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Promising effects of a new hat structure and double metal ring formechanical reinforcement of a REBaCuO ring-shaped bulk during fieldcooled magnetisation at 10 T without fracture
To cite this article: H Fujishiro et al 2019 Supercond. Sci. Technol. 32 065001

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Superconductor Science and Technology
Supercond. Sci. Technol. 32 (2019) 065001 (12pp)

https://doi.org/10.1088/1361-6668/ab0bed

Promising effects of a new hat structureand double metal ring for mechanicalreinforcement of a REBaCuO ring-shapedbulk during ﬁeld-cooled magnetisation at
10T without fracture
H Fujishiro1 , T Naito1 , Y Yanagi2, Y Itoh2 and T Nakamura3
1

Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, 4-3-5 Ueda, Morioka 020-8551, Japan
2
IMRA Material R&D Co., Ltd, 2-1 Asahi-machi, Kariya 448-0032, Japan
3
RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan
E-mail: fujishiro@iwate-u.ac.jp
Received 2 December 2018, revised 20 February 2019
Accepted for publication 1 March 2019
Published 23 April 2019
Abstract

We have investigated a new reinforcement hat structure for a RE-Ba-Cu-O (REBaCuO, RE: rareearth element or Y) ring-shaped bulk superconductor (ring bulk) shrink-ﬁtted with a double Al
alloy ring and set on the cold stage of a cryogenic refrigerator prior to ﬁeld-cooled magnetisation(FCM). With the hat structure, a ring bulk having an outer diameter of 64 mm, inner diameter of
40 mm, and height of 20.5 mm achieved a trapped ﬁeld as high as 6.8 T at 50 K during FCM
with an applied magnetic ﬁeld at Bapp=10 T without fracture; a similar ring bulk without
the hat structure broke during FCM at Bapp=8.8 T. Using a numerical simulation forelectromagnetic and mechanical properties, the potential beneﬁt of the hat structure was
conﬁrmed. The magnetic ﬁeld dependence of the average critical current density Jc(B) of the ringbulk used in the simulation was determined by experimental results of time step dependence ofthe trapped ﬁeld Bz at the bulk centre. As a result, the total hoop stress σtotal
in the ring bulk
θincluding the resulting stress of cooling from 300 K to 50 K and subsequent FCM at Bapp=10 T
is lower than the fracture strength of a typical Ag-doped REBaCuO bulk material. The effectof double Al alloy ring reinforcement was also analysed using a numerical simulation andcompared with that of the conventional single Al alloy ring reinforcement. These results suggestthat the double ring provides only a limited beneﬁt, whereas the hat structure is fairly effective in
reducing the electromagnetic hoop stress during FCM at Bapp=10 T. Using this reinforcement
method, a 400 MHz (9.4 T) nuclear magnetic resonance bulk magnet system could be realised.
Keywords: REBaCuO superconducting bulk, ﬁeld cooled magnetization, trapped ﬁeld,
electromagnetic hoop stress, mechanical reinforcement, numerical simulation(Some ﬁgures may appear in colour only in the online journal)ﬁeld trapped ﬁeld magnet is capable of generating several
Tesla. Although the trapped ﬁeld estimated from its criticalcurrent density Jc(B, T) could theoretically be over 30 T at
29 K [1], the mechanical strength of the brittle ceramic material restricts the maximum trapped ﬁeld experimentally. A

1. Introduction
It is important to enhance the trapped magnetic ﬁeld in REBa-Cu-O (REBaCuO, RE: rare earth element or Y) superconducting ring-shaped bulks (ring bulks) because the high0953-2048/19/065001+12$33.00

1© 2019 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et altrapped ﬁeld of 16 T has been achieved at 24 K in an Ag- andZn-doped YBaCuO disc bulk pair reinforced by austenite Cr–
Ni steel tubes [2], and a trapped ﬁeld of 17.24 T has beenachieved at 29 K in a YBaCuO disc bulk pair by mechanicalreinforcement of resin impregnation and carbon ﬁbre wrapping [1]. To date, the highest trapped ﬁeld of 17.6 T has beenachieved at 26 K in a GdBaCuO disc bulk pair reinforced byshrink-ﬁt stainless steel [3]. A REBaCuO ring-shaped bulk
superconductor is useful for practical applications such as
nuclear magnetic resonance (NMR) spectrometers andmagnetic resonance imaging (MRI) equipment [4–6]. Trapped ﬁeld enhancement in the ring bulk of an NMR/MRI
system is an ongoing challenge for improving resolution; atrapped ﬁeld of as high as 10 T is necessary to realise a
400 MHz (9.4 T) NMR bulk magnet system.
It should be noted that the above analytical investigationsof mechanical properties were reported only for disc- andring-shaped bulks of inﬁnite height during ﬁeld-cooledmagnetisation (FCM) or zero-ﬁeld-cooled magnetisation[7–9]; the results were primarily based on Bean’s critical statemodel with constant Jc characteristics independent of magnetic ﬁeld. Huang et al investigated the magnetostriction of asuperconducting disc or ring of ﬁnite height using the ﬁnite
element method (FEM), in which Bean’s model and a uniform applied ﬁeld from an inﬁnite magnetising solenoid coilwere assumed [10]. Furthermore, there have been no numerical simulations of the reinforcement effect of the metal ringﬁtting on the mechanical stresses in the superconducting discand ring-shaped bulks during FCM and the cooling process,except for the relevant work by Johansen et al [11].
Therefore, we have reported numerical simulation resultsof the mechanical stress behaviours in REBaCuO ring anddisc bulks of ﬁnite height and the metal ring reinforcement
during FCM [12–14], and we have also proposed a newreinforcement method to prevent bulk failure [15, 16]. The
following points have been clariﬁed so far.

