materials have been discovered or synthesized all over the world [1]. Based on Bi (Bi-Sr-Ca-Cu-O [2; 3]), Tl (Tl-Ba-Ca-Cu-O [4]), RE (RE-Ba-Cu-O [5], as well as with the participation of several RE [6]), and Y (Y-Ba-Cu-O [7]), high-temperature superconducting materials are becoming more and more popular in recent years. HTS-superconducting tapes are used without and with a stabilizer as windings in transformers and superconducting electromagnets and superconducting current limiters. Compared with HTSs, Nb\({}_{3}\)Sn low-temperature superconducting materials are widely used in the core of large high-field magnets because of their mature and simple preparation, high-critical magnetic field (\(H_{\mathrm{c2}}\approx 25\) T), critical current [8], and stable mechanical properties in low-temperature environments. However, even today, after thousands of papers devoted to the materials have been published, understanding their nonlinear mechanical behavior remains a challenge. The magnetic coils of the TOKAMAK of the International Thermonuclear Experimental Reactor (ITER) project and the Large Hadron Collider (LHC) of the European Organization for Nuclear Research all use Nb\({}_{3}\)Sn superconducting materials [9; 10]. The Nb\({}_{3}\)Sn superconducting strand works under strong magnetic fields, cryogenic environmental, and high electric potential when ITER superconducting magnets are operating at 4.2 K. In addition, the temperature difference between the preparation temperature of about 1000 K and the operating temperature of 4.2 K will also produce corresponding thermal stress in the strands, which obviously affects the axial mechanical behavior of the strands [11; 12]. These articles illustrate that cracks and fractures in the filaments are the essential reasons for the reduction in the current-carrying capacity that restrict the design and application of the Nb\({}_{3}\)Sn superconductor used in engineering. Numerous experimental investigations have been performed to describe the mechanical properties of Nb\({}_{3}\)Sn strands [13; 14; 15; 16; 17; 18; 19; 20; 21]. [PERSON] et al. [15] developed a test system (Tarsis) to procure the axial tensile stress-strain relation of Nb\({}_{3}\)Sn strands and measured the mechanical properties of four different kinds of superconducting strands at various temperatures to determine the tensile constitutive. It is found that the mechanical characteristics of the four different kinds of superconducting strands are all temperature dependent. Subsequently, [PERSON] et al. [11] developed a system to determine the constitutive relations of ITER-related CICC cables under axial tension and compression, cyclic loading, and cable radial deformation. [PERSON] et al. [18] used the quantum beam technique to quantify the local strain of the helical and non-helical Nb\({}_{3}\)Sn strands in the cryogenic environment. Both axial and transverse thermal stress components in Nb\({}_{3}\)Sn filaments are numerically measured. In addition, the ENEA Centro Ricerche Frascati group in Italy and the Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences group in Poland have carried out extensive research on ITER and achieved promising results [22; 23; 24; 25; 26; 27; 28]. Several theoretical and numerical models have been presented to predict the mechanical properties of Nb\({}_{3}\)Sn strands [11; 29; 30; 31; 32]. For instance, [PERSON] et al.