Superconductor Science and Technology
Supercond. Sci. Technol. 35 (2022) 045015 (10pp)

https://doi.org/10.1088/1361-6668/ac5785

Possibility of mechanical fracture ofsuperconducting ring bulks due tothermal stress induced by local heatgeneration during pulsed-fieldmagnetization
Motoki Shinden1, Hiroyuki Fujishiro1,∗, Keita Takahashi2 and Mark D Ainslie3
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 Physics, Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima,
Tokyo 171-8588, Japan3
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ,
United Kingdom
E-mail: fujishiro@iwate-u.ac.jp
Received 25 October 2021, revised 17 February 2022
Accepted for publication 22 February 2022
Published 4 March 2022
Abstract

During quasi-static magnetization of bulk superconductors using field-cooled magnetization(FCM) from high fields at low temperatures, such bulks are sometimes broken, which isbelieved to be mainly due to an electromagnetic force—and subsequent stress—larger than thefracture strength. However, a ring bulk can break, even during pulsed field magnetization(PFM), from relatively lower pulsed fields and at relatively higher temperatures. Previous
simulation results suggest that the ring bulk should not break due to the electromagnetic force
during PFM. In this paper, taking experimental and numerical results into consideration, we
propose the possibility of mechanical fracture of a ring bulk during PFM due to thermal stressinduced by local heat generation, which has not been considered and investigated to date. Twonumerical models with different sizes of heat-generating region were constructed for the ringbulk with a relatively large inner diameter (60 mm outer diameter, 36 mm inner diameter,
17 mm height). For Model-1, with a large heat region, the bulk fracture due to the thermal stressresults from the tensile stress along the radial direction in the neighboring heat region. The riskof bulk fracture is enhanced at the inner or outer edges of the bulk surface, compared with that
inside the bulk. For Model-2, with a small heat region inside the bulk, the bulk fracture due tothe thermal stress results from the compressive stress along the radial direction in theneighboring heat region. These results strongly suggest the possibility of mechanical fracture ofan actual ring bulk due to thermal stress induced by local heat generation. This idea is also
applicable more generally to the fracture mechanism during FCM of superconducting bulks.
Keywords: mechanical fracture, thermal stress, superconducting ring bulk, flux jump,
numerical simulation, pulsed field magnetization, local heat generation(Some figures may appear in colour only in the online journal)∗

Author to whom any correspondence should be addressed.

1361-6668/22/045015+10$33.00 Printed in the UK

1© 2022 IOP Publishing Ltd

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1. Introduction

the local overheating and the destruction of superconductivity
due to flux jumps. We have performed PFM experiments on
a GdBaCuO ring bulk with a relatively large I.D. (60 mm
in O.D., 36 mm in I.D., 17 mm in H) for an applied field
Bapp = 3.10 T at 65 K [22]. A flux jump took place during
the ascending stage of the pulse, and the final trapped field
had a negative value and exhibited a ‘C-shaped’ trapped field
profile. After the PFM experiment, a mechanical fracture wasconfirmed by examining the trapped field profile by FCM inliquid nitrogen.
The mechanical properties have been analyzed for superconducting bulks with an infinite height during FCM and zerofield-cooled magnetization [23–25]. We have also investigatedthe mechanical properties of disk- and ring-shaped REBaCuO
bulks with finite height during FCM using the finite element
method (FEM) and proposed optimal reinforcement structures
made of metal to avoid mechanical fracture [19, 26–29]. For
PFM, FEM analyses of mechanical properties have been performed such as quench and crack propagation in the disk bulk
[30–34]. We have also investigated the electromagnetic andthermal properties of REBaCuO ring bulks with an inhomogeneous critical current density, J c , profile during PFM using
numerical simulations, and compared these with those of aring bulk with a homogeneous J c profile [35], in which theelectromagnetic and thermal hoop stresses were lower than thefracture strength of the bulk material. These simulation results
suggest that the experimentally observed ‘C-shaped’ profile
does not result from the bulk fracture, but rather from the disconnect of the circulating supercurrent by a thermally-induced
flux jump. However, the ring bulk was actually broken during
PFM, even for lower Bapp and a higher operating temperature,T s [22]. We must consider another cause of the mechanicalfracture of the ring bulk during PFM. In a previous study, theflux penetration promotes the local temperature rise, which isestimated to be higher than T c [35]. The results strongly suggest the possibility of the mechanical fracture due not to theelectromagnetic hoop stress, but rather the local thermal stressrelated to the flux jump. Such a proposal, and related investigations, have not yet been considered in the literature.
In this paper, the possibility of mechanical fracture of asuperconducting ring bulk due to thermal stress induced by
local heat generation during the PFM process was investigated numerically. First, PFM experiments on a GdBaCuO ringbulk were performed, for which the mechanical fracture wasconfirmed again, similar to our previous experiments [22].
Second, to clarify the influence of local heat generation on themechanical fracture of the ring bulk, two numerical models
with different sizes of heat-generating region are constructed
and the thermal stress is investigated for each case, in which
the electromagnetic stress is not considered. As a result, thepossibility of mechanical fracture of a superconducting ringbulk specifically due to thermal stress during PFM is strongly
suggested. The proposed fracture mechanism due to thermal
stress is also applicable to the bulk fracture during FCM, forwhich only an electromagnetic origin has mainly been considered.

