Superconductor Science and Technology
Supercond. Sci. Technol. 32 (2019) 015007 (9pp)

https://doi.org/10.1088/1361-6668/aaee01

Thermal and magnetic strain measurements
on a REBaCuO ring bulk reinforced by ametal ring during ﬁeld-cooled magnetization
K Takahashi1, S Namba1, H Fujishiro1 , T Naito1 , Y Yanagi2, Y Itoh2 andT 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 14 July 2018, revised 26 October 2018
Accepted for publication 2 November 2018
Published 29 November 2018
Abstract

We have measured the mechanical strain, eq = dL q /L q , using strain gauges adhered along thecircumferential (θ) direction on the EuBaCuO ring bulk reinforced by an aluminum alloy ring
during the cooling process from 289 K to 50 K, and during ﬁeld-cooled magnetization (FCM) at
50 K under magnetic ﬁelds from Bapp=5.3 and 6.3 T. To discuss the mechanical reinforcement
effect of the aluminum alloy ring and the magnetic strain during FCM, we have performed
numerical simulations using the ﬁnite element method for the ring bulk assuming realistic
superconducting characteristics. The experimental results of the thermal strain during the coolingprocess, eqcool = dLqcool /L q , from 289 K to 50 K on the bulk surface validated our numerical results,
in which the eqcool value became smaller at the outer edge, compared to that at the inner edge of thebulk surface. These results strongly suggest an inhomogeneous reinforcement of the bulk due to thedifference in the thermal contraction along the height direction between the ring bulk and the outer
Al alloy ring with ﬁnite height. The experimental results of the time step dependence of themagnetic strain during FCM, eqFCM = dLqFCM /L q , were reproduced qualitatively by the numericalsimulation. The measurement of mechanical strain is effective to clarify the reinforcement effect ofthe metal ring during cooling and the mechanical stress during FCM.
Keywords: mechanical strain, ﬁeld cooled magnetization, REBaCuO bulk, shrink ﬁt steel,numerical simulation, strain gauge measurement(Some ﬁgures may appear in colour only in the online journal)
stainless streel [3]. A ring-shaped REBaCuO bulk superconductor is also considered useful for practical applications,such as nuclear magnetic resonance (NMR) spectrometersand magnetic resonance imaging (MRI) apparatus [4–8].
Trapped ﬁeld enhancement in a bulk superconducting NMR/MRI system is an ongoing challenge to improve its resolution.
To give the best indication of the trapped ﬁeld capabilityof a TFM, ﬁeld-cooled magnetization (FCM) is commonly
used. Although the trapped ﬁeld value of REBaCuO bulks,
which can be estimated from its Jc(B, T) characteristics, could
be over 20 T at 20 K [9], the mechanical strength of such

1. Introduction
RE-Ba-Cu-O (REBaCuO, RE: rare earth element or Y) bulk
superconductors have been developed from viewpoints oftrapped ﬁeld enhancement, including the critical current
density, Jc, and the enlargement of sample size [1, 2]. As aresult, such bulks with increased Jc can provide a compact
and strong magnetic ﬁeld source as a high-ﬁeld trapped ﬁeld
magnet (TFM) capable of generating several Tesla. To date,the highest trapped ﬁeld of 17.6 T has been achieved at 26 K
in a disk-shaped GdBaCuO bulk pair reinforced by shrink-ﬁt
0953-2048/19/015007+09$33.00

1© 2018 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 32 (2019) 015007

K Takahashi et al

Figure 1. (a) Measuring position of the central magnetic ﬁeld, Bz(t), using a Hall sensor. (b) Plan view of the bulk surface, on which four

strain gauges were adhered on the ring bulk and Al alloy ring in an array at r=17, 23, 30 and 35 mm to measure the strain along theθ-direction.

