IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 28, NO. 3, APRIL 2018

6800705

Numerical Simulation of Electromagnetic andThermal Stress in REBaCuO Superconducting
Ring and Disk Bulks Reinforced by StainlessSteel Ring With Various Widths During
Field-Cooled Magnetization
Keita Takahashi, Hiroyuki Fujishiro, Tomoyuki Naito, Yousuke Yanagi, Yoshitaka Itoh, and Takashi Nakamura

Abstract—The mechanical stress of the electromagnetic hoop
stress, σθFCM , during field-cooled magnetization (FCM) from 20 T,
and the thermal hoop stress, σθcool , under cooling, in REBaCuO
disk and ring bulks reinforced by stainless steel (SUS) ring withvarious widths, were calculated using numerical simulations. The
reinforcement effect of the SUS ring was investigated for each bulk.
For the ring bulk, the maximum of σθFCM was reduced with increasing SUS ring width, from 216 MPa for without SUS ring reinforcement, down to 93 MPa for using SUS ring with 20 mm in width,
was less than that of the disk bulk. The numerical results showedthat the stress concentration in the ring bulk could be alleviatedby the SUS ring reinforcement. We suggest that the width of SUS
ring needs to be at least 10 mm to prevent the fracture behavior of
the bulk. The optimal design of the reinforcement ring should beconstructed so that the total hoop stress, σθtotal (= σθFCM + σθcool ),could be less than the fracture strength of the bulk during FCM.

centric hole are also considered to be used as a TFM for practicalapplications such as nuclear magnetic resonance (NMR) spectrometer [2] and magnetic resonance imaging (MRI) apparatus[3]. Contrary to the superiority, TFMs are broken in the higher
magnetic field during FCM by Lorentz force due to the currentfield interaction. During its magnetization, the principal hoop
stress exceeds the fracture strength, and cracks propagate the radial direction of the bulk [4]. The fracture strength of REBaCuO
bulk is reported to be ∼50 MPa along the circumferential direction [5], and could be enhanced to 70 MPa by silver (Ag)addition [6]–[8]. Particularly, in the ring bulk, the hoop stressconcentrates at the inner edge of concentric hole, where the actual hoop stress is enhanced twice as high as that of the disk
bulk [9], and the bulk could break easier [10]. Numerical analysis is an effective way to understand the mechanical behavior
in the bulk during FCM. However, most of those considerations
have been based on a numerical equation for the infinite bulk
with Bean’s model ever [11]–[14]. Recently, we reported the numerical results of mechanical stresses during FCM in the finite
height REBaCuO ring bulk reinforced by a metal ring [15]. The
fracture strength of the ring bulk could be estimated using thesimulation combined with an actual experimental result [16]. Asa typical reinforcement technique, a metal ring has been usedcommonly [1], [2]. However, the reinforcement effect of suchmetal ring has not been investigated extensively, except for aconventional setup.
In this paper, we compared the numerical results of the mechanical stresses; the electromagnetic hoop stress, σθFCM , during
FCM from 20 T and the thermal hoop stress, σθcool , under cooling from 300 K to 50 K, in the finite height REBaCuO disk andring shaped bulks reinforced by a stainless steel (SUS) ring withvarious widths. It is shown that the stress concentration in thering bulk could be abbreviated to the same level as that of thedisk bulk using wider stainless steel ring.

Index Terms—Bulk superconductors, finite-element method,mechanical strength, numerical simulation, trapped field magnets.

