Electromagnetic strain measurements and two-directional mechanical stress estimation for a REBaCuO ring bulk reinforced by a metal ring during field-cooled magnetization

In this paper, simultaneous measurements of the electromagnetic strains along both the circumferential (θ) and radial (r) directions are reported for a large single-grain EuBaCuO ring bulk reinforced by an Al alloy ring during field-cooled magnetization (FCM) from 5 T at 50 K using several strain gauges adhered to the surface. To verify the experimental results and to understand the complex stress–strain behavior, mechanical analyses were carried out using a three-dimensional finite element model that closely represents the experimental setup. The simulation results of the electromagnetic strains along both directions showed excellent qualitative and quantitative agreement with the experimental ones. These results strongly suggest that the numerical model must include the exact same structure (size, shape and materials) of the mechanical support structure as the experimental setup in order to reproduce the experimental results both qualitatively as well as quantitatively. This also explains our previous research (SuST 2019 32 015007), where the measured circumferential strains were about 50% smaller those in the numerical simulation. Furthermore, the electromagnetic stresses along both directions during the FCM process are estimated from the obtained experimental strains. As a result, the estimated stresses were fairly consistent with those obtained by the numerical simulations, suggesting that our stress–strain simulation technique is both qualitatively and quantitatively reliable and useful to clarify the possibility of mechanical fracture of bulk superconductors.


Introduction
Large, single-grain REBaCuO (RE: a rare earth element or Y) bulk superconductors are a promising material for use as a compact, high-strength trapped field magnet (TFM) [1] for various practical applications, such as rotating machines, magnetic separation, flywheel energy storage systems and so on [2][3][4]. REBaCuO bulks can generate magnetic fields much higher than conventional Nd-Fe-B permanent magnets or other superconducting TFMs using MgB 2 [5] and iron-pnictide [6]. In general, the trapped magnetic field capability of the bulk superconductor is proportional to both the critical current density, J c , and bulk radius, r, based on the Bean's critical state model [7]. Until now, the increase of J c and the enlargement of r of the bulk have been carried out over the past three decades by the improved fabrication techniques, particle refinement of the secondary phases, suppression of defect formation, and the introduction of stronger pinning centers.
Besides the enhancement of the magnetic flux pinning performance in the bulks, the mechanical reinforcement of the bulk is also an important issue during field-cooled magnetization (FCM), especially for fields of several Tesla or more. A large Lorentz force is exerted in the bulk, which sometimes results in crack formation and propagation, leading to eventual mechanical failure. This is because the REBaCuO bulk material is intrinsically a brittle ceramic with relatively low mechanical strength. In previous studies, crack formation and/ or serious fracture occurred in a bulk without external reinforcement during ramped down from a high external magnetic field [8][9][10]. In this sense, the maximum trapped field capability is not restricted by the electromagnetic properties, but by their poor mechanical performance. Using mechanical reinforcement by shrink-fit stainless steel or epoxy resin impregnation, trapped magnetic fields over 17 T have been achieved below 30 K in disc bulk pairs [11,12]. In addition, a ring-shaped REBaCuO bulk with a large bore has been also applied for the compact and cryogen-free NMR/MRI magnet [13,14]. Trapped field enhancement with spatial homogeneity of ppm order is an ongoing challenge for improving resolution; a trapped field of as high as 10 T is necessary to realize a 400 MHz (9.4 T) NMR bulk magnet system. The mechanical strength of the bulk materials has usually been estimated using three-point or four-point bending tests [15][16][17][18] and tensile tests [19,20], from which the fracture strength of a typical Agdoped REBaCuO bulk is suggested to be as low as 50-70 MPa.
Analytical investigations of mechanical properties have been reported for disc-and ring-shaped bulks with infinite height during FCM or zero-field-cooled magnetization [8,[21][22][23][24]. On the other hand, we have reported numerical simulation results of the mechanical stress behavior for REBaCuO disc-and ring-shaped bulks with finite height and the metal ring reinforcement during FCM [25][26][27] using the finite element method (FEM). Furthermore, we have proposed a new reinforcement method to prevent bulk failure under higher fields [28][29][30].
