Coated inclusions of finite conductivity neutral to multiple fields in two dimensional conductivity or anti-plane elasticity

TL;DR

This paper characterizes the shape of neutral inclusions in 2D conductivity and anti-plane elasticity, showing they must be confocal ellipses if neutral to multiple fields, which aids in designing non-disturbing inclusions.

Contribution

It provides a geometric characterization of neutral inclusions for multiple fields, establishing that they are confocal ellipses in the 2D setting.

Findings

Neutral inclusions are confocal ellipses when neutral to two independent fields.

The shape of the inclusion does not disturb the external field.

The results apply to both conductivity and anti-plane elasticity problems.

Abstract

We consider the problem of neutral inclusions for two-dimensional conductivity and anti-plane elasticity. The neutral inclusion, when inserted in a matrix having a uniform field, does not disturb the field outside the inclusion. The inclusion consists of a core and a shell. We show that if the inclusion is neutral to two linearly independent fields, then the core and the shell are confocal ellipses.