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

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 a time-harmonic excitation is converted into a vector Wiener–Hopf equation using the Fourier transform. The matrix Wiener–Hopf kernel of this equation describes the dynamic phenomena engaged in the scattering process, which includes instances where localised interfacial waves can emerge along the structured defect. This information is exploited to identify the dependency of the existence of these waves on the incident wave parameters and the properties of inertial defect. Symmetry in the structure of the scattering medium allows us to convert the vectorial problem into a pair of uncoupled scalar Wiener–Hopf equations posed along the lattice row containing the defect. The solution embodies an exact representation of the 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 symmetry classes, which include 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 also provided to validate the results against our semi-analytical solution which involves, in particular, numerical computation of the contour integrals. Graphical illustrations demonstrate special dynamic responses encountered during the wave scattering process, including dynamic anisotropy, negative reflection and negative refraction.