Normal modes of a poroelastic plate and their relation to the reflection and transmission coefficients

A poroelastic plate that obeys the Biot theory is considered. Compact new forms of its reflection and transmission coefficients, similar to those of the resonance scattering theory for an elastic plate, are derived. A numerical comparison of the reflection coefficient modulus with the plate normal modes, at low frequency, shows that a study of the reflection or transmission coefficient does not provide the same kind of information on the poroelastic plate than an investigation of guided leaky waves propagation.     

Laser ultrasonic analysis of normal modes generated by a voltage pulse on an AT quartz sensor

Laser ultrasonic detection is a versatile and highly sensitive tool for the observation of surface waves. In the following study, laser ultrasonic detection is used for the experimental study of spurious normal vibration modes of a disk quartz sensor excited by a voltage pulse. The AT cut crystal (cut of the crystal relative to the the main crystallographic axis is 35.25 degrees) is optimal for generating mainly thickness-shear vibrations (central frequency 6 MHz) on the quartz surface. However, resulting from shear-to-longitudinal and shear-to-surface mode conversion, and from the weak coupling with the other crystallographic axes, other modes (thickness-compressional and bending modes) are always present in the plate response. Since the laser vibrometer is sensitive to normal displacements, the laser investigation shows waves that can be considered as unwanted for the AT quartz used as a shear sensor. The scanned three dimensional (3D) amplitude-space-time signals are carefully analysed using their representation in three dual Fourier domains (space-time, wave number-frequency). Results on the transient analysis of the waves, the normal bending modes and the dispersion curves are shown.     

Normal mode expansion method for laser-generated ultrasonic Lamb waves in orthotropic thin plates

A quantitative theory for modeling the laser-generated transient ultrasonic Lamb waves, which propagates along arbitrary directions in orthotropic plates, is presented by employing an expansion method of generalized Lamb wave modes. The displacement field is expressed by a summation of the symmetric and antisymmetric modes in the surface stress-free orthotropic plate, and therefore the theory is particularly appropriate for waveform analyses of Lamb waves in thin plates because one needs only to evaluate several lower modes. The transient waveforms excited by the thermoelastic expansion and the oil-coating evaporation are analyzed for a transversely isotropic thin plate. The results show that the theory provides a quantitative analysis to characterize anisotropic elastic stiffness properties of orthotropic plates by laser-generated Lamb wave detection.     

Laser-ultrasonic generation of Lamb waves in the reaction force range

The Laser-ultrasonic generation of Lamb waves in an elastic plate is investigated theoretically and experimentally for a laser source whose intensity is high enough to create reaction forces (normal tractions) on the illuminated surface of the specimen. The analytical solutions for transient waves are derived using the integral transform method first by considering an arbitrary source shape and time excitation function, and then specifically for circular and line source shapes. The simulation study allows us not only to predict the behavior of individual wave modes but also to construct the overall responses; thus it helps us better understand the wave excitation mechanisms. The dispersive and multi-modal nature of laser-generated Lamb waves is presented by showing the spatiotemporal Fourier transform of displacements obtained by the simulation study. The transform, displayed in the frequency-wave number domain, enunciates the characteristics of the propagating individual Lamb wave modes. The simulation results are then compared with the 2-D Fourier transform of a set of experimental data obtained by scanning an aluminum plate specimen.     

Ultrasonic waves propagation in absorbing thin plates application to paper characterization

Guided wave theory is applied to a thin orthotropic and absorbing plate for low frequency propagation of ultrasonic waves. The aim of this paper is to give some physical interpretations of the non-destructive characterization of paper materials, which are cellulosic fibrous networks. It is shown that the propagation problem reduces to two normal modes of propagation in the plane of the plate. Each of them depends on four complex and independent stiffnesses that are combinations of elementary complex stiffnesses of the media. The imaginary part of these stiffnesses corresponds to a possible mechanism of energy dissipation during the wave propagation for this kind of material. The reverse problem, which gives four complex values, is then numerically solved using a small attenuation assumption. The specially designed experimental set-up has led to the first measurements of tracing paper damping factors. The phase velocity measurements of the plate waves agree with the results already found by several paper researchers. As a particular and new result, the shear wave velocities are found to present a quasi-isotropic repartition in the plane of the paper sheet. It was found that the absorbing phenomenon can occur for each propagation mode in such a material. The attenuation values are small, except for one of them that corresponds to a coupling term in the propagation model. The anisotropy of their repartition is also shown in the case of quasi-longitudinal waves.     

