Superconductor Science and Technology
Supercond. Sci. Technol. 30 (2017) 085008 (12pp)

https://doi.org/10.1088/1361-6668/aa76a2

Simulation studies of mechanical stresses in
REBaCuO superconducting ring bulks with
inﬁnite and ﬁnite height reinforced by metalring during ﬁeld-cooled magnetization
H Fujishiro1,5, M D Ainslie2 , K Takahashi1, T Naito1, Y Yanagi3, Y Itoh3 andT Nakamura4
1

Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, 4-3-5 Ueda, Morioka 020-8551, Japan2
Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Trumpington
Street, Cambridge CB2 1PZ, United Kingdom
3
IMRA Material R&D Co., Ltd, 2-1 Asahi-machi, Kariya 448-0032, Japan
4
RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan
E-mail: fujishiro@iwate-u.ac.jp
Received 7 April 2017, revised 19 May 2017
Accepted for publication 2 June 2017
Published 10 July 2017
Abstract

We have performed the numerical simulation of mechanical stresses (hoop stress, σθ and radial stress,σr) in REBaCuO ring bulks with an inﬁnite and ﬁnite height reinforced by metal (aluminum alloy orstainless steel) ring during ﬁeld-cooled magnetization (FCM) using a solenoid coil with an inﬁnite andﬁnite height. The superconducting characteristics of the bulk material were assumed to follow Bean’s
critical state model. The electromagnetic hoop stress, sq FCM, of the ﬁnite height ring bulk during FCM
using the inﬁnite coil was larger than that for the inﬁnite ring bulk, and the time step dependence ofsq FCM was clearly different in each case. The sq FCM value was reduced by the reinforcement by themetal ring, and the stainless steel ring was more effective than the aluminum alloy ring. The sq FCM
value of the ﬁnite ring bulk magnetized using the ﬁnite coil was slightly reduced, compared tomagnetization using the inﬁnite coil. The thermal hoop stress, sq cool, which occurs in the ring bulk
when cooling down to operating temperature due to the difference of thermal contraction coefﬁcient
between ring bulk and metal ring, was also estimated. The compressive, sq cool , was reduced
comparatively at the uppermost surface of the ring bulk because of the larger thermal contraction of
the metal ring along the axial direction. The actual total hoop stress, sq (= sq FCM + sq cool ) was
analyzed for the ﬁnite ring bulk reinforced by the metal ring during FCM and the possibility ofmechanical fracture due to this hoop stress is also discussed.
Keywords: mechanical properties, ﬁeld-cooled magnetization, REBaCuO bulk, metal ring
reinforcement, hoop stress, trapped ﬁeld
(Some ﬁgures may appear in colour only in the online journal)or Y) and MgB2 is to realize a strong trapped ﬁeld magnet(TFM), which continues to attract increasing attention
because of improvements in superconducting material properties and sample size. When the superconducting bulk ismagnetized by ﬁeld-cooled magnetization (FCM), themagnetic ﬂux is trapped at pinning centers and an induced

1. Introduction
One of the most promising practical applications of superconducting bulks such as REBaCuO (RE: rare earth element
5

Author to whom any correspondence should be addressed.

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Supercond. Sci. Technol. 30 (2017) 085008

