The Next European Dipole (NED) activity is aimed at the development of a large-aperture, high-field superconducting magnet relying on high-performances Nb$_{3}$Sn conductors. Part of the NED program is devoted to the mechanical study of a new generation of Nb$_{3}$Sn wires and to predict and describe their behavior under the severe loading conditions of the cabling process. The deformation resulting from the cabling process was simulated through mechanical analyses by Finite Elements (FE). The ensuing results of FE analyses are presented, allowing the wire behavior under simple uni-axial loads to be described. They are compared to cross section micrographs of deformed wires

Boutboul, Thierry

Devred, Arnaud

Fabbricatore, P

Farinon, S

Leroy, Daniel

Oberli, Luc-Rene

CERN Document Server

1136 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007Finite Element Model to Study the Deformations ofNb3Sn Wires for the Next European Dipole (NED)Stefania Farinon, Thierry Boutboul, Arnaud Devred, Pasquale Fabbricatore, Daniel Leroy, and Luc OberliAbstract—The Next European Dipole (NED) activity is aimed atthe development of a large-aperture, high-field superconductingmagnet relying on high-performances Nb3Sn conductors. Partof the NED program is devoted to the mechanical study of a newgeneration of Nb3Sn wires and to predict and describe their be-havior under the severe loading conditions of the cabling process.The deformation resulting from the cabling process was simulatedthrough mechanical analyses by Finite Elements (FE). The ensuingresults of FE analyses are presented, allowing the wire behaviorunder simple uni-axial loads to be described. They are comparedto cross section micrographs of deformed wires.Index Terms—FE analysis, Nb3Sn wires, NED.I. INTRODUCTIONTHE aim of the Next European Dipole (NED) project [1],[2] is developing a high-field ( 15 T) and large aperturesuperconducting magnet made of high-performanceconductors. For achieving the highly demanding requirementsof NED, a high critical current density strand (at 4.2 K and 15 T) and a sub-element diameter of 50 are re-quested in order to ensure both high performance and stability.However, strands are known to be very sensitive to stressand strain. Their critical current degradation, during the cablingprocess to form a Rutherford cable, represents a genuine issue.In addition, the individual barriers of the sub-elements, whichhave typically a thickness of few , can easily be shearedduring cabling and not be anymore tight to avoid tin diffusioninto copper stabilizer during the reaction heat treatment. Sucha contamination is likely to cause a sharp drop in the RRR ofthe strand, thus dampening drastically its dynamic stability.Therefore it is of prime importance to adopt a strand designwhich will minimize the cabling damage to both sub-elementsand barriers. Important design parameters, like spacing betweensub-elements, barrier thickness or sub-element layout, can beconsidered for an optimal strand design. However, hundredsof meters of strand should be available to allow for crucialcabling tests. Instead, although a flat deformation of a strandManuscript received August 28, 2006. This work was supported in part by theEuropean Community-Research Infrastructure Activity under the FP6 “Struc-turing the European Research Area” program (CARE contract number RII3-CT-2003-506395).S. Farinon and P. Fabbricatore are with INFN-Sezione di Genova, Italy(e-mail: stefania.farinon@ge.infn.it).T. Boutboul, D. Leroy, and L. Oberli are with CERN, AT/MAS, CH-1211Genève 23, Switzerland.A. Devred is with CEA/Saclay, DSM/DAPNIA/SACM, 91191 Gif-sur-Yvette cedex, France, and with CERN, AT/MAS, CH-1211 Genève 23,Switzerland.Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TASC.2007.899138Fig. 1. Bilinear stress-strain curve used to represent the plastic behavior of thematerials composing the wires.by rolling is not equivalent to a real cabling test, the flat defor-mation approach was tried on NbTi [3] and [4] strands.When comparing filament distortion, a fair correlation betweendeformed and cabled strands was found for both materials.Considering that a flat deformation of a strand is much easierto realize than that of cabling, a simulation of a flat deformedstrand by a mechanical model could be an effective tool forprobing the expected mechanical behavior of a givenstrand design to cabling. It can then be used for selecting anadequate strand design. In this work, a