Mechanical strain measurement using a strain gauge is a conventional and effective method for the evaluation of the local mechanical properties of a large bulk without cutting it. The electromagnetic strain, ε, in the REBaCuO disc bulk caused by the Lorentz force during FCM was directly measured on the bulk surface, and the fracture strength was estimated [31]. Recently, we have reported the electromagnetic strain along the circumferential (θ) direction of a REBaCuO ring bulk reinforced by a metal ring during FCM, and the cooling strains during the cooling process from room temperature to 50 K [32]. These experimental results were compared with numerical simulation results, in which the simple three-dimensional (3D) model consisted of the ring bulk and the reinforcement metal ring. The numerical results of the time-step dependence of the strain during FCM reproduced the experimental ones qualitatively. However, the absolute value of the experimental strain was about 50% smaller than that of the simulated one. We concluded, in the paper, that the difference mainly comes from the simulation results, which was closely related with the assumed J c (B) characteristics, and/or from the experimental strain results, which result from the existence of the macroscopic and microscopic structural inhomogeneity. The quantitative difference in the electromagnetic strain between the experiments and simulations is a serious problem, because the agreement guarantees the reliability of our numerical simulation results. We hypothesized that the difference mainly comes from an insufficient numerical model; that is, the exact same structure (size, shape and materials) of the mechanical support structure as the experimental setup must be included in the model. In addition, to enhance the reliability of the simulation results, the simultaneous measurement and verification of the electromagnetic stresses along two directions is provided.
In this paper, we measured the electromagnetic strains along both circumferential (θ) and radial (r) directions of a EuBaCuO ring bulk reinforced by an aluminum (Al) alloy ring during FCM from 5 T at 50 K using several strain gauges. The electromagnetic stresses in both directions are estimated analytically at each position from the experimental results obtained by the strain measurement using Hooke's law. A full 3D finite element model based on the actual experimental setup was developed to understand the complex stress-strain behavior of the ring bulk. The quantitative agreement between the experiments and simulations can prove the validity of our experiment and simulation techniques and validates the simulation as a reliable tool to investigate the possibility of mechanical fracture of bulk superconductors. Figure 1(a) shows the experimental setup of the ring bulk with a metal ring for FCM. The EuBaCuO ring bulk (64 mm OD, 28 mm ID, and 20 mm height (H)) containing 10 wt% silver (Ag) was fabricated using a modified quench and melt growth method by Nippon Steel Corporation, Japan [33,34]. An Al alloy reinforcement ring (A7075-T6) with the same height as the ring bulk (74 mm OD, 64.1 mm ID, and 20 mm H) was bonded to the outer peripheral surface of the ring bulk using Stycast ™ 1266 epoxy resin (0.05 mm in thickness). This ring applies a compressive stress to the ring bulk during the cooling process due to the difference in the thermal contraction coefficient between two materials. The ring bulk with the Al alloy ring was tightly sandwiched by top (12 mm in thickness) and bottom (22 mm in thickness) Cu plates made of oxygen-free high-conductivity copper (OFHC Cu) using a bolt and nut. Thin indium sheets were inserted between the bulk and both Cu plates to obtain good thermal contact. The bottom Cu plate was thermally connected to the cold stage of a Gifford-McMahon helium refrigerator. The temperature of the cold stage was controlled by a Pt-Co thermometer and a heater, attached to bottom surface of the cold stage. A straight and long groove (64 mm in length, 10 mm in width and 5 mm in depth) was made on the bottom surface of the top Cu plate to prevent mechanical contact between the Cu plate and strain gauges and to set a CERNOX ™ thermometer to monitor the temperature of the bulk on the outer edge of the bulk surface. The trapped field, B z , at the center of the ring bulk (r =z =0) was measured by an axial-type Hall sensor (F. W. Bell, BHA-921). The bulk sheathed in a vacuum chamber was inserted into a cryocooled 10 T superconducting solenoid magnet (Japan Superconductor Technology, JMTD-10T100) and an external magnetic field of B app =5 T, parallel to z-axis, was applied to the bulk at 100 K, and then the bulk was slowly cooled to 50 K under the magnetic field of 5 T. Finally, the external magnetic field was ramped down to zero at a rate of −0.222 T min −1 .

