An effective thermal conductivity and thermomechanical homogenization scheme for a multiscale Nb3Sn filaments

A comprehensive study of the multiscale homogenized thermal conductivities and thermomechanical properties is conducted towards the filament groups of European Advanced Superconductors (EAS) strand via the recently proposed Multiphysics Locally Exact Homogenization Theory (LEHT). The filament groups have a distinctive two-level hierarchical microstructure with a repeating pattern perpendicular to the axial direction of Nb3Sn filament. The Nb3Sn filaments are processed in a very high temperature between 600 and 700°C, while its operation temperature is extremely low, −269°C. Meanwhile, Nb3Sn may experience high heat flux due to low resistivity of Nb3Sn in the normal state. The intrinsic hierarchical microstructure of Nb3Sn filament groups and Multiphysics loading conditions make LEHT an ideal candidate to conduct the homogenized thermal conductivities and thermomechanical analysis. First, a comparison with a finite element analysis is conducted to validate effectiveness of Multiphysics LEHT and good agreement is obtained for the homogenized thermal conductivities and mechanical and thermal expansion properties. Then, the Multiphysics LEHT is applied to systematically investigate the effects of volume fraction and temperature on homogenized thermal conductivities and thermomechanical properties of Nb3Sn filaments at the microscale and mesoscale. Those homogenized properties provide a full picture for researchers or engineers to understand the Nb3Sn homogenized properties and will further facilitate the material design and application.


