On subwavelength guided seismic waves in Matryoshka-type elastic media with ring defects

Abstract

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.