brittle ceramic materials restricts the realistic maximum
trapped ﬁeld value. When the bulk is magnetized by FCM, thebulk could sustain and trap some of the applied ﬁeld by an
induced supercurrent ﬂowing around the vortex pinning
centers within the bulk. Meanwhile, during its magnetization
process, large Lorentz forces (J×B) are generated due to theinteraction between induced current density (J) and magnetic
ﬁeld (B), and results in fracture of the sample when themagnetic stress exceeds the mechanical strength of the bulk
material. In any case, cracks propagate along the radial
direction [10, 11], and the principal hoop stress is appliedalong the circumferential direction of the bulk during FCM.
The tensile strength of REBaCuO bulks is as low as 50 MPa
along the ab-plane [12], which depends on the existence ofimperfections inside the bulk, such as grain boundaries,
cavities and micro-cracks [13]. Silver (Ag) addition is aneffective way as an internal reinforcement technique to reduce
numbers of cavities, and hence, improves the fracture strength
up to 70 MPa [14–16]. On the other hand, as an external
reinforcement technique, several kinds of metal ring ﬁtting
have been applied to the disk and ring bulks, which is
expected to apply a compressive stress to the bulk under thecooling process (before magnetization) due to the differenceof the thermal contraction coefﬁcient between the bulk andmetal ring [3, 5]. The acknowledgement of such a reinforcement effect has become more crucial to avoid the fracture of
the bulk, in parallel with the enhancement of the trapped ﬁeld.
We have reported the numerical simulation results of themechanical stress behaviors in the REBaCuO ring bulk withﬁnite height reinforced by a metal ring during FCM [17, 18].
We also mentioned the weakness in such conventional reinforcement techniques using a metal ring, in which an inhomogeneous stress proﬁle with a positive tensile stress occurs
at the outer edge of the bulk surface owing to the larger
thermal contraction of the metal ring with ﬁnite height along
both radial and height directions, compared to that of the bulk[18]. In these numerical models, the electromagnetic andmechanical parameters are assumed to be isotropic andhomogeneous inside the bulk for simplicity. In addition, theinterface between the outer bulk surface and inner metal ring
surface was assumed to be ﬁxed perfectly. Such numericalresults of mechanical behavior should be validated andcompared with actual experimental results.
A mechanical strain can be measured directly by a strain
gauge adhered on the bulk surface in actual experiments [11].
The mechanical stress can be estimated by the strain measurements using Hooke’s law and compared to the results of thenumerical simulations. We would mention that the direct strain
measurement is necessary to evaluate the reinforcement effect ofa metal ring and the electromagnetic strain on the bulk surface.
In this paper, we measured the thermal strain, eqcool =cooldLq /L q , along the circumferential (θ) direction on the bulk
surface during the cooling process from 289 K to 50 K toevaluate the reinforcement effect by aluminum alloy ring. We
also measured the magnetic strain, eqFCM = dLqFCM /L q , alongthe θ-direction on the bulk surface at 50 K during FCM undermagnetic ﬁelds from 5.3 and 6.3 T. The experimental results
were compared with the numerical results. The strain measurement technique using a strain gauge could be applicable for
evaluation of the mechanical reinforcement effect and themagnetic strain during FCM.

2. Experimental procedure and results
A EuBaCuO ring bulk with 10 wt% Ag of 64 mm in outer
diameter (OD), 28 mm in inner diameter (ID), and 20 mm inheight (H), was fabricated by the melt-processing method
under atmosphere by Nippon Steel and Sumitomo Metal[19, 20]. The EuBaCuO ring bulk can achieve higher trapped
ﬁeld and higher magnetic ﬁeld homogeneity with its low
relative magnetic permeability (m =1.0013), compared withthat of GdBaCuO bulk (m =1.0194) [5], which is preferable
for NMR/MRI bulk magnets. The ring bulk was mounted in
an aluminum (Al) alloy (A7075-T6) ring 5 mm in width(74 mm in OD and 64.1 mm in ID) with the same height as
the ring bulk using Stycast™ 1266 resin. Figure 1(a) showsthe experimental setup of the bulk and the measuring position2Supercond. Sci. Technol. 32 (2019) 015007

