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

Yun, KiHyun

doi:10.1090/qam/1472

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
MathSciNet review: 3686516
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References

Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Kihyun Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal. 208 (2013), no. 1, 275–304. MR 3021549, DOI https://doi.org/10.1007/s00205-012-0590-8
Habib Ammari, Hyeonbae Kang, Hyundae Lee, Jungwook Lee, and Mikyoung Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl. (9) 88 (2007), no. 4, 307–324 (English, with English and French summaries). MR 2384571, DOI https://doi.org/10.1016/j.matpur.2007.07.005
Habib Ammari, Hyeonbae Kang, and Mikyoung Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), no. 2, 277–286. MR 2178063, DOI https://doi.org/10.1007/s00208-004-0626-y
Ivo Babuška, Börje Andersson, Paul J. Smith, and Klas Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg. 172 (1999), no. 1-4, 27–77. MR 1685902, DOI https://doi.org/10.1016/S0045-7825%2898%2900225-4
Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 195–226. MR 2506075, DOI https://doi.org/10.1007/s00205-008-0159-8
Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Comm. Partial Differential Equations 35 (2010), no. 11, 1982–2006. MR 2754076, DOI https://doi.org/10.1080/03605300903564000
Eric Bonnetier and Michael Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. MR 1745481, DOI https://doi.org/10.1137/S0036141098333980

B. Budiansky, G. F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech. 51 (1984), 733–735.

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190

Hyeonbae Kang, Hyundae Lee, and KiHyun Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann. 363 (2015), no. 3-4, 1281–1306. MR 3412359, DOI https://doi.org/10.1007/s00208-015-1203-2

Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl. (9) 99 (2013), no. 2, 234–249. MR 3007847, DOI https://doi.org/10.1016/j.matpur.2012.06.013

Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math. 74 (2014), no. 1, 125–146. MR 3162415, DOI https://doi.org/10.1137/130922434

J. Lekner, Analytical expression for the electric field enhancement between two closely-spaced conducting spheres, J. Electrostatics 68 (2010), 299–304.

J. Lekner, Near approach of two conducting spheres: enhancement of external electric field, J. Electrostatics 69 (2011), 559-563.

John Lekner, Electrostatics of two charged conducting spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), no. 2145, 2829–2848. MR 2960928, DOI https://doi.org/10.1098/rspa.2012.0133

Haigang Li, Yanyan Li, Ellen Shiting Bao, and Biao Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), no. 3, 589–596. MR 3237564, DOI https://doi.org/10.1090/S0033-569X-2014-01339-0

Yanyan Li and Louis Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925. Dedicated to the memory of Jürgen K. Moser. MR 1990481, DOI https://doi.org/10.1002/cpa.10079

Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682, DOI https://doi.org/10.1007/s002050000082

Mikyoung Lim and Kihyun Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1287–1315. MR 2581974, DOI https://doi.org/10.1080/03605300903079579

Mikyoung Lim and KiHyun Yun, Strong influence of a small fiber on shear stress in fiber-reinforced composites, J. Differential Equations 250 (2011), no. 5, 2402–2439. MR 2756070, DOI https://doi.org/10.1016/j.jde.2010.12.024

M. Lim, S. Yu, Asymptotic analysis for superfocusing of the electric field in between two nearly touching metallic spheres, preprint, arXiv:1504.04691

Kihyun Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math. 67 (2007), no. 3, 714–730. MR 2300307, DOI https://doi.org/10.1137/060648817

KiHyun Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl. 350 (2009), no. 1, 306–312. MR 2476915, DOI https://doi.org/10.1016/j.jmaa.2008.09.057

KiHyun Yun, An optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions, J. Differential Equations 261 (2016), no. 1, 148–188. MR 3487255, DOI https://doi.org/10.1016/j.jde.2016.03.005