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Proposal of an effective mechanical reinforcement structure for a
REBaCuO disk bulk pair by full metal encapsulation to achieve a higher
trapped field over 20 T
To cite this article: Hiroyuki Fujishiro et al 2019 Supercond. Sci. Technol. 32 045005

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Superconductor Science and Technology
Supercond. Sci. Technol. 32 (2019) 045005 (9pp)

https://doi.org/10.1088/1361-6668/ab0017

Proposal of an effective mechanicalreinforcement structure for a REBaCuO disk
bulk pair by full metal encapsulation toachieve a higher trapped ﬁeld over 20T
Hiroyuki Fujishiro1 , Tomoyuki Naito1

and Satoshi Awaji2

1

Faculty of Science and Engineering, Iwate University, Morioka 020-8551, Japan
High Field Laboratory for Superconducting Materials (HFLSM), Institute for Materials Research, Tohoku
University, Sendai 980-8577, Japan2

E-mail: fujishiro@iwate-u.ac.jp
Received 19 October 2018, revised 16 January 2019
Accepted for publication 18 January 2019
Published 25 February 2019
Abstract

In this paper, we propose an effective mechanical reinforcement of a REBaCuO superconducting disk
bulk pair to avoid mechanical fracture due to a large hoop stress during ﬁeld-cooled magnetization(FCM), and conﬁrm the reinforcement effect using numerical simulation. In this reinforcement, thedisk bulk is fully encapsulated by an outer metal ring with upper and lower plates made by stainlesssteel (SUS316). The trapped ﬁeld, Bz, in the bulk pair with various critical current densities, Jc, wasnumerically simulated during FCM from Bapp=22 T, and the hoop stress, σθ, was also estimatedduring FCM after cooling from 300 to 20 K. As a result, the trapped ﬁeld of over 20 T can beachieved in the gap center of the bulk pair. A large compressive hoop stress, sqcool, due to thedifference of thermal expansion coefﬁcients between the bulk and stainless steel, was effectively
applied to the whole bulk during the cooling process, compared to that for the conventional
reinforcement using only a SUS316 outer ring. The electromagnetic hoop stress, σθFCM, during FCM
was also reduced, and the maximum of the total hoop stress, sqtotal (=σθFCM+sqcool ), can be reduced
below the fracture strength of the REBaCuO bulk. The possibility to achieve a higher trapped ﬁeld
over 20 T is suggested in the gap of the REBaCuO bulk pair without mechanical fracture.
Keywords: ﬁeld cooled magnetization, numerical simulation, mechanical reinforcement, higher
trapped ﬁeld, mechanical stress(Some ﬁgures may appear in colour only in the online journal)
1. Introductionbulk. Because the REBaCuO bulk is a brittle ceramic material, and has a lower mechanical strength than the larger
hoop stress due to the Lorentz force during the magnetization
process. Since the REBaCuO bulk superconductor is also acomposite material, in which the RE2BaCuO5 (RE211), Pt
and Ag particles are dispersed in the REBa2Cu3O7 (RE123)
quasi-single crystal matrix with a layered structure, the stressconcentration is likely to take place around the so called
secondary phases. In addition, the bulk contains a signiﬁcantnumber of imperfections such as grain boundaries, crystalline
defects, inclusions, voids, and so on, which may cause thestress concentration in local regions and may initiate small

The superconducting characteristics of REBa2Cu3Oy bulks(REBaCuO, RE: rare earth element or Y) have been recently
enhanced due to the introduction of strong pinning centers
and the improvement of the crystal growth technique, which
results in increased critical current densities, Jc [1, 2]. As aresult, the bulks have a signiﬁcant potential as high-ﬁeld
trapped ﬁeld magnets (TFMs). However, it is not an exaggeration to say that the maximum of the reported trapped ﬁeld
at lower temperatures was not determined by the superconducting characteristics, but by mechanical strength of the0953-2048/19/045005+09$33.00

