Prediction of effective properties for composite superconducting strand and multi-stage cables

Abstract

Superconducting multifilamentary ${Nb}_{3}Sn$ strand and cables have complex microstructure and macrostructure which are used in the International Thermonuclear Experimental Reactor (ITER) superconducting magnet system. In a generic form, the superconducting strand is made of copper, multifilamentary ${Nb}_{3}Sn$ and bronze which is a composite. The twisted composite strands form the superconducting cables with typical multi-stage structures. Here we propose a numerical model focusing on the problem of characterizing the effective properties of composite superconducting strand and cables at microscopic and macroscopic scales. The effective elastic constants of the composite superconducting strand can be derived by involving the combination of the representative volume element (RVE) and the Mori-Tanaka method. Subsequently, the proposed method is applied to capture the effects of temperature on the effective elastic constants of the superconducting multi-stage cables. Results show the elastic constants predicted by this model agree well with existing theoretical predictions and available experimental data. This numerical model may be utilized to optimize the mechanical properties of the composite superconducting strand by tuning the constituent fractions and to control the tensile stiffness of superconducting cables by tuning the winding laps of the cables.

Introduction

${Nb}_{3}Sn$ 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 ${Nb}_{3}Sn$ strand is made of ${Nb}_{3}Sn$ filaments, bronze and copper which is regarded as a multifilament superconducting composite [2]. For the multifilament superconducting composite, there are thousands ${Nb}_{3}Sn$ filaments inlaid in the bronze matrix. In practical application, the ${Nb}_{3}Sn$ cable-in-conduit conductors (CICCs) are often subject to transverse electromagnetic forces which may lead to the damage and fracture of ${Nb}_{3}Sn$ strand [[3], [4], [5], [6], [7]]. Therefore, the prediction for the mechanical properties of ${Nb}_{3}Sn$ 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 ${Nb}_{3}Sn$ 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 ${Nb}_{3}Sn$ 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 ${Nb}_{3}Sn$ 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.