Bloch Spectra for High Contrast Elastic Media

TL;DR

This paper develops analytic formulas and power series to describe the band structure of high contrast elastic media, providing explicit bounds and conditions for spectral separation in phononic crystals.

Contribution

It introduces new representation formulas and convergence bounds for Bloch spectra in high contrast elastic media, advancing understanding of phononic crystal band structures.

Findings

Explicit convergence bounds for Bloch wave spectra

Conditions for spectral branch separation

Representation formulas for elastic crystal spectra

Abstract

Analytic representation formulas and power series are developed to describe the band structure inside periodic elastic crystals made from high contrast inclusions. We use source free modes associated with structural spectra to represent the solution operator of the Lam\'e system inside phononic crystals. Convergent power series for the Bloch wave spectrum are obtained using the representation formulas. An explicit bound on the convergence radius is given through the structural spectra of the inclusion array and the Dirichlet spectra of the inclusions. Sufficient conditions for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum are identified.