A two-dimensional analysis of surface acoustic waves in finite elastic plates with eigensolutions

It is generally known that surface acoustic waves, or Rayleigh waves, have different mode shapes in infinite plates. To be precise, there are both exponentially decaying and growing components in plates appearing in pairs, representing symmetric and antisymmentric modes in a plate. As the plate thickness increases, the combined modes will approach the Rayleigh mode in a semi-infinite solid, exhibiting surface acoustic wave deformation and velocity. In this study, the two-dimensional theory for surface acoustic waves in finite plates is extended to include the exponentially growing modes in the expansion function. With these extra equations, we study the surface acoustic waves in a plate with different thickness to examine the coupling of the exponentially decaying and growing modes. It is found that for small thickness, the two groups of waves are strongly coupled, showing the significance of including the effect of thickness in analysis. As the thickness increases to certain values, such as more than five wavelengths, the exponentially decaying modes alone will be able to predict vibrations of surface acoustic wave modes accurately, thus simplifying the equations and solutions significantly. Supported by Qianjiang River Fund established by Zhejiang Provincial Government and Ningbo University and administered by Ningbo University and the National Natural Science Foundation of China (Grant No. 10572065)     

ANALYSIS OF VIBRATIONS AND ENERGY FLOWS IN SANDWICH PLATES BEARING CONCENTRATED MASSES AND SPRING-LIKE INCLUSIONS IN HEAVY FLUID-LOADING CONDITIONS

Vibrations of and the energy propagation in an infinitely long fluid-loaded sandwich beam (a plate of the sandwich composition in one-dimensional cylindrical bending) bearing concentrated masses and supported by springs are described in the framework of the sixth order theory of multilayered plates coupled with the standard theory of linear acoustics. A sandwich plate is loaded by a layer of a compressible fluid which is bounded opposite to a plate side by a rigid baffle. The dispersion equation for a fluid-loaded sandwich plate is derived. The wave numbers (complex, pure real and pure imaginary) and relevant normal modes (both the travelling and the evanescent ones) are obtained. Their dependence on the parameter of a fluid's depth is studied. Then the Green matrix is constructed analytically as a linear combination of normal modes to describe the response of a plate and an acoustic medium to the point loading by a force or a moment. Continuity conditions at the loaded cross-section of a plate and in a fluid are formulated. Attention is focused at the selection of roots of the dispersion relation for the formulation of the continuity condition for a fluid at the loaded cross-section. The convergence rate of an approximate solution based on the modal composition of the Green matrix is estimated. The parametric study of the “structural” and the “fluid” energy flows in a fluid-loaded sandwich plate without inclusions is performed for various excitation conditions. Then the Green matrix method is applied to analyze the influence of a pair of identical inclusions on localization of vibrations (modal trapping) and energy flows. Conditions of localization of flexural waves at these inhomogeneities are explored.     

Hopping transport in two-dimensional systems with spin-orbit interaction in external magnetic field

The theory of spin rotation waves (SRWs), representing excitations of a new type arising in twodimensional systems with spin-orbit interaction in an external electric field, has been developed. These intrinsic modes correspond to rotation of the magnetic moment vector in the plane formed by the electric field vector and the normal to the sample plate surface. An experimental method is proposed for detecting SRWs by measuring the frequency dependence of the magnetic susceptibility, which exhibits a resonance at the intrinsic mode frequency. A particular calculation is performed for a hopping conductivity model (for small-size polarons), but it is likely that intrinsic oscillations of the SRW type also take place for the band transport, since their appearance is related to the symmetry of the system.     

Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation

In this article, the propagation of guided thermoelastic waves in the circumferential direction of orthotropic cylindrical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green-Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic cylindrical plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions are shown to discuss the differences between the elastic modes and thermal modes. A thermoelastic cylindrical plate with a different ratio of radius to thickness is considered to discuss the influence of the ratio on the characteristics of circumferential thermoelastic waves.     

