Extensional Waves Research Articles

The influence of the initial stresses on Lamb wave dispersion in pre-stressed PZT/Metal/PZT sandwich plates

Within the scope of the plane-strain state, by utilizing the three-dimensional linearized theory of elastic waves in initially stressed piezoelectric and elastic materials, Lamb wave propagation and the influence of the initial stresses on this propagation in a sandwich plate with pre-stressed piezoelectric face and pre-stressed metal elastic core layers are investigated. Dispersion equations are derived for the extensional and flexural Lamb waves and, as a result of numerical solution to these equations, the corresponding dispersion curves for the first (fundamental) and second modes are constructed. Concrete numerical results are obtained for the cases where the face layers' materials are PZT-2 or PZT-6B, but the material of the middle layer is Steel (St) or Aluminum (Al). Sandwich plates PZT-2/St/PZT-2, PZT-2/Al/PZT-2, PZT-6B/St/PZT-6B and PZT-6B/Al/PZT-6B are examined and the influence of the problem parameters such as piezoelectric and dielectric constants, layer thickness ratios and third order elastic constants of the St and Al on the effects of the initial stresses on the wave propagation velocity is studied.

Propagation of extensional waves in a piezoelectric semiconductor rod

We studied the propagation of extensional waves in a thin piezoelectric semiconductor rod of ZnO whose c-axis is along the axis of the rod. The macroscopic theory of piezoelectric semiconductors was used which consists of the coupled equations of piezoelectricity and the conservation of charge. The problem is nonlinear because the drift current is the product of the unknown electric field and the unknown carrier density. A perturbation procedure was used which resulted in two one-way coupled linear problems of piezoelectricity and the conservation of charge, respectively. The acoustic wave and the accompanying electric field were obtained from the equations of piezoelectricity. The motion of carriers was then determined from the conservation of charge using a trigonometric series. It was found that while the acoustic wave was approximated by a sinusoidal wave, the motion of carriers deviates from a sinusoidal wave qualitatively because of the contributions of higher harmonics arising from the originally nonlinear terms. The wave crests become higher and sharper while the troughs are shallower and wider. This deviation is more pronounced for acoustic waves with larger amplitudes.

Extensional-wave stopband broadening across the joint of pipes of different thickness.

The stopband of pipe extensional waves is an interesting natural phenomenon. This study demonstrates an important extension of this phenomenon. That is, the stopband can be effectively broadened by transmitting the waves across the joint of pipes of different thickness. The theoretical and experimental results reveal the detailed process of stopband forming along the pipe and the band broadening across the pipe joint. The result can be utilized to provide a method for logging while drilling acoustic isolation design.

Thermo-acoustic engineering of silicon microresonators via evanescent waves

A temperature-compensated silicon micromechanical resonator with a quadratic temperature characteristic is realized by acoustic engineering. Energy-trapped resonance modes are synthesized by acoustic coupling of propagating and evanescent extensional waves in waveguides with rectangular cross section. Highly different temperature sensitivity of propagating and evanescent waves is used to engineer the linear temperature coefficient of frequency. The resulted quadratic temperature characteristic has a well-defined turn-over temperature that can be tailored by relative energy distribution between propagating and evanescent acoustic fields. A 76 MHz prototype is implemented in single crystal silicon. Two high quality factor and closely spaced resonance modes, created from efficient energy trapping of extensional waves, are excited through thin aluminum nitride film. Having different evanescent wave constituents and energy distribution across the device, these modes show different turn over points of 67 °C and 87 °C for their quadratic temperature characteristic.

