1 Introduction

The International Thermonuclear Experimental Reactor(ITER)magnet system is wound from cable-in-conduit conductors(CICC),made up of a multi-stage,rope-type cable inserted into a stainless steel conduit where circulates a flow of supercritical helium.The merit of the CICC is that it is easy to eliminate the heat raised by nuclear heating and other electromagnetic disturbances[1].The conductors for the toroidal field(TF)and central solenoid(CS)coils rely on Nb3Sn strands which are strain-sensitive and brittle[2–4].Moreover,it is well known that there is thermal mismatch between the stainless jacket and superconducting Nb3Sn strand,as a result,the jacket is in tension along the conductor axis while the cable is in compression at the end of cooling-down[5–7].What is more,for the ITER project requirements,the CS coils must be capable of driving inductively 30,000,15MA plasma pulses with a burn duration of 400s.That implies the CS coils will have to sustain severe and repeated electromagnetic(EM)cycles to high current/ field conditions during its life time.Thus,the investigation on the CICC behaviors under EM cycles is necessary and important for an ITER fusion reactor.

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In the years of2000–2002,the tests of several model coils showed gradual degradation of the current sharing temperature(Tcs)under several hundreds of EM cycles.They found it would be stable after 2000 EM cycles[8].In order to evaluate the effects of EM cycles on the Tcs degradation behavior,Mitchell[9–12],Nijhuis and Ilyin[13,14],Zhai and Bird[15,16],Bajas et al.[17,18]proposed several elaborate models in the last decade.From these models,the degradation of Tcs seems to be the result of the Nb3Sn strand’s bending and filament fracture[8–18],which can be considered as the contribution of the strand bending deformation[19–23].Subsequently,Sanabria et al.[24,25]thought that the filament fracture is not a decisive reason for the decline of Tcs.Moreover,some experimental results[26–30]revealed that the bending direction of the strand caused by the EM cycles is perpendicular to,rather than parallel to the direction of the electromagnetic force,contrary to the previous understanding.Recently,many studies[24,25,31–34]revealed that the factors like bending strain of the strand, filament fracture,stress concentration at the contact point were not sufficient to lead to Tcs degradation. Several additional factors which can affect the Tcs behavior such as strand surface roughness and longitudinal slip of bundles were reported in Refs.[24,25,35].It should be mentioned that most of the researchers put their focus on the deformation of single strand in the CICC earlier[11,13,15,19].They attempted to describe the strand behavior such as bending,buckling and fracture under thermal mismatch and the electromagnetic force to characterize the Tcs degradation behaviors. The natural aim of their investigations is to reveal the influences of the transverse stiffness,twist pitch,bending stiffness of strand,Cu/non Cu ratio,void fraction,and friction coefficient,etc.,on the Tcs behavior.From single strand to multi-strands,by using the statistical analysis to interpret the integral behavior of the CICC,it seems to be correct.But some absent key parameters will lead to the statistical bias,as a result,the experimental results cannot be interpreted qualitatively and quantitatively.So that Devred et al.[36]gave a new idea,and they thought the degradation of Tcs would be related to the magnetic field distribution feature of the SULTAN facility.The cable could slip from low field zone(LFZ)to high field zone(HFZ)due to the weaken stiffness of the HFZ,as a result of strand buckling and fracture.But their simplified model cannot give the interpretation of the Tcs increased behavior of the short twist pitch(STP)conductor samples in the SULTAN and CSMC experiments[37,38].

In this paper,we propose a mechanical model to describe the Tcs behavior of Nb3Sn CICC samples under EM cycles.The discussions on the calculated results and the mechanism of the Tcs degradation are presented.This paper is organized as follows.In Sect.2,we present the mechanical model and equations,and while in Sect.3,the calculated results are displayed.In Sect.4 we present the discussion of the proposed model.The conclusions of this work are presented in Sect.5.

Fig.1 Schematic illustration of the CS and TF CICC cross-section.The symbols R In and R denote the radius of the He channel and petal,respectively.r is on behalf of the sum of R In and R.Thatis,r=R In+R

Fig.2 a The deformation of the cable caused by the axial compression.b The cross-section of the cable

2 Mechanical analysis

2.1 Cable deformation caused by thermal mismatch and electromagnetic force

The TF and CS CICC are made up of a 5-stages formed around a central cooling spiral.Six 4th stage petals are wrapped with stainless steel tapes,respectively,which are integrally inserted into a stainless steel conduit as illustrated in the cross-sectional view in Fig.1.The outer diameter of the TF conductor is of 40 mm and the side length of CS conductor is 49 mm.The inner diameter of the TF conduit is equal to 37 and 36 mm for CS,respectively.Nb3Sn CICC consists of more than 1000 strands,in this paper,which can be fused into a cohesive whole with circle cross section,as displayed in Fig.1.Thus,the cable should be analyzed by using the wire rope model(Fig.2).