K Takahashi et alof the magnetic ﬁeld around the ring bulk. The bottom surfaceof the ring bulk with Al alloy ring was thermally connected tothe cold stage of a Gifford–McMahon cycle helium refrigerator using an indium sheet in a vacuum chamber. The
temperature of the bulk was controlled by a Pt–Co thermometer attached to the bottom surface of the cold stage. In the
FCM process, the ring bulk was cooled to the magnetizing
temperature of Ts=50 K under the initial applied ﬁelds ofBapp=5.3 and 6.3 T, using a cryo-cooled superconducting
solenoid magnet (JASTEC JMTD-10T100), and then, theexternal ﬁeld was decreased linearly at −0.222 T min−1 down
to zero. The time step (TS) of the descent of the magnetic
ﬁeld during FCM is deﬁned as follows
TS = 10

Bapp - Bex
Bapp

,

(1 )

where Bex is the actual external ﬁeld. The TS dependence ofthe local ﬁeld, Bz(t), along the z-direction was measured at thecenter of the bulk annuli using a Hall sensor (F. W. Bell,BHA 921). Figure 1(b) shows the schematic view of the top
surface of the ring bulk. Four strain gauges (Tokyo Sokki,
CFLA-1-350-11, (gauge length: 1 mm, gauge resistance: 350
Ω, gauge factor: F=2.09) were adhered in an array along theθ-direction at r=17, 23, 30 mm on the bulk surface and atr=35 mm on the Al alloy ring surface using an epoxy
adhesive (Tokyo Sokki, EA-2A) to measure the strain. The
mechanical strain, eq = dL q /dL, including the thermal contraction strain under the cooling process, eqcool = dLqcool /L q ,and the electromagnetic expansion strain during FCM,
eqFCM = dLqFCM /L q , is expressed by the following equation
eq =

dL q
L (T , B ) - L q
1 dR
,
= q
=
Lq
Lq
F R

Figure 2. Cross-sectional view of the three-dimensional numericalmodel of the ring bulk reinforced by Al alloy ring (A7075-T6),
magnetized using a solenoid coil.

3. Numerical simulation framework
To discuss the reinforcement effect of the metal ring and themagnetic strain during FCM in detail, we calculated both thethermal strain under cooling process, eqcool , and the magnetic
strain during FCM, eqFCM , using numerical simulations. We
constructed a three-dimensional (3D) ﬁnite element model
based on the actual experimental setup for FCM as shown inﬁgure 2. The ring bulk with the same size as that used for theexperiment was magnetized using a solenoid coil (170 mm in
OD, 120 mm in ID and 200 mm in H), which is similar toprevious work [18]. The ring bulk and the Al alloy ring wereassumed to be ﬁxed perfectly through an epoxy resin layer
with 0.1 mm in thickness to reproduce the experimental
condition.
Electromagnetic phenomena during FCM are describedby the fundamental equation as follows [8, 21](2 )where dL q = L q (T , B) - L q is the change of length alongthe θ-direction by temperature or magnetic ﬁeld, L q = 2pr is
the original length of the bulk along the θ-direction and r is
the radial measurement position on the bulk surface. The
eq = dL q /L q value on the bulk sample was calculated by
the ratio of dR/R shown in equation (2), where R (=350 Ω) is
the initial resistance of the strain gauge and dR is the change
in the resistance due to the mechanical strain under temperature or magnetic ﬁeld variation. The temperature dependenceof the electrical resistance, R(T), of the strain gauge adheredon the quartz plate was measured from 300 K to 50 K by thefour terminal method applying a constant current of 0.3 mA,
which were used as the calibrated value. The magnetic ﬁeld
dependence of the electrical resistance, R(B), of the strain
gauge adhered on the quartz plate was also measured up to
10 T at 50 K, in which there was no need to calibrate thegauge resistance under the magnetic ﬁeld. We have measured
the gauge resistance, R(T, B), adhered on the bulk several
times, which were conﬁrmed to be nearly the same values
under the identical condition. We have also measured thetemperature dependence of the thermal strain, ε(T), of highpurity copper from 300 K to 50 K using the same method, inwhich the deviation from the recommended ε0(T) value,(ε0(T)-ε(T))/ε0(T), was conﬁrmed to be within 2%.