on the ring bulk
(4) To restrict the reduction of σcoolθsurface, a new numerically simulated reinforcement
structure has been proposed, in which thick metal plates
are tightly moulded above and below the metal ring. As
a result, a 400 MHz (9.4 T) NMR bulk magnet can berealised without fracture [15].
(5) The mechanical strain has been directly measured on aring bulk surface during the cooling process and FCM
using a strain gauge, verifying that the simulation
results qualitatively reproduce the experimental
results [16].
For the numerical models used in the simulation, we
assumed that there was no slippage at the interface between
the outer ring bulk surface and the inner metal ring surface.
Such assumptions of mechanical behaviour and associatednumerical results should be validated and compared withactual experimental results; previous investigations have been
typically performed by simulation only. We performed several FCM experiments with an applied magnetic ﬁeld Bapp of
5–7 T at 50 K for REBaCuO ring bulks reinforced by a metal(Al alloy or stainless steel) ring of identical height; amechanical fracture frequently occurred. Although we haveproposed the new reinforcement method to prevent this ringbulk fracture [15], actual experimental conﬁrmation has notbeen performed.
In this paper, we report the results of FCM experiments
for a Bapp of up to 10 T at 50 K using a new reinforcement hat
structure for EuBaCuO ring bulk reinforced by a double Al
alloy ring and set on the cold stage of a cryogenic refrigerator.
The EuBaCuO ring bulk with this reinforcement can achieve
higher trapped ﬁeld and magnetic ﬁeld homogeneity with low
relative magnetic permeability (μr=1.0013), which is preferable for NMR/MRI bulk magnets [5]. As a result, theexperimental FCM procedure at Bapp=10 T succeeded
without fracture, and a trapped ﬁeld of 6.8 T was safely
achieved at the centre of the ring bulk. We analysed theeffects of the double Al alloy ring reinforcement and the hat
structure using numerical simulation and compared them withthose of single Al alloy ring reinforcement as well as those
without the hat structure.

(1) In REBaCuO ring bulks of ﬁnite height reinforced by ametal (Al alloy or stainless steel) ring, the compressive
stress is non-uniformly applied to the ring bulk during
the cooling process from room temperature to theoperating temperature of FCM. More speciﬁcally, theapplied compressive stress is reduced at the peripheryof the bulk surface and, in some cases, changes totensile stress because a larger thermal contraction of themetal ring takes place along both radial and height
directions relative to that of the bulk [12].
(2) The electromagnetic hoop stress σFCM
during FCM
θachieves a maximum at the inner peripheral surface ofthe ring bulk, which is larger than that for a disc bulk
with an identical outer diameter and height [12, 13].
(3) The thermal compressive stress σcool
increases in the ringθbulk during the cooling process from room temperature tooperating temperature, and σFCM
during FCM decreases
θwith the increasing width of the metal ring wr. As a result,
FCM
the total hoop stress σtotal(= σcool) can beθ
θ +σθreduced at the innermost surface of the ring bulk [14].

2. Experimental procedure
Two EuBaCuO ring-shaped bulks were prepared: bulk-A withan outer diameter (OD) of 64 mm, inner diameter (ID) of
32 mm, and height (ht) of 13 mm, and bulk-B with 64 mm
OD, 40 mm ID, and 20.5 mm ht; both were fabricated by
Nippon Steel & Sumitomo Metal, Japan. The double Al alloy
ring structure for each ring bulk was applied by the following
process. First, the inner Al alloy ring (A7075-T6) with
1.5 mm in width (67 mm OD and 64.1 mm ID for both bulk-Aand bulk-B) was moulded using 0.05 mm thick StycastTM
1266 resin at room temperature. Second, the outer Al alloy
ring (A7075-T6) 3.4 mm wide (74 mm OD and 66.70 mm ID
for bulk-A, and 74 mm OD and 66.84 mm ID for bulk-B,
which is slightly smaller than the OD of the inner ring) was2

Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

Figure 1. The experimental setup of (a) bulk-A and (b) bulk-B on the cold stage of the cryogenic refrigerator during FCM. Bulk-A wassandwiched by Cu plates using a nut and bolt, and the bottom Cu plate was thermally connected to the cold stage. Bulk-B was mechanicallyand thermally ﬁxed to the cold stage using a reinforcement hat structure.

