Importance of the near Lamb mode imaging of multilayered composite plates.

In recent years Lamb waves are being used for internal defect detection in multilayered composite plates. Different Lamb modes generate various stress levels in different layers. As a result, all Lamb modes are not equally sensitive to internal defects of various layers. A number of studies have been carried out to identify which Lamb mode is most effective for detecting defects in a specific layer. However, one shortcoming of the Lamb wave inspection technique is that in a symmetrically layered composite plate stress and displacement magnitudes and energy distribution profiles for all Lamb modes are symmetric about the central plane of the plate. As a result, the ability of a Lamb mode to detect defects in a specific layer of the plate is identical to its ability to detect defects in the corresponding layer of mirror symmetry. Hence, from the Lamb wave generated image one cannot distinguish between the defects in these two layers of mirror symmetry. In this paper it is investigated how by fine-tuning the frequency and the striking angle of the incident beam in the neighborhood of a Lamb mode one can separately detect internal defects in layers of mirror symmetry in the upper and lower halves of a plate.     

Propagation of guided elastic waves in 2D phononic crystals

The phononic band structure of two-dimensional phononic guides is numerically studied. A plane wave expansion method is used to calculate the dispersion relations of guided elastic waves in these periodic media, including 2D phononic plates and thin layered periodic arrangements. We show that, for any guided elastic wave, Lamb or generalised Lamb modes, stop bands appear in the dispersion curves, displaying a phononic band structure in both cases.     

Lamb waves in phononic crystal slabs: truncated plane parallels to the axis of periodicity

A theoretical study is presented on the propagation properties of Lamb wave modes in phononic crystal slabs consisting of a row or more of parallel square cylinders placed periodically in the host material. The surfaces of the slabs are parallel to the axis of periodicity. The dispersion curves of Lamb wave modes are calculated based on the supercell method. The finite element method is employed to calculate the band structures and the transmission power spectra, which are in good agreement with the results by the supercell method. We also have found that the dispersion curves of Lamb waves are strongly dependent on the crystal termination, which is the position of the cut plane through the square cylinders. There exist complete or incomplete (truncated) layers of square cylinders with the change of the crystal termination. The influence of the crystal termination on the band gaps of Lamb wave modes is analyzed by numerical simulations. The variation of the crystal termination leads to obvious changes in the dispersion curves of the Lamb waves and the widths of the band gaps.     

界面层对层状各向异性复合结构中Lamb波的影响

建立了各向异性界面层的弹簧模型,并将其应用于分析层状复合媒质的全局矩阵技术中.其 引入的机理是全局矩阵的程序结构、各层矩阵的排列不至于遭受大的破坏.因此,把弹簧界 面作为一个“材料层”.该层的材料常数用劲度常数来描述,其层厚为零.把该层放在全局矩 阵适当的位置而使系统中的其他层不发生任何变化.数值示例显示了刚性联接、滑移联接、 完全脱层三种不同界面条件下双层各向异性复合结构中Lamb波的频散特征,并对刚性联接和 滑移联接时质点沿板厚方向的位移分布进行了比较. 关键词： 各向异性界面 弹簧模型 全局矩阵 兰姆波频散     

Guided waves propagating in sandwich structures made of anisotropic,viscoelastic, composite materials

The propagation of Lamb-like waves in sandwich plates made of anisotropic and viscoelastic material layers is studied. A semi-analytical model is described and used for predicting the dispersion curves (phase velocity, energy velocity, and complex wave-number) and the through-thickness distribution fields (displacement, stress, and energy flow). Guided modes propagating along a test-sandwich plate are shown to be quite different than classical Lamb modes, because this structure does not have the mirror symmetry, contrary to most of composite material plates. Moreover, the viscoelastic material properties imply complex roots of the dispersion equation to be found that lead to connections between some of the dispersion curves, meaning that some of the modes get coupled together. Gradual variation from zero to nominal values of the imaginary parts of the viscoelastic moduli shows that the mode coupling depends on the level of material viscoelasticity, except for one particular case where this phenomenon exists whether the medium is viscoelastic or not. The model is used to quantify the sensitivity of both the dispersion curves and the through-thickness mode shapes to the level of material viscoelasticity, and to physically explain the mode-coupling phenomenon. Finite element software is also used to confirm results obtained for the purely elastic structure. Finally, experiments are made using ultrasonic, air-coupled transducers for generating and detecting guided modes in the test-sandwich structure. The mode-coupling phenomenon is then confirmed, and the potential of the air-coupled system for developing single-sided, contactless, NDT applications of such structures is discussed.     

