Elsevier

Cryogenics

Volume 50, Issue 5, May 2010, Pages 304-313
Cryogenics

Generalized stiffness coefficients for ITER superconducting cables, direct FE modeling and initial configuration

https://doi.org/10.1016/j.cryogenics.2009.11.006Get rights and content

Abstract

Superconducting coils are one of the key technical solutions used for generation of high magnetic field in modern tokamaks. Nb3Sn superconductivity depends not only on temperature and magnetic field as e.g. NbTi, but also on the strain state of the strands inside the conductor. It is hence very important to be able to predict the mechanical deformations due to manufacturing processes and operating conditions. The conductors for ITER, the International Thermonuclear Experimental Reactor currently under construction, have a complex structure that makes analytical estimations of stiffness applicable only for the first cabling stages. In this work, a wide range of numerical simulations has been performed, by using several types of finite element models. This paper shows some analytical estimations for stretching and twisting and compares them with the numerical results of the different models. Some comparisons with experimental tests are also presented. Furthermore, it is shown that direct finite element analyses are compulsory for higher cable stages, but need the knowledge of the initial configuration as precise as possible for meaningful simulations. This problem is also addressed in this paper.

Introduction

ITER (International Thermonuclear Experimental Reactor) [1], [2] is a worldwide project involving a great number of scientific and technical solutions for complex multi-disciplinary problems. Use of superconducting cables for strong magnetic field generation is one of the technical solutions in ITER [3], [4], [5], [6], [7]. The necessary condition for superconducting cable operation is the maintenance of temperature in a narrow range of extremely low temperatures. This condition can be easily broken by heat generation under pulsed loading in the conductors caused by their mechanical deformation. Other important aspects are the performance degradation due to excessive accumulated deformations and residual strains of the manufacturing process or energization [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. For this reason it is necessary to predict the mechanical deformations that occur in these cables down to the level of a single strand under external loads. This article presents two substantially different approaches for solution of this problem, compares obtained results and highlights the importance of the knowledge of the initial configuration. The problem of downscaling from strand to superconducting filament has already been dealt with in [18], [19].

A general view of a conductor for ITER considered in this paper is presented in Fig. 1.

The cable is made up of six petals wound around a cooling channel. The petals are made up of 237 strands arranged as 3 × 3 × 5 × 5 + 3 × 4. This means that the petal has a four-stage hierarchical structure. The first level (or, first macrostrand) is a triplet, i.e. three strands twisted around each other. Then three triplets are twisted around each other building the second-order macrostrand 3 × 3. The third level of the petal structure is the macrostrand 3 × 3 × 5, i.e. five second-order macrostrands twisted around each other. Finally five macrostrands 3 × 3 × 5 are twisted together and with an added core 3 × 4 form a petal. The four stages of the developed macrostrand CAD models are presented in Fig. 2.

Superconducting cables are subjected to various mechanical loadings during their manufacturing, installation and exploitation. From the structural point of view, four dominating types of deformation can be identified in an arbitrary deformed state of the cable: stretching, twisting, bending and transverse compression.

In this paper analytical and numerical approaches to the solution of cable stretching and twisting are considered for ITER superconducting cable macrostrands (from triplet up to the petal). Some comparisons with experimental results are also presented, for the 3 × 3 and 3 × 3 × 5 cable stages.

Two approaches are used to solve the problem of cable stress–strain state determination under tensile and twisting loads.

The first approach is based on using well-known theory of rope stretching and twisting [20], [21]. According to this theory, a wire rope is considered as a complex framed structure and the stress–strain state of a curvilinear rope subjected to simultaneous stretching, bending, twisting and contact loading is investigated. Some other simplifications are also introduced: the friction is neglected; the transverse compliance of the wire rope is neglected when contact forces acting on a wire are estimated and hypotheses of plain wire rope cross section and of unbent radii are assumed. Following this approach, linear equations, governing the stretching and twisting of the rope can be written as:A·ε+C·Θ=T,C·ε+B·Θ=M,where T and M are the applied tensile force and twisting moment and correspondingly, ε and Θ the longitudinal and angular rope deformations. AC are the generalized rope stiffness coefficients. A represents the stretching stiffness of the rope, B the twisting stiffness, and C is called the mutual influence coefficient.

Glushko [20] presents formulae for analytical estimation of stiffness coefficients for single and double lay ropes taking into account contact forces between strands. These formulae will be used for the determination of the stiffness coefficients of the hierarchical superconducting macrostrands.

The analytical approach allows to make a very quick estimation of the rope “macro” stress–strain state, but has rather strong limitations concerning the range of problems for which it can be applied. It also does not allow to obtain a precise strain distribution across the strand cross section and does not take into account the transverse deformation of strands.