During the calculations,the parameter R,R In,and twist pitch(h)are 6,6,450 mm,respectively.According to the geometric relation h=2πr tanα shown in Fig.3b,we can get α =80.5°,where α represents the initial helix angle.

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Fig.3 a Loads acting on the pet al.b Geometric relation of the petal centerline

Figure 3a also displays the loads acting on the petal and the geometric relation of the petal centerline.

If the six petals contact with each other in the initial state,the change in curvature Δk′and the change in twist per unit length Δτ can be expressed as[39]

where α and represent the original and final helical angle of the petal,r and stand for the original and final radius of the petal,Δα and Δr denote the change of helical angle and radius of the petal,respectively.The loads and moments become

where G′,H,T stand for the bend moment,twist moment,and axial force of the petal,respectively.v stands for the Poisson’s ratio.E is of the axial stiffness of the petal and ξ stands for the axial strain of the pet al.If the petals do not contact each other under the compression state,the contact force between the petals is equal to0.According to the model presented by Costello[39],the following formula will be satisfied:

where X stands for the resultant contact force per unit length of a pet al.N′is of the shear force acting on the pet al.From the Eqs.(1)–(3)and(5),the shearing force N′acting on the petal can be written as

From Eqs.(7)–(8)we can get

Substituting Eqs.(3)and(6)into Eq.(4),the axial strain of the petal ξ,which is only related to the Δr and Δα,can be written as

The deformed configuration of the petal in Fig.3 yields

DS18B20的典型用法如图4所示。一般来讲DS18B20都是充当从机的角色，而单片机就是主机，主机需要读取DS18B20的ROM指令，然后执行DS18B20的功能指令，即RAM指令。需要注意的是，本设计中采用的是多点测温，不像单点测温一样无需刻意读取ROM里的序列号后匹配，在软件设计中必须在初始化中添加读取每个DS18B20的ROM中的序列号，然后通过程序匹配哪个DS18B20进行读取和控制。ROM指令的字节为0xcc。DS18B20的温度读取函数参考以下步骤：

The angle of twist per unit length τ of the petal can be defined by the expression

That is,

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At the two ends of the petal per twist pitch, the rotation is zero,and the compression strain obeys:ε= −ε0,then Eqs.(9)and(11)become

The relation between axial strain ε0 and transverse strainan be expressed as

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It is found that the coefficient between transverse and axial strains is affected by the helical angle α and Poisson’s ratio υ.When the twist pitch of the 5th stage cable is of 427,450,476 mm,the corresponding helical angle of the petal equals 80°,80.5°,81°,respectively.Substituting these values into Eq.(13),one can see the transverse strain εTrans shows a linear relationship as a function of the axial strain ε0,which is illustrated in Fig.4.In order to validate this analytical analysis,we carry out a compression experiment by using the CS cable,which was fabricated in the Institute of Plasma Physics,Chinese Academy of Science.When the CS cable specimen is compressed along the axial direction,it will extend transversely with high resolution.The transverse extension can be measured by using a laser sensor.The experimental setup is schematically illustrated in the inset of Fig.4.Experimental results are displayed by using the green triangle symbol,as shown in Fig.4.One can find that,the theoretical model shows consistency with the experiment perfectly.It should be pointed out that the cable specimen uses six sub-cables(petals)consisting of Nb3Sn strands and cooper strands,which are wrapped with stainless-steel individually.In order to prevent the cable dispersing during the compression process,both ends of the sample are clamped by two rings.

Fig.4 Relationship between the axial strain ε0 and the transverse strain εT.The color lines are obtained by using the presented theoretical model.The green triangle symbols denote the experimental results

2.2 The axial strain of the cable at the HFZ

We now quantitatively discuss the influences of friction on the final strain.First,if the friction coefficient is 0,Eq.(19)can be simplified into:

Fig.5 a Axial thermal strain εT and radial pinching force P the conductor cooling down,b Magnetic field distribution along the sample,c,d the ideal extension of the cable compressed by the electromagnetic force,e,f the actual extension of the cable constrained by the LFZ.In this figure,L High stands for the length of the HFZ,L Low stands for the length of the LFZ,L1 stands for the length of the HFZ under EM loading,L2 stands for the sliding length at the LFZ under EM loading,L3 is the length of an unaffected area under EM loading.ΔL1 indicates the expansion of the HFZ’s cable L1 and it is the same quantity with the shortening in the L2 written by ΔL2.ΔL F stands for the compression of the HFZ’s cable by the interfacial force F

where ΔL2 is the cable axial compression.This relationship is as a result of the displacement continuous at the interface of the HFZ and LFZ.This process is shown in Fig.5e.If we take into account the friction between the cable and jacket with the friction coefficient of μ,there is a axial compression force F,the corresponding axial compression is of ΔL F.That leads to a strain of εF.It obeys:F=EAεF,where A denotes the cross-section area.The real extension of the cable at HFZ:ΔL1=L0(ε0 − εF).The axial compression force F satisfies:

where P denotes the pinching force and x is on behalf of the distance.If the friction force between the jacket and cable is lower,the cable at the LFZ can be totally influenced,so the total compression ΔL2 can be expressed by

where L High stands for the length of HFZ.

where L Low denotes the length of LFZ.It should be pointed out that the friction coefficient meets

If the friction between the jacket and cable is stronger,the length of the influencing region is less than L Low.Using the symbol L2 to stand for this length,schematically illustrated in Fig.5e,the upper limit of integral in Eq.(16)becomes L2.Now we will calculate this value by using the force balance condition at the interface.

Finally,the total axial strain of the cable caused by the electromagnetic force has the expression as follows

That can be simplified into:

Using Eq.(14),one can obtain:

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and the displacement continuous condition:

Substituting Eqs.(20)and(21)into Eq.(14),one can get:

Solving this equation,and substituting the real root of this equation into Eq.(18),the strain caused by the electromagnetic force has this form:

Figure 6 displays the normalized strain versus friction property.One can find that for the SULTAN conductor sample,its length at HFZ is of 0.4m,and it is 3.1m long in LFZ(black square symbol in Fig.6),with increase of the friction,the normalized tension strain turns into compression strain.Moreover,if the conductor length at LFZ is equal to 0,the total conductor is in compression state,its value is unchangeable.This figure can be easily understood according to the geometry property of the SULTAN conductor specimen.If the cable length at LFZ is equal to 0,the conductor is totally compressed by the electromagnetic force and thermal stress.That leads to the strain state that is in compression and stays invariable.

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Fig.6 The axial strain as a function of the parameters of the conductor under the loading of electromagnetic force

Because the thermal expansion coefficient of the stainless steel(jacket)between 933 and 4.2 K is about twice as large as that of Nb3Sn,as a result,in a real coil,the jacket is in tension along the conductor axis,while the superconducting cable inside is in compression at the end of cooling-down.Moreover,the transverse shrinkage of the jacket will pinch the cable,which is displayed in Fig.5a,where P denotes the pinching force,εT stands for the thermal mismatch strain,and E is Young’s modulus,respectively.Figure 5b shows the magnetic field distributions along the conductor sample in the SULTAN device.The length of a SULTAN conductor specimen is about 3.5 m long,with a 0.2 m HFZ of about 10.85 T plus about 0.75 T of self- field and one 0.8m field gradient region.Half of the HFZ is 1.0 m was shown in Fig.5b.The remaining 1m has a low field(due to self- field).When the conductor sample carries current as high as 68kA,the large electromagnetic force will compress the cable at the HFZ,and generate a void fraction,which is displayed in Fig.5c.This transverse compression will lead to the cable extension.The initial length of cable at the HFZ is of L0 and the ideal extension can be denoted byΔL0,the corresponding extension strain:ε0= ΔL0/L0.Therefore,the axial strain of the cable at the HFZ is of εT+ ε0,that is illustrated in Fig.5d.Because the cable has a large compression stiffness,the cable at LFZ will impede the axial extension of the cable at HFZ.As a result,the real axial extension ΔL1 is less than the ideal axial extension ΔL0.In addition,the axial extension ΔL1 has

where ε0 denotes the ideal axial strain of the cable caused by the transverse electromagnetic force.The cable is in axial tension in the case L Low＞L High,that leads to the compression strain caused by thermal mismatch releasing.On the contrary,it is in compression state.

Second,let μ→∞,one can obtain:

In this situation,the cable is totally in compression.