A superconducting bulk can be used as trapped field magnet inapplications such as motors and generators [1], magnetic separation [2] and nuclear magnetic resonance (NMR) [3]. In particular, REBaCuO superconducting bulks (RE: rare earth element or Y) exhibit a superior temperature and magnetic fielddependence of critical current density, J c (T, B), which haveresulted in trapped fields, BT FCM , over 17 T in disk bulk pairs
by field-cooled magnetization (FCM) [4–6].
In addition, pulsed-field magnetization (PFM) has been
intensively investigated as a compact, mobile and relatively
inexpensive magnetizing technique, in which a magnetic pulse
as short as several milliseconds is applied to the bulk cooled
below its critical temperature, T c . However, the trapped fieldby PFM, BT PFM , is often much lower than BT FCM because of alarge temperature rise due to the dynamical motion of the magnetic flux in the bulk. Various studies on PFM techniques havebeen performed to reduce the temperature rise and to enhance
the BT PFM value [7, 8]. At present, the maximum record of
BT PFM is 5.2 T at 29 K for an Ag-containing GdBaCuO disk
bulk 45 mm in diameter magnetized by the modified multipulse technique combined with stepwise cooling [9] and 5.3 T
at 30 K between a stack of two Ag-containing GdBCO diskbulks 30 mm in diameter using an optimized two-step pulse
sequence [10]. However, for a ring bulk, only a small BT PFM
has been observed to date [11, 12]. For example, BT PFM of
0.25 T was achieved at 60 K for the GdBaCuO ring bulk
(30 mm in outer diameter (O.D.), 12 mm in inner diameter(I.D.) and 12 mm in height (H)), because flux jumps readily
occur, even for relatively lower applied pulsed fields [13].
Since REBaCuO bulk materials are intrinsically a brittle
ceramic material, the mechanical strength, especially thetensile strength, is fairly low. The mechanical strength ofbulk materials has typically been measured by bending tests[14–16] and tensile tests [17, 18], from which the fracture strength of tensile stress is suggested to be as low as
50–70 MPa for a typical Ag-doped REBaCuO bulk [19]. The
fracture strength of the REBaCuO bulk for compressive stress
was reported to be at least −200 MPa [20, 21]. A large tensilestress (or hoop stress) due to the Lorentz force is generated
in the bulk during FCM, especially for fields higher than several tesla, which sometimes creates a crack and eventuallymechanical failure. To avoid mechanical fracture, mechanicalreinforcement is usually applied by shrink-fit stainless steel or
epoxy resin impregnation [4, 5].
On the other hand, for PFM, the compressive and tensilestresses are generated in the bulk during the ascending anddescending stages of PFM, respectively. The magnitude of theelectromagnetic stress is relatively low, compared with that for
the FCM process, because the strength of the applied field andresultant trapped field is lower and J c (T, B) is reduced dueto the heat generated. As a result, there is no report of mechanical fracture of a disk bulk during the PFM process, as far
as the authors’ knowledge. However, during the PFM of ringbulks, mechanical fracture has been observed, in addition to2

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2. Experimental procedure
2.1. Experimental setup and FCM process