 ´ (m-1 ´ A) = J0 - J,

(3 )where A is the magnetic vector potential, J0 is the coil current
density, J is the induced current density in the superconducting bulk, and m is the magnetic permeability for the
EuBaCuO ring bulk. The E–J power law was assumed todescribe the nonlinear electrical properties of the superconducting bulk as follows⎛ J ⎞nE = Ec ⎜ ⎟ ,⎝ Jc ⎠(4 )where Ec (= 10−4 V m−1) is the characteristic electric ﬁeld
and n (= 20) is an appropriate value for the bulk superconductor [21]. The Jc(B) characteristics of the bulk used in
the simulation were determined using the following equation[22–24], for bulk superconducting materials exhibiting a3

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K Takahashi et al

Table 2. Mechanical parameters (Young’s modulus, EY, and Poisson
ratio, ν) of EuBaCuO bulk, epoxy resin, and Al alloy (A7075-T6)
used in the numerical simulation.

EY (GPa)n

EuBaCuO bulk

Epoxy resin

Al alloy (A7075-T6)

100
0.33

3
0.37

78
0.34

In the numerical FCM process, the ring bulk was cooled
to 50 K in ﬁelds of Bapp=5.3 and 6.3 T, and then, theexternal ﬁeld, Bex, was decreased linearly to 0 T at
−0.222 T min−1 over ten steps. The temperature variation
during FCM was ignored for simplicity, i.e. Q=E·J=0.
The commercial FEM software package Photo-Eddy (Photon
Ltd, Japan) was adapted for the analysis of the trapped ﬁeld,
Bz, perpendicular to the bulk surface, and induced current
density, Jθ, along the θ-direction in the bulk. Elastic behavior
in an isotropic material can be explained by Hooke’s law, inwhich the stress tensor, sij, is linearly proportional to thestrain tensor, eij , as follows
sij = l · ekk · dij + 2G · eij ,EY · n,(1 + n )(1 - 2n )EYG=,
2 (1 + n )l=density, Jc(B), of the EuBaCuO bulk used in the numericalsimulations. (b) Temperature dependence of the thermal contraction
strain, ε=dL/L, of each material from 289 K to 50 K for
mechanical simulation during the cooling process, which was
linearly approximated in three temperature regions(289 K T 200 K, 200 K T 100 K, 100 K T 50 K).

Table 1. Numerical parameters for the Jc(B) characteristics of thepresent bulk for equation (5).

Presentbulk

BL (T)

Jc2 (A m−2)

Bmax
(T)

k

1.3×109

0.8

1.0×109

4.5

1.0

(7 )
(8 )

where l and G represent Lame’s constants, dij is the Kronecker
delta function, EY is the Young’s modulus, and n is the Poisson
ratio. The nodal force on each node of the meshed elements
calculated by Photo-Eddy was imported to the commercial
software package Photo-ELAS (Photon Ltd, Japan) for the
analysis of the magnetic strain during FCM, eqFCM =dLqFCM /L q , due to the current-ﬁeld interaction at each step ofFCM. The mechanical parameters, EY and ν, of each component
used in the mechanical simulation are summarized in table 2,
which are assumed to be isotropic and to be in the elastic region.
Figure 3(b) shows the temperature dependence of the thermal
contraction strain, ε=dL/L, of each material from 289 K to
50 K, in which the dL/L value was linearly approximated inthree temperature regions (289 K T 200 K, 200 K T
100 K, 100 K T 50 K). The thermal contraction coefﬁcient,α, of each material at each temperature region was estimatedfrom the gradient of the dL/L value shown in ﬁgure 3(b). The
thermal strain in the ring bulk and Al alloy ring, eqcool =dLqcool /L q , which was produced by the difference in α between
the bulk and the Al alloy ring, was calculated under coolingprocess, separately from the FCM process.