Strain measurement
is the original length of the bulk at B ex =0 along the θ-direction, r is the radial measurement position on the bulk surface. The electromagnetic strain along the r-direction, e = dL L 0 , r r r ( ) / is expressed by the following equation is the change of length along the r-direction under the magnetic field, and = L r 0 r ( ) at B ex =0. The e q and e r values on the surface were calculated by the ratio of dR/R shown in equations (1) and (2), in which R (=350 Ω) is the initial resistance of the strain gauge and dR=R(B) − R(0) is the change in resistance due to the electromagnetic strain under magnetic field variation. The R (B) value of the strain gauge adhered on the quartz plate was also measured up to 10 T at 50 K, in which there was no need to calibrate the gauge resistance under the magnetic field. We have measured the gauge resistance, R(B), adhered on the bulk several times, which were confirmed to be nearly the same values under identical conditions.

Numerical simulation framework
Based on the experimental setup shown in figure 1(a), a detailed 3D numerical model was constructed, in which OFHC Cu plates, indium sheets, bolt and nut made of stainless steel (SUS304) and Stycast ™ 1266 (epoxy resin) were also added, together with the EuBaCuO bulk and the Al alloy ring. The model was equally divided into 36 elements along the circumferential direction and into 1 mm intervals along the r-and z-directions, respectively. As boundary conditions, the outer surfaces of the Al-alloy ring and the Cu holder were assumed to be adiabatic and each part was fixed mechanically. The bulk was magnetized by a coil (170 mm OD, 120 mm ID, and 200 mm H), which was the same size as the superconducting solenoid coil used in the experiment.
The FCM simulation was carried out using a process identical to that described in section 2.1. The temperature of the system was set to 50 K because the maximum temperature rise during experimental FCM was as small as 1 K at the outer edge of the bulk surface. The commercial FEM software package, Photo-Eddy (Photon Ltd, Japan), was employed for the electromagnetic analysis during the FCM process. The details of fundamental equations for bulk superconductor magnetization simulations are described elsewhere [35,36].
The nonlinear electrical properties of the superconducting bulk was described using the E-J power law is the characteristic electric field and n (=20) is a typical value for bulk REBaCuO superconductors [37,38]. The magnetic field dependence of the critical current density, J c (B), for the bulk used in this simulation was determined using the following equation [39][40][41] = - We proposed a new method to estimate the J c (B) characteristics of the bulk [29], in which the time dependence of trapped field, B z , at the center of ring bulk during FCM was measured and fitted using the J c (B) characteristics shown in equation (4). This method can estimate the average and macroscopic J c (B) characteristics of the bulk without cutting and destroying the bulk. Figure 2 shows the experimental and simulation results of the time dependence of B z during FCM from 5 T at 50 K. In the experimental FCM, the external field, B ex , is ramped down linearly at a rate of −0.222 T min −1 . B z decreased gradually due to flux creep, and a final trapped field of 4.71 T was achieved successfully, when the B ex reached zero at t=1350 s. It can be seen that the simulation result of the time-dependent trapped field curve can reproduce the experimental curve by the fitting parameters shown in table 1.
The numerical simulation of the mechanical stress and strain in the bulk and Al alloy ring during FCM was carried out using the commercial FEM software, Photo-ELAS (Photon Ltd, Japan), coupled with the electromagnetic model by Photo-Eddy to incorporate Lorentz force. The bulk and Al alloy ring are assumed to be homogeneous and isotropic, and have a linear elastic response. Elastic behavior in an isotropic material can be explained by Hooke's law, in which the stress tensor, σ ij , is linearly proportional to the strain tensor, ε ij , as follows where λ and G represent the Lame's constants, δ ij is the Kronecker delta function, E is the Young's modulus, and v is the Poisson ratio of the material. In the two-dimensional model, where the mechanical stress, σ z , along the z-direction is ignored, the mechanical strains along the θand r-directions are expressed in the following where σ θ and σ r are the mechanical stress along the θand rdirection, respectively, and dr is the displacement along the rdirection.
