Physica C: Superconductivity and its applications 550 (2018) 52–56

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Physica C: Superconductivity and its applications
journal homepage: www.elsevier.com/locate/physc

New proposal of mechanical reinforcement structures to annular REBaCuO
bulk magnet for compact and cryogen-free NMR spectrometer

T⁎

H. Fujishiroa, , K. Takahashia, T. Naitoa, Y. Yanagib, Y. Itohb, T. NakamuracaFaculty of Science and Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, JapanIMRA Material R&D Co., Ltd., 2-1 Asahi-machi, Kariya 448-0032, Japanc
RIKEN, 2-1 Hirosawa, Wako 351-0198, Japanb

A B S T R A C T

We have proposed new reinforcement structures using an aluminum alloy ring to the annular REBaCuO bulks
applicable to compact and cryogen-free 400 MHz (9.4 T) nuclear magnetic resonance (NMR) spectrometer using
a numerical simulation of mechanical stress. The thermal compressive stress, σθcool, which was applied to theannular bulks during cooling due to the diﬀerence of thermal expansion coeﬃcient between bulk and aluminumalloy, became fairly enhanced at the surface of the uppermost bulk for the new reinforcement structures,compared to the conventional reinforcement with the same height as the annular bulk, in which the compressiveσθcool value was reduced. During ﬁeld-cooled magnetization (FCM), the electromagnetic hoop stress, σθFCM,
became the maximum at the innermost edge of the uppermost ring bulk at intermediate time step. The actual
total hoop stress, σθ (= σθcool + σθFCM), due to both cooling and FCM processes was also analyzed and the newring structures are fairly eﬀective to reduce the σθ value and became lower than the fracture strength of the bulk.
The new reinforcement structures have a possibility to avoid the fracture of the bulks and to realize a 400 MHz
NMR spectrometer.

1. Introduction
Nuclear magnetic resonance (NMR) spectrometer with high resolution using superconducting coils is a powerful apparatus to analyze
complex molecular structure or to develop new drugs. Recently, acompact and cryogen-free NMR spectrometer with a medium resolutionof 200 MHz (4.7 T) has been developed using annular REBaCuO superconducting bulks (RE: rare earth element), in which the NMR
spectra of ethanol with a full width at half of the maximum (FWHM) of
0.1 ppm (21 Hz) have been achieved [1]. The magnetic resonanceimaging (MRI) was also investigated using the same apparatus [2]. In
the apparatus, the annular REBaCuO bulks are magnetized by ﬁeldcooled magnetization (FCM), in which large hoop and radial stresses
are experienced and the bulks are sometimes fractured for higher applied ﬁeld. The metal ring support must be considered to avoid the bulk
fracture. We have investigated numerically and experimentally themechanical stresses (hoop stress, σθ, and radial stress, σr) in the ﬁnite
REBaCuO annular bulk reinforced by metal ring with the same height asthat of the bulk during FCM [3,4,5]. The thermal hoop stress, σθcool, wasalso estimated, which was applied to the bulk when cooling down tooperating temperature due to the diﬀerence of thermal expansion⁎

Corresponding author.
E-mail address: fujishiro@iwate-u.ac.jp (H. Fujishiro).

https://doi.org/10.1016/j.physc.2018.04.012
Received 25 February 2018; Received in revised form 11 April 2018; Accepted 17 April 2018
Available online 23 April 2018
0921-4534/ © 2018 Elsevier B.V. All rights reserved.

coeﬃcient between bulk and metal ring. As a result, it was found thatthe compressive σθcool was reduced fairly at the uppermost surface of
the ﬁnite bulk because of the larger thermal compression of the metalring along the axial direction. The total hoop stress during FCM undercooling, σθ (= σθcool + σθFCM), was larger than the results for the inﬁnite ring bulk reinforced by the metal ring with inﬁnite height [4].
These analytical results are valuable to compare to the mechanicalstrength of the bulk material and to judge the bulk fracture during FCM.
In this paper, we performed the numerical simulation of the electromagnetic properties (trapped ﬁeld, Bz, and induced persistent supercurrent density, Jθ) and the hoop stress, σθ, in the actual annular
REBaCuO bulks reinforced by new mechanical reinforcement structuresof aluminum alloy ring during FCM from Bex = 9.4 T at 50 K using theﬁnite element method (FEM) to realize the 400 MHz (9.4 T) NMR
spectrometer safely. The eﬀectiveness of the new reinforcement structure of the metal ring is deduced to reduce the total hoop stress.
2. Numerical simulation framework
Based on our experimental setup with eight stacked annular
REBaCuO bulks [6], we prepared the following framework of three-Physica C: Superconductivity and its applications 550 (2018) 52–56

