Superconductor Science and Technology

PAPER • OPEN ACCESSElectromagnetic strain measurements and two-directional mechanicalstress estimation for a REBaCuO ring bulk reinforced by a metal ring
during field-cooled magnetization
To cite this article: Sora Namba et al 2019 Supercond. Sci. Technol. 32 125011

View the article online for updates and enhancements.

This content was downloaded from IP address 160.29.130.194 on 05/11/2019 at 09:37

Superconductor Science and Technology
Supercond. Sci. Technol. 32 (2019) 125011 (10pp)

https://doi.org/10.1088/1361-6668/ab4a8bElectromagnetic strain measurements andtwo-directional mechanical stress estimation
for a REBaCuO ring bulk reinforced by ametal ring during ﬁeld-cooled magnetization
Sora Namba1,3 , Hiroyuki Fujishiro1 , Tomoyuki Naito1 ,Mark D Ainslie2 and Kai Y Huang2
1

Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, 4-3-5 Ueda, Morioka 020-8551, Japan2
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United
KingdomE-mail: g0318128@iwate-u.ac.jp
Received 1 April 2019, revised 20 September 2019
Accepted for publication 3 October 2019
Published 4 November 2019
Abstract

In this paper, simultaneous measurements of the electromagnetic strains along both thecircumferential (θ) and radial (r) directions are reported for a large single-grain EuBaCuO ring bulk
reinforced by an Al alloy ring during ﬁeld-cooled magnetization (FCM) from 5 T at 50 K using
several strain gauges adhered to the surface. To verify the experimental results and to understandthe complex stress–strain behavior, mechanical analyses were carried out using a threedimensional ﬁnite element model that closely represents the experimental setup. The simulation
results of the electromagnetic strains along both directions showed excellent qualitative andquantitative agreement with the experimental ones. These results strongly suggest that thenumerical model must include the exact same structure (size, shape and materials) of themechanical support structure as the experimental setup in order to reproduce the experimental
results both qualitatively as well as quantitatively. This also explains our previous research (SuST2019 32 015007), where the measured circumferential strains were about 50% smaller those in the
numerical simulation. Furthermore, the electromagnetic stresses along both directions during the
FCM process are estimated from the obtained experimental strains. As a result, the estimatedstresses were fairly consistent with those obtained by the numerical simulations, suggesting that
our stress–strain simulation technique is both qualitatively and quantitatively reliable and useful toclarify the possibility of mechanical fracture of bulk superconductors.
Keywords: bulk superconductors, trapped ﬁeld magnet, mechanical stress, strain gaugemeasurement, numerical simulation, ﬁnite element method(Some ﬁgures may appear in colour only in the online journal)
1. Introduction
3

Author to whom any correspondence should be addressed.

Large, single-grain REBaCuO (RE: a rare earth element or Y)
bulk superconductors are a promising material for use as acompact, high-strength trapped ﬁeld magnet (TFM) [1] forvarious practical applications, such as rotating machines,
magnetic separation, ﬂywheel energy storage systems and so

Original content from this work may be used under the termsof the Creative Commons Attribution 3.0 licence. Any
further distribution of this work must maintain attribution to the author(s) andthe title of the work, journal citation and DOI.
0953-2048/19/125011+10$33.00

