Time-Dispersive Behaviour as a Feature of Critical Contrast Media

TL;DR

This paper introduces a mathematical framework for understanding how critical-contrast composites exhibit time-dispersive behaviour, linking microstructure to macroscopic dispersive properties through asymptotic analysis of eigenmodes.

Contribution

It develops a novel homogenisation approach based on Dirichlet-to-Neumann maps to characterize time-dispersive features in critical-contrast media.

Findings

Soft components induce time-dispersive behaviour in stiff components

Asymptotic eigenmode analysis reveals dispersive properties

Method enables design of media with tailored dispersive features

Abstract

Motivated by the urgent need to attribute a rigorous mathematical meaning to the term "metamaterial", we propose a novel approach to the homogenisation of critical-contrast composites. This is based on the asymptotic analysis of the Dirichlet-to-Neumann map on the interface between different components ("stiff" and "soft") of the medium, which leads to an asymptotic approximation of eigenmodes. This allows us to see that the presence of the soft component makes the stiff one behave as a class of time-dispersive media. By an inversion of this argument, we also offer a recipe for the construction of such media with prescribed dispersive properties from periodic composites.