Scattering attenuation of elastic waves due to low-contrast inclusions

Seismic scattering attenuation due to random lithospheric heterogeneity has been theoretically modeled using two approaches. One approach is the Born approximation theory (BAT), which is primarily used to treat weak continuous heterogeneity, and the other approach is the Foldy approximation theory (FAT), which deals with sparsely distributed discrete inclusions. We apply the BAT to elastic wave scattering due to inclusions having low contrast with the matrix, and compare the results with those predicted by the FAT. We thus investigate the valid wavenumber range of the BAT based on a reasonable assumption that the inclusions are distributed so sparsely that the FAT is effectively correct for any wavenumber. For simplicity, we consider a specific type of round inclusion, which is either two- or three-dimensional and has a two-valued wave velocity and/or mass density. Both theories are confirmed to yield essentially equivalent results below a certain wavenumber limit, depending on the contrast. This is known as the Rayleigh-Gans scattering regime. Beyond the wavenumber limit, the BAT overestimates the attenuation for common-mode scattering due to wave-velocity contrast, but remains valid with respect to the attenuation for scattering due to mass-density contrast and/or conversion scattering. These conclusions are independent of the spatial dimensions of the media as well as the modes of the elastic waves (P or S). Some advantages of the BAT over the FAT for application to low-contrast inclusions are discussed.     

关于传递矩阵法分析饱和成层介质响应问题的讨论

水平成层土体的地震响应分析(自由场分析)是地震工程领域地震波散射问题的前提基础,由于饱和多孔方程的复杂性,以往的研究大多集中于干土情形,对于饱和土情形的研究相对较少.而实际工程中,地下水位以下,土体孔隙中充满流体,应考虑饱和多孔介质模型.基于Biot多孔介质模型,考虑饱和土中固液相对运动引起的衰减,采用Thomson--Haskell传递矩阵方法得到了饱和成层土体在地震波入射情形时的稳态反应,经傅里叶反变换,可得到时域暂态反应.通过SV波从基岩入射至上覆饱和土层的数值算例,验证了该方法的有效性.发现和初步阐明了计算中出现的两类违背因果律(即响应先于输入)的现象:(1)当SV波入射角度大于导致基岩中反射P波为非均匀波的临界角时,会使得计算结果违背因果律.因此,当入射角超过临界角时,非均匀波的表示尚需进一步完善;(2)由于P2波的衰减,当与稳态波衰减有关的渗透率、土层厚度、入射波频率等参数导致衰减系数超过计算机表示精度时,会出现结果违背因果律现象,并据此得到了满足因果律的参数范围,该范围可作为实际计算时的一个上界.该工作为采用传递矩阵法分析水平饱和土层自由场响应提供了指导依据,且地下水位以上可采用干土模型,水位以下采用饱和土模型,更符合实际情形.     

Shock pulse attenuation in a nonlinear viscoelastic solid

Thin, one-dimensional shock pulses were generated in a nonlinear viscoelastic material (polymethyl methacrylate) by a new experimental technique. The observed pulse attenuation was compared with an approximate theory based on the viscoelastic shock amplitude equation. The central assumption of this approximate theory is that the unloading wave propagates as a simple wave. Given an initial pulse shape it is shown that the attenuation and the pulse shape at any later time are accurately approximated. The calculated attenuation in polymethyl methacrylate agreed well with the experimental results.     

Phononic layered composites for stress-wave attenuation

The aim is to design a layered metamaterial with high attenuation coefficient and high in-plane stiffness-to-density ratio using homogenization to calculate and optimize the dynamic effective stiffness and mass density of layered periodic composites (phononic layers) over a broad frequency band. This is achieved by: (1) minimizing the frequency range of the first pass band, (2) maximizing the frequency range of the stop band, and (3) creating local resonance over the second pass band. To verify the theoretical calculation, laboratory samples were fabricated and their attenuation coefficient were measured and compared with the theoretical results. It is observed that over 4–20 kHz frequency range the attenuation per unit length in the optimally designed composite can exceed 500 dB/m; which increases with increasing frequency. A dynamic Ashby chart, depicting attenuation coefficient vs. in-plane stiffness-to-density ratio, is presented for various engineering materials and is compared with the fabricated metamaterial to show the significance of our design. This method can be used in variety of applications for stress wave management, e.g., in addition to match the impedance of the resulting composite to that of its surrounding medium to minimize (or essentially eliminate) stress wave reflection.     