1© 2019 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 32 (2019) 125011

S Namba et alon [2–4]. REBaCuO bulks can generate magnetic ﬁelds muchhigher than conventional Nd–Fe–B permanent magnets orother superconducting TFMs using MgB2 [5] and iron-pnictide[6]. In general, the trapped magnetic ﬁeld capability of the bulk
superconductor is proportional to both the critical current
density, Jc, and bulk radius, r, based on the Bean’s critical statemodel [7]. Until now, the increase of Jc and the enlargement of
r of the bulk have been carried out over the past three decades
by the improved fabrication techniques, particle reﬁnement of
the secondary phases, suppression of defect formation, and theintroduction of stronger pinning centers.
Besides the enhancement of the magnetic ﬂux pinning
performance in the bulks, the mechanical reinforcement of thebulk is also an important issue during ﬁeld-cooled magnetization (FCM), especially for ﬁelds of several Tesla or more. A
large Lorentz force is exerted in the bulk, which sometimes
results in crack formation and propagation, leading to eventual
mechanical failure. This is because the REBaCuO bulk material is intrinsically a brittle ceramic with relatively low
mechanical strength. In previous studies, crack formation and/or serious fracture occurred in a bulk without external reinforcement during ramped down from a high external magnetic
ﬁeld [8–10]. In this sense, the maximum trapped ﬁeld capability is not restricted by the electromagnetic properties, butby their poor mechanical performance. Using mechanicalreinforcement by shrink-ﬁt stainless steel or epoxy resin
impregnation, trapped magnetic ﬁelds over 17 T have beenachieved below 30 K in disc bulk pairs [11, 12]. In addition, aring-shaped REBaCuO bulk with a large bore has been also
applied for the compact and cryogen-free NMR/MRI magnet[13, 14]. Trapped ﬁeld enhancement with spatial homogeneityof ppm order is an ongoing challenge for improving resolution;a trapped ﬁeld of as high as 10 T is necessary to realize a
400 MHz (9.4 T) NMR bulk magnet system. The mechanicalstrength of the bulk materials has usually been estimated using
three-point or four-point bending tests [15–18] and tensile tests[19, 20], from which the fracture strength of a typical Agdoped REBaCuO bulk is suggested to be as low as 50–70 MPa.
Analytical investigations of mechanical properties havebeen reported for disc- and ring-shaped bulks with inﬁnite
height during FCM or zero-ﬁeld-cooled magnetization [8,
21–24]. On the other hand, we have reported numericalsimulation results of the mechanical stress behavior for
REBaCuO disc- and ring-shaped bulks with ﬁnite height andthe metal ring reinforcement during FCM [25–27] using theﬁnite element method (FEM). Furthermore, we have proposeda new reinforcement method to prevent bulk failure underhigher ﬁelds [28–30].
Mechanical strain measurement using a strain gauge is aconventional and effective method for the evaluation of the local
mechanical properties of a large bulk without cutting it. The
electromagnetic strain, ε, in the REBaCuO disc bulk caused by
the Lorentz force during FCM was directly measured on thebulk surface, and the fracture strength was estimated [31].
Recently, we have reported the electromagnetic strain along thecircumferential (θ) direction of a REBaCuO ring bulk reinforced
by a metal ring during FCM, and the cooling strains during thecooling process from room temperature to 50 K [32]. Theseexperimental results were compared with numerical simulation
results, in which the simple three-dimensional (3D) model
consisted of the ring bulk and the reinforcement metal ring. The
numerical results of the time-step dependence of the strain
during FCM reproduced the experimental ones qualitatively.
However, the absolute value of the experimental strain was
about 50% smaller than that of the simulated one. We concluded, in the paper, that the difference mainly comes from thesimulation results, which was closely related with the assumed
Jc(B) characteristics, and/or from the experimental strain results,
which result from the existence of the macroscopic and microscopic structural inhomogeneity. The quantitative difference inthe electromagnetic strain between the experiments and simulations is a serious problem, because the agreement guarantees
the reliability of our numerical simulation results. We hypothesized that the difference mainly comes from an insufﬁcient
numerical model; that is, the exact same structure (size, shapeand materials) of the mechanical support structure as theexperimental setup must be included in the model. In addition, toenhance the reliability of the simulation results, the simultaneous
measurement and veriﬁcation of the electromagnetic stresses
along two directions is provided.
In this paper, we measured the electromagnetic strains
along both circumferential (θ) and radial (r) directions of a
EuBaCuO ring bulk reinforced by an aluminum (Al) alloy ring
during FCM from 5 T at 50 K using several strain gauges. The
electromagnetic stresses in both directions are estimated analytically at each position from the experimental results obtainedby the strain measurement using Hooke’s law. A full 3D ﬁnite
element model based on the actual experimental setup wasdeveloped to understand the complex stress–strain behavior of
the ring bulk. The quantitative agreement between the experiments and simulations can prove the validity of our experimentand simulation techniques and validates the simulation as areliable tool to investigate the possibility of mechanical fractureof bulk superconductors.

2. Experimental procedure
2.1. Experimental setup and FCM process

Figure 1(a) shows the experimental setup of the ring bulk
with a metal ring for FCM. The EuBaCuO ring bulk (64 mm
OD, 28 mm ID, and 20 mm height (H)) containing 10 wt%
silver (Ag) was fabricated using a modiﬁed quench and melt
growth method by Nippon Steel Corporation, Japan [33, 34].
An Al alloy reinforcement ring (A7075-T6) with the sameheight as the ring bulk (74 mm OD, 64.1 mm ID, and
20 mm H) was bonded to the outer peripheral surface of thering bulk using Stycast™ 1266 epoxy resin (0.05 mm inthickness). This ring applies a compressive stress to the ringbulk during the cooling process due to the difference in thethermal contraction coefﬁcient between two materials. The
ring bulk with the Al alloy ring was tightly sandwiched by top(12 mm in thickness) and bottom (22 mm in thickness) Cu
plates made of oxygen-free high-conductivity copper (OFHCCu) using a bolt and nut. Thin indium sheets were inserted2Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

Figure 1. (a) Experimental setup of the EuBaCuO ring bulk on the cold stage of refrigerator during FCM, in which the ring bulk was

sandwiched between OFHC Cu plates using a bolt and nuts. (b) Positions and directions of the strain gauges adhered to the top surface of thering bulk and Al alloy ring.