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persistent current ﬂows concentrically inside the bulk, in which acurrent-ﬁeld interaction, that is, a Lorentz force exists in the bulk.
At present, REBaCuO disk bulks can trap magnetic ﬁelds over
17 T by the mechanical reinforcement using glass ﬁber reinforced
epoxy resin or shrink-ﬁt steel to reduce the electromagnetic hoop
stress [1, 2]. At such high ﬁelds, the performance of the bulk
superconductor as a magnet is limited not by its superconducting
properties, but its mechanical properties [1]. Recently, annular
REBaCuO bulks with a larger bore have been used for a compact
and cryogen-free nuclear magnetic resonance (NMR) spectrometer and a magnetic resonance imaging (MRI) apparatus, inwhich a uniform trapped ﬁeld as high as 4.7 T can be achieved in
the hollow space of the cylindrical bulks [3–7]. To enhance theresolution of the NMR signal, it is necessary to achieve a higher
trapped ﬁeld in the REBaCuO ring bulk. During the magnetizing
process for superconducting bulks, the mechanical strength of thebulk material must be considered to avoid fracture, because the
REBaCuO bulk is a brittle ceramic material and has a low
mechanical strength of 30–100 MPa, which is strongly related to
the fabrication process [8, 9]. The addition of the Ag and the
reduction of voids in the bulk during the growth process are also
effective to increase in the mechanical strength [10]. The ringbulk, in particular, suffers from higher mechanical stress than thedisk bulk because of the concentration of the hoop stress at the
innermost periphery [11].
Ren et al reported the experimental results and analyticalsolutions of the mechanical stresses of the TFM using the
expanded theory for the inﬁnite bulk cylinder [12]. Johansen et alpresented analytical solutions of the ﬂux-pinning-induced stressand strain distributions in the inﬁnite bulk cylinder and ring and athin disk, including the magnetostriction during FCM and zeroFCM [13–15]. These papers were based on Bean’s critical statemodel for a c-axis oriented, inﬁnite superconducting cylinder, inwhich the critical current density, Jc, is independent of themagnetic ﬁeld. We have performed numerical simulations of thehoop and radial stresses during the pulsed ﬁeld magnetization(PFM) of a GdBaCuO ring bulk reinforced by an aluminum alloy
ring using a ﬁnite element method (FEM) [11], in which theheight of the ring bulk was assumed to be inﬁnite and Bean’s
model was assumed. The compressive and tensile hoop stresses
were, respectively, concentrated at the innermost peripheral edgeof the inﬁnite ring bulk during the ascending and descending
stages of the PFM process, together with the compressive stressdue to the thermal contraction by cooling. Huang et al investigated the magnetostriction of superconducting cylinders andrings with ﬁnite height by FEM [16], in which Bean’s model anda uniform applied ﬁeld using an inﬁnite magnetizing solenoid
coil were assumed. They concluded that the mechanical responseof the ﬁnite height bulk is similar to that of an inﬁnitely longbulk, but there is also a difference due to boundary anddemagnetization effects. However, there has been no investigation of mechanical stresses for the ﬁnite ring bulk magnetizedby a solenoid coil of inﬁnite and ﬁnite height in previous studies,and no comparison has been made to the inﬁnite height bulk.
Furthermore, there have been no numerical simulations of thereinforcement effect of the metal ring ﬁtting on the mechanicalstresses in superconducting bulk rings and disks during FCM andcooling, except for the relevant work by Johansen et al [17].
In this paper, we perform numerical simulations of theelectromagnetic stresses (hoop stress, sq FCM, and radial stress,
sr FCM ), together with the superconducting characteristics(trapped ﬁeld, Bz, and induced persistent current density, Jθ)in the REBaCuO ring bulk of inﬁnite and ﬁnite height, andreinforced by an aluminum alloy ring, during FCM using
inﬁnite and ﬁnite solenoid magnetizing coils. The inﬂuencesof the bulk height and the coil height on the mechanicalstresses are investigated. We also investigate the reinforcement effect of a stainless steel ring on the hoop stress, which
is a typical material for mechanical reinforcement [1]. The
relationship between the mechanical stresses and the bulk
fracture during the FCM process is discussed, under thethermal compressive stress, sq cool, which occurs in the ringbulk when cooling, due to the difference of thermal contraction coefﬁcient between ring bulk and metal ring.

2. Numerical simulation framework
Based on our experimental setup for the FCM procedure [18],we prepared the following framework of three-dimensional
numerical simulation. Schematic views of three types ofnumerical models are presented in ﬁgure 1. The ring bulk
with inﬁnite height (64 mm outer diameter (O.D.) and 40 mm
inner diameter (I.D.)) reinforced by the metal ring (aluminumalloy) of 5 mm in thickness was magnetized by FCM using an
inﬁnite solenoid coil (150 mm O.D., 100 mm I.D. and inﬁnite
height), as shown in ﬁgure 1(a). We abbreviate this setup as‘case A’. The REBaCuO ring bulk with a ﬁnite height of
20 mm was mounted in the metal ring with the same height as
the ring bulk, as shown in ﬁgure 1(b), which was magnetizedby the inﬁnite solenoid coil. We abbreviate this setup as ‘case
B’. The same ﬁnite ring bulk with the metal (aluminum alloy
or stainless steel) ring was magnetized by a ﬁnite solenoid
coil (150 mm O.D., 100 mm I.D. and 100 mm height), as
shown in ﬁgure 1(c). We abbreviate this setup as ‘case C’.
Physical phenomena in the simulation during FCM aredescribed using electromagnetic equation [19–21], in which
the temperature of the REBaCuO bulk is assumed to beconstant at Ts = 50 K during FCM for simplicity, because theactual temperature rise during FCM from 6.3 T was as low as∼2 K under a descending speed of 0.1 T min−1 [22]. The
power-n model (n = 100) was used to describe the nonlinear
E–J characteristic of the superconducting bulk;
n

( ),
J

E = Ec J(1 )cwhere Ec (=10−4 V m−1) is the reference electric ﬁeld andJc (=4.8 × 108 A m−2) is the critical current density under zeroﬁeld, which is a typical Jc value at Ts = 50 K for the REBaCuO
bulk [7]. Jc is assumed to be independent of applied magnetic
ﬁeld [7]. For an ‘inﬁnite disk bulk’ with the same O.D., not the‘inﬁnite ring bulk’, the fully magnetized trapped ﬁeld is estimatedto be Bz* = μ0JcR = 19.30 T (R: radius of the disk bulk) by2

Supercond. Sci. Technol. 30 (2017) 085008

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Figure 1. Three types of models for the numerical simulation during FCM for the (a) inﬁnite ring bulk magnetized by solenoid coil with

inﬁnite height (‘case A’), (b) ﬁnite ring bulk magnetized by solenoid coil with inﬁnite height (‘case B’) and (c) ﬁnite ring bulk magnetized bysolenoid coil with ﬁnite height (‘case C’). For each case, the ring bulk was reinforced by a metal ring 5 mm in thickness and with the sameheight as the ring bulk. The dimensions of the ring bulk, metal ring and the magnetizing coil are also shown. (d) Time step of the reducing
applied ﬁeld during FCM from Bex, in which the applied ﬁeld is linearly decreased to zero in ten steps.

