Interaction of in-plane waves with a structured penetrable line defect in an elastic lattice

TL;DR

This paper develops an exact semi-analytical solution for in-plane wave scattering by a structured penetrable line defect in an elastic lattice, revealing complex wave interactions including localized interfacial waves and dynamic phenomena like negative refraction.

Contribution

It introduces a novel vector Wiener-Hopf approach reducing the problem to scalar equations, enabling detailed analysis of wave scattering and localized modes in a structured elastic lattice.

Findings

Exact representation of scattered fields including evanescent and propagating waves

Identification of conditions for localized interfacial wave existence

Observation of dynamic phenomena such as negative refraction and anisotropy

Abstract

We consider the scattering of in-plane waves that interact with an edge of a structured {penetrable (inertial)} line defect contained in a triangular lattice, composed of periodically placed masses interconnected by massless elastic rods. The steady state problem for time-harmonic excitation is converted into a vector Wiener-Hopf equation using Fourier transform. The matrix Wiener-Hopf kernel of this equation describes all dynamic phenomena engaged in the scattering process, which includes instances where localised interfacial waves can emerge along structured defect. This information is exploited to identify the dependency of the existence of these waves on the incident wave parameters and properties of the inertial defect. The symmetry in the structure of scattering medium allows us to convert the vectorial problem into a pair of scalar Wiener-Hopf equations posed along the lattice row containing the defect. The solution embodies the exact representation of scattered field, in terms of a contour integral in the complex plane, that includes the contributions of evanescent and propagating waves. The solution reveals that in the remote lattice, the reflected and transmitted components of incident field are {accompanied by dynamic modes from three distinct symmetry classes in addition to localised interfacial waves}. These classes correspond to tensile modes acting transverse to the defected lattice row, shear modes that act parallel to this row, and wave modes represented as a mixture of these two responses. Benchmark finite element calculations are provided to validate results against the obtained semi-analytical solution, which involve numerical computations of the contour integrals. Graphical illustrations demonstrate special dynamic responses encountered during the wave scattering process, including dynamic anisotropy, negative reflection and negative refraction.