H. Fujishiro et al.

Fig. 1. Schematic models of a conventional reinforcement structure ((a) type A)
and two new reinforcement structures ((b) type B and (c) type C).

dimensional (3D) numerical simulation for the FCM procedure. Schematic views of three types of numerical half models are presented inFig. 1. In Fig. 1(a), two end bulks named “A1” and “A2” (64 mm outer
diameter (O.D.), 32 mm inner diameter (I.D.) and 13 mm height (H))
and two inner bulks named “B1” and “B2” (64 mm O.D., 40 mm I.D. and
20 mm H) were vertically stacked along z-direction, all of which were
reinforced by the aluminum alloy ring 5 mm in width with the sameheight as the bulks. Aluminum alloy (A7075) is a suitable material forthe reinforcement of the bulk applied to NMR apparatus because ofnon-magnetic and higher mechanical strength, compared to pure aluminum. Indium thin foils 0.1 mm in thickness were inserted between
bulks with rings for better thermal contact. This model is abbreviated as“type A”, which is a conventional reinforcement structure [4]. We
propose two types of new reinforcement structures. In the “type B”
model, the ring plate made by aluminum alloy 3 mm in thickness is
integrally molded to the “A1” bulk of the “type A” model shown in
Fig. 1(b). In the “type C” model, the inner ring made by aluminum alloy
1 mm in thickness is further molded to the “A1” bulk of the “type B”
model, as shown in Fig. 1(c).
The bulk sets were magnetized by FCM using an inﬁnite solenoid
coil (150 mm O.D., 100 mm I.D. and inﬁnite H). Physical phenomena in
the FCM simulation are described using an electromagnetic equation
based on the axisymmetric coordinate system [7,8], in which thetemperature of the REBaCuO bulks is assumed to be constant at
Ts = 50 K during FCM. Because temperature rise is pretty small during
FCM in the experiment. The power-n model (n = 100) was used todescribe the nonlinear E–J characteristic of the bulk. The critical current density, Jc (= 9.6 × 108 A/m2) ﬂowing in the ab-plane of the bulk,
which is about twice as large as that at Ts = 50 K for the REBaCuO bulk[9], is assumed to be independent of magnetic ﬁeld to realize the approximate Bean's model [10]. In the simulation process of FCM, amagnetic ﬁeld of Bex = 9.4 T was applied above the critical temperature, Tc (= 92 K), and then the bulk was cooled to Ts = 50 K. Thereafter, the magnitude of the magnetic ﬁeld was monotonically decreased
from Bex to zero in 10 steps, as shown in the inset of Fig. 2(a).
In an isotropic material, Hooke's law is established, in which thestress is linearly proportional to the strain [4]. Since the REBaCuO bulk
consists of c-axis oriented REBa2Cu3O7 matrix phase, dispersed by nonsuperconductive ﬁne particles such as RE2BaCuO5 and Ag, the mechanical properties are anisotropic. The mechanical strength of the
REBaCuO bulk in the ab-plane is fairly larger than that along the c-axis
direction [11,12]. In the present numerical simulation, the REBaCuO
bulk is assumed to be isotropic and homogeneous and mechanicalparameters (Young's modulus and thermal contraction coeﬃcient) in
the ab-plane were used for simplicity. Because the electromagnetic andthermal hoop stresses, σθ, are applied in the ab-plane of the bulk. The

Fig. 2. (a) Time step dependence of the cross-sectional plots of the trapped
ﬁeld, Bz(r), for the annular REBaCuO bulks during FCM from Bex = 9.4 T.
Contour maps of the (b) trapped ﬁeld, Bz, and (c) persistent current density, Jθ,at the ﬁnal (10th) step are shown. (For interpretation of the references to colourin this ﬁgure legend, the reader is referred to the web version of this article.)electromagnetic hoop stress, σθFCM, was estimated for each step of FCM,
which acts circumferentially on each end face of the meshed element[13,14]. The thermal hoop stress, σθcool, by cooling from 300 to 50 K
was also estimated for each reinforcement structure. The mechanicalparameters (Young's modulus, E, Poisson ratio, ν, and thermal expansion coeﬃcient, α) of the REBaCuO bulk, aluminum alloy, and indium
sheet used in the elastic simulation are summarized in Table 1. Commercial software, Photo-Eddy and Photo-ELAS (Photon Ltd., Japan)
were used for the analyses of the electromagnetic properties and mechanical stresses.
3. Results and discussion
3.1. Trapped ﬁeld, Bz, and induced current density, jθFig. 2(a) shows the time step dependence of the cross-sectional plotsof the trapped ﬁeld, Bz(r), at z = 0 for the annular REBaCuO bulks
during FCM from Bex = 9.4 T. The external magnetic ﬁeld decreased
step by step without a ﬁeld slope due to the inﬁnite length of
Table 1
Mechanical properties (E: Young's modulus, ν: Poisson ratio, α: thermal contraction coeﬃcient) used in the numerical simulation [13].

