Physica C 469 (2009) 822–826

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Physica C

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Kim model for stress distribution in a hollow cylindrical superconductor

Jun Zeng, You-He Zhou *, Hua-Dong YongKey Laboratory of Mechanics on Western Disaster and Environment, Ministry of Education, People’s Republic of ChinaDepartment of Mechanics, Lanzhou University, Lanzhou 730000, People’s Republic of China

a r t i c l e i n f o

Article history:Received 10 March 2009Received in revised form 21 April 2009Accepted 6 May 2009Available online 10 May 2009

PACS:62.20.�D74.72.�h

Keywords:High-temperature superconductorKim modelFlux pinningTrapped-field

0921-4534/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.physc.2009.05.004

* Corresponding author. Address: Key Laboratory oEnvironment in Western China, Department of MechaCollege of Civil Engineering and Mechanics, Lanzhou730000, People’s Republic of China. Tel.: +86 9318914561.

E-mail addresses: zengjun06@lzu.cn, zjzj56384@@lzu.edu.cn (Y.-H. Zhou).

a b s t r a c t

In the present work, the distributions of the stress induced by flux pinning in a cylindrical superconductorwith a concentric elliptic hole are studied. The Kim model is considered for the critical state, and the ana-lytical expression of the stress in the cylinder is derived when the concentric hole is circular. Based on thefinite element method, at first, the validity of the calculation process and the accuracy of the numericalresults are proved by comparing with the analytical results. Subsequently, the distributions of the stressare obtained for different ratio of the major axis and the minor axis of the elliptic hole. It is found that dueto the effect of stress concentration, as the value of ratio the major axis and the minor axis becomes large,the radial and hoop stress in superconductor just increase significantly in the vicinity of the hole, and thevariation of the hoop stress is more sensitive. These results have significant effects on the safety in appli-cations of superconductor.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Bulk high-temperature superconductors (HTSs) enable theopportunity to develop several unique applications in engineering,such as magnetic bearings, motor components, magnetic separa-tors and magnetically levitated maglev [1–4]. Numerous of theseapplications made use of their ability to trap large magnetic field.High trapped magnetic field depends on large size of the currentloops and high critical current density. Melt growth of (RE)Ba2-Cu3Oy (RE: rare earth elements) has results in single-domain mate-rial with very strong critical current density and large size [5].Fuchs et al. [6] reported that high trapped fields up to 9.6 T at tem-perature around 50 K, and the trapped fields could be increased towell above 10 T at lower temperatures. Subsequently, Tomita andMurakami [7] reported that magnetic field of over 17 T could betrapped in bulk YBa2Cu3O7�y sample at 29 K by improving the ther-mal stability and the internal mechanical strength.

The inherently brittle melt-processed HTS materials have a lowtensile strength, which limits their ability of trapping large mag-

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f Mechanics on Disaster andnics and Engineering Science,

University, Lanzhou, Gansu8912340 (O); fax: +86 931

163.com (J. Zeng), zhouyh

netic fields. Therefore, considerable attentions have been given tothe mechanical response of superconductors to high magneticfields [8–11]. Johansen [8,9] exactly analyzed the mechanicsbehavior of solid superconductors by a series of researches before2000. Because the HTS materials produced by melt-texture pro-cessing contain lots of microscopic pores generating during thegrowth, some crack problems have also been investigated. Dikoand Krabbes [12] investigated the formation of the macrocracksin superconductor. Zhou and Yong [13] firstly studied the fracturebehavior for a superconductor slab with a center-situated crack,and then the crack problem in a long cylindrical superconductorwas investigated by them very recently [14]. Gao and Zhou [15] re-searched the crack growth for a long rectangular slab of supercon-ducting trapped-field. In addition, some researchers have paidattentions to the other topics of research on HTS [16–18].

