Strain redistribution effects on current-sharing measurements on straight samples of large Nb₃Sn cable-in-conduit conductors

The integrated thermal shrinkage coefficient of stainless steel between 933 and 4.2 K is about twice as large as that of Nb₃Sn (figure 4). As a result, at the end of cool-down, in a real coil as in a 'crimped' conductor sample, the jacket is in tension along the conductor axis while the superconducting cable inside is in compression.

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The magnetic field provided by SULTAN transverse to the conductor axis creates a force on each strand also transverse to the conductor. The strand current is 76 A so the individual force on each strand is about 800 N m⁻¹ (i.e. equivalent to 0.8 kg on each 10 m length). This load accumulates across the cable since the strands support each other. It leads to an accumulated pressure of up to about 20 MPa on the 'underside' of the cable (figure 2).

The improved manufacture/instrumentation of SULTAN samples implemented in 2007/2008 enabled the successful qualification of TF conductor suppliers [2]. However, most samples have shown a steady degradation in T_CS as they have undergone current cycling at peak field. Some samples have also shown a significant drop in T_CS (up to 0.5 K) following a warm-up and cool-down cycle.

It is known from measurements on full size coils [4] that Nb₃Sn conductors can undergo some degradation with magnetic load cycling due to the fracture of some of the superconducting filaments [7, 9]. Very high critical current strands (J_c > 1500 A mm⁻²) are also known to be particularly sensitive to such behaviour [10]. However, in ITER-type conductors with moderate J_c bronze and internal tin type strands, this degradation was expected to stabilize after about 1000 load cycles. Such stabilization occurred in the CS insert [6].

In some SULTAN samples, a similar stabilization is seen after about 1000 load cycles. However, in other samples, the degradation seems to continue almost linearly until at least 7000 load cycles, and is at least a factor of three larger than seen in the CS insert. Thermal cycles (to room temperature) on SULTAN samples also appear to be sometimes linked to a significant extra drop in T_CS, of up to a few tenths of a degree. This only appears to occur once the sample has been exposed to magnetic load cycles.

Furthermore, over the last year a set of three nominally identical samples, have been tested in SULTAN. Each sample is made up of a leg of conductor A and a leg of conductor B. Conductors A and B were cabled and jacketed by the same suppliers, they rely on the same tubes and only differ by their strands. The two strands are bronze-process strands and their designs and morphologies are very similar. The B strand has slightly lower critical current and is more strain-sensitive than the A strand, so that the performance of B legs is expected to be slightly below that of A legs.

One convenient way to look at the SULTAN test data is to consider the effective Nb₃Sn filament strain and the effective resistive transition index that can be derived by fitting the voltage data from the T_CS runs using critical current density versus temperature, field and strain parametrizations, J_c (T, B and ε), that are measured in dedicated experiments for each strand type. Figure 5 shows a summary plot of effective filament strain versus EM cycle derived from the test results. Two of the B legs exhibit a large initial drop of about 0.05% between N = 1 and 50, while the other legs made with A conductors show a comparable drop upon the final WUCD. Apart from these drops, the cycling behaviour of all six conductor legs appears to be similar, with an overall EM cycling degradation of ∼0.3 K over 1000 cycles and a warm-up cool-down degradation of ∼0.1 K.

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The variation of electric field E with temperature (T) is shown in figure 6, comparing the situation at 1 EM cycle and 1000. The degradation at 10 µV m⁻¹ is clear, but, surprisingly, the curves cross at 30 mV m⁻¹. This sort of behaviour suggests current non-uniformity, but the cause is unknown.

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The relatively thin jacket thickness of the conductor samples means that the mechanical stiffness of the jacket is not much higher than the stiffness of the cable. Changes in cable strain are therefore reflected in changes in jacket strain which can be measured by strain gauges placed on the jacket. Care is, however, required in interpreting the meaning of these strain measurements.