Introduction
A variety of high-performance superconducting wires [1][2][3][4] has been actively pursued and developed to meet the requirements of different engineering applications. Nb 3 Sn superconducting wire strand has superior critical temperature and field, which has been widely used for nextgeneration magnets for high-energy physics and fusion, including the KSTAR tokamak in South Korea [5], ITER CS model in Japan [6], a high-field Very Large Hadron Collider (VLHC) [7], ITER [1,2,8,9], as well as many other applications above 13 T [10][11][12]. Nb 3 Sn is brittle and strain-sensitive and is not suitable for extrusion [2,13,14]. The material is manufactured by first inserting uncompounded precursors of Nb 3 Sn into an ingot of bronze, then the ingot is drawn into a wire and finally the heat treatment is conducted to form Nb 3 Sn filaments. The typical heat treatment requires hundreds of hours between 600 and 700°C in a vacuum or inert gas to prevent bronze from oxidation [13]. The superconductor strand serves as the basic building block for the superconducting cables, such as cable-in-conduit conductors (CICC), Rutherford cables [7], which are designed and manufactured based on the specific commercial or scientific applications. For instance, CICC is utilized for ITER and over thousand strands are twisted together following a multistage packaging process and then inserted into a metal jacket which sustains the magnetic load and contains the liquid helium (4.2 K) during the operation stage [15].
Nb 3 Sn strand contains many Nb 3 Sn filaments, which possess complicated hierarchical microstructures [15][16][17][18]. Nb 3 Sn filaments are processed in a high temperature (600-700°C) and operate under an extremely low temperature (−269°C). An effective Multiphysics micromechanics model to predict the thermal conductivities and thermomechanical properties of the Nb 3 Sn strands would accelerate the material development, reduce the cost of material characterization, and further facilitate the cable analysis and design [19][20][21][22]. In general, the micromechanics methods can be categorized into two categories [19] ( Pindera et al. 2009). One is the microstructural detail-free estimates which do not consider the actual geometry of the material microstructures. In contrast, another one considers the real microstructures of the materials. The classic approaches fall into microstructural detail-free category, such as Reuss and Voigt estimates, self-consistent scheme, general self-consistent scheme, Mori-Tanaka scheme, Composite Spherical/Cylinder model (CCA and CSA), and Halpin-Tsai theory [19,[23][24][25]. For instance, general self-consistent scheme [15] and Mori-Tanaka scheme [26] are utilized to estimate the effective mechanical and thermal expansion properties for the Furukawa strand. In the category of considering material's real microstructures, Finite Element Approach (FEA) has become the main approach to calculate the homogenized properties of Nb 3 Sn filament groups via a 3D Representative Volume Element (RVE) [13,17,[27][28][29][30] due to its ability to consider complex microstructures in great detail and the popularity of general commercial FEA software represented by ABAQUS and ANSYS.
In finite element framework, the geometric microstructures are divided into subregions in which the local displacement fields are approximated by shape functions. A large system of algebraic equations for the unknown coefficients in the local approximations is constructed by satisfying the governing field equations using variational approach. In the field of multiscale homogenization analysis, the computational cost could increase dramatically because each material point at the upper scale is represented by a microstructure in the lower scale. The Multiphysics Locally Exact Homogenization Theory (MLEHT), an analytical approach, possesses great potential to determine the homogenized thermal conductivities and thermomechanical properties and recover the local fields in an efficient and accurate way. LEHT is initially developed by Pindera and coworkers for the hygro-thermomechanical responses of unidirectional composites [16,31,32] and has been extensively validated against FEA [31,32] and Finite Volume approach [16,32]. LEHT is a natural fit for the homogenization of Nb 3 Sn filament groups. First, LEHT is a Multiphysics homogenization approach which can cope with the Multiphysics phenomenon for Nb 3 Sn filaments, including Heat transfer and thermomechanical loading. Secondly, the Nb 3 Sn filaments usually possess a distinctive hierarchical microstructure and multiscale homogenization is required to predict the effective properties and LEHT has been proven itself as an idea candidate for the multiscale applications due to its distinctive features of user-friendliness, accuracy, and efficiency [32]. First, LEHT is an analytical approach without the need of timeconsuming preprocessing and post-processing of widely utilized commercial software. The method utilizes Fourier series expansions to represent the displacement fields. Secondly, the equilibrium equations and continuity conditions are satisfied in an exact way based on the method's analytical nature, so the method's accuracy is guaranteed from the fundamental level without the worry of mesh convergence. Thirdly, a balanced variational principle is applied for the periodic boundary condition of repeating unit cells to ensure rapid convergence of the Fourier series coefficients with a relatively small number of harmonic terms [31]. A full set of homogenized thermal conductivities and thermomechanical properties can be generated in less than 10 s on a personal computer.
The rest of the manuscript is organized as follows: Section 2 establishes a hierarchical model for the European Advanced Superconductors (EAS) Nb 3 Sn filament groups utilized for the ITER D-shaped magnet. Section 3 summarizes the key steps of Multiphysics LEHT, including Distinguishment of the Homogenized and Localized Responses, Derivation of Internal Expressions, Imposition of Continuity Conditions, Implementation of Periodic Boundary Conditions, Establishment of Homogenizations, and Encapsulation of Multiphysics LEHT. Section 4 validates the LEHT's effectiveness in predicating the homogenized thermal conductivities and thermomechanical properties against FEA prediction. Following the method validation, section 5 conducts a systematic investigation towards the effects of volume fraction and temperature on the two-scale homogenized thermal conductivities and thermomechanical properties. Section 5 concludes this presentation.

Model establishment
The hierarchical structure of an EAS strand, illustrated in Figure 1, can be analyzed from a multiscale perspective. First of all, a global cross section of an EAS strand is illustrated in Figure 1 (left) at the macroscale, containing the outer copper matrix, tantalum thick barrier, and around 55 groups of filaments, which are periodically arrayed in a bronze matrix. To facilitate the macroscale analysis, the thermal conductivities and thermoelastic properties of those mesoscale groups of filaments ( Figure 1 center) can be further characterized through the microscale unit cells that contain about 85 polygonal Nb 3 Sn filaments embedded in the bronze matrix, see Figure 1 (right). The single filament diameter is 2-5 μm, the diameter of a group of filaments is about 55-60 μm [13], and tantalum barrier diameter is 0.507 mm [15]. It is seen that the Nb 3 Sn filaments or filament groups are all periodically laid out in a hexagonal fashion, producing the isotropic in-plane properties, just shown as Figures 2 and 3 magnify the mesoscale structure of filament groups (Figure 1 center) and microscale structure of a group of filaments ( Figure 1 right) that are mapped in periodically hexagonal fashion (that are characterized by the yellow highlight). A multiscale LEHT framework is employed to generate the homogenized thermal conductivities and thermomechanical properties of unit cells of different scales.