The problem of the stress distribution in a cylindrical supercon-ductor with a concentric circular hole was first investigated byJohansen et al. [19], they obtained the results based on Bean model.Because of the oversimplified assumption of that the critical cur-rent density is a constant, the validity of the Bean model is ques-tionable. In addition, many holes in superconductor are notregular circular. Thus, it is worthwhile to study the stress in super-conductor with an elliptical hole based on the Kim model of thecritical state.

In this paper, the exact expressions of the stress for the Kimmodel are obtained first when the concentric hole is circular. Basedon the assumption that the hole forms a perfect barrier to the flow

0.4

0.5

0.6

0.7

σ0

Ba=0Ba=0Ba=0.3Bp

Ba=0.3Bp

Ba=0.6Bp

Ba=0.6Bp

(a)

J. Zeng et al. / Physica C 469 (2009) 822–826 823

of current, we obtain the numerical results of stress by using thefinite element method for circular and elliptical hole, respectively.The results show that the shape of the hole has significant effect onthe stress distributions in superconductor.

2. Analytical solution

In the present article, consider a cylindric superconductor con-taining a concentric elliptic hole placed in a magnetic field Ba ori-ented parallel to the cylinder axis (z axis). The long and shortaxis of ellipse and the radius of cylinder are 2a, 2b and R, respec-tively (as shown in Fig. 1). In this work, the cylinder is assumedto be isotropic, and the effects of demagnetization are neglected.

In order to calculate the radial and hoop stress in the cylinder, asimple model which accounts for the influence of the hole on thecritical current is presented. The hole does not disturb the currentdistributions outside the hole region a < r < R. Thus, the flux densitydistributions B(r) outside the hole region can be obtained analyti-cally based on the critical state model. For obtaining more reason-able results, the Kim model is adopted to describe the magneticbehavior. In the Kim model, the critical current density varies withthe magnetic flux density, and the expression is defined as follows:

JcðBÞ ¼ �a

jBj þ B0ð1Þ

The sign is determined by the slope of the magnetic flux densityin the cylinder, both of a and B0 are positive parameters.

The characteristic field equals to the full penetration field of asolid cylinder, is defined by:

Bp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2

0 þ 2l0aRq

� B0 ð2Þ

For the convenience of analyzing the problem, a dimensionlessparameter p is introduced to reduce the number of variables,

Fig. 1. A cylindric superconductor with a concentric elliptic hole placed in a parallelapplied magnetic field Ba.

Fig. 2. Finite element mesh of a quarter model.

P ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2l0aR

pB0

ð3Þ

and all the stresses in analytical and numerical results are normal-ized by the parameter,

r0 ¼B2

p

2l0ð4Þ

The inherently brittle melt-processed HTS materials have a lowtensile strength, which caused a superconductor sample is dam-aged easily during the applied field descent. Therefore, this articleconsiders only the field cooling stages generating tensile stress,and the applied field is decreased from Bfc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B20 þ 2l0aðR� aÞ

q� B0 to Ba. The parameter field Bfc equals to

the full penetration field of a hollow cylinder. According to theKim model, the distribution function of the flux density is

a < r < r0; BðrÞ ¼ Bfc ð5Þ

r0 < r < R; BðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðBa þ B0Þ2 þ 2l0aðR� rÞ

q� B0 ð6Þ

in which

r0 ¼ aþ BaðBa þ 2B0Þ2l0a

ð7Þ

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3σ r /

r /R

0.0 0.2 0.4 0.6 0.8 1.0r /R

0.0

0.1

0.2

0.3

0.4

0.5

0.6

RE

of σ

r / σ

0

Ba=0(b)

Fig. 3. (a) Numerical results of radial stresses in the cylinder during the appliedfield descent to Ba for the case of p = 1.0 and a/b = 1. The dash lines corresponding toanalytical results overlap the solid line corresponding to numerical results. (b) Therelative error between the numerical and the analytical results of radial stresses inthe cylinder with Ba = 0.

