Phononic plate waves

In the past two decades, phononic crystals (PCs) which consist of periodically arranged media have attracted considerable interest because of the existence of complete frequency band gaps and maneuverable band structures. Recently, Lamb waves in thin plates with PC structures have started to receive increasing attention for their potential applications in filters, resonators, and waveguides. This paper presents a review of recent works related to phononic plate waves which have recently been published by the authors and coworkers. Theoretical and experimental studies of Lamb waves in 2-D PC plate structures are covered. On the theoretical side, analyses of Lamb waves in 2-D PC plates using the plane wave expansion (PWE) method, finite-difference time-domain (FDTD) method, and finite-element (FE) method are addressed. These methods were applied to study the complete band gaps of Lamb waves, characteristics of the propagating and localized wave modes, and behavior of anomalous refraction, called negative refraction, in the PC plates. The theoretical analyses demonstrated the effects of PC-based negative refraction, lens, waveguides, and resonant cavities. We also discuss the influences of geometrical parameters on the guiding and resonance efficiency and on the frequencies of waveguide and cavity modes. On the experimental side, the design and fabrication of a silicon-based Lamb wave resonator which utilizes PC plates as reflective gratings to form the resonant cavity are discussed. The measured results showed significant improvement of the insertion losses and quality factors of the resonators when the PCs were applied.     

纵波脉冲回波法和兰姆波法在表征薄层结构中的应用比较

介绍了两种常用的材料薄层超声表征方法——纵波脉冲回波法和兰姆波法。阐述了其基本原理，并结合实例比较了它们的应用情况、优缺点和适用条件等。纵波脉冲回波法简单易行，能够获得薄层厚度、声速和衰减等信息。但实验结果受薄层介质的声衰减和表面粗糙度影响较大，对探头的性能要求较高。兰姆波法主要用于表征薄板类材料，无需较高检测频率，可以测得板中纵波和横波的声速、板厚和板与板之间的结合特性等，检测精度较高。缺点是发射和接收探头之间的距离和倾斜角度需要严格控制，影响因素较多。     

Dispersion relations and modes of wave propagation in inclusion-reinforced composite plates

This paper is intended to examine the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness on the dispersion relations and modes of wave propagation in inclusion-reinforced composite plates. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Mori–Tanaka mean-field theory is used to predict the effective elastic moduli of the composite plate explicitly. The effective elastic moduli are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior. The resulting moduli are then used to determine the dispersion relations and the modal patterns of Lamb waves using the dynamic stiffness matrix method. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions are symmetrical or antisymmetric about the midplane of the plate. Classifying the wave type in an anisotropic plate is not as simple as that in an isotropic plate, and has not received proper attention in the literature. The wave types and orders are identified by analyzing the dispersion curves and inspecting the calculated modal patterns, and the results indicate that the Lamb waves in an orthotropic composite plate can also be classified as either symmetric or antisymmetric waves. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns. Propagation speed is generally increased with the aspect ratio, e.g., using longer fibers generally results in a higher propagation speed.     

Shear horizontal wave propagation in periodically layeredcomposites

The transverse resonance approach to guided wave analysis is applied to shear horizontal (SH) wave propagation in periodically layered composites. It is found for SH waves that at high values of the guided wavevector β, the wave energy is trapped in the slower of the two media and propagates accordingly at the slower wavespeed. At low values of β, however, the modes demonstrate a clustering behavior, indicative of the underlying Floquet wave structure. The number of modes in a cluster is observed to correlate with the number of unit cells in the layered plate. New physical insights into the behavior of these systems are obtained by analyzing the partial waves of the guided SH modes in terms of Floquet waves. We show that the fast and slow shear waves in the periodically layered composite play an analogous role to the longitudinal and shear partial waves comprising Lamb waves in a homogeneous plate     

Dispersion of linearly polarized light propagating in a thin birefringent plate

We analyze theoretically the dispersion of linearly polarized light propagating in a uniaxial anisotropic medium where multibeam interference is present. Explicit expressions of the group-delay dispersion for transmitting waves are derived for the simplest situation, and the effect of dispersion on pulse broadening is analyzed for a few selected cases. Our results reveal that at normal incidence and in the situation where the optic axis is parallel to the surface of birefringent plate (in the x-y plane), the dispersion of the refracted wave decreases with the extent of birefringence. In particular, the dispersion for the electric field parallel to the polarization direction of the incident light changes with the rotation angle between the optic axis and the polarization direction of the incident field, whereas the dispersion for the refracted field whose direction is vertical to the polarization of incident light is independent of this angle. For oblique incidence, dispersion varies substantially for different incident angles. In the situation where the optic axis is in the x-z plane at either normal or oblique incidence, the dispersion increases in a periodically oscillating manner as a function of the relative thickness of the birefringent plate.     

