Study on internal strain distribution of the high-field Nb3Sn superconducting accelerator magnets with homogenization theory

To fabricate 10-16 Tesla high-field accelerator magnets, the high-Jc Nb3Sn superconductor is one of the most practical choices. However, it is a brittle material and is strain-sensitive. To reduce the potential Jc degradation of the conductor, it is crucial to accurately evaluate the stress of the coils owing to the large Lorentz force during magnet excitation. In the traditional stress analysis of magnets, the coils are usually regarded as homogenized structures using isotropic material coefficients; however, most superconducting (SC) coils have hierarchical and complicated structures. To obtain a more accurate estimation, it is necessary to consider the detailed internal structure of the coils during stress analysis. In this study, the homogenization method was applied to multi-scale stress analysis of a high-field accelerator SC magnet. The effective material coefficients of Nb3Sn coils were calculated using this method. Stress analysis of the magnet was performed using the effective material coefficients and isotropic material coefficients for comparison. Compared with the result obtained using isotropic material coefficients, the Von-Mises stress of the coil using effective material coefficients is larger at different load steps. The stress level of most Nb3Sn filaments in the strand was higher than that of the coil. The maximum stress of the Nb3Sn filaments appeared in the strand contact regions. In addition, there was a significant difference in the average stress of all materials in the strand. These findings will help develop more accurate analysis and simulation models for high-field SC magnets.


I. INTRODUCTION
Stress control is essential for the design and fabrication of high-field SC magnets. On the one hand, the SC materials used to fabricate the high-field SC coils, such as Nb3Sn, are stress-sensitive; on the other hand, the coils in the magnet sustain a large electromagnetic force during excitation. Presently, in the stress simulation of an SC magnet, the coil is regarded as a whole without an internal microstructure, and the material of the coils is regarded as an isotropic material. The material properties of the coil were obtained through experimental tests. However, the coils are complicated heterogeneous and hierarchical composites, including five scales, from a single SC filament to a full coil [1]. In addition, in a quench analysis of the magnet, it was found that the degradation of some coils was due to the fracture of filaments in the strand caused by excessive prestress [2]. Therefore, it is necessary to evaluate the stress of coils with internal microstructures. Usually, it is difficult to calculate the stress distribution of a magnet directly with these detailed microstructures in traditional methods, even with advanced computers.
Multi-scale analysis methods [1], [3]- [9] are appropriate solutions for the above problem. Multiple researchers [7], [9] have developed multi-scale models to study the mechanical behavior of Nb3Sn material between the strand and filament levels. The homogenization method [1], [4]- [6] is one such method, which can take into account micro-scale information at the macro-scale. The stress distribution in the composites can be accurately calculated using the effective coefficients with the homogenized structure. This method has been applied to calculate the effective coefficients of a variety of composites. It has also been used in the analysis of SC coils for nuclear fusion devices, and the simulation results are in good agreement with experimental results [5].
In this study, for the first time, the homogenization method has been applied to the stress analysis of an accelerator SC magnet, to study the mechanical behavior of Nb3Sn material between the magnet and filament levels. First, the related theory is introduced, and the effective material coefficient and stress recovery formulas are given. Secondly, based on the actual SC strand structure inside the SC coil, the homogenization finite element model of the strand is built, and effective material coefficients are calculated. Finally, a stress analysis of the SC magnet named LPF1 developed at IHEP [10], [11] is performed, which simulates three load steps: magnet assembly, cool down and excitation with effective material coefficients and isotropic material coefficients. The stress in the filament and strand is recovered and discussed based on the simulation results for the magnet during excitation.

