Crack problem for a bulk superconductor with nonsuperconducting inclusions under an electromagnetic force Feng Xue and Xiaofan Gou Citation: AIP Advances 5, 047128 (2015); doi: 10.1063/1.4918752 View online: http://dx.doi.org/10.1063/1.4918752 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/5/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetization and magnetoelastic behavior of a functionally graded rectangular superconductor slab J. Appl. Phys. 116, 023901 (2014); 10.1063/1.4887138 Fracture problems of a superconducting slab with a central kinked crack J. Appl. Phys. 114, 243907 (2013); 10.1063/1.4852495 Fracture properties of a cylindrical superconductor with a central cross crack J. Appl. Phys. 113, 203919 (2013); 10.1063/1.4808236 Edge-crack problem in a long cylindrical superconductor J. Appl. Phys. 109, 093920 (2011); 10.1063/1.3585830 Electromagnetic direct and inverse problems for a surface-breaking crack in a conductor at a high frequency J. Appl. Phys. 86, 3997 (1999); 10.1063/1.371319
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AIP ADVANCES 5, 047128 (2015)
Crack problem for a bulk superconductor with nonsuperconducting inclusions under an electromagnetic force Feng Xue and Xiaofan Goua Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu 210098, People’s Republic of China
(Received 6 February 2015; accepted 9 April 2015; published online 17 April 2015) In this paper, the flux-pinning-induced elastic stress analysis considering the crackinclusion interaction is carried out for a bulk superconductor in the magnetization process. A approximate model for the crack problem of a bulk superconductor with nonsuperconducting inclusions (particles) dispersed in a superconducting matrix is described. The crack is simulated as a continuous distribution of edge dislocations in the solution procedure. The obtained results show that, the shear modulus, inclusion-crack size, inclusion-crack distance, and also the magnetic field have obvious effects on the stress intensity factors (SIFs) at the crack tips of the superconductor. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4918752]
I. INTRODUCTION
Bulk high-temperature superconductors have been increasingly used as magnetic bearing flywheel energy storage systems, load transports, and trapped field magnets.1 It is known that artificial microscopic defects and inclusions in superconductors are effective in enhancing local supercurrents. Introduction of defects by irradiation in YBa2Cu3O7 has been shown2–5 to result in obvious increases in magnetization for temperatures and fields in the irreversible region. Using the top-seed-melt-textured growth process, Murakami et al.6 doped Y2BaCuO5(Y211) inclusions into the YBa2Cu3O7−x(Y123) matrix. They found that the dispersion of nonsuperconducting Y211 inclusions contributes to the improvement of flux-pinning force. Pt addition, which is found that can effectively reduce Y211 size and attain high Jc values in melt-textured YBCO superconductors.7 High trapped fields up to 14.35 T at 22.5 K has been achieved for a YBCO disk pair by adding silver and using a bandage made of stainless steel.8 Since the inclusion influence the magnetostriction of bulk superconductors under external fields, Yong9 studied the effect of nonsuperconducting particles on the effective magnetostriction of bulk superconductors. Owing to the wide use of bulk superconductors, more and more researchers pay attention to the fracture behaviors of them subjected to external magnetic fields. Yong et al.10 firstly studied the crack problem theoretically for a long rectangular slab of superconductor under an electromagnetic force. Thereafter, they made a series of work11–14 for various kinds of crack problems in superconductors. Gao et al.15–17 investigated the crack growth for superconducing trapped-field magnets and fracture behaviors induced by thermal stress. The effect of crack on the supercurrent streamlines and trapped fields was considered in two different ways by Zhou et al.18–21 Feng et al.22,23 investigated some crack problems for functionally graded thin superconducting films with transport currents or field dependent critical currents. Ceniga et al.24 pointed out that a crack will form in the matrix around the particle of a radius greater than a critical value in an anisotropic particle-matrix system. Expressly for the case that the
a Corresponding author, email:
[email protected].
2158-3226/2015/5(4)/047128/11
5, 047128-1
© Author(s) 2015
All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 92.222.237.21 On: Fri, 21 Aug 2015 08:28:13
047128-2
F. Xue and X. Gou
AIP Advances 5, 047128 (2015)
FIG. 1. A rectangular slab superconductor place in a applied field B a , with Jc flows in the slab.
size of inclusions is larger than the coherent length (ξ) of the superconductor, the crack problem for a inclusion-matrix system is even more important. Yi et al.25–27 have a serious of work about elastic-plastic stress analysis on crack problem using the dislocation method. In this article, we present a simple model for the inclusion-crack interaction of bulk superconductors. The critical Bean model is used for the electromagnetic relationship and the stress state is restricted to the plane strain problem. The physical problem is formulated as a set of singular integral equations by using the solution of a circular inclusion interacting with a single dislocation as the Green’s function. The stress intensity factors at the crack tips of the superconductor are obtained after the singular integral equations are solved numerically.
II. PROBLEM STATEMENT AND BASIC EQUATIONS
• Consider a linear crack of length 2a in an infinitely long slab of width 2h (the matrix) near a circular inclusion (the inhomogeneity) with reference to the rectangular coordinate system x, y, z (see Fig. 1). The slab is placed in a parallel field oriented parallel to the z axis, the crack and inclusion both lie in the x − y plane. The linear crack is oriented along the radial direction of the circular inclusion and the distance between the near crack tip and the inclusion is D. We denote the superconducting matrix as phase 1 with elastic properties κ 1 (Kolosov’s constant) and µ1 (shear modulus), occupying the region of r > r 0 (see Fig. 2). The nonsuperconducting circular inclusion (phase 2), with elastic properties κ 2 and µ2, occupying the region of r ≤ r 0. • The slab is assumed to be isotropic and infinite in the z direction, and the demagnetization effects are therefore negligible. The physical problem thus can be regarded as a plane problem (see Fig. 2). Besides, in order to investigate the inclusion-crack interaction, we impose a relatively simple model that the length of the crack and the radius of the circular inclusion are much smaller than the width of the slab, i.e., a