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The B leg of the last sample of the aforementioned series was equipped with strain gauges located in both the high-field and low-field zones of the jacket. The initial EM cycling and the final WUCD drops appear to be correlated with local strain relaxations in the high-field zone (HFZ) jacket section. As shown in figure 7, the LFZ strain gauges did not exhibit any change during EM cycling and WUCD, while some of the HFZ strain gauges recorded variations during the first 50 EM cycles and all of the HFZ gauges showed relaxation during the final WUCD [6]. These relaxations in the HFZ strain gauges are concomitant with the sharp T_CS drops observed in figure 5.

Similar strain relaxations had already been observed in the HFZ jacket section of another SULTAN sample (TFKO2) but the amplitude of the relaxation was much smaller [11].

The existence of non-uniform strain distribution along SULTAN samples is confirmed by residual strain measurements carried out at JAEA on sample jackets after test completion [12, 13]. The residual strain is measured at room temperature by cutting successive slices of conductor.

The data for a typical sample, summarized in figure 8, consistently show that the residual strain in the HFZ of the jacket is lower than in the LFZ. The measurements in figure 8 are in reasonable agreement with those in figure 7, indicating a difference in jacket strain in the 300–500 ppm range between high field and low field that develops progressively during sample testing.

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There are two possible interpretations for this, which may both be simultaneously applicable. Firstly, that the cable in the high-field region is softening in the longitudinal direction and/or secondly that the cable in the high-field region is slipping and relaxing its compression, leading to a higher compression in the low-field region. However, we note that the strain in the LFZ in figure 7 does not increase (as would be required by the second interpretation, although the expected increase would be small, about 50 ppm) but varies in the  − 50 to 0 ppm range, suggesting that the mechanism causing the strain reduction in HFZ is, in fact, cable softening. This is supported by previous modelling work, discussed in section 3.

Such a softening means that the cable is becoming more compressed in the HFZ, by the same amount as the jacket strain relaxes. The difference between HFZ and LFZ measured at RT on the jacket is 500–600 ppm and is of the same order of magnitude as the effective filament strain change estimated from the T_CS runs over the entire cold test of this sample ( − 0.06 to  − 0.07% ) as shown in figure 5.

A similar difference in residual jacket strain between LFZ and HFZ was already measured at CEA in 2007 during the disassembly of the TFMC-FSJS sample after SULTAN test completion [14].

Figure 7 also suggests that the jacket strain difference develops progressively with cycling. This could be due to interaction between LFZ and HFZ as discussed in section 3, where a slip of the cable within the jacket leads to increased longitudinal compression of the HFZ by the LZF, or it could be due to continuing transverse compaction of the cable and a resultant longitudinal softening of the HFZ as the magnetic load is cycled, controlled by friction and plastic deformation of the strands locally.

There is an extensive background to mechanical modelling of Nb₃Sn cables which, however, up to now has not been applied to the situation in SULTAN samples.

The first thorough attempt to model the strand-in-cable behaviour appears to have been in [15]. Although this analysis did not consider longitudinal variations in cable behaviour, it did address the link between longitudinal cable stiffness and the impact of the transverse magnetic loads. The analysis shows by both an analytical approach and a finite-element elasto/plastic analysis that transverse magnetic loads result in a reduction in the longitudinal elastic modulus of the cable. There are two possible mechanisms. In the first, the transverse loads create additional bending of the strands, as the cable compacts under the loads. This bending increases the longitudinal flexibility under compressive strain. In the second, the extra transverse loads on the strands create increased plastification of the copper fraction, weakening the strand stiffness in bending. Since the longitudinal stiffness of the cable depends on the strand bending stiffness, this also reduces the longitudinal modulus.

In the first case the longitudinal softening develops even though the cable remains in the fully elastic regime. By using a model for the transverse bending of the strands under magnetic loads such as developed in [16], an analytical prediction could be made of the longitudinal softening. However, the process is clearly also associated with friction between strands and plasticity of the copper component. It is not necessary to postulate that breakage of the superconducting filaments is contributing to progressive collapse of the cable, although filament fracture will increasingly develop as a result of tensile strains created by increased bending. This prediction of softening is now well supported by the strain measurements on SULTAN samples.

The subsequent modelling in this paper aims to examine the effect of this longitudinal softening in a sample where the magnetic loads are non-uniform along the length.

We first developed a simple mechanical model of the longitudinal sections of the cable as shown in figure 9, assuming full elastic behaviour.