824 J. Zeng et al. / Physica C 469 (2009) 822–826

The shielding currents J, flowing in circular loops, and a Lorentzforce f is generated on the flux lines. In the critical state the Lorentzforce is balanced by a distribution of pinning forces from numerousdefects embedded in the crystal lattice. As a result, the materialexperiences a radial body force equal to

f ðrÞ ¼ �ð2l0Þ�1dðB2Þ=dr ð8Þ

When the concentric hole is circular (a/b = 1), we can obtain theanalytical expression of the radial and hoop stresses as follows[19]:

rr ¼1

2l0B2 � B2

h þ1� ða=rÞ2

1� ða=RÞ2ðB2

h � B2aÞ þ

1� 2m1� m

(

� 1� ða=rÞ2

R2 � a2

Z R

ar0B2dr0 � 1

r2

Z r

ar0B2dr0

" #)ð9Þ

rh ¼1

2l0

m1� m

B2 � B2h þ

1þ ða=rÞ2

1� ða=RÞ2ðB2

h � B2aÞ þ

1� 2m1� m

(

� 1þ ða=rÞ2

R2 � a2

Z R

ar0B2dr0 þ 1

r2

Z r

ar0B2dr0

" #)ð10Þ

where Bh is the field in the concentric circular hole and m is thePoisson ratio.

According to the expressions of the magnetic flux density, Eqs.(5)–(7), we can analyze the flux-pinning-induced stress in thesuperconductor based on Eqs. (9) and (10) when the concentric

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4B

a=0

Ba=0

Ba=0.3B

p

Ba=0.3B

p

Ba=0.6B

p

Ba=0.6B

p

(a)

0.0000

0.0002

0.0004

0.0006

0.0008 Ba=0

(b)

0.0 0.2 0.4 0.6 0.8 1.0r /R

0.0 0.2 0.4 0.6 0.8 1.0r /R

σ θ /

σ 0R

E of

σθ / σ

0

Fig. 4. Numerical results of hoop stress in the cylinder during the applied fielddescent to Ba for the case of p = 1.0 and a/b = 1.

hole is circular. And the analytical results are shown as the dashedlines in Figs. 3–5. The parameter m = 0.3 is used in the analyticaland the following numerical calculations.

3. Numerical solution and discussion

As we know, because of the complexity of the problem, the ana-lytical results of tensile stress are not obtained for the case wherethe central hole is elliptical. In this section, we will obtain thenumerical results of tensile stress based on the finite elementmethod and linear elasticity theory. First of all, in order to verifythe correction of the numerical method, comparisons are made be-tween the analytical and numerical results when the concentrichole is circular. Then, the distributions of stress induced by the fluxpinning are investigated with an elliptical using this numericalmethod. It is noted the boundary conditions used in the numericalsolution of the elasticity problem are that both of inside and out-side surfaces of the hollow cylindrical superconductor are free sur-faces. In addition, the distributions of magnetic flux density areexpressed analytically as Eqs. (5) and (6) when the central hole iselliptic.

The parameters are given by m = 0.3 and a/R = 0.05 in thecalculations.

Fig. 2 shows the finite element mesh for a quarter of the model,and the element type of 8-node quadrangle is adopted to calculate.

Compare the analytical results (dashed curve) with the numer-ical results (real line) of stress in Figs. 3–5, we find that the twotypes of curves match well with each other. In fact, except several

0.0

0.1

0.2

0.3

0.4

0.5

0.6 p=0.1 p=0.1 p=1.0 p=1.0 p=10 p=10

(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

p=0.1 p=0.1 p=1.0 p=1.0 p=10 p=10

(b)

σ r /

σ 0

0.0 0.2 0.4 0.6 0.8 1.0r /R

0.0 0.2 0.4 0.6 0.8 1.0r /R

σ θ /

σ 0

Fig. 5. Numerical results of radial (a) and hoop (b) stresses as the applied field isdecreased to Ba = 0 for different values of p with a/b = 1.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

p=0.1 p=1.0 p=10

(a)

1.5

2.0

2.5

3.0

p=0.1 p=1.0 p=10

(b)

0.0 0.2 0.4 0.6 0.8 1.0r /R

σ r /

σ 0σ θ

/ σ 0

J. Zeng et al. / Physica C 469 (2009) 822–826 825

nodes which are close to the surface of cylinder, the relative errorsbetween the numerical and the analytical stresses are less than 1%at every node. The radial stresses near the surface of cylinder areclose to zero, so they have larger relative error. It is more importantthat, as the node density increases, the errors will become smaller.These results verify the accuracy of the numerical results.