mechanically pressed into the outer periphery of the inner ringusing a tapered die. The double ring reinforcement was
expected to add an initial compressive stress before theshrink-ﬁt process, which is a new method to reinforce thebulk. The double ring reinforcement is suitable for practicalapplications because it shows a comparable effect to that of a
thicker single ring, and the ring bulk can be safely magnetised
in a magnet bore with limited size.
Figure 1(a) shows the experimental setup for bulk-A
reinforced by the double Al alloy ring set on the cold stage ofa cryogenic refrigerator. Bulk-A was sandwiched by thick Cu
plates with 0.1 mm thin indium (In) sheets using a nut andbolt, and the bottom Cu plate was thermally connected to thecold stage. Figure 1(b) shows the experimental setup for bulkB reinforced with the double Al alloy ring. Bulk-B was
thermally and mechanically connected to the cold stage of thecryogenic refrigerator using the hat structure (74.26 mm ID,
84 mm OD, and 5 mm thickness of the upper plate), also
made of Al alloy, by the insertion of an In sheet. The temperature of each ring bulk was controlled by a Pt–Co thermometer attached to the bottom surface of the cold stage. In
the FCM process, bulk-A was cooled to the magnetisingtemperature of Ts=50 K under the applied ﬁeld of
Bapp=7.3, 8.3, and 8.8 T using a cryocooled superconducting solenoid magnet (JASTEC JMTD-10T100). The
external ﬁeld was then decreased linearly at −0.222 T min−1
down to zero. Bulk-B with the hat structure was also cooled
to Ts=50 K under the applied ﬁeld of Bapp=7.3, 8.3, 8.8,
9.4, and 10 T, and the external ﬁeld was decreased linearly at
the same ramp rate down to zero.
The time step (TS) of the decline of the magnetic ﬁeld
during FCM is deﬁned as TS = 10 (Bapp - Bex ) /Bapp , where
Bex is the actual applied ﬁeld at the bulk centre. The TS
dependence of the local ﬁeld Bz(t) along the z-direction wasmeasured at the centre of the bulk annuli (z=r=0) using a
Hall sensor (F. W. Bell, BHA921) for each case. For bulk-B,

the TS dependence of temperature T(t) was also measured bya CERNOXTM thermometer on the top of the hat structure.

3. Numerical simulation framework
To understand the effects of the hat structure and double ring
reinforcement, numerical simulation models were constructed. Figure 2 shows the cross-sectional view of thenumerical model. Based on the experimental setup of bulk-B
shown in ﬁgure 1(b), we constructed a three-dimensional
ﬁnite element model for a REBaCuO ring-shaped bulk
(64 mm OD, 40 mm ID, and 20 mm ht). In the previous
section, bulk-B was experimentally reinforced by a double Al
alloy ring using uniaxial pressing. Because it is difﬁcult toreproduce the mechanical pressing process of the outer ring
(Al alloy ring 2), the outer ring was instead heated to 540 K in
the numerical simulation and then smoothly inserted into theouter periphery of the inner ring (Al alloy ring 1), and thencooled to 300 K, as shown in ﬁgure 2(a). This numericalassumption seemed to reproduce the double ring effect forbulk-B. To verify the effect of the double Al alloy ring on
mechanical reinforcement, the ring bulk was also mounted
inside a single Al alloy ring of 5 mm in width (74 mm OD,
64.1 mm ID, and 20 mm ht) using an epoxy resin at 300 K, as
shown in ﬁgure 2(b).
The REBaCuO ring bulk reinforced by the double orsingle Al alloy ring as shown in ﬁgures 2(a) and (b),respectively, was connected to a Cu plate set on the cold stageof the cryogenic refrigerator using an In sheet with 0.1 mm inthickness; it was then reinforced by the hat structure as shownin ﬁgure 2(c). The numerical simulation of the cooling process from 300 to 50 K and that of the subsequent FCM using
a solenoid coil (170 mm OD, 120 mm ID and 200 mm ht)
were performed. In the experimental setup of the ring bulk
with the hat structure as shown in ﬁgure 1(b), the gap between
the OD of the outer Al alloy ring (74.0 mm) and ID of the hat
3Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

Figure 2. Cross-sectional view of the numerical model. REBaCuO ring bulk reinforced by the (a) double Al alloy ring and (b) single Al alloy

ring. (c) The numerical setup during FCM for each bulk using a reinforcement hat structure to the cold stage. (d) The numerical setup during
FCM for each bulk, which is ﬁxed to the cold stage without the hat structure.

structure (74.26 mm) was 0.26 mm at 300 K; therefore, theouter Al alloy ring connects to the hat structure thermally andmechanically during the cooling process from 300 K to 50 K
because the thermal shrinkage of the hat structure was greaterthan that of the outer Al alloy ring along the r-direction. In
this case, the width of the Al alloy ring increased from 5 mm
(initial width) to 10 mm (initial width plus the hat structure
width); thus, the reinforcement effect was expected toincrease [13]. To independently verify the effect of the hat
structure of the upper thick plate as shown in ﬁgure 2(c), theID of the hat structure was increased to 76.0 mm in the
numerical model, which was 2 mm larger than the experimental size. In this case, the Al alloy ring did not makecontact with the hat structure during the cooling process andsubsequent FCM. To verify the effect of the hat structure,
REBaCuO ring bulk reinforced a double ring or single ring
but without the hat structure was thermally connected to a Cu
plate and set on the cold stage using an In sheet, as shown inﬁgure 2(d). A similar cooling process from 300 K to 50 K andsubsequent FCM using the solenoid coil were numerically
performed.
It was assumed that there was no In sheet below andabove the Al alloy ring to prevent the thermal and mechanicalcontact of the Al alloy ring to the Cu plate and hat structure.
In the experimental cooling process, the Al alloy ring shrunk
signiﬁcantly along the z-direction as well as the r-direction.
Therefore, the absence of the In sheet below and above the Al
alloy ring was reasonable assumption for reproducing theactual experimental condition.
Electromagnetic phenomena during FCM were describedby fundamental equations [17, 18]. The E–J power law of thesuperconducting bulk was assumed to be E = Ec (J /Jc )n , inwhich Ec (= 10−4 V m−1) is the reference electric ﬁeld and n(= 20) is an appropriate value for a bulk superconductor [18].
The Jc(B) characteristics of the ring bulk were determinedusing the following equation [19–21], which exhibits a ﬁshtail
shape in the magnetisation loop.
⎡ ⎛⎛ B⎞⎛ B ⎞k ⎞ ⎤B1
Jc (B) = Jc1 exp ⎜ - ⎟ + Jc2
exp ⎢ ⎜⎜1 - ⎜⎟ ⎟ ⎥.
⎝ BL ⎠⎝ Bmax ⎠ ⎟⎠ ⎥⎦⎢⎣ k ⎝Bmax(1 )Three types of Jc(B) curves were examined to reproduce theexperimental results of the TS dependence of the trapped ﬁeld
Bz(r=z=0). Each parameter in equation (1) for the three
Jc(B) curves is shown in table 1.
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H Fujishiro et al

