Stress concentration via quasi-Minnaert resonance in bubble-elastic structures and applications

TL;DR

The paper addresses stress concentration caused by quasi-Minnaert resonance in bubble-elastic structures under subwavelength excitation. It develops a rigorous framework based on layer potential theory and spectral analysis of acoustic-elastic coupling, showing that boundary localization and surface resonance trigger pronounced stress amplification near the bubble boundary, with the quasi-Minnaert resonance forming a continuous spectrum rather than a discrete Minnaert frequency. A sharp quantitative lower bound for exterior stress is derived, and the phenomena are validated numerically on multiple geometries (2D disks, corners/apples, and 3D spheres) using high-contrast bubbles and tuned incident waves. The work advances understanding of stress concentration mechanisms and their potential uses in engineering blasting and medical therapies by linking wave pattern localization to boundary-driven stress amplification.

Abstract

Stress concentration in bubble-elastic scattering scenarios has significant applications in engineering blasting and medical treatments. This study provides a comprehensive mathematical analysis of stress concentration in bubbly-elastic structures, induced by the quasi-Minnaert resonance. The quasi-Minnaert resonance manifests as two distinct wave patterns near the bubble's boundary: boundary localization and high-oscillation phenomena. We demonstrate how to leverage the quasi-Minnaert resonance to induce stress concentration in the elastic total wave field near the air bubble's boundary by appropriately selecting the incident elastic wave and high-contrast structure. The interaction between the air bubble and the elastic background couples two physical wave fields-acoustic and elastic waves-across the bubble's boundary. The intricate transmission conditions, combined with the scalar nature of acoustic waves and the vectorial nature of elastic waves, present significant analytical challenges. To address these, we employ layer potential theory and asymptotic analysis to rigorously establish the stress concentration and quasi-Minnaert resonance phenomena in a radially geometry bubble-elastic model. Extensive numerical experiments are conducted to demonstrate the stress concentration phenomenon alongside quasi-Minnaert resonance for various bubble geometries, including a unit disk, a corner domain, an apple-shaped domain in $\mathbb{R}^2$, and a ball in $\mathbb{R}^3$. The findings of this study enhance the understanding of stress concentration mechanisms and their applications in engineering blasting and medical therapies.

Figures (17)

• Figure 1: The stress $\mathcal{E}(\mathbf{u})$ of the exterior total field $\mathbf{u}$ for the incident wave $\mathbf{u}^i$ with different indices $n$ ($n=5, 15, 25$).
• Figure 2: The $L^2$-norm of $u|_D$ for $\mathbf{u}^i$ with different indices $n$ ($n=5, 15, 25$), alongside the localized energy distribution near the point (1,0) and the corresponding contour plot.
• Figure 3: The $L^2$-norm of $\mathbf{u}^s|_{B_2\setminus\overline{D}}$ for $\mathbf{u}^i$ with different values $n$ ($n=5, 15, 25$), together with the localized energy distribution near (1,0) and the corresponding contour plot.
• Figure 4: $\| \nabla u \|_{L^2(D)^2}$ for the incident wave $\mathbf{u}^i$ with different indices $n$ ($n=5,15,25$).
• Figure 5: $\| \nabla \mathbf{u}^s\|_{L^2\left(B_2 \backslash \overline D\right)^2}$ for the incident wave $\mathbf{u}^i$ with different indices $n$ ($n=5,15,25$).

Theorems & Definitions (35)

• Remark 2.1
• Definition 2.1
• Remark 2.2
• Definition 2.2
• Definition 2.3
• Remark 2.3
• Definition 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.2