A new boundary integral equation for crack problems of the superconducting bulk in applied magnetic field Zhaoxia Zhang, Wen Chen, and Xiaofan Gou Citation: AIP Conference Proceedings 1648, 490012 (2015); doi: 10.1063/1.4912695 View online: http://dx.doi.org/10.1063/1.4912695 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1648?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Symposium: Boundary Value Problems and Integral Equations with Applications AIP Conf. Proc. 1281, 911 (2010); 10.1063/1.3498639 A NOVEL BOUNDARY INTEGRAL EQUATION FOR SURFACE CRACK MODEL AIP Conf. Proc. 1211, 329 (2010); 10.1063/1.3362412 Complete regularization of boundary integral equations in wave field diffraction problems on curved surfaces J. Acoust. Soc. Am. 117, 2483 (2005); 10.1121/1.4787729 Superconducting bulk magnets: Very high trapped fields and cracking Appl. Phys. Lett. 79, 3131 (2001); 10.1063/1.1413502 Hypersingular boundary integral equations for exterior acoustic problems J. Acoust. Soc. Am. 101, 3336 (1997); 10.1121/1.418349
A New Boundary Integral Equation for Crack Problems of the Superconducting Bulk in Applied Magnetic Field Zhaoxia Zhanga,b, Wen Chena and Xiaofan Goua,1 a
College of Mechanics and Materials, Hohai University, Nanjing, P.R. China, 210098 Faculty of Civil Engineering & Mechanics, Jiangsu University, Zhenjiang, P.R. China, 212013
b
Abstract. Rare-earth barium-copper-oxide (REBCO) bulk, as a high temperature superconductor having high currentcarrying capacity, is as typical brittle ceramic material. Crack problems are not only related to the mechanical strength, but also affect on the current-carrying capacity, especially in applied magnetic field. In this paper, a new boundary integral equation for crack problems has been derived. For a single micocrack of cylindrical REBCO bulk in magnetic field, the stress intensity factor (SIF) at microcracks has been obtained. The results show that the effect of the electromagnetic force on crack propagation is notable with respect to the variation of magnetic field intensity. Keywords: Boundary integral equation, Micro-crack, Stress intensity factor, Superconducting bulk. PACS: 74.25.Ld
INTRODUCTION Cracks in single grain REBCO bulk superconductor causes some serious safety problems, especially in applied magnetic field. There are many approaches to improve the properties of REBCO bulk to obtain high trapped field [1,2], but the cracking hinders the further magnetization process. Some researches [1-3] studied the crack problem and indicated that the different types of micocracks exist in REBCO bulk in melt-procedure[1]. For the II superconductor the defects are responsible for the higher trapped field, but simultaneously the microcracks incline to develop due to stresses at the tips of the cracks going up during magnetic procedure. The thermal stress is considered to distribute the cracking as well as the electromagnetic force does. In this paper, the effect of electromagnetic force on the microcrack propagation, and further the stress intensity factor (SIF) have been investigated.
CYLINDRICAL BULK SUPERCONDUCTOR IN MAGNETIC FIELD A cylindrical REBCO bulk is studied here (see figure 1), the field distribution is determined by the critical current density Jc. according to the Bean model in which the Jc is assumed as a constant, the magnetic field in the bulk is given as
FIGURE 1. The cylindrical REBCO bulk in the applied magnetic field Ba with the critical current density Jc. 1
Corresponding author. Email:
[email protected].
Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014) AIP Conf. Proc. 1648, 490012-1–490012-4; doi: 10.1063/1.4912695 © 2015 AIP Publishing LLC 978-0-7354-1287-3/$30.00
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B (r )
P0 J c r C .
0 r da
(1)
Then, the Lorentz force can be given as
FL (r )
J (r ) u B (r ) .
(2)
The distribution of the stress in the cylindrical coordinate can be obtained as follows
Vr
VT
3 2v 2 2 2v (a r ) k (a r ) , 8(1 v) 3(1 v) 1 v v ª º 1 ª º P0 J c2 «(3a 2 r 2 ) (a 2 3r 2 ) » k «(2a r ) ( a 2r ) » , 8 1 v 1 v ¬ ¼ 3 ¬ ¼
P0 J c2
(3)
in which, k=-μ0Jc(H+Jca). When the applied magnetic field Ba thoroughly penetrates the bulk, the penetrated field Bp is written as Bp=μ0Jca. Thus, the maximum stress can be obtained as
(9 6v) B p 2
V0
24(1 v) P0
,
(4)
in which, V0 is the stress at r=0, and Q is the Poisson ratio. Considering the symmetrical condition of cylindrical shape and the magnetic field distribution, above the 3D problem with microcracks can be simplified to a 2D problem. From the equations above, if the maximum intensity of the trapped field gets to 10-15T, the corresponding stress will reach 50-110Mpa.
