## 2 Numerical Analysis of the Constitutive Relationship of EAS-Nb\({}_{3}\)Sn Strands  
### The Hierarchical Homogenization Model of EAS-Nb\({}_{3}\)Sn Strands  
The overall properties of composite Nb\({}_{3}\)Sn strand not only depend on the components but also on the internal geometric structure [29]. Even though Nb\({}_{3}\)Sn fiber, bronze, and copper are all isotropic materials, Nb\({}_{3}\)Sn strands are constituted of orthotropic materials. As shown in Figure 1, the EAS-Nb\({}_{3}\)Sn strand contains around 8305 Nb\({}_{3}\)Sn fibers embedded in the bronze matrix, each fiber measuring just 2-3 \(\upmu\)m in diameter. As a result, complete numerical analyses of the mechanical properties of superconducting strands would be extremely computing intensive and unfeasible. To model the mechanical characteristics of the strand more effectively, the representative volume elements (RVEs) are employed in this study. Due to the clear hierarchical structure of EAS-Nb\({}_{3}\)Sn strands and the periodicity of fiber bundles [29], a three-level homogenization model could be utilized, as illustrated in Figure 2.  
Figure 1: Cross section of the EAS-Nb\({}_{3}\)Sn superconducting strand [32]. As shown in Figure 2, different hierarchical levels can be identified, from level 1 to level 3, starting from the fiber bundle composed of 55 Nb\({}_{3}\)Sn fibers and a bronze matrix (level 1), as shown in Figure 2c. The level 2 composite is composed of 151 groupings of level 1 fiber bundles embedded in bronze matrix, as shown in Figure 2b. Level 3 is the Nb\({}_{3}\)Sn strand compounded by the level 2 effective homogenous material surrounded by a copper and tantalum barrier layer, as shown in Figure 2a. The component materials and volume fractions of EAS-Nb\({}_{3}\)Sn strand are shown in Table 1.  
For the level 1 fiber bundle, considering the volume fraction and geometric structure of the Nb\({}_{3}\)Sn fibers, the level 1 RVE (geometric size of 5.2 \(\times\) 5.2 \(\times\) 1 \(\upmu\)m\({}^{3}\); fiber volume fraction \(V_{F1}\) = 0.5223), as shown in Figure 3c, would be selected to calculate its effective constitutiveness. Similarly, for the level 2 superconducting region, considering the volume fraction and the geometric structure of the Nb\({}_{3}\)Sn fiber bundle in the strands, the level 2 RVE (geometric size of 55 \(\times\) 100 \(\times\) 10 \(\upmu\)m\({}^{3}\); level 1 fiber bundle volume fraction \(V_{F2}\) = 0.7837) can be selected, as shown in Figure 3b, to calculate its effective stiffness matrix when the effective constituveness of level 1 is substituted, as depicted in the blue region in Figure 3b. The cross section of the homogenized EAS-Nb\({}_{3}\)Sn strand is shown in Figure 3a, and the 3D model is shown in Figure 4. The stress-strain behaviors of the EAS-Nb\({}_{3}\)Sn strand can be examined precisely from the micro- to macro-levels using the homogenization model. Calculating the effective elastic coefficients for level 1 and level 2 in the strand is important for mechanical analyses of the homogenization model. It is generally believed that Nb\({}_{3}\)Sn strands are an orthotropic material based on its internal structures (Figure 1). Therefore, the following calculation assumes that the stiffness matrix of level 1 and level 2 are both orthotropic. The constitutive relationship of the materials is as follows [32]:  
\[\mathbf{\sigma}^{(i)}=\mathbf{C}^{(i)}\mathbf{\varepsilon}^{(i)} \tag{1}\]  
where \(\mathbf{C}^{(i)}\) is the stiffness matrix of level \(i\), and the relationship with the flexibility matrix \(\mathbf{S}^{(i)}\) is  
\[\mathbf{S}^{(i)}=\left(\mathbf{C}^{(i)}\right)^{-1} \tag{2}\]  
The flexibility matrix of the orthotropic materials can be expressed through nine elastic constants, \(E_{1}^{(i)}\), \(E_{2}^{(i)}\), \(E_{3}^{(i)}\), \(v_{12}^{(i)}\), \(v_{13}^{(i)}\), \(v_{23}^{(i)}\), \(G_{12}^{(i)}\), \(G_{23}^{(i)}\), and \(G_{13}^{(i)}\). The subscripts 1, 2, and 3 denote