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Subwavelength Phononic Bandgaps in High-Contrast Elastic Media

Yuanchun Ren, Bochao Chen, Yixian Gao, Peijun Li

TL;DR

This work provides a rigorous mathematical framework for subwavelength bandgaps in elastic phononic crystals composed of high-contrast inclusions embedded in a soft matrix. By developing a quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form, subwavelength resonances are characterized as zeros of a $d\times d$ Hermitian determinant, and explicit asymptotic expansions for the resonant frequencies are derived. The analysis covers both general periodic configurations and the dilute 3D regime, with a detailed illustrative ball-resonator example giving explicit eigenvalues and bandwidths. The results establish the existence of subwavelength bandgaps and provide a rigorous foundation for designing low-frequency elastic metamaterials with tailored wave attenuation capabilities, including practical insights for ball-shaped resonators and dilute lattices.

Abstract

Inspired by [25], this paper investigates subwavelength bandgaps in phononic crystals consisting of periodically arranged hard elastic materials embedded in a soft elastic background medium. Our contributions are threefold. First, we introduce the quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form to characterize the subwavelength resonant frequencies, which are identified through the condition that the determinant of a certain matrix vanishes. Second, we derive asymptotic expansions for these resonant frequencies and the corresponding non-trivial solutions, thereby establishing the existence of subwavelength phononic bandgaps in elastic media. Finally, we analyze dilute structures in three dimensions, where the spacing between adjacent resonators is significantly larger than the characteristic size of an individual resonator, allowing the inter-resonator interactions to be neglected. In particular, an illustrative example is presented in which the resonator is modeled as a ball.

Subwavelength Phononic Bandgaps in High-Contrast Elastic Media

TL;DR

This work provides a rigorous mathematical framework for subwavelength bandgaps in elastic phononic crystals composed of high-contrast inclusions embedded in a soft matrix. By developing a quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form, subwavelength resonances are characterized as zeros of a Hermitian determinant, and explicit asymptotic expansions for the resonant frequencies are derived. The analysis covers both general periodic configurations and the dilute 3D regime, with a detailed illustrative ball-resonator example giving explicit eigenvalues and bandwidths. The results establish the existence of subwavelength bandgaps and provide a rigorous foundation for designing low-frequency elastic metamaterials with tailored wave attenuation capabilities, including practical insights for ball-shaped resonators and dilute lattices.

Abstract

Inspired by [25], this paper investigates subwavelength bandgaps in phononic crystals consisting of periodically arranged hard elastic materials embedded in a soft elastic background medium. Our contributions are threefold. First, we introduce the quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form to characterize the subwavelength resonant frequencies, which are identified through the condition that the determinant of a certain matrix vanishes. Second, we derive asymptotic expansions for these resonant frequencies and the corresponding non-trivial solutions, thereby establishing the existence of subwavelength phononic bandgaps in elastic media. Finally, we analyze dilute structures in three dimensions, where the spacing between adjacent resonators is significantly larger than the characteristic size of an individual resonator, allowing the inter-resonator interactions to be neglected. In particular, an illustrative example is presented in which the resonator is modeled as a ball.

Paper Structure

This paper contains 10 sections, 18 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminaries
  3. Problem formulation
  4. Quasi-periodic layer potentials
  5. Subwavelength bandgap
  6. Characterization based on quasi-periodic DtN map
  7. Subwavelength resonant frequencies
  8. Subwavelength bandgap
  9. Dilute structures
  10. Concluding remarks

Key Result

Lemma 2.1

For $\boldsymbol\alpha\neq 0$ and $k\rightarrow 0$, it holds that where

Figures (2)

  • Figure 1: A schematic representation of the problem geometry: (left) the elastic phononic crystal; (right) the unit cell.
  • Figure 2: The dilute configuration of a phononic crystal.

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 26 more