Elastic wave scattering in metamaterials with localized defects

Abstract

This work presents a semi-analytical multiple scattering formulation for elastic waves in periodic media with defects. We model the scattering of Bloch waves interacting with defects of arbitrary strength that are on a length scale much smaller than relevant wavelengths. The basis for the proposed framework is the far-field asymptotic approximation of the lattice Green’s function represented in terms of Bloch eigenfunctions. Thus, it is applicable to any periodic material whose dispersion relations and corresponding mode shapes are available either analytically or numerically. The local perturbation method is used to resolve the singularity in the far-field lattice Green’s function, enabling the derivation of a tractable multiple scattering system. Examples are presented showing agreement between the scattering model and high-fidelity transient simulations in the context of a mass-spring network and a beam-based metamaterial. The proposed modeling framework is computationally efficient, opening the door to inverse problems such as the design of defects for wave manipulation. We present the inverse design of a set of defects embedded in a periodic metamaterial for wave guiding, demonstrating that our scattering model is an effective tool for exploring the vast design space of defect engineering in metamaterials.