I. INTRODUCTIONEBACUO BULK superconductors (RE: rare earth elementor Y) have a capability to be a compact and strong magnetic field source as a trapped field magnets (TFMs), which havebeen shown to trap a magnetic field over 17 T in a GdBaCuO
disk bulk pair reinforced using shrink-fit stainless steel by fieldcooled magnetization (FCM) [1]. Ring shaped bulks with a con-

R

Manuscript received August 23, 2017; accepted January 22, 2018. Date of
publication January 30, 2018; date of current version February 20, 2018. This
work was supported in part by the “Development of Systems and Technologies
for Advanced Measurement and Analysis” from Japan Agency for Medical
Research and Development (AMED), and in part by the JSPS KAKENHI underGrant 15K04646. (Corresponding author: Keita Takahashi.)
K. Takahashi, H. Fujishiro, and T. Naito are with the Department of Physical Science and Materials Engineering, Faculty of Science and Engineering,
Iwate University, Morioka 020-8551, Japan (e-mail: t0212032@iwate-u.ac.jp;
fujishiro@iwate-u.ac.jp; tnaito@iwate-u.ac.jp).
Y. Yanagi and Y. Ito are with the IMRA Material R&D Company Ltd., Kariya
448-0032, Japan (e-mail: yanagi@ai-i.aisin.co.jp; y-itoh@ai-i.aisin.co.jp).
T. Nakamura is with RIKEN, Wako 351-0198, Japan (e-mail: takashi.
nakamura@riken.jp).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TASC.2018.2799578

II. NUMERICAL SIMULATION FRAMEWORKTo discuss the mechanical behavior during the magnetization,
we constructed the following framework of three-dimensional
numerical model. Fig. 1 presents schematic view of numerical1051-8223 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

6800705

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 28, NO. 3, APRIL 2018

Fig. 2. Time step dependence of the trapped field, B z , at the center of eachbulk for the disk bulk and the ring bulk during FCM from B app = 20 T.

Fig. 1. Schematic view of numerical models for (a) the disk bulk and (b) thering bulk.

half models (z ࣙ 0) for (a) the disk bulk (64 mm in O.D., 20 mm
in height), and (b) the ring bulk (40 mm in I.D., 64 mm in
O.D., 20 mm in height). Each bulk was reinforced by a stainlesssteel (SUS316L) ring holder of the same height as the bulk withvarious widths of 0 mm (without SUS ring), 5 mm, 10 mm or
20 mm in radial direction to compare the reinforcement effectfor each bulk. We abbreviate the width of SUS ring like as
WSUS = 5 mm, which is a conventional width used in actual
apparatus [2], [16].
Each bulk was magnetized by FCM using an infinite solenoid
coil (120 mm I.D., 170 mm O.D. and infinite height). Physicalphenomena during FCM are described using an electromagnetic equation based on the axisymmetric coordinate system[17], [18], in which the temperature of the REBaCuO bulks
is assumed to be constant at Ts = 50 K during FCM for simplicity. The power-n law (n = 20) was used to describe thenonlinear E–J characteristic of the bulk [19]. The magnetic fielddependence of critical current density, Jc (B), of the bulk was
described using the following equation [20], [21].
BB
Jc (B) = Jc1 exp −
+ Jc2
BL
Bm ax
 

k 
B
1
1−
× exp
.
(1)
k
Bm ax
Jc1 was supposed to be 8.0 × 108 A/m2 , which was a rational
value as to reproduce the actual experimental results of thetrapped field during FCM at 50 K [16]. In FCM process, amagnetic field of Bapp = 20 T was applied above the criticaltemperature, Tc (= 92 K), and then the bulk was cooled to Ts =50 K. Then, the magnitude of the magnetic field was decreased
linearly from 20 T to zero in 10 steps, as shown in Fig. 2. Elastic
behavior of an isotropic material can be explained by Hooke’s
law, in which the stress is linearly proportional to the strain[15]. The electromagnetic hoop stress, σθFCM , was estimated foreach step of FCM, which acts circumferentially on each end
face of the meshed element [11], [12]. The thermal hoop stress,

TABLE IMECHANICAL PARAMETERS USED IN THE NUMERICAL SIMULATION.

REBaCuO bulk
SUS316L

E (GPa)να (K−1 )

100
193

0.33
0.28

5.2e-61.27e-5(E: Young’s modulus, ν : Poisson ratio, α : thermal expansion coefficient.)

