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Numerical modelling of mechanical stresses in bulk superconductor
magnets with and without mechanical reinforcement
To cite this article: M D Ainslie et al 2019 Supercond. Sci. Technol. 32 034002

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Superconductor Science and Technology
Supercond. Sci. Technol. 32 (2019) 034002 (17pp)

https://doi.org/10.1088/1361-6668/aaf851

Numerical modelling of mechanical stresses
in bulk superconductor magnets with andwithout mechanical reinforcement
M D Ainslie1 , K Y Huang1, H Fujishiro2 , J Chaddock1, K Takahashi2 ,S Namba2, D A Cardwell1 and J H Durrell1
1

University of Cambridge, Department of Engineering, Trumpington Street, Cambridge, CB2 1PZ, United
Kingdom2
Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, Morioka 020-8551, Japan
E-mail: mark.ainslie@eng.cam.ac.uk
Received 10 September 2018, revised 21 November 2018
Accepted for publication 12 December 2018
Published 5 February 2019
Abstract

The magnetic ﬁeld trapping capability of a bulk superconductor is essentially determined by thecritical current density, Jc(B, T), of the material. With state-of-the-art bulk (RE)BCO (where
RE = rare earth or Y) materials it is clear that trapped ﬁelds of over 20 T are potentiallyachievable. However, the large Lorentz forces, FL=J×B, that develop during magnetisationof the sample lead to large mechanical stresses that can result in mechanical failure. The radial
forces are tensile and the resulting stresses are not resisted well because of the brittle ceramic
nature of (RE)BCO materials. Where ﬁelds of more than 17 T have been achieved, the samples
were reinforced mechanically using resin impregnation and carbon-ﬁbre wrapping or shrink-ﬁt
stainless steel. In this paper, two-dimensional (2D) axisymmetric and three-dimensional (3D)ﬁnite-element models based on the H-formulation, implemented in the commercial ﬁnite element
software package COMSOL Multiphysics, are used to provide a comprehensive picture of themechanical stresses in bulk superconductor magnets with and without mechanical reinforcement
during ﬁeld-cooled magnetisation. The chosen modelling framework couples together
electromagnetic, thermal and structural mechanics models, and is extremely ﬂexible in allowing
the inclusion of various magnetisation processes and conditions, as well as detailed and realistic
properties of the materials involved. The 2D model—a faster route to parametric optimisation—
is ﬁrstly used to investigate the inﬂuence of the ramp rate of the applied ﬁeld and any heatgenerated in the bulk. Finally, the 3D model is used to investigate the inﬂuence of
inhomogeneous Jc(B, T) properties around the ab-plane of the bulk superconductor on thedeveloped mechanical stress.
Keywords: high temperature superconductivity, bulk superconductors, trapped ﬁeld magnets,
numerical modelling, ﬁnite element method, mechanical stress, mechanical properties(Some ﬁgures may appear in colour only in the online journal)
1. Introduction
Bulk superconductors, acting as trapped ﬁeld magnets, cantrap magnetic ﬁelds over ten times larger than the maximum
ﬁelds produced by conventional permanent magnets, which
are limited practically to rather less than 2 T [1]. It has beenshown that (RE)BCO (where RE = rare earth or Y) bulk

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Figure 1. (a) 2D axisymmetric model of a disc-shaped bulk superconductor with a mechanical reinforcement ring in the r-z-Φ cylindricalcoordinate system, making additional use of symmetry along the r-axis to model half the cross-section. (b) 3D model in the x–y–z coordinate
system, where geometric symmetry is used to model only 1/8th of the bulk: 1/4 of the bulk around the ab-plane and 1/2 of the bulk along
the c-axis (z-axis). In this work, a diameter of 30 mm and thickness of 15 mm are assumed as the dimensions of the bulk, and thereinforcement ring is assumed to be 10 mm wide and the same thickness as the bulk.

Figure 2. Measured Jc(B, T) characteristics of a representative HTS
bulk (15 wt% Ag-containing GdBa2Cu3O7−δ), presented in [24, 25]
for ﬁelds up to 10 T over a temperature range of 40–85 K, extended
here to 20 T using the equation presented by Jirsa et al in [31]. The
data is input into the model using a two-variable, direct interpolation,
as described in [32, 33].

Figure 3. Time-dependence of the temperature, T(t), as well as theapplied magnetic ﬁeld, Bz(t), during the FCM process as simulated inthese models.

because of the brittle ceramic nature of (RE)BCO materials[7]. To achieve the record trapped ﬁelds >17 T, the samples
were reinforced mechanically using resin impregnation andcarbon-ﬁbre wrapping [2] and shrink-ﬁt stainless steel [3].
Extending numerical models developed to date toinvestigate the magnetisation of bulk superconductors, whichhave been primarily focused on the electromagnetic andthermal analyses [8], there has been a great deal of interest
recently in simulating and analysing the mechanical properties of bulk superconductors using numerical tools [9–14].
These studies have been predominately based on the
A-formulation used by Fujishiro et al [9–13], assuming a
constant temperature, but the H-formulation was recently
used to analyse the stresses developed during pulsed ﬁeld

superconductors can trap ﬁelds greater than 17 T: 17.24 T at
29 K [2] and 17.6 T at 26 K [3] have been demonstrated. The
magnetic ﬁeld trapping capability of a bulk superconductor isessentially determined by the critical current density, Jc(B, T),of the material. With state-of-the-art bulk (RE)BCO materials
it is clear that trapped ﬁelds of over 20 T are potentiallyachievable. However, the large Lorentz forces, FL = J × B,
that develop during magnetisation of the sample lead to large
mechanical stresses that can result in mechanical failure, withunreinforced samples typically failing for magnetic ﬁelds
greater than 7–9 T [4, 5]. The radial forces are tensile innature [6] and the resulting stresses are not resisted well2

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Table 1. List of assumed material properties for the coupled electromagnetic-thermal-mechanical modelling of mechanical stresses in bulk
superconductor magnets with and without mechanical reinforcement.