Figure 3. (a) Magnetic ﬁeld dependence of the critical current

Jc1 (A m−2)

(6 )ﬁshtail shape in their magnetization loop.
⎡ ⎛⎛ B ⎞k ⎞ ⎤⎛ B⎞B1
Jc (B) = Jc1 exp ⎜exp ⎢ ⎜⎜1 - ⎜⎟ ⎟ ⎥. (5 )⎟ + Jc2⎢⎣ k ⎝⎝ Bmax ⎠ ⎟⎠ ⎥⎦⎝ BL ⎠
Bmax

4. Experimental and numerical results

Each parameter, which was determined by the ﬁtting, isshown in table 1, and the Jc(B) curve used in the simulation isshown in ﬁgure 3(a). The parameters of Jc1 and Jc2 used forthe present simulation of the EuBaCuO bulk were determined
as to reproduce the TS dependence of the trapped ﬁeld, Bz(t),obtained from actual experiment, as shown in the later
section 4.2 in detail.

4.1. Cooling process

In this subsection, we focus on the external reinforcement effectby the Al alloy ring from the viewpoints of both the experimental
and numerical results. Figure 4(a) shows the experimental resultsof the temperature dependence of the thermal strain under thecooling process, eqcool , at each measurement position on the top4Supercond. Sci. Technol. 32 (2019) 015007

K Takahashi et al

Figure 5(a) shows the comparison of the cooling strain on
the surface of the bulk (z=10), eqcool , between experimental
and numerical results at 200, 100 and 50 K, as function ofradius, r, of the bulk. The results were extracted from ﬁgure 4.
Under the cooling process, both the bulk and the Al alloy ring
contracted and the eqcool value became a negative value from289 K to 200 K, 100 K and 50 K. The simulation results of ther-dependence of the eqcool proﬁle and the absolute eqcool value
reproduced the experimental ones at each temperature. The
eqcool value decreased with increasing r in the bulk region(14 mm<r<32 mm) at each temperature, and the slope ofthe dLqcool /dL q proﬁle also became steep with decreasing
temperature. And then, the eqcool value negatively increases on
the Al alloy ring region (32 mm<r<37 mm). Figure 5(b)shows the simulation results of the cooling strain, eqcool , at
50 K, as function of radius for various conditions. The
experimental and numerical eqcool proﬁles on the bulk surface(z=10 mm) with the epoxy resin layer at 50 K are also
shown again for reference. For the numerical eqcool proﬁlewithout the epoxy resin layer (w/o epoxy) in the gap with
0.1 mm between the outer ring bulk and inner Al alloy ring,
the eqcool value is −0.151% in the bulk region and this isindependent of the z value. This behavior indicates that thering bulk and Al alloy ring shrink freely and independently
during the cooling process, i.e. the thermal compressive stresssqcool =0. On the other hand, for the numerical eqcool proﬁle on
the central axis of the bulk (z=0 mm) with epoxy resin layer
in the gap, the eqcool value slightly increases to −0.21% atr=14 mm and −0.185% at r=30 mm. This result suggeststhat the thermal reinforcement is effectively applied by the Al
alloy ring. However, for the numerical eqcool proﬁle on thebulk surface (z=10 mm) with the epoxy resin layer, the eqcool
value exhibits a moderate slope and reproduces the experimental results. These results indicate that the bulk is difﬁcult
to shrink along the θ-direction on the top surface of the bulk
with increasing r.
Figures 5(c) and (d), respectively, show a schematic view ofthe displacement, dr and dz, along the r- and z-directions duringcooling from 289 to 50 K, where the epoxy resin layer exists in
the interface between the ring bulk and Al alloy ring. In
ﬁgure 5(c) along the r-direction, the shrink measured on the top
and bottom surfaces is smaller than that at the bulk center. On
the other hand, along the z-direction, as shown in ﬁgure 5(d), theshrink on the bulk surface is smaller than that on the Al alloy
ring. These results suggest that the thermal shrink in the bulk is
not homogeneous due to the larger thermal contraction of the Al
alloy ring with ﬁnite height, where the outer periphery of thering bulk was pulled toward z=0.
Figures 6(a) and (b), respectively, show the numericalresults of the thermal stress proﬁle after the cooling process,sqcool , along the θ-direction and srcool , along the r-direction
inside the bulk and the Al alloy ring for various height
positions from the center (z=0 mm) to the top surface(z=10 mm). In our previous study, we have reported thenumerical results of the thermal stress, sqcool , and the thermal
displacement proﬁle under the cooling process in the bulk