σθcool , by cooling from 300 K to 50 K was also calculated for eachmodel. Table I summarizes the mechanical parameters (Young’s
modulus, E, Poisson ratio, ν, and thermal expansion coefficient,α) of the REBaCuO bulk and the SUS ring. The REBaCuO bulk
is assumed to be not broken with an isotropic and homogeneous
properties for simplicity. Commercial software, Photo-Eddy andPhoto-ELAS (Photon Ltd., Japan), were used for analyses ofelectromagnetic properties and mechanical stresses.
III. RESULTS AND DISCUSSION
A. Trapped Field, Bz , and Lorentz Force, Fr
Fig. 2 shows the time step dependence of the trapped field,
Bz (t), at the center (r, z) = (0, 0) of each bulk. Bz (t) gradually
decreased with increasing time step, and settled in a certain
value of the trapped field, BT , at the final step after FCM;
BT = 13.8 T for the disk bulk and BT = 5.1 T for the ringbulk. We confirmed that the induced current flowed in the wholearea of each bulk, showing that the applied field is sufficient tomagnetize the bulk fully.
Fig. 3 shows the contour maps of the Lorentz force density
along r-direction, Fr (= Bz × Jθ ), at each time step of 2, 6 and
10 for the disk bulk (a)∼(c), and for the ring bulk (d)∼(f). Notethat the Lorentz force was not applied to the SUS ring due toJθ = 0. In Fig. 3(a) and (d), in the early stage at step 2, Fr wasapplied mainly on the outside of the bulk because the induced
current starts to flow from the outer side of the bulk. However,
the magnitude of Fr is relatively low at step 2.
In Fig. 3(b) and (e), Fr was enhanced in the whole regionof each bulk with progress of FCM due to the current-fieldinteraction between Bz and Jθ . After FCM, in Fig. 3(c) and(f), Fr turned on a negative value at the outside of the bulk

TAKAHASHI et al.: NUMERICAL SIMULATION OF ELECTROMAGNETIC AND THERMAL STRESS IN REBACUO SUPERCONDUCTING RING

6800705

Fig. 3. Contour maps of Lorentz force density along r-direction,
Fr , (= B y × Jθ ) at each time step of (a) Step 2, (b) Step 6 and (c) Step
10 for the disk bulk, and (d) Step 2, (e) Step 6 and (f) Step 10 for the ring bulk.
The SUS ring was supposed to be as a non-magnetic material and not involved
here.

Fig. 5. Cross-sectional plots of the electromagnetic hoop stress, σ θFCM , alongr-direction on the bulk surface (z = 10 mm) reinforced by SUS ring withvarious widths at step 6 during FCM for (a) the disk bulk and (b) the ring bulk.

Fig. 4. Time step dependence of cross-sectional plots of the electromagnetic
hoop stress, σ θFCM , along r-direction on the bulk surface (z = 10 mm) without
SUS ring reinforcement during FCM for (a) the disk bulk and (b) the ring bulk.
The contour maps of σ θFCM at step 6 are also shown in each panel. (a) Disk bulk.
(b)Ring bulk.

because the magnetic flux lines sneak around there to build aconical-shaped trapped field profile.
B. Electromagnetic Hoop Stress, σθFCM During FCM
Fig. 4 shows the time step dependence of cross-sectional plotsof the electromagnetic hoop stress, σθFCM , along r-direction on
the bulk surface (z = 10 mm) without SUS ring reinforcement
during FCM for each bulk. The contour maps of σθFCM at step
6 are also shown in each panel. For the disk bulk in Fig. 4(a),σθFCM had a peak value of +130 MPa in the center of the bulk
(r = 0 mm) at step 6 during FCM. Similar behavior was confirmed for the ring bulk in Fig. 4(b), but the maximum of σθFCM
was +216 MPa at step 6 due to the stress concentration at the
inner periphery (r = 20 mm), is much larger than that of thedisk bulk. The contour maps show that σθFCM profile is relatively
homogeneous along z-direction of the bulk.

