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Simulation of mechanical stresses in reinforced REBaCuO ring bulks
during pulsed-field magnetization
To cite this article: T. Hirano et al 2020 J. Phys.: Conf. Ser. 1559 012027

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14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012027 doi:10.1088/1742-6596/1559/1/012027

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

Department of Physical Science and Materials Engineering, Faculty of Science andEngineering, Iwate University, 4-3-5 Ueda Morioka, Japan2
Bulk Superconductivity Group, Department of Engineering, University ofCambridge, Trumpington Street, Cambridge CB2 1PZ, UK
E-mail: g0318132@iwate-u.ac.jp
Abstract. We have performed numerical simulations of the electromagnetic, thermal andmechanical properties of a REBaCuO ring-shaped bulk with various reinforcement structuresduring pulsed-field magnetization (PFM). Compressive and tensile electromagnetic stresses,σθmag, are developed in the ring-shaped bulk during the ascending and descending stages of PFM,
respectively. These stresses increase at lower operating temperatures and for higher applied
pulsed fields. In order to reduce these stresses, the ring-shaped bulk was fully encapsulated byouter and inner ring with upper and lower plates made by Al alloy. In particular, thisreinforcement structure can achieve the lowest electromagnetic compressive stress, whichcorresponds to about 54% of that for a conventional ring reinforcement structure, and theelectromagnetic tensile stress was also reduced. We also compared the simulation results of theelectromagnetic stresses for the ring-shaped bulk to those for a disk-shaped bulk.

1. Introduction
REBaCuO (RE: rare earth element or Y) superconducting bulks have promising potential to trap
magnetic fields of over 20 T. As a result, such bulks have been used as strong trapped-field magnets(TFMs) in practical applications such as a compact cryogen-free nuclear magnetic resonance (NMR)
spectrometer [1] and rotating machines [2]. However, TFMs suffer from mechanical fracture due to alarge Lorentz force, F = J x B (J represents the current density and B represents the magnetic field),that is generated during field-cooled magnetization (FCM) from higher applied magnetic fields, in which
only an electromagnetic tensile stress develops in the bulk [3]. We have investigated the mechanicalproperties of REBaCuO disk- and ring-shaped bulks during FCM using numerical simulations andproposed an optimal reinforcement structure by metal to prevent mechanical fracture [4 - 8]. On theother hand, the pulsed-field magnetization (PFM) process consists of an ascending (flux penetration)
stage on the order of milliseconds and a descending (flux flow) stage, during which electromagnetic
compressive (ascending stage) and tensile (descending stage) stresses are developed in the bulk [9, 10].
Accordingly, the optimal reinforcement structure for PFM, which may be different from that for FCM,
may need to be considered. As for a shape of the bulk, the risk of fracture for the ring-shaped bulk ishigher than that of the disk-shaped bulk due to the stress concentration at the inner edge of the concentric
hole [11]. The mechanical properties have been also investigated analytically for disk- and ring-shapedbulks with infinite height during FCM and zero-field-cooled magnetization (ZFCM) [11 - 13]. In
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1

14th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012027 doi:10.1088/1742-6596/1559/1/012027

addition, numerical simulations of the mechanical stresses have been performed using finite element
method (FEM) for disk-shaped bulks during PFM [9, 10, 14]. The final trapped field of the ring-shapedbulk during PFM experiment, which was negative value at the centre of the bulk, has been reported inprevious studies [15]. In order to realize the higher TFMs in ring-shaped bulk by PFM, in this study, weinvestigate the mechanical and thermal stresses during PFM for REBaCuO ring bulks using numericalsimulations and proposed an effective reinforcement structure. We also compare the mechanical stressfor the ring-shaped bulk to that for the disk-shaped bulk.
2. Numerical simulation Procedure
Based on our experimental setup for PFM [15], we constructed a three-dimensional model for a
REBaCuO ring-shaped bulk (60 mm in outer diameter (O.D.), 36 mm in inner diameter (I.D.) and 17
mm in height (H)), as shown in Fig. 1(a). The ring-shaped bulk was mounted in an Al alloy ring holder(70 mm in O.D. and 60 mm in I.D.) with the same height as the bulk. The Al alloy is a suitable material
for the reinforcement of the ring-shaped bulk for NMR apparatus because it is non-magnetic and has
higher mechanical strength, compared to pure aluminum. The bulk was cooled to Ts = 65, 45 and 25 K
from the bottom surface, and was magnetized by PFM using a solenoid coil (120 mm in O.D., 100 mm
in I.D. and 50 mm in H). A single magnetic pulse with a rise time of 13 ms and a duration time of 150
ms, and amplitudes ranging from Bex = 1.0 to 4.5 T, was applied to the bulk via a pulsed current in thecoil. The time evolution of trapped magnetic field, Bt(t), was monitored at the centre of the ring-shapedbulk (r = z = 0 mm). Physical phenomena during PFM are described by the fundamental electromagnetic
and thermal equations in [16]. The power-n law (n = 20) was used to describe the highly nonlinear E-J
characteristic of the bulk material [17]. The following magnetic field and temperature dependence of
critical current density, Jc(B, T), proposed by Kim [18], was used in the simulation.
3