The experimental PFM apparatus and procedure are described
elsewhere in detail [36]. The GdBaCuO superconducting ringbulk (60 mm in O.D., 36 mm in I.D., 17 mm in H), fabricated by Nippon Steel, Japan, was thermally connected to thecold stage of a GM-cycle helium refrigerator and then evacuated in a vacuum chamber. The temperature of the cold stage
was controlled at T s = 40 K using a Pt/Co thermometer andresistive heater. A magnetic pulse, Bex (t), was applied to thebulk using a solenoid-type copper coil set outside the vacuum
chamber and a capacitor bank. The peak of Bex (t) is defined as
Bapp , and the rise time and duration time are tr = 13 ms andtd = 200 ms, respectively. The time dependence of Bex (t) wasmonitored from the current I(t) flowing in a shunt resistor. The
time evolution of the magnetic field, Bz (t), was measured at
the bore center of the ring bulk using an axial-type Hall sensor(F. W. Bell, BHA-921). The time dependence of the temperature, T(t), was measured on the side surface of the stainlesssteel holder of the ring bulk using a Cernox thermometer.
After the PFM process at T s = 40 K, a two-dimensional
trapped field profile of Bz was mapped at 3 mm above the ringbulk surface in the vacuum chamber by scanning the same Hallsensor using an x–y stage controller. To confirm the mechanical fracture, the ring bulk was removed from the apparatus,then the bulk was magnetized by FCM from 1.5 T in liquid
nitrogen and a two-dimensional trapped field profile of Bz was
mapped at 1 mm above the ring bulk surface at 77 K.

Figure 1. Schematic views of the numerical models for the ringbulk: (a) top view of the 3D simulation model. (b) Model-1, where alarge heat region (red) is defined at 18 ⩽ r ⩽ 30 mm, 89 ⩽ φ ⩽ 92◦ ,and −8 ⩽ z ⩽ 8 mm. (c) Model-2, where a local and small heatregion (red) is defined at 22 ⩽ r ⩽ 25 mm, 89 ⩽ φ ⩽ 92◦ and−1 ⩽ z ⩽ 2 mm (see text).

anywhere, and to be thermally insulated (adiabatic) and mechanically ‘floating’.
Elastic behavior in an isotropic material is expressed by
Hooke’s law [26], in which the stress tensor, σ ij , is linearly
proportional to the strain tensor, εij , as follows,
σij = λ · εkk · δij + 2G · εij ,

(1)

E·ν
,
(1 + ν) (1 + 2ν)

(2)

E
,
2 (1 + ν)(3)λ=3. Numerical simulation frameworkG=

We constructed a three-dimensional (3D) numerical model for
a REBaCuO ring bulk (60 mm in O.D., 36 mm in I.D., 16 mmin H), of a similar size as the experiment. The commercial software package, Photo-Thermo-Elas (Photon Ltd, Japan), wasused for analyses of the thermal and mechanical properties.
The model was equally divided every 1◦ along the circumferential (φ) direction and every 1 mm along the radial (r)- andcentral axis (z)-directions. Figure 1(a) shows the top view ofthe simulation model. The azimuth angle, φ, was defined asshown in the figure and the heat region was set between 89◦
and 92◦ (totaling 3◦ ). In this study, two types of heat region
were assumed, named Model-1 and Model-2, as shown infigures 1(b) and (c), in which the heat power was induced forcibly. Model-1 was constructed based on the previous results
showing the ‘C-shaped’ trapped field profile [22], as shownin figure 1(b), where the heat region was 18 ⩽ r ⩽ 30 mm,
89 ⩽ φ ⩽ 92◦ and −8 ⩽ z ⩽ 8 mm. The number of meshelements within the heat region is 576, which is 1/120 of thetotal mesh. Model-2 was constructed as a local and small heatregion, as shown in figure 1(c), where the heat region was
22 ⩽ r ⩽ 25 mm, 89 ⩽ φ ⩽ 92◦ and −1 ⩽ z ⩽ 2 mm. The number of mesh elements in the heat region is 27, which is about
1/21 of that of Model-1. As thermal and mechanical boundary conditions, the modeled bulk was assumed not to contact

where λ and G are the Lame’s constants, δij is the Kronecker
delta function, E (=100 GPa) is the Young’s modulus, and ν(=0.33) is the Poisson ratio of the GdBaCuO bulk material.
The thermal expansion coefficient, α0 , of the bulk material is
assumed to be 5.2 × 10−6 K−1 . All the mechanical parameters
were used in our previous simulation studies [26, 37, 38].
In this study, the maximum temperature, T max , of the heatregion for each model was assumed as 90 K, 140 K, 200 K,
250 K and 300 K. The induced heat, Q, which was required
to raise up to T max from T s = 40 K, was calculated using thefollowing equation,
ˆTmax
Q=

C (T) Vheat dT,(4)

Ts =40 K

where V heat is the volume of the heat region for each model.
C(T) (J (m3 K)−1 is the temperature dependence of the specific heat, for which the measured C(T) [39] was approximated using the following 6th degree polynomial equation. The
parameters, a0 to a6 , in equation (5) are shown in table 1. The
temperature dependence of thermal conductivity, κ(T), wasreferred from [40].
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Table 1. Fitting parameters for equation (5).

