Article

Archive for Rational Mechanics and Analysis

, Volume 208, Issue 1, pp 275-304

Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity

  • Habib AmmariAffiliated withDepartment of Mathematics and Applications, Ecole Normale Supérieure
  • , Giulio CiraoloAffiliated withDipartimento di Matematica e Informatica, Università di Palermo
  • , Hyeonbae KangAffiliated withDepartment of Mathematics, Inha University Email author 
  • , Hyundae LeeAffiliated withDepartment of Mathematics, Inha University
  • , Kihyun YunAffiliated withDepartment of Mathematics, Hankuk University of Foreign Studies

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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.