Search Results (8)

This paper investigates the linear stability of the liquid film of Oldroyd-B fluid on an oscillating plate. The time-dependent Orr–Sommerfeld boundary-value problem is formulated through the assumption of a normal modal solution and the introduction of the stream function, which is further transformed into the Floquet system. A long-wavelength expansion analysis is performed to derive the analytical solution of the Orr–Sommerfeld equation. The results indicate that long-wave instability occurs only within specific bandwidths related to the Ohnesorge number ($Oh$). Fixing the elasticity parameter ($El$) and increasing the relaxation-to-delay time ratio ($\tilde{\lambda }$) from 2 to 4 or fixing ($\tilde{\lambda }$) and increasing ($El$) from 0.001 to 0.01 decreases the number of unstable bandwidths while enhancing the intensity of unstable modes. Increasing the surface-tension-related parameter (${\zeta }^{\ast }$) from 0 to 100 suppresses the wave growth rate, stabilizing the system. Additionally, increasing ($\tilde{\lambda }$) from 2 to 4 reduces the maximum values of the coupling of viscoelastic, gravitational, and surface-tension forces, as well as the maximum value of the Floquet exponent, further stabilizing the system. These findings provide supplements to the theoretical research on the stability of viscoelastic fluids and also offer a scientific basis for engineering applications in multiple fields. Full article

The unit slab track structure in high-speed railways exhibits multiple periodic characteristics, which result in bandgaps of elastic wave propagation within the track structure. Moreover, local defects inevitably occur in the ballastless track structure, disrupting its periodicity and leading to the generation of defect states. An analytical model for infinite periodic slab track structure was established using the Floquet transform and supercell method, accounting for local defects, to clarify the propagation of flexural waves in slab tracks. The formation mechanism of elastic wave bandgaps in periodic slab tracks can be explained by Bragg scattering and local resonance. In the low-frequency below 200 Hz, the local resonances of the slab interact with the flexural waves in the rail, forming an approximately broad coupling bandgap. The bandgaps expand significantly with the increasing fastening stiffness. Besides, when the stiffness of the isolating layer beneath the slab is within the range of 0.9 to 1.0 × 10⁹ N/m³, a broad coupled bandgap is generated in the frequency range of 180–230 Hz. Local damage caused by contact loss between the composite slab and baseplate leads to defect states, and the frequencies of the defect states correspond to unique wave modes, demonstrating the localization of elastic waves near the defect location. The formation mechanism of defect states can be elucidated by the local resonance of the structure at the defect. The frequency of the first-order defect state is significantly affected by the defect size, the second-order defect state exhibits unidirectional propagation characteristics, and the third-order defect state shows localized vibration characteristics, which can provide a reference for defect identification. Full article

The three-dimensional problem of the modelling of elastic wave propagation in a multi-layered acoustic metamaterial, a periodic elastic composite with periodic arrays of interface cracks or planar voids of arbitrary shape, is considered. The boundary integral equation method is extended for this purpose. The unknown crack-opening displacement vectors for each array are related using the Floquet theorem and solved using the Galerkin method at reference delaminations in the arrays. The developed method provides an efficient tool for fast parametric analysis of the influence of the periodic crack array characteristics on the transmission and diffraction of elastic waves. Two modifications to the boundary integral equation method are proposed and compared for rectangular cracks. To reduce computational costs, a preliminary analytical evaluation of the arising integral representations in terms of the Fourier transform of Green’s matrices and the crack-opening displacements are presented. Full article

The in-plane elastic waves in periodically multilayered isotropic structures, which are decoupled from the out-of-plane waves, are represented mainly by the frequency–wavenumber spectra and occasionally by the frequency–phase velocity spectra as well as being studied predominantly for periodic bi-layered media along and perpendicular to the thickness direction in the existing research. This paper investigates their comprehensive dispersion characteristics along arbitrary in-plane directions and in entire (low and high) frequency ranges, including the frequency–wavelength, wavenumber–phase velocity, wavelength–phase velocity spectra, the dispersion surfaces and the slowness curves with fixed frequencies, as well as the frequency–wavenumber and frequency–phase velocity spectra. Specially, the dispersion surfaces and the slowness curves completely reflect the propagation characteristics of in-plane waves along all directions. First, the method of reverberation-ray matrix (MRRM) combined with the Floquet theorem is extended to derive the dispersion equation of in-plane elastic waves in general periodic multilayered isotropic structures by means of the elastodynamic theory of isotropic materials and the state space formalism of layers. The correctness of the derivation and the numerical stability of the method in both low and high frequency ranges, particularly its superiority over the method of the transfer matrix (MTM) within the ranges near the cutoff frequencies, are verified by several numerical examples. From these demonstrations for periodic octal- and bi-layered media, the comprehensive dispersion curves are provided and their general characteristics are summarized. It is found that although the frequencies associated with the dimensionless wavenumber along thickness $ql=n\pi$ ($n$ is an integer) are always the demarcation between pass and stop bands in the case of perpendicular incident wave, but this is not always exist in the case of the oblique incident wave due to the coupling between the two modes of in-plane elastic waves. The slowness curves with fixed frequencies of Floquet waves in periodically multilayered isotropic structures, as compared to their counterpart of body waves in infinite isotropic media obtained from the Christoffel equation now have periodicity along the thickness direction, which is consistent to the configuration of the structures. The slowness curves associated with higher frequencies have a smaller minimum positive period and have more propagation modes due to the cutoff properties of these additional modes. Full article