Figure 4. (a) Experimental and (b) numerical results of thetemperature dependence of the thermal strain under the coolingprocess from 289 K to 50 K, eqcool = dLqcool /dL q, at each measurement position on the top surface of the ring bulk and the Al alloy
ring. The eqcool proﬁle for the ring bulk without reinforcement by the
Al alloy ring (w/o epoxy) was also shown in (b).

surface of the bulk and the Al alloy ring. The bulk and the Al
alloy ring start to shrink from 289 K gradually, and the eqcool
value becomes negative with decreasing temperature. The
negative eqcool value at r=17 mm is the largest, meaning thehighest compressive strain and decreases with increasing r in
the bulk. The negative eqcool value on the Al alloy ring(r=35 mm) showed slightly larger compressive strain than thatat r=23 mm. Figure 4(b) shows the numerical results of thetemperature dependence of the eqcool proﬁle at each position.
The position and temperature dependences of the eqcool proﬁlereproduce the experimental results well. In ﬁgure 4(b), thetemperature dependence of the eqcool proﬁle for the bulk is also
shown, in which there is no epoxy resin at the interface (w/oepoxy). This result shows the eqcool proﬁle of the bulk without
any reinforcement by the metal ring, which was independent of
the measurement position. The experimental results shown inﬁgure 4(a) suggest that the thermal strain under cooling isaffected by the Al alloy ring reinforcement and that the coolingstrain changes depending on the measurement position.

5Supercond. Sci. Technol. 32 (2019) 015007

K Takahashi et al

Figure 5. (a) Thermal strain proﬁle during the cooling process, eqcool = dLqcool /L q , on the top surface at each temperature, 200 K, 100 K and
50 K, cooling from 289 K, comparing the numerical simulation results with the experimental results. (b) The eqcool proﬁle under the coolingprocess, in which the no epoxy layer exists and both the ring bulk and Al alloy ring shrink independently. Schematic view of thedisplacement proﬁle of the ring bulk under the cooling process along the (c) r- and (d) z-direction.

6

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Figure 7. Experimental and numerical results of time step

dependence of the magnetic ﬁeld, Bz (z=r=0 mm) at 50 K at thecenter of the ring bulk during FCM from Bapp of 5.3 and 6.3 T.

There might be a slight difﬁculty to estimate the thermal
stress proﬁle inside the bulk, except when using numericalsimulations.
4.2. FCM process