FE mechanical model isproposed to assess the expected mechanical behavior ofstrands under a one-directional deformation. The simulationresults are compared to the micrographs of strand samplesflat-deformed by means of rollers.II. FE METHODOLOGYWith the aim of simulating via Finite Element (FE) analysisthe mechanical behavior of a wire subjected to a severeplastic deformation, it is very important to clearly understandthe choices made in the modeling phase, so that we can correctlyinterpret the results.The first key issue is related to the mechanical properties of thevarious materials composing the wires. Indeed, we verified thatthe results of the FE analyses are widely affected by the choiceof material properties, namely by the elasto-plastic stress-straincurves used to represent such deformations. For this reason,we decided to use bi-linear stress-strain curves (see Fig. 1), thesimplest description of a complex plastic behavior. Needingonly three parameters, the Young modulus (E), the yield strength(YS) and the second slope (H), it enables keeping the FE modelbehavior under control and isolating the basic influence of eachparameter on the final results. These mechanical propertieshave been experimentally assessed through microhardness andnanohardness measurements [5] and validated by tensile tests.Nanoindentation is a tool to measure not only hardness at a1051-8223/$25.00 © 2007 IEEEFARINON et al.: FINITE ELEMENT MODEL TO STUDY DEFORMATIONS OF WIRES FOR NED 1137very local scale (lateral resolution is in the nanometric rangeand total indentation depths as small as 20 nm are resolved bythe instrument), but also to measure local elastic properties ofthe individual phases based on the recorded force-displacementbehavior.The second key point is the choice of the most adequate 2DFE representation of the wire deformation. A 3D model, infact, would have been too time consuming for the calculationcampaign we intended to launch. In general, a 2D mechanicalmodeling of solid structures offers three options for solvingthe analysis: plane strain, plane stress or generalized planestrain. Let designate the wire axis. A plane strain 2D modelrepresents an infinitely long structure, in which the material isthen constrained orthogonally to the cross section ( plane),resulting in zero strain in the longitudinal direction .The volume being conserved, there is no cross section reductionduring deformation. Conversely, in a plane stress condition, thestress in the longitudinal direction has to be zero throughout thesection . This second option allows a free elongationof the wire normally to the cross section. The main drawbackis that the strain in the longitudinal direction is not constant,as it is in the real case, since different parts of the wire cannothave relative movements in that direction. As a consequence,the calculated area reduction due to longitudinal elongationcan be overestimated. What we found is that the cross sectionreduction calculated by FE analysis with the plane stress optionis much larger than the one measured. For instance, for a 28%deformation on diameter, the measured elongation is 0.5%,whilst the one inferred from the area reduction given by FEcalculations would lead to 10%. The conclusion is that theplane strain option gives better results than the plane stressone, that is is a better approximation of than.From this point of view, the best option would seem to be thethird one, the generalized plane strain option, which assumesa finite deformation domain length in the direction, as op-posed to the infinite value assumed for standard plane strain.This feature allows the 2D representation of a “fiber” in whichthe amount of deformation of all cross sections are identicalthroughout its length . Allowing the imposition ofa longitudinal strain, it correctly handles the reduction in sec-tion of wires during deformation. We found that this option isin slightly better agreement with measurements than the planestrain option, but this difference is not enough to justify themuch longer run time needed for the analysis (almost a factor10). Running all the FE analyses in plane strain option, we needto keep in mind that the transverse deformations and in-planestrain components will be slightly overestimated.The next step is to choose an element having plasticity, largedeflection, and large strain capabilities. All the FE calculationsshown in this paper have been carried out using the FE codeANSYS [6] and its PLANE182 element. Moreover, all themodels suppose a perfect bonding between different materials,and neither sliding nor detaching is allowed. Furthermore, nokind of breakage is taken into account: when rupture or mixingup of material occur, the mechanical system changes, as wellas the properties of the materials involved, and the FE model isno more representative of this new structure.Fig. 2. ALSTOM/CEA Nb Sn internal tin wire shown together with its FEmodel. White represents copper, dark grey is tantalum, light grey is tin and greyis a homogenized material representing the Nb-7.5 wt.