Abstract
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.
1 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. Nb3Sn superconducting wire strand has superior critical temperature and field, which has been widely used for next-generation 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]. Nb3Sn is brittle and strain-sensitive and is not suitable for extrusion [2,13,14]. The material is manufactured by first inserting uncompounded precursors of Nb3Sn into an ingot of bronze, then the ingot is drawn into a wire and finally the heat treatment is conducted to form Nb3Sn 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].
Nb3Sn strand contains many Nb3Sn filaments, which possess complicated hierarchical microstructures [15,16,17,18]. Nb3Sn 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 Nb3Sn 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 Nb3Sn 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 Nb3Sn filament groups. First, LEHT is a Multiphysics homogenization approach which can cope with the Multiphysics phenomenon for Nb3Sn filaments, including Heat transfer and thermomechanical loading. Secondly, the Nb3Sn 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 time-consuming 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) Nb3Sn 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.
2 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 Nb3Sn 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 Nb3Sn 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.
3 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 Nb3Sn 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 Nb3Sn strand are shown in Figure 4.
The general steps of LEHT are overviewed in this section for the readers’ interests:
3.1 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
Thermal conductivities [33]:
Thermomechanical behavior [16,32]:
where
3.2 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:
Thermomechanical behavior:
where the superscript “i” can be the “f” (fiber phase) or “m” (matrix phase).
3.3 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.
3.4 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
Thermomechanical behavior [34]:
where
From which all the remaining unknowns can be obtained. Then the internal thermal and mechanical distributions can be fully recovered.
3.5 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,
3.6 Step 6: Encapsulation of Multiphysics LEHT
The advantage of adopting the Trefftz concept in the LEHT is that the mesh discretization and pre- and post-processing 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.
4 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 in-house 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.
4.1 Thermal conductivity analysis
Figure 6 illustrates the comparison of homogenized axial thermal conductivity,
The volume fraction of Nb3Sn for unit cell at microscale is 0.24. It should be noticed that Boso et al. [13] utilized two different volume fractions at microscale for the calculation of axial thermal conductivity,
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,
4.2 Thermomechanical analysis
Figure 8 illustrates the comparison of homogenized mechanical properties and thermal expansion coefficients between LEHT’s and Boso et al. FEA predictions for unit cell with volume fraction of 0.42 for Nb3Sn filament at microscale. Nb3Sn’s main diagonal elasticity term is not sensitive 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 Nb3Sn 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,
After obtaining the Nb3Sn 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,
5 Results
5.1 Effects of volume fraction and temperature on homogenized thermal conductivities
Figure 11 illustrates the effect of volume fraction of a Nb3Sn filament at microscale on the homogenized axial thermal conductivity
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 Nb3Sn 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.
5.2 Effects of volume fraction and temperature on homogenized mechanical properties and thermal expansion coefficients
Figures 13 and 14 illustrate the effect of volume fraction of a Nb3Sn filament and temperature variation on the mesoscale homogenized mechanical properties,
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,
6 Conclusion
A comprehensive investigation of the homogenized thermal conductivity, mechanical properties, and thermal expansion coefficients has been conducted for the multiscale Nb3Sn 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 Nb3Sn 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:
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.
For the micro- and mesoscale thermal conductivity homogenization, the axial thermal conductivity decreases linearly as the increase of volume fraction of Nb3Sn filament or its group, while the transverse thermal conductivity decreases nonlinearly.
For microscale homogenization of a group of Nb3Sn 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 Nb3Sn 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.
The power of the LEHT technique is not only proved by its accuracy in the multiscale analysis of Nb3Sn 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.
Acknowledgment
Figures 1–3 in this paper are reprinted from: Boso DP et al. A multilevel homogenized model for superconducting strand thermomechanics. Cryogenics, 2005;45:257–271, [13], with permission from Elsevier.
Appendix
Herein, we list the governing/control equations that are solved through the present technique. The equilibrium equations in the cylindrical coordinate are expressed as [16,32,33]:
where
where the prime superscript stands for the fluctuating components of equations (4a–4c).
where
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Funding information: G. Wang acknowledges the support of National Natural Science Foundation of China (No. 12002303), the National Key Research and Development Program of China (No. 2020YFA0711701), and Fundamental Research Funds for the Central Universities (2020QNA4016). The corresponding author, W. Tu, acknowledges the support of Jiangsu University Research Initiation Fund for Senior Talents.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
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