Waves In Elastic Plates Research Articles

Simulation and measurement of the optical excitation of the S1 zero group velocity Lamb wave resonance in plates

Recent reports on the thermoelastic generation of Lamb waves in isotropic elastic plates show that a laser source efficiently excites a resonance that occurs at the minimum frequency of the first order symmetric (S1) Lamb mode. The group velocity of the Lamb wave goes to zero at this frequency while the phase velocity remains finite, and the resonance is referred to as the S1 zero group velocity (S1 ZGV) resonance. The S1 ZGV resonance can be employed for the nondestructive evaluation of the elastic properties of plates or plate thickness. A model for the generation of elastic waves in plates using an intensity-modulated continuous wave laser source is developed and used to study the behavior of the S1 ZGV resonance. The effects of the laser source parameters on the generation of the S1 ZGV resonance are explored, and the spatial distribution of the displacement produced at the resonance frequency is determined. The predicted displacement spectrum of Lamb waves generated in micron scale plates is found to compare well with experimental measurements. In addition, experimental measurements demonstrate that the S1 ZGV resonance can be used to map subsurface features in thin (4μm) membranes at high ultrasonic frequencies (700MHz).

Guided waves in elastic plates with Gaussian section variation: Experimental and numerical results

Experimental and numerical results are presented on the behavior of guided waves in elastic plates in plane strain that include a Gaussian variation of their section, located between two areas of constant thickness. The area of varying section is wide compared to the used wavelengths, which allows wave propagation inside this area. The experimental results show that an incident Lamb wave is indeed converted into an adiabatic wave inside the varying section domain. A trapped wave in the Gaussian domain is also observed, depending on the incident mode and on the Gaussian maximum height. Outside the varying section domain, conversion into different Lamb waves is observed. This conversion phenomenon is experimentally quantified by the measurement of the Lamb wave normal displacement and of its carried energy. A numerical study, based on the Finite Elements Method is performed, and successfully compared to the experimental results.

Ultrasonic wave transmission in periodically undulated plates

This study focuses on quantifying the change in phase speed of waves transmitting through periodically undulated plates under pass band interaction. A perturbation technique is used to analyze the transmission of horizontally polarized guided waves in elastic plates with sinusoidal periodicity at their outerfaces. Phase speed of transmitting modes is presented as a function of various parameters, including outerface wavenumber, undulation amplitude, degree of undulations symmetry about the periodically undulated plate midplane, plate average thickness, and frequency of propagation.

Theory of plane subsonic elastic waves in fluid-loaded anisotropic plates

Propagation of plane elastic waves in plates of unrestricted anisotropy in contact with an arbitrary non–viscous compressible fluid is studied. A general formalism is worked out based on the method of plate impedance/admittance. It is applied to the subsonic velocity interval (below the sound speed in fluid) which encloses the modes localized along the plate faces and decaying into the fluid depth. The cases analysed include a plate immersed into fluid, a plate loaded by two different fluids, and a plate subjected to fluid loading on one side (in addition, the formal limiting cases involving an absolutely rigid wall are examined). For each of these settings, use of the plate–admittance analytical properties enables us to identify subsonic solutions of the real–valued dispersion equation in a transparent graphical fashion and thereby to reveal unambiguously the layout of the velocity branches. Possible types of the subsonic spectrum configuration are established, depending on the relation between the sound speed in the loading fluid(s) and certain reference velocities characterizing the plate. Evolution of the subsonic spectrum with changing the fluid/plate density ratio and with varying the difference between parameters of the fluids loading the plate from opposite sides is analysed. The conditions are discussed for some specific features, like the plate–fluid uncoupling, the intersection of branches and others. The particular case of plate faces lying in a symmetry plane is specified. The closed–form explicit solutions for the low–frequency onset of the velocity branches are derived for an arbitrary anisotropic plate and various types of fluid loading.

PROPAGATION OF LOCALIZED VIBRATION MODES ALONG EDGES OF IMMERSED WEDGE-LIKE STRUCTURES: GEOMETRICAL-ACOUSTICS APPROACH

The theory of antisymmetric localized elastic modes propagating along edges of immersed wedge-like structures is developed using the geometrical-acoustics approach to the description of flexural waves in elastic plates of variable thickness. The velocities of these modes, often called wedge acoustic waves, are calculated using solutions of the dispersion equation of the Bohr-Sommerfeld type following from the geometrical-acoustics description of localized wedge modes. In a subsonic regime of wave propagation, i.e. for wedge modes slower than sound in liquid, the influence of liquid loading results in significant decrease of wedge wave velocities in comparison with their values in vacuum. This decrease is a nonlinear function of a wedge apex angle θ and is more pronounced for small values of θ. In a supersonic regime of wedge wave propagation, a smaller decrease in velocities takes place and the waves travel with the attenuation due to radiation of sound into the surrounding liquid. The comparison is given with the recent experimental investigations of wedge waves carried out by independent researchers.