1© 2019 IOP Publishing Ltd Printed in the UK

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H Fujishiro et al

local cracks. Additionally, micro-cracks along the ab-plane
formed during post-annealing treatment of oxygenation are acommon ingredient in the microstructure [3–7]. These weak
points should also severely reduce the mechanical strength ofthe material, as they are the starting points for further cracking
under tensile stress. As a result, the mechanical stress isconcentrated at the weak points, which has a lower mechanical strength, and the bulk would ultimately break and theactual destruction of the REBaCuO bulk is likely to happen
from the mechanically weak points in the bulk material during
the magnetizing and cooling process.
For YBaCuO bulks, a relatively low tensile strength ofabout σmax∼25 MPa has been found, which was limited by
the initial size of the small cracks produced by the crystal
growth [8]. The ring-shaped bulk, in particular, suffers fromhigher mechanical stress, compared to the disk bulk, becauseof the concentration of the hoop stress at the innermost periphery [9]. To avoid mechanical fracture, the increase ofinternal tensile strength by the addition of Ag in the YBaCuO
bulk was reported up to 30–40 MPa [10], up to 70 MPa [11]and even up to 115 MPa for the GdBaCuO bulk [12], all ofwhich were measured by bending tests. The epoxy resin
impregnation is also an effective way to enhance themechanical strength, in which voids near the bulk surface are
ﬁlled with the epoxy resin [13–15]. Another way to improve
the internal mechanical properties is the elimination of voids,
which were reduced by increasing the O2 pressure in the postannealing after the crystal growth [16]. Both the fracture
strength and Young’s modulus of the REBaCuO bulk wereincreased with decreasing contents of voids [17–19]. For thevoid-free and crack-free microstructure of YBaCuO bulk
superconductors, its mechanical strength was estimated to behigher than 300 MPa, suggesting a possible trapped ﬁeld ofover 30 T without fracture [20].
A complementary approach to avoid cracking during
magnetization process is to encapsulate the bulk in steel tube
externally [21], which leads to stress compensation by generating a compressive stress on the bulk after cooling from300 K to the magnetizing temperature due to the thermal
expansion coefﬁcient of the steel larger than the one of
REBaCuO in the ab-plane. Using the steel ring encapsulation,a trapped ﬁeld of 14.4 T was reported at 22 K in the gap
between Ag-doped YBaCuO disks [21] and of 16 T at 24 K
[22]. A trapped ﬁeld value of 17.24 T at 29 K was reported in
an arrangement of two YBaCuO bulks of 26 mm in diameterwith impregnated with wood’s metal and resin and reinforcedby carbon ﬁbre [13], in which a dome-like trapped ﬁeld
proﬁle can be seen. The ﬂat part of the proﬁle suggests thatthe bulk has a larger Jc value to achieve a trapped ﬁeld higher
than 20 T at 29 K under the sufﬁcient reinforcement. The
record-high trapped ﬁeld has been updated to 17.60 T at 26 K
in the gap between two GdBaCuO disks 25 mm in diameter,
reinforced by shrink-ﬁt steel [23], in which the reinforcement
effect of the shrink-ﬁt steel was simply estimated for the disk
bulk and steel ring with inﬁnite height. In this way, themechanical reinforcement of the bulk is the key technique toachieve higher trapped ﬁeld in the REBaCuO TFMs.