2.3 The relation between Tcs and axial strain of the cable at the HFZ

The magnetic field in the LFZ is much lower than that in the HFZ,so the sample’s Tcs is mainly controlled by the HFZ’s.As calculated in Sects.2.1and2.2,we translate the transverse strain into the axial strain,which is directly related to the Tcs behavior according to a scaling law for ITER Nb3Sn strand[40].The relation between Tcs and axial strain obeys:in which εT and εEM denotes the axial strain of the cable caused by the thermal mismatch and transverse electromagnetic force.γ is the fitting parameter for different conductors.With the increase of axial strain,the Tcs of the conductor decreases gradually.Moreover,the thermal strain εT is of−0.65%and stays invariable[40].

where

Fig.7 The cable’s axial strain ε0 at the HFZ versus the number of EM cycles

3 Calculation results

Figure 7 shows the axial strain versus the EM cycle numbers, where the red line with circle symbol displays the axial strain of CS specimen with void fraction of 36%, while the black line with black symbol shows that of the TF sample with void fraction of 29.7%. The parameters used in this calculation are cited from Ref. [41].

For the Inter-tin and Bronze TF conductor sample, the calculated Tcs degradation behaviors are displayed in Fig.8a,b,where the experimental results are cited from Refs.[42,43].During the calculation,the friction coefficient between the jacket and cable is assumed to be in finite(μ→ ∞),and in this case,the cable is pinching and stuck in the HFZ,which is similar to the experiments carried out by using TFEU6[42].As shown in Fig.8a,b,the theoretic calculations are coincident with the experimental results.Notably,at the primary stage(the cycle number is less than 100),the sharp degradation behavior of Tcs is also captured by the present model.

Fig.8 The calculated Tcs behavior versus the number of load EM cycles for conductor samples.a Inter-tin Nb3Sn conductors.b.Bronze Nb3Sn conductors.The colored dot and solid lines are the experimental results,while the black squares and dot lines are the theoretical results.The degradation with cyclic loading of bronze conductors is smaller than that of IT based conductors.The TFRF 2 left sample even shows practical enhancement.One can find the calculated results are in good accordance with the experimental results

For CS Nb3Sn conductor samples with baseline and long twist pitch(LTP),the Tcs shows degradation behavior with increase of the number of EM cycles,while it becomes constant or enhanced for the samples with STP. These experimental results presented in Ref.[26]have resisted theoretical prediction for many years.Using the model proposed in this paper,the Tcs rise and degradation behaviors can be simulated quantitatively.Figure 9 displays a comparison of experimental and calculated results,where the dot lines show the theoretical results.It is found that the quick increase of axial strain during the primary stage of the EM cycle,that leads to the Tcs dropping dramatically at the same time.The conductor samples with LTP and baseline show the degradation behavior of Tcs,this is the same as the TF conductors.But several CS conductor samples with STP show an increase in behavior of Tcs,which implies that the axial compression strain has been released by squeezing the cable at HFZ into LFZ.The length of extrusion is dependent on not only the parameters of the SULTAN device such as L High and L Low,but also the character parameters including friction coefficient μ between cable and jacket,Young’s modulus and cross-section area,etc.We now discuss two limitations of the cable length at the HFZ.One is that the L High is of 400 mm,the corresponding L Low is equal to 3100 mm.As a result,the conductor sample’s Tcs becomes greater,which is drawn by an orange line with circle symbol in Fig.9.The other is that L High is of 1000mm,that also leads to conductor sample’s Tcs enhancement,which is displayed by a purple line with a triangle symbol in Fig.9.

Fig.9 A comparison of the experimental results and calculated results based on the presented model.When the expansions of the HFZ’s cable get stacked,the Tcs will drop as the black dash line in this figure.While,if the friction ratio is negligible,the upper boundary of the Tcs is the orange dash line with circle symbol and the length of the HFZ is 400 mm.when the HFZ’s length is 1000 mm,then the lower boundary of the enhancing Tcs is given with a purple line with triangle symbol.We can find that the theoretical results agree with the experimental results very well

