Direct measurement of Nb₃Sn filament loading strain and stress in accelerator magnet coil segments

The samples studied are stacks made of ten reacted Nb₃Sn Rutherford cables used for the 11 T dipole coils (so-called ten-stack samples). The 11 T dipole cables with a nominal width of 14.7 mm and a mid-thickness of 1.25 mm, are stacked alternatingly in order to compensate for the keystone angle of 0.79° [10]. The Rutherford cables are made of 40 Restacked-Rod Process (RRP) type Nb₃Sn strands from Oxford Instruments Superconducting Technology, now Bruker-OST. The Nb₃Sn heat treatment was performed with a ramp rate of 50 °C h⁻¹ and comprised three isothermal plateaus of 210 °C-48 h, 400 °C-48 h and 665 °C-75 h. The cables with a 25 μm thick stainless steel core are surrounded by a 0.15 mm thick cable insulation made of a mica tape and S2/E-glass fibre. A non-impregnated and an impregnated cable stack were used for the experiments in the transverse and axial load directions, respectively. The height of the non-impregnated and impregnated ten-stack samples was 16.6 mm and 15.6 mm, respectively. For more information about the samples see [18].

Loading strains were measured at ambient temperature in situ under a neutron beam as a function of externally applied stress. Compressive stresses were applied either in the transverse or in the axial direction using a load frame combined with an Eulerian cradle that enables rotation of the sample load axis with respect to the scattering geometry (figure 2). The maximum force that can be applied with the load frame is 50 kN, corresponding to a maximum compressive stress of 220 MPa on the samples with a cross section of 225 mm². The effect of friction on the stress–strain results was limited by adding lubricant (PTFE spray) on the sample to pressing tool contact surfaces. In order to limit buckling effects, the sample height does not exceed the sample width. Force measurements were performed with an HBM Type 03 50 kN load cell, and the sample strain was measured with an Instron 2620-602 extensometer with a gauge length of 13.1 mm (figure 2).

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All experiments were started with a compressive preload of about −0.2 kN, corresponding with a pre-stress of about 1 MPa. Measurements were performed with applied stress steps of 15 MPa, and the load rate between the stress plateaus was always 50 N s⁻¹.

A bent Si (400) monochromator provides a neutron beam with an approximate wavelength of λ = 1.672 Å. Lattice strain measurements were performed in the assumed principal stress directions, which are in the following referred to as axial, radial and transverse directions. The transverse direction corresponds to the azimuthal direction in a coil. A diffraction angle range 70.5° < 2Ɵ < 83.5° was recorded, and the Nb₃Sn (321) and Cu (220) reflections were fitted by Gaussian functions. In addition to the pure Cu stabiliser, the reacted RRP type conductor contains α-bronze in the Nb₃Sn subelement cores. Because of the Sn in solid solution the Cu lattice parameter of α-bronze is larger than the pure Cu lattice parameter, and both diffraction peaks could be distinguished in the experiment. Only the pure Cu diffraction peak has been fitted and taken into account in the elastic strain calculations.

Previous stress dependent synchrotron x-ray diffraction measurements where a large d-spacing range could be recorded [6] have shown that the Nb₃Sn (321) reflection represents the overall Nb₃Sn strain behaviour well. In face-centred cubic materials like Cu the effect of intergranular stresses on peak position is comparatively strong on (200) reflexes [19], therefore, these peaks have been avoided. Prior to peak fitting the diffractograms were corrected for the detector efficiency using incoherent scattering from a vanadium scan. The diffraction angles are an average inside the analysed gauge volume of nominally 5 × 5 × 5 mm³, which is defined by a 5 × 5 mm² slit that shapes the beam and by a radial collimator in front of the detector. The acquisition time for the recording of the Nb₃Sn (321), and Cu (220) diffraction peaks in the three directions is about 45 min per measurement point. The main Nb (100) peak is not present in this configuration (beam parallel to wire drawing axis) because of the strong Nb texture [20].

