Degradation of I c due to residual stress in high-performance Nb3Sn wires submitted to compressive transverse force

Future particle colliders in search for new physics at the energy frontier require the development of accelerator magnets capable of producing fields well beyond those attainable with Nb-Ti. As the next generation of high-field accelerator magnets is presently planned to be based on Nb3Sn, it becomes crucial to establish precisely the mechanical limits at which this brittle and strain sensitive superconductor can operate safely. This paper reports on the stress dependence and the permanent reduction of the critical current under transverse compressive loads up to 240 MPa in state-of-the-art restacked-rod-process (RRP®) and powder-in-tube Nb3Sn wires. Single-wire experiments were performed at 4.2 K in magnetic fields ranging between 16 T and 19 T on resin-impregnated samples to imitate the operating conditions of a wire in the Rutherford cable of an accelerator magnet. Depending on the wire technology, we measured irreversible stress limit values—defined as the transverse stress value, leading to a permanent reduction in the critical current of 5%, assessed by convention at 19 T—ranging between 110 MPa and 175 MPa. This permanent reduction of the critical current after mechanical unload can occur for two reasons, which can be concomitant: the plastic deformation of the Cu matrix that produces residual stresses on the Nb3Sn lattice and the formation of cracks. We developed a method to identify the dominant degradation mechanism in our experiments that allowed us to predict the fraction of critical current lost due to residual stresses. Interestingly, we found that in the RRP® wires the measured reduction of Ic after unload from stresses as high as 240 MPa can be fully ascribed to residual stresses. An independent confirmation of this conclusion coming from a study combining x-ray tomography and deep learning Convolutional Neural Networks is also reported.

Future particle colliders in search for new physics at the energy frontier require the development of accelerator magnets capable of producing fields well beyond those attainable with Nb-Ti. As the next generation of high-field accelerator magnets is presently planned to be based on Nb 3 Sn, it becomes crucial to establish precisely the mechanical limits at which this brittle and strain sensitive superconductor can operate safely. This paper reports on the stress dependence and the permanent reduction of the critical current under transverse compressive loads up to 240 MPa in state-of-the-art restacked-rod-process (RRP ® ) and powder-in-tube Nb 3 Sn wires. Single-wire experiments were performed at 4.2 K in magnetic fields ranging between 16 T and 19 T on resin-impregnated samples to imitate the operating conditions of a wire in the Rutherford cable of an accelerator magnet. Depending on the wire technology, we measured irreversible stress limit values-defined as the transverse stress value, leading to a permanent reduction in the critical current of 5%, assessed by convention at 19 T-ranging between 110 MPa and 175 MPa. This permanent reduction of the critical current after mechanical unload can occur for two reasons, which can be concomitant: the plastic deformation of the Cu matrix that produces residual stresses on the Nb 3 Sn lattice and the formation of cracks. We developed a method to identify the dominant degradation mechanism in our experiments that allowed us to predict the fraction of critical current lost due to residual stresses. Interestingly, we found that in the RRP ® wires the measured reduction of I c after unload from stresses as high as 240 MPa can be fully ascribed to residual stresses. An independent confirmation of this conclusion coming from a study combining x-ray tomography and deep learning Convolutional Neural Networks is also reported. * Author to whom any correspondence should be addressed.
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Introduction
The development of accelerator magnets that produce fields beyond those attained in the Large Hadron Collider (LHC) is among the key technologies required to expand the scientific reach of particle colliders in the search for new physics [1]. LHC, upgraded in luminosity with the ongoing highluminosity LHC (HL-LHC) project, ready for beam commissioning in early 2029, will remain the most powerful accelerator in the world for at least the next two decades. This upgrade relies, among other innovative technologies, on cutting-edge quadrupole Nb 3 Sn magnets reaching a peak field in the conductor of about 12 T [2]. After the latest update of the European Strategy for Particle Physics, released in June 2020 [3], and with the goal of increasing the potential to discover new particles and phenomena by pushing at the energy frontier, plans are going ahead to conceive a future proton-proton collider with a center-of-mass energy of at least 100 TeV, i.e. more than 7 times the design energy of the LHC. A first analysis of the general parameters of this 100 TeV Future Circular Collider (FCC) leads to a baseline configuration requiring dipole magnets of 16 T in a 100 km-long tunnel [1]. The above projects and studies are turning towards superconducting materials other than Nb-Ti, with Nb 3 Sn being the preferred solution for magnetic fields of up to 16 T at present. A special grade Nb 3 Sn wire was industrialized for HL-LHC, with values of the non-copper critical current density J c , i.e. the critical current, I c , divided by the wire cross-section area minus the Cu area, of at least 2450 A mm −2 at 4.2 K, 12 T, corresponding to about 1000 A mm −2 at 4.2 K, 16 T [4]. These values are about three times higher than those of the Nb 3 Sn wires developed for the International Thermonuclear Experimental Reactor [5], but are still ∼40% below the performance requirement for the 16 T dipoles of the FCC [6]. Achieving the projected target of 1500 A mm −2 at 4.2 K, 16 T in a wire with proven scalability to long-length industrial production requires a major innovation step and this is the main drive of the Conductor Development Program launched by CERN in 2015 [7].
