AI 中文总结
研究任意形状套娃型弹性超材料的亚波长共振与地震波局域化，通过位移 - 牵引映射等方法推导共振条件，利用相关理论确定共振存在及特征，证明多层结构中环形缺陷可作地震波导管，实现波的局域化与引导传播。
AI 中文摘要
地震表面波，尤其是低频瑞利波，具有很强的破坏性且难以用传统工程方法控制。弹性超材料的进展为亚波长尺度下操纵和引导地震波开辟了新途径。本文对任意形状的套娃型弹性超材料中的亚波长共振和地震波局域化进行了严格数学研究。通过位移 - 牵引映射和变分公式，推导了表征亚波长共振的充要条件。利用Gohberg - Sigal理论和多值代数函数的Puiseux级数，确定了亚波长共振的存在并获得其渐近特征。在同心球情形下，刚度张量呈现块对角结构。还证明了对于多层结构，分层亚波长谐振器中的环形缺陷可作为有效的径向层状地震波导管，支持波的局域化和沿缺陷的引导传播，特定放置的环形缺陷能诱导本征模在多个缺陷位置同时指数局域化且具有精确量化的振幅比。
英文摘要
Seismic surface waves, particularly low-frequency Rayleigh waves, are notoriously destructive and remain difficult to control using conventional engineering methods. Recent advances in elastic metamaterials have opened new avenues for manipulating and guiding seismic waves at subwavelength scales. In this work, we present a rigorous mathematical study of subwavelength resonances and seismic-wave localization in arbitrarily shaped Matryoshka-type elastic metamaterials, which are composed of nested high-contrast resonators made of materials much stiffer than the background medium. By employing the displacement-to-traction map and a variational formulation, we derive necessary and sufficient conditions that characterize subwavelength resonances. Using the Gohberg--Sigal theory and Puiseux series of multivalued algebraic functions, we establish the existence of subwavelength resonances and obtain their asymptotic characterization in terms of the eigensystem of the generalized stiffness tensor, which serves as the elastic analogue of the capacitance matrix in the celebrated Minnaert acoustic-cavitation systems. Furthermore, in the concentric spherical case, the stiffness tensor exhibits a block-diagonal structure separating translational and rotational components, with each block being tridiagonal and possessing positive, simple eigenvalues. Based on this matrix formulation,we rigorously demonstrate that, for structures with a sufficiently large number of layers, ring defects in layered subwavelength resonators can act as effective radially laminated seismic waveguides, supporting both wave localization and guided propagation along the defects. In particular, appropriately placed ring defects can induce eigenmodes that are exponentially localized at multiple defect sites simultaneously,with precisely quantified amplitude ratios.
