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Simulation of mechanical stresses in reinforced REBaCuO disk bulks
during pulsed-field magnetization
To cite this article: F. Shimoyashiki et al 2020 J. Phys.: Conf. Ser. 1559 012029

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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

Simulation of mechanical stresses in reinforced REBaCuO
disk bulks during pulsed-field magnetization
F. Shimoyashiki1, H. Fujishiro1, T. Hirano1, T. Naito1, M. D. Ainslie2
1

Department of Physical Science and Materials Engineering, Faculty of Science andEngineering, Iwate University, Morioka 020-8551, Japan2
Bulk Superconductivity Group, Department of Engineering, University ofCambridge, Trumpington Street, Cambridge CB2 1PZ, UK
E-mail: fujishiro@iwate-u.ac.jp
Abstract. We have performed numerical simulations of the electromagnetic hoop stress, 𝜎! , ina REBaCuO disk bulk reinforced by a metal ring during pulsed-field magnetization (PFM) using
a solenoid coil, in which the superconducting characteristics of the bulk material were assumed
to have realistic Jc-B-T ones. The compressive and tensile 𝜎! stresses were applied in the bulk
during the ascending and descending stages of PFM, respectively. The time and position
dependences of the mechanical stresses were estimated. The possibility of mechanical fracture
due to these hoop stresses and the effect of the metal ring reinforcement were discussed.

1. Introduction
It is important to enhance the trapped magnetic field in RE-Ba-Cu-O (REBaCuO, RE: rare earth
element or Y) superconducting bulks because the high-field trapped field magnet (TFM) is capable ofgenerating several Tesla. Although the trapped field estimated from its critical current density Jc(B, T)
could theoretically be over 30 T at 29 K [1], the mechanical strength of the brittle ceramic material
restricts the maximum trapped field experimentally. A trapped field of 17.24 T has been achieved at 29
K in a YBaCuO disk bulk pair by mechanical reinforcement of resin impregnation and carbon fiber
wrapping [1]. To date, the highest trapped field of 17.6 T has been achieved at 26 K in a GdBaCuO disk
bulk pair reinforced by shrink-fit stainless steel [2], both of which were magnetized by field-cooledmagnetization (FCM). Analytical investigations of mechanical properties were reported for disk- andring-shaped bulks with infinite height during FCM or zero-field-cooled magnetisation (ZFCM) [3 – 6];
the results were primarily based on Bean’s critical state model with constant Jc characteristics. We have
performed the numerical simulation of electromagnetic, thermal properties and mechanical stresses(hoop stress, 𝜎𝜃 , and radial stress, 𝜎𝑟 ) in REBaCuO disk and ring bulks with finite height reinforced bymetal ring during FCM using finite element method (FEM) [7 - 11], in which the mechanical stresses
and reinforcement of disk bulks during FCM and the cooling process were investigated [9].
The pulsed-field magnetization (PFM) is a practical magnetizing technique to realize TFMs without
the need for a superconducting magnet, in contrast to FCM, because of its relatively compact,
inexpensive, and mobile experimental setup [12 - 16]. However, the trapped field by PFM is much lower
than that by FCM because of a large temperature rise due to the rapid and dynamic motion of magnetic
flux. The trapped field of 5.20 T has been achieved on a GdBaCuO bulk 45 mm in diameter at 30 K
using a modified multi-pulse technique with stepwise cooling (MMPSC) [15], which is a record-highContent from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