The electromagnetic strain along the θ-direction,
eq = dL q /L q (0), is expressed by the following equation [32]between the bulk and both Cu plates to obtain good thermal
contact. The bottom Cu plate was thermally connected to thecold stage of a Gifford-McMahon helium refrigerator. The
temperature of the cold stage was controlled by a Pt–Co
thermometer and a heater, attached to bottom surface of thecold stage. A straight and long groove (64 mm in length,
10 mm in width and 5 mm in depth) was made on the bottom
surface of the top Cu plate to prevent mechanical contact
between the Cu plate and strain gauges and to set a
CERNOX™ thermometer to monitor the temperature of thebulk on the outer edge of the bulk surface. The trapped ﬁeld,
Bz, at the center of the ring bulk (r =z =0) was measured byan axial-type Hall sensor (F. W. Bell, BHA-921). The bulk
sheathed in a vacuum chamber was inserted into a cryocooled
10 T superconducting solenoid magnet (Japan Superconductor Technology, JMTD-10T100) and an external
magnetic ﬁeld of Bapp=5 T, parallel to z-axis, was applied tothe bulk at 100 K, and then the bulk was slowly cooled to50 K under the magnetic ﬁeld of 5 T. Finally, the external
magnetic ﬁeld was ramped down to zero at a rate of−0.222 T min−1.

eq =

dL q
L (B ) - L q (0 )
1 dR
,
= q
=
L q (0 )L q (0 )F R(1 )where dL q = L q (B) - L q (0) is the change of length along theθ-direction by magnetic ﬁeld, L q (0) = 2pr is the original
length of the bulk at Bex=0 along the θ-direction, r is theradial measurement position on the bulk surface. The
electromagnetic strain along the r-direction, er = dL r /L r (0),is expressed by the following equation
er =

dL r
L (B ) - L r (0 )
1 dR
= r
=
,
L r (0 )L r (0 )F R(2 )where dL r = L r (B) - L r (0) is the change of length along ther-direction under the magnetic ﬁeld, and L r (0) = r at
Bex=0. The eq and er values on the surface were calculatedby the ratio of dR/R shown in equations (1) and (2), in which
R (=350 Ω) is the initial resistance of the strain gauge anddR=R(B) − R(0) is the change in resistance due to theelectromagnetic strain under magnetic ﬁeld variation. The R(B) value of the strain gauge adhered on the quartz plate wasalso measured up to 10 T at 50 K, in which there was no needto calibrate the gauge resistance under the magnetic ﬁeld. We
have measured the gauge resistance, R(B), adhered on thebulk several times, which were conﬁrmed to be nearly thesame values under identical conditions.

2.2. Strain measurement

Figure 1(b) shows the measurement positions of the strain
gauges adhered to the top surface of the EuBaCuO ring bulk
and the Al alloy ring during the FCM process. Eight strain
gauges (Tokyo Sokki, CFLA-1-350-11, (gauge length: 1 mm,
gauge factor: F=2.09)) were adhered linearly on the top
surface using an epoxy adhesive (Tokyo Sokki, EA-2A). Fourgauges were adhered at r=17, 23, 30 mm on the bulk and atr=35 mm on the Al alloy ring from the center of the bulk
annuli in the growth sector region to measure the electromagnetic strain, εθ, along the θ-direction. The other fourgauges were also adhered at symmetric positions of r=−17,−23, −30 and −35 mm to measure the electromagnetic
strain, εr, along the r-direction.

3. Numerical simulation framework
Based on the experimental setup shown in ﬁgure 1(a), a
detailed 3D numerical model was constructed, in which
OFHC Cu plates, indium sheets, bolt and nut made of stainless steel (SUS304) and Stycast™ 1266 (epoxy resin) werealso added, together with the EuBaCuO bulk and the Al alloy
ring. The model was equally divided into 36 elements along3Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

Table 1. Numerical parameters for the Jc(B) characteristics of the
EuBaCuO bulk in equation (4).

EuBaCuObulk

BL (T)

Jc2 (A m−2)

Bmax
(T)

k

1.30×109

0.80

1.00×109

4.50

1.00

macroscopic Jc(B) characteristics of the bulk without cutting
and destroying the bulk.
Figure 2 shows the experimental and simulation resultsof the time dependence of Bz during FCM from 5 T at 50 K.
In the experimental FCM, the external ﬁeld, Bex, is ramped
down linearly at a rate of −0.222 T min−1. Bz decreased
gradually due to ﬂux creep, and a ﬁnal trapped ﬁeld of 4.71 T
was achieved successfully, when the Bex reached zero att=1350 s. It can be seen that the simulation result of thetime-dependent trapped ﬁeld curve can reproduce the experimental curve by the ﬁtting parameters shown in table 1.
The numerical simulation of the mechanical stress andstrain in the bulk and Al alloy ring during FCM was carriedout using the commercial FEM software, Photo-ELAS(Photon Ltd, Japan), coupled with the electromagnetic modelby Photo-Eddy to incorporate Lorentz force. The bulk and Al
alloy ring are assumed to be homogeneous and isotropic, andhave a linear elastic response. Elastic behavior in an isotropic
material can be explained by Hooke’s law, in which the stresstensor, σij, is linearly proportional to the strain tensor, εij, as
follows