The Nb₃Sn and Cu loading strains ε_hkl in the transverse, axial, and radial directions have been determined from the Nb₃Sn (321) and Cu (220) scattering angles θ_hkl according to equation (1). The assumed stress free scattering angles are 2θ_{0_}Nb₃Sn(321) = 72.783° and 2θ_{0_}Cu(220) = 81.937°.

$$\begin{eqnarray}&&{\varepsilon }_{hkl}=\displaystyle \frac{{d}_{hkl}-{d}_{0,hkl}}{{d}_{0,hkl}}=\displaystyle \frac{\sin ({\theta }_{0,hkl})}{\sin ({\theta }_{hkl})}-1.\end{eqnarray} \tag{ 1 }$$

Loading stresses have been calculated according to equation (2), assuming that the transverse, axial and radial directions are the principal stress directions in the sample, and that there are no shear stresses. Nb₃Sn (321) and Cu (220) elastic constants (E_hkl) and Poisson ratios (ν_hkl) have been calculated from single crystal elastic constants [21, 22] using the Kröner coupling model [23] for the interaction between crystallites (E_Nb3Sn(321) = 131 GPa, ν_Nb3Sn(321) = 0.363,E_Cu(220) = 138.9 GPa, ν_Cu(220) = 0.333).

$$\begin{eqnarray}&&{\sigma }_{ii}=\displaystyle \frac{{E}_{hkl}}{1\,+\,{\nu }_{hkl}}\left({\varepsilon }_{ii}\,+\,\displaystyle \frac{{\nu }_{hkl}}{1\,-\,2{\nu }_{hkl}}({\varepsilon }_{11}\,+\,{\varepsilon }_{22}+{\varepsilon }_{33})\right).\end{eqnarray} \tag{ 2 }$$

The evolution of the Nb₃Sn and Cu loading strains and stresses in a non-impregnated 11 T dipole cable stack in the three principal directions are presented in figure 3. Before application of external stress, Nb₃Sn is under slight axial compression and Cu under slight axial tension. This is the result of the thermal expansion mismatch of the different conductor constituents during cooling from the Nb₃Sn processing temperature of 650 °C. Upon application of external transverse pressure, a similar compressive Cu loading stress evolution is observed in the three principal directions (figure 3(d)). The transverse Cu pressure in the axial and radial directions imposes an axial and radial tensile stress on the Nb₃Sn filaments (figure 3(c)).

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In figure 4 the Nb₃Sn and Cu loading stresses in the non-impregnated 11 T dipole cable stack are compared. In the transverse direction (i.e. the loading direction), the Nb₃Sn and Cu stress evolutions are nearly identical, and the transverse stresses increase linearly with increasing external pressure. This indicates that the composite exhibits iso-stress behaviour when it is loaded perpendicularly to the filament (wire) direction. The Nb₃Sn and Cu loading stress increase is about 20% higher than the external pressure increase. This difference can be partly explained by the presence of about 25% porosity in the sample that does carry load. When the external load is released the transverse Nb₃Sn and Cu stresses are almost completely relaxed as well. In the radial direction Nb₃Sn and Cu stresses have similar magnitude, Nb₃Sn being in tension and Cu in compression. In the axial direction the higher Nb₃Sn tensile stress with respect to the Cu compressive stress can be explained by a comparatively smaller Nb₃Sn cross section with respect to the Cu cross section. When the external transverse pressure is released the axial and radial stresses are only partly released.

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The Nb₃Sn loading stresses shown above are average values inside the gauge volume. The maximum Nb₃Sn stresses are likely higher, as indicated by the Nb₃Sn diffraction peak broadening when the external load is increased (figure 5). Nb₃Sn diffraction peak broadening indicates that the strain inhomogeneity increases, which occurs first in the transverse and radial directions. When the external transverse pressure on the non-impregnated cable stack exceeds 130 MPa, Nb₃Sn (321) peak broadening is also observed in the axial direction.

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Figure 6 presents the Nb₃Sn and Cu loading stress evolution in an impregnated 11 T dipole cable stack under axial compressive loading. At 150 MPa externally applied axial pressure the axial Nb₃Sn (321) and Cu (220) loading stresses are about 300 MPa and 100 MPa, respectively. This stress evolution is compatible with the assumption of iso-strain conditions when the conductor is loaded in the wire and Nb₃Sn filament direction.

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The assumption of iso-strain conditions is further confirmed by the comparison of the macroscopic sample strain measured with an extensometer, and the Nb₃Sn filament strain determined from the Nb₃Sn diffraction angles (figure 7(a)). Under axial loading the radial and transverse lattice parameter changes can be explained almost entirely by the Poisson effect (figure 7(b)).

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