Overview of multiscale LEHT
Relative to other micromechanics models, the theoretical development of the LEHT is mainly based on the Trefftz concept, where the internal expressions of thermal conductivities or mechanical fields of the composites' constituents are first obtained through solving the partial differential equations, whose solutions are represented through series expansions. The internal expressions with unknown coefficients are solved through imposing the continuity conditions at the fiber-matrix interface, as well as the weak-form periodic boundary conditions. The effective coefficients of a composite are finally derived through the homogenization theory. The macroscale properties of an EAS strand with two-level hierarchical microstructures can be obtained through a two-level multiscale homogenization process. First, the effective properties of a group of Nb 3 Sn filaments can be obtained through a homogenization process at microscale. Then the calculated homogenized properties are fed into the analysis of mesoscale unit cell, from which the effective properties of the EAS strand at macroscale are predicted. In contrast, the localization process passes down loads from macroscale to microscale. The local stress and strain fields of a   mesoscale unit cell are predicted once the macroscale strain/stress is specified. Then the local stress/strain fields of a microscale unit cell are determined by applying the corresponding mesoscale stress/strain. The date flow of multiscale homogenization and location of hierarchical microstructures of Nb 3 Sn strand are shown in Figure 4.
The general steps of LEHT are overviewed in this section for the readers' interests:

Step 1: Distinguishment of the homogenized and localized responses
In this presentation, we establish a two-scale homogenization framework with microstructures. Thus, herein we define each upper scale as the homogenized scale, while its lower scale as the localized microstructure. The homogenized properties of the upper scale are obtained through homogenization of the lower-scale microstructure, while the homogenized response of upper scale is transmitted to stimulate the lower-scale reaction. For instance, the governing equations (1) and (2) for temperature T x y , ( ) and the displacement fields u x y , i ( ) can be decomposed into the homogenized and fluctuating parts: Thermal conductivities [33]: Thermomechanical behavior [16,32]: = ( ) and y y y y , , 1 = ( ) are the upper-scale homogenized and lower-scale localized coordinates, respectively. i = 1 is longitudinal direction, while i = 2,3 is transverse cross section. The average and fluctuating parts denote the homogenized and localized responses, respectively.

Step 2: Derivation of internal expressions
To recover the localized response within lower-scale microstructure, the solutions of the governing partial differential equations for fluctuating fields presented in the Appendix are obtained by adopting the series functions [16,32,33]: Thermal conductivities: where the superscript "i" can be the "f" (fiber phase) or "m" (matrix phase). r is the parameter of fiber radius and can be utilized as a dimensional parameter, β nj and p nj are the eigenvalues and eigenvectors that are obtained in solving the differential governing equations listed in the Appendix. Note that several unknown coefficients, I nj , H nj , F nj , G nj in the equations (3) and (4) are introduced and need to be obtained through the continuity and boundary conditions in the following steps. What's more, the expressions of heat fluxes (e.g., q r , q θ ) and stress-strain (e.g., σ zr , σ rr , σ rθ ) fields are also readily derived.

Step 3: Imposition of continuity conditions
The aforementioned solutions are explicitly expressed in the cylindrical coordinates for the easiness of imposing continuities. For a perfectly bonded interface, the continuities are established at the fiber-matrix interface: Thermal conductivities: Thermomechanical behavior: from which the unknown fiber coefficients are expressed in terms of matrix coefficients, where a denotes the radius of inhomogeneity.

Step 4: Implementation of periodic boundary conditions
For unit cells that are periodically arranged within the lower levels, the periodic boundary conditions need to be imposed. In this work, weak-form boundary conditions are employed to guarantee the numerical stability: Thermal conductivities [33]: where Q S qn S d and T T 0 = are the periodicities of heat flux and temperature components. n i is the unit normal of surface S j 1, 2, , 6 j ( = … ) of hexagonal unit cells. Thermomechanical behavior [34]: From which all the remaining unknowns can be obtained. Then the internal thermal and mechanical distributions can be fully recovered.