Extensional Waves in a Sandwich Plate With Interface Slip

The two-dimensional (2D) equations for thin elastic plates are used to study extensional motions of a sandwich plate with weak interfaces. The interfaces are governed by the shear-slip model that possesses interface elasticity and allows for a discontinuity of the tangential displacements at the interfaces. Equations for the individual layers of the sandwich plate are coupled by the interface conditions. Through a procedure initiated by Mindlin, the layer equations can be written into equations for the collective motion of the layers representing the extensional motion of the sandwich plate, and equations for the relative motions of the layers with respect to each other representing the symmetric thickness-shear motion of the sandwich plate. The use of plate equations results in relatively simpler models compared to the equations of three-dimensional (3D) elasticity. Solutions to a few useful problems are presented. These include the propagation of straight-crested waves in an unbounded plate with weak interfaces, the reflection of extensional waves at the joint between a perfectly bonded sandwich plate and a sandwich plate with weak interfaces, and the vibration of a finite sandwich plate with weak interfaces.

Resonance of an imperfect joint of plates by the lowest-order symmetric Lamb mode.

Resonance behavior of an imperfect joint of elastic plates subjected to the incidence of the lowest-order symmetric (S0) Lamb mode is numerically analyzed in the frequency domain by the hybrid finite element method. To this purpose, the reflection and transmission characteristics of the S0 mode are calculated for the frequency range in which the S0 mode is the only symmetric mode that can propagate in the plates. The imperfect joint is modeled as a linear spring-type interface characterized by the normal and tangential stiffnesses. As a result, it is shown that the imperfect joint of plates has two resonance frequencies at which the out-of-plane displacement amplitudes at the joint are remarkably increased. One resonance frequency depends only on the normal stiffness, and the other only on the tangential stiffness. Each resonance frequency coincides with that of a free edge of a plate, and monotonically increases with the corresponding joint stiffness. Furthermore, it is also shown that the reflection and transmission behavior of the S0 mode at the imperfect joint of the plates can be well reproduced by the one-dimensional thin-plate approximation of extensional waves when the frequency is sufficiently small compared to the resonance frequencies.

Acoustic isolation of a monopole logging while drilling tool by combining natural stopbands of pipe extensional waves

Acoustic isolation of a monopole logging while drilling tool by combining natural stopbands of pipe extensional waves

Modal response of a simply supported plate driven by a single, asymmetric piezoelectric actuator for use as a flat-panel loudspeaker

An approximate analytical solution for the mechanical response of a thin plate to a single, asymmetrically bonded piezoelectric driver has been found. Previous work has focused only on the response of plates driven by symmetrically aligned actuators or for beams with drivers bonded to one side. The relative magnitudes of extensional and flexural waves produced by the single-sided driver configuration were determined as a function of the physical parameters of the plate and driver. The modal response of flexural waves for a simply supported thin plate also was determined. Interactions at the plate boundaries with the support structure may convert extensional waves into sound-producing flexural waves, which can lead to audio distortion. A non-moment inducing support scheme with high damping is proposed to minimize this effect.

Optimal Energy Transfer Pipe Arrangement for Acoustic Drill String Telemetry

Drill string acoustic telemetry is an effective transmission method to retrieve downhole data. Finite-difference simulations produce the comb-filter-like channel response (patterns of pass bands and stop bands) due to the presence of coupling joints in the metallic drill string. Practical pipes used for drilling deep wells have slight variation in length. The selection and arrangement of downhole pipes is important for improving the transmission efficiency of extensional waves transmitted through the drill string. Downhole drill string channel is studied using the transmission coefficients calculated from the transmission matrix method, and the resultant transfer function produces identical results similar to the finite-difference simulations. Reciprocity of the drill string structure is proved by comparing the pass band responses using the ascend-only (AO) and descend-only pipe arrangements. Transferred energies calculated up to 180 pipes of random length at the end of the drill strings using transmission coefficients for the three different pipe arrangements, namely, AO, descend-then-ascend, and ascend-then-descend (ATD), are compared to find the optimal pipe arrangement for single measurement. For the situations when pipes are distributed in sets, multiple measurements are required. In this paper, two sets of AO and two sets of ATD arrangements are analyzed for multiple measurements. ATD and $n$ xATD arrangements are proposed as optimal pipe arrangements to produce the best possible telemetry performance in terms of optimal acoustic energy transfer via one- and two-way acoustic communication for single and multiple measurements, respectively.