Effective wave propagation along a rough thin-elastic beam

Two methods for computing the complex-valued effective wavenumber of a rough beam in the context of linear time-harmonic theory are presented. The roughness of the beam is modelled as a continuous random process of known characteristic length and root-mean-square amplitude for either the beam mass or the beam rigidity. The first method is based on a random sampling method, with the effective wave field calculated as the mean of a large ensemble of wave fields for individual realisations of the roughness. The individual wave fields are calculated using a step approximation, which is validated for a deterministic problem via comparison to results produced by an integral equation approach. The second method assumes a splitting of the length scale of the fluctuations and an observation scale, employing a multiple-scale approximation to derive analytical expressions for the effective attenuation rate and phase change. Numerical comparisons show agreement of the results of the random sampling method and the multiple-scale approximation for a wide range of parameters. It is shown that the effective wavenumbers only differ by a real constant between the cases of varying beam mass and rigidity.     

In-plane waves in an elastic solid containing a cracked slab region

Propagation of longitudinal and transverse waves in an elastic solid that contains a cracked slab region is investigated. The cracks have a uniform probability density in the slab region, are parallel to the boundaries of the slab, and the solid is uncracked on either side of the slab. The waves are normally incident on the cracks. It is shown that the resulting average total motion in the solid is governed by a pair of coupled integral equations. These equations are solved under the special assumption that the average exciting motion near a fixed crack is equal to the average total motion. In this case, one finds that in the cracked region, where multiple scattering occurs, there is a forward motion and a backward motion. The two motions have identical frequency-dependent velocity and attenuation, for which simple closed-form formulae are obtained. Simple formulae are also obtained for the wave amplitudes outside the slab. Numerical results corresponding to the velocity, attenuation, reflection amplitude, and transmission amplitude are presented for several values of crack density and slab thickness.     

Radiation and attenuation of low- and high-frequency waves in a random medium

The wave field radiated by a point source in a weakly inhomogeneous, weakly dissipative, one-dimensional random medium is considered. Approximate expressions for the mean intensity and mean energy flux as a function of propagation range, based on general results obtained previously, are derived for the limiting cases of low and high frequencies. These expressions show that, in both cases, the mean intensity and mean energy flux decrease more rapidly with range than would be the case in the absence of randomness. This more rapid rate of decrease with range can be interpreted as an excess attenuation. These results are similar to those obtained previously for the intermediate-frequency case. The excess-attenuation results are found to be consistent with observations.     

Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities

The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis.The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem.The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.     

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.     

Frequency dependence of elastic (SH-) wave velocity and attenuation in anisotropic two phase media

The propagation and attenuation of elastic waves in a random anisotropic two-phase medium is studied using statistical averaging procedures and a self-consistent multiple scattering theory. The specific geometry and orientation of the inhomogeneities (second phase) are incorporated into the formulation via the scattering matrix of each inhomogeneity. The anisotropy of the composite medium is due to the specific orientation of the non-symmetric inclusions. At low frequencies, analytical expressions are derived for the effective wave number in the average medium as a function of the geometry and the material properties and the angle of orientation of the inclusions. The results for the special cases of oriented cracks may find applications in geophysics and material science. The formulation is ideally suited for numerical computation at higher frequencies as evidenced by the results presented for composites reinforced by fibers of elliptical cross section.     

水平光面爆破激发地震波的成分及衰减特征

借助极化偏振分析方法,针对一组现场爆破实验,分析了水平光面爆破激发地震波的成分构成及特性,比较了不同波的衰减特征及各自对爆破振动的影响,并探讨了水平光面爆破的内在力学机理。结果表明,爆破振动中不同波的相对量值及主导波的类型均会随测点位置的改变而变化,爆源特性和沿传播路径的不同衰减共同决定波的成分构成及演化,各测点的优势振动方向也与波的成分构成密切相关。对于水平光面爆破,在光爆孔平面上,P波的影响可忽略,S波主要在竖直向振动,R波对水平及竖直向的振动均有贡献,其中水平向的振动主要由R波引起,而S波的竖直向振速在近区远高于R波,但归因于S和R波的不同衰减,R波在距离爆源22.5 m/kg^1/2(58~67 m)处开始主导竖直向的振动;在光爆孔平面外,P波的影响不可忽略,且在特定位置会成为优势波型。     

Precursors for waves in random media

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty–Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully three-dimensional situation.     