The mechanical parameters, E and v of the REBaCuO bulk, Al alloy ring (A7075-T6), epoxy resin, indium, copper (OFHC Cu) and stainless steel (SUS304) in the stress-strain analysis are summarized in table 2. The mechanical parameters of the REBaCuO bulk were assumed to be isotropic and homogeneous for simplicity.   (4).

Figures 3(a)-(c)
, respectively, show simulation results of the cross-sectional profiles of the magnetic field, B z , induced persistent current density, J θ , and Lorentz force density, F r , in the bulk and Al alloy ring at B ex =4, 1 and 0 T during FCM from B app =5 T. At B ex =4 T, the magnitude of B z in the whole region of the ring bulk is nearly 5 T except for its outer periphery, where J θ flows along the θ-direction only in outer region and top and bottom surfaces of the bulk. As a result, F r is applied along the r-direction by the current-field interaction (i.e. J θ ×B z ). In the Al alloy ring, F r equals to zero because J θ =0. As the external field is ramped down, e.g. B ex =1 and 0 T, the region through which the persistent supercurrent flows became wider, compared to that at B ex =4 T. As a result, B z in the outer region of the bulk decreased, the region with higher B z became narrower and the B z value at the center slightly decreased. It is obvious that the peak position of F r shifts from the periphery to the center of the bulk with decreasing B ex . When B ex reaches 0 T, B z =5 T is still present in the inner central region of the bulk (r=17 mm, z=0 mm), and B z =4.8 T can be achieved at the center of the ring bulk (r=z=0 mm), as shown in figure 2. These results suggest that the ring bulk was not fully magnetized during FCM from B app =5 T at 50 K due to the larger J c (B) value. Figure 4(a) shows the experimental results of the external field dependence of the electromagnetic strain along the θdirection, ε θ , during FCM from 5 T at 50 K at each position, in which the magnetic field decreased from 5 to 0 T. The initial strain of each gauge was assumed to be zero, after the bulk was cooled from 300 to 50 K under the magnetic field of 5 T. The ε θ value at each position monotonically increases during the ramp down of the external field, takes a maximum around B ex =1 T, and then slightly decreases. This trend was qualitatively similar to our previous study [32]. The ε θ value at r=17 mm is the largest, which is +0.027%, meaning the highest tensile strain in the innermost position. The ε θ value decreases with increasing r in the bulk, e.g. ε θ =+0.013% at r=30 mm. The ε θ on the Al alloy ring (r=35 mm) also see positive values during FCM, nevertheless the Lorentz force is not produced in this region (32<r<37 mm) as shown in figure 3(c). This behavior implies that the Al alloy ring was pushed outwards from the inner bulk due to the Lorentz force and, as a result, expanded in the θ-direction. Figure 4(b) shows the results of numerical simulation of the external field dependence of the electromagnetic strain along the θ-direction, ε θ , during FCM from 5 T at 50 K. The simulated ε θ data well reproduced the experimental data shown in figure 4(a) qualitatively and quantitatively. This agreement results from the reduction of the simulated ε θ values and strongly suggests that such a realistic numerical model is necessary for the simulation to reproduce the experimental data accurately. Because the mechanical stress and strain are related three-dimensionally to each other and the electromagnetic strain is reasonably influenced by the size, shape and assumed mechanical properties of the surrounding parts. Figure 5(a) shows the experimental results of the external field dependence of the electromagnetic strain along the rdirection, ε r , at each measurement position, for which the magnetic field decreased from 5 to 0 T. The ε r values at r=−17, −30 and −35 mm increased negatively, took a minimum, and then showed a slightly positive increase with decreasing B ex , which are in clear contrast to the ε θ values at the corresponding positions. It can be considered that the inner periphery of the ring bulk (r =14 mm) and the outer periphery of the Al alloy (r =37 mm) are in a 'stress-free' state, and are not restricted mechanically since there are no external stresses in the r-direction from the inner surface of   the bulk and outer surface of the Al ring. Thus, the negative ε r can be