Lamb wave scattering by a surface breaking crack in a plate : Datta, S.K.; Al-Nassar, Y. Review of Progress in Quantitative Nondestructive Evaluation, La Jolla, California (United States), 15–20 Jul. 1990. Vol. 10 A, pp. 97–104. Edited by D.O. Thompson and D.E. Chimenti. Plenum Press (1991). ISBN 0-306-43903-4

An NDE method based on finite-element representation and modal expansion has been developed for solving the scattering of Lamb waves in an elastic plate waveguide. This method is very powerful for handling discontinuities of arbitrary shape, weldments of different orientations, canted cracks, etc. The advantage of the method is that it can be used to study the scattering of Lamb waves in anisotropic elastic plates and in multilayered plates as well.

The local algorithms of autoresonant absorption of the energy of bending waves in elastic plates

For the simplest model—an infinite elastic plate—three control algorithms (local in space and time) are studied by a normal point shift to achieve the maximum of absorbed power. They are: (a) the algorithm of half-return, (b) the algorithm of the random search for the maximum of the instantaneous absorbed power, (c) the continuous algorithm of the resonant absorption. These algorithms provide the exit to the trajectory of the resonant absorption during the time much less than the minimum time scale of the damped wave, i.e., knowing either its period, or its length. The considered algorithms differ in the necessary volume of the apriori information on the plate parameters and in the power of the ‘‘technological’’ high-frequency radiation caused by rapid manipulations with the boundary condition. The speed of response of sensors and final control elements is supposed to allow the formation of arbitrary kinematic characteristics of the boundary. The conditions for the effective operation of the algorithms in the case of plates with finite dimensions are formulated. The algorithms may be used to decrease the quality of mechanical systems made up of elastic plates.

Elastic waves in plates with periodically placed inclusions

We calculated the frequency ω versus the wave vector K for elastic waves propagating in both thick and thin plates consisting of solid inclusions placed periodically in the host material. We were particularly interested in the possible creation of spectral gaps (stop bands) in all directions of propagation for bending waves. We found that Mo, Fe, steel, or Pb inclusions forming a two dimensional hexagonal lattice in a Lucite host give rise to spectral gaps.

Numerical modelling of the scattering of elastic waves in plates

This paper describes the application of finite difference methods to the calculation of the scattering of elastic waves. The emphasis is on cracklike defects in plates, and it is shown that a common numerical technique can span a range of wavelengths from Lamb waves to ultrasonic waves with many reflections from the surfaces of the plate. Quantitative results are given for the scattering of Lamb waves and ultrasonic shear waves from surface-breaking cracks.

Spectra of transient waves in elastic plates

Frequency spectra of transient Rayleigh–Lamb waves in a plate are evaluated by two methods, a direct numerical integration (FFT), and an asymptotic calculation of the theoretical responses. A simple formula is developed for the interference pattern of the spectra of two or more modes of progressing waves, which is then applied to interpret the complex frequency spectra of the waves in a plate over a wide range of frequency. These interpretive techniques may be applicable to source characterization of acoustic emission.

Transient and time harmonic waves in elastic plates

This paper deals with transient wave propagation in elastic homogeneous and isotropic plates, in terms of displacement discontinuities of all order ⩾ 1 at the wave front. The problem of steady state time harmonic waves is dealt with in terms of an asymptotic series expansion. The possible wave types along with the general transport-induction equations for each type are given, and the interrelationship between transient and time harmonic waves is discussed. Special constrained wave motions which allow uncoupling of the various possible wave types are defined. Several illustrative examples of the theory developed are given.

Propagation of Elastic Waves in Plates and Bars of Solid Material with Low Shear Modulus

Sound propagation in solid media of finite cross section is very complicated in the general case, due to the fact that the boundary conditions can be fulfilled only by superposition of both longitudinal and transverse waves. It can be shown that this picture will be much clearer for rubber-like materials (with low shear modulus, the Poisson constant being nearly 0.5), because the velocity of the transverse waves is much smaller than that of the longitudinal waves. As long as all dimensions normal to the axis are small compared with half the wavelength of the longitudinal wave, the dispersion of the phase velocity is given by the shear modulus as in an incompressible medium with finite shear modulus. Above this frequency limit the influence of the bulk modulus overcomes that of the shear modulus; and if energy losses exist, it will alone determine the behavior. The phase velocity then will nearly be equal to that of the different modes of the longitudinal wave in a liquid column. Measurements of phase velocity and attenuation confirm that below this limiting frequency the damping is determined by the loss factor of the shear modulus, while above the limit a much lower damping exists, a result of the smaller influence of the shear modulus.

Propagation of Elastic Waves in Plates and Cylinders

Some additional solutions have recently been given for the problem of elastic waves in plates as originally formulated by Lamb. Using recent solutions for the reflection of elastic waves at a free surface, it is possible to construct a wave system that satisfies the boundary conditions for the plate. The construction of these waves surfaces make it possible to understand Holdens results in geometrical terms. This is analogous to the elementary explanations of electromagnetic wave guides in terms of the reflections of plane waves. The utility of the construction is that it makes it possible to gain an intuitive understanding of mathematically intractable problems. It is also possible to better understand the recent solutions for elastic wave propagation in solid circular cylinders by this method.