However, the superior superconducting properties of Nb 3 Sn with respect to Nb-Ti are impaired by its mechanical fragility, which poses critical challenges to the magnet mechanics. As a result of the large fields and current densities, the superconducting coils of an accelerator magnet experience large electromagnetic forces. In dipoles, the azimuthal component of these forces accumulates at the midplane of the coil, with a magnitude of many hundreds kN m −1 . In the same way, the radial component pushes the coil outwards with a maximum displacement localized again at the midplane, while the axial component tends to elongate the coil. The mechanical design of the magnets aims to avoid deformation or movement of the conductor during powering, which may lead to premature quenches, and this is achieved by applying a mechanical pre-compression to the superconducting coils when assembling the magnet. However, the combination of pre-compression, thermal stresses during heat treatment and during the cooldown at cryogenic temperature, and electromagnetic forces arising in the operation regime exposes the brittle and strain sensitive Nb 3 Sn to the risk of irreversible reduction of its critical current performance. Evidences of these permanent damages have been documented, for instance, for a two-in-one-aperture model dipole magnet wound with Nb 3 Sn Rutherford cables, which was built in the frame of the HL-LHC project [8]. All four coils constituting the magnet exhibited a reduction in I c distributed over a large fraction of the inner layer turn at the level of the midplane and it was suspected that degradation occurred due to too high levels of stress during the magnet's production.
Currently, a large majority of the proposed designs for the production of Nb 3 Sn-based accelerator magnets entail peak stresses in the coils that are in the range of 150-200 MPa during magnet assembly and operation [9]. These peak stresses are compressive and act in the transverse direction of the Nb 3 Sn Rutherford cables used to wind the coils. As the local stresses approach and, in some cases, exceed the currently understood limits of the conductor, it becomes crucial to establish precisely the mechanical limits to be adopted in any of the magnet assembly, cooling and operation phases. Hence, several laboratories have developed dedicated experiments aimed at testing the mechanical sensitivity of Nb 3 Sn, in which a conductor-a full Rutherford cable, a sub-scale cable or a single wire-is exposed to mechanical loads relevant to accelerator magnet applications [10][11][12][13][14][15][16][17]. In particular, CERN has carried out extensive measurement campaigns on Rutherford cables to assess the effect of transverse loads exerted either at room temperature, to determine the maximum level of acceptable pre-compression during the magnet assembly process [11,18,19], or at 4.2 K to reproduce the electromagnetic forces acting on the coils at operating conditions [12,13]. At the University of Geneva, we perform tests of the critical current on single Nb 3 Sn wires submitted to transverse compressive loads at 4.2 K [17]. Wires are resin-impregnated to imitate the working conditions in the Rutherford cables of accelerator magnets [14][15][16]. During our experiments, samples are exposed to load/unload cycles at low temperature under increasing transverse stress values and this allows us to determine the reversible and irreversible components of the measured reduction of I c . The reversible component is fully recovered when removing the load and is associated with the reversible reduction under stress of the upper critical field, B c2 [20]. On the other hand, two distinct mechanisms can contribute to the irreversible reduction of the critical current performance after unload: residual stresses and filament breakages [13,15,19,21]. First, the superconducting filaments in a Nb 3 Sn wire are embedded in a soft Cu matrix, whose yield strength ranges between 40 MPa and ∼90 MPa [22,23]. The plastic deformation of the Cu matrix determined by an external load imposes a residual stress to Nb 3 Sn, after unload that in turn results in a reduction of B c2 . Second, Nb 3 Sn is a brittle intermetallic compound characterized by a strong propensity to fracture. The formation of filament breakages at sufficiently high pressures leads to the reduction of the current carrying section of the wire.
