E. Anderssen

Superconducting magnet coils are subject to enormous electro-magnetic forces which push the cables away from the winding pole, and against the surrounding structure. This structure is usually optimized trying to limit the overall motion and the strains experienced by the superconducting elements. To achieve this, preload forces are applied both in the coil cross-section and along its length. If the e.m. forces overcome these preloads, separation between the coil and the pole occurs, resulting in an overall loss of rigidity. During the magnet design process, it is often tempting to treat the optimization of the azimuthal and longitudinal preload systems separately. However, the two are inextricably related: as the cross-section preload increases, friction can prevent any motion in the longitudinal plane, and decreasing it can instead allow dangerous motions in the conductor ends. The latter can result in very high strains and, in Nb₃Sn conductors, damage that can prevent the magnet to reach the desired performances. In an attempt to define design guidelines, in this paper we use simplified numerical models to compute, as a function of the in-plane prestress, the variation of the peak strains in the end region of the coils. Finally, we investigate the impact of the azimuthal prestress on a real magnet case, the High-Luminosity Nb₃Sn Quadrupole MQXF.

A large aperture dipole magnet for testing inserts and cables at high field is under development at LBNL. Its design targets a 15 T field in a 144 mm by 94 mm rectangular aperture, and is based on block coils with flared ends. The coils are inserted in an aluminum shell based structure and prestressed using the bladder and key technology. The quench protection relies on energy extraction. Measurements and computations on cos( $\theta$ ) magnets have shown that the temperature rise after quench events and its gradient within the windings can significantly increase the mechanical stresses in the superconducting coils. In this study, we couple STEAM-LEDET 2-dimensional (2D) quench simulations to the 2D electro-thermo-mechanical ANSYS model of the magnet, predicting the stress acting on the coils during a quench discharge after activating the energy extraction system. The model is then used to optimize the quench protection system, in terms of hot-spot temperature, peak voltage, and limiting the peak stress reached during an energy discharge below the cooldown and powering one.

Superconducting magnet coils are subject to large thermo-mechanical loads applied during magnet assembly, cooldown and operation. These loads can cause the reduction of their critical current due to mechanical strains or local filament failures. Measurements on longitudinally stretched strands and Rutherford cables under transverse pressure have allowed exploration of material limits in two directions. However, no systematic study of the effect of multi-axial loading conditions has been done. Finite Element (FE) models show that, indeed, the actual limits of the material are strongly dependent on the nature of the applied load and that the strength under multi-axial loading can be significantly higher with respect to uniaxial loading conditions. In this paper, we try, for the first time, to measure the effect of multi-direction loading conditions on Nb ₃ Sn Rutherford cables. The experiments are performed on impregnated cable stacks under transverse, lateral, and longitudinal constraints. The integrity of the cables is verified by destructive metallography inspection, evaluating the damage as a function of the applied loading condition.

Stress managed magnet designs allow to limit the strain and stresses applied to the conductor during assembly and operation. In canted cos(θ) (CCT) designs, the conductor is wound around a mandrel: the impregnation process creates a bonding between the two, that can fail during magnet powering. The energy releases due to debonding are considered a potential cause of training quenches. In this study, we investigate these events modeling the mandrel-conductor interfaces by means of cohesive zone material models. The material properties were calibrated by means of measurements performed on representative interfaces, and the models were validated comparing the results with strain gauge measurements. A thermal model was used to compute the local temperature increase in the strands as a function of the energy released by debonding and frictional sliding across the newly formed interfaces. The result was then used to define a quench condition for the model, allowing to simulate the full training process of the CCT magnet. The obtained training curve is in reasonable agreement with the experimental results.

Stainless steel vessels see widespread use in superconducting magnets for particle accelerator applications. Their function varies in different magnet designs: they always provide the necessary liquid helium containment, but in some cases are also used to provide azimuthal prestress and can also be welded to the magnet end plate to provide additional longitudinal stiffness. A magnet designed with the bladder and key technology does not rely on the structural role of the vessel. They are structurally supported using azimuthally prestressed aluminum shells, and the longitudinal constraint by rods. In this case, the magnet designer would generally like to minimize the interaction between the magnet and the stainless-steel vessel and to minimize the coil stress variation due to the vessel. The stress state in the vessel and in the coil is a function of the circumferential interference, defined as the vessel azimuthal length minus the magnet circumference. The vessel and the magnet azimuthal length machining tolerances are relatively large resulting in significant stress variations in the superconducting coils. In this paper we introduce an interference-control shim, which can significantly limit the stress variation of the coils for a given variation of the interference. The effectiveness of the interference-control shim is evaluated numerically on the MQXF, the low- $\beta$ quadrupole for the High Luminosity LHC.