It is important to analyze the mechanical stress numericallyin the bulk during magnetization process. The analytical solutions of the trapped ﬁeld and mechanical stress of the TFM werereported using the expanded theory for the bulk cylinder with
inﬁnite height [8]. The similar analytical solutions of the ﬂuxpinning-induced stress and strain distributions in the inﬁnite bulk
cylinder and ring bulk were presented during ﬁeld-cooled magnetization (FCM) and zero-ﬁeld cooled magnetization [24–26].
The magnetostriction of superconducting cylinders and rings
with ﬁnite height was also reported during FCM by the ﬁnite
element method [27]. However, these papers were based on
Bean’s critical state model for a c-axis oriented, inﬁnite superconducting cylinder, in which the critical current density, Jc, is
independent of the magnetic ﬁeld.
Recently, we have investigated the numerical simulationof the trapped ﬁeld, Bz, and the mechanical stresses (hoopstress, σθ, and radial stress, σr) in the REBaCuO ring bulk
with ﬁnite height reinforced by a metal ring with the sameheight during FCM [28, 29]. The thermal compressive stress,
sqcool, which occurs in the ring bulk when cooling down tooperating temperature due to the difference of thermal
expansion coefﬁcient between the bulk material and metal
ring, was effectively applied at the center of the bulk. However, the sqcool value was reduced comparatively at theuppermost surface of the ring bulk because of a larger thermal
contraction of the metal ring along the height direction
besides the radial direction. These results suggest that thereinforcement effect by metal ring becomes weak at the surface of the bulk with ﬁnite height, and the bulk might break
due to the large Lorentz force during FCM from the higher
magnetic ﬁeld. We have recently proposed a new reinforcement structure using an aluminum alloy ring to the annular
REBaCuO bulks applicable to compact and cryogen-free
400 MHz (9.4 T) nuclear magnetic resonance spectrometerusing a numerical simulation, in which the surface metal plate
molded to the metal ring at the uppermost ring bulk iseffective to reduce the hoop stress [30]. We believe that ahigher trapped ﬁeld over 20 T can be also achieved in the bulk
pair reinforced by a suitable reinforcement structure.
In this paper, we propose an effective mechanical reinforcement structure for the REBaCuO disk bulk and performthe numerical simulation for the trapped ﬁeld and the hoop
stress, σθ, in the bulk pair, compared to those of the bulk pair
reinforced by conventional metal ring with the same height as
the disk bulk. The possibility to achieve a higher trapped ﬁeld
over 20 T is suggested in the gap of the REBaCuO disk bulks
without fracture.

2. Numerical simulation procedure
Based on our experimental setup for the FCM procedure at theHigh Field Laboratory for Superconducting Materials, Institutefor Materials Research, Tohoku University, we prepared thefollowing framework of three-dimensional numerical simulation for electromagnetic and mechanical properties. A schematic view of the numerical model using a new reinforcement
structure is presented in ﬁgure 1(a). Two REBaCuO disk bulks
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Figure 1. Schematic view of the numerical model of (a) the proposed reinforcement structure named ‘full ring’ and (b) the conventional

reinforcement structure named ‘normal ring’.

(20 mm in diameter and 10 mm in height) were fully surrounded by a stainless steel (SUS316) ring 5 mm in width(w=5 mm) and upper and lower SUS316 plates 1.5 mm thick.
We abbreviate the fully encapsulated disk bulks as ‘full ring’.
Then, two bulk modules were stacked with a gap of 0.1 mm
along the z-direction and were magnetized by FCM at
Ts=20 K using an inﬁnite solenoid coil (150 mm outer diameter, 100 mm inner diameter and inﬁnite height), in which the
distance between two bulks is 3.1 mm. For comparison of thereinforcement effect, a schematic model for a conventional
reinforcement is presented in ﬁgure 1(b), in which two bulk
disks are bundled by SUS316 ring 5 mm in width and stacked
with a gap of 3.1 mm. We abbreviate this disk bulks as ‘normal
ring’. In these numerical models, the interface between the bulk
and SUS316 part was ﬁxed perfectly. Physical phenomena inthe simulation during FCM are described using an electromagnetic equation [28–30], in which temperature is assumed tobe constant for simplicity. The simulation results of the trapped
ﬁeld, Bz, depend strongly on the temperature and magnetic ﬁeld
dependence of critical current density, Jc(T, B), of the superconducting bulk [31]. To achieve the trapped ﬁeld of
Bz* = 22 T in the inﬁnite height bulk 20 mm in diameter,Jc* = 1.752 ´ 109 A m-2 is necessary under the relation of
Bz* = m0 Jc*R (R: radius of the disk bulk) by Bean’s criticalstate model. However, for the actual ﬁnite height bulk, the Jc of
the bulk must be higher than Jc* to achieve Bz=22 T at
Bapp=22 T, because the induced superconducting current
ﬂows fully in the upper and lower bulk surface.
Kii et al measured the temperature and magnetic ﬁeld
dependence of critical current density, Jc(T, B), of the
GdBaCuO bulk at low temperatures, which was determined