𝑇 2 2 𝐵0𝐽c (𝐵, 𝑇) = 𝛼 {1 − (𝑇 ) } |𝐵|+𝐵
,
c
0

(1)

where Tc = 92 K is the transition temperature of the REBaCuO bulk, and B0 = 1.3 T and α = 3.45 × 109
A/m2 were assumed. The REBaCuO bulk is assumed to be isotropic and homogeneous for simplicity.
The commercial software package, Photo-Eddy, combined with Photo-Thermo and Photo-Elas (Photon
Ltd, Japan), were used for analyses of electromagnetic, thermal and mechanical properties. The
numerical simulation procedure and the parameters used in the simulation were described elsewhere indetail [16]. Elastic behaviour of an isotropic material can be explained by Hooke’s law, in which thestress is linearly proportional to the strain [4]. The electromagnetic hoop stress, σθmag, and the thermalhoop stress, σθtherm, were calculated for each case during PFM. Table 1 summarizes the assumed
mechanical parameters (Young’s modulus, E, Poisson ratio, ν, and thermal expansion coefficient, α0) of
the REBaCuO bulk and the Al alloy.
The σθmag and σθtherm values for the bulk reinforced by a simple Al alloy ring, denoted as “Ring”, inFig. 1(a), were compared to those for the bulks with various reinforcement structures. An Al alloy innerring of 5 mm in width and an upper Al alloy plate 2 mm in thickness were added to the “Ring” setup, asshown in Fig. 1(b). We denote this setup as “Cap Cover”. The ring-shaped bulk with the “Cap Cover”
was further reinforced by a lower Al alloy plate 2 mm in thickness, as shown in Fig. 1(c). We denote
this setup as “Full Cover”. The width of the inner ring for the “Full Cover” was increased from 5 mm to
9 mm, as shown in Fig. 1(d). We abbreviate this setup as “Full Cover wide”.

214th European Conference on Applied Superconductivity (EUCAS 2019)
IOP Publishing
Journal of Physics: Conference Series
1559 (2020) 012027 doi:10.1088/1742-6596/1559/1/012027

Figure 1. Schematic view of the numerical models of the various reinforcement structures for the
ring-shaped bulk: (a) “Ring”, (b) “Cap Cover”, (c) “Full Cover” and (d) “Full Cover wide”.
Table 1. Mechanical parameters of REBaCuO bulk and Al alloy used in the simulation.
E (GPa)ν
α0 (K-1)
REBaCuO bulk
100
0.33
5.20 x 10-6
Al alloy78
0.34
1.48 x 10-5
(E: Young’s modulus, ν: Poisson ratio, α0: thermal expansion coefficient)

3. Numerical simulation results and discussion
3.1. Electromagnetic hoop stress, σθmag
Figure 2(a) shows final trapped field, Bt(t = 7 s), at the centre of the ring-shaped bulk with the “Ring”
reinforcement (r = z = 0 mm), as a function of the applied pulsed field, Bex, at Ts = 65, 45 and 25 K. The
Bt value increases moderately with increasing Bex, takes a maximum of 1.07 T at Bex* = 2.5 T at 65 K,
and then subsequently decreases. We define the activation field as Bex*, at which the bulk achieved the
maximum trapped field, Bt,max. The Bex* and Bt,max values increased with decreasing Ts, which originatesfrom the enhancement of Jc at lower temperatures and subsequent increased Lorentz force. It is therefore
harder for magnetic flux to penetrate into centre of the bulk at lower Ts. Figure 2(b) shows maximum
temperature rise, ΔTmax, of the outer surface of the bulk (r = 30 mm, z = 0 mm), as a function of Bex at
Ts = 65, 45 and 25 K. ΔTmax increased with increasing Bex and with lowering Ts for all cases due to theincrease of the dynamical motion of the magnetic flux against the vortex pinning force, Fp, and theviscous force, Fv, and also due to the lowering the specific heat for the bulk [19].