Effect of the surface free energy on the behaviour of surface and guided waves

The aim of this paper is to evaluate the influence of the surface free energy upon the propagation of the eigenmodes of structures, by studying successively (a) the Rayleigh wave for an elastic half-space, (b) the Lamb waves for an elastic layer, and (c) the guided modes for a tri-layer structure (e.g., metal/adhesive/metal). The surface free energy is a parameter which appears in the jump conditions of stresses and displacements at each interface, and which consequently modifies the eigenmodes, solutions of the boundary conditions system. As expected, the Rayleigh wave is dispersive and its velocity increases when the surface free energy increases. In the same way, the velocity of Lamb waves also increases except at normal angle of propagation where the surface free energy does not arise. Moreover, near the Rayleigh angle, the behaviour of the A₀ and S₀ Lamb modes varies strongly according to the surface free energy. Similar results are observed for the tri-layer structure.     

Band structures of bilayer graphene superlattices

We formulate a low energy effective Hamiltonian to study superlattices in bilayer graphene (BLG) using a minimal model which supports quadratic band touching points. We show that a one dimensional (1D) periodic modulation of the chemical potential or the electric field perpendicular to the layers leads to the generation of zero-energy anisotropic massless Dirac fermions and finite energy Dirac points with tunable velocities. The electric field superlattice maps onto a coupled chain model comprised of "topological" edge modes. 2D superlattice modulations are shown to lead to gaps on the mini-Brillouin zone boundary but do not, for certain symmetries, gap out the quadratic band touching point. Such potential variations, induced by impurities and rippling in biased BLG, could lead to subgap modes which are argued to be relevant to understanding transport measurements.     

Excitations of thermoelastic waves in plates by a pulsed laser

The method of the eigenfunction expansion, also known as the expansion in normal modes, is employed to study numerically the axisymmetric excitation of the thermoelastic waves in plates by a pulsed laser. This method gives a systematic treatment and allows one to investigate not only the quasistatic and dynamic thermoelastic responses of pulsed photothermal deformation on the time scale of 1 s, but also the thermoelastic generation of longitudinal, transverse, and surface acoustic waves in thick materials, as well as the excitations of the Rayleigh-Lamb wave modes in thin plates. The formalism is particularly suitable for waveform analyses of the excitations of transient Lamb waves in thin plates because one needs only to calculate the contributions of several lower eigenmodes. The numerical technique provides a quantitative tool for the experimental determination of material properties, especially the mechanical and elastic properties of free-standing films and thicker sheet materials by thermoelastic detection.     

The analysis and interpretation of some special properties of higher order symmetric lamb waves: the case for plates

Results derived from exact linear homogeneous elastodynamic theory are used for two-dimensional unloaded plates in order to understand certain features generated by proper symmetric Lamb modes. It is shown that S1 modes for all elastic materials have a phase velocity defined below the usual critical frequency and which initially exhibits anomalous dispersion (has a negative slope with respect to frequency). Over a certain range, it has a phase velocity that is double valued. In addition, there are an infinite number of proper symmetric Lamb modes that have this characteristic for materials with a Poisson ratio equal to 1/3. It also appears that all A3n modes are anomalous when V(L) < or = 2 V(T). The cause and implication of these effects are examined, including an associated negative group velocity over a small frequency zone for these modes. Further, it is noted that all proper symmetric Lamb modes have a plateau region in phase velocity with respect to wave number. It is shown that this always occurs for a phase velocity corresponding to the longitudinal bulk velocity of the elastic material. These issues are examined along with how one may obtain material parameters and possibly plate thickness from their dispersion curves.     

The effect of surface stress on the propagation of Lamb waves

This work investigates the possibility of the propagation of Lamb waves in thin solid layers with external traction free surfaces, in the presence of surface elasticity, inertia and residual stress. It is demonstrated that such waves do exist and that their characteristics can be quite different from their classical counterparts. The governing equations with non-classical boundary conditions involving the bulk and surface stress are solved exactly in the frequency-wavenumber domain. This solution is utilized to compute the Lamb wave modes for different layer thicknesses. An efficient strategy to capture all the modes of Lamb waves within a given frequency window is outlined. It is shown that the effect of surface elasticity and inertia becomes significant with increasing frequency and decreasing layer thickness, where the number of modes participating within a given frequency window is more than that permitted by the classical theory. Further, it is observed that the nature of the Lamb wave modes (in terms of negative dispersion) in the presence of surface stress is similar to what predicted by the nonlocal theory and microstructure based continuum theory.     

