A high-contrast composite with annular inclusions: Norm-resolvent asymptotics

TL;DR

This paper analyzes the operator-norm resolvent asymptotics of a high-contrast composite with annular soft inclusions, revealing different wave propagation behaviors and deriving dispersion functions using an operator framework.

Contribution

It introduces a novel analysis of high-contrast composites with annular inclusions, applying an operator framework to derive asymptotics and dispersion functions for wave behavior.

Findings

Different effective wave propagation behaviors identified.

Derived dispersion functions for the composite.

Established a foundation for studying wave propagation in such materials.

Abstract

We investigate the operator-norm resolvent asymptotics of a high-contrast composite, consisting of a "stiff" material, with annular "soft" inclusions (a "stiff-soft-stiff" setup). This setup is derived from two models with very different effective wave propagation behaviors. Our analysis is based on an operator-framework proposed by Cherednichenko, Ershova, and Kiselev in [Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I. Commun. Math. Phys. 375, p. 1833-1884]. Then, as a first step towards studying wave propagation on the stiff-soft-stiff composite, we use the effective description to derive analogous "dispersion functions".