1362 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007 Thermo-Mechanics of the Hierarchical Structure of ITER Superconducting Cables D. P. Boso, M. J. Leﬁk, and B. A. Schreﬂer Abstract—Recent experimental tests on the Model Coils have shown that the behavior of based cables is not as good as expected on the basis of the characteristics evaluated for the un- cabled strands. This degradation of the cable performance seems to be due to various factors, among which the strain state of the ﬁlaments due to bending and contact phenomena inside the cable. After these results it was decided that high performance strands will be used for ITER magnets, even if they have never been tested on a full size cable. Therefore the capability of evaluating the coil performance from the strands characteristics becomes a crucial point in the Research and Development (R&D) activity. Recently some extrapolations were attempted, but the mechanical model used is rather simpliﬁed and needs some ﬁtting parameters which are not known a priori. In this work we present a thermo-me- chanical model suitably developed to evaluate the strain state of a strand inside a superconducting (SC) cable. It is based on the idea of multiscale modeling, starting from a enriched formula- tion of the beam kinematics to take into account the ﬁbrous nature of a multiﬁlamentary strand. The method consists in performing a successive substitution of discrete models involving many beams with a single equivalent beam model, which behavior is identiﬁed from the preceding cabling stage. This recursive substitution allows to perform the analysis within a reasonable computational time. Once the stress and strain ﬁelds are obtained at the higher level, a suitable unsmearing technique gives the strain till the ﬁrst level, on the scale of the SC ﬁlament. The method is applied to the real case of the and CICC sub-size samples tested at FZK in Germany. Index Terms—Composite beam model, ITER superconducting cable, multiscale analysis, thermal and bending strain. I. INTRODUCTION D URING the last decade an extensive Research and Devel- opment (R&D) program has been performed to demon- strate the feasibility of the magnet system of the future Inter- national Thermonuclear Experimental Reactor (ITER) [1], [2]. The major elements of this program have been the construction and test of real scale coils (the Central Solenoid Model Coil CSMC, [3] and the Toroidal Field Model Coil TFMC [4]) as well as solenoid prototypes (the various Insert Coils [5]–[7]). In Manuscript received August 25, 2006. This work was supported in part by PRIN 2004094015_002 “Thermo-hydraulic-mechanical and Electro-Mechanical Modeling of ITER Superconducting Magnets” and by “KMM-NoE—Knowledge-based multicomponent materials for durable and safe performance—Network of Excellence”. D. P. Boso and B. A. Schreﬂer are with the Dipartimento di Costruzioni e Trasporti, Università di Padova, Via Marzolo 9, 35131 Padova, Italy (e-mail: boso@dic.unipd.it; bas@dic.unipd.it). M. J. Leﬁk is with Geotechnical Engineering and Engineering Structures, Technical University of Lódz, Al. Politechniki 6, 93-590 Lodz, Poland (e-mail: emleﬁk@p.lodz.pl). Digital Object Identiﬁer 10.1109/TASC.2007.897759 addition a series of tests is being completed on short samples, i.e. 2–3 m long straight segments of superconducting (SC) ca- bles. The testing of the model coils has provided valuable informa- tion to ﬁnalize the design of the ITER magnetic system. How- ever the behavior of based cables was not as good as expected on the basis of the characteristics evaluated for the un- cabled strands [4], [8]. This degradation in performance seems to be due to various factors, among which the strain state of the ﬁlaments due to bending and contact phenomena inside the cable. It is worth to mention that critical current de- pends upon the strain state, in addition to the applied magnetic ﬁeld and temperature. The effects of bending strain on the critical characteristics of a SC strand have recently been investigated both experimen- tally at the University of Twente (The Netherlands) and at JAEA (Japan) [9], [10] and numerically [11]. The possibility of evaluating the coil performance from the strand characteristics is a crucial point in the R&D activity, since after model coil results it was decided that high perfor- mance strands will be used for ITER magnets, and they have never been tested on a full size cable [12]. In some recent studies an extrapolation is attempted, from single strand or short sample information to magnet performances. The mechanical model used is however rather simple, and that approach needs some ﬁtting parameters, which are not known before the exper- imental tests themselves [13], [14]. In this work a thermo-mechanical model is presented, suitable to estimate the strain state of a strand inside a full size cable. It is based on the idea of multiscale modeling, starting from an en- riched formulation of the beam kinematics to take into account the ﬁbrous nature of the multiﬁlamentary strand. II. THE HIERARCHICAL BEAM MODEL The hierarchical beam model consists in performing a suc- cessive substitution of discrete models involving many beams with a single equivalent beam model, which behavior is identi- ﬁed from the preceding cabling stage analysis. At the level of the strand a beam-type element is used, endowed with an enriched kinematics to take into account its ﬁbrous structure [15]–[18]. In fact the cross section is not homogeneous and this composite nature plays a fundamental role on the strain state. Going up with the cabling stages, it would be impossible to model all the strands of the SC cable and their possible contact points in a reasonable computational time, so that we substitute, level by level, an equivalent beam element to the bundle of strands. For each cabling stage we consider a discrete model (called ) and an equivalent single-beam 1051-8223/$25.00 © 2007 IEEE