Figure 2. Experimental and simulation results of the timedependence of the trapped ﬁeld, Bz, at the center of the ring bulk(r=z=0) and the external ﬁeld, Bex, during FCM, ramped downfrom Bapp=5 T at 50 K.

the circumferential direction and into 1 mm intervals along
the r- and z-directions, respectively. As boundary conditions,the outer surfaces of the Al-alloy ring and the Cu holder wereassumed to be adiabatic and each part was ﬁxed mechanically. The bulk was magnetized by a coil (170 mm OD,
120 mm ID, and 200 mm H), which was the same size as thesuperconducting solenoid coil used in the experiment.
The FCM simulation was carried out using a process
identical to that described in section 2.1. The temperature of
the system was set to 50 K because the maximum temperaturerise during experimental FCM was as small as 1 K at the outeredge of the bulk surface. The commercial FEM software
package, Photo-Eddy (Photon Ltd, Japan), was employed forthe electromagnetic analysis during the FCM process. The
details of fundamental equations for bulk superconductor
magnetization simulations are described elsewhere [35, 36].
The nonlinear electrical properties of the superconducting bulk was described using the E–J power law⎛ J ⎞nE = Ec ⎜ ⎟ ,⎝ Jc ⎠

Jc1 (A m−2)

sij = l · ekk · dij + 2G · eij ,E·n,(1 + n )(1 + 2n )EG=,
2 (1 + n )

l=

(5 )
(6 )
(7 )

where λ and G represent the Lame’s constants, δij is the
Kronecker delta function, E is the Young’s modulus, and v isthe Poisson ratio of the material. In the two-dimensionalmodel, where the mechanical stress, σz, along the z-direction
is ignored, the mechanical strains along the θ- and r-directions
are expressed in the following(3 )where Ec (=10−4 Vm−1) is the characteristic electric ﬁeld andn (=20) is a typical value for bulk REBaCuO superconductors[37, 38]. The magnetic ﬁeld dependence of the critical current
density, Jc(B), for the bulk used in this simulation was determined using the following equation [39–41]

dr(s - nsr )
,
= q
r
E
d (dr )(s - nsq )
,
er =
= r
dr
E
eq =

⎡ ⎛
⎛ B⎞
⎛ B ⎞k ⎞ ⎤B1
Jc (B) = Jc1 exp ⎜ exp ⎢ ⎜⎜1 - ⎜⎟ + Jc2⎟ ⎟ ⎥.
⎝ BL ⎠⎝ Bmax ⎠ ⎟⎠ ⎥⎦⎢⎣ k ⎝
Bmax

(8 )
(9 )

where σθ and σr are the mechanical stress along the θ- and rdirection, respectively, and dr is the displacement along the rdirection.
The mechanical parameters, E and v of the REBaCuO
bulk, Al alloy ring (A7075-T6), epoxy resin, indium, copper(OFHC Cu) and stainless steel (SUS304) in the stress–strain
analysis are summarized in table 2. The mechanical parameters of the REBaCuO bulk were assumed to be isotropic
and homogeneous for simplicity.

(4 )We proposed a new method to estimate the Jc(B) characteristics of the bulk [29], in which the time dependence oftrapped ﬁeld, Bz, at the center of ring bulk during FCM wasmeasured and ﬁtted using the Jc(B) characteristics shown inequation (4). This method can estimate the average and4Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

Figure 3. Simulation results of the cross-section of the ring bulk and Al alloy ring at Bex=4, 1 and 0 T during FCM from Bapp=5 T. The
external magnetic ﬁeld dependent distributions of (a) the trapped ﬁeld in the z-direction, Bz, (left), (b) the induced current density in the θdirection, Jθ, (middle) and (c) the Lorentz force density in the r-direction, Fr, (right), are shown.

(r=17 mm, z=0 mm), and Bz=4.8 T can be achieved at
the center of the ring bulk (r=z=0 mm), as shown inﬁgure 2. These results suggest that the ring bulk was not fully
magnetized during FCM from Bapp=5 T at 50 K due to thelarger Jc(B) value.

