Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Two types of electric field enhancements by infinitely many circular conductors arranged closely in two parallel lines

Author: KiHyun Yun
Journal: Quart. Appl. Math. 75 (2017), 649-676
MSC (2010): Primary 35J25, 35Q60, 78M35
DOI: https://doi.org/10.1090/qam/1472
Published electronically: May 10, 2017
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Abstract: In stiff fiber-reinforced composites, high shear stress concentrations occur in narrow regions between neighboring fibers. Electric fields and conductors correspond to shear stresses and cross-sections of fibers respectively in the anti-plane shear model. Due to material failure of composites, there have been intensive studies so far to estimate an electric field in between only a finite number of conductors. Indeed, the composites contain a large number of stiff fibers, and the concentration can be strongly enhanced by some combinations of inclusions. Two types of enhancements in parallel and perpendicular directions by a combination of infinitely many inclusions in a line are the main subjects of this paper. We thus consider the electric fields in between a infinite number of perfectly conducting unit disks arranged closely and regularly in two parallel lines. Asymptotes and optimal blow-up rates for the fields in two kinds of narrow regions are obtained in terms of the distances between conductors. In particular, very strong enhancement in the parallel direction is exhibited to have the blow-up rate substantially different from the existing result in the case of finite inclusions.

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KiHyun Yun
Affiliation: Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do 17035, Republic of Korea
Email: kihyun.yun@gmail.com

DOI: https://doi.org/10.1090/qam/1472
Received by editor(s): February 19, 2017
Received by editor(s) in revised form: April 6, 2017
Published electronically: May 10, 2017
Additional Notes: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education NRF-2015R1D1A1A01059212
Article copyright: © Copyright 2017 Brown University

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