Parameter

Description

Value

n
E0
Jc(B, T)
Tc
dBup/dt
dBdown/dt
dT/dt
ρbulk
ρsus
κab
κc
κsus
Cbulk
Csus
αbulk
αsus
Ebulk
Esus
νbulk
νsus

n value (E–J power law)
Characteristic voltage (E–J power law)
In-ﬁeld, temperature-dependent critical current density
Superconducting transition temperature (GdBa2Cu3O7−δ)Ramp rate of applied external magnetic ﬁeld (ascending stage)
Ramp rate of applied external magnetic ﬁeld (descending stage)
Time rate of change of temperature during cooling (in-ﬁeld) phase
Density (bulk)
Density (stainless steel)
Thermal conductivity (bulk, ab-plane)
Thermal conductivity (bulk, c-axis)
Thermal conductivity (stainless steel)Heat capacity (bulk)
Heat capacity (stainless steel)
Coefﬁcient of thermal expansion (bulk)
Coefﬁcient of thermal expansion (stainless steel)Young’s modulus (bulk)
Young’s modulus (stainless steel)Poisson’s ratio (bulk)
Poission’s ratio (stainless steel)20
1×10−4 V m−1
Interpolation (see ﬁgure 2)
92 K
0.4 T s−1
25 mT s−1
–1 K s−1
5900 kg m−3
8000 kg m−3
20 W m−1 K−1
4 W m−1 K−1
Measured data (see ref.)Measured data (see ref.)Measured data (see ref.)5.2×10−6 K−1
1.27×10−5 K−1
1×1011 Pa
1.93×1011 Pa
0.33
0.28

Reference(s) (ifapplicable)[8][8][24, 25][8][35][8, 35][8, 34]
[24]
[24]
[24]
[9–13][10–12][9–13][10–12][9–13][10–12]modelling framework couples electromagnetic, thermal andstructural mechanics models, and is extremely ﬂexible toinclude various magnetisation processes and conditions, aswell as detailed and realistic properties of the materials
involved. In section 2, the modelling framework is introduced
and the results of the 2D and 3D models are compared todemonstrate the consistency of the two models, as well as toshow the effect that the reinforcement has on the hoop stressin the bulk during FCM. In section 3, the 2D model—a fasterroute to parametric optimisation—is ﬁrstly used to investigate
the inﬂuence of varying the ramp rate of the applied ﬁeld andany heat generated in the bulk. Finally, the 3D model is used
to investigate the inﬂuence of inhomogeneous Jc(B, T) properties around the ab-plane of the bulk superconductor on thedeveloped mechanical stress.
Figure 4. Trapped ﬁeld at the centre of the bulk and at the centre of
the top surface at z=+0.5 mm above the bulk during the ramp
down of the applied ﬁeld in the FCM process (t=1250 s), for the
2D axisymmetric and 3D models, under the following magnetisingconditions: Top=50 K, Bapp=20 T, dBdown/dt=25 mT s−1.

2. Modelling framework and preliminary results
2.1. Modelling framework

Numerical techniques based on the ﬁnite element method(FEM) have been applied to many superconducting material
problems using a variety of formulations [8]. Each of theseformulations are equivalent in principle, i.e. the choice of
formulation should not result in a different physical meaningof the solution, but the solutions of the corresponding partial
differential equations can be very different [16]. For moredetailed information, including techniques not based on FEM,
the reader may refer to recent review papers on numericalmethods for calculating AC losses in high-temperature
superconducting (HTS) materials [17], the modelling of bulk
superconductor magnetisation [8] and HTS applications [18].

magnetisation (PFM) by Wu et al [14], which was coupledwith a thermal model to include the inﬂuence of heat generated during the PFM process.
In this paper, two-dimensional (2D) axisymmetric andthree-dimensional (3D) ﬁnite-element models based on the
H-formulation, implemented in the commercial ﬁnite element
software package COMSOL Multiphysics [15], are used toprovide a comprehensive picture of the mechanical stresses inbulk superconductor magnets with and without mechanicalreinforcement during ﬁeld-cooled magnetisation (FCM). The
3

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Figure 5. Comparison of the magnetic ﬂux density, |B|, and current density, |J|, distributions throughout the cross-section of the bulk at theend of the FCM process (t=1250 s), for the 2D and 3D models.

Figure 6. Comparison of the Lorentz force in the radial direction, FL,r, and hoop stress, σj, distributions throughout the cross-section of thebulk at the end of the FCM process (t=1250 s), for the 2D and 3D models.

ab-plane. However, a 3D geometry (x, y, z coordinates) is
required when including any inhomogeneous material properties around the ab-plane, e.g. differences in Jc between
growth sector boundaries (GSBs) and growth sector regions(GSRs) [19], and for shapes without cylindrical symmetry,

A bulk superconductor is usually fabricated in the formof a cylindrical disc and this geometry lends itself well tosimpliﬁcation using a 2D axisymmetric model (r, z, Φ in acylindrical coordinate system), assuming that its properties
are homogeneous in the Φ-direction, i.e. around the bulk’s
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the permeability of free space, and for the superconducting
and air sub-domains, the relative permeability can be assumed
as simply μr=1.
The E–J power law [29, 30] is used to simulate the highly
non-linear resistivity of the superconductor, where E is proportional to Jn and n=20 is assumed as a typical value for
HTS materials and a good approximation of Bean’s criticalstate model [26].
The results of the numerical simulation depend strongly
on the assumed Jc(B, T) characteristics of the material, and in
this paper, the measured Jc(B, T) characteristics of a representative bulk HTS (15 wt% Ag-containing GdBa2Cu3O7−δ),presented in [24, 25] for ﬁelds up to 10 T over a temperature
range of 40–85 K, is extended to 20 T using the equation
presented by Jirsa et al in [31]:Figure 7. Time-dependence of the hoop stress, σj, across the centreof the bulk (z = 0 mm) at discrete points every 4 T during the ramp
down of the applied ﬁeld in the FCM process.

