Formulas for the H/V ratio (ellipticity) of Rayleigh waves in orthotropic elastic half-spaces

ABSTRACT

This paper concerns the propagation of Rayleigh waves in compressible and incompressible orthotropic elastic half-spaces. The main aim of the paper is to derive formulas for the H/V ratio (ellipticity) of Rayleigh waves. First, the expression of the H/V ratio in terms of the Rayleigh wave velocity for compressible orthotropic elastic half-spaces is derived using the surface impedance matrix. From this expression and the secular equation of Rayleigh waves, the equation for the H/V ratio is obtained. It is a cubic equation. Employing the theory of cubic equations, the formula for the H/V ratio for compressible orthotropic elastic half-spaces has been derived. The H/V ratio formula for incompressible orthotropic elastic half-spaces has been obtained using the incompressible limit method. Since the H/V ratio of Rayleigh waves is a good tool for nondestructively evaluating mechanical properties of structures before and during loading, the obtained formulas will be very significant in practical applications.     

On Rayleigh waves in incompressible orthotropic elastic solids

In this paper the secular equation for the Rayleigh wave speed in an incompressible orthotropic elastic solid is obtained in a form that does not admit spurious solutions. It is then shown that inequalities on the material constants that ensure positive definiteness of the strain-energy function guarantee existence and uniqueness of the Rayleigh wave speed. Finally, an explicit formula for the Rayleigh wave speed is obtained.     

The explicit secular equation for surface acoustic waves in monoclinic elastic crystals

The secular equation for surface acoustic waves propagating on a monoclinic elastic half-space is derived in a direct manner, using the method of first integrals. Although the motion is at first assumed to correspond to generalized plane strain, the analysis shows that only two components of the mechanical displacement and of the tractions on planes parallel to the free surface are nonzero. Using the Stroh formalism, a system of two second order differential equations is found for the remaining tractions. The secular equation is then obtained as a quartic for the squared wave speed. This explicit equation is consistent with that found in the orthorhombic case. The speed of subsonic surface waves is then computed for 12 specific monoclinic crystals.     

Forced oscillations of an axisymmetric structure in contact with an elastic half-space by a version of global local finite elements

A version of the global local finite element method (GLFEM) is presented for determining the steady state forced response of an axisymmetric structure in contact with an elastic, homogeneous, isotropic half-space. In this GLFEM version, conventional finite elements are used to model the structure and some portion of the sorrounding medium, and global functions in the form of a complete set of outgoing spherical harmonic waves for the entire space are used to capture the behavior in the half-space region beyond the finite element mesh. An arbitrary distribution of steady state normal loads may be applied to the structure. Full traction and displacement continuity are enforced at the finite element mesh interface with the outer region. On the half-space surface of the outer region, traction-free surface conditions are met by requiring a sequence of weighted-average integrals of the tractions to vanish. Comparison of the present results for the problem of a rigid plate in frictionless contact with the half-space with those obtained by krenk and Schmidt [1] show excellent agreement over a wide frequency range. Other examples on circular plates under various contact conditions with the half-space are presented as illustrations of the forced response.     

Sound transmission across orthotropic laminates with a 3D model

This investigation examines the propagation of elastic waves in orthotropic materials to explain the sound insulation of FRP (Fiber Reinforced Plastics). The mechanical characteristics of an orthotropic material generally require nine independent parameters: three Young’s moduli, three shear moduli and three Poisson’s ratios. Three-dimensional analysis is performed with the elastic wave equations of an orthotropic material. The transfer matrix method which expresses the relationship between stress and velocity is adopted to calculate the sound transmission loss across an orthotropic material. Further, the transfer matrix method can only be calculated under the continuous boundary condition in the interface of each FRP layer. The boundary conditions which are indicated above are velocity and stress. The numerical results are compared with the experimental results. Additionally, along with varying material properties such as Young’s modulus, the acoustical properties of the orthotropic material are explained and discussed later.     

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.     

Dispersion dependences of elastic waves in an ice-covered shallow sea

The dispersion dependences of acoustic waves propagating under conditions close to an ice-covered shallow sea are investigated. A dispersion equation is derived for a layered medium consisting of an elastic layer (the ice cover), a liquid layer (the water column), and a homogeneous elastic half-space (the bottom). The possibilities of estimating certain parameters of the studied layered structure from an analysis of the dispersion curves are discussed. The results of numerical calculations are compared with the data of a full-scale experiment on detecting seismoacoustic signals in an ice-covered sea region.     

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.     

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.     

The effect of a surface impedance load on the properties of quasi-Rayleigh waves

The propagation of quasi-Rayleigh waves along an impedance-loaded plane boundary of an isotropic elastic half-space is studied theoretically. The dispersion equation of these waves is derived with allowance for the fact that an impedance load has both normal and tangential components. The conditions for the existence of such waves are analyzed depending on the magnitude and nature of each of these components. Specific examples of calculating the quasi-Rayleigh wave velocities are considered: for the models of surface and bulk cracked media, for a fluid layer in an elastic medium, and for a resonant load.     

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.     

Ultrasonic plate waves in paper

A theoretical and experimental study of ultrasonic plate waves in machine-made paper is described. The paper is assumed to behave as a homogeneous orthotropic plate. The dispersion equation for orthotropic plate waves in principal directions is then developed analytically. It is shown that at low frequencies the orthotropic dispersion equation depends only on in-plane parameters. Using appropriate elastic constants, a computer is used to construct the dispersion curves and describe the internal motion for plate waves in paper. Finally, a portion of the dispersion curve is checked experimentally.     