Figure 2. The reported Jc(T, B) proﬁles of GdBaCuO bulk at 20 K
and 30 K by Kii et al [32]. The Jc values used in the present
simulation; Jc=2Jc*=3.503×109 A m−2, Jc=3Jc*
=5.254×109 A m−2, and Jc=4Jc*=7.006×109 A m−2,which are assumed to be constant under magnetic ﬁeld.

using the extended Bean’s model for magnetic moment hysteresis (M–H) loops measured by a SQUID magnetometer
under the applied ﬁeld parallel to the c-axis of a small sample
without cracks [32]. Figure 2 shows the ﬁtted Jc(T, B) curves
at 20 and 30 K by Kii et al. The Jc increases with decreasing
temperature and decreases with increasing magnetic ﬁeld,
which shows weaker magnetic ﬁeld dependence, compared tothe one at higher temperature [32]. The trapped ﬁeld, Bz, on
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Table 1. Mechanical parameters used in the numerical simulation (E:Young’s modulus, ν: Poisson ratio, α: thermal expansion
coefﬁcient) [9, 28].

REBaCuO bulk
SUS316E (GPa)να (K−1)

100
193

0.33
0.28

5.2×10−6
1.27×10−5

FCM, a magnetic ﬁeld of Bapp=22 T was applied above thecritical temperature, Tc (= 92 K), and then the bulk wascooled to Ts=20 K. Thereafter, the magnitude of themagnetic ﬁeld was monotonically decreased from 22 T tozero by ten steps with −0.15 T min−1, which is a ramp rateused in our experiments. The time step (TS) of the descent ofmagnetic ﬁeld during FCM is deﬁned as follows
TS = 10

Bapp

,

(2 )

where Bex is the actual external magnetic ﬁeld.
In an isotropic material, Hooke’s law is established, inwhich the stress is linearly proportional to the strain. Although
the actual REBaCuO bulk is mechanically anisotropic andinhomogeneous, as already mentioned in section 1, themechanical properties of the REBaCuO bulk are assumed to be
isotropic and homogeneous for simplicity. The electromagnetic
hoop stress, s qFCM, was calculated for each step of FCM, whichacts circumferentially on each end face of the meshed element[8, 24]. The thermal hoop stress, sqcool, by cooling from 300 to
20 K was also calculated, which was generated by the difference of the thermal expansion coefﬁcient, α, between the
REBaCuO bulk and the SUS316 ring. The mechanical parameters (Young’s modulus, E, Poisson ratio, ν, and thermal
expansion coefﬁcient, α) of the REBaCuO bulk and SUS316
used in the elastic simulation are summarized in table 1, in
which E and ν values are those at 300 K and are assumed to be
independent of temperature [9, 29], although the mechanicalstrength such as bending strength of the REBaCuO bulk slightly
increases with decreasing temperature due to the decrease of theinter-atomic distance by cooling [16]. Commercial software
package, Photo-EDDY (Photon Ltd, Japan) was used for the
analysis of the distributions of magnetic ﬁeld, Bz, along zdirection and the induced persistent current density, Jθ, thecircumferential (θ) direction during FCM. The s qFCM and sqcool
values were calculated for each step by Photo-ELAS (Photon
Ltd, Japan) using the nodal force at each node calculated by
Photo-EDDY. In the model, the bulk enclosed by the SUS316
part was equally divided into 36 elements along the θ-direction
and by the 0.5 mm pitch along the r- and z-directions.

Figure 3. (a) Time step dependence of the cross section of thetrapped ﬁeld proﬁle during FCM from Bapp=22 T in the gap centerof two bulks along the r-direction, Bz(z=0 mm), for the case of
Jc = 4Jc*. The ﬁnal Bz(z=0 mm) proﬁles of 10th step for the casesof Jc=3Jc* and 2Jc* were also shown. (b) Bz(r=0 mm) proﬁles of
the 10th step for each Jc case along the z-direction.