trapped field for REBaCuO bulk by PFM to date. However, there were experimentally few evidencesof the mechanical fracture during PFM. The stress analyses by numerical simulation during PFM havebeen reported, in which the disk bulk was not reinforced by metal ring [17]. However, it is necessary to
clarify the mechanical stress for the bulk reinforced by metal ring, which is a usual setup during
magnetization process.
In this paper, based our experience for the mechanical analysis during FCM, the time and position
dependences of the electromagnetic hoop stress, 𝜎𝜃 , along the circumferential direction are estimatedfor the disk bulk reinforced by a stainless steel ring during PFM. The possibility of mechanical fracture
due to the compressive and tensile stresses and the effect of the metal ring reinforcement are discussed.
2. Numerical simulation procedure
Based on our experimental setup [12 - 16], a three dimensional (3D) numerical model coupling theelectromagnetic, thermal and mechanical properties was constructed. The REBaCuO disk bulk (65 mm
in outer diameter (O.D.) and 20 mm in height (H)) was mounted in a stainless steel (SUS) ring holderof 5 mm in thickness (75 mm in O.D. and 65 mm in I.D.) with the same height and was thermallyconnected to the cold stage using a thin indium sheet, as shown in Fig. 1(a). The disk bulk wasmagnetized using a coil (121 mm in O.D., 95 mm in I.D. and 50 mm in H). The applied pulsed field,
Bex(t), with the rise time of 13 ms and the duration time of 200 ms was applied, in which the magnitudeof pulsed field, Bex, was changed from 3.5 to 6.0 T. The initial temperature, Ts, was set to Ts = 65 K. The
following magnetic field dependence of Jc(B), proposed by Jirsa, was used in the simulation [18],𝐽! (𝐵) = 𝐽!" exp *−

𝐵
𝐵
1
𝐵 (
, + 𝐽!$
exp . 11 − *
, 23.
𝐵#
𝐵%&'
𝑘
𝐵%&'

(1)

The experimental Jc(B, T) data were fit up to 10 T between 60, 65, 70 and 77 K using eq. (1), as shownin Fig. 1(b), and the parameters were determined at each temperature [19]. For example, each parametervalue at 65 K was Jc1 = 1.17 × 109 A/m2, BL = 0.57 T, Jc2 = 7.56 × 108 A/m2, Bmax = 3.0 T and k = 1.3.
The Jc(B, T) profiles at intermediate magnetic fields and temperatures are interpolated based on eachparameter [19]. The commercial software packages, Photo-Eddy, combined with Photo-Thermo (Photon
Ltd, Japan), were used for the analysis.

Figure 1. (a) A schematic view of the disk bulk and the stainless steel ring, which ismagnetized by the solenoid coil. (b) The Jc(B, T) profiles of the REBaCuO bulk at 60, 65, 70
and 77 K, used in the numerical simulation.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

Table 1. Mechanical parameters (Young’s modulus, 𝐸# , Poisson ratio, ν, and average thermal
expansion coefficient, 𝛼$% ) of REBaCuO bulk, stainless steel and indium used in the simulation.
REBaCuO bulk
stainless steel (SUS)
indium

