Abstract
When holes or hard elastic inclusions are closely located, stress which is the gradient of the solution to the anti-plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient of such an equation. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann–Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blow-up of the gradient in terms of explicit functions. In electrostatics, our results apply to the electric field, which is the gradient of the solution to the conductivity equation, in the case where perfectly conducting or insulating inclusions are closely located.
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Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.: Spectral theory of a Neumann–Poincaré-type operator and analysis of the anomalous localized resonance, submitted. arXiv:1109.0479
Ammari H., Kang H.: Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. Applied Mathematical Sciences, Vol. 162. Springer-Verlag, New York (2007)
Ammari H., Kang H., Lee H., Lee J., Lim M.: Optimal bounds on the gradient of solutions to conductivity problems. J. Math. Pures Appl. 88, 307–324 (2007)
Ammari H., Kang H., Lee H., Lim M., Zribi H.: Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions. J. Differ. Equ. 247, 2897–2912 (2009)
Ammari H., Kang H., Lim M.: Gradient estimates for solutions to the conductivity problem. Math. Ann. 332(2), 277–286 (2005)
Babuska I., Andersson B., Smith P., Levin K.: Damage analysis of fiber composites. I. Statistical analysis on fiber scale. Comput. Methods Appl. Mech. Eng. 172, 27–77 (1999)
Bao E., Li Y.Y., Yin B.: Gradient estimates for the perfect conductivity problem. Arch. Ration. Mech. Anal. 193, 195–226 (2009)
Bao E.S., Li Y.Y., Yin B.: Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions. Commun. Part. Differ. Equ. 35, 1982–2006 (2010)
Bonnetier E., Triki F.: Pointwise bounds on the gradient and the spectrum of the Neumann–Poincaré operator: the case of 2 discs. Contemp. Math. 577, 81–92 (2012)
Bonnetier E., Vogelius M.: An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section. SIAM J. Math. Anal. 31, 651–677 (2000)
Budiansky B., Carrier G.F.: High shear stresses in stiff fiber composites. J. Appl. Mech. 51, 733–735 (1984)
Gorb Y., Novikov A.: Blow-up of solutions to a p-Laplace equation. SIAM Multiscale Model. Simul. 10, 727–743 (2012)
Kang H., Seo J.K.: Layer potential technique for the inverse conductivity problem. Inverse Probl. 12, 267–278 (1996)
Kang, H., Seo, J.K.: Recent progress in the inverse conductivity problem with single measurement. In: Inverse Problems and Related Fields. CRC Press, Boca Raton, FL, 69–80, 2000
Khavinson D., Putinar M., Shapiro H.S.: Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal. 185, 143–184 (2007)
Keller J.B.: Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J. Appl. Phys. 34(4), 991–993 (1963)
Li Y.Y., Nirenberg L.: Estimates for elliptic system from composite material. Commun. Pure Appl. Math. LVI, 892–925 (2003)
Li Y.Y., Vogelius M.: Gradient estimates for solution to divergence form elliptic equation with discontinuous coefficients. Arch. Ration. Mech. Anal. 153, 91–151 (2000)
Lim M., Yun K.: Blow-up of electric fields between closely spaced spherical perfect conductors. Commun. Part. Diff. Equ. 34, 1287–1315 (2009)
Lim M., Yun K.: Strong influence of a small fiber on shear stress in fiber-reinforced composites. J. Differ. Equ. 250, 2402–2439 (2011)
Yun K.: Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape. SIAM J. Appl. Math. 67, 714–730 (2007)
Yun K.: Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross sections. J. Math. Anal. Appl. 350, 306–312 (2009)
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Ammari, H., Ciraolo, G., Kang, H. et al. Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity. Arch Rational Mech Anal 208, 275–304 (2013). https://doi.org/10.1007/s00205-012-0590-8
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DOI: https://doi.org/10.1007/s00205-012-0590-8