## 2 Numerical Analysis of the Constitutive Relationship of EAS-Nb\({}_{3}\)Sn Strands
### Nb\({}_{3}\)Sn Strand Elastoplastic Uniaxial Tensile Study with Fiber Damage  
In the homogenization EAS-Nb\({}_{3}\)Sn strand model shown in Figure 4, the superconducting region was replaced by an effective homogeneous material, making it difficult to account for the plasticity of bronze at a specific strain, \(\epsilon_{z}\). One approach for considering matrix plasticity in a hierarchical homogenization model is the utilization of the secant modulus theory. It may be used to determine the degraded effective stiffness of a Nb\({}_{3}\)Sn strand homogenization model when fiber damage is present. The effective plastic stress and effective plastic strain of the elastic-plastic deformation materials can be expressed through the deviator stress and strain components, respectively:  
\[\overline{\sigma}=\left[\frac{3}{2}\left({\sigma^{\prime}}_{x}^{2}+{\sigma^{ \prime}}_{y}^{2}+{\sigma^{\prime}}_{z}^{2}+2{\tau^{\prime}}_{xy}^{2}+2{\tau^{ \prime}}_{yz}^{2}+2{\tau^{\prime}}_{xz}^{2}\right)\right]^{\frac{1}{2}} \tag{12}\]  
\[\overline{\epsilon}=\frac{1}{1+\mu}\left[\frac{3}{2}\left({\epsilon^{\prime}} _{x}^{2}+{\epsilon^{\prime}}_{y}^{2}+{\epsilon^{\prime}}_{z}^{2}+\frac{1}{2} {\gamma^{\prime}}_{xy}^{2}+\frac{1}{2}{\gamma^{\prime}}_{yz}^{2}+\frac{1}{2} {\gamma^{\prime}}_{xz}^{2}\right)\right]^{\frac{1}{2}} \tag{13}\]  
The secant modulus of the elastic-plastic deformation material is  
\[E_{s}=\frac{\overline{\sigma}}{\overline{\epsilon}} \tag{14}\]  
Assessed using the relationship between applied loads and fiber fracture, Section 2.2 effectively accounts for the influence of fiber damage in level 1 and level 2. The plastic behavior of the bronze matrix should also be incorporated in the analysis by considering the relationship between axial loads and the secant modulus. On the other hand, as an elastic-plastic material, the parameters of bronze depend on the stress state at different positions. It may consider more than only the effect of axial deformation on material properties. Therefore, it is necessary to determine the partial strain component of bronze to calculate the secant modulus using Equations (12)-(14). For this reason, we need to employ the iterative approach shown in Figure 6 to determine the actual relationship between strand stress and strain. The main steps are as follows:  
1. Apply uniaxial tensile strain, \(\varepsilon_{z}\), to the EAS-Nb\({}_{3}\)Sn strand by fixing the axial displacements pointwise at the left end: \[u_{z}=0\] (15) and the axial displacements pointwise at the left end: \[u_{z}=L\times\varepsilon_{z}\] (16) While considering the fiber damage and plasticity of copper, then calculate and output the average strain tensor, \(\varepsilon_{L2}\), of level 2 homogenization part of the strand. 2. Apply \(\varepsilon_{L2}\) to level 2 RVE (Figure 3b) by setting the displacements of the points on the level 2 RVE surface: \[u_{i}=(\varepsilon_{L2})_{ji}x_{j}\] (17) in which \(x_{j}\) (\(j=1\), 2, 3) are the \(x\), \(y\), and \(z\) coordinate values. \(u_{i}\) (\(i=1\), 2, 3) represents the displacements along the \(x\), \(y\), and \(z\) directions. Calculate and output the average strain, \(\varepsilon_{L1}\), of the level 1 homogeneous part of the strand and the deviant strain, \(\varepsilon^{\prime}{}_{B2}\), of the bronze matrix. 3. Apply \(\varepsilon_{L1}\) to the level 1 RVE (Figure 3c) by setting the displacements of the points on the level 2 RVE surface: \[u_{i}=(\varepsilon_{L2})_{ij}x_{j}\] (18) where, \(x_{j}\) and \(u_{i}\) are the \(x\), \(y\), and \(z\) coordinate values and displacements in the corresponding directions, respectively. Calculate and output the deviant strain, \(\varepsilon^{\prime}{}_{B1}\), of the first-class bronze matrix. Through the above stepwise calculations, the deviator strain components of the different levels of the bronze matrix of strands could be obtained.