Fig. 5 shows the cross-sectional plots of the electromagnetic
hoop stress, σθFCM , along r-direction on the bulk surface(z = 10 mm) reinforced by SUS ring with various widths at
step 6 during FCM for each bulk. In Fig. 5(a), σθFCM was reducedwith increasing WSUS in the whole area of both the disk bulk andthe SUS ring. The maximum σθFCM of the disk bulk was reduced
from +130 MPa for without SUS ring down to +108 MPa
for WSUS = 20 mm. On the other hand, for the ring bulk in
Fig. 5(b), the maximum σθFCM was fairly reduced from
+216 MPa to +97 MPa. These results indicate that the stressconcentration in the ring bulk could be alleviated to the samelevel as the disk bulk. Thus, the reinforcement effect of wider
SUS ring is more effective to reduce σθFCM during FCM in thering bulk.
C. Thermal Hoop Stress, σθcool , Under CoolingFig. 6 shows the cross-sectional plots of the thermal hoop
stress, σθcool , along r-direction on the bulk surface (z = 10 mm)
reinforced by SUS ring with various widths under cooling from300 K to 50 K for each bulk. Note that the bulk without SUS
ring (WSUS = 0 mm) shrink freely, that is, σθcool = 0 MPa. We
have reported that the difference of the thermal contraction between the ring bulk and the SUS ring might cause a tensilestress under cooling process [11], [12]. Also here, for the case

6800705

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 28, NO. 3, APRIL 2018

Fig. 6. Cross-sectional plots of the thermal hoop stress, σ θcool , along r-direction
on the bulk surface (z = 10 mm) reinforced by SUS ring with various widths
under cooling from 300 K to 50 K for (a) the disk bulk and (b) the ring bulk.

Fig. 7. Cross-sectional plots of the total hoop stress, σ θtotal , along r-direction
on the bulk surface (z = 10 mm) reinforced by SUS ring with various widthsfor (a) the disk bulk and (b) the ring bulk.

of WSUS = 5 mm, σθcool had a maximum value at the outer edge(r = 32 mm) of the bulk, +47 MPa in Fig. 6(a), and +10 MPa
in Fig. 6(b). These tensile σθcool values were reduced with increasing WSUS , which results in a negative value in the wholearea of each bulk for WSUS = 10, and 20 mm. The reinforcement using wider SUS ring was effective for each bulk undercooling process as well.
D. Total Hoop Stress, σθtotal , During FCM Under Cooling
To discuss the mechanical behavior of the bulk reinforced
by SUS ring in the actual FCM, the total hoop stress, σθtotal ( =σθFCM + σθcool ) should be considered, including σθFCM and σθcool
mentioned in previous subsections. Fig. 7 shows the crosssectional plots of the total hoop stress, σθtotal , along r-direction
on the bulk surface (z = 10 mm) reinforced by the SUS ringwith various widths, were calculated from Figs. 5 and 6 for eachbulk. For the case of WSUS = 0 mm, σθFCM was imported asσθtotal value directly. When the SUS ring with WSUS = 5 mm
was used for both cases, the tensile σθtotal higher than +50 MPa
is applied, is as high as the fracture strength of the REBaCuO
bulk (∼= 59 MPa) [16]. The σθtotal was reduced with increasing WSUS , resulting in the minimum value for WSUS = 20 mm
in the whole region of the bulk and the SUS ring, due to thestress relaxation by the reinforcement effect. These results indicate that the SUS ring alleviates the tensile strain in thebulk during FCM. Meanwhile, under cooling process, the SUS

Fig. 8. The SUS ring width, W SUS , dependence of the maximum of theelectromagnetic hoop stress, σ θFCM , and the total hoop stress, σ θtotal , for(a) the disk bulk and (b) the ring bulk, which were extracted from Figs. 5
and 7, respectively.