Parameter

a0

a1

a2

a3

a4

a5

a6

1.513 × 103−4.431 × 103

4.955 × 102−5.565 × 10◦

2.869 × 10−2

−7.153 × 10−5

6.943 × 10−8

C (T) = a0 + a1 T + a2 T2 + a3 T3 + a4 T 4 + a5 T5 + a6 T 6 (5)center of the ring bulk for Bapp = 3.49 T. The magnetic
shielding effect is not observed at t < 5 ms and Bex (t) and
Bcenter (t) overlapped fully. Bcenter (t) at t = 200 ms was negative(= −0.05 T) similar to figure 2(a). These results also indicate
the disconnection of circulating supercurrent and suggest thatthe ring bulk was broken during the first pulse application. An
FCM experiment was performed from Bapp = 1.5 T at 77 K.
Figure 2(d) presents the trapped field profile at 1 mm above thering bulk. The ‘C-shaped’ trapped field profile was confirmed,
which suggests the bulk fracture with good certainty. We concluded that the present ring bulk was fractured at aroundt = 5 ms during the first pulse application (Bapp = 3.88 T)
at 40 K.
Our group previously investigated the electromagnetic andthermal properties of a REBaCuO ring bulk with an inhomogeneous J c profile during PFM using numerical simulations[35], in which Bapp = 4.0 T was applied at T s = 65 K. The calculated compressive electromagnetic stress and tensile thermal
stress were lower than the fracture strength of the bulk material, suggesting that the experimentally observed ‘C-shaped’profile resulted from a (thermomagnetic) flux jump, rather thanthe bulk fracture. However, the present ring bulk was actuallybroken. We must consider a cause of this fracture other thanthat of an electromagnetic origin.

As shown in the next section, a flux jump, i.e. an abrupt flux
intrusion and related heat generation, took place at t = 5 ms.
For t < 5 ms, a shielding current, I shield , flows in the ringbulk and increases monotonically with increasing time [41],at which there is no heat generation. In the present numerical simulation, to realize the convergence of the calculation,
we assume that the heat power q (W m−3 ) is applied to theheat region and then the temperature increases between 0 and
5 ms and reaches T max , which may be slightly different tothe actual phenomenon. As a result, a local thermal stressmust be generated due to the local thermal expansion difference between heat region and its neighboring region. The
thermal stress increases with time for 0 < t < 5 ms, and thendecreases for t > 5 ms because q = 0 for t > 5 ms. In thisstudy, the thermal stress along the radial (r) direction, σ r heat ,and that along the circumferential (φ) direction, σ φ heat , were
investigated at each T max at t = 5 ms for the heat region(89◦ < φ < 92◦ ) and neighboring regions (φ < 89◦ , φ > 92◦ )for each model. Korotokov and Brazhnik calculated the shielding current, I shield , from which the electromagnetic energy,
Q = LI shield 2 /2, accumulated in the bulk was released to theheat region by the flux jump [41–43], where L is the inductance of the ring bulk. Based on the results of Korotokov andBrazhnik [41, 44], we calculated I shield for each T max and foreach model, where the L value in the present ring bulk was
estimated to be L = 1.048 × 10−7 H.

5. Numerical results and discussion
5.1. Induced heat Q to reach the maximum temperature Tmax

First, the induced heat, Q, to reach a maximum temperature, T max , of 90 K, 140 K, 200 K, 250 K and 300 K ineach model was estimated. The heat power q (W m−3 ) wasapplied to the heat region for 5 ms to reach each T max .
Figure 3(a) shows the time dependence of the temperature,
T, to reach T max = 140 K and 300 K for Model-1. When aheat power of q = 8.21 × 1010 (W m−3 ) was applied for 5 ms
to the heat region (V heat = 2.56 × 10−7 m3 ), the temperature of the heat region increased monotonically with time andbecame T max = 300 K at t = 5 ms. In this case, the total
induced heat was Q = qV heat t = 105 J. When a heat powerof q = 2.20 × 1010 (W m−3 ) was applied for 5 ms to the heatregion, the total induced heat was Q = qV heat t = 25.8 J and thetemperature became T max = 140 K at t = 5 ms. The mechanical results after this are shown at T max (t = 5 ms).
Figure 3(b) shows the relationship between T max and Q
for each model. T max increases with increasing Q for eachmodel with a nearly square root dependence of T max on Q, i.e.
T max ∼ Q1/2 . The Q value in Model-1 is 21 times as large as thatin Model-2 to reach the same T max because of the same ratioof V heat for the two models. The temperature was confirmed torise from 40 K to 46 K for PFM with Bapp = 3.88 T, as shown infigure 2(a). In this case, the estimated heat generation—using