4. Results and discussion
4.1. Numerical simulation results of electromagnetic properties

Figures 3(a)–(c), respectively, show simulation results of thecross-sectional proﬁles of the magnetic ﬁeld, Bz, induced
persistent current density, Jθ, and Lorentz force density, Fr, inthe bulk and Al alloy ring at Bex=4, 1 and 0 T during FCM
from Bapp=5 T. At Bex=4 T, the magnitude of Bz in thewhole region of the ring bulk is nearly 5 T except for its outer
periphery, where Jθ ﬂows along the θ-direction only in outerregion and top and bottom surfaces of the bulk. As a result, Fr
is applied along the r-direction by the current-ﬁeld interaction(i.e. Jθ×Bz). In the Al alloy ring, Fr equals to zero becauseJθ=0. As the external ﬁeld is ramped down, e.g.
Bex=1 and 0 T, the region through which the persistentsupercurrent ﬂows became wider, compared to that at
Bex=4 T. As a result, Bz in the outer region of the bulk
decreased, the region with higher Bz became narrower and theBz value at the center slightly decreased. It is obvious that thepeak position of Fr shifts from the periphery to the center ofthe bulk with decreasing Bex. When Bex reaches 0 T, Bz=5 T
is still present in the inner central region of the bulk

4.2. Measurement and simulation of electromagnetic strain

Figure 4(a) shows the experimental results of the externalﬁeld dependence of the electromagnetic strain along the θdirection, εθ, during FCM from 5 T at 50 K at each position,in which the magnetic ﬁeld decreased from 5 to 0 T. The
initial strain of each gauge was assumed to be zero, after thebulk was cooled from 300 to 50 K under the magnetic ﬁeld of
5 T. The εθ value at each position monotonically increases
during the ramp down of the external ﬁeld, takes a maximum
around Bex=1 T, and then slightly decreases. This trend was
qualitatively similar to our previous study [32]. The εθ value
at r=17 mm is the largest, which is +0.027%, meaning thehighest tensile strain in the innermost position. The εθ value
decreases with increasing r in the bulk, e.g. εθ=+0.013% atr=30 mm. The εθ on the Al alloy ring (r=35 mm) also see
positive values during FCM, nevertheless the Lorentz force is5Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

Table 2. Mechanical parameters (Young’s modulus, E, and Poisson ratio, ν) of the EuBaCuO bulk, Al alloy (A7075-T6), epoxy resin,
indium, copper (OFHC Cu) and stainless steel (SUS304) used in the simulation.

E (GPa)v

EuBaCuO bulk

Al alloy (A7075-T6)

Epoxy

IndiumCopper (OFHC Cu)

Stainless steel (SUS304)

100
0.33

78.0
0.34

3.00
0.37

12.7
0.45

125
0.34

193
0.28

Figure 5. (a) Experimental and (b) simulation results of the

Figure 4. (a) Experimental and (b) simulation results of theelectromagnetic strain along the r-direction, εr, as a function of theexternal ﬁeld, Bex, at each position on the top surface of the ring bulk(r=−17, −23 and −30 mm) and Al alloy ring (r=−35 mm)
during FCM from 5 T at 50 K.

electromagnetic strain along the θ-direction, εθ, as a function of theexternal ﬁeld, Bex, at each position on the top surface of the bulk(r=17, 23 and 30 mm) and Al alloy ring (r=35 mm) during FCM
from Bapp=5 T at 50 K.

the electromagnetic strain is reasonably inﬂuenced by thesize, shape and assumed mechanical properties of the surrounding parts.
Figure 5(a) shows the experimental results of the externalﬁeld dependence of the electromagnetic strain along the rdirection, εr, at each measurement position, for which themagnetic ﬁeld decreased from 5 to 0 T. The εr values atr=−17, −30 and −35 mm increased negatively, took aminimum, and then showed a slightly positive increase withdecreasing Bex, which are in clear contrast to the εθ values atthe corresponding positions. It can be considered that the
inner periphery of the ring bulk (r =14 mm) and the outer
periphery of the Al alloy (r =37 mm) are in a ‘stress-free’state, and are not restricted mechanically since there are no
external stresses in the r-direction from the inner surface of

not produced in this region (32<r<37 mm) as shown inﬁgure 3(c). This behavior implies that the Al alloy ring was
pushed outwards from the inner bulk due to the Lorentz force
and, as a result, expanded in the θ-direction.
Figure 4(b) shows the results of numerical simulation of
the external ﬁeld dependence of the electromagnetic strain
along the θ-direction, εθ, during FCM from 5 T at 50 K. The
simulated εθ data well reproduced the experimental data
shown in ﬁgure 4(a) qualitatively and quantitatively. Thisagreement results from the reduction of the simulated εθvalues and strongly suggests that such a realistic numericalmodel is necessary for the simulation to reproduce theexperimental data accurately. Because the mechanical stressand strain are related three-dimensionally to each other and6Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

