Analytical solution for the fracture problem in superconducting tapes with oblique cracks under the electromagnetic force

doi:10.1007/s10483-025-3227-6

摘要/Abstract

Abstract:

The fracture behavior of superconducting tapes with central and edge oblique cracks subject to electromagnetic forces is investigated. Maxwell's equations and the critical state-Bean model are used to analytically determine the magnetic flux density and electromagnetic force distributions in superconducting tapes containing central and edge oblique cracks. The distributed dislocation technique (DDT) transforms the mixed boundary value problem into a Cauchy singular integral equation, which is then solved by the Gauss-Chebyshev quadrature method to determine the stress intensity factors (SIFs). The model's accuracy is validated by comparing the calculated electromagnetic force distribution for the edge oblique crack and the SIFs for both crack types with the existing results. The findings indicate that the current and electromagnetic forces are significantly affected by the crack length and oblique angle. Specifically, for central oblique cracks, a smaller oblique angle enhances the risk of crack propagation, and a higher initial magnetization intensity poses greater danger under field cooling (FC) excitation. In contrast, for edge oblique cracks, a larger angle increases the likelihood of tape fractures. This study provides important insights into the fracture behavior and mechanical failure mechanisms of superconducting tapes with oblique cracks.

Key words: superconducting tape, electromagnetic body force, integral equation, distributed dislocation technique (DDT), oblique crack, stress intensity factor (SIF)

中图分类号:

O346.1

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(3): 485-500.

Jinjian XIE, Zhaoxia ZHANG, Pengpeng SHI, Xiaofan GOU. Analytical solution for the fracture problem in superconducting tapes with oblique cracks under the electromagnetic force[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(3): 485-500.

