Wave Propagation In Elastic Plate Research Articles

Controlling elastic wave propagation in plates using magic squares

A magic square is defined as a square array of distinct positive numbers arranged such that the sum along the horizontal, vertical, and main diagonal directions are the same, which is called a magic number. Previously, we leveraged the magic square concept to create a new class of phononic structures whose dispersion behavior can be tuned without altering their global mass and stiffness. In this study, we study the dispersion behavior of a thin plate with non-structural mass arranged in a magic square pattern. The plate unit cell is partitioned into nine portions and individual point masses are embedded at the center of each portion. Our results show that the magic square mass-embedded plate provides a low-frequency out-of-plane bandgap whose widths can be controlled while maintaining the global mass of the plate structure. Additionally, we investigate a magic plate concept by applying different densities at each portion associated with the magic square patterns. Our preliminary results show that the magic density distribution results in the emergence of polarized bandgaps and provides a possibility of altering the plate modal frequency without altering the mode shape or the global mass.

Localized waves in elastic plates with perturbed honeycomb arrays of constraints.

In this paper, we study wave propagation in elastic plates incorporating honeycomb arrays of rigid pins. In particular, we demonstrate that topologically non-trivial band-gaps are obtained by perturbing the honeycomb arrays of pins such that the ratio between the lattice spacing and the distance of pins is less than 3; conversely, a larger ratio would lead to the appearance of trivial stop-bands. For this purpose, we investigate band inversion of modes and calculate the valley Chern numbers associated with the dispersion surfaces near the band opening, since the present problem has analogies with the quantum valley Hall effect. In addition, we determine localized eigenmodes in strips, repeating periodically in one direction, that are subdivided into a topological and a trivial section. Finally, the outcomes of the dispersion analysis are corroborated by numerical simulations, where a time-harmonic point source is applied to a plate with finite arrays of rigid pins to create localized waves immune to backscattering.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

Solitary wave propagation in elastic plate on the Winkler foundation

A new problem to find the solitary solution for interaction of elastic plate with the Winkler foundation is considered. It is assumed that the plate is characterized by elastically nonlinear properties which are described by the geometrically nonlinear deformations. It is assumed that longitudinal deformations are not taken into account, the Lagrangian corresponding to the Kirchhoff classical model of transverse bending vibrations of plate is obtained. For corresponding resolving equation a solution is found in the class of the amplitude modulated signals. By means of regularizations of formal asymptotic decompositions, the Schrodinger nonlinear equation, determining the amplitude in the first order of smallness, is obtained. We obtain the solitory solution in correspondence with considerations and results obtained by Abblowitz et al., Newel, Zakharov and Shabat. There are a lot of systems of extensive applications in various fields which include such elements. In connection with it our communication considers the problem of the interaction of an elastic plate with the Winkler foundation. A linear problem was first very clearly considered by S. P. Timoshenko as STRENGTH OF MATERIALS, vol. 1 and 2, during his work at the KPI. Since then no course, better than this one, has been developed.

Steering in-plane shear waves with inertial resonators in platonic crystals

Numerical simulations shed light on control of shear elastic wave propagation in plates structured with inertial resonators. The structural element is composed of a heavy core connected to the main freestanding plate through tiny ligaments. It is shown that such a configuration exhibits a complete band gap in the low frequency regime. As a byproduct, we further describe the asymmetric twisting vibration of a single scatterer via modal analysis, dispersion, and transmission loss. This might pave the way to functionalities such as focusing and self-collimation in elastic plates.

Analytical study of dispersion relations for shear horizontal wave propagation in plates with periodic stubs

The coupled mode theory with coupling of diffraction modes and waveguide modes is usually used on the calculations of transmission and reflection coefficients for electromagnetic waves traveling through periodic sub-wavelength structures. In this paper, I extend this method to derive analytical solutions of high-order dispersion relations for shear horizontal (SH) wave propagation in elastic plates with periodic stubs. In the long wavelength regime, the explicit expression is obtained by this theory and derived specially by employing an effective medium. This indicates that the periodical stubs are equivalent to an effective homogenous layer in the long wavelength. Notably, in the short wavelength regime, high-order diffraction modes in the plate and high-order waveguide modes in the stubs are considered with modes coupling to compute the band structures. Numerical results of the coupled mode theory fit pretty well with the results of the finite element method (FEM). In addition, the band structures’ evolution with the height of the stubs and the thickness of the plate shows clearly that the method can predict well the Bragg band gaps, locally resonant band gaps and high-order symmetric and anti-symmetric thickness-twist modes for the periodically structured plates.

On the role of fluid viscosity in wave propagation in elastic plates under heavy fluid loading

This paper addresses free wave propagation and attenuation in elastic plates loaded by a quiescent viscous compressible fluid. To validate the adopted fluid model, an elementary problem of wave propagation in a layer of fluid is considered. Then wave propagation in a conventional Kirchhoff plate and in a sandwich plate loaded by a layer of viscous compressible fluid is analysed. The location of dispersion curves in each of these three cases is determined and compared with each other. The viscosity-induced attenuation of otherwise propagating ‘fluid-originated’ and ‘structure-originated’ waves is quantified. The parametric studies are conducted to explore the influence of, e.g., the width of a fluid layer or a stiffness ratio of the materials of skin and core plies in a sandwich plate on the wave attenuation.