tensor σij and strain tensor, eij, in the following,

Bean’s model. We conﬁrmed that the larger n value than 80 andconstant Jc values realized the approximate Bean’s model. In thesimulation process of FCM, a magnetic ﬁeld of Bex = 4.7 T or
9.4 T was applied above the critical temperature, Tc (= 92 K), andthen the ring bulk was cooled to Ts = 50 K. Thereafter, themagnitude of the magnetic ﬁeld was monotonically decreased
from Bex to zero in ten steps, as shown in ﬁgure 1(d). Note thatthe magnetic ﬁeld of Bex = 4.7 T and 9.4 T corresponds to 1H
resonance frequency of 200 and 400 MHz in the NMR spectrometer, respectively.
In an isotropic material, Hooke’s law is established, inwhich the stress is linearly proportional to the strain. For
simplicity, the REBaCuO bulk is assumed to be isotropic
and homogeneous. The relation is expressed using the stresssij = l ⋅ ekk ⋅ dij + 2G ⋅ eij ,E⋅nl = (1 + n )(1-2n ) ,E

G = 2(1 + n ) ,

(2 )
(3 )
(4 )

where λ and G represent the Lame’s constants, δij is the
Kronecker delta function, E is the Young’s modulus, and ν
is the Poisson ratio. The electromagnetic hoop stress, sq FCM,
and the electromagnetic radial stress, sr FCM, were calculated
for each step of FCM based on the axisymmetric coordinate
system, in which sq FCM acts circumferentially on each end

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superconducting bulk, despite using the inﬁnite solenoid coil.
In the ring bulk region (20  r  32 mm), the induced persistent current ﬂows only in the periphery of the ring bulk at
the early step up to the 6th step as shown in ﬁgure 2(d). The
current ﬂowing region moves into the inside and the persistentcurrent ﬂows fully in the ring bulk after the sixth step due tothe reduction of the external ﬁeld. As a result, the trapped
ﬁeld at the center in the hole (r = z = 0) decreases step bystep, and the ﬁnal trapped magnetic ﬁeld reaches 2.70 T. It
should be noted that the electromagnetic properties for theﬁnite ring bulk drastically change, compared to those for theinﬁnite ring bulk using the same inﬁnite magnetizing coil,which can be clearly observed for the ﬁrst time.
Figures 2(e) and (f) show similar plots of Bz and Jθ,respectively, at z = 0 for the ‘case C’ (ﬁnite bulk magnetizedby ﬁnite coil). The Bz and Jθ proﬁles are nearly the same as
those for the ‘case B’ shown in ﬁgures 2(c) and (d), except for
the initial Bz(r) proﬁle at step 0 with a slight gradient becauseof the use of the ﬁnite coil. The trapped ﬁeld decreases step bystep, and the ﬁnal magnetic ﬁeld at the center in the hole of
the ring bulk (r = z = 0) reaches 2.70 T, which was the sameas that for ‘case B’, because a persistent current with the sameJc ﬂows over the entire cross-section of the ring bulk (asshown in ﬁgure 2(f), which is the same result as ‘case B’shown in ﬁgure 2(d)). The Bz(r) proﬁle at the bulk surface(z = 9.9 mm) is also shown in ﬁgure 2(e), for which themagnitude of Bz at z = 9.9 mm is smaller than that at z = 0
because of the edge effect of the ﬁnite ring bulk. There is little
inﬂuence of the coil height on the electromagnetic properties
for the ﬁnite ring bulk, as observed when comparing
ﬁgures 2(c) and (e) and ﬁgures 2(d) and (f). Because thesolenoid coil height in ‘case C (100 mm)’ is much higher thanthe bulk height (20 mm).
Figure 3 shows the contour maps of the trapped ﬁeld, Bz,
in the ring bulk at the (a) fourth step for ‘case A’, (b) 2nd stepfor ‘case C,’ and (c) fourth step for ‘case C’. The contour
maps of the persistent current density, Jθ, for each case arealso shown in the lower panels from (d) to (f). The Bz proﬁleof the fourth step in ‘case A’ shown in ﬁgure 3(a), which is
the inﬁnite bulk magnetized by the inﬁnite coil, is axisymmetric along the z-direction due to the axisymmetric Jθ proﬁleshown in ﬁgure 3(d). On the other hand, the Bz proﬁle of the
2nd step in ‘case C’ shown in ﬁgure 3(b), which is ﬁnite bulk
magnetized by ﬁnite coil, is rounded around the surface of the
ﬁnite bulk, in which the persistent current similar to Jc ﬂows
even around the surface region as shown in the red region inﬁgure 3(e). The region of the persistent current increases withincreasing the time step, such as the fourth step as shown inﬁgure 3(f) and, as a result, the Bz proﬁle clearly becomesrounded at the fourth step shown in ﬁgure 3(c).
Figure 4 shows the time step dependence of the trapped
ﬁeld, Bz, at the center (r = z = 0) in the hole of the inﬁnite
and ﬁnite ring bulks during FCM from Bex = 4.7 or 9.4 T for
all three cases. For ‘case A’, the Bz value of the inﬁnite bulk
maintains 9.4 T up to the seventh step for Bex = 9.4 T, atwhich the induced persistent current with the same value to Jc
ﬂows in the whole part of the bulk, and then Bz decreases withincreasing time steps. The ﬁnal Bz value was 7.5 T. For

Table 1. Mechanical properties used in the numericalsimulation [11].