In this paper, we report on the results of a measurement campaign of the critical current under transverse load in stateof-the-art restacked-rod-process (RRP ® ) and powder-in-tube (PIT) Nb 3 Sn wires. We focused our study on wires extracted from production billets used at CERN for the development of magnets. In addition, we present here a method that allows us to identify the dominant mechanism behind the irreversible reduction of I c and to quantify the fraction of I c lost after unload due to residual stresses and filament breakages, respectively.

Sample characteristics and experimental details
Three Nb 3 Sn wires have been studied in this work. Two of them are RRP ® wires composed of 108 sub-elements embedded in a high-purity Cu matrix following the 108/127 hexagonal restack, with a nominal copper to non-copper ratio (Cu/non-Cu) equal to 1.2. They differ in diameter: the wire at 0.7 mm belongs to one of the production billets developed for the 11 T dipoles of HL-LHC [8], while that at 0.85 mm was extracted from a billet produced for use in the new interaction region quadrupoles of HL-LHC, identified by the acronym MQXF [24]. The third wire is a PIT wire with 192 filaments and a Cu/non-Cu volume ratio of approximately 1.22. It has a diameter of 1.0 mm and was produced for the FRESCA2 dipole project at CERN [25]. Dedicated measurement campaigns of the electromechanical properties of the RRP wire at 0.85 mm and of the PIT wire at 1.0 mm are reported in [16] and [15], respectively.
Technical parameters and main properties of the wires are resumed in table 1.
Wires were tested for the stress dependence of the critical current under transverse compressive load using a specially developed measurement device with a geometry similar to a Walters spring. The design and operation of the device are well documented in the literature [14][15][16][17]21]. The wire sample is confined in a U-shaped groove, whose width, w, is adapted to the sample diameter: for the presented cases, it is w = 1.0 mm when the wire diameter is 0.7 mm and w = 1.15 mm when the wire diameter is 0.85 mm or 1.0 mm. A force of up to 35 kN is transmitted by an upper anvil that fits the groove. Samples are resin-impregnated to reproduce working conditions close to those of a Nb 3 Sn wire in a Rutherford cable. Impregnation is performed using a de-gassed mixture of epoxy resin (type L + hardener L provided by R&G Faserverbundenwerkstoffe) and filler (thixotropic agent provided by SCS-Füllstoffe) in a 100:40:2 weight ratio. The choice of this type of impregnation relies on its favorable characteristics for manipulation and on its elastic modulus, which is very similar to the one of the CDT-101 K resin used for the impregnation of HL-LHC Nb 3 Sn coils [26,27]. Figure 1 shows the results of a typical I c vs transverse stress measurement performed at T = 4.2 K on the RRP wire at 0.7 mm. I c was measured at B = 19 T for increasing values of the applied transverse force and the stress on the wire was calculated as the force divided by the groove area. The force was increased in regular steps and in every other step the sample was fully unloaded to monitor the irreversible reduction of I c . In addition, the magnetic field dependence of I c under load and after unload was measured at given values of the applied force. The I c values recorded for applied fields ranging between B = 16 T and B = 19 T were then used to determine the upper critical field, B c2 , via the well-known Kramer extrapolation [28]. No correction for the self-field generated by the current passing through the wire was applied. All I c values reported in this paper have been determined with the 0.1 µV cm −1 criterion over a gauge length of 126 mm. The behavior of the wire in the field of interest for HL-LHC, i.e. at 12 T, could not be measured because of the high I c values. However, the analysis reported in the following section provides a method to extrapolate to lower fields the results of the electromechanical tests. (σ → 0) divided by I c0 . The examined wires exhibit a reduction of the critical current under transverse stress, which at σ = 150 MPa ranges between 20% and 50% with respect to I c0 . When the applied stress exceeds a certain threshold that is wire-dependent, a monotonic decrease of I c unload (σ → 0) is also observed, indicating a permanent reduction of the current carrying capability of the wire. By convention, we define as the irreversible stress limit, σ irr , of the wire, the stress value leading to a reduction of I c unload (σ → 0) by 5% with respect to I c0 . The experiment revealed substantial differences in the value of σ irr for the examined wires. We measured at 19 T σ irr = 155 MPa for the RRP ® wire at 0.70 mm, 175 MPa for the RRP ® wire at 0.85 mm and 110 MPa for the PIT wire at 1.0 mm. There is a clear indication of a higher tolerance to transverse stress of RRP ® wires compared to the PIT one, which