Guided elastic waves in a pre-stressed compressible interlayer

The propagation of guided elastic waves in a pre-stressed elastic compressible layer embedded in a different compressible material is examined. The waves propagate parallel to the planar layer interfaces as a superposed dynamic stress state on the statically pre-stressed layer and host material. The underlying stress condition in the two materials is characterized by equibiaxial in-plane deformations with common principal axes of strain, one of the axes being perpendicular to the layering. Both materials have arbitrary strain energy functions. The dispersion equation is derived in explicit form. Analysis of the dispersion equation reveals the propagation characteristics and their dependence on frequency, material parameters and stress parameters. Combinations of these parameters are also defined for which guided waves cannot propagate.     

Influence of imperfectly bonded micropolar elastic half-space with non-homogeneous viscoelastic layer on propagation behavior of shear wave

A mathematical model is developed in which the effect of imperfect bonding between the constituents of layer and half-space on the phase velocity and damped velocity of SH-wave is discussed. The model consists of a micropolar elastic half-space bonded imperfectly with a heterogeneous viscoelastic layer. The dispersion equation and damping equation of SH-wave propagation in the said model is obtained in the closed form analytically. The effects of imperfect bonding, internal friction, heterogeneity, micropolarity, and complex interface stiffness parameters highlighted through numerical computation and graphical demonstrations. Standard Love-wave equation and dispersion equation as well as damping equation for perfectly bonded micropolar half-space with heterogeneous viscoelastic layer is obtained as a special case of the problem. Through comparative study of homogeneity with heterogeneity in the layer; imperfect bonding of layer and half-space with their welded (perfect) contact; and presence of micropolarity in half-space with its absence in half-space are compared meticulously.     

Cancellation of acoustic scattering from an elastic sphere

Recent research has suggested the possibility of creating acoustic cloaks using metamaterial layers to eliminate the acoustic field scattered from an elastic object. This paper explores the possibility of applying the scattering cancellation cloaking technique to acoustic waves and the use of this method to investigate its effectiveness in cloaking elastic and fluid spheres using only a single isotropic elastic layer. Parametric studies showing the influence of cloak stiffness and geometry on the frequency dependent scattering cross-section of spheres have been developed to explore the design space of the cloaking layer. This analysis shows that an appropriately designed single isotropic elastic cloaking layer can provide up to 30 dB of scattering reduction for ka values up to 1.6. This work also illustrates the importance of accounting for the elasticity of the object and the relevant limitations of simplistic quasi-static analyses proposed in recent papers.     

柱面Love波频散分析与SH波场的数值计算

对贴井壁环型剪切源在柱状双层弹性介质中激发的SH波场进行了理论求解,导出了柱面Love波频散方程,讨论了柱面Love波存在的条件及其区域.通过数值计算考察了柱面Love波的频散特性和激发强度,发现最低阶柱面Love波具有截止频率,这与平面半空间双层弹性介质模型下的Love波无截止频率的特征不同.渐近分析与数值考察都表明,井径r₁→∞时,柱面Love波频散方程趋向平面双层半空间的Love波方程,柱面Love波的截止频率趋于零.全波计算还显示用激发SH波来探测侵入带外原状地层的横波信息是一个十分简洁的途径..     

Note on the forced vibration of an orthotropic plate on an elastic foundation

An analysis is presented of the response of an infinite, linearly elastic, orthotropic plate, resting on an elastic foundation, to a forcing function which may vary in both time and space. Some numerical results are given for a case of concentrated blast loading.     

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.     

Effect of periodic corrugation,reinforcement, heterogeneity and initial stress on Love wave propagation

This paper investigates the impact of corrugated boundary surfaces, reinforcement on the propagation of Love-type wave in prestressed corrugated heterogeneous fiber-reinforced layer resting over a void pores half-space. The heterogeneity in the upper corrugated layer is caused due to exponential variation in the elastic constants with respect to the space variable pointing positively downwards. The dispersion equation in the complex form has been derived using method of separation of variables. The real and imaginary parts of the complex dispersion equation were separated and found in well agreement with the classical Love wave equation. Also, the attenuation of the Love waves has been discussed. The study reveals that such a medium transmits two fronts of Love waves. The first front depends upon the change in volume fraction of the pores and the second front depends upon the modulus of rigidity of the elastic matrix of the medium. The substantial influence of corrugation parameters, reinforcement, undulatory parameter, initial stress, heterogeneity parameter and position parameter on the phase velocity, and attenuation of Love-type wave have been observed and depicted by means of graph. It has been observed that the phase velocity decreases with the increase in initial stress parameters, heterogeneity, and reinforcement in upper layer.     

谱有限元模拟正交各向异性黏弹性材料中导波传播特性

研究了导波在正交各向异性黏弹性复合板中传播的色散特性、波结构及功率流密度。基于二维平面运动方程,采用谱有限元方法得到了导波色散的特征方程。分析了正交各向异性黏弹性板中各向异性和黏性对能量速度和波结构的影响,以及基底对导波功率流密度的影响。数值研究结果表明:导波沿纤维方向传播的能量速度大,材料的黏性对速度影响较小,但会减小波结构的幅度;在高频时,基底的存在使两个基本模态的功率流密度分别集中到波导的上下表面,形成弱色散、高衰减及无色散、零衰减的表面波。数值模拟结果为导波用于复合材料定量无损检测和性能评价提供理论依据。