the bulk and outer surface of the Al ring. Thus, the negative εrcan be simply explained by Poisson ratio v =−εθ/εr, and theouter boundary of superconductor bulk is pressed by the Al
alloy ring. As a result, the εr value at r=−30 mm shrinks
largely because the displacement at outer periphery of the ringbulk (r=32 mm) is restricted due to the external compressive stress from the Al alloy ring. On the other hand, the εrvalue at r=−23 mm, which is an intermediate position on
the ring bulk surface, shows a slight positive expansion, takes
a peak, and then decreases slightly with decreasing Bex. It
should be noted that the maximum variation of εr at eachposition is several times smaller than εθ.
Figure 5(b) shows the numerical simulation results of theexternal ﬁeld dependence of the electromagnetic strain along
the r-direction, εr, during FCM from 5 T at 50 K. The simulated εr versus Bex curves at r=−23, −30 and −35 mm
reproduce the experimental data well qualitatively andquantitatively. On the other hand, the simulated εr versus Bex
curve at r=−17 mm exhibits a convex proﬁle, but theexperimental εr versus Bex curve at r=−17 mm shown inﬁgure 5(a) exhibits a concave one. This discrepancy may
come from a difference in the load condition for the stress–
strain analysis; in the actual experimental setup shown inﬁgure 1(a), a stronger fastening torque was applied to theinner periphery of the bulk though the Cu plates along the zdirection produced from a central bolt and nut, rather than tothe outer periphery, which suggest that inhomogeneous
mechanical fastening pressure might be applied to the ringbulk. On the other hand, in the numerical simulation, thetorque was not considered and the mechanical and thermal
contact was assumed to be homogeneous. These discrepancies inﬂuence on the strain proﬁles, especially at the
inner periphery region. However, these results of simulations
and experiments are in good agreement, which suggests thatthe construction of a realistic FEM model and the use ofrealistic average and macroscopic Jc(B) characteristics for thebulk are necessary to reproduce the external ﬁeld dependenceof the electromagnetic strains along both directions qualitatively and quantitatively. This allows us to better understandthe complex physical phenomena during magnetization process and make accurate predictions.

assumed in the Al alloy region, as shown in table 2. Theseequations are obtained from Hooke’s law for an isotropic
material within its elastic range, in which the stress along thez-direction, σz, in the materials is assumed to be zero. This
assumption was used previously in the analytical calculations
for the stress–strain relationship for a ring-shaped bulk
superconductor [22].
Figure 6(a) shows the estimated results of the electromagnetic stress along the θ-direction, σθ, as a function of theexternal ﬁeld, Bex, at each measurement position during FCM
from 5 T at 50 K, which were obtained using equation (10)and the experimental strain values shown in ﬁgures 4(a) and
5(a). Estimated σθ values at each position increase, take amaximum at an intermediate Bex, and then slightly decrease as
the external ﬁeld is ramped down. The σθ value at r=17 mm
is higher, compared to those at the other positions and at any
Bex. Its maximum is estimated to be +29 MPa at Bex=1 T,
meaning that the tensile stress concentration occurs at the
inner periphery, which are similar to the results for εθ shownin ﬁgure 4(a). These trends have been reported in our previous
works [25–29, 42].
Figure 6(b) shows the simulation results of the electromagnetic stress along the θ-direction, σθ, as a function of theexternal ﬁeld, Bex, at each measurement position during FCM
from 5 T at 50 K, for which the numerical simulation was
performed only using the electromagnetic and mechanicalanalyses. The numerical simulation results are in good
agreement with the estimation using the measured results
qualitatively and quantitatively. A maximum stress of+28 MPa was observed at r=17 mm, which is in excellent
agreement with estimated results for the maximum stress at
same position. The fracture strength of a typical Ag-doped
REBaCuO bulk is considered to be 50–70 MPa [43–45]. In
this sense, the present FCM conditions can avoid themechanical fracture. Figure 6(c) shows the σθ distribution in
the ring bulk (14<r<32 mm) and Al alloy ring(32<r<37 mm) at Bex=1 T. The σθ distribution in thering bulk is inhomogeneous and σθ is concentrated at the
innermost edge of the ring bulk surface. These results suggestthat the electromagnetic stress takes a maximum at the
innermost edge of the ring bulk surface, and that theelectromagnetic strain also takes a maximum at the same
position.
Figures 7(a) and (b), respectively, show the estimatedand simulated results of the electromagnetic stresses along ther-direction, σr, as a function of the external ﬁeld, Bex, at eachmeasurement position during FCM from 5 T at 50 K, forwhich the estimated results were obtained using equation (11)and the experimental strain values shown in ﬁgures 4(a) and
5(a). The estimated σr values at r=−23, −30, and −35 mm
were reproduced well by the numerical simulation. The estimated σr versus Bex curve at r=−17 mm was not necessarily consistent with the simulation curve, which may result
from the similar cause as that for the strain measurement asshown in ﬁgure 5(a). Figure 7(c) shows the σr distribution in
the ring bulk (14<r<32 mm) and the Al alloy ring(32<r<37 mm) at Bex=1 T. The σr distribution in thering bulk is inhomogeneous and σr is concentrated at the

4.3. Estimation of electromagnetic stress using measured
strains along both directions

In this subsection, we estimate the electromagnetic stresses
applied to the ring bulk and Al alloy ring during FCM using
the experimentally obtained strains. To estimate the stresses
along both the θ- and r-directions in the bulk and Al alloy ringfrom the experimental strain results, the following stress–
strain relationships are adoptedE(eq + n · er ) ,1 - n2Esr =(er + n · eq ) ,1 - n2

sq =

(10)
(11)

where Ebulk=100 GPa and vbulk=0.33 are assumed in thering bulk region, and EAl=78.0 GPa and vAl=0.34 are
7

Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

Figure 6. (a) Estimated and (b) numerical simulation results for themechanical stress, σθ, along the θ-direction, as a function of theexternal ﬁeld, Bex, at each of the measurement positions during FCM
from 5 T at 50 K. (c) The σθ distribution in the bulk and Al alloy ringat Bex=1 T.