REBaCuO bulk
Al alloy ring
SUS304 ring

E (Pa)να (K−1)1.0e11
7.8e101.93e11

0.33
0.34
0.28

5.2e-6
1.48e-5
1.27e-5(E: Young’s modulus, ν: Poisson ratio, α: thermal
contraction coefﬁcient.)

face of the meshed element and sr FCM acts radially on the
inner and outer curved surfaces of the element [12, 13]. The
thermal hoop stress, sq cool, by cooling from 300 to 50 K was
also calculated, which was generated by the difference of thethermal contraction coefﬁcient, α, between the superconducting bulk and the metal ring material. The mechanicalparameters (E, ν, and thermal contraction coefﬁcient, α) of
the bulk and ring materials used in the elastic simulation aresummarized in table 1 [11]. Commercial software, PhotoEddy (Photon Ltd, Japan) was used for the analysis of the
distributions of magnetic ﬁeld, Bz, along z-direction and theinduced persistent current density, Jθ, along the circumferential direction during FCM. The sq FCM and sr FCM values
were calculated for each step by Photo-ELAS (Photon Ltd,
Japan) using the nodal force at each node calculated by
Photo-Eddy. In a half model (z0), the ring bulk withmetal ring was equally divided into 36 elements along thecircumferential (θ) direction and 17 elements along ther-direction and ten elements along the z-direction.

3. Results and discussion
3.1. Time step dependence of trapped field, Bz, and induced
current density, JθFigures 2(a) and (b), respectively, show the time step
dependence of the cross-section of the trapped ﬁeld along thez-direction, Bz, and the induced superconducting current
density in the θ-direction, Jθ, for the ‘case A’ (inﬁnite bulk
magnetized by inﬁnite coil). In ﬁgure 2(a), the external
magnetic ﬁeld decreased step by step without a ﬁeld slope dueto the inﬁnite length of magnetizing solenoid coil. In the ringbulk periphery, the penetrated magnetic ﬁeld monotonicallydecreases with increasing time step and has a constant ﬁeld
slope, which is equivalent to a constant critical current density, Jc, based on Bean’s model. The Jθ, which is equal to Jc,
ﬂows only in the outer region of the ring bulk (24  r 
32 mm) up to the ﬁnal step, as shown in ﬁgure 2(b), becauseof the lower applied ﬁeld, compared to Bz* that results in fullmagnetization. As a result, the magnetic ﬁeld, Bz, in the holeat r  20 mm is the same as the applied ﬁeld, Bex = 4.7 T.
Figures 2(c) and (d), respectively, show similar plots of
Bz and Jθ at z = 0 for ‘case B’ (ﬁnite bulk magnetized byinﬁnite coil). It should be noted that, in ﬁgure 2(c), the initial
Bz proﬁle at step 0 is independent of r because of the inﬁnite
magnetizing coil. The external magnetic ﬁeld decreased stepby step with a ﬁeld slope due to the existence of4Supercond. Sci. Technol. 30 (2017) 085008

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Figure 2. Time step dependence of the cross-sectional plots of the trapped ﬁeld, Bz(r), along the z-direction for (a) ‘case A’ (inﬁnite bulk

magnetized by inﬁnite coil), (c) ‘case B’ (ﬁnite bulk magnetized by inﬁnite coil), and (e) ‘case C’ (ﬁnite bulk magnetized by ﬁnite coil)during FCM from Bex = 4.7 T. The time step dependence of the cross-sectional plots of the induced superconducting current density, Jθ(r), inthe θ-direction for each case is shown in (b), (d) and (f).

reduces to the ﬁnal step to Bz = 2.75 T, at which the Jθ = Jc
ﬂows across the whole bulk cross-section. For Bex = 4.7 T, Bz
gradually decreased up to the sixth step and then decreased
more rapidly to the ﬁnal value of Bz = 2.74 T, which is nearly
the same as that for Bex = 9.4 T. It should be noted that thetime step dependence of the Bz value for the ﬁnite bulk can beobtained only by FEM, although the Bz value is analytically