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The total length of the sample is L = x₁ + x₂ + x₃ and the differential thermal contraction between jacket and cable that arises during cool-down is △L. △L is negative (i.e. the cable would be longer than the jacket if free to move independently). To maintain compatibility between cable and jacket, a linear mechanical strain results. If the strain along the cable is distributed uniformly then it is

$$\begin{eqnarray} &&{\varepsilon }_{\mathrm{o}}=\bigtriangleup L/({x}_{1}+{x}_{2}+{x}_{3}). \end{eqnarray} \tag{ 1 }$$

The elastic length changes of the three sections of cable are △L₁, △L₂ and △L₃. In addition high-field region 2 may undergo a permanent plastic length change due to the magnetic loads of △L_p.

Therefore

$$\begin{eqnarray} &&\bigtriangleup L=\bigtriangleup {L}_{1}+\bigtriangleup {L}_{2}+\bigtriangleup {L}_{3}+\bigtriangleup {L}_{\mathrm{p}}. \end{eqnarray} \tag{ 2 }$$

The effective area of the cable is A. Let E₁, E₂ and E₁ be the respective elastic moduli of the three cable sections 1–3 (so the high field has a different modulus). By neglecting friction between the cable and the jacket (so the whole force reaction takes place at the ends), the cable force balance gives

$$\begin{eqnarray} &&{A E}_{1}\bigtriangleup {L}_{1}/{x}_{1}={A E}_{2}\bigtriangleup {L}_{2}/{x}_{2}={A E}_{1}\bigtriangleup {L}_{3}/{x}_{3}. \end{eqnarray} \tag{ 3 }$$

The increase in strain in the high field Δε₂ if the strain is non-uniform is then

$$\begin{eqnarray} &&{\bigtriangleup \varepsilon }_{2}=\left(\frac{\bigtriangleup {L}_{2}}{{x}_{2}}-\frac{\bigtriangleup {L}_{\mathrm{p}}}{{x}_{2}}\right)-\frac{\bigtriangleup L}{{x}_{1}+{x}_{2}+{x}_{3}}. \end{eqnarray} \tag{ 4 }$$

Putting △E₂ = E₂–E₁ and manipulating the equations we get

$$\begin{eqnarray} &&\bigtriangleup {\varepsilon }_{2}={\varepsilon }_{\mathrm{o}}\left[{\left(1+\frac{\Delta {E}_{2}}{{E}_{2}}\frac{{x}_{1}+{x}_{2}}{L}\right)}^{-1}-1\right]-\frac{\bigtriangleup {L}_{\mathrm{p}}}{{x}_{2}}. \end{eqnarray} \tag{ 5 }$$

For the case where △E₂/E₂ ≪ 1 this can be written approximately as

$$\begin{eqnarray} &&{\bigtriangleup \varepsilon }_{2}=-{\varepsilon }_{\mathrm{o}}\left(\frac{\bigtriangleup {E}_{2}}{{E}_{2}}\frac{{x}_{1}+{x}_{2}}{L}\right)-\frac{\bigtriangleup {L}_{\mathrm{p}}}{{x}_{2}}. \end{eqnarray} \tag{ 6 }$$

This enables some deductions to be made about the likely limiting behaviour of the cable strain in SULTAN samples. The model does not include friction and so represents only the final state. Since achieving this state requires frictional slip between cable and jacket, it is likely to be achieved only after load cycling and will be sensitive to the exact nature of the cable–jacket interface as regards residual compressive stresses and tolerances.

Assuming that ΔL_p is zero for the moment, we note that (obviously), if x₁ and x₃ are small compared to x₂, there is no longitudinal strain non-uniformity. This would be the ideal condition in a SULTAN sample, achieved by crimping the cable and jacket at the edges of the high field instead of at the ends of the sample. If, however, x₂ is small compared to x₁ + x₃ (as is approximately the case with the present SULTAN sample configuration) then the strain non-uniformity is proportional to the reduction in modulus in the high-field region. The thermal strain in cables with different jacket materials has been extensively investigated in [17], including a survey of the available material database. If △E₂/E₂ is  − 0.1 (i.e. the cable becomes softer) and ε_o is  − 0.7% (as suggested as a likely value in [17]) then Δε₂ is  − 0.07%, giving a strain in the high-field region of  − 0.77%.