Modeling of a functionally graded micro-ring segment for the analysis of coupled extensional–flexural waves

Established in this study is a microstructure-dependent continuum model of a micro-ring with a functionally graded material composition in the thickness direction. The developed model is employed to furnish the nature of elastic waves propagation within a segment of the micro-ring. The development of the model is grounded on the modified couple stress theory, the Hamilton’s principle, and the classical rule of mixture. For validation purposes, patterns of wave propagation within homogenous rings and homogeneous micro-rings are analyzed from the reduced forms of the model. Key effects incorporated in the new model include small-scale parameter, power-law index, Winkler–Bach and rotary inertia effects. Dispersion analyses revealed a profile of frequency spectra characterized by distinct frequency zones, each of which is engendered by a bifurcation point. Explicit closed-form expressions are provided for high and low frequency limits as well as the long and short wavelength limits of the coupled waves. Phase speed dispersion curves for the micro-ring are found to have different branches that signify the distinctly flexural and extensional waves. Higher values of the power-law index are found to lead to a marked decrease in the phase speed and a noticeable increase of the group speed of the extensional wave.

Effects of Delamination on Guided Waves in a Symmetric Laminated Composite Beam

For successful structural health monitoring and structural integrity evaluation of a laminated composite structure, it is important to study the effects of delamination on the propagations of the guided waves in a delaminated composite beam by using an accurate and computationally efficient method. Thus, we developed a “frequency-domain” spectral element model for the symmetric composite beams. First-order-shear-deformation-theory (FSDT) based Timoshenko beam theory and Mindlin-Herrmann rod theory are adopted for the flexural (bending) waves and axial (extensional) waves, respectively. A spectral element model is derived from the governing equations of motion by using the variation method in the frequency domain. After validating the accuracy of the proposed spectral element model, the model is used to investigate the effects of delamination on the propagation of guided waves in examples of composite beams.

Acoustic Emission Monitoring on Crack Growth of Wind Turbine Blade Made of Fiber Glass Epoxy Plastic

Abstract. To extract the AE signal feature effectively, an automatically operating blind source separation method is presented. The blind deconvolution algorithm is discussed for the non-stationary AE signals to eliminate the noise through minimizing the mutual information to get the separating matrix. Finally, the experimental result is indicated that the mixing noise of the AE signals in the collecting process can be eliminated by this method, and the individual extensional and flexural wave modes can be separated, which can provides the theory analysis basis for the crack recognition of wind turbine blade.

A method for multidimensional wave propagation analysis via component‐wise partition of longitudinal and shear waves

SUMMARYAn explicit integration algorithm for computations of discontinuous wave propagation in two‐dimensional and three‐dimensional solids is presented, which is designed to trace extensional and shear waves in accordance with their respective propagation speeds. This has been possible by an orthogonal decomposition of the total displacement into extensional and shear components, leading to two decoupled equations: one for the extensional waves and the other for shear waves. The two decoupled wave equations are integrated with their CFL time step sizes and then reconciled to a common step size by employing a previously developed front‐shock oscillation algorithm that is proven to be effective in mitigating spurious oscillations. Numerical experiments have demonstrated that the proposed algorithm for two‐dimensional and three‐dimensional wave propagation problems traces the stress wave fronts with high‐fidelity compared with existing conventional algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

Parameter estimation from wave propagation tests on a tube perforated by helical slots