is related to the differences in the internal architecture of the wire composites. In particular, the mechanical properties of the materials left at the center of the Nb 3 Sn sub-elements or filaments after reaction are very different for RRP ® and PIT wires and are expected to play a role in the electromechanical properties: the core of an RRP ® sub-element consists  of a solid low-Sn bronze matrix embedding a large number of micrometric voids [29], while the reacted filaments of PIT wires contain a poorly connected powder core where the reaction remainders are aggregated. Numerical simulations performed by Baffari and Bordini in [30] support also this conclusion. On the other hand, the difference in the σ irr values of the two RRP ® wires could be considered a priori an unexpected result, as the layout of their sub-elements is identical. The reduction of the transverse stress tolerance observed in the 0.70 mm wire may be influenced by the heat treatment schedule. Following the recipe used at CERN, the final step of the heat treatment for the 0.70 mm wire was performed at 650 • C for 50 h, i.e. at a lower temperature compared to the 665 • C used for the reaction of the 0.85 mm wire keeping the same duration. The lower temperature allows preserving a high residual resistivity ratio, RRR, of the Cu-matrix also in the smaller wire diameter, which is needed to ensure the stability of magnets against quenching. Nonetheless, a reduction of the electromechanical limits when decreasing the heattreatment temperature was documented in the literature and confirmed in various wire designs of RRP ® wires [31].

Results and discussion
It was already pointed out in section 1 that the permanent reduction of the critical current observed when the load is removed occurs for two reasons, which can be concomitant: (a) the residual stress imposed to Nb 3 Sn by the plastically-deformed Cu matrix and (b) the formation of cracks in the Nb 3 Sn layer. Even if both mechanisms can lead to the irreversible reduction of I c unload (σ → 0), their effects are expected to be different, as already pointed out in [15] and [13]. Similarly to what is observed when the wire is under load, the distortion of the Nb 3 Sn lattice due to residual stress determines a reduction of the upper critical field after unload, B c2 unload (σ → 0), with respect to the value measured on the virgin wire, B c2,0 . On the other hand, when cracks are present in the Nb 3 Sn layer, even if the critical current density of the superconductor is in principle preserved, the total current-carrying cross-section and, thus, I c unload (σ → 0) can be reduced.
The following equation proposes an expression to describe the dependence of the critical current after stress release on the magnetic field, B, and on the maximum transverse stress applied to the wire, σ: The magnetic field dependence follows the wellestablished scaling relation proposed in [32], with the constant prefactor C being independent of σ [13]. f (σ → 0) ranges between 0 and 1 and accounts for the reduction of the currentcarrying cross-section due to cracks. It follows from (a) that the percentage of critical current drop due to cracks must be constant in field. The field dependence of I c unload (σ → 0) is governed by B c2 unload (σ → 0). As the residual stresses on Nb 3 Sn drive the reduction of B c2 unload (σ → 0), the percentage of critical current drop due to residual stresses increases with the applied field.
In figure 3 the stress dependence of the upper critical field under load, B c2 load (σ), and after unload, B c2 unload (σ → 0) for the three examined wires is reported. Following the same convention used in figure 2, each point of B c2 unload (σ → 0) is represented at the value of σ from which the unload started. The most important result of figure 3 is the continuous decrease of B c2 unload (σ → 0) after unload from increasing stress values. A reduction of more than 1 T with respect to B c2,0 is measured after the last unload from 210 MPa in the PIT wire, while the final unload from 240 MPa leads to a drop of about 0.5 T for the two RRP wires. These experiments provide a clear indicator of the residual stresses acting on Nb 3 Sn. However, this result is    Very interestingly, the measured reduction of I c unload (σ → 0) from stress values as high as 240 MPa can be fully accounted for with the reduction of B c2 unload (σ → 0) due to residual stress, regardless of the type of wire. Cracks and breakages in Nb 3 Sn seem to have a negligible effect on the permanent degradation of the critical current in the geometry of our experiment, i.e. for an impregnated wire confined in a groove and submitted to transverse compressive load. Similar conclusions have been drawn by De Marzi et al [13] from the analysis of the critical current reduction under transverse stress of Rutherford cables based on the same type of PIT Nb 3 Sn wires presented in this work.