New method of change in temperature coefficient delay of acoustic waves in thin piezoelectric plates

As is well-known, the development of high-effective and thermostable acoustic devices assumes using the acoustic waves with high coefficient of electromechanical coupling (K2) and low temperature coefficient of delay (TCD). At present, it also is well-known that fundamental shear horizontal (SH0) acoustic waves in thin piezoelectric plates possess by significantly more electromechanical coupling compared to surface acoustic waves (SAW) in the same material. However, although the value of TCD of SH0 waves is insignificantly less than for SAW, this is not enough for development of thermostable devices. This paper suggests a new way of decreasing TCD of SH0 waves in piezoelectric plates at a high level of electromechanical coupling. This way assumes to use the structure containing the piezoelectric plate and liquid with the special dependence of permittivity on temperature. Theoretical and experimental investigation showed that, for SH0 wave in YX LiNbO3 plate at hf = 700 m/s (h = plate thickness, f = wave frequency) the presence of butyl acetate can decrease the value of TCD by six times at K2 = 30%. In a whole the obtained results open the wide prospect of using SH0 wave in thin piezoelectric plate for development of high effective and thermo-stable acoustic devices.     

识别薄板中兰姆波和体波的阶梯板方法

斜探头在某些频率下激励出的兰姆波,其群速度与体波的传播速度相近,所以通过判断传播速度不易区分出兰姆波和体波。通过数值模拟和实验,分别研究了激励频率为2 MHz的纵波和S0模态兰姆波在阶梯板上的反射特性,发现:在阶梯板上入射S0模态兰姆波时,有反射回波;而入射纵波时,无反射回波。基于这种反射特性的差别,提出了一种利用阶梯板区别薄板中兰姆波和体波的方法,该方法可用于确认探头的激励特性。     

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.     

Guided thermoelastic waves in functionally graded plates with two relaxation times

Functionally graded material (FGM) is a promising heat insulation material. Wave propagation in FGM structures has received much attention for the purpose of non-destructive testing and evaluation. Few literatures dealt with the thermoelastic wave in FGM structures although the thermal effect would cause attenuations of elastic waves. In this paper, guided thermoelastic waves in FGM plates subjected to stress-free, isothermal boundary conditions are investigated in the context of the Green–Lindsay (GL) generalized thermoelastic theories (with two relaxation times). Coupled wave equations and heat conduction equation are solved by the Legendre polynomial approach. Dispersion curves for a pure elastic graded plate are calculated to make a comparison with the published data. For the thermoelastic graded plate, dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Attenuation curves for graded plates with different relaxation times are compared. The influences of different material gradient shapes are discussed. Two homogeneous thermoelastic plates with different volume fractions are obtained to show their differences from graded plates. Finally, thermoelastic wave dispersion curves for a homogeneous plate and a graded plate are calculated in the context of the classical coupled thermoelastic theory (CT) to show its differences and similarities to the generalized theory.     

DPSM modeling for studying interaction between bounded ultrasonic beams and corrugated plates with experimental verification

Periodically corrugated structures play an important role in the field of vibration control and for designing structures with desired acoustic band gaps. Analytical solutions for corrugated plates are available for well-defined, smooth corrugations, such as sinusoidal corrugations that are not very common in the real world. Often corrugated plates are fabricated by cutting grooves at regular intervals in a flat plate. No analytical solution is available to predict the wave propagation behavior in such a periodically corrugated plate in which the equation of the plate surface changes periodically between a planar fiat surface and a nonplanar parabolic groove. This problem is solved here for steady-state case by a newly developed semianalytical technique called distributed point source method (DPSM), and the theoretical predictions are compared with the experimental results generated by reflecting a bounded 2.25 MHz ultrasonic beam by a fabricated corrugated plate. The main difference that is observed in the reflected beam profile from a flat plate and a corrugated plate is that the back-scattering effect is much stronger for the corrugated plate, and the forward reflection is stronger for the flat plate. The energy distribution inside the corrugated plate also shows backward propagation of the ultrasonic energy.     

旋转层合圆板的行波动力学特性分析

基于一阶剪切变形理论和哈密顿原理,建立了旋转层合圆板动力学运动方程和相应的边界条件。运用伽辽金法对旋转层合圆板横向振动的行波动力学特性进行了模拟,给出了其前行波、后行波振动频率随层合圆板铺层几何特征、材料参数、旋转速度等的变化规律,并对旋转层合圆板几何和材料参数对其动力失稳临界转速的影响进行了讨论。     

Extensional thermoelastic waves in transversely isotropic plate according to a higher order theory

The objective of this work is to provide a rigorous analysis of thermoelastic ultrasonic waves in transversely isotropic plates. Characteristic features such as dispersion curves of thermoelastic waves of plates are investigated and the influence of coupling in the heat equation on these features is critically examined. If the propagation of the waves is along the axis of symmetry of the plate, then it is possible to decouple the antisymmetric modes from the symmetric ones. This is conveniently done in approximate theories by retaining and omitting various terms in the expansions for the displacement and temperature. In this work, it is assumed that the wave propagation is along the axis of symmetry of an infinite anisotropic plate. Hence, extensional (symmetric) modes can be investigated apart from the antisymmetric modes. Displacement and temperature are expanded across the thickness of the plate using Legendre polynomials. Obviously, such a theory best fits those applications where a low frequency pulse is employed. Further, keeping only the leading terms in the expansion of displacement and temperature gives rise to a lower order theory, which predicts well the correct behavior of symmetric modes in relatively smaller frequency range. Results also show that the effect of coupling in the heat equation is insignificant for thermoelastic waves and can be ignored.     