For a tube with doubly symmetric cross section and perforations by helical slots there is a coupling between extension and torsion. In this paper a one dimensional (1D) model structure for a tube with such a helical slot segment (HSS) is established, and parameters accounting for the coupling between extension and torsion are estimated from wave propagation experiments. In these experiments incident extensional waves were generated through axial impact by strikers of different lengths, causing reflected and transmitted waves of extensional and torsional type which were measured in terms of surface strains on either side of the HSS part of the tube. A statistical test on the experimental data shows that the output residuals (the difference between modeled and experimental output) cannot be explained by measurement noise alone. This is not surprising since the 1D model structure is based on some simplifying assumptions concerning the geometry of the HSS. Parameters for two different geometries of the HSS are estimated, and the models are assessed in terms of model fit, simulations and wave energy distribution. It turns out that for one case, where the geometrical assumptions are valid, the 1D model is adequate, while for another case, where the validity of the assumptions is questionable, it is not. It is concluded that the 1D model structure provides a simple and efficient description of the HSS if the geometrical assumptions are valid.

Reduction Techniques of Vibration in Curved Beams Using Negative Group Velocity

The dispersion curve of an extensional wave of a curved beam has negative slope at a restricted frequency region. A wave packet which is synthesized by more than two extensional waves at the region of negative slope has negative group velocity. This paper addresses an issue of engineering application of the negative group velocity. Firstly, the horizontal axis of dispersion curve was normalized as a dimensionless frequency to specify the frequency region of negative slope. Then, a structure to allow negative group velocity was designed as a U-shaped beam based on the normalized dispersion curve, and waves on the beam was verified using transient response analysis of a finite element model. Validation was carried out from the standpoint of both negative group velocity and backward movement of structural intensity. The results show that wave energy in a curved beam does not propagate in the positive direction along the axial direction.

Ultrasonic contact pulse transmission for elastic wave velocity and stiffness determination: Influence of specimen geometry and porosity

Elasticity determination by means of ultrasonic pulse transmission requires experimental realization of non-dispersive, i.e. frequency-independent, wave propagation, be it in form of bulk waves propagating in an (approximately) infinite medium, or of extensional waves propagating through a 1D bar system. While it is conceptually known that wavelengths need to tend towards zero (as compared to the specimen dimensions perpendicular to the pulse propagation direction) in the 3D case, and towards infinity in the 1D case, we here report on a new systematic experimental assessment of the influence of the sample geometry on wave type: tests on solid isotropic aluminum samples reveal that the extensional (or bar) wave propagation mode requires transmission of truly slender samples (required slenderness ratio of 20 or larger for wavelengths equal to the wave travel distance; this minimum slenderness ratio is increasing with increasing travel distance-over-wavelength ratio). After a transition zone with dispersive wave propagation, non-dispersive bulk waves are detected once the slenderness ratio is reduced to 5 or lower (at wavelengths equal to the wave travel distance; this maximum slenderness ratio is increasing with increasing travel distance-over-wavelength ratio). On the other hand, it is conceptually known from continuum mechanics that the wavelength needs to be larger than the investigated material volume or representative volume element (RVE), as to reveal the material’s elastic properties, while corresponding quantitative data are rare. As a remedy, we here report on new experiments on transversely isotropic, porous aluminum samples, which reveal that minimum pore dimension-over-wavelength ratios of 1 and 10, respectively, relate to detection of normal and shear stiffnesses, respectively, of the solid material between the pores, while these ratios need to be smaller than 0.01 and 0.1, as to detect the normal and shear stiffnesses of the overall porous materials. The latter can be quantified through various homogenization techniques.

