Pre-Bending Effect of Nb3Sn conductor

Masayoshi Wake( )

The critical temperature of superconductors is affected by the lattice of the superconductor. If there is some distortion due to the external stress, the critical current density of a superconductor is degraded by this effect. Empirically, this degradation is proportional to the square of the elongation. In the case of compound conductor such as Nb3Sn multi filament conductor, internal stresses due to the thermal contraction differences cause a natural degradation of the current density. Since Nb3Sn is formed at 925K and brought down to the operation temperature of 4.2K, Nb3Sn filaments are compressed because the surrounding bronze contracts more than the copper. Therefore, if the conductor is given a tension, the current density of the conductor is increased due to the compensation of such residual stress in the beginning. Then, of course, if the tension becomes greater, it degrades again due to the tensile distortion. Prof. Awaji and his group, Tohoku University, found the way of removing such residual stress. If the conductor is bent several times at room temperature, due to the creeping of the bronze and copper, the internal stress of the Nb3Sn filaments can be removed. His group experimentally confirmed this effect. Confirming this effect by the FEM analysis was first made by prof. Murase of Okayama University. This is an attempt to seek further vivid simulation of the pre-bend effect.
Von Mises strains are displayed in the following pictures through the process of forming ofNb3Sn, cooling and tensioning. The color scales are common to all the figures except the ones for the strains under bent stage. Scales for such figures should be multiplied by surrounding bronzes are not displayed to show the filament status in most of pictures. The simulation was made for a 1mm diameter Nb3Sn conductor with bronze matrix and stabilizing coppr.
The conductor is heat treated at 650C for the reaction. At this stage, the conductor is fully annealed and there is no stress in the conductor (left). Then the temperature goes down to room temperature. Thermal contraction difference between filaments and matrix develops the internal stress as in the right figure. Filaments show green color indicating they are under compression of 0.15%. This compression deforms the Nb3Sn lattice resulting the decrease of critical current.
Vertical bending is applied as the "pre bending"(left). Actually this is a force proportional to the y-coordinate of 0.06MPa/mm. The stress in the bottom part turns into tension and top part receives large compression.
The bending force is applied in the opposite direction and then original direction again with a little force to compensete the imbalance. As a result, the in ternal stress is greatly reduced as shown in the right picture. Note that the colour scale is difernt in the left and right pictures.
Next is the application of horizontal bending to the right direction.
Left is during the bend and the right is the stress after the bending force is removed. Bending to the opposite direction was also applied and removed. Note that the color scale is different in the left and right pictures. The internal stress is further reduced.
Then the cooling of the conductor started. These pictures are at the temperature 150K. Left and right are with and without pre-bending, respectively. The internal stress of the conductor increases again but the stress level in the case with pre-bending remain less than the case without pre-bending.
Further cooling of the conductor started. The internal stress of the conductor increases again. Left is without pre-bend and right is with pre-bend at 4.2K. There is a significant difference of the stress level. Also, it is interesting that the peripheral filaments in the case without pre-bending have higher stress due to the radial compression. This is the cause of additional current density degradation than the stress offset.
When electromagnetic force works on the conductor, the tensile stress is generated and the compressive stress in the filament is reduced. Left (without pre-bending) and right (with pre-bending) still has different stress level.
The tensile force is further increased. Left (without pre-bending) is still under compression but the right (with pre-bending) reaches to the minimum stress. Forces are 300N/node
The internal stresses in all the fialments are further reduced.
The tensile force is further increased. Left (without pre-bending) reaches the minimum and right (with pre-bending) turns into the tensile state. Forces are 400N/node
The internal stress in all the fialments are further reduced.
The tensile force is further increased. Left (without pre-bending) and right (with pre-bending) both goes into tensile state. The difference becomes less evident. Forces are 500N/node
The tensile force is further increased. Left (without pre-bending) and right (with pre-bending) are under tensile state. Both behave not much different. Forces are 500N/node
The stress distribution in longitudinal direction through out the process is summarized as the following graph. The dashed lines are for pre-bended conductor.

Since the current density of the conductor is proportional to the square of the stress, the curve to predict the Jc behavior of conductors with and without pre-bend can be estimated as follows. The difference in the current density especially at the low stress region is evident. This result is close to the measured current density results.

Structure dependence of the pre-bend effect is further studied and reported here.