II. BRIEF REVIEW OF THE HOMOGENIZATION THEORY
The homogenization theory developed in the 1970s was a limit theory [12]- [14]. In theory, asymptotic expansion and periodic assumptions are used to simplify differential equations. For some specific equations, the solutions are almost equivalent to those of the initial equation [15]. The macro-scale and micro-scale properties of the composites can be predicted using this theory. The following is a brief introduction to the basic concepts and formulas. The detailed derivation is described in [1], [4]- [6].
Here, two different scales are considered: one is the macro-scale x representing the composite, and the other is the micro-scale y representing the cell of periodicity of the composite Y. The ratio of the actual sizes of the two scales is the characteristic inhomogeneity dimension ε, which is a very small value. In this way, the material properties inside the composite are piecewise uniform [1].
where D is the Y-periodic fourth-order elastic tensor of the material. The governing equations are as follows: where superscript ε indicates that the problem is based on the characteristic inhomogeneity dimension, and superscript H represents the homogenization values. The meaning of the other symbols is that σ is the stress tensor, e is the strain tensor, α is the coefficient of thermal expansion, and θ is the temperature. When ε is close to be zero, Eqs. (3) becomes a limit form of Eqs. (2). The stress and strain in the equation depend on macro-variables and satisfy the general equilibrium equation. The new constitutive equation contains the effective material coefficient, which ensures the energy conservation of the equation [1].
For Eqs. (2), the displacement is expressed by the doublescale asymptotic expansion, as shown in Eqs. (4). u is the displacement tensor. The first-order component represents the macro-displacement field, and the second-order component represents the micro-displacement disturbance. Then, the first-order and second-order displacement are solved using the volume average and global integral. Finally, the effective elastic modulus, effective coefficient of thermal expansion, and local stress recovery formulas are obtained, as shown in Eqs. (5)-(7).
where χ is the displacement homogenization equation of the cell of periodicity. By solving these equations, the effective elastic modulus, effective coefficient of thermal expansion, and local stress of the material can be solved. Based on the above equation and the finite element software, a homogenization finite element program has been developed, and then the effective coefficient and local stress can be solved by the program. As stated, the SC coils are the above composite with a periodic structure, which is composed of strands and epoxy resin. Here, the macro-scale is the coil and the micro-scale is the strand. The homogenization method can be applied to calculate the effective coefficient of the coil. The effective coefficients mean the effective elastic modulus and effective coefficient of thermal expansion. The cell of periodicity is an impregnated strand.

A. THE CHARACTERIZATION OF THE SC STRAND FOR ACCELERATOR SC MAGNETS
To build a homogenization finite element model, it is necessary to obtain the geometric structure of the strand. In this study, a tested Nb3Sn SC coil was used as the experimental sample to obtain the geometric dimensions of the strand. First, the straight part of the coil was cut into coil segments. Then, the coil segment was made into the strand stack sample. The cross-section of the sample was mechanically polished using SiC pads. The grit size of the pad ranges from 320 to 1200. Then it was polished using diamond slurries of particle size (5μm, 3μm). Finally, the sample was characterized using an optical microscope. The optical image of the strand is shown in Fig. 1. The geometric dimensions of the internal structures of the strand are listed in Table I.

B. THE CALCULATION OF EFFECTIVE MATERIAL COEFFICIENTS
Based on the geometric dimensions, a homogenization finite model of the strand was built, as shown in Fig. 2. This model contains all microstructures of the strand, including the epoxy resin, OHFC Cu matrix, Nb3Sn filaments, and Sn rods. The properties of all materials in the impregnated strand are listed in Table II [16,17]. After the geometric model was built, periodic boundary conditions were applied to the opposite sides of the model. The prestress load derived from homogenization theory was applied to all domains of the model. The homogenization model was obtained and the homogenization equations were solved. The effective elastic modulus and effective coefficient of thermal expansion were calculated with the equations.  Table III lists the components of the effective elastic modulus and the effective coefficient of thermal expansion. Note that the subscript representation of the effective elastic modulus adopts the Voigt notation. The non-zero components of the material coefficients are listed. The components that are not listed are zero. Besides, the isotropic material coefficients of the Nb3Sn coil are listed in Table IV.

C. THE STRESS ANALYSIS OF THE SC MAGNET AND THE STRAND
Stress analysis of an accelerator SC magnet named LPF1 was performed. In this analysis, the stress in the central Nb3Sn coil was calculated with effective material coefficients and isotropic material coefficients respectively. The results were compared and discussed. Based on the symmetrical structure of the magnet, a 1/4 2D finite element model is built to calculate the stress distribution of the magnet. The crosssection of the magnet is shown in Fig. 3. The magnet is composed of two Nb3Sn coils and four NbTi coils. The detailed mechanical structure and design of the magnet are described in [10,11]. The stress simulations of the magnet include three load steps: assembly, cool down, and magnet excitation. By using the above simulation, the stress components at point A of the Nb3Sn coil were obtained during the excitation. Point A was located at the middle of the straight part of the Nb3Sn coil, where the highest field is located. Then the stress distribution of the strand at point A was recovered with the homogenization method. The Von-Mises stress of the Nb3Sn coil using effective material coefficients and isotropic material coefficients at different load steps are shown in Fig. 4. At the step of the magnet assembly, the stress distribution and maximum Von-Mises stress in the coil using different material coefficients shows a little difference, as shown in Fig. 4 left column. After the cool down and excitation, the difference becomes more and more significant. The maximum Von-Mises stress or strain calculated by using the effective material coefficients is larger than the value calculated with the isotropic material coefficients. The discrepancy in maximum Von-Mises stress or strain during the excitation is larger than that at the other load steps. It shows that the different material coefficients have a relatively large impact on the maximum Von-Mises stress or strain evaluation in the coil during the magnet excitation.