4. Experimental results
Figure 2(a) presents the time dependence of the applied pulsed
field, Bex (t), and trapped field, Bcenter (t), at the bore center ofthe ring bulk for T s = 40 K and Bapp = 3.88 T. Bcenter (t) was
zero for 0 ⩽ t ⩽ 5 ms due to flux shielding and then abruptly
increased with time for t > 5 ms. After that, Bcenter (t) was larger than Bex (t), took a maximum, and then became negative(= −0.08 T) at t = 200 ms. The final negative value suggests the disconnection of the circulating supercurrent in thebulk [22, 35, 42]. The characteristic result of Bcenter (t) > Bex (t)comes from the so-called ‘magnetic lens effect’ [45, 46],
which appeared due to the disconnection of circulating supercurrent as pointed out by Brazhnik [44]. Figure 2(b) shows
the trapped field profile 3 mm above the bulk surface after
PFM. The ‘C-shaped’ trapped field profile was observed. At
this stage, we cannot conclude the origin of the ‘C-shaped’trapped field profile, i.e. whether it is due to thermal disconnection of the circulating supercurrent or mechanical fractureof the ring bulk.
After that, the bulk was warmed above T c to release thetrapped field, and then cooled to T s = 40 K again. Figure 2(c)shows the time dependence of Bex (t) and Bcenter (t) at the bore
4

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Figure 2. (a) Time dependence of the applied pulsed field, Bex (t), and trapped field, Bcenter (t), at the bore center of the ring bulk for

Bapp = 3.88 T (1st pulse). (b) Trapped field profile 3 mm above the bulk surface after PFM. (c) Time dependence of Bex (t) and Bcenter (t) at
the bore center of the ring bulk for Bapp = 3.49 T (2nd pulse). (d) Final trapped field profile of the ring bulk 1 mm above the bulk surface by
FCM from Bapp = 1.5 T at 77 K.

Figure 3. (a) Time dependence of the temperature, T, to reach maximum temperatures, T max = 140 K and 300 K for Model-1. (b) The

relationship between T max and induced heat, Q, for each model.

T max for Model-1. In the heat region (φ = 89 ∼ 92◦ ), both
thermal stresses were negative, i.e. compressive stress. Eachthermal compressive stress increases with increasing T max ;σ r heat was over −100 MPa at T max = 300 K. However, σ φ heatis about one order of magnitude smaller than σ r heat . This may
come from the possibility of displacement in the heat region
in Model-1 with open ends along the r- and z-directions. On
the other hand, σ r heat and σ φ heat for the neighboring regions(φ < 89◦ , φ > 92◦ ) are positive, i.e. tensile stress, and increase
with increasing T max .

equation (4) and C(T) of the ring bulk and its support ring–was
141 J. These results suggest that the local temperature in thebulk could possibly increase above 300 K, if most of the heat
power is released in the local heat region.
5.2. Angular dependence of thermal stress for Model-1

Figures 4(a) and (b) show the angular (φ) dependence ofthe thermal stress along the r-direction, σ r heat , and along theφ-direction, σ φ heat , in the bulk (r = 24 mm, z = 0 mm) for each5Supercond. Sci. Technol. 35 (2022) 045015

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Figure 4. Angular (φ) dependence of thermal stress along (a) the r-direction, σ r heat , and (b) the φ-direction, σ φ heat , in the bulk (r = 24 mm,

z = 0 mm) for each T max for Model-1. (c) The relationship between the maximum thermal compressive stress, σ heat (comp.), and the
maximum thermal tensile stress, σ heat (tensile), for σ r heat and σ φ heat for each T max . The data above 300 K are extrapolated to higher
temperatures. The fracture strength in tension, σ c (tensile) = 50 MPa, and in compression, σ c (comp.) = 250 MPa, are indicated.

tensile stress, σ heat (tensile), along the r-direction in the neighboring region, if such fracture happens inside of the ring bulk.

Figure 4(c) shows the relationship between the maximum thermal compressive stress, σ heat (comp.), in the heatregion and the maximum thermal tensile stress, σ heat (tensile),in the neighboring region for σ r heat shown in figure 4(a)and σ φ heat shown in figure 4(b) for each T max . The data
above 300 K are extrapolated to the higher temperatures.
The fracture strength in tension, σ c (tensile), for a typicalAg-doped REBaCuO bulk was as low as 50–70 MPa [19],and that in compression, σ c (comp.), was reported to beat least −200 MPa [20, 21]. In this study, the fracture
strength in tension, σ c (tensile) = 50 MPa, and in compression, σ c (comp.) = 250 MPa, were adopted as typical values. σ heat (tensile) is linearly proportional to σ heat (comp.) along
both directions and the maximum thermal stress along ther-direction is larger, compared with that along the φ-direction.
In the case that σ r (tensile) reaches σ c (tensile) = 50 MPa, it isnecessary to increase the temperature as high as 400 K. In thecase that σ r (comp.) reaches σ c (comp.) = 250 MPa, it is necessary to increase the temperature higher than 500 K. For thethermal stress along the y-direction, it is difficult to reach thefracture strength for tensile and compressive stresses. Whenthe flux jump and the subsequent bulk fracture happened for
REBaCuO bulk during FCM, the trace of burning was confirmed at the fracture position [47]. The results suggest thatthe temperature could raise several hundreds of kelvin and theestimation of the temperature rise seems reasonable. In Model1, the bulk fracture due to the thermal stress results from the