NEW BOUNDARY INTEGRAL EQUATION FOR CRACK PROBLEM IN APPLIED MAGNETIC FIELD The new boundary integral equation(BIE) for co-linear cracks is based on Muskhelishvili’s formulation, Somigliana formula is adopted for reduction. The new boundary integral equation has been proved to be excellently agree with existing results [4,5]. The basic equations are given as below, where Į(t) is the angle between the global coordinate axis ox1 and the tangent at t on crack boundary, where o is the origin of the coordinate system. ın and ıns are the normal and shear stresses on the boundary. H(t) is a complex unknown function that related to the boundary force and displacement density, q(t) is the surface force acted on the crack boundary. “+” and “-’’ represent upper and lower crack surfaces, respectively.
S if t0
H (t )
H (t ) 2iD ( t ) e t t0 0 , t t0 2 iD ( t ) H (t ) q(t ) 4 iD ( t ) e H (t ) e ]dt t t0 (t t 0 )2
³ [t t *
(5)
Since ın or ıns has its maximum stress at the center of the cylindrical bulk, the cracks located on the symmetry ax is center are supposed to be the most dangerous. In the cylindrical Coordination the stresses are expressed as Vn=Vr, Vns=VT.
ª¬V T t V T t º¼ i ª¬V r t V r t º¼ , f (t0 ) ª¬V T t0 V T t0 º¼ i ª¬V r t0 V r t0 º¼ . q( t )
if t * , and if t0 * ,
0D , t t , t0 t0 , Eq. (5) is simplified as 2 H (t ) ³* t t0 dt S if (t 0 ) ,
For the crack located at the symmetry center,
(6) (7)
D
(8)
The single displacement condition is given as
p H t ds t ³
r
pr i pT pT
*
N 1
where p is the surface force acted on the crack, for the plane strain problem, N
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,
3 4v .
(9)
EXAMPLE AND DISCUSSION As shown in figure 2, a single crack with the length 2c is located at the center of the cross section. The diameter of the bulk is 50mm, the height 12mm. The crack size in a-b plane is less than 1μm, and the size of the inclusion crack is 1-10μm. The variation of the KI0/KIc with Bin are calculated as shown in figure 3. KIc is the fracture toughness of REBCO bulk, and KI0=V0(Sc)1/2. According to the experimental research, the Ag additional inclusion is helpful for higher trapped field due to decreasing the thermal tensile stresses in a-b plane and a-c plane. But it causes big crack and becomes danger under the electromagnetic force. Whether the microstructure crack on a-b plane or c-axis macrocrack with Ag additional inclusion, the crack can be regarded as one located at a infinite body, the crack problem can be solved with the same BIE. For microstructure cracks the fracture toughness KIc is 0.32 in a-b plane and 0.8 in a-c plane. It is clearly found that the variations of the KI0/KIc with Bin for cracks and Ag additional inclusion monotonically increase.
y r o 2c
x J
B
FIGURE 2. The cross section of the REBCO bulk with a single microcrack located at the symmetrical center in applied magnetic field.
FIGURE 3. The SIF of the single micocrack in applied magnetic field. Bin is the maximum of the penetrated magnetic field intensity. In calculation, 2c=10ȝm for Ag additional inclusion crack, and 2c=1ȝm for microstructure crack in REBCO.
In conclusion, for the microcrack problem of REBCO superconducting bulk, the new boundary integral equation was obtained. Using these equations, the fracture toughness can be calculated when the REBCO bulk in applied magnetic field. In the future, the new boundary integral equation can also be used for solving the multiple cracks problems in the REBCO bulk.
ACKNOWLEDGMENTS This work was financially supported by the Fund of the National Science Foundation of China (No. 11372096),
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and Program for Research Fund for the Doctoral Program of Higher Education of China. The authors gratefully acknowledge these financial supports.
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