图/表 11

参考文献 45

[1]	MAEDA, H. and YANAGISAWA, Y. Recent developments in high-temperature superconducting magnet technology. IEEE Transactions on Applied Superconductivity, 24(3), 4602412 (2014)
[2]	PAIDPILLI, M. and SELVAMANICKAM, V. Development of RE-Ba-Cu-O superconductors in the US for ultra-high field magnets. Superconductor Science and Technology, 35(4), 043001 (2022)
[3]	SAKAI, N., LEE, S., CHIKUMOTO, N., IZUMI, T., and TANABE, K. Delamination behavior of Gd123 coated conductor fabricated by PLD. Physica C: Superconductivity and its Applications, 471, 1075–1079 (2011)
[4]	VAN DER LAAN, D. C., EKIN, J., CLICKNER, C. C., and STAUFFER, T. C. Delamination strength of YBCO coated conductors under transverse tensile stress. Superconductor Science and Technology, 20, 765–770 (2007)
[5]	YANAGISAWA, Y., NAKAGOME, H., TAKEMATSU, T., TAKAO, T., SATO, N., TAKAHASHI, M., and MAEDA, H. Remarkable weakness against cleavage stress for YBCO-coated conductors and its effect on the YBCO coil performance. Physica C: Superconductivity and its Applications, 471, 480–485 (2011)
[6]	DIKO, P. Cracking in melt-grown RE-Ba-Cu-O single-grain bulk superconductors. Superconductor Science and Technology, 17(11), R45 (2004)
[7]	TOMITA, M. and MURAKAMI, M. High-temperature superconductor bulk magnets that can trap magnetic fields of over 17 tesla at 29 K. nature, 421, 517–520 (2003)
[8]	HAHN, S., KIM, K., KIM, K., HU, X., PAINTER, T., DIXON, I., KIM, S., BHATTARAI, K. R., NOGUCHI, S., JAROSZYNSKI, J., and LARBALESTIER, D. C. 45.5-tesla direct-current magnetic field generated with a high-temperature superconducting magnet. nature, 570, 496–499 (2019)
[9]	HARTNETT, W. N., RAMIREZ, J., OLSON, T. E., HOPP, C. T., JEWELL, M. C., KNOLL, A. R., HAZELTON, D. W., and ZHANG, Y. F. Characterization of edge damage induced on REBCO superconducting tape by mechanical slitting. Engineering Research Express, 3(3), 035007 (2021)
[10]	MIYAMOTO, T., NAGASHIMA, K., SAKAI, N., and MURAKAMI, M. Direct measurements of mechanical properties for large-grain bulk superconductors. Physica C: Superconductivity and its Applications, 340, 41–50 (2000)
[11]	ZHOU, Y. H., LIU, C., SHEN, L., and ZHANG, X. Y. Probing of the internal damage morphology in multilayered high-temperature superconducting wires. Nature Communications, 12, 3110 (2021)
[12]	YONG, H. D., JING, Z., and ZHOU, Y. H. Crack problem for superconducting strip with finite thickness. International Journal of Solids and Structures, 51(3-4), 886–893 (2014)
[13]	HE, A., XUE, C., YONG, H. D., and ZHOU, Y. H. Fracture behaviors of thin superconducting films with field-dependent critical current density. Physica C: Superconductivity and its Applications, 492, 25–31 (2013)
[14]	YANG, Z. R., LI, Y., SONG, P., GUAN, M. Z., FENG, F., and QU, T. M. Effect of edge cracks on critical current degradation in REBCO tapes under tensile stress. Superconductivity, 1, 100007 (2022)
[15]	JIANG, Z. F., HUANG, Y. Q., JIANG, D. H., CHEN, W. G., and KUANG, G. L. Influence of edge crack propagation on critical current degradation of REBCO tape studied by phase-field fracture method and current shunting model. IEEE Transactions on Applied Superconductivity, 33(2), 1–11 (2023)
[16]	JIANG, Z. F., ZHOU, X. X., and JIANG, D. H. The Fracture behavior of REBCO tape with multiple oblique edge cracks. Journal of Superconductivity and Novel Magnetism, 36, 477–485 (2023)
[17]	MA, J. T. and GAO, Y. W. Numerical analysis of the mechanical and electrical properties of (RE)BCO tapes with multiple edge cracks. Superconductor Science and Technology, 36(9), 095013 (2023)
[18]	ZENG, J., YONG, H. D., and ZHOU, Y. H. Edge-crack problem in a long cylindrical superconductor. Journal of Applied Physics, 109, 093920 (2011)
[19]	ZENG, J., ZHOU, Y. H., and YONG, H. D. Fracture behaviors induced by electromagnetic force in a long cylindrical superconductor. Journal of Applied Physics, 108, 033901 (2010)
[20]	FENG, W. J., GAO, S. W., and LI, Y. S. An inner annular crack in a superconducting cylinder and its crack front properties under applied magnetic field. Applied Mathematical Modelling, 40, 2529–2540 (2016)
[21]	FENG, W. J., GAO, S. W., and LIU, L. L. Fracture properties of a cylindrical superconductor with a central cross crack. Journal of Applied Physics, 113, 203919 (2013)