Jc(T, B) value is generally overestimated, and the Jc also
decreases along the bulk radius because the quality decreases
with the growth front progression [33]. From this evidence,
the magnitude of Jc(B) should be adjusted to one third of its
small specimen value to adequately reproduce the trapped
ﬁeld by FCM [34, 35]. In the present simulation, the trapped
ﬁeld, Bz, is estimated in the bulk pair to eliminate thedemagnetization effect and the Jc value is assumed to be 2Jc*,
3Jc* and 4Jc*, as shown in ﬁgure 2, which is independent ofapplied magnetic ﬁeld.
The power-n model was used to describe the nonlinear
E–J characteristic of the superconducting bulk:⎛ J ⎞nE = Ec ⎜ ⎟ ,⎝ Jc ⎠

Bapp - Bex

3. Numerical simulation results(1 )3.1. Electromagnetic properties

where Ec (= 10−4 V m−1) is the reference electric ﬁeld and Jc
is the critical current density. To realize the approximate
Bean’s model, a large n value (n=100) and a constant Jc areassumed in equation (1) [28]. In the simulation process of

Figure 3(a) shows the TS dependence of the cross section of
the trapped ﬁeld proﬁle, Bz(r), during FCM from Bapp=22 T
in the gap center of two bulks along the r-direction at z=0,
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Figure 4. The ﬁnal (10th step) contour maps of the (a) trapped ﬁeld,
Bz, and (b) persistent current density, Jθ, in the pair of bulk module
for Jc = 4Jc* during FCM from Bapp=22 T. The similar contour
maps of (c) Bz and (d) Jθ proﬁles for Jc = 2Jc* during FCM from
Bapp=22 T.

Figure 5. The radius (r) dependence of the thermal hoop stress,
sq cool, for the (a) ‘full ring’ and (b) ‘normal ring’ at various zpositions under the cooling from 300 K to 20 K without FCM. The
inset of each ﬁgure shows the contour map of the sq cool proﬁle.

for the case of Jc=4Jc*=7.006×109 A m−2. The external
magnetic ﬁeld decreased step by step without a ﬁeld slope dueto the inﬁnite length of magnetizing solenoid coil. The Bz(r)value in the bulk also decreased step by step, but Bz(r) forr<2 mm kept a constant value of 21.8 T, which was nearly
the same value as Bapp. These results suggest that the bulk
was not fully magnetized due to the larger Jc (= 4Jc*) value,that is, the induced supercurrent partially ﬂows in the periphery region. In the ﬁgure, the ﬁnal Bz(r) proﬁles of the 10th
step at z=0 mm for the cases of Jc=3Jc* and 2Jc* are also
shown, in which the constant Bz(r) region became narrow andBz(z=0 mm) decreased with decreasing Jc, because thepersistent supercurrent ﬂowing region became wide.
Figure 3(b) shows the ﬁnal (10th step) trapped ﬁeld proﬁle,Bz(z), along the z-direction (r=0 mm) for each Jc value. For
Jc = 4Jc*, the decrease of Bz at the gap center, ΔBz, is as
small as 0.3 T (Bz(r=z=0 mm)=21.5 T). However, ΔBz
was 0.74 T (Bz(r=z=0)=21.06 T) for Jc = 3Jc* and2.98 T (Bz(r=z=0 mm)=18.82 T) for Jc=2Jc*. To
achieve higher Bz in the central gap, the Jc value should be
increased and the width of the gap between the bulks should
be minimized.