4 Discussions

According to the mechanical model presented in Sect.2,one can see that the axial strain of the cable is the determining factor for the Tcs behavior during the EM cycles.The relationship between Tcs and the axial compression strain of the cable showed a simple law:Tcs degrades with the increase of axial strain.The major source of the cable’s axial strain is from two aspects.One is the thermal mismatch compression strain εT.Its value comes from the thermal expansion coefficient difference between the stainless steel jacket and the superconducting strands,the cable’s axial stiffness,and cross-section area.The initial value of Tcs can be decided by this thermal strain and hoop strain as represented by the preceding two items in Eq.(26).It is found that the initial values of Tcs in CSMC experiments with hoop strain of 0.1%is 0.6K higher than the test result in SULTAN facility[37,38].The ratio of Tcs vs axial strain is 0.0006 K/ppm in CS insert coils which shows consistency with the scaling law of short samples.The other is the electromagnetic force directed normal to the cable axial,which compresses the cable at HFZ and generates a void fraction.This transverse compression usually makes the cable extend along the cable axis.If there is enough friction coefficient between the cable and jacket(μ → ∞),or there are rings at both sides of the HFZ zone,this EM compression strain could be released by the elasticity deformation of the superconducting strands in the axial direction.The corresponding Tcs shows degradation during the EM cycles(see the calculated results displayed in Figs.8 and 9).Moreover,the indentation of the Nb3Sn strands has plasticity and its depth becomes deeper as the number of EM cycles increases,this process is irreversible and may lead to the filament fracture.This viewpoint can be confirmed from Fig.8a,b,in which the inter-tin based strand exhibits softer than the Bronze.Therefore,the physical source of the Tcs gradual degradation behavior is of the electromagnetic force,while the thermal strain only has a significant effect on the initial Tcs values.Although the presented model gives an average strain of the cable,it is not for the single superconducting strand.It is sufficient to show that the cable axial strain can predict the cable Tcs during the EM cycles.The strain contributed by the EM cycle is represented by the third item in Fig.7.

Experiments revealed that the Tcs for STP showed an increase and be came almost constant through the EM cycles,while the baseline(BL)and pseduo long twist pitch(PLTP)samples showed unsaturated decrease.It is not difficult to describe that,the cable with shorter twist pitch leads to a higher stiffness and more compact structure.That would mean the tight STP cable could overcome the friction between cable and conduit,and then released the axial strain in the HFZ, as a result, the Tcs increases. In addition, the tight STP conductor could prevent the strands movement,bending and filament fracture in the cable,these could be considered as another factors to enhance the Tcs value.On the contrary,because the structure of LTP cable is loose,the Tcs of which displays a continued decrease during EM cycles.Notably,if the cable is stiff enough,its Tcs does not degrade even it has a long twist pitch.This phenomenon has been observed in the experiments presented by Ref.[43](see the TFRF 2 Left sample drawn in Fig.8b).

It should be emphasized that the influences of ratio of length for HFZ and LFZ on the Tcs behavior could not be neglected.With the increase of the ratio of HFZ/LFZ,the cable’s strain has a process from decrease to increase(Seen in Fig.6),the related Tcs behavior shows enhancement to degradation.Compared to the length of the entire samples,the short length of the HFZ in the SULTAN facility should be responsible for the rising Tcs of the STP conductor.This viewpoint was also confirmed in the CSMC experiment(seen in Fig.7 in Ref.[37].),as the length of the HFZ of the CS insert sample is half of the coil length.

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5 Conclusions

A mechanical model for analysis of the Tcs behavior of the ITER Nb3Sn CICC under EM cycle is proposed and validated.This model presents a linear relationship between the Tcs and cable’s axial strain,which includes thermal mismatch strain and electromagnetic strain caused by electromagnetic Lorentz force.The calculated results based on the present model agree with the experimental results quantitatively.This implies that this mechanical model can capture the important strain characteristics of the cable during the EM cycles taking into account the thermal mismatch during the cooling process.Moreover,the influences of the length ratio of the HFZ/LFZ in SULTAN and CSMC facility on the Tcs behavior are discussed.For the CICC sample with STP,the observed Tcs enhancement phenomenon could be considered as the contribution of the decrease of elastic strain along the axis.But in the case of the baseline and the LTP,they have opposite effects,revealed both in theSULTAN and CSMC facility.This is because the longer cable twist pitch has the lower cable stiffness.In addition,we are apprehensive that the fretting wear and indentation under the transverse EM force(up to 60,000 cycles)in the poor stiffness cable may lead to filament breakage and unsaturated degradation,this threat to the CICC Tcs performance cannot be ignored.If confirmed,that would mean the CICC design at present could meet the requirement of the ITER fusion.

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Acknowledgements This work was supported by the National Natural Science Foundation of China(Grant 11622217),the National Key Project of Scientific Instrument and Equipment Development(Grant 11327802),National Program for Special Support of Top-Notch Young Professionals.This work was also supported by the Fundamental Research Funds for the Central Universities(Grants lzujbky-2017-ot18,lzujbky-2017-k18).

推行《青苗法》是王安石农业改革的主要措施，其实质就是国家以贷款为名，用高达40%至60%的年利率强迫向农民征收新税，穷苦百姓是难以承受这种高息贷款的。

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