simply explained by Poisson ratio v =−ε θ /ε r , and the outer boundary of superconductor bulk is pressed by the Al alloy ring. As a result, the ε r value at r=−30 mm shrinks largely because the displacement at outer periphery of the ring bulk (r=32 mm) is restricted due to the external compressive stress from the Al alloy ring. On the other hand, the ε r value at r=−23 mm, which is an intermediate position on the ring bulk surface, shows a slight positive expansion, takes a peak, and then decreases slightly with decreasing B ex . It should be noted that the maximum variation of ε r at each position is several times smaller than ε θ . Figure 5(b) shows the numerical simulation results of the external field dependence of the electromagnetic strain along the r-direction, ε r , during FCM from 5 T at 50 K. The simulated ε r versus B ex curves at r=−23, −30 and −35 mm reproduce the experimental data well qualitatively and quantitatively. On the other hand, the simulated ε r versus B ex curve at r=−17 mm exhibits a convex profile, but the experimental ε r versus B ex curve at r=−17 mm shown in figure 5(a) exhibits a concave one. This discrepancy may come from a difference in the load condition for the stressstrain analysis; in the actual experimental setup shown in figure 1(a), a stronger fastening torque was applied to the inner periphery of the bulk though the Cu plates along the zdirection produced from a central bolt and nut, rather than to the outer periphery, which suggest that inhomogeneous mechanical fastening pressure might be applied to the ring bulk. On the other hand, in the numerical simulation, the torque was not considered and the mechanical and thermal contact was assumed to be homogeneous. These discrepancies influence on the strain profiles, especially at the inner periphery region. However, these results of simulations and experiments are in good agreement, which suggests that the construction of a realistic FEM model and the use of realistic average and macroscopic J c (B) characteristics for the bulk are necessary to reproduce the external field dependence of the electromagnetic strains along both directions qualitatively and quantitatively. This allows us to better understand the complex physical phenomena during magnetization process and make accurate predictions.

Estimation of electromagnetic stress using measured strains along both directions
In this subsection, we estimate the electromagnetic stresses applied to the ring bulk and Al alloy ring during FCM using the experimentally obtained strains. To estimate the stresses along both the θand r-directions in the bulk and Al alloy ring from the experimental strain results, the following stressstrain relationships are adopted s n e n e = -+ where E bulk =100 GPa and v bulk =0.33 are assumed in the ring bulk region, and E Al =78.0 GPa and v Al =0.34 are assumed in the Al alloy region, as shown in table 2. These equations are obtained from Hooke's law for an isotropic material within its elastic range, in which the stress along the z-direction, σ z , in the materials is assumed to be zero. This assumption was used previously in the analytical calculations for the stress-strain relationship for a ring-shaped bulk superconductor [22]. Figure 6(a) shows the estimated results of the electromagnetic stress along the θ-direction, σ θ , as a function of the external field, B ex , at each measurement position during FCM from 5 T at 50 K, which were obtained using equation (10) and the experimental strain values shown in figures 4(a) and 5(a). Estimated σ θ values at each position increase, take a maximum at an intermediate B ex , and then slightly decrease as the external field is ramped down. The σ θ value at r=17 mm is higher, compared to those at the other positions and at any B ex . Its maximum is estimated to be +29 MPa at B ex =1 T, meaning that the tensile stress concentration occurs at the inner periphery, which are similar to the results for ε θ shown in figure 4(a). These trends have been reported in our previous works [25][26][27][28][29]42]. Figure 6(b) shows the simulation results of the electromagnetic stress along the θ-direction, σ θ , as a function of the external field, B ex , at each measurement position during FCM from 5 T at 50 K, for which the numerical simulation was performed only using the electromagnetic and mechanical analyses. The