314th European Conference on Applied Superconductivity (EUCAS 2019)
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Journal of Physics: Conference Series
1559 (2020) 012027 doi:10.1088/1742-6596/1559/1/012027

Figure 2. (a) Final trapped field, Bt(t = 7 s), at the centre of the bulk (r = z = 0 mm) and (b) maximum
temperature rise, ΔTmax, of the outer surface of the bulk (r = 30 mm, z = 0 mm), as a function of theapplied pulsed field, Bex, at Ts of 65, 45 and 25 K for the ring-shaped bulk with the “Ring”
reinforcement.
Figure 3(a) shows maximum electromagnetic compressive hoop stress, σθmagmax, in the ascending
stage, at the inner surface of the ring-shaped bulk (r = 18 mm, z = 0 mm) with the “Ring”
reinforcement, as a function of Bex at Ts = 65, 45 and 25 K. The compressive σθmagmax value gradually
increases with increasing Bex and approaches a constant value when Bex overcomes the magnetic
shielding. The compressive σθmagmax also increases at lower Ts, which again originates from theenhancement of Jc at lower temperatures and subsequent increased Lorentz force. The maximum
electromagnetic tensile hoop stress, σθmagmax, in the descending stage is also shown in Fig. 3(b), which
is about half of the maximum compressive stress. During FCM, the electromagnetic tensile stressdevelops in the bulk, in which the mechanical fracture sometimes happens in the case that the tensilestress is larger than the fracture strength of the bulk [20 - 22]. On the other hand, not only a tensilestress, but also a compressive stress develops during PFM. The fracture strength of REBaCuO bulks
under compressive stress has been reported to be several times larger than that under tensile stress[23, 24]. However, the stress distribution in the ring-shaped bulk is different from that in the diskshaped bulk. Since the stress concentration occurs around the bore of a ring-shaped bulk, the stressincreases locally. The risk of fracture where this stress concentration occurs also increases withdecreasing width of the ring-shaped bulk. It is, therefore, necessary to reinforce the ring-shaped bulk
for the compressive stress as well as tensile stress during PFM.
Figure 4 shows time evolution of the electromagnetic hoop stress, σθmag(t), at the inner surface ofthe ring-shaped bulk (r = 18 mm, z = 0 mm), for various reinforcement structures for Bex = 4.0 T at
Ts = 25 K. The time evolution of the applied pulsed field, Bex(t), is also shown. Increasing the appliedfield, σθmag starts to increase negatively (compressive) and takes a peak value of -32.2 MPa at t = 9
ms for the ring-shaped bulk with the “Ring” reinforcement. After that, σθmag increases positively, andchanges to a tensile stress of +6.5 MPa at t = 40 ms, which result from the change of the direction of
the induced superconducting current. The maximum compressive σθmagmax was reduced for the otherreinforcement structures, corresponding to approximately 67% (-21.5 MPa) for the “Cap Cover”, 65%(-20.9 MPa) for the “Full Cover” and 54% (-17.4 MPa) for the “Full Cover wide”, compared to thatfor the “Ring” reinforcement. The improved reinforcement structures, including the inner ring weremore effective to reduce the compressive stress for the ring-shaped bulk during PFM.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
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Figure 3. Maximum electromagnetic (a) compressive stress in the ascending stage and (b) tensilestress, σθmag,max, in the descending stage at the inner surface of the bulk (r = 18 mm, z = 0 mm) of thering-shaped bulk with the “Ring” reinforcement, as a function of Bex at Ts = 65, 45 and 25 K.

Figure 4. Time evolution of the electromagnetic hoop stress, σθmag, at the inner surface of the ringshaped bulk (r = 18 mm, z = 0 mm) for the various reinforcement structures for Bex = 4.0 T and at Ts= 25 K.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
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Journal of Physics: Conference Series
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Figure 5 shows contour maps of the electromagnetic hoop stress, σθmag, during PFM (Bex = 4.0 T,
Ts = 25 K, t = 9 ms, reaches maximum compressive stress) for the ring-shaped bulk with the various
reinforcement structures. Here, the σθmag value was the maximum at the inner periphery of the ringshaped bulk. The results clearly indicated that the highest electromagnetic compressive stressdevelops in the bulk with the “Ring” reinforcement. In the case of the “Cap Cover”, as shown in Fig.
5(b), the reinforcement at the bottom of the bulk is not enough, compared to those of the bulks withthe “Full Cover” in Fig. 5(c) and with the “Full Cover wide” in Fig. 5(d).