⎡1 ⎛
⎛ B⎞
⎛ B ⎞y ⎞ ⎤B
Jc (B) = Jc1 exp ⎜ - ⎟ + Jc2
exp ⎢ ⎜1 - ⎜⎟ ⎟ ⎥.
⎢⎣ y ⎝⎝ BL ⎠⎝ Bmax ⎠ ⎠ ⎥⎦
Bmax
(3 )

such as rectangular-shaped bulks [20]. It can also be useful to
exploit symmetry, where possible, in a 3D model by applying
appropriate boundary conditions to model, for example, one
half (2D or 3D) or one quarter of the bulk. This can reduce thenumber of mesh elements required and improve both computational memory requirements and speed, while retaining
the necessary 3D effect. Figure 1(a) shows a 2D axisymmetricmodel of a disc-shaped bulk superconductor in the r-z-Φ
cylindrical coordinate system, making additional use ofsymmetry along the r-axis to model half the cross-section.
The cross-section (Φ-direction; rz-plane) corresponds to theab-plane of the bulk and the z-axis corresponds to the c-axis.
Figure 1(b) shows a 3D model in the x–y–z coordinate system, where geometric symmetry is used to model only 1/8thof the bulk: 1/4 of the bulk around the ab-plane and 1/2 ofthe bulk along the c-axis (z-axis). In this work, a diameter, D,of 30 mm and thickness, H, of 15 mm are assumed as thedimensions of the bulk, which is a typical size of those fabricated by our research group, and the reinforcement ring isassumed to be 10 mm wide and the same thickness as
the bulk.
The bulk’s electromagnetic properties are simulatedusing the H-formulation, implemented in the commercial
software package COMSOL Multiphysics 5.3a [15]. This
framework has been used previously by the authors tosimulate bulk (RE)BCO materials under various magnetisation conditions [19, 21–26], as well as FCM of MgB2 [27]and iron-pnictide [28] bulks. In the H-formulation, the governing equations are derived from Maxwell’s equations—
namely, Ampere’s (1) and Faraday’s (2) laws:
´H=J
´E+

d (m 0 m r H )
dB
=´E+
= 0,
dt
dt

Figure 2 shows the assumed Jc(B, T) characteristics,
which are input into the model using a two-variable, directinterpolation, as described in [32, 33]. Since FCM is being
simulated, there are three distinct steps in the numericalmodel [27]: (1) applying a ramped external magnetic ﬁeld
parallel to the c-axis of the bulk up to a maximum magnitude,
Bapp, by setting appropriate magnetic ﬁeld boundary conditions [34] such that Hz(t)=Happ(t/tramp) for ttramp, whereHapp=Bapp/μ0 and tramp is the duration of the ramp, while
the temperature is held at T>Tc=92 K (T=300 K in thispaper); (2) cooling the bulk to an appropriate operating
temperature, Top=Tc, while the external ﬁeld is held at Bapp;and (3) once the temperature has stabilised, ramping theexternal ﬁeld down from Bapp to zero. Hence, Jc must also bedeﬁned for T Tc=92 K. Here, Jc is assumed to be1×106 A m−2 for T 92 K. This is also true for whenJc<1×106 A m−2 for T between 60 K and 85 K in ﬁgure 2.
Since the temperature of the bulk changes signiﬁcantly
during FCM, the electromagnetic model is coupled with athermal model, based on the following thermal transient
equation:r·CdT=  · (k T ) + Q.
dt

(4 )

The models in this section are coupled through the Jc(B,T) characteristics described earlier and isothermal conditions are assumed while ramping down the ﬁeld, i.e. Q=0,
because the magnetisation process is slow. It is possible tofurther couple the electromagnetic and thermal models by
including any heat generated during the magnetisation
process by assuming Q = Ej · Jj (2D axisymmetricmodel) or Q = Enorm · Jnorm (3D model), where
Enorm = Ex2 + Ey2 + Ez2 and Jnorm = Jx2 + Jy2 + Jz2 .
This was recently performed in [14] to analyse the stressduring PFM and will be used in section 3.1.
Figure 3 shows the time-dependence of the temperature,
T(t), as well as the applied magnetic ﬁeld, Bz(t), during the

(1 )
(2 )

where H = [Hr, Hz] represents the magnetic ﬁeld components,
J=[Jj] represents the current density and E=[Ej] represents the electric ﬁeld in the 2D axisymmetric model. In 3D,H=[Hx, Hy, Hz], J=[Jx, Jy, Jz] and E=[Ex, Ey, Ez]. μ0 is

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Figure 8. Comparison of the hoop stress, σj, distributions, when including a stainless-steel reinforcement ring that provides a thermal
compressive stress when cooled during FCM, for the 2D and 3D models, at the end of the FCM process (t = 1250 s). The same magnetisingconditions as section 2.2.1 are assumed: Top=50 K, Bapp=20 T, dBdown/dt=25 mT s−1.