REBaCuO bulk
Al alloy ring
Indium sheet

53

E (GPa)να (K−1)

100
78
110

0.33
0.34
0.44

5.2 × 10−6
1.66 × 10−5
3.25 × 10−5Physica C: Superconductivity and its applications 550 (2018) 52–56

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3.2. Thermal hoop stress, σθcool, under cooling from 300 to 50 K
Fig. 3 shows the radius (r) dependence of the thermal hoop stress,σθcool, in the annular bulks reinforced by three types of aluminumalloy rings at various z-positions under the cooling from 300 to 50 K
without FCM. The contour map of the σθcool is shown in the inset ofeach panel. In Fig. 3(a) for the conventional “type A”, a thermallycompressive stress (σθcool < 0) is eﬀectively applied to the “A2”, “B1”and “B2” bulks, and a tensile stress (σθcool > 0) is applied to the aluminum alloy ring region (32 ≤ r ≤ 37 mm), because the thermal expansion coeﬃcient, α, of the aluminum alloy is about three times
larger than that of the REBaCuO bulk, as shown in Table 1. However,
it is surprising that the αθcool value at the top surface of the “A1” bulk(z = 66 mm) is positive (tensile stress), which is as large as+100 MPa. Because of the larger α value of the aluminum alloy, thealuminum alloy rings shrink along both z- and r-directions. Duringcooling, the annular bulks were fastened along the r-direction and, atthe same time, were pulled along the z-direction toward z = 0, assuming not to slip between bulk and metal ring. As a result, thecompressive hoop stress of the “A1” bulk became fairly weak andchanged to the large positive value [4].
In Fig. 3(b) for the new “type B” reinforcement, the large positive
αθcool value of “A1” bulk in “type A” was drastically reduced andchanged to weak negative value. This result strongly suggests that theupper plate eﬀectively acts to reduce the expansion of the aluminumalloy ring along the r-direction. In Fig. 3(c) for another new “type C”
reinforcement, the reduction of the positive σθcool value becomes more
eﬀective by the additional thin ring support in the inner periphery of
the “A1” bulk, compared to the “type B” reinforcement. The compressive σθcool value is further enhanced in the “A1” bulk. As a result, thecompressive σθcool, which is larger than −70 MPa is applied for all theannular bulks during cooling procedure. Note that the σθcool value of theinner ring support is positive and is as high as +250 MPa, which is,
however, smaller than the fracture strength of the aluminum alloy.

3.3. Electromagnetic hoop stress, σθFCM during FCM
During FCM, the electromagnetic hoop stress is applied to the annular bulks. Fig. 4(a) shows the cross-sectional plots of the electromagnetic hoop stress, σθFCM, for the annular bulks with the “type A”
reinforcement at the 6th step of FCM, in the case that there is no
cooling. The contour map of the σθFCM is shown in the inset. The
maximum σθFCM is achieved at the innermost periphery edge of eachannular bulk. The σθFCM value gradually decreases with increasing r,
but steeply drops at the interface (r = 32 mm) between bulk and aluminum alloy ring, which results from the smaller Young's modulus, E, of
the aluminum alloy ring. Note that the σθFCM depends on the Bex and Jc
values. Fig. 4(b) and (c), respectively, show the same plots of σθFCM forthe annular bulks with the “type B” and “type C” reinforcement at the
6th step of FCM. The similar characteristics are observed to that for the“type A”, but the magnitude of the σθFCM value decreased and theσθFCM(r) proﬁles changed depending on the reinforcement structure of
the aluminum ring.
Fig. 5 shows time step dependence of the maximum electromagnetic hoop stress, σθFCM(max), in each annular bulk with the three
types of reinforcements during FCM from 9.4 T. For each case,σθFCM(max) increases with increasing FCM time step, takes a maximum at the intermediate step, and then decreases with increasingtime step for each bulk and each reinforcement. However, the magnitude and the time step dependence of σθFCM(max) change dependingon the reinforcement structure. In Fig. 5(a) for the conventional “typeA” reinforcement, the σθFCM(max) value is larger, compared to thosefor the “type B” and “type C” reinforcements. The “type C” reinforcement is the most eﬀective to reduce the electromagnetic hoop
stress during FCM.