Bex = 4.7 T in ‘case A’, the Bz value of the inﬁnite bulk
maintains 4.7 T up to the ﬁnal step. These results for ‘case A’
can be reproduced using analytical equations based on Bean’s
model [21]. On the other hand, for ‘case B’ and ‘case C’ forthe ﬁnite ring bulk at Bex = 9.4 T, Bz gradually decreased up
to the third step, at which the region of current ﬂow reaches
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Figure 3. Contour maps of the trapped ﬁeld, Bz, in the ring bulk at the (a) fourth step in ‘case A’, (b) second step in ‘case C’ and (b) fourth
step in ‘case C’ during FCM from 4.7 T. The cross sections of the persistent current density, Jθ, for each case are also shown in the lower
panels from (d) to (f), in which the persistent current similar to Jc (= 4.8 × 108 A m−2) ﬂows in the ‘red region’ perpendicular to the paper
space.

the electromagnetic hoop stress, sq FCM, and electromagnetic
radial stress, sr FCM, for inﬁnite ring bulk during FCM, inwhich Ren’s analytical equations for the inﬁnite disk bulk
[12] were modiﬁed for the inﬁnite ring-shaped bulk [23].
Figure 5 shows the comparison between the numericalsimulation results and analytical ones of the cross-sectionalplots of sq FCM and sr FCM for the ‘case A’ without any
reinforcement by metal ring at the ﬁnal step of FCM from Bex= 4.7 T. The obtained values by the present FEM method
showed an excellent agreement with those by the analyticalequations, which could give the exact solution.
Figures 6(a) and (b) present the time step dependences of
the cross-sectional plots of the electromagnetic hoop stress,
sq FCM, and the electromagnetic radial stress, sr FCM, respectively, in the inﬁnite bulk reinforced by an aluminum alloy ring
during FCM from Bex = 4.7 T for ‘case A’. In this subsection,
we assume that there is no thermal stress in the bulk. In
ﬁgure 6(a), the electromagnetic hoop stress, sq FCM, increases
with increasing FCM time step, takes a maximum of 23 MPa at
the innermost periphery edge of the ring bulk (r = 20 mm) at
the ﬁnal step. At each time step, the sq FCM value decreases
with increasing r, but steeply drops at the interface(r = 32 mm) between bulk and aluminum alloy ring, which
results from the smaller Young’s modulus, E, of the aluminumalloy ring, as shown in table 1. The time step and the rdependence of sr FCM shown in ﬁgure 6(b) are clearly different
to the sq FCM behaviors; the magnitude of sr FCM is fairly
smaller and the time step dependence is not monotonous.
Figures 6(c) and (d) show similar plots of sq FCM andFCM
sr, respectively, at z = 0 in the ﬁnite bulk reinforced by analuminum alloy ring for ‘case B’ (magnetized by inﬁnite

Figure 4. Time step dependence of the trapped ﬁeld, Bz, at the center(r = z = 0) of the hole for three cases during FCM from Bex= 4.7 and 9.4 T.

calculated for the inﬁnite bulk. In this way, the time step
dependence of the trapped ﬁeld and induced persistent current
density in the ﬁnite and inﬁnite ring bulks can be estimatedduring the FCM process using the FEM method.

3.2. Mechanical stresses during FCM
3.2.1. Electromagnetic hoop stress, σθ FCM, and radial stress,
σr FCM for the finite and infinite ring bulks without cooling. To

conﬁrm the reliability of the present numerical simulation forthe mechanical stresses, we calculated the r dependences of6Supercond. Sci. Technol. 30 (2017) 085008

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Figure 5. The comparison between the numerical simulation results and analytical ones of the cross-sectional plots of (a) electromagnetic

hoop stress, sq FCM, and (b) electromagnetic radial stress, sr FCM, for the inﬁnite ring bulk without any reinforcement, magnetized by inﬁnite
magnetizing coil (‘case A’) during FCM from Bex = 4.7 T.

solenoid coil). The time step dependence of sq FCM and sr FCM
is clearly different to ‘case A’. The sq FCM value increases
with increasing time steps, takes a maximum of 37 MPa at the
innermost periphery of the ring bulk at the sixth step, which isabout 1.6 times as large as the maximum value in ‘case A’,and then decreases with increasing time steps. This difference
results from the Lorentz force, F = Jθ × B, that exists over alarger region of the ﬁnite ring bulk, where the induced
persistent current, Jθ, ﬂows over a wider area of the ﬁnite ringbulk cross-section than for the inﬁnite ring bulk. The time
step dependence of the sr FCM behavior is also different to thatof the inﬁnite bulk in ‘case A’. It should be noted that themechanical stresses for the ﬁnite ring bulk drastically change,
compared to those for the inﬁnite ring bulk for the inﬁnite
magnetizing coil, which can be clearly observed for theﬁrst time.
Figures 6(e) and (f) show similar plots of sq FCM andFCM
sr, respectively, at z = 0 in the ﬁnite ring bulk reinforcedby an aluminum alloy ring for ‘case C’ (magnetized by theﬁnite solenoid coil). The time step dependence of the sq FCM
and sr FCM closely resembles to that for ‘case B’ with a veryslight increase in magnitude. These results suggest that theinﬂuence of the magnetizing coil height on the electromagnetic stresses for the ﬁnite ring bulk is not so large fromﬁgures 6(c)–(f).
Figures 7(a) and (b) show the cross-sectional plots of theelectromagnetic hoop stress, sq FCM, and the electromagnetic
radial stress, sr FCM, respectively, in the ﬁnite ring bulk
reinforced by an aluminum alloy ring at the sixth step during
FCM from Bex = 4.7 T for ‘case C’, at which the sq FCM takes
a maximum. The sq FCM is axisymmetric and is independent of
the z-position due to the uniform ﬂow of a persistent current
with the same Jc. On the other hand, sr FCM is axisymmetric,
but does depend on the z-position, which is about one order ofmagnitude smaller than sq FCM. These results suggest that thebulk fracture might start from innermost periphery of the ringbulk by means of the electromagnetic hoop stress and that theelectromagnetic radial stress has little inﬂuence on the bulk
fracture.