Figure 9. Time-dependence of the hoop stress, σj, across the centreof the bulk and reinforcement ring (z=0 mm) at the beginning(20 T) and end (0 T) of the ramp down of the applied ﬁeld in the
FCM process when the bulk is mechanically reinforced. The results
at 4 T, corresponding to the maximum σj in the unreinforced case(see ﬁgure 7), are also shown for comparison.

Figure 10. Time-dependence of the maximum hoop stress, σj,max,anywhere in the cross-section/volume of the bulk during the ramp
down of the applied ﬁeld in the FCM process, with and without
mechanical reinforcement.

where σs is the Cauchy stress tensor and FL is the Lorentz
force (described below). Thus, the structural transient behaviour is assumed to be quasi-static and the second order timederivatives of the displacement variables, u—the inertial
terms, ρd2u/dt2—are zero.
The strain–displacement relationship is given by

FCM process as simulated in these models. It should be notedthat in section 3.1, the effect of heat generation during ramp
down of the ﬁeld is investigated by including Q=E·J andvarying the ramp rate.
Finally, the mechanical stresses during FCM are derived
from the following, principal governing equation, an
expression of Newton’s second law in direct tensor form:
 · sS + FL = 0,e=1[u + (u)T ] ,2(6 a )

where ε is the inﬁnitesimal strain tensor. The bulk is assumed
to be an isotropic, linear elastic material, obeying Hooke’s law,

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the trapped ﬁeld at the centre of the top surface(z = +0.5 mm), Bt, is approximately 7.6 T, and the trapped
ﬁeld at the centre, Bc, is approximately 11.4 T.
Figure 5 shows a comparison of the magnetic ﬂux
density, |B|, and current density, |J|, distributions throughout
the cross-section of the bulk at the end of the FCM process(t=1250 s), for the 2D axisymmetric and 3D models, under
the same magnetising conditions. Figure 6 shows a similar
comparison of the Lorentz force in the radial direction, FL,r,and hoop stress, σj, distributions. The results show clear
consistency between the two models and the cross-sectionalplots give a clear indication of the location of maximum
Lorentz force, as well as maximum hoop stress, which ishighest at the centre of the bulk.
Figure 7 shows the time-dependence of the hoop stress,
σj, across the centre of the bulk (z=0) at discrete points
every 4 T from during the ramp down of the applied ﬁeld in
the FCM process. The results clearly show the dynamic
nature of the generated hoop stress, which is directly related
to the Lorentz force generated by the distribution of thesimultaneously reducing magnetic ﬁeld, B, and increasing
induced current, J, which takes a maximum when the ﬁeld
has ramped down to around 4 T under the given magnetisingconditions and material properties. The inﬂuence of thereinforcement on the dynamic nature of the hoop stress, aswell as its maximum value, is investigated in the next
section.

Figure 11. Trapped ﬁeld at the centre of the top surface atz=+0.5 mm above the bulk (without reinforcement) during theramp down of the applied ﬁeld in the FCM process for ramp-down
rates of 2.5, 25 and 250 mT s−1, for Top=50 K and Bapp = 20 T.
The heat source, Q=E · J, is introduced to include the inﬂuence ofany heat generated and the bulk is cooled from its bottom surface.

such that σs=C(E, ν): ε; C is the fourth-order stiffness tensor.
Here the mechanical properties of the bulk are deﬁned by its
Young’s modulus, Ebulk=1×1011 Pa, and Poisson’s ratio,νbulk=0.33 [9–13], and a density, ρbulk=5900 kg m−3, isassumed [35]. The stainless-steel reinforcement ring, where
present, is also assumed to be an isotropic, linear elastic
material with Esus=1.93× 1011 Pa, νsus=0.28 andρsus=8000 kg m−3 [10–12] and, by default, a perfectmechanical contact between the ring and the bulk is assumed.
The Lorentz force, FL=J×B, that develops during the FCM
process, derived from the electromagnetic model, is used as the
input to calculate the mechanical stresses. Table 1 lists all of
the assumed material properties for the coupled electromagnetic-thermal-mechanical modelling, including applicablereferences to other work in the literature.

2.2.2. With ring reinforcement. In this section, the samemagnetising conditions are assumed (Top=50 K, Bapp=20 T,
dBdown/dt=25 mT s−1), but a stainless-steel ring of width
10 mm (inner radius, rinner, of 15 mm, to match the radius of thebulk; outer radius, router, of 25 mm) and the same thickness as
the bulk (15 mm) is added around the periphery of the bulk toprovide mechanical reinforcement. The resultant plots of thetrapped ﬁeld (see ﬁgure 4), as well as |B| and |J| (see ﬁgure 5)
and FL,r (see ﬁgure 6) are the same as for the unreinforcedcase in section 2.2.1; however, the resultant hoop stresschanges because of the contribution from the thermal
compressive stress applied to the bulk by the reinforcement
ring when cooled during FCM (before ramping down theapplied ﬁeld). This occurs due to the difference in the thermal
coefﬁcient of expansion between the two materials. A
comparison of the hoop stress distributions throughout thecross-sections of the bulk and reinforcement ring at the end of
the FCM process (t=1250 s), for the 2D and 3D models, isshown in ﬁgure 8. Again, the results show clear consistency, and
it is also clear that the bulk is in compression (negative σj),whereas the reinforcement ring is in tension (positive σj), at theend of the FCM process.
Figure 9 shows the time-dependence of the hoop stress,
σj, across the centre of the bulk and reinforcement ring(z=0) at the beginning (20 T) and end (0 T) of the ramp
down of the applied ﬁeld in the FCM process when the bulk ismechanically reinforced. The results at 4 T, corresponding to
the maximum σj in the unreinforced case (see ﬁgure 7), are2.2. Preliminary results (2D axisymmetric and 3D models)In this section, the results of the 2D axisymmetric and 3Dmodels are compared, both with and without mechanicalreinforcement with a stainless-steel ring, to demonstrate theconsistency of the two models, as well as to show the effectthat the reinforcement has on the hoop stress in the bulk
during FCM.
2.2.1. Without ring reinforcement. Figure 4 shows thetrapped ﬁeld at the centre of the bulk and at the centre of
the top surface at z=+0.5 mm above the bulk during
the ramp down of the applied ﬁeld in the FCM process, forthe 2D axisymmetric and 3D models, under the following
magnetising conditions: Top=50 K, Bapp=20 T, dBdown/dt= 25 mT s−1. At the end of the FCM process (t=1250 s),

7Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 12. Comparison of the temperature distribution within the bulk (without reinforcement) at the end of the FCM process for ramp-down

rates of 2.5, 25 and 250 mT s−1, for Top=50 K and Bapp=20 T and including the heat source, Q=E·J. The bulk is cooled from its
bottom surface.

also shown for comparison. It should be noted that σj in thiscase includes both contributions from σj(FCM), the hoop
stress related to the Lorentz force, and σj(COOL), the thermal
compressive stress applied to the bulk by the ring whencooling to Top. The results suggest that for these particularmagnetising conditions, there is adequate reinforcement, as
the centre of bulk (where the largest hoop stress occurs
in the unreinforced case) is in compression, as is the rest of
the bulk.
Figure 10 shows the time-dependence of the maximum
hoop stress, σj,max, anywhere in the cross-section/volume of
the bulk, with and without mechanical reinforcement, as theﬁeld is ramped down. For the unreinforced bulk, σj,max
corresponds to the maximum value at the centre of the bulk asshown in ﬁgure 7: σj,max = 63.3 MPa at approximately 4 T.
However, for the reinforced bulk, σj,max (4.9 MPa atapproximately 6 T) is not located at the centre of the bulk,
but at the edge of the bulk near the surface, which can be seen
in ﬁgure 8. This was also observed in numerical models forreinforced ring-shaped bulks in [12], and careful optimisationof the ring geometry compared to the bulk is necessary to
avoid local stresses and improve the uniformity of the effectof the mechanical reinforcement, which could be carried out
easily using this numerical framework.

assumption that the material properties are homogeneousaround the bulk’s ab-plane holds. In the following sections,the effect of varying the ramp rate of the applied ﬁeld and any
heat generated in the bulk is investigated using the 2D axisymmetric model, and then the 3D model is used to investigate the inﬂuence of an inhomogeneous Jc(B) distribution
around the bulk’s ab-plane.
3.1. Influence of ramp rate and heat generated
3.1.1. Without ring reinforcement. In the simulations so far,isothermal conditions have been assumed, i.e. Q=0 inequation (4), neglecting the inﬂuence of any heat generated
during the ramp down of the applied ﬁeld. In general, theramp-down rate in the FCM process is usually slow and theheat generated is nowhere near as extreme as that experienced
during PFM [14]. However, heat generated during any
magnetisation process limits the trapped ﬁeld capability ofthe material and it is important to understand how much heat
may be generated, as well as its inﬂuence on the mechanicalstresses in the material. In this section, the heat source,
Q=E·J, is introduced to the thermal model (equation (4)),
and the inﬂuence of three ramp down rates (2.5, 25 and
250 mT s−1) on the FCM process is investigated. It is
assumed, as done in section 2.2, that Top=50 K andBapp=20 T.
In the previous analyses, the temperature of the bulk was
set by a temperature constraint across the centre of the bulk;
however, when including the heat generated during FCM, onemust apply a more realistic assumption regarding cooling and3. Inﬂuence of system parameters
The 2D axisymmetric model provides a faster route to parametric optimisation in comparison to the 3D model, if the8Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 13. Comparison of the magnetic ﬂux density, |B|, distribution within the bulk (without reinforcement) at the end of the FCM process

for ramp-down rates of 2.5, 25 and 250 mT s−1, for Top=50 K and Bapp=20 T and including the heat source, Q=E·J.

Figure 14. Comparison of the current density, |J|, distribution within the bulk (without reinforcement) at the end of the FCM process for
ramp-down rates of 2.5, 25 and 250 mT s−1, for Top=50 K and Bapp=20 T and including the heat source, Q=E·J.

whole cross-section of the bulk (in the 2D axisymmetricmodel). It is also possible to cool the sample from theperiphery of the bulk for a split coil magnetisation ﬁxture[24]. For simplicity, perfect thermal contact between the bulk

here it is assumed that the bulk is cooled from its bottom
surface (a typical scenario when magnetised in solenoidal
high ﬁeld magnets), necessitating the removal of thesymmetry along the centre of the bulk and modelling of the9Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 15. Hoop stress, σj, at various positions within the bulk (without reinforcement), z=0, ±6 mm, at the end of the FCM process
for Top=50 K and Bapp=20 T and including the heat source, Q=E·J: (a) ramp-down rate of 25 mT s−1 (isothermal conditions),(b) 25 mT s−1, (c) 2.5 mT s−1 and (d) 250 mT s−1.