%Ta filaments in Cu ma-trix. The wire diameter is 0.825 mm.Finally, the loading of the FE model is very simple: usingthe ANSYS contact tool, two rigid and parallel planes in con-tact with the external surface of the wire have been generatedthrough CONTA172 and TARGE169 elements. Whilst one ofthe planes is kept fixed, the other is gradually lowered, resultingin a uni-axial compression of the wire in the direction.III. WIRE MODELSA. ALSTOM/CEA Internal Tin WireThe model was first applied to an old internal tin wire man-ufactured by Alstom/MSA for CEA/Saclay [7] and shown inFig. 2. This wire is made up of 19 sub-elements and 6 addi-tional Sn sources surrounded by a Ta/Nb diffusion barrier iso-lating the stabilizing copper on the outside. Each sub-elementis made up of Nb-7.5 wt.%Ta filaments arranged in a pure-Cumatrix around a Sn pool. Fig. 2 shows also the FE model forthis analysis; it is made up of 4 different materials: copper isrepresented in white, the Nb/Ta barrier in dark grey, tin in lightgrey, and finally grey is a homogenized material representingthe filamentary areas. The properties of this homogenized ma-terial were found by running a sort of virtual experiment, thatis: a plastic analysis simulating a stress-strain measurement ona significant part of the sub-element system where a uni-axialforce in tension is applied to the system and the correspondingdisplacement is retrieved. Each step of this analysis correspondsthen to a point in the stress-strain plane, allowing outlining a newbi-linear stress-strain curve for the homogenized material.B. SMI/NED PIT WireThe layout of the SMI/NED PIT wire is shown in Fig. 3. It ismade up of hexagonal cells regularly arranged within a coppermatrix. Each cell is made up of a hexagonal copper tube sur-rounding a niobium tube which is hexagonal on the outside andcircular on the inside, filled up with a powder mixture. Fig. 3shows also the FE model for this analysis: white represents thepowder core, light grey is copper and dark grey is niobium. Themain difficulty in this modelization was to assess relevant prop-erties to the powder core, as they cannot be directly measured.To do so, we ran several models changing the parameters of the1138 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007Fig. 3. SMI/NED PIT wire shown together with its FE model. White representsSn/Nb powder core, light grey is copper and dark grey is niobium. The wirediameter is 1.25 mm.Fig. 4. ALSTOM/CEANb Sn internal tin wire after a 24% reduction in diam-eter. The observed deformation is shown together with the FE analysis result.bilinear curve of this material until we found the best agreementbetween calculations and observed deformations.IV. RESULTS AND DISCUSSIONFirst, let us consider the deformation of the ALSTOM/CEAwire. Fig. 4 shows the wire after a 24% reduction in diameter to-gether with the deformed picture coming from the FE analysis.Note that the left and right sides are not symmetrical becausethe starting position of the wire is rotated by 15 with respect tothe model in Fig. 2. The agreement is very good, the calculatedshape of the external barrier is very similar to the observed oneand also the overall dimensions of FE analysis and micrographmatch pretty well. One of the mostly critical parameters is thestrain in the direction (i.e. perpendicular to the direction ofcompression). When compressing the wire, the thickness in thedirection reduces and strain is mainly negative (only smallregions in the external copper are in tension). The main conse-quence is that, due to Poisson’s ratio, most parts of the wire willbe in tension in the direction. At a certain value of diameterreduction, this tension overcomes the elongation at fracture andparts of the wire break. This effect is clearly visible in Fig. 5,corresponding to a 48% reduction in diameter of the wire. TheFE analysis, which does not take into account any kind of rup-tures, presents a plastically deformed shape which is no moreFig. 