Figure 8(a) shows the time step dependence of the
maximum electromagnetic hoop stress, sq FCM (max), for the
ring bulks with (w/) and without (w/o) the aluminum alloy
ring during FCM for each case from Bex = 4.7 T. In all cases,
sq FCM (max) initially increases with increasing time step, takes
a maximum, and then decreases, and the location ofsq FCM (max) occurs at the innermost surface of the ring bulk.
However, the time step at which the sq FCM (max) value isachieved, and the maximum sq FCM (max) value change
depending on the height of the ring bulk; sq FCM (max) takes
a maximum at the tenth (ﬁnal) step for the inﬁnite bulk, but
for the ﬁnite bulk, it takes a maximum at the sixth step and islarger than that for the inﬁnite bulk. The sq FCM value
decreased about 20% by the reinforcement with the aluminumalloy ring in all cases, which results from the relative
compressive stress from the aluminum alloy ring, because thealuminum alloy ring restricts the expansion of the ring bulk
due to the Lorentz force during FCM.
Figure 8(b) shows the same plots during FCM for eachcase from Bex = 9.4 T. The maximum sq FCM value and thetime step at which this maximum takes place are different tothat of the 4.7 T case. The maximum sq FCM value for ‘caseC’, which takes place at the third (or fourth) step, increased to
100 MPa, which is larger than that for the inﬁnite ring bulk,
which takes a maximum at the eighth step. These results
suggest that the sq FCM (max) value for the inﬁnite ring bulk,
which can be calculated by analytical equations [23], might
be underestimated, compared to that of the actual ﬁnite ringbulk. Therefore, estimation of the hoop stress must be
performed for the actual ﬁnite bulk reinforced by the metalring using numerical simulations to avoid the mechanicalfracture of the bulk during FCM.
3.2.2. Influence of thermal contraction on the hoop stress in thering bulk reinforced by metal (aluminum alloy or SUS) ring. In

this subsection, we investigate the inﬂuence of thermal
contraction on the hoop stress, sq cool, in the ring bulk
reinforced by a metal ring. Figure 9(a) shows the radius (r)dependence of the thermal hoop stress, sq cool, in the ﬁnite
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Figure 6. Time step dependence of the cross-sectional plots of (a) electromagnetic hoop stress, sq FCM, and (b) electromagnetic radial stress,
sr FCM, for the inﬁnite ring bulk magnetized by inﬁnite coil (‘case A’) during FCM from Bex = 4.7 T. (c) and (d) show similar sq FCM andsr FCM plots for the ﬁnite ring bulk magnetized by the inﬁnite coil (‘case B’), and (e) and (f) show similar plots for the ﬁnite ring bulk
magnetized by the ﬁnite coil (‘case C’). In all cases, the ring bulk was reinforced by an aluminum alloy ring.

tensile stress (σθ > 0) is applied to the aluminum alloy ring
(32  r  37 mm), because the thermal contraction
coefﬁcient, α, of the aluminum alloy is about three times
larger than that of the superconducting bulk, as shown inring bulk reinforced by an aluminum alloy ring at various
z-positions under the cooling from 300 to 50 K without FCM.
In the ring bulk region (20  r  32 mm), a thermallycompressive stress (σθ < 0) is applied to the ring bulk, and a8Supercond. Sci. Technol. 30 (2017) 085008