intermediate ramp rate (25 mT s−1), there is a temperature riseof around 3 K towards the upper surface of the bulk. In thecase of the fastest ramp rate (250 mT s−1), there is asigniﬁcant temperature rise of up to 18 K in the same region.
The impact of this temperature rise on the magnetic ﬂux
and current density distributions is shown in ﬁgures 13
and 14, respectively, which accounts for the reduced trapped
ﬁelds in ﬁgure 11, except for the 2.5 mT s−1 case. In this case,although the temperature rise is negligible, because the ramp
down occurs over a longer time scale (t=450→8450 s),
there is more ﬂux creep, resulting in a slightly lower trapped
ﬁeld at the end of the FCM process. The 250 mT s−1 rampdown rate results in a signiﬁcantly distorted current density/
trapped ﬁeld distribution, and although there is a small
temperature rise for the 25 mT s−1 case, which deviates

and the cryocooler is also assumed; however, as shown in[24, 25, 35], it is possible to adapt the model to include more
realistic assumptions regarding this thermal contact, as well as
the ﬁnite cooling power of the cryocooler.
Figure 11 shows the trapped ﬁeld at the centre of the top
surface at z=+0.5 mm above the bulk during the ramp
down of the applied ﬁeld in the FCM process for ramp-down
rates of 2.5, 25 and 250 mT s−1, for Top=50 K andBapp=20 T. As the ramp-down rate is increased, there is areduction in the trapped ﬁeld due to a reduction in Jc, which ismost noticeable for the fastest ramp rate. Figure 12 shows thetemperature of the bulk at the end of the FCM process att=for 450, 1250 and 8450 s for the three ramp rates, 2.5, 25
and 250 mT s−1, respectively. There is almost no temperaturerise for the slowest ramp rate (2.5 mT s−1), and for the10Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al450, 1250 and 8450 s for the three ramp rates, 2.5, 25 and
250 mT s−1, respectively. There is almost no temperature risefor the slowest ramp rate (2.5 mT s−1), and for theintermediate ramp rate (25 mT s−1), there is a temperaturerise of around 1.2 K towards the upper surface of the bulk
(3 K without reinforcement). In the case of the fastest ramp
rate (250 mT s−1), the temperature rise is less than 10 K (18 Kwithout reinforcement) in the same region. The presence ofthe reinforcement ring provides an additional thermal pathway that allows cooling through the ab-plane of the bulk, forwhich the thermal conductivity is several times larger thanalong the c-axis [9].
The impact of the reduced temperature rise on themagnetic ﬂux and current density distributions is shown inﬁgures 18 and 19, respectively, and it is clear that there is lessdistortion of the current density/trapped ﬁeld distribution.
There is then signiﬁcantly less distortion of developed hoop
stress, σj, within in the bulk, which is shown in ﬁgure 20 at
various positions within the bulk, z=0 and ±6 mm, for thethree ramp-down rates, including 25 mT s−1 under theassumption of isothermal conditions. It should be noted againthat σj in this case includes both contributions fromσj(FCM), the hoop stress related to the Lorentz force, andσj(COOL), the thermal compressive stress applied to the bulk
when cooling to Top. There is only a perceptible difference in
σj for the fastest ramp rate (250 mT s−1), meaning that thereinforcement ring not only provides compression of the bulk
during the FCM process to avoid mechanical fracture, but
provides an additional thermal pathway to remove the heatgenerated during the magnetisation process, which isparticularly useful in the case of PFM.

Figure 16. Trapped ﬁeld at the centre of the top surface atz=+0.5 mm above the bulk (without reinforcement) during theramp down of the applied ﬁeld in the FCM process for ramp-down
rates of 2.5, 25 and 250 mT s−1, for Top=50 K and Bapp = 20 T.
The heat source, Q = E · J, is introduced to include the inﬂuence ofany heat generated.

slightly from the previous assumption of isothermal conditions, a signiﬁcant impact is not observed.
The impact on the developed hoop stress, σj, within in
the bulk is shown in ﬁgure 15 at various positions within thebulk, z=0 and ±6 mm, for the three ramp-down rates,
including 25 mT s−1 under the assumption of isothermal
conditions. The faster ramp rates, resulting in a higher
temperature rise and subsequently reduced Jc, see lower
stresses developed overall, which agrees well with theanalysis in the previous section. In addition, there is a larger
distortion of the stresses between the top and bottom regionsof the bulk. In the next section, the impact of thereinforcement ring on these ﬁndings is explored.

3.2. Inhomogeneous Jc distribution around the ab-plane

Although the 2D axisymmetric model provides a faster routeto parametric optimisation, this assumes that the material
properties are homogeneous around the bulk’s ab-plane. In
this section, the 3D model is used to investigate an example ofthe inﬂuence of inhomogeneous material properties aroundthe ab-plane, such as those that occur during the growth
process of c-axis seeded, single grain (RE)BCO bulk superconductors [19]. The 3D model is also useful for shapes
without cylindrical symmetry, such as rectangular-shapedbulks [20].
Following the same method presented in [19, 36], theJc(B, T) characteristics are varied using a cosine function
around the ab-plane, supposing some difference of ±10% in
Jc between the GSBs and GSRs, such that

With
ring
reinforcement. The
stainless-steel
reinforcement ring is now added around the periphery of
the bulk and the same magnetising conditions as the previous
section are assumed: Top=50 K, Bapp=20 T and three
ramp-down rates (2.5, 25 and 250 mT s−1), including the heatsource, Q=E·J. It is also assumed again that the thermal
contact between the bulk, ring and cryocooler is ideal.
Figure 16 shows the trapped ﬁeld at the centre of the top
surface at z=+0.5 mm above the bulk during the ramp
down of the applied ﬁeld in the FCM process for ramp-down
rates of 2.5, 25 and 250 mT s−1, for Top=50 K andBapp=20 T. As seen in the previous section, as the rampdown rate is increased, there is a reduction in the trapped ﬁeld
due to a reduction in Jc, which is most noticeable for the
fastest ramp rate. However, there is notably less reduction intrapped ﬁeld with the reinforcement ring, compared towithout one. Figure 17 shows the temperature of the bulk
with reinforcement at the end of the FCM process at t=for

3.1.2.

