Scattering of elastic waves in damaged media

The scattering of elastic waves in a medium with damage from microcracking is discussed. The influence of damage from penny-shaped microcracks within a homogeneous medium is considered. The microcracks are assumed to be randomly oriented and uniformly distributed. Explicit expressions are derived for the attenuation of longitudinal and shear elastic waves in terms of the damage parameter and the effective elastic moduli of the medium. A generalized tensor-based approach is used such that the results are coordinate free. The derivation is based upon diagrammatic methods. The problem is formulated in terms of the Dyson equation, which is solved for the mean field response within the limits of the first-order smoothing approximation. The longitudinal and shear attenuations are discussed in terms of their frequency dependence and damage dependence. In particular, the attenuations are shown to scale linearly with the damage parameter.     

各向异性渗流条件下弹性波的传播特征

研究了弹性波在非均匀裂纹孔隙介质中的传播特性,建立了各向异性喷射流模型.当弹性波通过裂纹孔隙介质时,由于波的扰动及裂纹和孔隙几何结构的不一致,导致在裂纹内部及裂纹与周边孔隙之间同时存在着流体压力梯度.此时的弹性波波动响应中包含着裂纹内连通性特征和背景孔隙渗透率信息.流体的动态流动过程使得介质的等效弹性参数为复数(非完全弹性),并且具有频率依赖性.当弹性波为低频和高频极限时,介质为完全弹性;当处于中间频段时,波有衰减和频率依赖.裂纹孔隙介质的各向异性连通性(渗透率)对应着各向异性特征频率(当渗流长度等于非均匀尺度时的弹性波频率),波的传播受到裂纹内连通性的影响.在一定频段内,随着裂纹厚度的增加,将出现第二峰值,峰值大小同时受到裂纹厚度和半径的影响.     

SH-wave propagation and scattering in periodically layered composites with a damaged layer

Plane SH-wave propagation in periodically layered elastic composites with a damaged layer is investigated. Two different models are developed to approximate the damaged layer, namely, a periodic array of cracks and continuously distributed springs in the layer. In the first model, the total wave field in the elastic stack of layers with cracks is described as a sum of incident wave field modeled by the transfer matrix method and the scattered wave field governed by an integral representation in terms of the crack-opening-displacements on the crack-faces. The integral equation derived from the boundary conditions on the crack-faces is solved numerically by a Galerkin method. By using Bloch–Floquet theorem the crack-opening-displacements for a periodic array of cracks are expressed by the crack-opening-displacement on a reference crack. In the spring model, the spring constant is estimated by the material properties and the crack density and the modified transfer matrix method is used to compute the wave reflection and transmission coefficients. Numerical results obtained by both models are presented and discussed. Special attention of the analysis is devoted to wave transmissions and reflections, band gaps, wave localization and resonance phenomena due to damages. The influences of the damage types (periodic cracks and stochastic cracks approximated by distributed springs) on the wave field pattern and the band gaps are analyzed.     

Resonance blocking of traveling waves by a system of cracks in an elastic layer

Wave processes that occur in an elastic layer when waves traveling in it are diffracted by a system of horizontal cracks are investigated. Integral representations of wave fields are constructed in terms of the convolution of Green’s matrices and unknown jumps of displacements at the cracks. The displacement jumps are determined from the boundary integral equations, which are obtained from the initial boundary-value problem with the boundary conditions at crack faces being satisfied. The spectrum of the integral operator is studied for different variants of mutual crack arrangement and is compared with the spectrum of the corresponding operators for individual cracks; the relationship between the spectrum and the blocking effects is analyzed. The possibility of obtaining an extended frequency band of waveguide blocking in the case of groups of cracks is demonstrated.     

Elastic stress of moving dislocations in finite crystals

The elastic stress of a single crystal with finite dimensions is investigated by means of the linear continuum theory; external forces and dislocation motion are assumed to be known. A formal solution is derived with a Green's dyadic. The dyadic is expanded in a series, the first term of which delivers the well-known solution for infinite crystals, the following terms describe successive reflections of sound at the surface of the finite crystal. They can be computed by iteration. This expansion corresponds to the physical situation of pulse-echo-experiments. Fourier transforms lead to analogous expressions describing the influence of the surface on time-periodic dislocation experiments.     

