Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener–Hopf factorization

Abstract

Scattering of time-harmonic plane wave by two parallel semi-infinite rows, but with staggered edges, is considered on square lattice. The condition imposed on the semi-infinite rows is a discrete analogue of Neumann boundary condition. A physical interpretation assuming an out-of-plane displacement for the particles arranged in the form of a square lattice and interacting with nearest-neighbours, associates the scattering problem to lattice wave scattering due to the presence of two staggered but parallel crack tips. The discrete scattering problem is reduced to the study of a pair of Wiener-Hopf equation on an annulus in complex plane, using Fourier transforms. Due to the offset between the crack edges, the Wiener-Hopf kernel, a 2 × 2 matrix, is not amenable to factorization in a desirable form and an asymptotic method is adapted. Further, an approximation in the far field is carried out using the stationary phase method. A graphical comparison between the far-field approximation based on asymptotic Wiener-Hopf method and that obtained by a numerical solution is provided. Also included is a graphical illustration of the low frequency approximation, where it has been found that the numerical solution of the scattering problem coincides with the well known formidable solution in the continuum framework.

Scattering on a square lattice from a crack with a damage zone

“…In the context of the matrix kernel (4.17), with the distinguished presence of the off-diagonal factors z −2M and z 2M , the reduction to the linear algebraic equation obtained above is reminiscent of that proposed for the Wiener-Hopf kernel with exponential phase factors that appear in several continuum scattering problems in fluid mechanics and fracture mechanics [52][53][54][55], and their discrete analogues in the form of scattering due to a pair of staggered cracks and rigid constraints [34,56,78]; both of these are based on an exact solution of the corresponding staggerless case [35,[79][80][81].…”

“…In this paper, it is shown that a reduction to a scalar problem is possible with the additional clause that it involves solving an auxiliary linear system of N × N equations, where N represents the size of the cohesive zone. Such a reduction resembles the one proposed for the Wiener-Hopf kernel with exponential phase factors in the continuum case [52][53][54][55], and its recently investigated discrete analogue of scattering due to a pair of staggered crack tips [34,56]. It is also relevant to recall for such kernels an asymptotic factorization-based alternative, but approximate, approach [34,57].…”

Wave scattering on lattice structures involving an array of cracks

“…After taking the Fourier transform along x, the general solution of the discrete Helmholtz equation (3) for the scattered wave field in the lattice sites sandwiched between the two edges of S 0 , i.e., y = 0 and y = N − 1, is given by u F y = aλ y + bλ −y , y ∈ Z N−1 0 , where λ is defined in (26). After solving for a and b in terms of u F 0 , u F N−1 , it is easy to see that…”

“…Suppose that the discrete Fourier transform of a sequence {u m } m∈Z is denoted by u F and defined by u F (z) = +∞ m=−∞ u m z −m . Using the discrete Fourier transform (see also [20,26]), the transformed Green's function can be written as (suppressing z dependence for brevity)…”

Discrete scattering by a pair of parallel defects

“…The assumption of complex frequency, analogous to above, holds for the incidence from the waveguide when a wave mode inside the waveguide formed by the two defects replaces the ansatz (2.5). Taking cue from the continuum model [20,42], with some effort for the discrete model, it is easy to recognize the presence of a 2 × 2 matrix WH kernel [39,44]; the details are omitted in this paper [39]. Intuitively, the 2 × 2 matrix WH kernel arises as the two sequences of sources on a pair of semi-infinite rows, induced by the defects interacting with incident wave and scattered wave, cannot be de-coupled from each other in the presence of stagger.…”