Figure 5. Contour maps of the electromagnetic hoop stress, σθmag, during PFM (Bex = 4.0 T, Ts = 25
K, t = 9 ms) for the ring-shaped bulk with the various reinforcement structures.
3.2. Thermal hoop stress, σθtherm
During PFM, a large temperature rise occurs in the bulk, compared with that during FCM [25]. The
inhomogeneous temperature distribution results in a thermal stress due to the inhomogeneous thermal
displacement in the bulk. Figure 6 shows the contour map of the temperature during PFM (Bex = 4.0 T,
Ts = 25 K, t = 11 ms, reaches maximum tensile stress) for the various reinforcement structures. In Fig.
6(a), for the “Ring” reinforcement, the temperature rise mainly takes place at the outer periphery of thering-shaped bulk, which resulted in a temperature as high as 45 K. The temperature rise decreasedslightly for the bulk for the other reinforcement structures due to heat diffuse to the Al alloy holder.
Figure 7 shows the contour maps of the thermal hoop stress, σθtherm, during PFM (Bex = 4.0 T, Ts = 25
K, t = 11 ms) for the various reinforcement structures. The inhomogeneous temperature distribution
results in a thermal stress. In Fig. 7(a) for the “Ring” reinforcement, the compressive stress develops at
the narrow region of the outer periphery of the ring-shaped bulk, which was as small as -8 MPa.
Although the bulk periphery thermally expands along the r-direction due to the large temperature rise,as shown in Fig. 6(a), the Al alloy ring cannot expand due to the small temperature rise. As a result, athermal compressive stress develops in the outer region of the ring-shaped bulk and a thermal tensilestress develops in the inner region of the ring-shaped bulk. For the other reinforcement structures, as
shown in Figs. 7(b) – 7(d), the thermal tensile stress can be reduced, compared to the “Ring”
reinforcement. On the other hand, the thermal compressive stress, σθtherm, increases for the improved

614th European Conference on Applied Superconductivity (EUCAS 2019)
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Journal of Physics: Conference Series
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reinforcement structures. However, the thermal compressive stress, σθtherm, in the bulk is much smaller
than the electromagnetic compressive stress, σθmag, and the region where the compressive stress ismaximum is different. These results suggest that the influence of the thermal stress is relatively small.

Figure 6. Contour maps of the temperature during PFM (Bex = 4.0 T, Ts = 25 K, t = 11 ms) for thevarious reinforcement structures.

Figure 7. Contour maps of the thermal hoop stress, σθtherm, during PFM (Bex = 4.0 T, Ts = 25 K, t =11 ms) for the various reinforcement structures.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
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3.3. Comparison to disk-shaped bulk.
Finally, we compare the electromagnetic hoop stress, σθmag, of the ring-shaped bulk with that of the
disk-shaped bulk during PFM. Figure 8 shows the electromagnetic hoop stress, σθmag, for the ascending
stage of PFM (Bex = 4.0 T, Ts = 25 K, t ≤ 13 ms) for the disk-shaped bulk (60 mm in O.D. and 17 mmin H) as shown in Fig. 8(c), compared to the ring-shaped bulk, as shown in Fig. 1(a). The disk-shapedbulk is also mounted in an Al alloy ring holder 5 mm in width. The compressive stress takes a maximumof -32.2 MPa at the inner periphery of the ring-shaped bulk, as shown in Fig. 4. On the other hand, inthe case of the disk-shaped bulk of the same O.D., a smaller compressive stress takes a maximum of 16.9 MPa in the bulk. Wu et al. described the maximum compressive hoop stress of the disk-shapedbulk (46 mm in O. D. and 15 mm in H) during PFM (Bex = 6.25 T, Ts = 30 K) was -25.5 MPa [10]. The
difference of the compressive stress values results mainly from the Bex value, because the compressive
stress increases with increasing Bex. These results indicate that the disk-shaped bulk is tough and is lesslikely to break during PFM, compared with the ring-shaped bulk, even if the applied pulse field isstronger and the operating temperature is lower.

Figure 8. The electromagnetic hoop stress, σθmag, for the ascending stage of PFM (Bex = 4.0 T, Ts =25 K, t ≤ 13 ms) for the (a) ring-shaped bulk and (b) disk-shaped bulk (60 mm in O.D. and 17 mm inH), each of which were reinforced by an Al alloy ring 5 mm in width and (c) schematic view ofnumerical model for disk-shaped bulk.
4. Conclusion
We have performed numerical simulations of the electromagnetic, thermal and mechanical properties
of REBaCuO ring-shaped bulk with various reinforcement structures during PFM. Compressive andtensile electromagnetic stresses, σθmag, develop in the ring-shaped bulk during the ascending anddescending stages of PFM, respectively. These stresses increased with lower operating temperatures andhigher applied fields. In particular, in order to reduce the compressive stress, the fully encapsulated
reinforcement was optimal, which can achieve the lowest electromagnetic compressive stress andcorresponds to about 54% of that for the conventional ring reinforcement structure. Compared with the
disk-shaped bulk, the ring-shaped bulk with the “Ring” reinforcement has a risk to break by theelectromagnetic compressive stress during the ascending stage during PFM. As such, suitablereinforcement must be considered.

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14th European Conference on Applied Superconductivity (EUCAS 2019)
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Journal of Physics: Conference Series
1559 (2020) 012027 doi:10.1088/1742-6596/1559/1/012027

Acknowledgements
This research is partially supported from JSPS KAKENHI Grant No. 15K04646. 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 ofthis paper.
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