AN ANALYTICAL APPROACH TO DETERMINING THE DYNAMIC CHARACTERISTICS OF A CYLINDRICAL SHELL WITH CLOSING CRACKS

The paper is devoted to developing mathematical models of the elastic oscillations of a cylindrical shell with surface closing cracks. The respective forms of shell vibrations have been chosen to represent various types of damage of the shell. In the case of dispersed and single-surface damage, the transverse shell vibrations are simulated. The cycle of vibrations is assumed to be subdivided into two parts, in one of them the damaged surface fibers are compressed so closing the cracks and negating their influence. For the second part, the cracks are open, so their influence is taken into account. The problem is solved in a piecewise linear with different frequencies and amplitudes at each vibrations cycle interval. The vibration parameters are calculated by means of Relay's energy conservation method and are represented by analytical expressions, the system being assumed to be conservative. The functions determining the vibration process are decomposed by a Fourier analysis using the averaged frequency, the coefficients of the resulting series being obtained as analytical expressions. Vibrodiagnostic functions, which enable the geometrical parameters of the cracks to be determined depending on the geometry of the shell and type of damage, have been plotted.     

冲击波极端条件下玻璃的细观结构破坏

 研究了冲击波极端条件下玻璃介质的细观结构破坏问题，指出在低于Hugoniot弹性极限的应力区内，按照细观结构损伤程度的不同，在受压玻璃介质中可以划分出两个区域，即压缩区和破坏区。以K9和ZF1玻璃为例，通过双层结构样品实验，确认了玻璃样品的表面效应（即表面原生微裂纹的扩展）是破坏区形成的第一位原因。其次，基于对破坏区内细观结构损伤和破坏特性的分析，进一步提出：由于玻璃内部散布的不均匀相与其基体介质之间的压缩率不同，冲击波压缩造成了众多的局域变形点，当表面裂纹扩展到不均匀相与基体的边界处，会出现裂纹扩展路径拐折或分叉，造成介质的分割甚至粉碎，这是破坏区生成的第二位原因。     

The three-nucleon problems with weak NN interaction

We consider the three-nucleon system in which each two-nucleon pair interacts both strongly and weakly. The strong part is assumed to be separable but an addition of small non-separable term is possible. The weak two-nucleon interaction (which is absolutely non-separable) is assumed to be given in terms of its partial wave components in momentum space. These elements have been calculated by Lassey and McKellar starting from recent models of non-relativistic (strangeness conserving non-leotonic) NN weak interaction. An expression for the weak part of the elastic nucleon-deuteron scattering amplitude is derived within the framework of two potential scattering theory. The weak three-body driving terms are explicitly evaluated. They reflect weak form factors responsible for the weak dissociation of two nucleon isobars (including parity impurity of the deuteron wave function) as well as weak NN scattering. The expressions appearing below are numerically tractable and can be used to elaborate on previous calculations.     

Rayleigh wave at the boundary of an inhomogeneous nonlinear viscoelastic half-space

In the approximation of weak nonlinearity and weak viscosity of the medium, we obtain an equation describing the spectral density of the particle horizontal velocity for a Rayleigh wave propagating along the boundary of a half-space. The coefficients of nonlinear interaction between the wave harmonics are found on the assumption that the third-order elastic moduli arbitrarily depend on the depth. We find expressions for the complex correction to the wave frequency due to small relaxation corrections to the elastic moduli and small variations in the medium density, which arbitrarily depend on the depth as well. The imaginary part of this correction to the frequency determines the decay of the linear Rayleigh wave due to small relaxation corrections to the elastic moduli arbitrarily dependent on the depth. Using numerical simulation (with allowance for the interaction of 500 harmonics), we study distortions of an initially harmonic Rayleigh wave for a particular dependence of variations in the nonlinear moduli on the depth. An integral equation is derived for the nonlinear elastic moduli as functions of the depth. It is shown that for independent spatio-temporal distributions of the viscous moduli, functions determining the dependence of the viscosity on the depth are described by an analogous integral equation. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 3, pp. 212–226, March 2007.     

Macroscopic theory of elasticity of fourth order elastic constants of cubic structure solids

The macroscopic theory of elasticity is extended to determine the fourth order elastic constants. The expressions for the small amplitude sound wave velocity and for a natural velocity, in statically stressed media, are derived in terms of second, third and fourth order elastic constants.     