Jc (B , T , q ) = Jc (B , T )(1 + 0.1 cos (4q )).

(6 b )

Given the relevant material properties are known, by
measuring a number of sub-specimens from a bulk superconductor, any position-dependent Jc could be introduced in
the model.
The same magnetising conditions as the previous
sections are assumed: Top=50 K, Bapp = 20 T and a ramp-

11Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 17. Comparison of the temperature distribution within the bulk (with reinforcement) at the end of the FCM process for ramp-down

rates of 2.5, 25 and 250 mT s−1, for Top=50 K and Bapp=20 T and including the heat source, Q=E·J.

down rate of 25 mT s−1 under isothermal conditions (Q=0).
At the end of the FCM process (t=1250 s), the trapped ﬁeld
at the centre of the top surface (z=+0.5 mm), Bt, of theinhomogeneous bulk is approximately 7.4 T, and the trapped
ﬁeld at the centre, Bc, is approximately 11 T (see 7.6 T and
11.4 T, respectively, for the homogeneous bulk described in
section 2.2.1).
Figure 21 shows a 3D picture of the hoop stress, σj,distributions for the homogeneous (left) and inhomogeneous(right) Jc assumptions at the end of the FCM process andﬁgure 22 shows a 2D surface plot of σj across the centreof the bulk (xy-plane, where the c-axis is oriented along the zaxis), where the supposed GSBs and GSRs are highlighted
by dotted and dashed white lines, respectively. It is clear thatthe inhomogeneous ab-plane Jc distorts σj with lower stressin low-Jc regions and higher stress in high-Jc regions.
Figure 23 shows a 2D plot of the σj distributions forthe homogeneous and inhomogeneous Jc assumptions atz=0 mm, i.e. across the centre of the bulk. The plots for the
inhomogeneous bulk correspond to the supposed GSBs and
GSRs highlighted in ﬁgure 22. This further emphasises thedifference between the stresses developed in an inhomogeneous bulk compared to a homogeneous one and shows how
the 3D model is useful for incorporating more realistic
assumptions when a 2D axisymmetric model is insufﬁcient.
Figure 24 shows the 2D surface plot of σj across thecentre of the bulk (xy-plane) and ﬁgure 25 shows a 2D plot of
σj across the centre of the bulk at z=0 mm along the supposed GSBs and GSRs at the end of the FCM process, in thecase that the bulk is reinforced by the stainless-steel ring. In
order to better highlight the differences within the homogeneous and inhomogeneous bulks, the ring is not included inthese plots. Here σj includes both contributions fromσj(FCM), the hoop stress related to the Lorentz force, andσj(COOL), the thermal compressive stress applied to the bulk
when cooling to Top. The trends for σj are identical to thoseshown in ﬁgures 22 and 23 for the unreinforced case; however, there is an offset owing to σj(COOL), such that σj isentirely negative (and less than 60 MPa). Thus, the 3D model
provides a useful tool to analyse the inﬂuence of inhomogeneous material properties and design adequate reinforcement
so that mechanical fracture is avoided in high magnetic ﬁelds.

4. Conclusion
In this paper, 2D axisymmetric and 3D ﬁnite-element models
based on the H-formulation, implemented in the commercial
ﬁnite element software package COMSOL Multiphysics,
are used to analyse the mechanical stresses in bulk superconductor magnets during FCM with and without mechanicalreinforcement using a stainless-steel ring.
1. Under identical magnetising conditions and assuming
the same material properties, the 2D axisymmetric and3D ﬁnite-element models produce consistent results: the2D model provides a faster route to parametric
optimisation, but the 3D model is required for bulk
superconductor shapes that do not have cylindricalsymmetry or for inhomogeneous properties in the Φdirection, i.e. around the ab-plane. The models can beused to examine the stresses developed during themagnetisation process, which depend on a number ofimportant factors, including the Jc(B, T) characteristics,

12

Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 18. Comparison of the magnetic ﬂux density, |B|, distribution

Figure 19. Comparison of the current density, |J|, distribution within

within the bulk (with reinforcement) at the end of the FCM process
for ramp-down rates of 2.5, 25 and 250 mT s−1, for Top=50 K andBapp=20 T and including the heat source, Q=E·J.

the bulk (with reinforcement) at the end of the FCM process for
ramp-down rates of 2.5, 25 and 250 mT s−1, for Top = 50 K and
Bapp = 20 T and including the heat source, Q = E · J.

sample geometry, ramp-down rate and so on. It isshown how the use of a mechanical reinforcement ring,with a higher coefﬁcient of thermal expansion thanthe bulk superconductor, signiﬁcantly reduces themechanical stresses developed during FCM owing toσj(COOL), the thermal compressive stress applied tothe bulk by the ring when cooling to the operating
temperature, Top.
2. The 2D axisymmetric model, assuming homogeneous
properties around the ab-plane, is used as a fast
optimisation tool to investigate the inﬂuence of any
heat generated during the ramp down of the appliedﬁeld by including the heat source, Q=E·J. A faster
ramp-down rate of the applied ﬁeld results in a larger
temperature rise, resulting in an uneven temperature
distribution within the bulk, depending on the methodof cooling. This will distort the magnetic ﬂux andcurrent density distributions, which, in turn, affect thedeveloped stresses. The presence of a reinforcement
ring not only provides compression of the bulk during
FCM to avoid mechanical fracture, but also provides anadditional thermal pathway to remove the heatgenerated during the magnetisation process, which isparticularly useful to consider in the case of PFM.
3. The 3D model is used to investigate the inﬂuence of
inhomogeneous Jc(B, T) properties around the ab-plane
and it shown how the developed hoop stress is
distorted, with lower stress in low-Jc regions and higher
stress in high-Jc regions. When the bulk is reinforced
with a stainless-steel ring, the trends for σj are identicalto those in the unreinforced case; however, there is anoffset owing to σj(COOL), such that σj is entirely
negative under the conditions analysed.
The modelling framework couples together electromagnetic, thermal and structural mechanics models to provide
a complete and detailed picture of the mechanical stresses
developed in a bulk superconductor during the FCM process
and a useful and ﬂexible design tool to adequately reinforce
the bulk to avoid mechanical fracture at high magnetic ﬁelds,
taking into account many of the practical situations faced
when carrying out such high ﬁeld experiments.