Research on the propagation of guided waves in multi-layered plates

The dispersion characteristics and excitation mechanisms of the guided waves in multilayered plates are studied in this paper. Firstly, the dispersion equation is obtained by the propagator matrix method. Then, the bisection technique is employed to find all the roots of the dispersion equation. The dispersion characteristics of the guided waves are investigated and analyzed. For the multilayered plates in which the S-wave velocity increases or decreases from top to bottom, it is found that the velocity limits in high frequency of the first and high modes are equal to the Rayleigh wave and S-wave velocities of the low-velocity layer, respectively. It is also found that the velocity limits in the high frequency of all modes are equal to the S-wave velocity of the low-velocity layer for the plate with a low-velocity middle layer. The normal displacement spectrum of all the modes excited by the normal force source with a definite width on the surface of the plate is also investigated. It is proved that the dominant mode is the first mode when the S-wave velocity increases from top to bottom layer and the dominant mode is different in different frequency range for the plate with a low-velocity middle layer.     

Generalized thermoelastic Lamb waves in a plate bordered with layers of inviscid liquid

The propagation of thermoelastic waves in a homogeneous isotropic, thermally conducting plate bordered with layers of inviscid liquid or half-space of inviscid liquid on both sides is investigated in the context of generalized theories of thermoelasticity. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases. The different regions of secular equations are obtained and special cases, such as Lame modes, thin plate waves and short wavelength waves of the secular equations are also discussed. The secular equations for thermoelastic leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for an aluminum-epoxy composite and aluminum materials plate bordered with water. The dispersion curves for symmetric and antisymmetric thermoelastic wave modes and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement. The results have been deduced and compared with the relevant publications available in the literature at the relevant stages of the work.     

Influence of the slow wave on the relation between the angular resonances and the leaky guided modes properties for a poroelastic plate embedded in water

A numerical study of the guided modes in a water-saturated poroelastic plate that obeys the Biot theory is presented. In the first part, we study the leaky guided modes and the angular resonances when the slow wave does not propagate. Two types of guided modes exist. The first ones occur from coupling of the fast longitudinal wave with the shear wave; most of them propagate whatever the frequency is, provided that it is not close to their cut-off frequencies. The leaky guided modes of the second type occur from coupling of the two longitudinal waves and the shear wave. These modes do not propagate (they are highly damped) as long as the slow wave remains diffusive. We show that the characteristics of the angular resonances can be linked to the leaky guided waves of the first type in the same way as for an elastic plate. The guided modes of the second type may not be associated to angular resonances. In the second part, we consider a thinner plate in a higher frequency range so that the slow wave can propagate. Once again its influence is studied both on the leaky guided modes and on the angular resonances.     

分层固体板中导波的激发与频散特性

针对无限大弹性分层固体板,研究了结构中导波的频散和激发特性。首先使用传递矩阵法推导分层板模型中导波的频散方程,然后用二分法求取导波各模式的频散曲线,进而分析结构中导波的频散特性。结果表明:在速度递增或递减的分层板中,基阶模式和高阶模式的高频极限分别等于低速层的瑞利波速和横波波速。对于含低速夹层的分层板,所有模式的高频极限都等于低速层的横波速度。在导波激发特性方面,研究了在具有一定宽度的法向力源作用下的分层板中导波各模式在结构中的法向位移谱。发现在速度递增的分层板结构中基阶模式是主导模式,而对于速度递减和含低速夹层模型,主导模式在不同的频段范围内对应不同的导波模式。      

光纤中的电磁对偶变换与导波的模式分析

讨论了麦克斯韦方程在各向同性均匀电介质中的电磁对偶变换不变性,定义了电磁混合比,求出了对偶场的充要条件.把上述不变性和充要条件应用于阶跃光纤,分析其导波模式,证明在非轴对称情况下,阶跃光纤中不存在具有色散的受导简正模.存在的解只有临界折射模(n=n₂时的特解).求出了它的分立频谱.受导简正模只存在于轴对称情况中.以上结论与实验事实不矛盾,并由此解释了基模的特性 关键词： 光纤光学 光波导 电磁对偶变换 受导波     

