Natural resonance frequencies, wave blocking, and energy localization in an elastic half-space and waveguide with a crack

Abstract

Sharp stopping of time-harmonic wave transmission in elastic structures with defects is considered as a manifestation of the well-known trapped mode effect. It is associated with natural resonance poles lying close to the real axis in the complex frequency plane. Nonresonant wave blocking may also occur due to antiphase combination of the incident and scattered waves. The present paper is aimed to give an insight into such phenomena using an analytically based computer model which strictly takes into account all wave interactions in a cracked structure. Numerical examples are restricted to the case of a line horizontal crack in a half-plane or in a layer (2D in-plane motion), that is, nevertheless, quite enough to demonstrate two kinds of the Rayleigh wave stopping mechanisms (resonant and nonresonant) as well as a possibility of pure real natural resonance frequencies and of a full blocking effect with energy localization.

The propagation of in-plane P-SV waves in a layered elastic plate with periodic interface cracks: exact versus spring boundary conditions

“…[16]). Equation (7) implies a connection between the unknown vector functions q(x) and v m (x) set at the interface.…”

Resonance blocking and passing effects in two-dimensional elastic waveguides with obstacles

“…Usually it exhibits strong localization of the oscillation amplitude near the obstacles, indicating the accumulation of wave energy in energy vortices. 7 Such vortices block up the energy flow transferred along the waveguide by the incident field u 0 . Figure 2 gives examples of energy streamlines and power density distribution in the case of resonant (top) and nonresonant (bottom) waveguide blocking by a single horizontal obstacle.…”

“…The incident field u 0 generated by a surface load q obeys the elastodynamic Lamé-Navier equations and boundary conditions at the plane-parallel sides of the waveguide without obstacles. It can be represented via the convolution of the Green matrix k with the source vector q: 7,13 Furthermore, the incident field in the form of traveling waves u 0 ¼ P j a j ðzÞe if j x is derived from the integral representation using the residue technique. In this context, the wave numbers f j are poles of the Green matrix Fourier symbol Kða; zÞ ¼ F x ½kðx; zÞ in the complex a-plane (a is the parameter of the Fourier transform F x with respect to the longitudinal coordinate x); the amplitude vectors a j are expressed through the residues of K at these poles.…”

“…In the case of cracks or thin inclusions the BIE is discretized by expanding the unknown vector function in terms of splines or orthogonal polynomials. 7,13 For volumetric voids and inclusions the discretization is based on the laminate element approximation. 14 In all cases the problem is reduced to a linear algebraic system…”

“…5,6 This effect features the capture and localization of time-averaged wave energy of a time-harmonic wave field ue Àixt in the form of energy vortices. 7 The trapped mode effect is closely connected with the allocation of spectral points x n of the corresponding boundary value problem. They appear in the solution as complex poles of the frequency spectrum of the diffracted field u sc analytically continued into the complex frequency plane x.…”