H Fujishiro et alprocess and the FCM process. Figure 10(a) shows the radius(r) dependence of the maximum sq (= sq FCM + sq cool ) value
at the 4th step at various z-positions in the ﬁnite ring bulk
reinforced by an aluminum alloy ring, under both the FCM
process (sq FCM ) of Bex = 9.4 T (‘case C’) and the coolingprocess (sq cool ) from 300 to 50 K. The σθ value increased
positively in the ring bulk by the superposition, and was+50 MPa at r = 20 mm and +95 MPa at r = 31 mm and atz = 9.9 mm (near the top surface of the ring bulk). These
results suggest that the ring bulk might be broken by thelarger electromagnetic hoop stress during the actual FCM
process from 9.4 T. Figure 10(b) shows the contour map of
the maximum sq (= sq FCM + sq cool ) in the ﬁnite bulk at the
4th step from 9.4 T (‘case C’) under cooling from 300 to
50 K. The σθ distribution can be seen in the ring bulk and alarge positive hoop stress exists at the surface of the bulk
periphery.
Figure 11(a) shows the r dependence of the sq cool value
in the ﬁnite ring bulk reinforced by a SUS ring at various
z-positions under cooling from 300 to 50 K without FCM.
The maximum compressive sq cool value at the center (z = 0)
is −91 MPa at r = 20 mm and −140 MPa at r = 31 mm,
which is larger than that for the ring bulk reinforced by thealuminum alloy ring (ﬁgure 9(a)), and gradually decreases
with increasing z. The larger compressive sq cool value in thering bulk reinforced by the SUS ring results from the larger
Young’s modulus, E of the SUS ring, compared to that of thealuminum alloy ring, in spite of smaller thermal contraction
coefﬁcient, α, of the SUS ring. These results suggest that thering bulk could actually be broken during only cooling,
because the REBaCuO bulk has a lower mechanical strengthof +30 ∼ +100 MPa [8, 9]. Figure 11(b) presents the time
step dependence of the electromagnetic hoop stress, sq FCM,
for the ﬁnite ring bulk reinforced by the SUS ring at z = 0
during FCM from Bex = 9.4 T (‘case C’). The r dependenceof sq FCM is in clear contrast to that for the ring bulk reinforcedby an aluminum alloy ring shown in ﬁgure 6(e) because of therelative relationship of the Young’s modulus between the ringmaterials. The inset of ﬁgure 11(b) shows the time step
dependence of sq FCM (max) for the ﬁnite ring bulk with andwithout the SUS or aluminum alloy ring during the FCM
process from 9.4 T, in which the SUS ring restricts the
expansion of the ring bulk more effectively than thealuminum alloy ring because of larger Young’s modulus.
Figure 11(c) shows the r dependence of thesq (= sq FCM + sq cool ) value of the fourth step at various zpositions in the ﬁnite ring bulk reinforced by the SUS ringunder FCM process with Bex = 9.4 T (‘case C’) and combinedwith cooling from 300 to 50 K. A small positive (tensile) σθvalue can be observed only at the ring bulk surface(z = 9.9 mm), which suggests that the compressive effect of
the SUS ring is much more effective, compared to thealuminum alloy ring due to its larger E.
From the present numerical simulation shown inﬁgure 11(c), the actual hoop stress, sq (= sq FCM + sq cool ), of
the ring bulk reinforced by the SUS ring during FCM fromBex = 9.4 T was lower than the mechanical strength of
REBaCuO bulk [8, 9]. These results suggest that the ﬁnite

Figure 7. Contour maps of (a) electromagnetic hoop stress, sq FCM,

and (b) electromagnetic radial stress, sr FCM, at the sixth step during
FCM from Bex = 4.7 T in ‘case C’ (ﬁnite bulk magnetized by theﬁnite coil).

table 1. The compressive stress for the bulk at the center(z = 0) is −56 MPa at r = 20 mm and −115 MPa at
r = 31 mm, and gradually decreases with increasing z. It issurprising that the sq cool value of the ring bulk surface(z = 9.9 mm) changed to positive (tensile stress) at the bulk
periphery (r > 27 mm). Figure 9(b) shows the contour map of
the displacement, dz, along the z-direction in the ring bulk
during the cooling. Because of the larger α value of thealuminum alloy compared to the bulk, the aluminum alloy
ring shrinks along both z- and r-directions. The ring bulk was
fastened along the r-direction and, at the same time, theperiphery region of the ring bulk was pulled toward z = 0. As
a result, the compressive hoop stress around the bulk
periphery became weak and changes to positive. Figure 9(c)shows the contour map of the displacement, dr, along ther-direction in the ring bulk during the cooling. The magnitudeof the thermal shrink at the bulk surface (z = ±10 mm) islarger than that at the bulk center (z = 0), which suggests thatthe surface region of the ring bulk is expanded like a trumpet.
Note that there is no mechanical stress of thermal origin in thering bulk during cooling, if the ring bulk is not reinforced by ametal ring.
In the actual FCM process, both the thermal andmechanical stress must be considered from both the cooling9Supercond. Sci. Technol. 30 (2017) 085008

H Fujishiro et al

Figure 8. Time step dependence of the maximum electromagnetic hoop stress, sq FCM (max), in the ring bulk with (w/) and without (w/o) thealuminum alloy ring during FCM from (a) Bex = 4.7 T and (b) 9.4 T for all cases.