Step 5: Establishment of homogenizations
The effective thermal and elastic coefficients at higher levels can be generated through generalized homogenized equations: Fourier's law of heat conduction: Homogenized constitutive relation: where, q i and H j are the effective heat flux component and temperature gradient, respectively, while σ ij and ε kl are the stress and strain components, respectively. T Δ is the change of temperature, the K ij * , C ijkl * , and α ij * are the effective thermal conductivities, elastic, and thermal expansion matrices, respectively, from which the thermal conductive and thermoelastic coefficients in all directions can be readily obtained.

Step 6: Encapsulation of Multiphysics LEHT
The advantage of adopting the Trefftz concept in the LEHT is that the mesh discretization and pre-and postprocessing are avoided in the in-house MATLAB programs. Based on this point, we encapsulated the program into "black-boxes" with only input/output (I/O) connections [32]. Users only need to input the geometric and material properties of the composite materials and the programs will automatically generate the effective coefficients and localized responses within a few seconds. Based on the theoretical derivations, we developed our own in-house programs to conduct numerical validations and parametric investigations.

Validation
To validate Multiphysics LEHT's effectiveness to predict the homogenized thermal conductivities, mechanical and thermal expansion coefficient generated via Multiphysics LEHT has been compared against the results generated by ABAQUS-based FEA [13] and an in-house FEA calculation [35][36][37]. Figure 5 shows the mesh utilized in our inhouse FEA analysis. A fully anisotropic two-dimensional Q8-type Multiphysics quadratic element has been constructed under the generalized plane strain constraint and the element number is 1152. Figure 6 illustrates the comparison of homogenized axial thermal conductivity, K 11 ⁎ , and transverse thermal conductivities, K 22 ⁎ , K 33 ⁎ , between LEHT and Boso's FEA predictions [13] for the hexagonal unit cell at microscale, in which Nb 3 Sn is surrounded by the bronze matrix, as shown in Figure 3. Nb 3 Sn's thermal conductivity is insensitive to the temperature variation and its value is exceedingly small. The Poisson's ration of Nb 3 Sn is 0.3 [27]. In contrast, bronze possesses an interesting characteristic regarding its thermal conductivity towards the temperature variation. In general, the thermal conductivity's variation caused by the temperature change could be divided into three stages. Initially, the thermal conductivity increases dramatically from 0 K and reaches a peak value, 2,400 W/(m K) at 33.3 K, and then decreases sharply to 1,000 W/(m K) at 55 K. In the second stage between 55 and 200 K, bronze's thermal conductivity decreases gradually and reach a stable value of 386 W/(m K) at 200 K. During the rest of temperature variation, the thermal conductivity almost maintains a constant value, 386 W/(m K). And the Poisson's ratio of Bronze is 0.34 [27].

Thermal conductivity analysis
The volume fraction of Nb 3 Sn for unit cell at microscale is 0.24. It should be noticed that Boso et al. [13]  Meanwhile, in Boso's estimation [13], a quasi-hexagonal unit cell is utilized for the analysis, but K 22 ⁎ 's and K 33 ⁎ 's predictions are almost identical. Out of simplicity, a hexagonal unit cell is utilized in our analysis and the homogenized transverse thermal conductivity K 22 ⁎ is equal to K 33 ⁎ . Figure 6 shows excellent agreement between LEHT's prediction and Boso et al. FEA [13] estimation for the transverse thermal conductivities, K 22 ⁎ and K 33 ⁎ , and between LEHT's prediction and Rule of Mixture's (RoM) estimation for the axial thermal conductivity, K 11 ⁎ .
After obtaining the homogenized thermal conductivities for the RUC representing a filament group at microscale, the homogenization of all filament groups at mesoscale could be conducted for a hexagonal unit cell in which a homogenized filament group is embedded in the bronze matrix, as illustrated in Figure 2. Figure 7 illustrates the comparison of transverse thermal conductivities, K 22 ⁎ and K 33 ⁎ , generated LEHT and Boso et al. FEA predictions [13] and the comparison of homogenized axial thermal conductivity, K 11 ⁎ , calculated by LEHT and the Rule of Mixture (RoM) with volume fraction of 0.72 at mesoscale. Good agreement for both axial and transverse conductivity coefficients is obtained. Therefore, the accuracy of the LEHT in predicting the homogenized thermal conductivity is validated for both a filament group at microscale and multiple filament groups at mesoscale. Also, its worth to mention there is no laborious work involved in LEHT's prediction, such as mesh discretization, periodic boundary imposition, and post-processing for homogenization properties as in Boso's calculation in ABAQUS with 3D FEA model and only unit cell geometric information and constituents' properties are required. Meanwhile, LEHT's calculation is highly efficient due to its trait of an analytical solution and only 0.7 s is required to generate a full set of homogenized thermal conductivities on a PC with Windows 10 64-bit operation system, Intel 2.9 GHz i7-10700 CPU, and 16 GB RAM. to temperature change, but the elasticity term of matrix material, Bronze, decreases significantly as the increase of temperate till 500 K and then maintain a constant value of 80 GPa. Regarding the thermal expansion coefficients, both Nb 3 Sn and filament phases increase linearly as the temperature increases. Figure 8 (Left) shows good agreement between LETH's and Boso et al. [13] prediction for the homogenized diagonal elasticity terms, Figure 8 (Right) shows good agreement for the homogenized axial thermal expansion coefficient, α 11 ⁎ , and the transverse homogenized thermal expansion coefficients, α 22 ⁎ , α 33 ⁎ , generated by LEHT and Boso et al. [13].

