Materials Today Communications
Prediction of effective properties for composite superconducting strand and multi-stage cables
Introduction
strand is a kind of superconductor material used in high fields poloidal field (PF) coils and central solenoid (CS) coils which are employed in International Thermonuclear Experimental Reactor (ITER) superconducting magnet system [1]. Typical strand is made of filaments, bronze and copper which is regarded as a multifilament superconducting composite [2]. For the multifilament superconducting composite, there are thousands filaments inlaid in the bronze matrix. In practical application, the cable-in-conduit conductors (CICCs) are often subject to transverse electromagnetic forces which may lead to the damage and fracture of strand [[3], [4], [5], [6], [7]]. Therefore, the prediction for the mechanical properties of strand is very important issues in the design of superconducting cables which are made of twisted strands. In recent years, some models have been proposed to predict the mechanical properties of superconducting strand. For example, Zhou et al used an equivalent model which can instead of the thousands filaments to give the effective moduli of the strand along axial direction [5]. A micromechanical model is adopted to characterize the mechanical behavior of the superconducting strand with twisted filaments [8]. Gao et al used the homogenization theory and Costello’s wire rope theory to determine the effective Young’s moduli of the hierarchical superconducting cable [9,10]. The equivalent Young’s modulus and Poisson’s ratio of the coil conductor are predicted by Zhou et al [11]. In addition, the axial thermal expansion coefficient of twist cable is predicted by experimental and modelling researches by Zhang et al [12]. These models focus on the single material of the superconducting strand, but the composite structures have not been considered in their studies. In this work, through considering the microscopic composite structures, a numerical model was established to model the effective properties in superconducting composite strand and multi-stage cables. As for the study of the effective properties of composite materials, there are many theoretical and numerical models have been proposed to investigate the composite elastic moduli due to the significance in predicting effective elastic constants of composite material. Prediction of the effective elastic moduli can be traced back to 1960s. Hashin and Rosen [13] developed the upper and lower bounds for composite elastic moduli based on the energy variational principles. Whitney and Riley [14] obtained closed-form analytical expressions for a composite’s elastic moduli based on the energy balance approach. Sun and Vaidya [15] developed a vigorous mechanical analysis using a representative volume element (RVE) to obtain the complete set of elastic constants for a unidirectional composite. Xia [16,17] established an explicit unified form of boundary conditions for a periodic RVE to produce all elastic moduli for unidirectional or angle-ply laminates. In the present paper, an appropriate micromechanics RVE is presented to calculate the elastic constants of superconducting composite strand firstly. Then, a numerical model is addressed to predict the tensile stiffnesses of the multi-stage cables which are made of twisted strands. In addition, the composite strand is brittleness which requires a special manufacturing condition [18,19]. Usually, the strand is heated up to the reaction temperature (900-950 K), then the strand is kept for several days at high temperature. Finally, it is cooled down to room temperature or to its operating conditions (about 4.2 K). Based on the above facts, a numerical model is developed to obtain the effective thermal properties due to the different thermal expansion properties of the strand material. Furthermore, the effect of the temperature on the tensile stiffnesses of multi-stage cables structures also are studied. The remainder of the paper is organized as follows. In section 2, we formally defined the problem under study and present a numerical model. Section 3 details a computational model which is used to validate the validity and applicable range of the numerical model. In section 4, the numerical results are detailed. Finally, we conclude with discuss and conclusions in section 5.
Section snippets
Micromechanical model and representative volume element
The strand is a composite material which contains three components: filaments, copper stabilizer and bronze matrix [20]. Generally, thousands of filaments are grouped into several plies and embedded in bronze matrix. Then, by using barrier material with copper to form the strand. According to the different barrier material, strand can be divided into different forms. For example, bronze route strand and internal tin strand use the Tantalum and unreacted and
Validate
To validate the proposed model, two examples are displayed to make a comparison with the results of literatures. The validity of the proposed scheme is demonstrated by comparisons with Sun’s theory and Boso’s results [15,19]. In Section 3.1, a unidirectional fiber composite is considered, the effective elastic moduli of the composite are determined by finite element analysis of the RVE. This is an excellent choice to study how to select the RVE of composites. In Section 3.2, a micro-macro scale
Numerical results and discussion
For the sake of predicting the effect of temperature on the effective properties of the whole FUR strand and the multi-stage cables, the elastic constants in the filament phase and the whole FUR strand phase must be determined before calculating the tensile stiffness of multi-stage cables varying with temperature. Fig. 9, Fig. 10 illustrate the Young’s modulus of filaments phase in the FUR strand versus temperature obtained from previous established model. Results of literature [19]
Conclusions
A numerical model was developed to investigate the effective elastic constants of the composite superconducting strand and cables at microscopic and macroscopic scales. This model is based on the RVE and Mori-Tanaka method which are typical composite approach. The effective elastic constants of the FUR strand and the effect of temperature on the effective elastic constants of multi-stage cables were predicted. The simulated results agree well with the results of existing literatures and
Declaration of Competing Interest
The authors report no declarations of interest.
Acknowledgments
The authors gratefully acknowledge the financial support provided by the Natural Science Foundation of China (NSFC 11772142, 11872196), the Fundamental Research Funds for the Central Universities (lzujbky-2015-176) and the 111 Project, B14044.
References (33)
For the ITER project team, An overview of the ITER project
Fusion Engineering and Design
(2007)Analysis of the effect of Nb3Sn strand bending on CICC superconductor performance
Cryogenics
(2002)- et al.
Mechanical behavior of Nb3Sn strands under transverse electromagnetic loads
Fusion Engineering and Design
(2016) - et al.
Fracture behavior of filament in Nb3Sn strands with crack-bridging model
Fusion Engineering and Design
(2016) - et al.
Electro-mechanical behaviors of composite superconducting strand with filament breakage
Physica C: Superconductivity and its applications
(2016) - et al.
Strain distributions in superconducting strands with twisted filaments
Composite Structures
(2017) - et al.
Study on the effective Young’s moduli of CICC strand with multi-stage structures
Fusion Engineering and Design
(2019) - et al.
A theoretical model for characterizing the internal contact of the CICC wires under axial strain
Acta Mechanica Solida Sinica
(2016) - et al.
Prediction of composite properties from a representative volume element
Composites Science and Technology
(1996) - et al.
A unified periodical boundary conditions for representative volume elements of composites and applications
International Journal of Solids and Structures
(2003)