Dissipation and dispersion properties of microinhomogeneous media

The propagation of a harmonic elastic wave in a microinhomogeneous (defect-containing) medium is considered in the framework of the rheological model that reperesents the medium in the form of a one-dimensional chain of masses connected by purely elastic elements and by Kelvin-Voigt viscoelastic elements. Analytical expressions are derived for the dissipation and dispersion characteristics of this medium for various distributions of the parameters of the viscoelastic elements. The dissipation and dispersion properties are found to obey the Kramers-Kronig relations. It is also shown that the damping decrement of the wave is almost constant, and the phase velocity monotonically increases in a sufficiently wide range of parameters of the viscoelastic elements in a wide frequency band. The derived expressions for the dispersion and dissipation are used to simulate the propagation of broadband pulses in this kind of medium.     

Observation of the "Luxemburg-Gorky effect" for elastic waves

An experimental observation of a new nonlinear-modulation effect for longitudinal elastic waves is reported. The phenomenon is a direct elastic wave analogy with the so-called Luxemburg-Gorky (L-G) effect known over 60 years for radio waves propagating in the ionosphere. The effect consists of the appearance of modulation of a weaker initially non-modulated wave propagating in a nonlinear medium in the presence of an amplitude-modulated stronger wave that produces perturbations in the medium properties on the scale of its modulation frequency. The reported transfer of modulation from one elastic wave to another was observed in a resonator cut of a glass rod containing a few small cracks. Presence of such a small damage drastically enhances the material nonlinearity compared to elastic atomic nonlinearity of homogeneous solids, so that the pronounced L-G type cross-modulation could be observed at strain magnitude in the stronger wave down to 10(-7) and smaller. Main features of the effect are pointed out and physical mechanism of the observed phenomena is discussed.     

Reliability assessment of different plate theories for elastic wave propagation analysis in functionally graded plates

The importance of elastic wave propagation problem in plates arises from the application of ultrasonic elastic waves in non-destructive evaluation of plate-like structures. However, precise study and analysis of acoustic guided waves especially in non-homogeneous waveguides such as functionally graded plates are so complicated that exact elastodynamic methods are rarely employed in practical applications. Thus, the simple approximate plate theories have attracted much interest for the calculation of wave fields in FGM plates. Therefore, in the current research, the classical plate theory (CPT), first-order shear deformation theory (FSDT) and third-order shear deformation theory (TSDT) are used to obtain the transient responses of flexural waves in FGM plates subjected to transverse impulsive loadings. Moreover, comparing the results with those based on a well recognized hybrid numerical method (HNM), we examine the accuracy of the plate theories for several plates of various thicknesses under excitations of different frequencies. The material properties of the plate are assumed to vary across the plate thickness according to a simple power-law distribution in terms of volume fractions of constituents. In all analyses, spatial Fourier transform together with modal analysis are applied to compute displacement responses of the plates. A comparison of the results demonstrates the reliability ranges of the approximate plate theories for elastic wave propagation analysis in FGM plates. Furthermore, based on various examples, it is shown that whenever the plate theories are used within the appropriate ranges of plate thickness and frequency content, solution process in wave number-time domain based on modal analysis approach is not only sufficient but also efficient for finding the transient waveforms in FGM plates.     

材料非线性衰减系数的二次谐波测量方法研究

采用有限幅值法测量材料在基波和非线性引起的二次谐波作用下的衰减系数:利用准线性下的KZK方程推导基波和二次谐波的声压分布,并提取波束修正系数;采用短纯音信号进行非线性实验,对检测得到的基波和二次谐波声压进行衍射修正处理,有效抑制衍射对衰减系数测量的不利影响,继而通过线性拟合的方法计算得到更精确的基波和二次谐波的衰减系数。以水为例进行实验,研究了实验测量所得衰减系数的频率依赖关系,结果表明在非线性条件下水的衰减系数与频率间存在较强的线性关系,而线性条件下衰减系数随频率呈现二次方增长的特性则不适用于非线性条件。该研究提出了准确测量非线性声波衰减系数的方法,为更有效地应用非线性超声检测提供理论依据。      