The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

AbstractThis paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four‐noded element for the Reissner–Mindlin plate is derived in this paper, which is free of shear locking. Such a locking‐free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency‐dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright © 2006 John Wiley & Sons, Ltd.

On the nature of shear horizontal wave propagation in elastic plates coated with viscoelastic materials

The possibility of using long–range ultrasonic guided–wave non–destructive testing for the inspection of metallic structures coated with attenuative layers is extremely attractive. However, the material damping may cause severe attenuation of the acoustic signal, resulting in reduction of the test range. This paper addresses the dispersion of shear horizontal waves propagating in metallic plates coated with viscoelastic layers. It is shown that the bilayer modes can be viewed as an interaction between the free elastic plate modes and the modes of the viscoelastic layer if it were clamped on its lower surface. For low–attenuation materials the interaction is strong and the trajectories of the bilayer dispersion curves jump between the two families as the frequency increases. However, as the material damping increases the interaction becomes weak and most of the energy is confined in either the metallic plate or in the attenuative layer, and modes no longer jump. The guided–wave attenuation is related to the strain energy in the viscoelastic layer per unit in–plane power flow by means of an energy–balance argument. The guided–wave attenuation of the modes whose energy travels primarily in the elastic plate exhibits periodic peaks in the frequency domain, which occur at roughly equally spaced critical frequencies. These frequencies are close to the through–thickness resonance frequencies of the clamped–free viscoelastic layer if it is considered to be elastic. Minima of the guided–wave attenuation occur around the Love transition frequencies.

Lamb wave propagation in elastic plates coated with viscoelastic materials

This paper addresses the effects of attenuative coatings on the dispersion characteristics of Lamb wave propagation in elastic plates. The topology of phase velocity and guided wave attenuation spectra is analyzed as a function of the coating internal damping (longitudinal and shear bulk attenuations) and it is proved that in contrast to elastic plates, all modes are propagating albeit with large attenuation in some cases. An energy-based correspondence between the dispersion of the attenuative bilayer and that of a related elastic bilayer is derived in order to investigate separately the effects of the longitudinal and shear bulk attenuations on the attenuation of the guided modes. It is shown that at low frequency the guided wave attenuation is only slightly affected by the longitudinal bulk attenuation, while the contribution of the shear bulk attenuation is substantial. The attenuation characteristics of shear horizontal modes are compared with those of Lamb modes in order to identify the mode and the frequencies which result in minimum guided wave attenuation.

Adaptive computation for elastic wave propagation in plate/shell structures under moving loads

An adaptive remeshing method tailored to computations for elastic wave propagation is proposed and its application to the elastic wave propagation in plates or shells under moving loads is presente...

On approximating guided waves in plates with thin anisotropic coatings by means of effective boundary conditions

In this paper, effective boundary conditions for elastic wave propagation in plates with thin coatings are derived. These effective boundary conditions are used to obtain an approximate dispersion relation for guided waves in an isotropic plate with thin anisotropic coating layers. The accuracy of the effective boundary conditions is investigated numerically by comparison with exact solutions for two different material systems. The systems considered consist of a metallic core with thin superconducting coatings. It is shown that for wavelengths long compared to the coating thickness there is excellent agreement between the approximate and exact solutions for both systems. Furthermore, numerical results presented might be used to characterize coating properties by ultrasonic techniques.

On the Nonlocal Theory of Wave Propagation in Elastic Plates

Propagation of longitudinal waves in isotropic homogeneous elastic plates is studied in the context of the linear theory of nonlocal continuum mechanics. To determine the nonlocal moduli, the dispersion equation obtained for the plane longitudinal waves in an infinite medium is matched with the parallel equation derived in the theory of atomic lattice dynamics. Using the integroalgebraic representation of the stress tensor and the Fourier transform, the system of two coupled differential field equations is solved in the standard manner giving the frequency equations for the symmetric and antisymmetric wave modes. It is found that the short wave speed in the Poisson medium differs by about 13 percent from the speed established in the classical theory. A numerical example is given.

Wave propagation in elastic plates

The problem of wave propagation within the framework of a complete linear theory of elastic plates is treated using the method of wave curves. A complete classification of the various extensional and bending waves is obtained, along with the corresponding speeds of propagation. These are shown to correspond to the phase velocities of harmonic waves for infinite wave number. The decay and coupling equations are found, and the problem of waves due to a punch applied to the plate surface is treated.

Wave Propagation in Elastic Plates: Low and High Mode Dispersion

This is an analysis of dispersive properties of elastic plates in vacuo. For low modes there exists conclusive experimental verification of these properties. Model studies show prominent arrivals having the proper spectra and velocities for group velocity maxima and minima corresponding to several symmetric and anti-symmetric modes. In addition, detailed calculations based upon exact formulas predict some new and as yet unconfirmed properties of plates, e.g., negative phase velocities. New results concerning high modes of propagation are also displayed. These modes are of considerable theoretical interest since they belong to the transition region between the domains of validity of the wave and ray theories. The structure of the symmetric and antisymmetric fiftieth mode dispersion curves illustrates that for a given angle of incidence the propagating energy will be either chiefly shear or chiefly compressional, passage from one to the other corresponding to energy transfer between symmetrical and antisymmetrical modes or propagation. Comparison with multilayered liquid wave guides are mentioned.