Dispersion analysis of stabilized finite element methods for acoustic fluid interaction with Reissner–Mindlin plates

The application of stabilized finite element methods to model the vibration of elastic plates coupled with an acoustic fluid medium is considered. A complex‐wavenumber dispersion analysis of acoustic fluid interaction with Reissner–Mindlin plates is performed to quantify the accuracy of stabilized finite element methods for fluid‐loaded plates. Results demonstrate the improved accuracy of a recently developed hybrid least‐squares (HLS) plate element based on a modified Hellinger–Reissner functional, consistently combined with residual‐based methods for the acoustic fluid, compared to standard Galerkin and Galerkin gradient least‐squares plate elements. The technique of complex wavenumber dispersion analysis is used to examine the accuracy of the discretized system in the representation of free waves for fluid‐loaded plates. The influence of fluid and coupling matrices resulting from consistent implementation of pressure loading in the residual for the plate equation is examined and clarified for the different finite element approximations. Copyright © 2001 John Wiley & Sons, Ltd.     

Five-mode frequency spectra of x3-dependent modes in AT-cut quartz resonators

We study straight-crested waves and vibration modes with variations along the x(3) direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlin's two-dimensional equations for anisotropic crystal plates are used. Dispersion relations and frequency spectra of the five relevant waves are obtained. It is found that, to avoid unwanted couplings between the resonator operating mode and other undesirable modes, in addition to certain known values of the plate length/thickness ratio that need to be avoided, an additional series of discrete values of the plate length/thickness ratio also must be excluded.     

Propagation of Iamb waves in adhesively bonded multilayered media

The effect of introducing attenuation on Lamb wave dispersion curves is studied in this paper. Attenuation is introduced to a three-layered composite plate by an adhesive bond layer with viscous behavior. No changes are required to the transfer matrix formulation for the propagation of elastic waves. By introduction of a complex wavenumber, the model can be used to the propagation of attenuative Lamb waves. Numerical examples for a three-layered aluminium-epoxy-aluminium plate show that attenuation values of each mode in plates are related not only to attenuation, but also to the thickness of the bonded layer, which is in agreement with practical situations.     

The peculiarities of propagation of the backward acoustic waves in piezoelectric plates

The characteristics of backward acoustic waves in piezoelectric plates under different electrical boundary conditions were investigated. It has been shown that electrical shorting of the plate leads to increasing and decreasing the phase velocity for backward and forward branches, respectively. The peculiarities of the hybridization effect of backward waves were studied.     

2-2压电复合材料面板的压力放大作用

２－２压电复合材料应用普遍。材料中聚合物对压电陶瓷的应力作用，使性能得到施工改善。在复合材料两电极面上复盖薄面板，由于板的应力传递作用会使使接收国和辐射力产生放大效应，从而提高了接收灵敏度和发射响应。     

Calculations of Lamb wave band gaps and dispersions for piezoelectric phononic plates using mindlin's theory-based plane wave expansion method

Based on Mindlin's piezoelectric plate theory and the plane wave expansion method, a formulation is proposed to study the frequency band gaps and dispersion relations of the lower-order Lamb waves in two-dimensional piezoelectric phononic plates. The method is applied to analyze the phononic plates composed of solid-solid and airsolid constituents with square and triangular lattices, respectively. Factors that influence the opening and width of the complete Lamb wave gaps are identified and discussed. For solid/solid phononic plates, it is suggested that the filling material be chosen with larger mass density, proper stiffness, and weak anisotropic factor embedded in a soft matrix in order to obtain wider complete band gaps of the lower-order Lamb waves. By comparing to the calculated results without considering the piezoelectricity, the influences of piezoelectric effect on Lamb waves are analyzed as well. On the other hand, for air/solid phononic plates, a background material itself with proper anisotropy and a high filling fraction of air may favor the opening of the complete Lamb wave gaps.     

Piezoelectromagnetic waves in a ceramic plate

The propagation of shear horizontal (SH) waves in an infinite plate of polarized ceramics is studied from the three-dimensional equations of linear piezoelectro-magnetism with full electromagnetic coupling. Exact solutions are obtained for plates with electroded and unelectroded surfaces.     

Edge waves in a thermoelastic plate

Temperature-rate-dependent thermoelasticity theory is employed to study waves propagating along the edges of a thin flat plate of infinite length, which is in a state of plane stress. Governing equations of the plane stress problem are derived and it is found that the speed of heat waves in this problem is, in general, less than that in the plane strain and the general three-dimensional problems. The waves of the desired type are analysed for isothermal and insulated edge conditions, and the paths of particles as well as the cut-off frequencies are determined. Waves in a thin bar and in a semi-infinite plate are discussed as limiting cases. The effects of the finiteness of the neat propagation speed are analysed in some detail.