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.     

Experimental investigation of vibration power flow in thin technical orthotropic plates by the method of vibration intensity

Vibration intensity technique is used to measure vibration power transmission in thin single layer technical orthotropic plates for flexural waves. Measurement of flexural wave power is carried out in far-field conditions. All measurements are undertaken in the frequency domain using the cross-spectra of acceleration signals, facilitating the use of FFT analyzer. The two-transducer technique applicable to these plates is used for these measurements. Technical orthotropic (rectangular corrugation) plates of steel are used for the measurements. One isotropic plate of steel is also considered for comparison. Method of elastic equivalence technique is used. Both input power and vibration power transmission through the plates are estimated. Far-field power is normalized with the input power for flexural wave. Influence of flexural rigidity on vibration energy transfer is also investigated.     

Detection and localization of inclusions in plates using inversion of point actuated surface displacements

A numerical simulation is carried out demonstrating the use of plate surface vibration measurements for detecting and locating inclusions within the structure. A finite element code is used to calculate normal surface displacement for both steel and mortar plates subjected to a monochromatic point force. The data is generated for the homogeneous plate and the identical plate within which exists a small rectangular inclusion. It is observed that when the elastic modulus of the inclusion is orders of magnitude lower than the base material, resonances of the inclusion produce large local displacements that are readily observed in the raw displacement data. For more modest moduli differences, there are no such directly observable effects. In this case, three inverse algorithms are used to process the displacement data. The first two are local inversion techniques that each yield a spatial map of the elastic modulus normalized by density. These algorithms successfully detect and localize the inclusion based on its modulus difference from that of the base plate. The third technique uses a form of the inhomogeneous equation of motion to obtain the induced force distribution connected with the inclusion. The spatial mapping of this force also successfully detects and localizes the inclusion.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

Directionality of flexural intensity in orthotropic plates

This work investigates composite plates and their ability to direct flexural intensity, which has important implications for noise and vibration control. It is well known that a composite plate supports a flexural wave whose wavenumber depends strongly on its angle of propagation. This suggests that a composite plate will direct more flexural intensity in some directions than others. The present work considers a thin multi-layered plate in which each layer is constructed from an orthotropic material and has a chosen orientation relative to the other layers. Such an approach may be used to design highly directive structures. An analysis is presented in which a two-dimensional Fourier transform is analytically applied to the equation of motion, yielding algebraic expressions for displacements and stress resultants. Next, a two-dimensional discrete inverse Fourier transform is applied to compute displacements and stress resultants at discrete locations. Flexural intensity is computed at these locations.     

Flexural edge waves in semi-infinite elastic plates

This paper presents a detailed analysis of the dispersion for flexural edge waves in semi-infinite isotropic elastic plates. A solution to the dynamic equations of motion is constructed by the superposition of two partial solutions, each providing zero shear stresses at the plate faces. A dispersion equation is expressed via the determinant of an infinite system of linear algebraic equations. The system is reduced to a finite one by taking into account the asymptotic behaviour of unknown coefficients. The accuracy of the solution is confirmed by a good agreement with the available experimental data and by a proper satisfaction of the prescribed boundary conditions.A detailed analysis of dispersion properties for the edge wave and corresponding displacements at various frequencies is carried out. In addition to the well-known results it is shown that the plate height does not influence the existence of the edge wave at high frequencies and, as the frequency increases, the phase velocity of the edge wave in a semi-infinite plate asymptotically approaches the velocity of an edge wave in a right-angled wedge. The performed analysis allows evaluating the plate theories such as the Kirchhoff theory or other refined plate theories developed for modeling edge waves in semi-infinite elastic plates at low frequencies.     

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.     

A new orthogonality relationship for orthotropic thin plate theory and its variational principle

While Hamiltonian system was led to solution of elastic theory a symplectic system-atic methodology for theory of elasticity was established and a symplectic orthogonality relationship was presented[1,2]. For two-dimensional theory of elasticity a new dual vec-tor and a new dual differential matrix were presented by putting the old dual vector[1] in a new order. It was discovered for isotropic materials that the symplectic orthogonality relationship may be decomposed into two independent and s…     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

Localized bending waves in a transversely isotropic plate

The problem of bending waves localized near the free edge of a transversely isotropic plate is investigated using the Ambartsumian higher-order plate theory which takes account of the transverse shears generated by flexural deformation. Unlike the first-order Reissner-Mindlin theory, which also takes account of transverse shears, Ambartsumian's analysis does not demand that plane normal cross-sections remain plane during bending. Within this analysis the existence of localized bending waves in transversely isotropic plates is established, and solutions of the dispersion equation obtained for different values of the elastic parameters.The analysis of frequencies of localized bending waves shows that for thick plates the effect of anisotropy can be considerable. For the particular case of vibrations of a narrow plate, from the long wave approximation a new beam vibration equation of the Timoshenko type is obtained for a transversally isotropic plate.     

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.     

Two-dimensional analysis of the effect of an electrode layer on surface acoustic waves in a finite anisotropic plate

The effect of a metal layer over an elastic substrate on surface acoustic wave propagating in the structure can be evaluated precisely for semi-infinite solids and infinite plates, but there is no accurate analytical solution if the finite size of the plate has to be considered. By expanding displacements with eigensolutions of surface acoustic waves in a semi-inifite solid, a set of two-dimensional equations similar to the Mindlin plate theory are obtained. Then for a thin electrode layer, the effect is considered through the approximation of displacements in the metal layer with the ones in the substrate, and an integration over the thickness incorporated the properties of the metal layer into equations through the modification of material properties with the decaying indices of surface acoustic waves and the thickness of the metal layer. Using AT-cut quartz crystal as the substrate, we present the effect of silver electrode layers of finite thickness on the phase velocity of propagating surface acoustic waves.     