Figures 4(a) and (b), respectively, show the contour mapsof the ﬁnal (10th step) trapped ﬁeld, Bz, and the persistentcurrent density, Jθ, in the bulk pair for Jc=4Jc* during FCM
from Bapp=22 T. In ﬁgure 4(b), the persistent supercurrent,
which is as large as 4Jc*, ﬂows along the θ-direction in outerregion and top and bottom surfaces of the bulk pair. As aresult, the magnetic ﬁeld was proﬁled as shown in ﬁgure 4(a),which was closely correlated with the Jθ proﬁle. Figures 4(c)and (d), respectively, show the similar contour plots of the10th step Bz, and Jθ in the bulk pair for Jc = 2Jc* during FCM
from Bapp=22 T. The region, at which the persistentsupercurrent with 2Jc* ﬂows, became wider, compared to thatfor the Jc = 4Jc* case. As a result, the region with higher Bzvalue became narrow and the Bz value at the gap centerdecreased.

3.2. Mechanical properties

Figures 5(a) and (b), respectively, show the radius (r)dependence of the thermal hoop stress, sqcool, for the ‘full ring’and the ‘normal ring’ cases at various z-positions under thecooling from 300 to 20 K without FCM (only cooling). The
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Figure 6. The time step (TS) dependence of the sq FCM values at the positions of (a) (r, z)=(0.5 mm, 11.4 mm) and (b) (r, z)=(0.5 mm,

6.5 mm) for the ‘full ring’ and the ‘normal ring’ cases under Jc=2Jc*, 3Jc* and 4Jc* during FCM from 22 T without cooling (only FCM).
The radius (r) dependence of the sq FCM, for the (c) ‘full ring’ and (d) ‘normal ring’ at various z-positions during for the ﬁnal step (10th step)of FCM from 22 T.

inset of each ﬁgure shows the contour map of the sqcool proﬁle.
In ﬁgure 5(a), a large compressive stress (σθ <0) is thermallyapplied to the disk bulk region (r10 mm) and the tensilestress is applied to the SUS316 ring, because the thermal
expansion coefﬁcient, α, of the SUS316 is about 2.5 times
larger than that of the REBaCuO bulk, as shown in table 1.
The sqcool values for the bulk center (z=6.5 mm, r=0 mm) andbulk surface (z=1.6 or 11.4 mm, r=0 mm) are −195 and−137 MPa, respectively. On the other hand, in ﬁgure 5(b), thesqcool values of the ‘normal ring’ case at the bulk center(r=0 mm) are smaller than those of the ‘full ring’ case. It issurprising that the compressive sqcool value near the bulk surface(z=1.6 and 11.4 mm) drastically decreases at the bulk periphery(r=9 mm). Because of the larger α value of the SUS316
compared to the bulk, the SUS316 ring shrinks along both z- andr-directions. The bulk was fastened along the r-direction and, atthe same time, the periphery region of the bulk was pulled
toward the bulk center (z=6.5 mm). As a result, the compressive hoop stress around the bulk periphery became weak [28].
The electromagnetic hoop stress, s qFCM, for the ‘full ring’and the ‘normal ring’ cases was simulated for each step of
FCM from 22 T without cooling (only FCM). The s qFCM value
distributes in the bulk and changes with increasing TS.
Figure 6(a) shows the TS dependence of the s qFCM values atthe position of (r, z)=(0.5, 11.4 mm) for the ‘full ring’ andthe ‘normal ring’ cases under Jc=4Jc*, 3Jc* and 2Jc* during
FCM from 22 T. Since the s qFCM value on the z-axis(r=0 mm) cannot be strictly obtained due to the criticalpoint in the numerical model, s qFCM values at (r, z)=(0.5,
11.4 mm) were substituted. The s qFCM value took a maximum
at the center of the bulk surface for all cases. These tendencies
were also obtained for the similar simulation [8, 28]. The
s qFCM value for the ‘full ring’ case gradually increases withincreasing TS, and with increasing Jc value of the bulk. For
the Jc = 2Jc* case, the s qFCM value took a maximum at theseventh step and decreased with the further increase in TS,
which result from the decrease of the trapped ﬁeld. On theother hand, the s qFCM value for the ‘normal ring’ case is about
twice as large as that for the ‘full ring’ case because the‘normal ring’ reinforcement effect is weaker at the bulk surface [28]. Figure 6(b) shows the similar TS dependence of thes qFCM value at the position of (r, z)=(0.5, 6.5 mm) of thebulk center. The s qFCM value hardly changes depending on theJc value due to the sufﬁcient reinforcement at the bulk center,and the difference between the ‘full ring’ and the ‘normal
ring’ cases was relatively small.
Figures 6(c) and (d), respectively, show the radius (r)dependence of the electromagnetic hoop stress, s qFCM, for the‘full ring’ and the ‘normal ring’ cases at various z-positions
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Figure 7. The TS dependence of the sq total values at the positions of (a) (r, z)=(0.5 mm, 11.4 mm) and (b) (r, z)=(0.5 mm, 6.5 mm) for the‘full ring’ and the ‘normal ring’ cases under Jc=2Jc*, 3Jc* and 4Jc* during FCM from 22 T. The radius (r) dependence of the sq total, for the(c) ‘full ring’ and (d) ‘normal ring’ at various z-positions during for the ﬁnal step (10th step) of FCM from 22 T.

