Resonant modes of two hard inclusions within a soft elastic material and their stress estimate

TL;DR

This paper analyzes subwavelength resonant modes of two hard inclusions in soft elastic materials, deriving explicit frequencies, classifying modes, and examining stress distribution to aid the design of elastic metamaterials with negative properties.

Contribution

It provides explicit formulas for resonant frequencies, categorizes modes, and analyzes stress behavior, advancing the understanding of elastic metamaterials with negative parameters.

Findings

Explicit resonant frequencies for convex resonators

Classification of resonant modes into dipolar, quadrupolar, and hybrid

Stress estimates showing bounded stress with proper curvature design

Abstract

In this paper, we are concerned with subwavelength resonant modes of two hard inclusions embedding in soft elastic materials to realize negative materials in elasticity. All the $12$ subwavelength resonant frequencies are derived explicitly for general convex resonators. In addition, the resonant modes are categorized into dipolar, quadrupolar, and hybrid groups, facilitating the effective realization of negative mass density, negative shear modulus and double-negative properties (both mass density and shear modulus) in elastic metamaterials. Moreover, we analyze the stress distribution between two hard inclusions when they are closely touching. We also precisely derive the sharp blow-up rates of the gradient estimates of the resonant modes. Our findings show that certain resonant modes have bounded stress estimates when the curvature of the hard inclusions is appropriately designed. These results provide valuable insights into the stability of the design and fabrication of elastic metamaterials. Lastly, we express the scattering wave fields explicitly in terms of resonant modes, offering a clear understanding of their impact on the overall scattering behavior.