5.3. Influence of an open-end in the heat region on thethermal stress in Model-1

In Model-1, open ends exist in the heat region along ther- and z-directions. In this subsection, the thermal stress atthe open ends in the heat region is investigated. As shown infigure 5(a), the cross-sections of the center of the heat region(φ = 90◦ ), Ωheat , and of the neighboring region (φ = 95◦ ),Ωcool , were defined. The thermal stress profiles, σr heat along
the r-direction and σ φ heat along the φ-direction, were mapped
at T max = 300 K for each cross-section. Figures 5(b) and (c)show the σ r heat and σ φ heat profiles in Ωheat and figures 5(d) and(e) show the σ r heat and σ φ heat profiles in Ωcool , respectively.
Let us consider the thermal stress along the r-direction foreach cross-section first. σ r heat in Ωheat in figure 5(b) indicatesa compressive stress in all regions of Ωheat (σ r < 0) and negatively increases as we approach the bulk center. On the other
hand, σ r heat in Ωcool in figure 5(d) indicates a tensile stress inall regions of Ωcool (σ r > 0). This trend reproduces the result in figure 4(c). Next, we consider the thermal stress along
the circumferential direction. At the outer surface of Ωheat , atensile stress exists (σφ > 0), as shown in figure 5(c), andσφ = 80 MPa was concentrated at the outer top surface (r = 18,
30 mm, φ = 90◦ , z = ±8 mm). In Ωcool , as shown in figure 5(e),6Supercond. Sci. Technol. 35 (2022) 045015

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equation (4). The shielding current, I shield , was also estimatedto be 3.1 × 104 A using the relation of Q = LI shield 2 /2. I shieldflowing in the GdBaCuO ring bulk was experimentally measured to be 1.6–2.1 × 104 A at 77 K [26]. Since I shield for thepresent case, which is for a lower T s and larger cross sectionof the ring bulk, is larger than that for the reported case in [41],such local heat generation due to inductance loss at the surface
could take place during PFM.
5.4. Angular dependence of thermal stress for Model-2

In this subsection, the angular dependence of the thermal stressfor Model-2 is described, in a similar way as for Model-1.
Figures 7(a) and (b), respectively, show the φ dependenceof the thermal stress along the r-direction, σ r heat , and along
the y-direction, σ φ heat , in the bulk (r = 24 mm, z = 0 mm)
for each T max . Each thermal compressive stress increases withincreasing T max in the heat region (φ = 89◦ –92◦ ). The compressive σr heat along the r-direction shown in figure 7(a) issimilar to that for Model-1 in figure 4(a). However, the compressive σφ heat along the φ-direction in the heat region shownin figure 7(b) is six times larger than that in figure 4(b)for Model-1. σφ heat along the φ-direction in the neighboring
region (φ < 89◦ , φ > 92◦ ) was nearly zero. The heat region
in Model-2 exists inside the ring bulk, where the mechanicaldisplacement due to thermal stress is restricted in all directions, which results in the difference in thermal stress between
Model-1 and Model-2.
Figure 7(c) shows the relationship between the maximum thermal compressive stress, σ heat (comp.), in theheat region and the maximum thermal tensile stress,
σ heat (tensile), in the neighboring region for σ r heat shownin figure 7(a) and for σ φ heat shown in figure 7(b) for each
T max . Again, the data above 300 K are extrapolated tohigher temperatures. σ heat (tensile) is nearly proportional toσ heat (comp.) along the r-direction. In the case that σ heat (comp.)
reaches σ c (comp.) = 250 MPa and σ heat (tensile) reaches
σ c (tensile) = 50 MPa for each direction, it is necessary
to increase the temperature higher than 500 K. However,
σ heat (tensile) along the y-direction is nearly zero in theneighboring region, contrary to Model-1. The relationshipin figure 7(c) suggests that the σ heat (comp.) value along ther-direction reaches σ c comp. = 250 MPa in the heat region
with increasing T max , before the σ heat (tensile) value along
the r-direction reaches σ c tensile = 50 MPa in the neighboring
region. That is, in Model-2, the bulk fracture due to thermal
stress results from the compressive stress, σ heat (comp.), along
the r-direction in the neighboring region.
We consider the thermal stress along the r- and φ-directions
for each cross-section in Model-2 similar to that shown for
Model-1 in figure 5. Figure 8(a) shows the cross-sectionsof the center of the heat region (φ = 90◦ ), Ωheat , and of
the neighboring region (φ = 95◦ ), Ωcool . The thermal stressprofiles, σ r heat along the r-direction and σ φ heat along theφ-direction, were mapped at T max = 300 K for each crosssection. Figures 8(b) and (c) show the σ r heat and σ φ heat profiles in Ωheat and figures 8(d) and (e) show the σ r heat and σ φ heat
profiles in Ωcool , respectively.