during for the ﬁnal step (10th step) of FCM from 22 T without
cooling (only FCM). The inset of each ﬁgure shows thecontour map of the s qFCM proﬁle. In ﬁgure 6(c), the s qFCM
value was larger at the central region and the z dependence ofthe s qFCM value is relatively small. On the other hand, for the‘normal ring’ case, the s qFCM values at the bottom(z=1.6 mm) and top (z=11.4 mm) surfaces were enhanced
because of the reduction of the reinforcement effect at thebulk surface. The r dependences of the s qFCM are quite different and asymmetric between z=1.6 and 11.4 mm, which
result from the inﬂuence of other magnetized bulk.
Figure 7(a) shows the TS dependence of the maximum
total hoop stress, sq total (max) (=s qFCM +sqcool ), for the ‘full
ring’ and the ‘normal ring’ cases under Jc=4Jc*, 3Jc* and2Jc* during FCM from 22 T at the position of (r, z)=(0.5,
11.4 mm), at which the maximum hoop stress is applied
during FCM process. The sqtotal (max ) value is smaller than
the s qFCM value as shown in ﬁgure 6(a) because of the compressive effect of the metal ring by cooling. For the ‘normal
ring’ case, the sqtotal (max ) value was as high as +100 MPa at
the ﬁfth step even for the Jc = 2Jc* case and increased withincreasing Jc value, in which the bulk of the ‘normal ring’must be broken if the fracture strength of the typical Agdoped REBaCuO bulk is 50–70 MPa [16–18]. On the other
hand, for the ‘full ring’ case, the sqtotal (max ) value was only

+10 MPa even for the Jc = 4Jc* case, which suggests that thebulk might not break because of the effective mechanicalreinforcement.
Figure 7(b) shows the similar TS dependence of thesqtotal (max ) value at the position of (r, z)=(0.5 mm, 6.5 mm)of the bulk center. The sqtotal (max ) value hardly changes
depending on the Jc value due to the sufﬁcient reinforcement
effect at the bulk center. The sqtotal (max ) value for the ‘full
ring’ case is negative (compressive stress) during FCM step.
For the ‘normal ring’ case, the maximum sqtotal value wasdrastically reduced to be +45 MPa by the cooling reinforcement, compared with the maximum s qFCM value of +180 MPa
shown in ﬁgure 6(b).
Figures 7(c) and (d), respectively, show the radiusdependence of the total hoop stress, sqtotal (= s qFCM +sqcool ),for the ‘full ring’ and the ‘normal ring’ cases at various zpositions during for the ﬁnal step (10th step) of FCM from
22 T. The inset of each ﬁgure shows the contour map of thesqtotal proﬁle. In ﬁgure 7(c), the sqtotal value was larger at thecentral region (r=0 mm) and the z dependence of the sqtotal
value is relatively small. The similar sqtotal proﬁle can be seenfor each TS. On the other hand, for the ‘normal ring’ caseshown in ﬁgure 7(d), the sqtotal value at the bottom(z=1.6 mm) and top (z=11.4 mm) surfaces was enhanced
because of the reduction of the reinforcement effect of the7Supercond. Sci. Technol. 32 (2019) 045005