In the case of the FCM process, we ﬁrstly describe theelectromagnetic behavior of the trapped ﬁeld, and then referto the mechanical behavior. Figure 7 shows experimental
results of the TS dependence of the ﬁeld, Bz (z=r=0 mm),
at 50 K at the center of the ring bulk during FCM from
Bapp=5.3 and 6.3 T. For each Bapp, the Bz value slightly
decreased with increasing TS and then settled to the ﬁnal
value. Final trapped ﬁelds of BT=5.02 and 5.87 T weresuccessfully achieved at the tenth step after FCM from
Bapp=5.3 and 6.3 T, respectively. The numerical results of
the TS dependence of the Bz (z=r=0 mm) are also shownin ﬁgure 7, in which the parameters for the Jc(B) proﬁleshown in table 1 were used in the simulation. The experimental Bz (z=r=0 mm) proﬁles can be reproduced by thenumerical simulation. These results suggest that the magnetic
ﬁeld dependence of the average critical current density, Jc(B),can be decided by the numerical ﬁtting of the TS dependenceof the trapped ﬁeld, Bz(TS), to the experimentally obtained
Bz(TS), which was measured at the center of the ring bulk.
The ring bulk was not broken for applied ﬁelds up to 6.3 T
and the ﬁnal trapped ﬁeld almost linearly increased withincreasing Bapp. These results suggest that the ring bulk was
not fully magnetized during FCM and that the induced
supercurrent ﬂowed only in the outer and top and bottom
regions of the ring bulk.
Figures 8(a) and (b) show experimental results of the TS
dependence of the magnetic strain, eqFCM = dLqFCM /L q , at
50 K during FCM from Bapp=5.3 and 6.3 T, respectively, ateach measurement position on the bulk surface. The numerical results of the TS dependence of the eqFCM value on thebulk surface (z=10 mm) are also shown in each ﬁgure. BothFigure 6. Numerical results of the thermal stress proﬁles inside thebulk and Al alloy ring after the cooling process from 289 K to 50 K,
for (a) s qcool, along the θ-direction and (b) s rcool, along the r-direction
for various height (z-) positions (z=0–10 mm).

reinforced by a metal ring, in which the stress proﬁle might beinhomogeneous inside the bulk due to the ﬁnite height of thebulk and metal ring [17, 18]. In ﬁgure 6(a), it was also conﬁrmed that the inhomogeneity of sqcool exists along thez-direction inside the bulk, in which the compressive sqcool
decreased with increasing z and changed to the positive(tensile) value of +32 MPa at the outer edge of the bulk
surface (r=30 mm, z=10 mm). In ﬁgure 6(b), the tensile
srcool stress is also enhanced at the bulk surface (z=10 mm),which is as large as +100 MPa. The present experimental
results shown in ﬁgures 4 and 5 are partially reﬂected by such
inhomogeneous stresses in conventional reinforcement using
a relatively thin metal ring with the same height as the bulk.
Recently, we have reported the wider metal ring is effective to
reduce the electromagnetic expansive stress during FCM andto enhance the thermal compressive stress under cooling [18].
Note again that the present numerical simulation assumes thatthe epoxy resin layer is in the elastic region and is not broken
or cracked. The experimental measurement of eqcool , shown inﬁgures 4 and 5, was limited only along the θ-direction andonly on the bulk surface (z=10 mm), although the actual
stress distribution must be obtained from 3D strain variation.
7

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K Takahashi et al

region has relatively lower Jc(B) [1, 24]. In this case, the eqFCM
values estimated by the average Jc(B) might be overestimated,compared to the experimental values.
The ring bulk was assumed to be homogeneous in thissimulation, in which the strain exists axisymmetrically in thebulk. However, in the present ring bulk, there exists aninhomogeneous strain distribution, which comes from theexistence of growth sector regions and growth sector
boundaries, and also voids, small cracks, Ag and RE211
particles. In an inhomogeneous material, a region with a large
strain might exist and, as a result, the region with a small
strain also exists [11]. The ﬁnite size of the actual strain gauge
may also inﬂuence on the accuracy of the eqFCM value.
In this study, we measured magnetic strain, eqFCM , only
along the θ-direction. When we also measure the magnetic
strain, erFCM , along the r-direction at the same positions as theeqFCM measured positions, we can experimentally estimate themagnetic stresses, s qFCM and s rFCM, along the θ- and r-direction, respectively, and better conﬁrm the validity of thenumerical simulation.