Figure 9. (a) The radius (r) dependence of the thermal hoop stress, sq cool, for the ﬁnite ring bulk reinforced by an aluminum alloy ring foreach z-position under cooling from 300 to 50 K. The contour maps of the (b) displacement, dz, along z-direction, and (c) dr, along r-direction
under cooling from 300 to 50 K are also shown.

addition, the bulk contains a signiﬁcant number of imperfections such as grain boundaries, crystalline defects, inclusions,voids, and so on. The actual destruction of the REBaCuO
bulk during FCM is likely to happen from the mechanicallyweak points in the bulk material such as voids and crystal
imperfections. These imperfections may cause the stressconcentration in local regions and may initiate small local
cracks. Additionally, micro-cracking along the ab-plane
formed during post-annealing treatment is a common

ring bulk would not break during the FCM process, even fromBex = 9.4 T. However, actual disk and ring bulks of ﬁnite
height reinforced by a SUS ring break frequently during FCM
under the magnetic ﬁelds of several Tesla. Since the
REBaCuO bulk superconductor is a composite 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 stress concentration is
likely to take place around the so called secondary phases. In
10Supercond. Sci. Technol. 30 (2017) 085008

H Fujishiro et al

Figure 10. (a) The r dependence of the total hoop stress, sq (= sq FCM + sq cool ), for the ﬁnite ring bulk reinforced by an aluminum alloy ring(‘case C’) at the fourth step of FCM from Bex = 9.4 T under the cooling from 300 K to 50 K. (b) The contour map of the hoop stress, σθ, atthe fourth step.

Figure 11. (a) The radius (r) dependence of the thermal hoop stress, sq cool, for the ﬁnite ring bulk reinforced by stainless steel (SUS) ring(‘case C’) under cooling from 300 to 50 K. (b) The r dependence of the electromagnetic hoop stress, sq FCM, for the ﬁnite ring bulk reinforced
by SUS ring (‘case C’) during FCM from Bex = 9.4 T without cooling. The inset shows the time step dependence of the sq FCM (max) for theﬁnite ring bulk reinforced by the SUS ring for Bex = 9.4 T, compared to those with and without Al alloy ring. (c) The r dependence of thehoop stress, sq (= sq FCM + sq cool ), for the ﬁnite ring bulk reinforced by SUS ring (‘case C’) during FCM from Bex = 9.4 T under the coolingfrom 300 to 50 K.

11

Supercond. Sci. Technol. 30 (2017) 085008

H Fujishiro et al

and voids, at which the mechanical stress is concentrated to
the weak points and the bulk breaks ﬁnally. We must
consider the numerical simulation of the mechanicalproperties of superconductors including such imperfectionsin the future.

ingredient in the microstructure. These weak points should
severely reduce the mechanical strength of the material, as
they are the starting points for further cracking under tensilestress. The tiniest hole in the bulk cylinder doubles the tensile
hoop stress [13]. As a result, the mechanical stress isconcentrated at the weak points, which has a lower
mechanical strength, and the bulk would ultimately break.
We must consider the numerical simulation of the mechanicalproperties for the superconductors with such imperfections in
the future, and this numerical simulation framework can beeasily extended to different superconducting and reinforcing
material properties, for various geometries and magnetization
processes, such as PFM.

Acknowledgments
The authors thank Prof. F Nishimura of Iwate University forthe valuable discussions of the mechanical stresses. This
research is partially supported by a ‘Development of Systems
and Technologies for Advanced Measurement and Analysis’from Japan Agency for Medical Research and development,
AMED, and by JSPS KAKENHI Grant No. 15K04646. Dr
Mark Ainslie would like to acknowledge the support of a
Royal Academy of Engineering Research Fellowship.

4. Summary
We have performed numerical simulations of the mechanicalstresses (hoop stress, σθ, and radial stress, σr) in a REBaCuO ringbulk of inﬁnite and ﬁnite height, reinforced by a metal ring(aluminum alloy or stainless steel), during FCM from Bex = 4.7
and 9.4 T using inﬁnite and ﬁnite solenoid coils by the FEM. The
important analytical results and conclusions are summarized asfollows.

ORCID
M D Ainslie

https://orcid.org/0000-0003-0466-3680

References(1) The mechanical stresses of the ﬁnite ring bulk reinforced
by a metal ring during the FCM process have been
investigated by FEM for the ﬁrst time. The electromagnetic
hoop stress in the ﬁnite ring bulk is larger than that in theinﬁnite ring bulk magnetized by an inﬁnite solenoid coil.
The time step dependence of the hoop stress is different foreach case. These results suggest that the mechanical stress,
especially for the electromagnetic hoop stress, which canbe calculated by analytical equations for simple situations,is likely to be underestimated.
(2) The metal ring ﬁtting for the ﬁnite ring bulk is
conﬁrmed to be effective to reduce the electromagnetic
hoop stress during FCM. The reduction effect by the
SUS ring is much improved compared with thealuminum alloy ring.
(3) In the actual FCM process, in which a compressive
stress is applied to the ring bulk by cooling down to theoperating temperature, and then an electromagnetic
hoop stress is developed in the ring bulk during FCM,
the thermal compressive stress due to the shrink-ﬁtmetal ring is effective to reduce the electromagnetic
hoop stress and to avoid fracture of the bulk.
(4) However, the thermal hoop stress around the surface ofthe bulk periphery changes from compressive to tensilestress because of the larger shrink of the metal ringalong the axis direction, where the total hoop stressbecomes large positively and the bulk is likely to break.
(5) The actual REBaCuO disk or ring bulks of ﬁnite heightreinforced by a metal ring break frequently under magnetic
ﬁelds of several Tesla, although the bulk is safe from thesimulation results, which assumes homogenous properties.
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