ring enhances the compressive strain to countermeasure σθFCM
during FCM. The width of SUS ring needs to be at least 10 mm
to prevent the fracture behavior of the bulk in present simulation.
Fig. 8 shows the SUS ring width, WSUS , dependence ofthe maximum of the electromagnetic hoop stress, σθFCM , during FCM and the total hoop stress, σθtotal , during FCM undercooling,, for each bulk, which were extracted from Figs. 5 and7. During FCM, the ring bulk was subject to a large maximum

TAKAHASHI et al.: NUMERICAL SIMULATION OF ELECTROMAGNETIC AND THERMAL STRESS IN REBACUO SUPERCONDUCTING RINGσθFCM of +216 MPa at the inner periphery around the bulk annuli
for the case of WSUS = 0, without SUS ring reinforcement, due
to the stress concentration as shown in Fig. 8(b). Surprisingly,
compared to the disk bulk, σθFCM during FCM in the ring bulk
was more susceptible to the reinforcement effect of wider SUS
ring, which results in a lower σθFCM value at WSUS = 20 mm.
A decrease of σθFCM by the wider SUS ring was within a few
MPa in the disk bulk, and that reinforcement effect was fairly
small during FCM. On the other hand, under cooling process, it
was confirmed that the wider SUS ring was desirable to enhance
the compressive stress for each bulk. The achieved lowest σθtotal
value at WSUS = 20 mm was +9 MPa for the disk bulk, and−29 MPa for the ring bulk, respectively. The magnitude of themechanical stress depends on the magnetizing condition aroundthe bulk, such as geometry, superconducting characteristics ofthe bulk, and the applied field. An optimal design of the reinforcement ring could be useful to alleviate the magnetic stress,
and should be constructed so that the σθtotal could be less than
the fracture strength of the bulk during FCM.
IV. SUMMARY
In this study, the numerical results of the mechanical stressof the electromagnetic hoop stress, σθFCM , during FCM from20 T at 50 K, and the thermal hoop stress, σθcool , under cooling,in REBaCuO disk and ring bulks reinforced by SUS ring withvarious widths, were compared. For the ring bulk, the maximumof σθtotal at step 6 during FCM was reduced from +216 MPa
without SUS ring reinforcement, down to -29 MPa for WSUS =20 mm, is less than that of the disk bulk. The numerical results
showed that the stress concentration in the ring bulk could be
alleviated by the wider SUS ring. The width of SUS ring needs to
be at least 10 mm to prevent the fracture behavior of the bulk. An
optimal design of the reinforcement ring should be constructed
so that the total hoop stress, σθtotal (= σθFCM + σθcool ), could beless than the fracture strength of the bulk during FCM.
REFERENCES[1] J. H. Durrell et al., “A trapped field of 17.6 T in melt-processed, bulk
Gd-Ba-Cu-O reinforced with shrink-fit steel,” Surpercond. Sci. Technol.,vol. 27, 2014, Art. no. 082001.
[2] T. Nakamura et al., “Development of a superconducting bulk magnet for
NMR and MRI,” J. Mag. Reson., vol. 259, pp. 68–75, 2015.
[3] D. Tamada, T. Nakamura, Y. Itoh, and K. Kose, “Experimental evaluationof the magnetization process in a high Tc bulk superconducting magnet
using magnetic resonance imaging,” Physica C, vol. 492, pp. 174–147,
2013.