Figure 5. (a) Schematic view of the cross-sections of the center ofthe heat region (φ = 90◦ ), Ωheat , and of the neighboring region(φ = 95◦ ), Ωcool , for Model-1. The thermal stress profiles of (b)σ r heat and (c) σ φ heat in Ωheat . The thermal stress profiles of (d) σ r heatand (e) σ φ heat in Ωcool at T max = 300 K for Model-1.

Figure 6. The circumferential tensile stress, σ φ heat , at the inner edge(r = 18 mm, φ = 90◦ , z = 8 mm) and the outer edge (r = 30 mm,
φ = 90◦ , z = 8 mm) of the heat region in Model-1, as a function of
T max . σr heat for the neighboring region (r = 24 mm, φ = 95◦ ,z = 0 mm) is also shown, which was extracted from figure 4(c).

the strength of σ φ was moderate at the top/bottom and outer/inner surface regions.
Figure 6 shows the circumferential tensile stress, σ φ heat ,at the inner edge (r = 18 mm, φ = 90◦ , z = 8 mm) andouter edge (r = 30 mm, φ = 90◦ , z = 8 mm) of the heatregion in Model-1, as a function of T max . σ r heat for the neighboring region (r = 24 mm, φ = 95◦ , z = 0 mm), which is
extracted from figure 4, is also shown. It should be noted thatσ φ heat at the inner and outer edges of the heat region reaches
σ c tensile even around 200 K. In figure 4(c), we concluded thatthe bulk fracture due to the thermal stress, σ r heat , results fromthe tensile stress in the neighboring region, where a temperature rise higher than 400 K was necessary to break the bulk
from inside. These results suggest that the risk of bulk fractureis enhanced at the edge of the bulk surface, rather than inside
the bulk.
The induced heat Q was estimated to be 53 J in the case thatthe temperature rises from T s = 40 K to T max = 200 K using
7

Supercond. Sci. Technol. 35 (2022) 045015

M Shinden et al

Figure 7. Angular (φ) dependence of thermal stress along (a) the r-direction, σ r heat , and (b) the φ-direction, σ φ heat , in the bulk (x = 24 mm,

z = 0 mm) for each T max for Model-2. (c) The relationship between the maximum thermal compressive stress, σ heat (comp.), and the
maximum thermal tensile stress, σ heat (tensile), for σ r heat and σ φ heat for each T max . The data above 300 K are extrapolated to higher
temperatures. The fracture strength in tension, σ c (tensile) = 50 MPa, and in compression, σ c (comp.) = 250 MPa, are indicated.

stress for the small heat region and a small tensile stress forthe neighboring region are generated at φ = 90◦ , as shownin figure 8(c). These tendencies are weakened at φ = 95◦ ,as shown in figure 8(e). It should be noted that, contrary to
Model-1 in figure 5, the stress increase was not observed at
the corner of the heat region, because there are no open ends
in the heat region in Model-2.
Finally, we consider which model reflects the real situation
for the bulk fracture. Early in the ascending stage of PFM,
magnetic flux intrudes from the outer surface of the bulk withsmall heat generation. After that, the dominant heat generation
can take place due to a flux jump from a starting point suchas voids, impurities and crystal defects. The realistic situation
may be somewhere between Model-1 and Model-2, dependingon the exact microstructure of the sample. On the other hand,
Brandt pointed out numerically that the magnetic flux intrudes
also from inner edge of a ring bulk of finite thickness [48].
This may create a sufficiently large temperature difference andstress near the inner edge for a crack to propagate from a local
defect, resulting in the ring bulk fracture. In our study, the possibility of ring bulk fracture by thermal stress during PFM wassuggested for a ring bulk with a large I.D, in which the ratio,
ID/OD is 0.6. If the I.D. of the ring bulk is smaller, in which thebulk fracture was not reported for ID/OD = 0.4, for example
in [13], the ‘C-shaped’ profile may arise as the result of local
overheating and the destruction of superconductivity due toa flux jump. This idea is also applicable to superconducting