without the hat structure during FCM at Bapp=7.3, 8.3, and
8.8 T at 50 K. For Bapp=7.3 and 8.3 T, Bz gradually
decreased with increasing TS; at TS=10, Bz increased withincreasing Bapp. The Bz value continued to decrease slightly
after TS=10 (Bex=0 T) due to ﬂux creep. However, at
Bapp=8.8 T, Bz suddenly dropped after TS=4, and the TS
dependence of Bz was lower than that of Bex; ﬁnally, Bz
reached a negative value at TS=10. These results suggestthat bulk-A without the hat structure was broken during FCM
at Bapp=8.8 T.
Figure 3(b) shows the TS dependence of Bz(z=r=0 mm) for bulk-B with the hat structure during FCM
at Bapp=7.3, 8.3, 8.8, 9.4 T, and 10 T at 50 K. The Bz value
gradually decreased with increasing TS for each Bapp. The
value of Bz at Bapp=8.3 T was larger than that at
Bapp=7.3 T. However, the ﬁnal Bz value at TS=10 with aBapp higher than 8.3 T was nearly identical, which suggeststhat the ring bulk was fully magnetised at Bapp 8.3 T;
therefore, a supercurrent as large as Jc(B) ﬂowed in all of thering bulk area. It should be noted that a Bz value of 6.8 T wastrapped at the centre of bulk-B without fracture. These results
suggest that the hat structure is effective for achieving higher
trapped ﬁelds without fracture compared with the results ofbulk-A without the hat structure, as shown in ﬁgure 3(a).
However, the ID and ht values of bulk-A and bulk-B are
different from one another; therefore, this is not necessarily an
exact comparison between identical cases with and without
the hat structure. A detailed discussion is given in a later
section.
Figure 3(c) shows the TS dependence of the temperaturechange for bulk-B with the hat structure during FCM at a Bappof up to 10 T at 50 K. The temperature increased withincreasing TS at each Bapp, and the temperature rise stopped
around TS=10; this is when the magnetic ﬂux movement
driven by the Lorentz force FL=Jc × B against the pinning
force Fp stopped. Therefore, the cooling power of the cryogenic refrigerator was stronger than the heat generation. As aresult, the maximum temperature rise was as small as 1.5 K
on the upper surface of the hat structure for FCM at
Bapp=10 T.
Figure 3(d) shows the temperature dependence of thetrapped ﬁeld Bz (z = r = 0 mm) at the centre of bulk-B after
FCM was performed at 50 K for each Bapp. The Bz versus T
curve for Bapp=7.3 T was lower than that of other Bapp atT<55 K; however, the curves above 55 K were identical foreach Bapp, which is closely reﬂected in the results ofﬁgure 3(b).

Table 1. Numerical parameters for the three types of Jc(B)characteristics for equation (1) used in the simulation.

Higher Jc
Best ﬁt Jc
Lower Jc

Jc1 (A m−2)

BL (T)

Jc2 (A m−2)

Bmax
(T)

k

2.3×109
2.3×109
2.3×109

0.8
0.8
0.8

2.0×109
1.57×109
1.3×109

4.5
4.5
4.5

1.0
1.0
1.0

Table 2. Mechanical parameters (Young’s modulus EY , Poisson ratio
ν, and average thermal expansion coefﬁcient αav) of the REBaCuO
bulk, epoxy resin, Al alloy (A7075-T6), In and Cu used in the
numerical simulation.

REBaCuO bulk
Epoxy resin
Al alloy (A7075-T6)
InCu

EY (GPa)naav (K−1)

100
3
78
12.7
125

0.33
0.37
0.34
0.45
0.34

6.80×10−6 [22]4.61×10−5
1.72×10−5
2.78×10−5
1.60×10−5

In the numerical FCM, the ring-shaped bulk was cooled
to 50 K in the magnetic ﬁeld of Bapp=7.3, 8.3, and 10 T; theexternal ﬁeld Bex was then decreased linearly to 0 T at
−0.222 T min−1 by 10 steps. The temperature variation during FCM was ignored for simplicity. The commercial FEM
software package Photo-Eddy (Photon Ltd, Japan) was
adapted for the analysis of the trapped ﬁeld Bz.
Elastic behaviour in an isotropic material can be explainedby Hooke’s law, in which the stress tensor sij is linearly proportional to the strain tensor eij. The detailed procedure of themechanical analysis used in this study has been shown elsewhere[12, 13]. The thermal stress under the cooling process sqcool from540 K to 50 K and the total hoop stress s qtotal = sqcool + s qFCM
after the cooling process and subsequent FCM at Bapp=7.3 T,
8.3 T, and 10 T were numerically simulated. The nodal force oneach node of the meshed element calculated by Photo-Eddy was
exported to the commercial software package Photo-ELAS(Photon Ltd, Japan) for analysis of the magnetic and thermal
stresses. The mechanical parameters (Young’s modulus EY,
Poisson ratio ν, and the average thermal contraction coefﬁcientαav) of the REBaCuO ring bulk, Al alloy ring (A7075-T6),
epoxy resin, In and Cu used in the elastic simulation aresummarised in table 2. The αav value of each material
was deﬁned as the average between 300 and 50 K,
aav = (L 300 - L 50 ) /L 300 / (300 - 50), where L 300 and L 50
are the lengths of the material at 300 K and 50 K, respectively.
The mechanical parameters of the REBaCuO bulk were assumed
to be isotropic and homogeneous for simplicity, and typicalvalues of the ab-plane were used, in which the αav value wasmeasured [22].