5. Conclusion
In this study, we have measured the mechanical strain along
the circumferential direction of the EuBaCuO ring bulk
reinforced by an aluminum alloy ring, during both the coolingprocess from 289 K to 50 K and during FCM from 5.3 and
6.3 T at 50 K. To discuss the mechanical reinforcement effectof the aluminum alloy ring, we have performed numericalsimulations for the ring bulk with realistic superconducting
characteristics. The important results and conclusions are
summarized as follows.

Figure 8. Experimental and numerical results of the time step

dependence of the magnetic strain, eqFCM = dLqFCM /L q, during FCM
from (a) Bapp=5.3 T and (b) 6.3 T at each measurement position on
the bulk surface (z=10 mm).

(1) The experimental results of the thermal strain under thecooling process, eqcool = dLqcool /L q , from 289 K to 50 K
on the bulk surface validated our numerical results, forwhich the eqcool value was getting smaller at the outeredge of the bulk surface, compared to that at the inneredge of the bulk surface. These results strongly suggest
an inhomogeneous reinforcement on the bulk due to thedifference in thermal contraction along the z-direction
between the bulk and the outer Al ring with ﬁnite
height.
(2) Using the electromagnetic simulation, the magnetic
ﬁeld dependence of the average critical current density,
Jc(B), can be deduced by the numerical ﬁtting of the TS
dependence of the trapped ﬁeld, Bz(TS), to theexperimentally obtained Bz(TS), which was measured
at the center of the ring bulk.
(3) The experimental results of the TS dependence of themagnetic strain during FCM, eqFCM = dLqFCM /L q , andthe applied ﬁeld and position dependences of eqFCM werereproduced qualitatively by the numerical simulation.
However, the experimental eqFCM values were about
50% smaller than those of the simulation. The
difference mainly comes from the simulation resultsin the experiments and numerical simulations, the eqFCM
values rise with increasing TS, take a peak value aroundTS=6–7, and then decrease with increasing TS. The maximum eqFCM value increases with increasing Bapp. These
results strongly suggest magnetic strain can be measured
using strain gauges.
The absolute eqFCM values obtained experimentally wereabout 50% smaller than those of the simulations at eachmeasuring position and Bapp. The difference between theexperiments and numerical simulations is larger than that in
the eqcool value without magnetic ﬁeld shown in ﬁgure 5(a).
These results suggest that the difference mainly comes fromthe simulation results of the magnetic strain, which is closely
related with the used Jc(B) characteristics. In the present
study, the used Jc(B) characteristics shown in ﬁgure 3(a) were
determined as an ‘average Jc(B) value’ in the bulk by theﬁtting to the experimental results shown in ﬁgure 7. The
electromagnetic stress and strain depend on the Jc(B), that is,
the stress and strain decrease with decreasing Jc(B). In actual
REBaCuO bulks, there exists the Jc(B) distribution, in whichhigher Jc(B) region exists only near the seed crystal and other8Supercond. Sci. Technol. 32 (2019) 015007

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of the magnetic strain, which is closely related with theused Jc(B) characteristics. The existence of the macroscopic and microscopic structural inhomogeneity may
also affect the eqFCM value.
(4) When both the magnetic strains, eqFCM and erFCM , alongthe θ- and r-directions at the same positions are measured
using this technique, the magnetic stresses, s qFCM ands rFCM, can be experimentally estimated and the validity of
the numerical simulation can be conﬁrmed.

Acknowledgments
This research is partially supported by a ‘Development of
Systems and Technologies for Advanced Measurement andAnalysis’ from Japan Agency for Medical Research anddevelopment, AMED and by JSPS KAKENHI grant no.
JP15K04646. The Authors thank Dr Mark D Ainslie ofUniversity of Cambridge and Professor Fumihito Nishimuraof Iwate University for valuable suggestions about the numerical 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
T Nakamura https://orcid.org/0000-0002-8278-2688

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Durrell J H and Cardwell D A 2016 Supercond. Sci.
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