Influence of adhesive layer properties on laser-generated ultrasonic waves in thin bonded plates

This paper studies quantitatively the generation of Lamb waves in thin bonded plates subjected to laser illumination, after considering the viscoelasticity of the adhesive layer. The displacements of such plates have been calculated in the frequency domain by using the finite element method, and the time domain response has been reconstructed by applying an inverse fast Fourier transform. Numerical results are presented showing the normal surface displacement for several configurations: a single aluminum plate, a three-layer bonded plate, and a two-layer plate. The characteristics of the laser-generated Lamb waves for each particular case have been investigated. In addition, the sensitivity of the transient responses to variations of material properties (elastic modulus, viscoelastic modulus, and thickness) of the adhesive layer has been studied in detail.     

Time-frequency analysis of the dispersion of Lamb modes.

Accurate knowledge of the velocity dispersion of Lamb modes is important for ultrasonic nondestructive evaluation methods used in detecting and locating flaws in thin plates and in determining their elastic stiffness coefficients. Lamb mode dispersion is also important in the acoustic emission technique for accurately triangulating the location of emissions in thin plates. In this research, the ability to characterize Lamb mode dispersion through a time-frequency analysis (the pseudo-Wigner-Ville distribution) was demonstrated. A major advantage of time-frequency methods is the ability to analyze acoustic signals containing multiple propagation modes, which overlap and superimpose in the time domain signal. By combining time-frequency analysis with a broadband acoustic excitation source, the dispersion of multiple Lamb modes over a wide frequency range can be determined from as little as a single measurement. In addition, the technique provides a direct measurement of the group velocity dispersion. The technique was first demonstrated in the analysis of a simulated waveform in an aluminum plate in which the Lamb mode dispersion was well known. Portions of the dispersion curves of the A0, A1, S0, and S2 Lamb modes were obtained from this one waveform. The technique was also applied for the analysis of experimental waveforms from a unidirectional graphite/epoxy composite plate. Measurements were made both along and perpendicular to the fiber direction. In this case, the signals contained only the lowest order symmetric and antisymmetric modes. A least squares fit of the results from several source to detector distances was used. Theoretical dispersion curves were calculated and are shown to be in good agreement with experimental results.     

Interaction of interface plasmons and longitudinal optical phonons in a multilayer structure

The interaction between two-dimensional interface plasmons and longitudinal optical phonons in multilayer structures is analysed in this communication. The dispersion relations for the mixed modes are obtained for periodic boundary conditions in the direction normal to the layer planes. The energy gap between the two bands of mixed modes can be seen in the density of states ?_k(ω) at fixed k parallel to the layers. ?_k(ω) also exhibits singularities at the band edges characteristics of the one-dimensional periodic array of parallel layers.     

Lamb wave dispersion in a PZT/metal/PZT sandwich plate with imperfect interface

Abstract

The Lamb wave dispersion in a PZT/Metal/PZT sandwich plate is investigated by employing the exact linear equations of electro-elastic waves in piezoelectric materials within the scope of the plane-strain state. It is assumed that at the interfaces between the piezoelectric face layers and metal core layer, shear-spring and normal-spring type imperfect conditions are satisfied. The degree of this imperfectness is estimated through the corresponding shear-spring and normal-spring type parameters which appear in the contact condition characterizing the transverse and normal displacements’ discontinuity. The corresponding dispersion equation is derived, and as a result of the numerical solution to this equation, the dispersion curves are constructed for the first and second lowest modes in the cases where the material of the face layers is PZT and the material of the middle layer is Steel (St). Consequently, for the PZT/St/PZT sandwich plate, the study of the influence of the problem parameters such as the piezoelectric and dielectric constants, layer thickness ratios, non-dimensional shear-spring, and normal-spring type parameters, is carried out. In particular, it is established that the imperfectness of the contact between the layers of the plate causes a decrease in the values of the wave propagation velocity.     

Symmetric and Anti-Symmetric Lamb Waves in a Two-Dimensional Phononic Crystal Plate

It is weft known that Lamb waves in a plate with a mirror plane can be separated into two uncoupled sets： symmetric and anti-symmetric modes. Based on this property, we present a revised plane wave expansion method （PWE） to calculate the band structure of a phononie crystal （PC） plate with a mirror plane. The developed PWE method can be used to calculate the band structure of symmetric and anti-symmetric modes separately, by which the depending relationship between the partial acoustic band gap （PABG）, which belongs to the symmetric and anti-symmetric modes alternatively, and the position of the scatterers can be determined. As an example of its application, the band structure of the Lamb modes in a two-dimensional PC plate with two layers of void circular inclusions is investigated. The results show that the band structure for the symmetric and anti-symmetric modes can be changed by the position of the scatterers drastically, and larger PABGs will be opened when the scatterers are inserted into the area of the plate, where the elastic potential energy is concentrated.     

