# Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media

## TL;DR

This paper extends two-scale homogenisation theory to high-contrast media with disconnected stiff components, revealing complex spectral behavior and spatial dispersion effects due to quasimomentum dependence.

## Contribution

It introduces an extended two-scale convergence framework to accurately capture spectral properties in high-contrast media with disconnected stiff phases.

## Key findings

- Persistent waves of all periods are shown to exist asymptotically.
- Homogenised equations exhibit non-trivial quasimomentum dependence.
- Spectral limits reveal rich dispersion phenomena.

## Abstract

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasimomentum dependence are found as resolvent limits of the original operator family, resulting in limiting spectral behaviour with a rich dependence on quasimomenta.