H Fujishiro et als qFCM and sqtotal values for the w=3 mm bulk, compared tothose for the w=5 mm bulk. These values for the w=3 mm
are slightly larger than those for the w=5 mm bulk. However,the sqtotal value is fairly smaller than the fracture strength of the
REBaCuO bulk material of 50∼70 MPa [16–18]. These results
suggest that the width of w=3 mm is sufﬁcient to reinforce thebulk and to prevent the bulk fracture during FCM up to 22 T, andthat the diameter of the REBaCuO bulk could be slightly
increased, in which the restriction with higher Jc(B) of the bulk
can be relaxed to achieve higher trapped ﬁeld over 20 T.

4. Conclusion
We proposed an effective mechanical reinforcement structure
for REBaCuO superconducting disk bulk pair to avoid
mechanical fracture during FCM from 22 T, and conﬁrmed
the reinforcement effect using numerical simulation. The
important results and conclusions are summarized as follows.
(1) A higher trapped ﬁeld, Bz, over 20 T can be achieved in thegap center of the bulk pair, which was fully encapsulated
by outer ring and upper and lower plates made by SUS316.
(2) A large compressive hoop stress, sqcool, due to thedifference of thermal expansion coefﬁcient between thebulk and SUS316, was effectively applied to the wholebulk during the cooling process from 300 K to 20 K,
compared to that for the conventional reinforcement
using only a SUS316 ring.
(3) The electromagnetic hoop stress, σθFCM, during FCM wasalso reduced and the maximum of the total hoop stress,
sqtotal (max) (= σθFCM+sqcool ) can be reduced to+10 MPa by the proposed reinforcement structure, which
was lower than the mechanical strength of the typical Agdoped REBaCuO bulk (50–70 MPa). The possibility toachieve higher trapped ﬁeld over 20 T was suggested in
the gap in the REBaCuO bulk pair without fracture.

Figure 8. (a) The radius (r) dependence of the thermal hoop stress,
sq cool, for the ‘full ring’ with w=3 mm at various z-positions under
the cooling from 300 K to 20 K without FCM (only cooling). The time
step dependence of the (b) sq cool and (c) sq total values at z=11.4 mm
for the w=3 mm bulk, compared with those for the w=5 mm bulk.

normal ring. The 0.2% proof stress of SUS316 is about
600 MPa at 77 K, which changes depending on the carbon
and nitrogen contents in the material [36]. The maximum
stress applied at the inner periphery of the SUS316 ringshown in ﬁgures 7(c) and (d) was smaller than that value. The
SUS316 part used in the study can reinforce the bulk within
the 0.2% proof stress safely.

Acknowledgments
This work was performed at the High Field Laboratory for
Superconducting Materials (HFLSM), Institute for Materials
Research, Tohoku University (Project No. 18H0055), and waspartially supported by JSPS KAKENHI grant No. JP15K04646.

3.3. Effect of the width of outer SUS316 ring

In the previous subsection, we presented the effect of the newreinforcement structure of the metal ring, in which the width ofthe outer SUS316 ring was w=5 mm, as shown in ﬁgure 1.
Finally, we investigate the inﬂuence of the outer ring width, w,
on the compressive stress during cooling, sqcool, and total hoop
stress, sqtotal during FCM. Figure 8(a) shows the radius (r)dependence of the thermal hoop stress, sqcool, for the ‘full ring’
with w=3 mm at various z-positions under the cooling from300 K to 20 K without FCM (only cooling). The tensile stress in
the SUS316 ring with w=3 mm was sqcool = 250 Mpa atr=10.5 mm, which is larger than that the tensile stresssqcool = 200 Mpa obtained for w=5 mm, as shown inﬁgure 5(a). However, the compressive stress in the bulk part withw=3 mm is nearly the same as the w=5 mm case.
Figures 8(b) and (c), respectively, show the TS dependence of the

ORCID iDs
Hiroyuki Fujishiro https://orcid.org/0000-0003-1483-835X
Tomoyuki Naito https://orcid.org/0000-0001-7594-3466

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