Fig. 3. The radius (r) dependence of the thermal hoop stress, σθcool, for theannular REBaCuO bulks reinforced by three types of aluminum alloy rings foreach z-position under cooling from 300 to 50 K. Inset of each ﬁgure shows thecontour map of the σθcool.

magnetizing solenoid coil. In the annular bulk periphery, the magnetic
ﬁeld monotonically decreases with increasing time step and has aconstant ﬁeld slope, which is equivalent to a constant critical current
density, Jc, based on Bean's model. The Bz value at z = r = 0 was
maintained to 9.4 T up to the ﬁnal (10th) step due to the large Jc value.
Fig. 2(b) and (c), respectively, show the contour maps of the trapped
ﬁeld, Bz, and the persistent current density, Jθ, at the ﬁnal step. The Bz
proﬁle is rounded at the top surface of the “A1” bulk, where the persistent current similar to Jc ﬂows even at the surface of the “A1” bulk, asshown in the blue region in Fig. 2(c).

54Physica C: Superconductivity and its applications 550 (2018) 52–56

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Fig. 4. Cross-sectional plots of the electromagnetic hoop stress, σθFCM, for theannular bulks with the (a) “type A”, (b) “type B” and (c) “type C” reinforcements at the 6th step of FCM from Bex = 9.4 T, without cooling.

Fig. 5. Time step dependence of the maximum electromagnetic hoop stress,σθFCM(max), in each annular bulk with the three types of aluminum alloy ring
reinforcements ((a) conventional “type A”, (b) new “type B” and (c) new “typeC”) during FCM from Bex = 9.4 T.

3.4. Total hoop stress, σθ during FCM under coolingelectromagnetic hoop stress than the mechanical strength of
30–100 MPa [15,16], during the actual FCM process from 9.4 T. For
the “type B” reinforcement shown in Fig. 6(b), the larger tensile σθvalue was fairly reduced to +116 MPa only at the innermost periphery of the surface of the “A1” bulk. Furthermore, for the “type C”
reinforcement shown in Fig. 6(c), the σθ value was further reduced tobe smaller than +17 MPa even for the “A1” bulk, in which all annularbulks will never break during FCM even from 9.4 T. However, thelarge positive σθ value is applied in the inner ring support of the “typeC” reinforcement, which is, however, lower than the fracture strengthof the aluminum alloy.

In the actual FCM process, both the thermal and electromagnetic
stresses must be considered from both the cooling and the FCM process. Fig. 6 shows the radius (r) dependence of the total hoop stress, σθ(= σθcool + σθFCM), at the 6th step at various z-positions in the annularbulks reinforced by three types of the reinforcements, under both thecooling process (σθcool) from 300 to 50 K and the FCM process (σθFCM)
from 9.4 T. In the Fig. 6(a) for the “type A” reinforcement, the σθ value
increased positively in the annular bulks by the superposition; σθ was+235 MPa at r = 16 mm and z = 66 mm (near the top surface of the“A1” bulk), but the σθ was negative for other annular bulks. These
results suggest that the “A1” bulk might be broken by the fairly larger
55

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the actual annular REBaCuO bulks reinforced by three types of reinforcements using aluminum alloy ring during FCM from Bex = 9.4 T
at 50 K. The important analytical results and conclusions are summarized as follows.
(1) The thermal hoop stress, σθcool, which was applied to the annularbulks during cooling, changed depending on the shape of the aluminum alloy reinforcement. For the conventional metal ring reinforcement (“type A”), a large tensile stress was applied at thesurface of the end bulk. The new proposed “type B” reinforcement
structure with a ring plate to the end bulk of “type A” is eﬀective to
reduce the tensile stress. Furthermore, another new “type C” reinforcement structure with an inner ring to the end bulk of “type B”
is more eﬀective.
(2) The electromagnetic hoop stress, σθFCM, also changed depending on
the shape of the aluminum alloy reinforcement and became themaximum at the innermost edge of the uppermost ring bulk at intermediate step in FCM.
(3) The actual total hoop stress, σθ (= σθcool + σθFCM), due to bothcooling and FCM processes was analyzed for three types of aluminum alloy reinforcements. As a result, the “type C” reinforcement
was conﬁrmed to be fairly eﬀective to reduce the total hoop stressand to have a possibility to avoid the mechanical fracture of theannular bulks.
Acknowledgements
This research is partially supported by a “Development of Systems
and Technologies for Advanced Measurement and Analysis” from Japan
Agency for Medical Research and development, AMED, and from JSPS
KAKENHI grant no. 15K04646.
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Fig. 6. The radius (r) dependence of the total hoop stress, σθ (= σθcool + σθFCM),
value at the 6th step at various z-positions in the ﬁnite ring bulk reinforced bythree types of the reinforcements ((a) conventional “type A”, (b) new “type B”,
and (c) new “type C”), under both the cooling process (σθcool) from 300 to 50 K
and the FCM process (σθFCM) of Bex = 9.4 T.

4. Conclusion
We have performed numerical simulation of the hoop stress, σθ, in56