Prolonged acousto-optic interaction with Lamb waves in crystalline plates

The propagation and acousto-optic interaction of Lamb modes in an anisotropic plate of tellurium dioxide (TeO2) are studied numerically and analytically. In the case of a Y-cut X-propagating TeO2 plate, the very high elastic anisotropy of the crystal greatly modifies the dispersion curves, giving rise to their multiple oscillations. The existence ranges of backward Lamb modes increase with the mode order contrary to the case of isotropic plates. The quasi-collinear light scattering by Lamb waves is considered. Owing to the structure of Lamb wave field, a simultaneous light diffraction at two different optical frequencies can take place while Lamb waves are excited only at the single frequency. It is demonstrated with the Z-cut (110)-propagating plate that a small change in the acoustic frequency can result in a significant shift in the frequency of the scattered light.     

Numerical stable determination of Floquet-modes and the application to the computation of band structures

Periodic structures like Bragg-gratings are important components of optical circuits. The analysis of these devices can be done very efficiently by combining eigenmode propagation methods with Floquet's theorem. A particular problem is the determination of the Floquet modes. Transfer matrix formulas are stable only in case of low losses and when the length of the periods is not too big. A stable method, also in the mentioned cases, is presented in this paper. Reflection coefficients are transformed from the output of a periodic segment to its input and the fields are computed in opposite direction. By this, the exponential increasing terms, which lead to the numerical problems are avoided. The formulas are applied to determine eigenmodes in various waveguide structures. Particular periodic structures are photonic crystals (PC), who have very promising features. For tailoring these PCs the knowledge of the band structure is required. With the Floquet modes that have been determined before this band structure has been calculated. A comparison with the literature showed a very good agreement.     

Instability of a charged layer of a viscous liquid on the surface of a solid spherical core

The dispersion relation for the spectrum of capillary waves of a spherical layer of a viscous liquid coating a solid spherical core with a layer of finite thickness is introduced and analyzed. It is shown that the existence of two mechanisms for the viscous dissipation of the energy of the capillary-wave motions of the liquid, viz., damping in the bulk of the layer and on the solid core, leads to restriction of the spectrum of the realizable capillary waves of the liquid on both the high-and low-mode sides. At a fixed value of the system charge which is supercritical for the first several capillary modes, the maximum growth rates in the case of a small solid core are possessed by modes from the middle of the band of unstable modes, while in thin liquid layers the highest of the unstable modes have the largest growth rates. This points out differences in the realization of the instability of the charged surface of the spherical layer for small and large relative sizes of the solid core. Zh. Tekh. Fiz. 67, 8–13 (September 1997)     

面向模分复用的偏振保持领结型椭圆芯少模光纤的设计

提出了一种弱耦合领结型椭圆芯应力保偏少模光纤(PM-FMF),通过使用高折射率纤芯,所提出的光纤可在1505～1585 nm波段下,支持32个独立的本征模式。椭圆纤芯和领结型应力区的引入,有效地分离了相邻的本征模式。采用有限元法对领结型椭圆芯应力PM-FMF的纤芯及领结型应力区的结构参数进行优化。评估了光纤参数对模式数量、模式间的最小有效折射率差、模态双折射、应力双折射以及弯曲损耗的影响。此外还分析了该光纤的带宽性能,包括模式间的有效折射率、有效折射率差、差分模式时延(DMD)。经数据分析,在1505～1585 nm波段下,该光纤支持的32个本征模式是完全分离的,相邻模式之间的最小有效折射率差大于1.295×10?4。所提出的弱耦合保偏少模光纤能够提高传输容量,在本征模式复用传输中具有潜在的应用前景。     

Band gap structures in underwater screens of periodically spaced porous plates

Acoustic properties of different periodic structures composed of alternating fluid and fluid-saturated porous layers obeying Biot’s theory are investigated. At first, the network of modes and the transmission coefficients of finite structures of six plates are studied in the frequency-angle of incidence plane. It is shown that the network of modes concentrates in localized domains of the plane where the transmission coefficients will take the greatest values. With this minimum of six plates, the structures exhibit the main features as for structures containing more plates, especially those with an infinite number of plates. Then, considering infinite structures the band gap calculations are led using the Bloch–Floquet theorem. The evanescent and propagative zones in the frequency-angle of incidence plane are determined. What is proposed here is a class of underwater porous screens that exhibits band gaps extending over great angular domains and enlarging in the frequency domain when the pores at the interfaces of the porous plates are sealed. The effect of porosity on the band gaps is also investigated.