An estimate of the residual stress values can be inferred by comparing the I c data under load and after unload reported in figure 2. In particular, one can look for the values of applied stress that lead to the same I c reduction measured after unload. The PIT wire exhibits an I c value after the final unload from σ unload = 180 MPa that matches the I c value measured under load at σ load = 75 MPa, whereas for the RRP wires, the I c value after unload from σ unload = 240 MPa corresponds to the I c value measured under σ load in the 135-145 MPa range. When the stress σ load is applied to the wire, the resulting stress distribution on Nb 3 Sn, which depends on the mechanical properties of the wire composite, leads to the same overall change in the superconducting properties as the residual stresses left after releasing σ unload , even though the local stress distributions may differ. However, a mechanical analysis based on finite element simulations is required to extract a quantitative correlation between applied load, plastic deformation of the Cu matrix, and residual stress distribution in the Nb 3 Sn phase. We plan to conduct such an analysis in future work. Notably, we observe that for both types of wires, the relative reduction of I c remains consistent for a given difference in stress ∆σ = σ unload -σ load , with I c decreasing to approximately 83% of I c0 when ∆σ is approximately 100 MPa, and to 95% of I c0 when ∆σ is approximately 65 MPa.
An independent confirmation of the dominant role of residual stresses on the degradation of I c came from a study combining x-ray tomography and deep learning Convolutional Neural Networks, reported in [33]. Tomography images were taken at the European Synchrotron Radiation Facility on the RRP wire sample at 0.7 mm extracted from the I c vs transverse stress probe at the end of the experiment after the final unload from 240 MPa. Figure 5(a) shows a tomography cross section of the wire. Figure 5(b) provides a 3D perspective of the voids, which are formed during the reaction heat treatment, in cyan and of the cracks in yellow, as detected by deep learning. The wire volume examined by tomography corresponds to a scan length of about 1.5 mm. More details about the methods of detection and image recognition are reported in the [29,33,34]. Only very few cracks are present in the examined sample after unload from 240 MPa. These cracks are up to 30 µm long in the transverse direction, with a maximum width of ∼5 µm, while they propagate beyond the scan length of 1.5 mm in the longitudinal direction. However, none of the observed cracks creates interruptions in the subelements, which could lead to a reduction of the active cross-section. This observation supports the conclusion that the irreversible reduction under transverse compression of the critical current is only marginally influenced in our experiment by the presence of cracks. Other work also reports on micro-cracks, generated in the subelements of Nb 3 Sn RRP ® Rutherford cables subjected to transverse pressure at room temperature, which did not impact the measured critical current [18].

Conclusions
This manuscript presents a comparison of the electromechanical response to transverse compressive loads of state-ofthe-art Nb 3 Sn wires produced by two different technologies, RRP ® and PIT. The scope was to establish the limit of irreversible critical current degradation in wires used in high-field accelerator magnets, whose conductors are exposed to large mechanical loads during the assembly, cooling and operation phases. The experiments revealed marked differences in terms of tolerance to transverse stress between the measured RRP ® and PIT wires, which follow from the different layout, composition and mechanical properties of the wire composites. The irreversible stress limit, defined as the stress level leading to a permanent reduction of the critical current by 5% at 19 T, was found to be 110 MPa for the PIT wire and in the 155-175 MPa range for the RRP ® wires examined in this work. The degradation of I c occurs generally from the combination of the effects of the residual stresses on Nb 3 Sn, which result from a plastic deformation of the Cu matrix, and the formation of cracks. We developed a method to identify the dominant degradation mechanism following the evolution of the upper critical field after unload from increasing stress values. This analysis allowed us to conclude that the irreversible reduction of I c measured in our experiments can be mainly ascribed to the effects of residual stresses. Complementary studies combining x-ray tomography and deep learning Convolutional Neural Networks were performed on the exact same samples tested for the I c vs transverse stress dependence, after the final unload from 240 MPa. Interestingly, only very few cracks were detected in these samples exposed to very high stresses, and none of these cracks were creating interruptions to the flow of the current in the Nb 3 Sn cross section. Therefore, this result provides an independent confirmation of our conclusions. The presented analyses and methodologies can be extended to evaluate the degradation mechanism of superconducting wires submitted to any type of mechanical load and suggest directions for the development of methods to enhance the mechanical limits of magnet conductors.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.