Thermomechanical analysis
After obtaining the Nb 3 Sn homogenized mechanical properties and thermal expansion coefficients for the RUC representing a filament group at microscale, the homogenization of multiple filament groups at mesoscale could be conducted for the hexagonal RUC in which a homogenized filament group is embedded in the bronze matrix. It should be noticed that in Boso's FEA estimation, volume fraction 0.72 is utilized to generate all homogenized mechanical properties, but volume fraction 0.90 is utilized to generate all the homogenized thermal expansion, α 22 ⁎ , α 33 ⁎ , except α 11 ⁎ , Figure 9 (Right). To maintain consistency, volume fraction 0.72 is utilized to calculate all homogenized thermomechanical properties in LEHT. As shown in Figure 9 (Left), good agreement is obtained for the homogenized mechanical properties at mesoscale. To further validate LEHT's thermomechanical homogenization capability, an in-house FEA program [35][36][37] is utilized to calculate the homogenized main diagonal elasticity terms and the thermal expansion coefficients, as shown in Figure 10, and excellent agreement is achieved. Therefore, the effectiveness of the LEHT in predicting the homogenized mechanical properties and thermal expansion coefficients at both microscale and mesoscale levels is validated. Meanwhile, as the homogenization of thermal conductivity, the thermomechanical homogenization is very efficient and only 17 s is utilized to calculate the homogenized thermomechanical properties on a PC with Windows 10 64-bit operation system, Intel 2.9 GHz i7-10700 CPU, and 16 GB RAM. The efficiency facilitates the proposed technique in the material's design and practice, especially when combined with the optimization technique. It should also be pointed out that when validating against the FE-based ABAQUS simulations, we employ a different strategy. For instance, here we didn't employ the mesh discretization, but directly obtain the analytical internal solutions. We also utilize the weak-form variational principle to enforce the boundary periodicities, while ABAQUS directly impose periodic boundary conditions in a node-tonode fashion. Figure 11 illustrates the effect of volume fraction of a Nb 3 Sn filament at microscale on the homogenized axial  After generating homogenized thermal conductivities of a filament group at microscale, the homogenized thermal conductivities of multiple filament groups can be obtained by analyzing a hexagonal RUC with a circular homogenized filament group embedded in the  bronze matrix. Figure 12 shows the effects of mesoscale volume fraction of the homogenized Nb 3 Sn filament group on the global homogenized thermal conductivities between 5 and 800 K. The mesoscale homogenized thermal conductivity coefficients are first calculated with a given volume fraction of 0.72 between 5 and 800 K. Both the global homogenized axial and transverse thermal conductivities decrease as volume fraction of a homogenized filament group increases. The upper bound of thermal conductivity converges to the thermal conductivity of bronze and the lower bound of thermal conductivity converges to the thermal conductivity of mesoscale homogenized filament group at volume fraction 1. In contrast with the linear relation with the volume fraction for the axial thermal conductivity at a prescribed temperature, the homogenized transverse thermal conductivity varies nonlinearly with the volume fraction at a prescribed temperature. respectively. The maximum homogenized elasticity terms, D 11 ⁎ , D 22 ⁎ , D 33 ⁎ , occur at the lowest temperature level with only filament material Nb 3 Sn and the minimum value occurs at the highest temperature level with only the bronze matrix. As filament volume fraction increases, the temperature influence on the homogenized mechanical properties decreases due to temperature insensitivity of Nb 3 Sn. The maximum homogenized thermal expansion coefficients, α 11 ⁎ , α 22 ⁎ , α 33 ⁎ , occur at the highest temperature level with only the bronze matrix and the minimum value occurs at the lowest temperature with only filament phase, Nb 3 Sn. Meanwhile, as the filament volume fraction increases, the temperature influence on the homogenized thermal expansion coefficients decreases because the filament phase Nb 3 Sn is less sensitive to the temperature change.