Figure 7. (a) Estimated and (b) numerical simulation results of themechanical stress, σr, along the r-direction, as a function of theexternal ﬁeld, Bex, at each measurement position during FCM from
5 T at 50 K. (c) The σr distribution in the bulk and Al alloy ring at
Bex=1 T.

innermost edge of the ring bulk surface. The maximum σrvalue is smaller than 10 MPa, which is much smaller than thefracture strength. Thus, the ring bulk did not fracture during
FCM from 5 T.

which cannot be directly measured. We have simultaneously
measured electromagnetic strains along both the θ- and rdirections on a large single-grain EuBaCuO ring bulk, reinforced by an Al alloy ring, during FCM from Bapp=5 T at
50 K using several strain gauges. A 3D ﬁnite element modelthat represents the exact experimental setup was constructed,
including not only the ring bulk and Al alloy ring, but also thesample holder and mechanical support structure (OFHC Cu

5. Conclusion
Two-directional strain measurement has been shown to be avaluable method to estimate precise mechanical stresses,
8Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al

References

plates, indium sheets, bolt and nut made of stainless steel(SUS304) and Stycast™ 1266 (epoxy resin)), with excellent
qualitative and quantitative agreement. The important results
and conclusions of this study are summarized as follows.

[1] Gotoh S, Murakami M, Fujimoto H and Koshizuka N 1992
J. Appl. Phys. 72 2404–10[2] Zhou D, Izumi M, Miki M, Felder B, Ida T and Kitano M 2012
Supercond. Sci. Technol. 25 103001[3] Hayashi H, Tsutsumi K, Saho N, Nishijima N and Asano K
2003 Physica C 392–396 745–8[4] Werfel F N, Floegel-Delor U, Rothfeld R, Riedel T, Goebel B,
Wippich D and Schirrmeister P 2012 Supercond. Sci.
Technol. 25 014007[5] Fuchs G et al 2013 Supercond. Sci. Technol. 26 122002[6] Weiss J D, Yamamoto A, Polyanskii A A, Richardson R B,
Larbalestier D C and Hellstrom E E 2015 Supercond. Sci.
Technol. 28 112001[7] Bean C P 1962 Phys. Rev. Lett. 8 250[8] Ren Y, Weinstein R, Liu J, Sawh R P and Foster C 1995
Physica C 251 15–26[9] Fuchs G, Schätzle P, Krabbes G, Gruß S, Verges P, Müller K-H,
Fink J and Schultz L 2000 Appl. Phys. Lett. 76 2107–9[10] Nariki S, Sakai N and Murakami M 2005 Supercond. Sci.
Technol. 18 S126–30[11] Durrell J H et al 2014 Supercond. Sci. Technol. 27 082001[12] Tomita M and Murakami M 2003 Nature 421 517–20[13] Nakamura T, Tamada D, Yanagi Y, Itoh Y, Nemoto T,
Utumi H and Kose K 2015 J. Magn. Reson. 259 68–75[14] Ogawa K, Nakamura T, Terada Y, Kose K and Haishi T 2011
Appl. Phys. Lett. 98 234101[15] Nariki S, Sakai N, Tomita M and Murakami M 2002 Physica C
378–381 779–82[16] Fujimoto H, Murakami A, Teshima H and Morita M 2013
Cryogenics 57 6–11
[17] Konstantopoulou K, Shi Y H, Dennis A R, Durrell J H,
Pastor J Y and Cardwell D A 2014 Supercond. Sci. Technol.
27 115011[18] Huang K Y et al 2018 IEEE Trans. Appl. Supercond 28 6801505[19] Sakai N, Mase A, Ikuta H, Seo S J, Mizutani U and
Murakami M 2000 Supercond. Sci. Technol. 13 770–3[20] Murakami A et al 2003 Cryogenics 43 345–50[21] Johansen T, Wang C, Chen Q Y and Chu W–K 2000 J. Appl.
Phys. 88 2730–3[22] Johansen T 2000 Supercond. Sci. Technol. 13 R121–37[23] Huang C, Yong H and Zhou H 2013 Supercond. Sci. Technol.
26 105007[24] Johansen T H, Chen Q Y and Chu W-K 2001 Physica C 349
201–10[25] Fujishiro H, Ainslie M D, Takahashi K, Naito T, Yanagi Y,
Itoh Y and Nakamura T 2017 Supercond. Sci. Technol. 30
085008[26] Takahashi K, Fujishiro H, Naito T, Yanagi Y, Itoh Y andNakamura T 2017 Supercond. Sci. Technol. 30 115006[27] Takahashi K, Fujishiro H, Naito T, Yanagi Y, Itoh Y andNakamura T 2018 IEEE Trans. Appl. Supercond. 28
6800705[28] Fujishiro H, Takahashi K, Naito T, Yanagi Y, Itoh Y andNakamura T 2018 Physica C 550 52[29] Fujishiro H, Naito T, Yanagi Y, Itoh Y and Nakamura T 2019
Supercond. Sci. Technol. 32 065001[30] Fujishiro H, Naito T and Awaji S 2019 Supercond. Sci.
Technol. 32 045005[31] Miyamoto T, Nagashima K, Sakai N and Murakami M 2000
Physica C 340 41–50[32] Takahashi K, Namba S, Fujishiro H, Naito T, Yanagi Y,
Itoh Y and Nakamura T 2019 Supercond. Sci. Technol. 32
015007[33] Morita M, Sawamura M, Takebayashi S, Kimura K,
Teshima H, Tanaka M, Miyamoto K and Hashimoto M 1994
Physica C 235–240 209–12(1) The electromagnetic strain along the θ-direction, εθ,takes a maximum at an intermediate external ﬁeld and is
the largest at the innermost position of the bulk surface,decreasing with increasing radius. The electromagnetic
strain along the r-direction, εr, is several times smaller
than the εθ, and has a contrasting Bex dependence ateach position along the bulk surface.
(2) The simulation results of the electromagnetic strains
along both directions show excellent qualitative andquantitative agreement with the experimental ones. In
the numerical simulation, the construction of a realistic
model based on all parts of the actual experimental
setup is necessary to reproduce the experimental results
precisely. The quantitative agreement between theexperiments and simulations validate both our experimental and simulation techniques, and this numericalsimulation framework can be used as a reliable tool toinvestigate the possibility of mechanical fracture of bulk
superconductors for any bulk superconducting material
and geometry or magnetizing process.
(3) The electromagnetic stresses, σθ and σr, along both
directions are estimated during the FCM process from theexperimentally obtained strains, εθ and εr, using Hooke’s
law. The consistency of the experimental and numericalresults suggests that this kind of strain measurement canestimate electromagnetic stresses precisely and is aneffective technique for the evaluation of the mechanicalproperties of bulk superconductors during FCM to clarify
the possibility of mechanical fracture.