13

Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 20. Hoop stress, σj, at various positions within the bulk (with reinforcement), z=0, ±6 mm, at the end of the FCM process

for Top=50 K and Bapp=20 T and including the heat source, Q=E·J: (a) ramp-down rate of 25 mT s−1 (isothermal conditions),(b) 25 mT s−1, (c) 2.5 mT s−1 and (d) 250 mT s−1. σj in this case includes both contributions from σj(FCM), the hoop stress related
to the Lorentz force, and σj(COOL), the thermal compressive stress applied to the bulk when cooling to Top.

Acknowledgments
Dr 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 areprovided in full in the results section of this paper. Mr Kaiyuan
Huang would like to acknowledge ﬁnancial support from Siemens AG, Corporate Technology, Munich, Germany. Professor
Hiroyuki Fujishiro would like to acknowledge ﬁnancial support
from JSPS KAKENHI Grant No. JP15K04646.
Figure 21. 3D volume plot of the hoop stress, σj, distributions of an
unreinforced bulk superconductor at the end of the FCM process,assuming homogeneous Jc properties around the ab-plane (left) andinhomogeneous properties (right) supposing some difference of±10% in Jc between the growth sector boundaries (GSBs) andgrowth sector regions (GSRs).

AppendixSome comments on the implementation in COMSOL Multiphysics of the modelling framework described in section 2.2
are provided as follows.
14

Supercond. Sci. Technol. 32 (2019) 034002

M D Ainslie et al

Figure 22. 2D surface plot, cut across the centre of the bulk
(xy-plane), of the hoop stress, σj, distributions of an unreinforced
bulk superconductor at the end of the FCM process for the
homogeneous (left) and inhomogeneous (right) Jc assumptions.
The supposed GSBs and GSRs are highlighted by dotted anddashed white lines, respectively.
Figure 25. 2D plot of the hoop stress, σj, distributions of areinforced bulk superconductor at the end of the FCM process for the
homogeneous and inhomogeneous Jc assumptions at z=0 mm. The
plots for the inhomogeneous bulk correspond to the supposed GSBs
and GSRs highlighted in ﬁgure 24.

1. The electromagnetic model is implemented using the‘Magnetic Field Formulation’ interface in COMSOL’s
AC/DC module.
2. In order to make use of the symmetry in the 3D model(see ﬁgure 1(b)), additional boundary conditions mustalso be set for the sides (yz-plane, x=0; xz-plane,y=0) and bottom plane of the entire geometry: (xyplane, z=0): this is achieved using the ‘Magnetic
Insulation’ node for the former (setting the tangential
component of the magnetic potential to zero) and the‘Perfect Magnetic Conductor’ node for the latter(setting the tangential component of the magnetic ﬁeld
to zero).
3. The thermal model is implemented using COMSOL’s‘Heat Transfer in Solids’ interface and the temperature,
T(t), is set by applying a constraint across the centre of
the bulk in the z=0 plane.
4. The electromagnetic and thermal models are coupledwith COMSOL’s ‘Solid Mechanics’ interface in its
Structural Mechanics module to include analyses of themechanical stresses during FCM. A ‘roller’ constraint(equivalent to a symmetry condition) is applied to thebulk (and ring when applicable) in the z=0 plane,making the displacement in the direction normal to thisplane zero.
5. The Lorentz force, FL=J×B, is implemented as aforce per unit volume using the ‘Body Load’
node where Fr=Bz·Jj and Fz=–Br·Jj (2Daxisymmetricmodel)
orFx=Jy·Bz – Jz·By,
Fy=Jz·Bx – Jx·Bz and Fz=Jx·By – JyBx (3Dmodel).
6. A thermal coefﬁcient of expansion is included for boththe bulk and reinforcement ring (see table 1) using
COMSOL’s ‘Thermal Expansion’ multiphysics interface to simulate the thermal compressive stress applied

Figure 23. 2D plot of the hoop stress, σj, distributions of an
unreinforced bulk superconductor at the end of the FCM process forthe homogeneous and inhomogeneous Jc assumptions at z=0 mm.
The plots for the inhomogeneous bulk correspond to the supposed
GSBs and GSRs highlighted in ﬁgure 22.

Figure 24. 2D surface plot, cut across the centre of the bulk
(xy-plane), of the hoop stress, σj, distributions of a reinforced bulk
superconductor (ring not shown) at the end of the FCM process forthe homogeneous (left) and inhomogeneous (right) Jc assumptions.
The supposed GSBs and GSRs are highlighted by dotted and dashedwhite lines, respectively.
15

Supercond. Sci. Technol. 32 (2019) 034002

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to the bulk by the reinforcement ring when cooledduring FCM because of the difference in this coefﬁcient
between the two materials [9].

ORCID iDs
M D Ainslie https://orcid.org/0000-0003-0466-3680
H Fujishiro https://orcid.org/0000-0003-1483-835X
K Takahashi https://orcid.org/0000-0002-8278-2688
D A Cardwell https://orcid.org/0000-0002-2020-2131
J H Durrell https://orcid.org/0000-0003-0712-3102

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