Figure 8. (a) Schematic view of the cross-sections of the center ofthe heat region (φ = 90◦ ), Ωheat , and of the neighboring region(φ = 95◦ ), Ωcool , for Model-2. The thermal stress profiles of (b)σ r heat and (c) σ φ heat in Ωheat . The thermal stress profiles of (d) σ r heatand (e) σ φ heat in Ωcool at T max = 300 K for Model-2.

In figure 8(b), σr heat in the small heat region (φ = 90◦ ),Ωheat , indicates a compressive stress (σr < 0), but that outside of Ωheat shows a small tensile value. On the other hand,
σr heat in the center of the neighboring region (φ = 95◦ ), Ωcool ,in figure 8(d) indicates a tensile stress, which reflects theexistence of the small heat region at φ = 89◦ –92◦ . For thethermal stress along the φ-direction, the small compressive8Supercond. Sci. Technol. 35 (2022) 045015

M Shinden et albulks magnetized by FCM, for which the cause of bulk fracture has been mainly considered only to be due to the electromagnetic hoop stress.

Acknowledgments
This new concept of thermal stress origin started from discussion for the mechanical fracture of the disk bulk during FCM
between the author (H.F.) and Prtof. Tomoyuki Naito (T.N.)
of Iwate University in 2018. H.F. thanks T.N. for valuablesuggestions. The authors thank Mr Michael Beck of the University of Cambridge for valuable discussions. This researchis supported by JSPS KAKENHI Grant No. 19K05240, andby Adaptable and Seamless Technology transfer Program
through Target-driven R&D (A-STEP) from Japan Scienceand Technology Agency (JST), Grant Nos. VP30218088419
and JPMJTM20AK. M D Ainslie would like to acknowledge financial support from an Engineering and Physical Sciences Research Council (EPSRC) Early Career Fellowship,
EP/P020313/1. All data are provided in full in the results
section of this paper.

6. Conclusion
We have constructed two numerical models (Model-1 andModel-2) with different sizes of heat-generating region in
a REBaCuO superconducting ring bulk and investigated thepossibility of the mechanical fracture of the ring bulk due tothermal stress induced by local heat generation during PFM.
Model-1 considers a large heat region and Model-2 considers asmall heat region inside the bulk, in which the number of meshelements is about 1/21 of that of Model-1. The valuable results and important conclusions of this study are summarized
as follows.
(a) For Model-1, the thermal stress along the radial and circumferential directions was compressive in nature in theheat region and tensile in nature in the neighboring region.
The thermal stress increases with increasing induced heat
Q (or maximum temperature, T max ). The numerical analyses suggest that bulk fracture inside the bulk due tothe thermal stress results from the tensile stress along ther-direction in the neighboring region.
(b) In Model-1, open ends exist in the heat region along thez- and r-directions. The thermal stress at the open ends inthe heat region was also investigated. The risk of bulk fracture is enhanced at the inner or outer edges of the bulk surface, compared with inside the bulk. The ring bulk is easy
to break from the surface edge under the assumptions ofModel-1.
(c) For Model-2, the thermal stress, σ r heat , along the radial direction was compressive in the heat region and tensile in theneighboring region. However, the thermal stress, σ φ heat ,along the circumferential direction was also compressive,
which is larger than that of Model-1, in the heat region, but
was nearly zero along the circumferential direction in theneighboring region. The numerical analyses using Model2 suggest that the fracture inside the bulk due to the thermal
stress results from the compressive stress along the redial
direction in the neighboring region.
(d) Up to now, the bulk fracture in the ring bulk during magnetizing process has been mainly considered to result fromthe electromagnetic stress. However, the electromagnetic
stress alone may not account for the ring bulk fractureduring PFM, as suggested by our numerical simulations:the possibility of mechanical fracture of a ring bulk due
to thermal stress induced by local heat generation. This
concept is also applicable more generally to the fracture mechanism during FCM of ring/disk superconducting
bulks.

ORCID iDs
Hiroyuki Fujishiro  https://orcid.org/0000-0003-1483835X
Keita Takahashi  https://orcid.org/0000-0002-8278-2688
Mark D Ainslie  https://orcid.org/0000-0003-0466-3680
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Data availability statement
All data that support the findings of this study are included
within the article (and any supplementary files).
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