Shown in Figs. 3a and 4a are the radial and hoop stresses dis-tributions as the applied field is decreased from Bfc to Ba forthe case of p = 1.0 and a/b = 1. The distributions of the radialand the hoop stress are shown for Ba/Bp = 0.6, 0.3 and 0. InFig. 3a, the radial stress increases with increasing r/R first, andthen decreases after it reaches a certain peak value. As can beseen in Fig. 4a, the maximum of the hoop stress increases whendecreasing the field, and the hoop stress descends with increas-ing r/R at all times.

Fig. 5 shows the distributions of the stress as the applied field isdecreased to Ba = 0 for different values of p with a/b = 1. The distri-butions of the radial and the hoop stress are shown for p = 0.1, 1and 10. When the value of p increases, both of the radial stressand the hoop stress increase also too, but the peak values of stres-ses are very close for different values of p. In Fig. 5a and b, the trendof variation in the radial stress and the hoop stress with respect tor/R are the same as the curves in Figs. 3a and 4a, respectively.

Fig. 6a and b shows the stress distributions in the cylinder dur-ing the applied field descent to Ba with p = 1.0 and a/b = 2. The dis-tributions of stress are shown for Ba/Bp = 0.6, 0.3 and 0. In addition,shown in Fig. 7 is the stress distributions as the field is decreased toBa = 0 for p = 0.1, 1 and 10 with a/b = 2. By comparing with the re-

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ba=0

Ba=0.3B

p

Ba=0.6B

p

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Ba=0

Ba=0.3B

p

Ba=0.6B

p

(a)

(b)

σ r /

σ 0

0.0 0.2 0.4 0.6 0.8 1.0r /R

0.0 0.2 0.4 0.6 0.8 1.0r /R

σ θ /

σ 0

Fig. 6. Numerical results of radial (a) and hoop (b) stresses in the cylinder duringthe applied field descent to Ba for the case of p = 1.0 and a/b = 2.

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0r /R

Fig. 7. Numerical results of radial (a) and hoop (b) stresses as the applied field isdecreased to Ba = 0 for different values of p with a/b = 2.

sults obtained from the cases in Figs. 3–5, we find that the trend ofvariation of the stresses with respect to r/R is similar. As the ratio ofthe major axis and the minor axis a/b become large, the radialstress and the hoop stress increase significantly in the vicinity ofthe hole, and this is due to the increasing of the effect of stress con-centration. However, the variation of the radial and hoop stressesare little away from the hole.

4. Conclusions

In summary, this article presents the stress distributions causedby flux pinning in a cylindrical superconductor with a concentricelliptic hole. The exact expressions of the radial and the hoop stres-ses with the Kim model are obtained when the concentric hole iscircular (a/b = 1). Numerical results obtained show that due tothe effect of stress concentration, as the value of a/b becomes large,the radial and hoop stresses in superconductor just increase signif-icantly in the vicinity of the hole, but the variation of stresses arenot obvious away from the hole. This conclusion may be importantand useful in production design and engineering application of thesuperconductor.

Acknowledgements

This work is supported by the Fund of Natural Science Founda-tion of China (No. 10472038), and the Fund of Ministry of Educa-tion of the program of Changjiang Scholars and Innovative

826 J. Zeng et al. / Physica C 469 (2009) 822–826

Research Team in University (No. IRT0638). The authors gratefullyacknowledge this financial support.

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