Modeling Lamb Wave Propagation in Damaged Structures Based upon Spectral Element Method

Structural health monitoring based on Lamb wave attracts great attention in large-span structures. Lamb wave propagation in complex structures is very complicated due to multiple reflection and mode conversion at geometrical and material features. For effectively inspecting structural integrity, numerical simulation is employed to for extract damage features. It is essential to develop fast and low-cost simulating methods to study Lamb wave propagation in damaged structures. Spectral element method (SEM) is one of the most attractive methods, which is employed to study wave propagation in damaged structures. A massless spring, coupling the longitudinal and rotational vibration, is proposed to model a transverse crack and analyze wave propagation in a composite cracked beam based on SEM. Cracked spectral element formulation is derived by modeling the crack as the spring, whose stiffness is obtained from laws of fracture mechanics. Due to asymmetry of the crack, extensional and flexural wave modes are reflected and transmitted from an incident flexural wave mode. The proposed model is verified by comparing with conventional finite element analysis. Power reflection and transmission varying with the crack depth is also calculated. The results indicate that power reflection/transmission ratio of a single mode is monotonic, which may provide some quantitative foundations for structural health monitoring.

A logging while drilling acoustic isolation technology by varying thickness of drill collars at a distance greater than wavelength

A key technology for logging while drilling (LWD) acoustic measurements is the design of an acoustic isolator to suppress tool waves propagating along the drill collar, such that acoustic signals from earth formations can be effectively measured under LWD conditions. Up to now, the LWD acoustic isolation is achieved by periodically cutting grooves along the drill collar between acoustic transmitter and receivers. Such a technique, although it is effective, reduces the mechanical strength of the drill collar and adds cost to the manufacturing and maintenance of the LWD tool, hindering the application of the LWD acoustic technology. We have developed an LWD acoustic technology that does not use the groove-cutting design. We utilize the inherent frequency stopband for extensional wave propagation along a cylindrical pipe and effectively broaden the stopband by combining drill pipes of different cross-section areas whose lengths are greater than a wavelength but are shorter than the transmitter-to-receiver distance. After propagation through the combined drill collar system, the stopband in the collar extensional wave is significantly widened and the wave amplitude in the stopband is substantially reduced. Making LWD acoustic measurements in this widened stopband allows for recording acoustic signals from the surrounding formation.

Extensional and Transversal Wave Motion in Transversely Isotropic Thermoelastic Plates by Using Asymptotic Method

The present investigation is concerned with the study of extensional and transversal wave motions in an infinite homogenous transversely isotropic, thermoelastic plate by using asymptotic method in the context of coupled thermoelasticity, Lord and Shulman (1967, “The Generalized Dynamical Theory of Thermoelasticity,” J. Mech. Phys. Solids, 15, pp. 299–309), and Green and Lindsay (1972, “Thermoelasticity,” J. Elast., 2, pp. 1–7) theories of generalized thermoelasticity. The governing equations for extensional, transversal, and flexural motions have been derived from the system of three-dimensional dynamical equations of linear thermoelasticity. The asymptotic operator plate model for extensional motion in a homogeneous transversely isotropic thermoelastic plate leads to sixth degree polynomial secular equation that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion. The Rayleigh–Lamb frequency equation for the plate is expanded in power series in order to obtain polynomial frequency equation and velocity dispersion relations. Their validation has been established with that of asymptotic method. The special cases of short and long wavelength waves are also discussed. The expressions for group velocity of extensional and transversal modes have been derived. Finally, the numerical solution is carried out for homogeneous transversely isotropic plate of single crystal of zinc material. The dispersion curves of phase velocity and attenuation coefficient are presented graphically.

Extensional edge waves in pre-stressed incompressible plates

Consider a semi-infinite layer of an incompressible elastic material subjected to a finite static homogeneous pre-deformation. This paper describes the long-wave low-frequency limiting behaviour of extensional edge waves that may propagate along the free edge of such a layer. The analysis is done by constructing an asymptotic plate theory that describes the extensional motion of thin pre-stressed incompressible plates. The secular equation for edge waves propagating along one of the principal axes of the pre-stress is studied for bi-axial deformations and several specific material models. We show that certain combinations of material models and configurations of pre-stresses do not support the propagation of edge waves, which mirrors the situation with surface waves in pre-stressed media. More unusually, we present a seemingly first explicit example of the non-unique edge wave solutions existing in thin pre-stressed plates.