numerical simulation results are in good agreement with the estimation using the measured results qualitatively and quantitatively. A maximum stress of +28 MPa was observed at r=17 mm, which is in excellent agreement with estimated results for the maximum stress at same position. The fracture strength of a typical Ag-doped REBaCuO bulk is considered to be 50-70 MPa [43][44][45]. In this sense, the present FCM conditions can avoid the mechanical fracture. Figure 6(c) shows the σ θ distribution in the ring bulk (14<r<32 mm) and Al alloy ring (32<r<37 mm) at B ex =1 T. The σ θ distribution in the ring bulk is inhomogeneous and σ θ is concentrated at the innermost edge of the ring bulk surface. These results suggest that the electromagnetic stress takes a maximum at the innermost edge of the ring bulk surface, and that the electromagnetic strain also takes a maximum at the same position. Figures 7(a) and (b), respectively, show the estimated and simulated results of the electromagnetic stresses along the r-direction, σ r , as a function of the external field, B ex , at each measurement position during FCM from 5 T at 50 K, for which the estimated results were obtained using equation (11) and the experimental strain values shown in figures 4(a) and 5(a). The estimated σ r values at r=−23, −30, and −35 mm were reproduced well by the numerical simulation. The estimated σ r versus B ex curve at r=−17 mm was not necessarily consistent with the simulation curve, which may result from the similar cause as that for the strain measurement as shown in figure 5(a). Figure 7(c) shows the σ r distribution in the ring bulk (14<r<32 mm) and the Al alloy ring (32<r<37 mm) at B ex =1 T. The σ r distribution in the ring bulk is inhomogeneous and σ r is concentrated at the innermost edge of the ring bulk surface. The maximum σ r value is smaller than 10 MPa, which is much smaller than the fracture strength. Thus, the ring bulk did not fracture during FCM from 5 T.

Conclusion
Two-directional strain measurement has been shown to be a valuable method to estimate precise mechanical stresses, which cannot be directly measured. We have simultaneously measured electromagnetic strains along both the θand rdirections on a large single-grain EuBaCuO ring bulk, reinforced by an Al alloy ring, during FCM from B app =5 T at 50 K using several strain gauges. A 3D finite element model that represents the exact experimental setup was constructed, including not only the ring bulk and Al alloy ring, but also the sample holder and mechanical support structure (OFHC Cu  plates, indium sheets, bolt and nut made of stainless steel (SUS304) and Stycast ™ 1266 (epoxy resin)), with excellent qualitative and quantitative agreement. The important results and conclusions of this study are summarized as follows.
(1) The electromagnetic strain along the θ-direction, ε θ , takes a maximum at an intermediate external field and is the largest at the innermost position of the bulk surface, decreasing with increasing radius. The electromagnetic strain along the r-direction, ε r , is several times smaller than the ε θ , and has a contrasting B ex dependence at each position along the bulk surface. (2) The simulation results of the electromagnetic strains along both directions show excellent qualitative and quantitative agreement with the experimental ones. In the numerical simulation, the construction of a realistic model based on all parts of the actual experimental setup is necessary to reproduce the experimental results precisely. The quantitative agreement between the experiments and simulations validate both our experimental and simulation techniques, and this numerical simulation framework can be used as a reliable tool to investigate the possibility of mechanical fracture of bulk superconductors for any bulk superconducting material and geometry or magnetizing process. (3) The electromagnetic stresses, σ θ and σ r , along both directions are estimated during the FCM process from the experimentally obtained strains, ε θ and ε r , using Hooke's law. The consistency of the experimental and numerical results suggests that this kind of strain measurement can estimate electromagnetic stresses precisely and is an effective technique for the evaluation of the mechanical properties of bulk superconductors during FCM to clarify the possibility of mechanical fracture.