[22]	GAO, S. W., FENG, W. J., FANG, X. Q., and ZHANG, G. L. Fracture analysis for a penny-shaped crack problem of a superconducting cylinder in a parallel magnetic field. Physica C: Superconductivity and its Applications, 506, 6–11 (2014)
[23]	GAO, S. W., FENG, W. J., and LIU, J. X. Fracture problems of a superconducting slab with a central kinked crack. Journal of Applied Physics, 114, 243907 (2013)
[24]	AN, D. M., SHI, P. P., and GOU, X. F. Fracture behavior of superconductor REBCO tapes with multiple edge cracks under electromagnetic force. Engineering Fracture Mechanics, 295, 109794 (2024)
[25]	HILLS, D. A. Solution of Crack Problems: the Distributed Dislocation Technique, Kluwer Academic Publishers, Dordrecht, 29–64 (1996)
[26]	JIN, X. Q. and KEER, L. M. Solution of multiple edge cracks in an elastic half plane. International Journal of Fracture, 137, 121–137 (2006)
[27]	LI, X. T., LI, X., and JIANG, X. Y. Influence of a micro-crack on the finite macro-crack. Engineering Fracture Mechanics, 177, 95–103 (2017)
[28]	LI, X. T., LI, X., YANG, H. D., and JIANG, X. Y. Analysis of the effect of a micro-crack on plastic zone of the edge macro-crack tip by macroscopic and microscopic methods. Engineering Fracture Mechanics, 201, 1–12 (2018)
[29]	ZHANG, J., QU, Z., HUANG, Q. Q., XIE, L. C., and XIONG, C. B. Solution of multiple cracks in a finite plate of an elastic isotropic material with the distributed dislocation method. Acta Mechanica Solida Sinica, 27, 276–283 (2014)
[30]	YONG, H. D., XUE, C., and ZHOU, Y. H. Thickness dependence of fracture behaviour in a superconducting strip. Superconductor Science and Technology, 26(5), 055003 (2013)
[31]	GAO, Z. W., ZHENG, Z. Y., and LI, X. Y. Fracture problem of a nonhomogeneous high temperature superconductor slab based on real fundamental solutions. Physica C: Superconductivity and its Applications, 519, 5–12 (2015)
[32]	LIU, Q. F., YAN, Z., and FENG, W. J. Fracture behaviors for a functionally graded superconducting cylinder with a cylindrical crack. Mechanics of Advanced Materials and Structures, 24, 176–182 (2017)
[33]	XUE, F. and GOU, X. F. Crack problem for a bulk superconductor with nonsuperconducting inclusions under an electromagnetic force. AIP Advances, 5, 047128 (2015)
[34]	XUE, F., ZHANG, Z. X., and GOU, X. F. Fracture behavior of an inclined crack interacting with a circular inclusion in a high-TC superconductor under an electromagnetic force. AIP Advances, 5, 117141 (2015)
[35]	ZHANG, Z. X. and GOU, X. F. Interaction among multiple parallel cracks in REBCO superconductor bulks under applied magnetic fields: a boundary element analysis. IEEE Transactions on Applied Superconductivity, 34(6), 1–10 (2024)
[36]	FENG, W. J., LIU, Q. F., and HAN, X. Crack problem for a functionally graded thin superconducting film with field dependent critical currents. Mechanics Research Communications, 61, 36–40 (2014)
[37]	FENG, W. J., ZHANG, R., and DING, H. M. Crack problem for an inhomogeneous orthotropic superconducting slab under an electromagnetic force. Physica C: Superconductivity and its Applications, 477, 32–35 (2012)
[38]	ZHOU, Y. H. and YONG, H. D. Crack problem for a long rectangular slab of superconductor under an electromagnetic force. Physical Review B, 76, 094523 (2007)
[39]	KELLY, P. and O'CONNOR, J. Transmission of rapidly applied loads through articular cartilage, part 2: cracked cartilage. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 210(1), 39–49 (1996)
[40]	KELLY, P., O'CONNOR, J., and HILLS, D. The stress field due to a dislocation in layered media. Journal of Physics D: Applied Physics, 28(3), 530 (1995)
[41]	ERDOGAN, F., GUPTA, G. D., and COOK, T. S. Numerical Solution of Singular Integral Equations, Springer, Dordrecht, 368 (1973)
[42]	KAYA, A. C. and ERDOGAN, F. On the solution of integral equations with strongly singular kernels. Quarterly of Applied Mathematics, 45(1), 105–122 (1987)
[43]	KRENK, S. On the use of the interpolation polynomial for solutions of singular integral equations. Quarterly of Applied Mathematics, 32, 479–484 (1975)
[44]	YONG, H. D., YANG, Y., and ZHOU, Y. H. Dynamic fracture behavior of a crack in the bulk superconductor under electromagnetic force. Engineering Fracture Mechanics, 158, 167–178 (2016)
[45]	Chinese Aeronautical Establishment. Handbook of Stress Intensity Factors(in Chinese), Science Press, Beijing, 212 (1993)