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 Nos.
15K04646 and 19K05240. M D Ainslie would like toacknowledge ﬁnancial support from an Engineering andPhysical Sciences Research Council (EPSRC) Early Career
Fellowship, EP/P020313/1. All data are provided in full in
the results section of this paper.

ORCID iDs
Sora Namba https://orcid.org/0000-0001-7268-5326
Hiroyuki Fujishiro https://orcid.org/0000-00031483-835X
Tomoyuki Naito https://orcid.org/0000-0001-7594-3466
Mark D Ainslie https://orcid.org/0000-0003-0466-3680
Kai Y Huang https://orcid.org/0000-0001-7476-305X
9

Supercond. Sci. Technol. 32 (2019) 125011

S Namba et al[40] Muralidhar M, Sakai N, Jirsa M, Koshizuka N and
Murakami M 2004 Appl. Phys. Lett. 85 3504
[41] Kii T et al 2012 IEEE Trans. Appl. Supercond. 22 4100904[42] Ainslie M D, Huang K Y, Fujishiro H, Chaddock J,
Takahashi K, Namba S, Cardwell D A and Durrell J H 2019
Supercond. Sci. Technol. 32 034002[43] Lee D and Salama K 1990 Japan. J. Appl. Phys. 29 L2017–9[44] Matsui M, Sakai N and Murakami M 2002 Supercond. Sci.
Technol. 15 1092–8[45] Katagiri K, Murakami A, Kan R, Kasaba K, Noto K,
Muralidhar M, Sakai N and Murakami M 2003 Physica C
392-396 526–30[34] Nariki S, Teshima H and Morita M 2016 Supercond. Sci.
Technol. 29 034002[35] Fujishiro H, Itoh Y, Yanagi Y and Nakamura T 2015
Supercond. Sci. Technol. 28 095018[36] Ainslie M D and Fujishiro H 2015 Supercond. Sci. Technol. 28
053002[37] Ainslie M D, Srpcic J, Zhou D, Fujishiro H, Takahashi K,
Cardwell D A and Durrell J H 2018 IEEE Trans. Appl.
Supercond. 28 6800207[38] Zhou D et al 2018 Supercond. Sci. Technol. 31 105005[39] Jirsa M, P̊ ust L, Dlouhý D and Koblischka M R 1997 Phys.
Rev. B 55 3276–84

10