6800705[4] T. Miyamoto, K. Nagashima, N. Sakai, and M. Murakami, “Direct measurement of mechanical properties for large-grain bulk superconductors,”
Physica C, vol. 340, pp. 41–50, 2000.
[5] A. Murakami et al., “Mechanical properties of Gd123 bulk superconductors at room temperature,” Cryogenics, vol. 43, pp. 345–350,
2003.
[6] D. Lee and K. Salama, “Enhancements in current density and mechanicalproperties of Y-Ba-Cu-O/Ag composites,” Jpn. J. Appl. Phys., vol. 29,
pp. L2017–L2019, 1990.
[7] M. Matsui, N. Sakai, and M. Murakami, “Effect of Ag2 O addition on thetrapped field and mechanical properties of Nd-Ba-Cu-O bulk superconductors,” Surpercond. Sci. Technol., vol. 15, pp. 1092–1098, 2002.
[8] K. Katagiri et al., “Effects of Ag content on the mechanical properties of(Nd,Eu,Gd)-Ba-Cu-O bulk superconductors,” Physica C, vol. 392–396,
pp. 526–530, 2003.
[9] T. H. Johansen, C. Wang, Q. Y. Chen, and W. -K. Chu, “Enhancementof tensile stress near a hole in superconducting trapped-field magnets,” J.
Appl. Phys., vol. 88, pp. 2730–2733, 2000.
[10] H. Mochizuki, H. Fujishiro, T. Naito, Y. Itoh, Y. Yanagi, and T. Nakamura,“Trapped field characteristics and fracture behavior of REBaCuO bulk ring
during pulsed field magnetization,” IEEE Trans. Appl. Supercond., vol. 26,
no. 4, Jun. 2016, Art. no. 6800205.
[11] Y. Ren, R. Weinstein, J. Liu, R. P. Sawh, and C. Foster, “Damage caused bymagnetic pressure at high trapped field in quasi-permanent magnets composed of melt-textured Y–Ba–Cu–O superconductor,” Physica C, vol. 251,
pp. 15–26, 1995.
[12] T. H. Johansen, “Flux-pinning-induced stress and strain in superconductors: Case of a long circular cylinder,” Phys. Rev. B, vol. 60, pp. 9690–9703,
1999.
[13] T. H. Johansen, “Flux-pinning-induced stress and magnetostriction in bulk
superconductors,” Supercond. Sci. Technol, vol. 13, pp. R121–R137, 2000.
[14] W. J. Feng, P. Ma, Q. H. Li, and J. X. Liu, “Flux-pinning-induced magnetostriction and stress in a transversely isotropic thin superconducting
disk with a concentric small hole,” J. Supercond. Novel Magn., vol. 26,
pp. 539–543, 2013.
[15] H. Fujishiro et al., “Simulation studies of mechanical stresses in REBaCuO
superconducting ring bulks with infinite and finite height reinforced bymetal ring during field-cooled magnetization,” Supercond. Sci. Technol.,vol. 30, 2017, Art. no. 085008.
[16] K. Takahashi, H. Fujishiro, T. Naito, Y. Yanagi, Y. Itoh, and T. Nakamura,“Fracture behavior analysis of EuBaCuO superconducting ring bulk reinforced by a stainless steel ring during field-cooled magnetization,” Supercond. Sci. Technol., vol. 30, 2017, Art. no. 115006.
[17] H. Fujishiro and T. Naito, “Simulation of temperature and magnetic field
distribution in superconducting bulk during pulsed field magnetization,”
Supercond. Sci. Technol., vol. 23, 2011, Art. no. 105021.
[18] H. Fujishiro, H. Mochizuki, M. D. Ainslie, and T. Naito, “Trapped fieldof 1.1 T without flux jumps in an MgB2 bulk during pulsed field magnetization using a split coil with a soft iron yoke,” Supercond. Sci. Technol.,vol. 29, 2016, Art. no. 084001.
[19] M. D. Ainslie and H. Fujishiro, “Modelling of bulk superconductor magnetization,” Supercond. Sci. Technol., vol. 28, 2015, Art. no. 053002.
[20] M. Jirsa, L. P̊ust, D. Dlouhý, and M. R. Koblischka, “Fishtail shape in themagnetic hysteresis loop for superconductors: Interplay between different
pinning mechanisms,” Phys. Rev. B, vol. 55, pp. 3276–3284, 1997.
[21] M. Muralidhar, N. Sakai, M. Jirsa, N. Koshizuka, and M.
Murakami, “Direct observation and analysis of nanoscale precipitatesin (Sm,Eu,Gd)Ba2 Cu3 Oy ,” Appl. Phys. Lett., vol. 85, pp. 3504–3506,
2004.