Harmonic circular inclusions for non-uniform fields through the use of multi-coating

Abstract

© 2014 Brown University. We propose a novel method for rendering a circular inclusion harmonic even in the presence of non-uniform loading. Most significantly, the condition that the inclusion be harmonic is shown to be independent of the specific form of the loading. In addition, we demonstrate that the harmonicity condition obtained actually leads to the stronger property of neutrality when the loading takes a particular form. Our method is based on the idea of 'multi-coating' (surrounding the inclusion with a specified number of coatings each with its own separate elastic properties) used in the design of cloaking structures for the conductivity problem. Consequently, the harmonic inclusions designed here can also be thought of as special kinds of 'near-cloaking' structures in plane elasticity (in the sense that they are invisible to any changes in mean stress in the structure).