FIGURE 5. (A) Maximum Von-Mises stress and (B) Maximum Von-Mises strain in the Nb3Sn coil at different load steps with effective material coefficients and isotropic material coefficients.
In addition, compared with the maximum Von-Mises strain, the discrepancy in the maximum Von-Mises stress is relatively larger during excitation. In a traditional way of a study model, the maximum Von-Mises stress is used to evaluate the current-carrying capacity of the coil, the coil performance would be greatly overestimated in such type of model. Furthermore, the discrepancy between the maximum and average Von-Mises stress in the Nb3Sn coil is shown in Fig. 6. It also becomes more and more significant after the cool down and excitation using effective material coefficients. It shows that the stress distribution in the coil is more uneven after cool down and excitation. The overall stress of the coil is low but the local stress is high. Since the effective material coefficients are closer to the actual material coefficients, which reflects more accurately the actual mechanical behavior of materials. The maximum Von-Mises stress and the stress distribution are essential in the stress analysis of the magnet. Evaluating them accurately could better optimize the design of the magnet. The strain and stress components of the coil at point A are obtained from the simulation results at the excitation. The strain components εxx, εyy, and εxy are -3.75×10 -3 , -7.16×10 -3 , and -5.86×10 -6 respectively. The stress components σxx, σyy, and σxy are -0.954 MPa, -122.55 MPa, and -0.095 MPa respectively. With the strain components, the stress of the strand at this point was recovered. The stress distributions of the filaments inside the strand are shown in Fig. 7. It shows that the distribution of each stress component is quite different and the stress range is large. There are both tensile stress and compressive stress in the Nb3Sn filaments. The stress of most Nb3Sn filaments is at the same level, however, the stress of a few filaments at the edge of the strand is significantly larger than the other parts, possibly because it is the contact region between adjacent strands. This phenomenon was also observed in [3]. Considering the stress and strain sensitive nature of the high-field superconductors like Nb3Sn, it indicates that the current-carrying capacity of these filaments will degrade first when the stress level of the strand is high. The Maximum stress appears in the conjunction of the Nb3Sn filament and the Sn rod. Because the material properties of the Nb3Sn filaments are different from that of Sn rods. Thus, stress evaluation for the SC material will help us better understand the mechanical behavior of the SC material.
The difference in the average stresses of all materials of the impregnated strand is significant as shown in Fig. 8. Stress component σxx of epoxy resin, OHFC Cu, and Sn are tensile stress, but that of Nb3Sn is compressive stress. Note that stress component σxx at the macro-scale level is compressive. It indicates that the stress state of the materials in the strand is quite different from that of the coil. Meanwhile, the average stress of stress components σxx and σyy of the epoxy resin are larger. Epoxy resin will be cracked by high stress, which may become the origin of the quench. Stress evaluation of all materials will help find the origin of the quench. Furthermore, the stress level in the Nb3Sn filaments is much larger than that in the coil at point A. Because the stress in the other materials is lower than that in the filaments. Meanwhile, the stress at the macro-scale level is the average stress of the strand. In addition, the average stress of Nb3Sn is larger in all materials. The volume proportion of Nb3Sn in the strand is the largest. So, the stress state of Nb3Sn determines the stress state of the coil at the point A. In conclusion, the mechanical behavior of all materials inside the coil will directly affect the performance of the coil. An accurate stress evaluation of these materials will help to optimize the stress distribution in the SC strands and coils, to further improve the performance of the coil and avoid some quench origins.

V. CONCLUSION
In this study, a study model was developed for the multiscale stress analysis of the accelerator SC magnets by applying the homogenization theory. An Nb3Sn coil was cut open to make a strand block to obtain the geometric dimension of the strand for building the homogenization finite model. The effective material coefficients of the coil were calculated from the model. The stress analysis of the magnet was performed using different material coefficients. The results show that at the macro-scale, the maximum Von-Mises stress in the coil at three load steps using effective material coefficients is higher than that using isotropic material coefficients. The distribution of Von-Mises stress of the coil is more uneven using effective material coefficients. At the micro-scale, all stress components of the impregnated strand in the coil were recovered. The stress state, stress level, and stress distribution of all materials in the impregnated strand are obtained. This information will help us more accurately evaluate the stress state of the Nb3Sn filaments. It will also help the SC conductor and magnet designers to optimize the stress distribution in the SC strands and coils to further improve their performance.