𝐸# (GPa)
100
78
12.7

ν
0.33
0.34
0.34

𝛼$% (K-1)
6.80 × 10-6
1.72 × 10-5
1.60 × 10-5

Elastic behaviour in an isotropic material can be explained by Hooke’s law. The electromagnetic
hoop stress, 𝜎& , was numerically simulated. The detailed procedure of the mechanical analysis used inthis study has been shown elsewhere [7, 10]. The nodal force on each node of the meshed element
calculated by Photo-Eddy was exported to the commercial software package Photo-ELAS (Photon Ltd.,
Japan) for analysis of the electromagnetic stress. The mechanical parameters (Young’s modulus, 𝐸# ,Poisson ratio, ν, and the average thermal contraction coefficient, 𝛼$% ) of the REBaCuO bulk, SUS ring,and indium are summarised in table 1. The mechanical parameters of the REBaCuO bulk were assumed
to be isotropic and homogeneous for simplicity, and typical values of the ab-plane were used.
3. Results and discussion
Fig. 2(a) shows the time dependence of the applied pulsed field, Bex(t), of 6.0 T used in the simulationand the simulation result of the trapped field, Bz(t), at the center of the bulk (r = z = 0 mm) at Ts = 65 K.
The magnetic flux intruded into the bulk center a little later and then the trapped field gradually
decreased due to the flux creep. Figs. 2(b) and 2(c), respectively, show the measured and calculated final
trapped field, Bz, on the center of the bulk surface (r = 0 mm, z = 10.5 mm) at t = 7 s, and the maximum
temperature rise, DTmax, on the edge of the bulk surface (r = 32 mm, z = 10.5 mm), as a function of
applied pulsed field, Bex. The experimental results are also shown [14], which were reproduced by thesimulation quantitatively. The Bz value took a maximum and then decreased with increasing Bex. The
DTmax value monotonically increased with increasing Bex.
Figs. 3(a) and 3(b) show the simulation results of ascending and descending stages of the trappedfield profiles, Bz(z = 0 mm), after applying the pulsed field of Bex = 5.5 T. Figs. 3(c) and 3(d), respectively,
show the trapped field profiles at t = 13 ms (pulse peak) and t = 9 s (final). In the ascending stage, themagnetic flux gradually penetrated from the bulk periphery and surface as shown in Fig. 3(c). In thedescending stage, the magnetic flux intruded abruptly like a flux jump to the bulk center at t = 20 ms
due to the reduction of the pinning force by the temperature rise. Finally, the cone-shaped trapped field
profile was obtained as shown in Fig. 3(d).

Figure 2. (a) Time dependence of the applied pulsed field, Bex(t), of 6.0 T, and of the trappedfield, Bz(t), at the center of the bulk (r = z = 0) at Ts = 65 K. (b) The experimental and numericalresults of the trapped field, Bz, on the surface of the bulk center and (c) the maximum
temperature rise, DTmax, in the bulk, as a function of applied pulsed field, Bex.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

Figure 3. The simulation results of time evolution of the (a) ascending and (b) descending
stages of the trapped field profiles, Bz(z = 0 mm), after applying the pulsed field of Bex = 5.5 T
at 65 K. The cross sections of the trapped field profiles at (c) t = 13 ms (pulse peak) and (d) t= 9 s (final).
Figs. 4(a) and 4(b) show the time evolution of the temperature profiles, T(z = 0 mm), in the ascending
and descending stages, after applying the pulsed field of Bex = 5.5 T. Figs. 4(c) and 4(d), respectively,
show the temperature profiles at t = 13 ms and t = 9 s. In the ascending stage, the temperature increased
gradually from the bulk periphery due to the flux penetration, as shown in Fig. 3(c). In the descending
stage, the generated heat mainly diffused toward the bulk center and was removed by the cold stage. As
a result, the temperature was decreased and equalized at t = 9 s, as shown in Fig. 4(d).
Figs. 5(a) and 5(b) show the time evolution of the electromagnetic hoop stresses, 𝜎& (z = 0 mm), in
the ascending and descending stages, after applying the pulsed field of Bex = 5.5 T at 65 K. Figs. 5(c)and 5(d), respectively, show the 𝜎& profiles at t = 13 ms and t = 9 s. In the ascending stage, theelectromagnetic stress negatively increased with increasing time and the compressive stress, which was
as large as -15 MPa at t = 13 ms, was the largest at the bulk center, as shown in Fig. 5(c). The
compressive stress was also applied to the SUS ring. In the descending stage, the compressive stresschanges to the tensile stress with increasing time. The maximum tensile stress of + 7 MPa was applied
at t = 40 ms and then the final tensile stress at t = 9 s was lower than +5 MPa, as shown in Fig. 5(d).

Figure 4. The time evolution of the (a) ascending and (b) descending stages of the temperature
profiles, T(z = 0 mm), after applying the pulsed field of Bex = 5.5 T. The cross sections oftemperature profiles at (c) t = 13 ms (pulse peak) and (d) t = 9 s (final).