5. Numerical simulation results for double Al ring
reinforcement
5.1. Determination of average Jc(B) for bulk-B

4. Experimental results

Electromagnetic and mechanical properties strongly dependon the Jc(B, T) characteristics of the superconducting bulk.
To precisely estimate the hoop stress in the ring bulk
during FCM using simulation, it is necessary to determine the

Figure 3(a) shows the TS dependence of the trapped ﬁeld at
the centre of the ring bulk Bz (z=r=0 mm) for bulk-A
5

Supercond. Sci. Technol. 32 (2019) 065001

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Figure 3. (a) TS dependence of the trapped ﬁeld Bz(z=r=0) at the centre of (a) bulk-A without the hat structure and (b) bulk-B with the hat

structure during FCM at various Bapp at 50 K. (c) TS dependence of the temperature, T, on the hat structure above the bulk-B during FCM atvarious Bapp at 50 K. (d) Temperature dependence of the trapped ﬁeld, Bz(z=r=0 mm), at the centre of the bulk-B, after the FCM
procedure was performed at 50 K for each Bapp.

Jc(B, T) characteristics of the bulk; this is because the stress is
closely related to the Lorentz force FL. In a previous simulation paper [23], the Jc(B) characteristics, which were estimated by the magnetic moment hysteresis (M–H) loop using asmall piece cut from the bulk, were typically used. Bean’s
critical state model [24] or the Kim model [25] were also
used. Since the Jc(B) determined by the M–H loop occasionally overestimates the trapped ﬁeld by FCM, Jc(B) had tobe adjusted to one third of its small specimen value to adequately reproduce the FCM results [21]. In this study, a new
method has been proposed to estimate the average Jc(B)characteristics of the bulk, in which the TS dependence ofBz(z=r=0 mm) measured experimentally was ﬁtted using
the Jc(B) characteristics shown in equation (1).
Figure 4(a) shows the numerical simulation results of TS
dependence of Bz(z=r=0 mm) during FCM at Bapp=10 T
at 50 K for the three Jc(B) curves shown in table 1; these results
were compared to the experimental result. The three Jc(B) curves
are shown in the inset. It was found that the numerical result

using the best ﬁt Jc well reproduced the experimental curve,
compared to the cases for lower and higher Jc. Figure 4(b)shows the numerical simulation results of the TS dependence ofBz(z=r=0 mm) during FCM at Bapp=7.3, 8.3, and 10 T at
50 K, in which the best ﬁt Jc was used; these results were also
compared with the experimental results. The parameters used in
the best ﬁt Jc reproduced the experimental curves for FCM withdifferent Bapp values. This suggests that the average Jc(B)characteristics can be estimated using this method, which produces reasonable values for a typical REBaCuO bulk at 50 K
[21, 23], even though the Jc(B) characteristics cannot be determined uniquely. In a later section, the numerical simulation
results of the hoop stress during FCM are estimated using thebest ﬁt Jc characteristics.
5.2. Shrink-fit effect of the outer Al alloy ring from 540 K
to 300 K

In this subsection, the mechanical reinforcement effect of adouble ring structure was investigated using numerical6

Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

Figure 4. (a) The numerical results of TS dependence of Bz(z=r=0 mm) during FCM at Bapp=10 T at 50 K for three Jc(B) curves shownin table 1, compared to that the experimental result which was recited from ﬁgure 3(b). Three Jc(B) curves at 50 K are shown in the inset.
(b) The numerical results of the TS dependence of Bz(z=r=0 mm) during FCM at Bapp=7.3 T, 8.3 T and 10 T at 50 K using the best ﬁt
Jc, compared to the experimental results.

5.3. Overall shrink-fit effect from 540 K to 50 K

The ring bulk reinforced by the double Al alloy ring was set
on the cold stage with and without the hat structure at 300 K
and then cooled to 50 K. Figures 6(a) and (b) show thesimulation results of the radius (r) dependence of the total
thermal hoop stress σcool
for the ring bulk with and withoutθthe hat structure, respectively, at the bulk centre (z=0 mm)
and the bulk surface (z=±10 mm); the outer ring wascooled from 540 K to 50 K, and the inner ring was cooled
from 300 K to 50 K. For the case with the hat structure asshown in ﬁgure 6(a), a large compressive stress was applied
over the bulk area; the compressive stress gradually decreased
from the bottom surface to the top surface. For the casewithout the hat structure as shown in ﬁgure 6(b), a larger
tensile stress was applied at the top surface (z=+10 mm),compared to that of the shrink-ﬁt from 540 K to 300 K shownin ﬁgure 5. At the bulk centre (z=0 mm) and the bottom
surface (z=−10 mm), a large compressive stress wasapplied during the overall shrink ﬁtting process from 540 K to50 K. The inhomogeneous thermal stress for the double alloy
ring reinforcement without the hat structure, which introduced
a shear stress to the ring bulk, might affect the bulk failure
only during cooling. If this shear stress were applied in thebulk, the bulk could peel along the c-direction; this is because
the fracture strength along the c-axis is about one order ofmagnitude smaller than that of the ab-plane [26, 27].