Superconducting magnets experience significant thermo-mechanical loads throughout their life cycle. These are introduced by the electro-magnetic forces during powering, but also by the prestress applied in many magnet designs. Further to this, the large thermal excursion that components of different materials experience can generate significant internal forces. The loads are also experienced by the superconducting coils, whose critical current can decrease as a consequence of the applied strain. It is then crucial to predict the overall mechanical behavior and conservatively design a magnet, avoiding failure of the mechanical components and of the superconducting coils. Finite Element Analysis (FEA) is generally used to perform these tasks, but its results rely heavily on the material properties and models used. This is in particular true for the coil composite, which is simplified to allow reasonable model sizes in full magnet models. In this paper, we present the state-of-art knowledge of the mechanical properties of the materials mostly used in superconducting magnet construction. We review elastic and plastic properties at room and cryogenic temperature, thermal contraction, and summarize the state-of-art failure criteria for these materials. Finally, the paper summarizes the present understanding of the mechanical behavior and limits of Nb ₃ Sn coils. For the first time, an orthotropic failure criteria is proposed.

A test facility dipole is being developed at LBNL, targeting a 16 T field in a 144 mm wide aperture. The magnet uses a block design, with two double-pancake coils. In order to minimize motion under the large Lorentz forces, the coils are preloaded against a thick aluminum shell and iron yoke using bladder and key technology. It is then crucial to verify that the performance of the magnet is not degraded due to strain induced on the Nb ₃ Sn conductor during assembly, cool-down and powering. The critical current of extracted strands was measured in a varying background magnetic field and as a function of the applied longitudinal strain. Finite element analysis was used to extract the strain state inside the superconducting strands during magnet assembly and operation. This strain was then compared to the measurements to evaluate potential reversible and irreversible effects on the magnet performances. The results suggest that the magnet can reach 16 T with sufficient margin, with no irreversible degradation in the high field region.

In the quest for higher field accelerator superconducting magnets, essential parts of their design are the so called yokes, which are traditionally made of low – carbon magnetic steel. In currently used magnets, they are typically found in the form of fine – blanked laminations, or machined from laminated heavy plates. The material’s choice is made based on a compromise between the high saturation field, providing a return path for the magnetic flux, and the mechanical robustness conferred to the magnets’ cold masses. This paper describes the mechanical characterization of low – carbon steel, and applies several approaches for the design and validation of the material from the structural point of view, applicable to a Nb $_{3}$ Sn quadrupole: MQXF. Tensile tests at room and cryogenic temperatures, together with fatigue and fracture toughness at cryogenic temperature have been performed. Calculations based on the obtained material properties and results of extensive non-destructive examination (ultrasonic testing) have been implemented in order to ascertain the structural limits of low – carbon steel for its use in the fabrication of high field accelerator superconducting magnets.

The High-Luminosity LHC Accelerator Upgrade Project (HL-LHC AUP) is approaching the production phase of the US-contributed Q1 and Q3 Interaction Region Quadrupoles (MQXFA). The structures for the MQXFA prototypes were designed and inspected by the US-LARP (LHC Accelerator Research Program), AUP developed criteria, which will be used for the pre-series structures. The first two full-length prototypes with 4.2 m magnetic length, MQXFAP1 and MQXFAP2, were designed and assembled at Lawrence Berkeley National Laboratory (LBNL), and tested at Brookhaven National Laboratory (BNL). The end aluminum short shell of MQXFAP2 was fractured along the shell length during the test, and tests were stopped. Analytical and Finite Element analysis were performed in light of the graded procedure defined in the Structural Design Criteria to investigate the fracture failure for MQXFAP2. In this paper, we report the fracture analysis of the current shell design, including the elasto-plastic simulations with sub-model technique, and calculations with Linear Elastic Fracture Mechanics (LEFM). Test material properties are also presented. The results of this analysis explain why the end shell of MQXFAP2 failed, and suggest fillets on the end shell notches to meet the margin specified in the Structural Design Criteria.