414th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

Figure 5. The time evolution of the (a) ascending and (b) descending stages of theelectromagnetic stress profiles, 𝜎& (z = 0 mm), during the pulsed field of Bex = 5.5 T at 65 K.
The cross sections of the 𝜎& profiles at (c) t = 13 ms (pulse peak) and (d) t = 9 s (final).
Fig. 6(a) shows the time evolution of the maximum electromagnetic hoop stress, 𝜎&'$( , at z = 0 and
10 mm in the disk bulk in the ascending and descending stages, during the applied pulsed field of Bex =5.5 T. The time evolution of the applied field, Bex(t) is also shown. The compressive stress took amaximum at t = 13 ms, and then changed to the tensile stress at t > 40 ms. The compressive stress at thebulk center (z = 0) is negatively larger than that at the bulk surface (z = 10 mm). Fig. 6(b) shows the
maximum electromagnetic hoop stress, 𝜎&'$( , in the disk bulk in the ascending and descending stages,
as a function of the applied pulsed field, Bex. In both stages, the 𝜎&'$( value increases with increasing
Bex and the 𝜎&'$( value was as large as -17 MPa and +8 MPa in the ascending and descending stages,
respectively. These 𝜎&'$( values are fairly smaller than the mechanical strength of the ReBaCuO bulk,
which is as large as several hundred MPa for the compressive stress [21, 22] and 50 to 70 MPa for thetensile stress [22, 23]. When the operating temperature, Ts, decreases and the applied field, Bex, increases,
the 𝜎&'$( value would increase more.

Figure 6. (a) The time evolution of the maximum electromagnetic stress, 𝜎&'$( , at z = 0 and
10 mm in the disk bulk in the ascending and descending stages, during the applied pulsed fieldof Bex = 5.5 T at 65 K. (b) The maximum electromagnetic stress, 𝜎&'$( , in the disk bulk in theascending and descending stages, as a function of the applied pulsed field, Bex.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

We have also performed the numerical simulation of the mechanical stress along the radial direction,𝜎) . Figs. 7(a) and 7(b) show the time evolution of the electromagnetic hoop stresses, 𝜎) (z = 0 mm), in
the ascending and descending stages, after applying the pulsed field of Bex = 5.5 T at 65 K. The trends
are similar to 𝜎& along the circumferential direction. In the actual PFM experiments, the bulk with SUS
ring is cooled from room temperature to the starting temperature, Ts. An additional thermal compressive
stress is applied to the bulk from the SUS ring due to the difference of thermal expansion coefficient
between REBaCuO bulk and SUS, which also protects the mechanical fracture. Actually, we have
performed the PFM experiments several thousands of times for the disk bulks. However, we have neverexperienced the mechanical fracture of the disk bulk during PFM experimentally. The simulation resultsin this study are reasonable and support the experimental evidences.

Figure 7. The time evolution of the (a) ascending and (b) descending stages of theelectromagnetic stress profiles, 𝜎) (z = 0 mm), during the pulsed field of Bex = 5.5 T at 65 K.
4. Summary
We have performed numerical simulations of the electromagnetic hoop stress, 𝜎𝜃 , in a REBaCuO
disk bulk reinforced by a SUS ring during PFM, in which the superconducting characteristics of the bulk
material were assumed to have realistic Jc-B-T ones. The compressive and tensile 𝜎𝜃 were applied in thebulk during the ascending and descending stages of PFM, respectively. The maximum 𝜎𝜃 values werereasonably smaller than the fracture strength of the ReBaCuO bulk for the present experimental
condition. If the operating temperature, Ts, decreases to 40 – 20 K and the applied field, Bex, increases
to 10 T, the 𝜎𝜃 value will increase, but the disk bulk should never break during PFM. The simulation
results in this study are reasonable and support the experimental evidence.
Acknowledgments
This research is partially supported by the Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant No. 15K04646. Mark 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.
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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012029 doi:10.1088/1742-6596/1559/1/012029

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