5. ALSTOM/CEANb Sn internal tin wire after a 48% reduction in diam-eter. The observed deformation is shown together with the FE analysis result.Fig. 6. Total strain in the x direction of the ALSTOM/CEA Nb Sn internaltin wire after a 48% reduction in diameter.in agreement with what is observed. Evidently, the regions ofthe wire which are the most strained in tension break up andmix with one another. It is clear that the FE model cannot ade-quately represent this new wire layout with different mechanicalproperties. In any case, the FE analysis is still helpful in under-standing what are the regions having the highest risk factor, i.e.,the ones having the largest values. Fig. 6 shows the total strainin the direction of the ALSTOM/CEA wire after a 48% reduc-tion in diameter. The tin of the central sub-elements is the moststrained material, having values of well over 100%, and upto a maximum value of 176%. These conditions are obviouslynot sustainable by tin, which appears to break. Also the Ta/Nbbarrier can be expected to break. Already, after a 24% reductionin diameter it is strained up to 30%, the maxima being locatedat the top and bottom parts, where the initial undulation seen inFig. 2 appears to be straightened. Even if these values are a bitoverestimated, it was predictable that further reducing the diam-eter would result in wire breakages in these areas, as observedon the deformed wire.The PIT/SMI wire can also be very well simulated by our FEanalysis. Fig. 7 shows the wire after a 24% reduction in diam-eter. The match is not perfect, but the calculations give a suitabledescription of the overall behavior of the real wire. Two shearplanes tilted by about 45 with respect to the axis can be easilyidentified. They divide up the plane into four regions: theregions on both sides are almost unaffected by the deformation,FARINON et al.: FINITE ELEMENT MODEL TO STUDY DEFORMATIONS OF WIRES FOR NED 1139Fig. 7. SMI/NED PIT wire after a 24% reduction in diameter. The observeddeformation is shown together with the FE analysis result.Fig. 8. SMI/NED PIT wire after a 24% reduction in diameter enlarged in thearound the center of the wire. The observed deformation is shown together withthe FE analysis result.whilst the top and bottom regions are the most strained, espe-cially near the center of the wire. In Fig. 8 an enlargement ofthe previous picture around the center of the wire is shown. It ispossible to appreciate that there is a sort of “bumping” effect onthe central tubes which is well predicted by the FE analysis.V. CONCLUSIONWe demonstrated that FE analysis is a powerful tool to sim-ulate severe plastic deformations of wires. It can be adopted topredict the mechanical behavior during cabling of different wiredesigns, with the aim to find an optimum design minimizing ca-bling damages. The only limit lies in the fact that possible rup-tures are not included in the FE model: we can only determinethe regions which, having overcome their elongation at fracture,are possible candidates for breakages.ACKNOWLEDGMENTThe authors thank C. Scheuerlein and S. Sgobba forhaving measured the relevant material properties of our wires,J. L. Servais for wire deformations, and A. Desirelli for fruitfuldiscussions.REFERENCES[1] A. Devred et al., “Status of the Next European Dipole (NED) activityof the Collaborated Accelerator Research in Europe (CARE) project,”IEEE Trans. Appl. Supercond., vol. 15, no. 2, June 2005.[2] A. Devred et al., “Overview and status of the next european dipolejoint research activity,” Supercond. Sci. Technol., vol. 19, pp. S67–S83,2006.[3] V. Previtali et al., “Critical current studies on Nb-Ti deformed strands,”IEEE Trans. Appl. Supercond., vol. 15, no. 2, June 2006.[4] E. Barzi et al., “Instabilities in transport current measurements ofNb Sn strands,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, June2005.[5] A. Devred et al., “Tensile properties of the individual phases in super-conducting multifilament Nb Sn wires,” Advances in Cryogenic En-gineering, accepted for publication.[6] ANSYS, Inc., Southpointe, 275 Technology Drive, Canonsburg, PA15317 U.S.A. [Online]. Available: http://www.ansys.com[7] M. Durante et al., “Development of a multifilamentary Nb Sn wirefor accelerator magnet applications,” Physica C, vol. 354, pp. 449–453,2001.

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