Localization of nonlinear waves in elastic bodies with inclusions

By the example of the problem of the motion of a semi-infinite string lying on an elastic base, a method for describing wave localization near inclusions is proposed for the case of a cubic nonlinearity of the base. The method applies the perturbation technique to the amplitude of a localized mode. The nature of the divergences is revealed, and the secular terms are found to belong to one of two types: inphase or antiphase with the localized wave. It is shown that a combination of the renormalization method and multiscale method provides a convergence of the solutions, which are sought for in the form of power series in the amplitude of the localized mode. It is found that the localization process is determined by the type of the discrete spectrum, type of the nonlinearity, and type of dispersion. The nonlinearity of the elastic base produces two characteristic effects. First, the frequency of the localized wave becomes dependent on the wave amplitude. Second, the system can generate traveling waves at multiple frequencies, which withdraw energy from the localized wave and cause it to decay. The decay behavior is determined by the minimum frequency of these traveling waves (because it must be higher than the cutoff frequency). The lifetime of the localized wave as a function of the mass of a dynamic inclusion exhibits a number of maxima. In particular, the first maximum corresponds to the minimum amplitude of the traveling wave at the triple frequency.     

Elastic nonlinearity parameters of tetragonal crystals

The equations of motion for the propagation of finite amplitude elastic waves in crystals of tetragonal symmetry have been derived starting from the expression for the elastic strain energy. The equations have been solved for a finite amplitude sinusoidal wave propagating along the pure mode directions which are [100], [110] and [001] for the tetragonal group TI. The solutions corresponding to longitudinal wave propagation yield expressions for the amplitudes of the fundamental and generated second harmonic for these directions in terms of certain combinations of second and third order elastic constants of the medium. The results will aid the experimenter to determine these constants using ultrasonic harmonic generation technique.     

Floquet wave homogenization of periodic anisotropic media

A dynamic homogenization method based on Floquet wave theory is developed. The theory is based on the equivalency within the homogenization domain of Floquet waves in a periodic anisotropic medium and plane waves in a dispersive homogeneous anisotropic medium. A simple procedure has been developed to estimate analytically critical angles and the upper frequency bound of this homogenization domain. Using this method, the frequency dependent effective elastic constants are obtained and examples for [0/90] and [0/45/90/-45] composites are given. By comparison with an exact theory, it is shown that the time domain signal propagation in a periodic laminate is well described by the Floquet wave homogenization theory in the homogenization domain. It is also shown that in the static limit the results are identical to those calculated by static homogenization theory (the generalized method of cells). The potential applications of the method are discussed.     

Properties of surface waves in an elastic cylinder filled with a liquid

The properties of harmonic surface waves in an elastic cylinder filled with a liquid are studied. The case of elastic material for which the shear wave velocity is higher than the sound velocity in a liquid is considered. The wave motion is described based on the complete system of equations of the dynamic theory of elasticity and the equation of motion of an ideal compressible liquid. The asymptotic analysis of the dispersion equation in the region of large wave numbers and qualitative analysis of the dispersion spectrum showed that in such a waveguiding system there exist two surface waves, the Stoneley and the Rayleigh waves. The lowest normal wave forms the Stoneley wave on the internal surface of the cylinder. In this waveguide phase, velocities of all normal waves, except for the lowest one, have the velocity of sound in the liquid as their limit. Therefore, the Rayleigh wave on the external surface of the cylinder is formed by all normal waves in the range of frequencies and wave numbers in which phase velocities of normal waves of the composite waveguide and the lowest normal wave of the elastic hollow cylinder coincide.     

声波在一维易膨胀介质中传播的计算机模拟

建立了一种新的介质模型，其弹性模量在声波处于压缩状态时较大，而在膨胀状态时较小. 在这种介质中，纯的压缩或膨胀波的传播特性与一般弹性介质类似，只是它们分别以压缩声 速或膨胀声速传播.但当它们在某一区域中同时存在时，它们之间会相互耦合，产生非常强 的非线性效应.对这两种波在对行和追赶两种情况的耦合特性作了详细地模拟计算.结果显示 在两种情况下，压缩波和膨胀波的耦合均会造成体系的膨胀.体系的膨胀与压缩弹性模量和 膨胀弹性模量的相对差有关.此外，还对弹性模量随声波压力连续变化的体系进行了模拟计 算.结果证实由非连续变化的弹性模量所得的结论可由连续变化的弹性模量的极限情况得到. 关键词： 颗粒物质 声波 弹性介质