Loss of pre-stress in impregnated superconducting magnets, experimental results and numerical analysis

The SMC (Short Model Coil) R&D program was started at CERN around 2007 to develop the Nb₃Sn technology. The small magnet structure allowed relatively cheap and fast testing of various superconducting coils. One of the key questions to be answered, was related to the relation between the pre-stress and the magnet’s performance. To measure this dozens of strain gauges were installed on the coils, the axial tie-rods and the external shell. The experimental results of the strain measurements during all stages of the load: room temperature (RT) pre-stress, cool-down, powering, warm-up were analyzed in an extensive report [56]. A repeatable pattern of a decreasing strain after the warm-up, compared to the value before the cool-down, was observed on the external cylinder for all the tested coils. Values from 2 % to 50 % were reported.

In this work a viscoelastic model was used to explain this effect. The Nb₃Sn coil was treated as a composite material with decreasing stiffness due to mechanical damage. The Generalized Maxwell Solid model (Prony series model) was employed, including one spring and one damper, leading to a relatively simple model characterized by only two parameters. The two constants of the viscoelastic model were found: 1st – the relative relaxation moduli $\alpha$ based on a calibration curve derived from the experimental results of the SMC program and the 2nd one – relaxation time $\tau$ – based on minimizing the computational cost, by finding the asymptotic solution in one integration step. The model showed the capability of explaining the strain drop (loss of pre-stress) of more than 80 %. In addition to the viscoelastic effects, the role of friction coefficient was studied revealing the possibility of explaining up to 14 % of the strain drop. Yet, to fit with the experimentally measures strains on the SMC cylinder, especially during the RT pre-load, the most-probable value of the friction coefficient should be $\mu <0.4$. The strong impact of the stiffness of the G-10/G-11 laminate used to spread the load on the coil was found, indicating the need of knowing the elastic properties of this material very precisely. In addition, the experimentally measured strain values showed strong asymmetric, both in plane and along the magnet’s axis, revealing the potential sensitivity to the geometric imperfections and the need for 360° magnet models.