It can also be noted that the length of the high-field region has no effect on the strain non-uniformity once x₂ is a small fraction of the total length.

As will be seen at the end of the section on finite-element modelling, each 0.01% of extra compressive strain decreases the T_CS of the cable by about 0.1 K. So 10% softening of the cable due to the transverse magnetic loads results in a T_CS drop in the high-field region of 0.7 K. This is at least as large as the observed degradation due to filament fracture and suggests that strain non-uniformity is easily a potential source of distortion of the results obtained from SULTAN testing.

If there is a plastic deformation in the longitudinal direction in the high field that leads to a lengthening of the cable (i.e. ΔL_p is positive, as would occur if the cable had a positive Poisson's ratio as a result of the transverse compression) then the increase in magnitude of the negative elastic strain is further increased. However, due to the void fraction, it seems likely that the Poisson's ratio of the cable will be close to zero. In any case, since any plastic deformation is likely also to contribute to the total strand strain, the net effect (elastic plus plastic strain) is unchanged. The plastic strain appears in the finite-element model described in section 3.3.

The aim is to find a simple finite-element mesh and material representation of the cable and jacket system that allows a simulation of the potential mechanism of the degradation seen in SULTAN, including frictional effects. At this stage, we are interested in scoping calculations and can tolerate some approximation in the geometrical accuracy and physical rigour of the material model. This model will assume the cable 'softening' is due to increased plastification of the strands due to the transverse loads rather than a lower elastic modulus. The intent is to model the longitudinal friction interaction with the jacket and the extent to which sliding can be a progressive process.

For this modelling, we have chosen to use elasto-plastic material models that are already available in the ANSYS code as a standard tool for the cable to be constrained. Special-purpose material models would be better but require programming and at the moment the particular features of such models are uncertain. The features that the material model should show are:

• (1)  
  that bending of strands under magnetic load leads to a softening or weakening of the cable in the longitudinal direction (due either to filament fracture or simply to further plasticity of the copper in the strand);
• (2)  
  the friction between cable and jacket;
• (3)  
  the transverse compression of the cable under the magnetic loads.

It is known from the measurement of stress–strain on reacted Nb₃Sn filaments that the strands display a bilinear hardening curve [18] and that the bending of isolated strands under transverse loads is nonlinear. We do not have data to quantify how transverse and longitudinal loadings could interact but it is clear that the mechanism exists and that the plastic modulus of a strand (under axial load) can be as low as 20% of the initial elastic modulus.

We have chosen to represent the cable and jacket system using two-dimensional plane stress elements, as shown in figure 10. The elements have a thickness normal to the plane, representative of the actual cable and jacket dimensions. The top and bottom jacket elements are linked by a 2D side plate which does not contact the cable. The cable and jacket are linked at both ends (simulating the crimping rings). The cable–jacket interface is modelled by contact surface elements. The Poisson's ratio of the cable is set to 0 because the cable contains void and can be compressed in volume. The friction factor between cable and jacket is varied from 0–0.4, and 0.2 was used as the baseline value.

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There is a generic issue with the material model. With any elasto-plastic model, yielding is determined by Tresca or von Mises combined stress. In the SULTAN sample with a continuum isotropic or orthotropic model of the cable, the application of magnetic load on top of a uniaxial thermal compression from the differential contraction with the jacket results in no change or a decrease in the combined stress since the stress normal to the plane is zero. So the material model cannot be used to simulate cable weakening in the high-field region. A special material model could be developed that links transverse pressure to longitudinal modulus. However, there is a simpler approach for these calculations: artificially putting the cable in tension by simulating that, during cool-down, the cable shrinks longitudinally relative to the jacket instead of expanding. In this case, the application of magnetic pressure creates a higher Tresca stress and local yielding can be modelled.