Analysis of stress field in a single anisotropic diamond/square shaped inclusion using the volume integral equation method and the numerical equivalent inclusion method

A volume integral equation method (VIEM) is used to study elastostatic problems in an unbounded elastic solid containing a single diamond/square shaped inclusion subject to uniform tensile stress at infinity. The inclusion is assumed to be a long parallel diamond/square cylinder composed of isotropic or anisotropic elastic materials and perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinder. A detailed analysis of the stress field at the interface between the isotropic matrix and the single isotropic/orthotropic diamond/square shaped inclusion is carried out. The effects of a single isotropic/orthotropic diamond/square shaped inclusion on the stress field at the interface between the matrix and the inclusion are investigated in detail. The accuracy of the volume integral equation method for the interfacial stress field is validated and compared by the numerical equivalent inclusion method (NEIM) and the finite element method (FEM) using ADINA. Through detailed analysis of plane elastostatic problems using the parallel volume integral equation method (PVIEM) in an unbounded isotropic matrix with multiple isotropic diamond shaped inclusions under uniform remote tensile loading, it is demonstrated that the volume integral equation method can also be applied to solve general two- and three-dimensional elastostatic problems involving multiple isotropic/anisotropic inclusions whose shape and number are arbitrary.     

Non-linear flexural waves in thin layers

Non-linear flexural waves in thin plates or layers have been analyzed in this paper. The equation of motion of the plate is derived assuming that the motion is antisymmetric about the mid-plane of the plate and that the plate is thin. The plate is considered to be elastic. The Von Karman non-linear strains and Landau elastic constants have been used to model geometric and material non-linearities, respectively. An asymptotic analysis of wave motion is presented using the method of multiple scales. Evolution equations are derived for small amplitude traveling flexural elastic waves. Numerical results show waveform distortion, amplitude amplification, and harmonic generation.     

Rayleigh surface waves in a plate

In the article the analysis of a monochromatic elastic surface wave in an unlimited plate of limited thickness whose opposite surfaces are planar and mutually parallel, made of an elastic isotropic material of a constant density is described. Equations are calculated for the wave number calculation — speed of surface wave propagation, and an equation for the calculation of the vector components deformation (component trajectory) of the elastic medium.     

Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations

A general model of dynamic bending of isotropic micropolar elastic thin plates with independent fields of displacements and rotations is presented. The model has been justified asymptotically based on the solutions for special cases subject to simplifying assumptions. The model incorporates transverse shear deformations. Neglecting transverse shear, a model of the dynamics of micropolar elastic thin plates is also constructed. Then, we study free and forced oscillations and derive the natural frequencies, the amplitudes of the forced oscillations and the resonance conditions for micropolar elastic hinge-supported rectangular and circular plates. Finally, the basic characteristic features of micropolar plates are numerically analysed for different values of various elastic and inertial constants of the micropolar material.     

Experimental study of the A0 and S0 Lamb waves interaction with symmetrical notches

The aim of this work is to study the fundamental Lamb modes interaction with defects in isotropic plates. For these experimental investigations, symmetrical notches with various depths milled in aluminum plates are considered. Moreover, the incident Lamb wave of a specific mode is generated by means of two identical thin piezoceramic transducers placed at the opposite sides of the plate. The waves scattered by the notch are recorded with conventional transducers located on the plate surface in front and behind the defect. The selection of the A₀ or the S₀ modes is obtained by exciting the transducers with anti-phased or in-phased signals, respectively. Furthermore, a calibration process is investigated to correct errors caused by the presence of the receiver between the emitters and the defects. The power reflection and transmission coefficients are then obtained and the power balance is verified. Finally, these measurements are compared successfully with those obtained by a numerical method using the finite-element modeling described in a previous work.     

The flexural vibration of rectangular plates with point supports

Orthogonally generated polynomial functions are used in the Lagrangian multiplier method to study the free, flexural vibration problem of point supported, thin, flat, rectangular plates. The analysis applies to isotropic and specially orthotropic plates having any combination of clamped, simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges. Numerical results are presented for a number of specific problems, illustrating the accuracy and versatility of the approach, and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

基于模式分离的兰姆波逆时偏移成像

应用频率域逆时偏移方法实现各向同性和各向异性板中缺陷的兰姆波成像.由于缺陷引起的多模态散射信号会在重建图像中形成伪像,根据基本导波模式振动对称性的差别进行了模式分离预处理.基于多元阵列超声技术,开展了铝板和复合板内缺陷频率域逆时偏移超声成像方法的数值仿真研究.首先,建立有限元模型,采用环形传感器数值采集由缺陷引起的兰姆波散射信号,然后,将采集到的多模式散射信号进行模式分离处理,再将模式分离后的兰姆波散射信号经时间反转后并在相应的接收器处重新激励,在频域中运用格林函数反向传播兰姆波散射信号,获取监测区域的声场信息,与正向传播声场进行互相关,重建缺陷图像.首先对铝板中单缺陷以及复合材料板中相邻的两个相同缺陷进行数值仿真,对比有无模式分离处理的缺陷逆时偏移成像效果,体现出模式分离的重要性.在此基础上,采用逆时偏移方法对复合板材内位置邻近、深度不同的双缺陷进行识别.数值结果表明,模式分离预处理后的缺陷重建图像能够有效去除多模式干扰产生的伪像.文中提出的成像方法对各向同性板和各向异性板内缺陷的检测和成像具有很好的发展潜力,可以准确地探测多个缺陷的形状、尺寸和深度.