cool
Figure 5. The radius (r) dependence of the thermal hoop stress, σθ ,at each z-position for the ring bulk reinforced by the double Al alloy
ring, in which the outer Al ring was shrink-ﬁtted from 540 K
to 300 K.

simulation. Figure 5 shows the radius (r) dependence of thethermal hoop stress σcool
at the bulk centre (z=0 mm) and theθbulk surface (z=±10 mm) for ring bulk reinforced by a double
Al alloy ring; the outer Al alloy ring (Al alloy ring 2) was heated
to 540 K, ﬁtted to the inner Al alloy ring (Al alloy ring 1),
and then cooled to 300 K. For the bulk centre (z=0 mm),a compressive stress greater than −100 MPa was applied to
the ring bulk (20 mm<r<32 mm); for the bulk surface (z=
±10 mm), a tensile stress of up to +75 MPa was applied to theouter region of the ring bulk because of the larger thermal
contraction of the Al alloy ring along the z-direction as well as
the r-direction [12]. These results suggest that the compressive
stress is enhanced by the double ring structure at the bulk centre(z=0 mm), but is weakened at the bulk surface; this is compared to ring bulk reinforced by a single ring, where the stress is
not applied to the ring bulk at 300 K.

5.4. Total hoop stress σtotal
during the cooling process andθ
FCM (Bapp=8.3 T)

Figure 7 shows the TS dependence of the cross-sectional
FCM
proﬁles of total hoop stress σtotal(= σcool) at the (a)θθ +σθ
top surface (z=+10 mm) and (b) bulk centre (z=0 mm) of
the ring bulk reinforced by the double Al alloy ring with the
hat structure during FCM at Bapp=8.3 T. Similar plots ofσtotalat the (c) top surface (z=+10 mm) and (d) bulk centreθ7Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

Figure 6. The radius (r) dependence of the total thermal hoop stress, σcoolθ , for the ring bulk reinforced by double Al ring (a) with and (b)without the hat structure at the bulk centre (z=0 mm) and bulk surface (z=±10 mm), in which the outer Al alloy ring was cooled from540 K to 50 K and then the bulk module was cooled from 300 K to 50 K.

(z=0 mm) of the ring bulk reinforced by the double Al alloy
ring without the hat structure are also shown. In each case, theproﬁle at the 0 step was recited from ﬁgure 6. The σtotal
σtotalθθvalue at each position within the ring bulk region gradually
increased along the positive (tensile) direction with increasing
TS due to the increase in the electromagnetic hoop stress; thenslightly decreased with further increase of TS. A disσtotal
θ
proﬁle between the case withtinctive difference in the σtotal
θand without the hat structure was seen at the bulk surface(z=+10 mm). For the case without the hat structure asvalue was +160 MPa at
shown in ﬁgure 7(c), the σtotalθr=20 mm and +195 MPa at r=31 mm for TS=6, which
was fairly larger than the fracture strength (50–70 MPa) of atypical Ag-doped REBaCuO bulk [26–28]. Therefore, thering-shaped bulk might break, even if the ring bulk were
reinforced with the double Al alloy ring. In the experimentwith bulk-A, as shown in ﬁgure 3(a), bulk-A actually broke
after TS=4 of FCM at Bapp=8.8 T. For the case with thewas
hat structure shown in ﬁgure 7(a), the maximum σtotalθ−100 MPa (compressive) on the bulk surface, which suggeststhat the ring bulk might not break, as shown in ﬁgure 3(b). At
the bulk centre (z=0 mm), as shown in ﬁgures 7(b) and (d),the compressive stress could be safely applied to the ringshaped bulk with and without the hat structure during thecooling process and FCM.
The hat structure may be similar to a Yoroi-coil structure,
which has been proposed for reinforcement of the pancake
coil wound by REBaCuO tape on the Hastelloy substrate, sothat it may withstand the electromagnetic hoop stress [29, 30].
The outer plates made of glass ﬁbre reinforced plastics(GFRP) or carbon ﬁbre reinforced plastics (CFRP) in the
Yoroi-coil structure exist on the pancake coil. However, there
is no shrink-ﬁt effect in the reinforcing outer plates, because
the thermal expansion coefﬁcient of GFRP and CFRP is muchsmaller than that of the Hastelloy substrate. The hat structure
especially demonstrates this important effect during thecooling process by reducing the inhomogeneous compressive
stress at the bulk surface. In addition, the reinforcement effectfor the electromagnetic hoop stress is also enhanced by the hat
structure.
Figures 8(a) and (b) summarise the TS dependence ofat typical positions of the ring bulk with and without theσtotalθ
hat structure, respectively, during FCM at Bapp=8.3 T. The
σtotalvalue at each position increased with increasing TS,θreached a maximum at around TS=6, and then decreased
with the increase in TS. The maximum σtotal
value was
θ
observed at the outer and top surface (z=10 mm,r=31 mm), which was +28 MPa for the ring bulk with the
hat structure and +200 MPa for the ring bulk without the hat
structure. Therefore, the fracture of bulk-A during FCM at
Bapp=8.8 T as shown in ﬁgure 3(a) can be understoodqualitatively from these simulation results; however, σtotalofθ
bulk-A may be slightly smaller than that of bulk-B because of
the smaller height of bulk-A (ht=13 mm).