Faraday instability has great application value in the fields of controlling polymer processing, micromolding colloidal lattices on structured suspensions, organizing particle layers, and conducting cell culture. To regulate Faraday instability, in this article, we attempt to introduce an elastic polymer film covering the surface of a viscous fluid layer and theoretically study the behaviors of the Faraday instability phenomenon and the effect of the elastic polymer film. Based on hydrodynamic theory, the Floquet theory is utilized to formulate its stability criterion, and the critical acceleration amplitude and critical wave number are calculated numerically. The results show that the critical acceleration amplitude for Faraday instability increases with three increasing bending stiffness of the elastic polymer film, and the critical wave number decreases with increasing bending stiffness. In addition, surface tension and viscosity also have important effects on the critical acceleration amplitude and critical wave number. The strategy of controlling Faraday instability by covering an elastic polymer film proposed in this paper has great application potential in new photonic devices, metamaterials, alternative energy, biology, and other fields. Full article

The transient scattering of in-plane elastic waves from a finite-sized periodic structure, comprising a regular grid of Swiss-cross holes arranged according to a square lattice, is considered. The theoretical and numerical modelling focuses on the unexplored ultrasonic frequency regime, well beyond the first, wide, locally resonant band-gap of the structure. Dispersive properties of the periodic array, determined by Bloch–Floquet analysis, are used to identify candidates for high-fidelity GPU-accelerated transient scattering simulations. Several unusual wave phenomena are identified from the simulations, including negative refraction, focusing, partial cloaking, and wave trapping. The transient finite element modelling framework offers insights on the lifetimes of such phenomena for potential practical applications. In addition, nonideal counterparts with rough edges are modelled using characteristic statistical parameters commonly observed in additive manufacturing. The analysis shows that the identified wave effects appear likely to be robust with respect to potential manufacturing uncertainties in future studies. Full article

In this work we investigate the properties of elastic waves propagating in gyroid lattices. First, we rigorously characterize the lattice from the point of view of crystallography. Second, we use Bloch–Floquet analysis to compute the dispersion relations for elastic waves. The results for very long wavelengths are then compared to those given by classic elasticity for a cubic material. A discrepancy is found in terms of the polarization of waves and it is related to the noncentrosymmetry of the gyroid. The gyroid lattice results to be acoustically active, meaning that transverse waves exhibit a circular polarization when they propagate along an axis of rotational symmetry. This phenomenon is present even for very long wavelengths and is not captured by classic elasticity. Full article

Phononic crystals can be used to control elastic waves due to their frequency bands. This paper analyzes the passive and active control as well as the dispersion properties of longitudinal waves in rod-type piezoelectric phononic crystals over large frequency ranges. Based on the Love rod theory for modeling the longitudinal wave motions in the constituent rods and the method of reverberation-ray matrix (MRRM) for deriving the member transfer matrices of the constituent rods, a modified transfer matrix method (MTMM) is proposed for the analysis of dispersion curves by combining with the Floquet–Bloch principle and for the calculation of transmission spectra. Numerical examples are provided to validate the proposed MTMM for analyzing the band structures in both low and high frequency ranges. The passive control of longitudinal-wave band structures is studied by discussing the influences of the electrode’s thickness, the Poisson’s effect and the elastic rod inserts in the unit cell. The influences of electrical boundaries (including electric-open, applied electric capacity, electric-short and applied feedback control conditions) on the band structures are investigated to illustrate the active control scheme. From the calculated comprehensive frequency spectra over a large frequency range, the dispersion properties of the characteristic longitudinal waves in rod-type piezoelectric phononic crystals are summarized. Full article