Effects of volume fraction and temperature on homogenized mechanical properties and thermal expansion coefficients
Once the homogenized mechanical properties and thermal expansion coefficients of a filament group on the microscale are available, the global homogenized mechanical properties and thermal expansion coefficients of all the filament groups can be obtained by conducting thermomechanical homogenization towards the hexagonal RUC with a circular homogenized filament group embedded in the bronze matrix. Figures 15 and 16 illustrate the effect of mesoscale volume fraction and temperature variation on the global homogenized mechanical properties, D 11 ⁎ , D 22 ⁎ , D 33 ⁎ , and thermal expansion coefficients, α 11 ⁎ , α 22 ⁎ , α 33 ⁎ , respectively. During above calculation, the utilized volume fraction of a Nb 3 Sn filament group at microscale is 0.42. The homogenized elasticity terms decrease as temperature increases due to the temperature    softening effects on the bronze matrix and increase as the increase of volume fraction due to the increase of harder phase of Nb 3 Sn. The global homogenized thermal expansion coefficients increase as the temperature increases due to the temperature effects on the bronze matrix and decrease as the increase of volume fraction due to the smaller mesoscale homogenized thermal expansion coefficients.

Conclusion
A comprehensive investigation of the homogenized thermal conductivity, mechanical properties, and thermal expansion coefficients has been conducted for the multiscale Nb 3 Sn filament groups centered in an EAS strand utilized in ITER D-Shape magnet. The filament groups are modelled by two-scale repeating hexagonal unit cells via the Multiphysics LEHT, including thermal conductivity analysis and thermomechanical analysis. First, LEHT's results are validated against the Finite Element Analysis at both microscale and mesoscale homogenization. Then, a comprehensive parametric study is conducted for Nb 3 Sn filament groups with the consideration of various volume fractions and temperature on both microscale and mesoscale thermal conductivities, mechanical properties, and thermal expansion coefficients. The principle conclusions of this investigation are as follows: 1. A comprehensive parametric study is conducted for the first time towards the effects of volume fraction and temperature variation on both mesoscale (a filament group) and global scale (multiple filament group) homogenized thermal conductivities, mechanical properties, and thermal expansion coefficients. 2. For the micro-and mesoscale thermal conductivity homogenization, the axial thermal conductivity decreases linearly as the increase of volume fraction of Nb 3 Sn filament or its group, while the transverse thermal conductivity decreases nonlinearly. 3. For microscale homogenization of a group of Nb 3 Sn filament, all elasticity terms increase, and all thermal expansion coefficients decrease as the increase of volume fraction, and both variations are much more sensitive to high temperature. For mesoscale homogenization of Nb 3 Sn filament groups, all elasticity terms increase, and all thermal expansion coefficients decrease as the increase of volume fraction, but the variation is not sensitive to temperature change. 4. The power of the LEHT technique is not only proved by its accuracy in the multiscale analysis of Nb 3 Sn filaments through a bottom-up procedure. As is also indicated in the text, the LEHT is also advantageous in its efficiency, which can be utilized and combined with reliable optimization techniques for the materials' design through a "top-down" procedure. Aiming at certain effective engineering requirements, the micromechanics-oriented optimization would provide a robust tool in varying the microstructural details, such as candidate materials and volume fractions. Most of those analytical scenarios can be even achieved through a PC instead of large-scale workstations.