The other approach for the cable investigation consists of solving numerically the general elasticity theory equations with appropriate boundary conditions. The elasticity theory equations for anisotropic heterogeneous media can be written as:·(C(r)u)+fv=0,where the symbols have their usual meaning: r is the position vector, u=ukek is the displacement vector, =ekxk denotes the gradient operator, C(r)=Cijkleiejekel is the constitutive tensor (heterogeneous anisotropic for the strands in the general case) and fν=fνκeκ is the body force vector. Three scalar boundary conditions should be defined at the boundary except for the regions of contact interaction, where the static and kinematic contact conditions [22] should be satisfied. The finite element (FE) analysis [23] is used to find the numerical solution of this problem. The procedure of the implementation of contact conditions in FE method is described in [24]. This approach requires significant preparatory work (creation of geometrical and FE models), but it allows to obtain a detailed stress–strain state of each strand, besides the generalized cable stiffness coefficients. The other important benefit of this approach is the possibility of expanding the analysis to the elasto-plastic case. In this case instead of Eq. (2) the next more general equation is to be solved:·σ(r)+fv=0,where σ(r) is the stress tensor and the relation between stress and strain tensors is governed by theory of plasticity equations [25]. The implementation of the plasticity model in FE codes is described in [26]. Besides the elasto-plastic material behavior the direct FE modeling allows also to take into account contact interaction with friction between the strands, large deformations, any structure of the cable and any loading.

Section snippets

Preliminary investigations

Every strand of an ITER conductor is a complex composite structure, where generally Nb3Sn filaments are arranged in groups and embedded in a bronze matrix. Since direct modeling of the composite structure in a cable analysis is actually not feasible, homogenization procedures are in general used to obtain effective strand properties [18], [19]. For example, performing a two-level homogenization procedure for the VAC superconducting strand in the elastic case with components properties

Stretching and twisting of the triplet

The analytical and numerical approaches are applied first to the analysis of the mechanical behavior of a triplet (the first order macrostrand, consisting of three strands). The geometrical and mechanical characteristics taken into consideration for the triplet are: strand diameter d = 0.81 mm (all strands are assumed to be identical and have a circular cross section), twist pitch h = 45 mm, Young’s modulus of strand material E = 117.7 GPa, Poisson’s ratio ν = 0.3 [28], [29]. For the non linear case, an

Stretching and twisting of the second-order macrostrand 3 × 3

The next level structure in the superconducting cable is the second-order macrostrand 3 × 3 (Fig. 2). Analytical estimations of stiffness coefficients for the double lay rope according to [20] are:A=mA0cos3β+2C0rsin3βcos3β+B0sin2β+g0cos2βr2sin4βcosβ,B=mA0r02sin2βcosβ+2C0r0sinβcos4β+B0cos7β+g0(1+cos2β)2sin2βcosβ,C=m·A0rcos2βsinβ+B0cos4βrsin3β-g0(1+cos2β)cos2βrsin3β+C0(1+tg4β)cos5β,where A0, B0 and C0 are the generalized stiffness coefficients of the single lay rope (the triplet that was analyzed

Stretching and twisting of the macrostrand 3 × 3 × 5

Five macrostrands 3 × 3 considered above are twisted around each other to form the third cable stage, the macrostrand 3 × 3 × 5 (Fig. 2). Despite of the fact that the analytical estimations of generalized stiffness coefficients were developed in [20] for single and double lay ropes, they can be also applied to higher order macrostrands. For the macrostrand 3 × 3 × 5 expressions (11), (12), (13) will be used again with assumption that A0, B0 and C0 are the stiffness coefficients of the macrostrand 3 × 3,

Stretching and twisting of the petal (3 × 3 × 5 × 5 + 3 × 4)

The petal is formed by twisting five 3 × 3 × 5 macrostrands around each other and adding a 3 × 4 bundle as a core (Fig. 2). The analytical estimations presented in [20] can provide just a very rough estimation of the stiffness coefficients for such a bundle. The reason is that they were developed for cables with much simpler structure and the underlying simplifications can lead to significant errors for more complex cables. So for higher order cable stages, the finite element modeling becomes the

Comparison with experimental tests

Purely mechanical tests are not easy to be found, and data are even scarcer for mechanical experiments on sub-size cables. So far, most model validations have been performed indirectly, involving critical current measurements. However, recently more attention is being paid to mechanical aspects, see e.g. [33], [34].

Particularly, in [34] some purely mechanical data are reported, performed by the group of University of Twente, concerning both the single wires and a 36-strand cable. As stated

Conclusions

In this paper a wide range of numerical simulations of various ITER cabling stages is shown, till the last but one (the petal). We report also some analytical estimations for the stretching and twisting of a bundle of wires, and compare them with the numerical results.

We have not only systematically tested our numerical models one against the other, since they should give similar results for the same problem, but also against the available experimental tests, in this case for the 3 × 3 × 4 strand

Acknowledgements

The financial support of CISM, the International Centre for Mechanical Sciences in Udine, and the University of Padova, research Grant: STPD08JA32_004, is gratefully acknowledged.

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