Mathematical Justifications of Dipolar Resonances with Hard Inclusions Embedded in a Soft Elastic Material
Hongjie Li, Jun Zou · SIAM Journal on Applied Mathematics · 2025
Abstract. The current study is motivated by the work [Z. Liu, et al., Science, 289 (2000), pp. 1734–1736], which investigated the incorporation of hard inclusions into a soft elastic matrix (HISE) to achieve a negative mass density through some subwavelength dipolar resonances. This work offers a comprehensive and mathematically rigorous justification of the subwavelength dipolar resonances within the HISE structure. First, an explicit formula for computing the resonant frequencies for arbitrarily shaped hard inclusions is derived for the first time. Then it is shown that the resonances are generated by the high contrast in Lamé parameters between different materials, while the subwavelength nature of the resonances is attributed to the significant disparities in material densities. Furthermore, the wave fields inside the hard inclusion are derived explicitly when the frequencies of the incident wave are located within different regimes. In addition, the dipolar characterization of the resonances is justified. To validate our findings, we study resonant phenomena within a spherical geometry via two methods. One is to apply the results derived in this work for arbitrarily shaped hard inclusions. The other is through directly solving the original model system. The investigation within the spherical geometry offers not only an alternative method to calculate resonant frequencies, but also a confirmation of the results presented in this work.