Maxima and minima of the displacement components for the Lamb modes

This paper revisits the vanishing of the transverse component of the particle displacement vector in free surfaces of an isotropic homogeneous plate, for both symmetric and antisymmetric Lamb waves. Drawing on well-known analytical expressions from Viktorov's book [(1967) Rayleigh and Lamb Waves: Physical Theory Applications, Chap. II, pp. 67-121], two distinct frequency-thickness product expressions, in cases where this vanishing occurs, are derived: one for the symmetric modes and another for the antisymmetric modes. At these frequency-thickness products, phase and group velocities have appreciable values which are discussed herein. It appears that these velocities depend on the transverse bulk wave velocity only. This is the specific condition of the Lame? modes. Moreover, theoretical and experimental investigations of displacements in the surface of a plate in air have been carried out. The theoretical part shows that the normal and transverse displacements have, respectively, a local maximum and a local minimum in the vicinity of these frequency-thickness products. The experimental part corroborates the presence of the local maximum of the S(0) Lamb mode for various materials.     

Lamb mode reflections at the end of a plate loaded by a viscoelastic material

This paper deals with a study of the conversion of a Lamb wave at the loaded edge of a plate. An experimental study is performed to know the interaction of surface waves with a viscoelastic material. A Lamb wave is generated by a wedge. At the end of the plate, this wave gives rise to several reflected waves. A first study makes it possible qualitatively to know the Lamb modes present in reflection. Secondly, conversion phenomena are studied to know if surface waves are sensitive to evolution in the time of the viscoelastic material structure. The selected material is a sol-gel containing silica.     

Guided waves in a plate with linearly varying thickness: experimental and numerical results

The aim of this work is to give experimental and numerical results on the behaviour of guided waves that propagate downslope in a free elastic plate with slowly linearly varying thickness. We show experimentally the propagation of adiabatic modes, which are guided waves that adapt to the varying thickness of the plate. As the thickness is decreasing, a given guided wave will reach its thickness cut-off. When this happens, we show that two phenomena occur: the reflection of this wave and its propagation backward in the plate, its conversion into a different guided wave which goes on propagating downslope in the plate. The numerical study is done with the software Ansys, based on the finite element method. The results obtained confirm the experimental ones.     

Reflection of normal acoustic waves in thin plates from a grating with a periodically distributed mechanical load

Reflection of zero-order normal acoustic waves excited in a thin piezoelectric plate from a set of conducting strips of a finite thickness is studied both theoretically and experimentally. The analysis shows that the effects produced by the short-circuiting of the plate surface and by the elastic load on the impedance ratio of adjacent plate segments are in opposition to each other. These effects can be commensurable, and, hence, for each wave type, there is a certain value of the strip thickness at which the reflection coefficient becomes equal to zero. The experimental results obtained for a shear horizontal normal wave (an SH ₀ wave) propagating in a lithium niobate plate are in good agreement with the theory and justify the use of the equivalent-circuit model in analyzing the properties of reflectors of the type under study.     

两侧有固体层负载时板中Lamb波的传播

本文研究了薄板二面有固体导负载时板中Ｌａｍｂ波的传播，从弹性波理论出发并结合应的边界条件，导出板中Ｌａｍｂ波的色散方程，数值计算表示，不管作为自由状态时板中Ｌａｍｂ波相速（板厚取定时）是大于或小于外层固体的声表面波波速，板中对称及反对称模式的Ｌａｍｂ波相速都随着外层固体层厚度增加而变化并且渐近于外层固体的声表面波波速，数值计算又表明，对很薄的板，板中对称及反对称模式的相速均随负载板的厚度呈线性变化     

Prediction and measurement of nonpropagating Lamb modes at the free end of a plate when the fundamental antisymmetric mode A0 is incident

Reflection of Lamb waves when the fundamental mode A0 is incident at the free end of a plate is studied, in order to identify the extent to which the generation of nonpropagating modes influences the field local to the end of the plate. Semi-analytical predictions, finite element simulations, and experimental measurements are presented for frequencies below the A2 cutoff. First it is shown, for frequencies below the A1 cutoff, that reflection of the A0 mode is accompanied by a delay in phase, and that there is significant additional motion due to nonpropagating modes within about five plate thicknesses of the end. The extend of this additional motion in the vicinity of the end of the plate is demonstrated by subtracting the contribution of the propagating modes from the displacement field. The wave field at frequencies above the A1 cutoff is more complex because the A1 as well as the A0 propagating modes are present at the end of the plate. Nevertheless, it has still been possible, using semi-analytical predictions and finite element simulations, to demonstrate the additional motion due to the nonpropagating modes.