6. Numerical simulation results of ring bulk
reinforced by a single Al alloy ring
6.1. Shrink-fit effect from 300 K to 50 K

In section 4, the ring bulk reinforced by the double Al alloy
ring and the hat structure was successfully magnetised at
Bapp=10 T, implying that both the double ring and hat
structure might simultaneously contribute the success. To
analyse the double ring effect independently, the hoop stresses during the cooling process and FCM were numericallyinvestigated for ring bulk reinforced by a single Al alloy ring,as shown in ﬁgure 2(d).
Figure 9 shows the simulation results of the radius (r)dependence of the thermal hoop stress σcool
at the bulk centreθ(z=0 mm) and the top and bottom surfaces of the bulk(z=±10 mm) for ring bulk reinforced by a single Al alloy
ring with the hat structure; it was then cooled from 300 to
50 K. The compressive stress in the ring bulk region was
reduced relative to that in the case of the double Al alloy ring,as shown in ﬁgure 6(a). For example, at the position (z=08Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

FCM
Figure 7. Time step (TS) dependence of the cross-sectional proﬁles of total hoop stress, σtotal(= σcool), at (a) top surfaceθθ +σθ(z=+10 mm) and (b) bulk centre (z=0 mm) of the ring bulk with the hat structure during FCM at Bapp=8.3 T. The similar proﬁles of
σtotalat (c) top surface (z=+10 mm) and (d) bulk centre (z=0 mm) of the ring bulk without the hat structure during FCM at Bapp=8.3 Tθare also shown.

Figure 8. TS dependence of the σtotal
value at the typical positions of the ring bulk (a) with and (b) without the hat structure during FCM atθBapp=8.3 T.

9Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

Figure 9. The simulation results of the radius (r) dependence of thethermal hoop stress, σcoolθ , for the ring bulk reinforced by single Al
alloy ring with the hat structure at the bulk centre (z=0 mm) andthe top and bottom surfaces of the bulk (z=±10 mm) duringcooling from 300 K to 50 K.

Figure 10. TS dependence of the cross-sectional plots of total hoop

FCM
stress, σtotal(= σcool), at the top surface (z=10 mm) of theθθ +σθring bulk reinforced by single Al alloy ring with the hat structureduring FCM at Bapp=8.3 T.

and r=25 mm), the compressive σcool
was slightly reduced
θfrom −250 MPa for the double ring to −180 MPa for the
single ring. However, a large compressive thermal stress was
still applied throughout the ring bulk region during thecooling process by the hat structure, even when the single Al
alloy ring was used.

6.2. Total hoop stress σtotal
during the cooling process andθ
FCM (Bapp=8.3 T)

Figure 10 shows the TS dependence of the cross-sectional
FCM
proﬁles of total hoop stress σtotal(= σcool) at the topθθ +σθsurface (z=+10 mm) of the ring bulk reinforced by a single
Al alloy ring with the hat structure during FCM at
Bapp=8.3 T. The σtotal
in the ring bulk was the compressiveθstress and slightly changed along the positive (tensile)
direction with increasing TS due to the increase in theelectromagnetic hoop stress. Similar σtotal
behaviour with theθdouble Al alloy ring can be seen in ﬁgure 7(a). These results
suggest that the double Al alloy ring has a limited effect andthat the hat structure is the most effective for applying thecompressive stress needed to avoid fracture during FCM.
Figure 11 shows the TS dependence of σtotal
at typicalθ
positions of the ring bulk reinforced by a single Al alloy ringwith the hat structure during FCM at Bapp=8.3 T, which was
extracted from ﬁgure 10. The σtotal
value at each position
θ
increased with increasing TS, reached a maximum atapproximately TS=6∼7, and then decreased withincreasing TS. A maximum σtotalof +28 MPa was seen at theθinner top ring bulk surface (z=10 mm, r=20 mm), which
was slightly larger than that of the bulk reinforced by the
double Al alloy ring, as shown in ﬁgure 8(a). The maximum
σtotalwas not determined by the structure of the Al alloy ring,θ
but was primarily governed by the existence of the hat
structure.

total
Figure 11. TS dependence of the σθ value at the typical positionsof the ring bulk reinforced by single Al alloy ring with the hat
structure during FCM at Bapp=8.3 T.

7. Optimal reinforcement structure for ring bulk
In the ﬁnal section, we summarise the mechanical reinforcement effects of the double and single Al alloy rings with the
hat structure during FCM, as shown in sections 5 and 6. Wealso overview the optimal and practical reinforcement structure for REBaCuO ring-shaped bulk, including our previous
investigations.
Figures 12(a) and (b) show the results of the numericalsimulation of the maximum σtotal
value, σtotalθθ (max), at typical
positions in the ring bulk with and without the hat structure,
respectively, as a function of Bapp during FCM. The results
for the ring bulk reinforced by only the double or single Al
alloy ring are also shown. The σtotal
θ (max) value at eachposition linearly increased with increasing Bapp for all cases.
In the experimental setup similar to the conditions shown inﬁgure 12(a), a σtotal
θ (max) of +50 MPa was applied at the
periphery of the bulk surface (z=10 mm, r=31 mm) at
10Supercond. Sci. Technol. 32 (2019) 065001

H Fujishiro et al

Figure 12. The numerical simulation results of the σtotal
θ (max) value at the typical positions in the ring bulk reinforced by the double andsingle Al alloy ring (a) with and (b) without the hat structure, as a function of the applied magnetic ﬁeld, Bapp, during FCM.