It has to be emphasized that this reversal of strain direction does not affect the validity of the analysis but is an artefact to utilize an existing material model. It means that longitudinal x-direction stresses need to be interpreted as negative, as do the x-direction strain and slip directions. Since the Poisson's ratio is zero, the x-tension/compression is decoupled from the y direction.

Load conditions are applied as follows.

• (1)  
  Longitudinal thermal strain of 0.7% applied between cable and jacket; minor transverse shrinkage (to stabilize model) since the cable is largely copper in the transverse direction. This simulates the cool-down of the steel-Nb₃Sn system from the reaction heat treatment (about 650 °C).
• (2)  
  Magnetic load applied as surface pressure to cable: 45 kA and 11 T. Figure 10 defines the orientation of the magnetic loads, using figures 2 and 16.
• (3)  
  Warm-up and cool-down. The thermal strain between cable and jacket due to a warm-up to room temperature is 0.12%, so that the 4 K thermal strain of 0.7% is reduced to 0.58% and then reapplied.

The loads are applied in the following sequence:

• (a)  
  (1) is applied;
• (b)  
  200 applications of (2);
• (c)  
  one application of (3);
• (d)  
  100 applications of (2).

The elasto-plastic model is shown in figure 11. We use an initial elastic modulus of 50 GPa, followed by a plastic modulus in the range of 2.5–20 GPa, 2.5 GPa being used as the baseline. The yield stress is 100 MPa (0.2% strain) so that, after the first cool-down, the cable is in the fully yielded condition. The transverse modulus is 2 GPa and is fully elastic.

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In terms of yielding, this model implies that the extra strain in the high-field region is caused by a plastic deformation (the term △L_p in the analytical model of section 3.2). The bilinear hardening model does not create a cable with a reduced modulus, but a cable that work-hardens with load cycles. The yield envelope is determined by a Tresca stress of 100 MPa. In the transverse direction the modulus is unaffected by yield and the response remains elastic. In the longitudinal direction, the addition of the magnetic loads creates a plastic yielding. Expressed in terms of compressive strain (rather than the tension used in the finite-element model) the cable gets shorter (△L_p is negative) and the extra strain in the high-field region is △L_p/x₂. This develops slowly since the cable yielding is controlled by the lengths 1 and 3 on each side, which need to slip against the jacket for yielding to occur.

The results of the analysis are presented in figures 12–26 and are largely self-explanatory.

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The thermal longitudinal strain after the first cool-down is 0.006 56 (0.656%). The application of the first magnetic load cycle changes the longitudinal strain (figure 12) due to plastic yielding and a subsequent strain adjustment. The transverse stress (figure 13) reflects the magnetic loads, with a peak pressure of about 15 MPa in the cable.

The contact gap between cable and jacket (figure 16) is about 0.24 mm, which corresponds to estimates from the model coil programme. The pattern reflects the magnetic loads accurately. Relative sliding in the longitudinal direction is about 0.002 mm and is symmetric in the high-field region. The sliding is low since it is prevented by the magnetic pressure acting on one surface.

Following removal of the transverse load, a non-uniform longitudinal strain system is created, as shown in figure 18 (compare with thermal longitudinal strain of 0.656% after the first cool-down), with a reduction in strain magnitude at the ends and an increase in the high-field region. The relative sliding of the jacket–cable interface increases to 0.03 mm (so removing the transverse load allows relative slip to occur). After 200 cycles, the relative slip increases to 0.11 mm (see figure 20) and the total longitudinal strain magnitude in the high-field region further increases to match.

The warm-up/cool-down cycle causes the strain magnitude to further increase (figure 22) and the next load cycle (figure 23) increases it again. Figure 24 shows the longitudinal strain after a further 100 load cycles and figure 25 shows the relative slip (now 0.15 mm).

Figure 26 shows the thermal strain in the cable after the first 200 load cycles and after warm-up to room temperature.

Figure 27 summarizes the overall changes in longitudinal strain according to the number of load cycles. We have also transformed these strain changes into an equivalent drop in sample T_CS using the ITER Nb₃Sn strand scaling [19] and assuming an initial sample T_CS of 6.1 K. The first 200 load cycles produced a drop in T_CS of about 0.4 K, and the warm-up/cool-down cycle, a further drop of about 0.05 K.

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