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On quasi-Minnaert resonances in elasticity and their applications to stress concentrations

Huaian Diao, Ruixiang Tang, Hongyu Liu

TL;DR

This work introduces and analyzes quasi-Minnaert resonance in elasticity, a boundary-localized, incident-wave–dependent resonance for a hard elastic inclusion in a soft medium in the sub-wavelength regime. It develops a rigorous framework based on layer potential theory and spectral analysis of the Neumann-Poincaré operator to obtain boundary localization and surface resonance results, with asymptotic expansions in small frequency $\\omega$ and high-contrast parameters $\\delta$ and $\\tau$. A key finding is that certain incident waves and material contrasts yield a continuous spectrum of resonance frequencies, in contrast to classical Minnaert resonances which are discrete; this is accompanied by rigorous stress concentration estimates for both the internal and scattered fields. The results provide insights for metamaterial design and potential medical applications, demonstrating how incident-wave tuning and material contrast control boundary localization, surface resonance, and localized stress amplification.

Abstract

This paper unveils and investigates a novel quasi-Minnaert resonance for an elastic hard inclusion embedded in a soft homogeneous medium in the sub-wavelength regime. The quasi-Minnaert resonance consists of boundary localization and surface resonance for the generated internal total and external scattered wave fields associated with the hard inclusion. It possesses similar quantitative behaviours as those for the classical Minnaert resonance due to high-contrast material structures, but occurs for a continuous spectrum of frequencies instead of certain discrete Minnaert resonant frequencies. We present a comprehensive analysis to uncover the physical origin and the mechanism of this new physical phenomenon. It is shown that the delicate high-contrast material structures and the properly tailored incident waves which are coupled together in a subtle manner play a crucial role in ensuring such phenomena. The stress concentration phenomena in both the internal total field and the scattered field components are also rigorously established. The analysis in this paper is deeply rooted in layer potential theory and intricate asymptotic analysis. We believe that our findings can have a significant impact on the theory of composite materials and metamaterials.

On quasi-Minnaert resonances in elasticity and their applications to stress concentrations

TL;DR

This work introduces and analyzes quasi-Minnaert resonance in elasticity, a boundary-localized, incident-wave–dependent resonance for a hard elastic inclusion in a soft medium in the sub-wavelength regime. It develops a rigorous framework based on layer potential theory and spectral analysis of the Neumann-Poincaré operator to obtain boundary localization and surface resonance results, with asymptotic expansions in small frequency and high-contrast parameters and . A key finding is that certain incident waves and material contrasts yield a continuous spectrum of resonance frequencies, in contrast to classical Minnaert resonances which are discrete; this is accompanied by rigorous stress concentration estimates for both the internal and scattered fields. The results provide insights for metamaterial design and potential medical applications, demonstrating how incident-wave tuning and material contrast control boundary localization, surface resonance, and localized stress amplification.

Abstract

This paper unveils and investigates a novel quasi-Minnaert resonance for an elastic hard inclusion embedded in a soft homogeneous medium in the sub-wavelength regime. The quasi-Minnaert resonance consists of boundary localization and surface resonance for the generated internal total and external scattered wave fields associated with the hard inclusion. It possesses similar quantitative behaviours as those for the classical Minnaert resonance due to high-contrast material structures, but occurs for a continuous spectrum of frequencies instead of certain discrete Minnaert resonant frequencies. We present a comprehensive analysis to uncover the physical origin and the mechanism of this new physical phenomenon. It is shown that the delicate high-contrast material structures and the properly tailored incident waves which are coupled together in a subtle manner play a crucial role in ensuring such phenomena. The stress concentration phenomena in both the internal total field and the scattered field components are also rigorously established. The analysis in this paper is deeply rooted in layer potential theory and intricate asymptotic analysis. We believe that our findings can have a significant impact on the theory of composite materials and metamaterials.
Paper Structure (4 sections, 10 theorems, 125 equations)

This paper contains 4 sections, 10 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Mathematical Setup
  3. Boundary localization of the interior total field and exterior scattered wave field
  4. Surface resonance and stress concentration

Key Result

Lemma 3.1

The set $\left(\mathcal{I}_n^m, \mathcal{T}_n^m, \mathcal{N}_n^m\right)$, denoting the vectorial spherical harmonics of order $n$, constitutes an orthogonal basis of $L^2(\mathbb{S}^2)^3$, where Denote $D_R$ by the ball centered at the origin with radius $R \in \mathbb{R}_{+}$ in $\mathbb{R}^3$. We have the following spectral property of the single-layer potential operator ${\mathcal{S}}_{\partia

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • ...and 19 more