Bapp=10 T, which is smaller than the fracture strength of atypical Ag-doped REBaCuO bulk [26–28]. The σtotal
θ (max) atother positions was a large negative (compressive) or slightly
positive (tensile) value. For the case using a single Al alloy
ring with the hat structure, a σtotal
θ (max) of +65 MPa wasapplied at the innermost position of the bulk surface(z=10 mm, r=20 mm) at Bapp=10 T, which was slightly
larger than that for the double Al alloy ring case. These results
suggest that for both double and single Al alloy ring reinforcement with the hat structure, the ring bulk would notbreak at a Bapp up to 10 T.
As for single and double Al alloy ring reinforcement
without the hat structure as shown in ﬁgure 12(b), σtotal
θ (max)exceeded +200 MPa at Bapp=10 T; this suggests that theσtotalθ (max) value was larger than the fracture strength[26–28]. These results indicate that the hat structure effectively reinforced the ring bulk, but the mechanical reinforcement effect of the double Al alloy ring was limited, evenwhen the inner surface of the hat structure did not makecontact with the outer surface of the double Al alloy ring after
the cooling process, as explained in the section 3. If the hat
structure contacts the double Al alloy ring after the coolingprocess, the width of the Al alloy ring would effectively
increase by 5 mm, thus increasing width reinforcement effectof the Al alloy ring. Combining the Al alloy ring width
reinforcement effect with the reinforcement effect of the hat
structure [14], the compressive stress becomes large and thering bulk can be magnetised more safely. Therefore, the ringbulk reinforced by a double Al alloy ring and hat structureand set on the cold stage of a cryogenic refrigerator can becooled to 50 K and magnetised under a magnetic ﬁeld up to
10 T without fracture; this correlates reasonably well with theexperimental results.
We have determined the optimal reinforcement structurethat can prevent mechanical fracture for REBaCuO ring bulk
during FCM with an applied magnetic ﬁeld up to 10 T. We
have also investigated the mechanical reinforcement effect ofa metal ring on REBaCuO ring bulk during FCM. The reinforcement enhancement of increasing the width of the metalring was found to be effective [14]. Although SUS316 is
better than Al alloy as a reinforcement ring material due tolarger Young’s modulus [14], the non-magnetic Al alloy(A7075-T6) ring must be used for the NMR bulk magnet. The
hat structure was found to be effective, as shown in this study
and in [15]; however, the double ring effect was limited. The
hoop stress and reinforcement effect also change dependingon the ID, OD and ht of the ring bulk with identical Jc(B)[31]. Therefore, we can conclude that the proposed hat
structure is the most reliable and effective reinforcement
structure to realise a 400 MHz (9.4 T) NMR bulk magnet
system.

8. Conclusion
We have experimentally and numerically investigated a newreinforcement hat structure for the ring-shaped REBaCuO
bulk superconductor, which was shrink-ﬁtted by a double Al
alloy ring and set on the cold stage of a cryogenic refrigerator
prior to FCM. The important results and conclusions in thisstudy are summarised as follows.
(1) In the experiments, the ring bulk (64 mm OD, 40 mm
ID and 20 mm ht) with the hat structure trapped amagnetic ﬁeld as high as 6.8 T at 50 K during FCM at
Bapp=10 T without fracture; a similar ring bulk
(64 mm OD, 32 mm ID and 13 mm ht) without the
hat structure broke during FCM at Bapp=8.8 T.
(2) The magnetic ﬁeld dependence of the average criticalcurrent density Jc(B) was experimentally determined by
the results of TS dependence of the trapped ﬁeld Bz at
the centre of the ring bulk; this is one of the practicalmethods for estimating Jc(B) for the numerical11Supercond. Sci. Technol. 32 (2019) 065001

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simulation of the mechanical properties of the bulk.
Using numerical simulation, the total hoop stress σtotalθFCM(= σcool) in the bulk reinforced by the doubleθ +σθAl alloy ring and hat structure was analysed during
FCM with a Bapp of up to 10 T after cooling to 50 K; themaximum σtotal
was lower than the typical fractureθstrength of the REBaCuO bulk (50–70 MPa). Therefore, the results of the simulation verify the experimental FCM results at Bapp=10 T.
(3) The effect of the hat structure on mechanical reinforcement was veriﬁed using numerical simulation andcompared with the case without the hat structure, inwhich the thermal compressive stress σcool
provides theθgreatest contribution for preventing mechanicalfracture.
(4) The effect of the double Al alloy ring reinforcement
was compared with that of the conventional single Al
alloy ring using numerical simulation. The double ringeffect was limited, and the reinforcement method withthe hat structure was fairly effective in reducing thehoop stress for the ring bulk during FCM at
Bapp=10 T without fracture. These results suggestthat it is possible to realise a 400 MHz (9.4 T) NMR
bulk magnet system.

Acknowledgments
This research is partially supported by Development of Systems
and Technologies for Advanced Measurement and Analysis
from the Japan Agency for Medical Research and Development(AMED) and by JSPS KAKENHI Grant No. JP15K04646.
The authors would also like to thank Dr Mark D Ainslie of the
University of Cambridge for his valuable suggestions about thenumerical simulation of the mechanical properties.

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

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