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 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. Bycomparing the singular function with the one corresponding to two disks osculatingto the inclusions, we quantitatively characterize the blow-up of the gradient in termsof 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.
|Numero di pagine||30|
|Rivista||Archive for Rational Mechanics and Analysis|
|Stato di pubblicazione||Published - 2013|
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)
- Mechanical Engineering