Analytical dynamic displacement response of rigid pavements to moving concentrated and line loads

This paper presents the formulation of a plate of infinite dimensions (without boundary conditions) on an elastic foundation, subjected to a moving concentrated and line load of constant amplitude and speed, using a triple Fourier transform. The solution is carried out integration by residues. A closed-form solution of displacement field has been obtained for a moving load with subsonic, transonic and supersonic speeds. It is found that the maximum response of the slab occurs beneath the moving load and travels with the load at the same speed. It is also shown that a critical speed exists. If the moving load travels at critical speed, slab displacement becomes infinite in amplitude.     

脆性材料中应力波衰减规律与层裂实验设计的数值模拟

对混凝土、岩石类脆性材料的层裂实验进行了有限元模拟,研究了应力波在此类材料中传播的衰减规律,包括两类机制:弹性波因大尺寸试样的几何弥散产生的小幅度线性衰减、与应变率相关的黏塑性波因本构关系导致的指数衰减。在此基础上,提出了包含常数项的指数型应力波峰值拟合公式。建议采用可以忽略应力波衰减影响的细长形试样进行层裂实验。混凝土类脆性材料层裂破坏模拟结果显示,有限元模拟得到的层裂片厚度与一维应力波理论得到的结果非常吻合,验证了按一维应力波理论确定层裂强度的实验方法的有效性。通过对比3种不同入射波形下层裂片的形状和净拉应力波形,发现不对称的入射波形状更有利于实验获得平直的层裂断面和较准确的层裂强度。     

Correlation of elastic wave attenuation and scattering with volumetric grain size distribution for polycrystals of statistically equiaxed grains

This study establishes an explicit relation between spatial two-point correlation function (TPCF) and volumetric (or three-dimensional) grain size distribution for aggregates of statistically equiaxed grains by extending a prior study (Sha, JASA, 2018). This relation is further validated by applying it to available TPCF and volumetric grain size distribution in the literature. Based on this relation, analytical attenuation coefficients for longitudinal and transverse waves, accounting for volumetric grain size distribution, are derived under Born approximation for macroscopically isotropic polycrystals of equiaxed triclinic grains. These attenuation models are applicable for whole frequency range except geometric region. Moreover, scattering coefficients for a polycrystal of equiaxed triclinic grains with a volumetric grain size distribution are obtained. Finally, the analytical attenuation model for the longitudinal wave is verified by comparison with existing 3D finite element simulation results in the literature. This theoretic study has practical applications to the inverse determination of volumetric grain size distribution from ultrasonic measurements.     

Apparent emissivity of an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index

A backward Monte Carlo curved ray-tracing method is developed to analyze the apparent emissivity of one-dimensional absorbing-emitting-scattering semitransparent slab with variable spatial refractive index. An isothermal semitransparent slab with linear variable spatial refractive index is taken as an example to examine the accuracy of the proposed method. The directional apparent emissivity is determined by the proposed method and compared with the data in references, and the effects of the single scattering albedo and the scattering phase function on the directional apparent emissivity are analyzed. The results show that the backward Monte Carlo curved ray-tracing method has a good accuracy in solving the apparent emissivity of semitransparent slab with variable spatial refractive index, and converges more quickly than the forward Monte Carlo method. The effect of single scattering albedo is significant and the influence of the scattering phase function is similar to that of the slab with constant refractive index.     

Classical and generalized coupled thermoelasticity analysis in one-dimensional layered media

The behavior of thermoelastic waves at the interface of layered medium and distributions of these waves through the domain are examined by applying the direct finite element method to obtain the field variables directly within the spatial and temporal domains. The analysis is performed in a one-dimensional domain with two different layers to provide a means to follow the behavior of the reflected thermoelastic waves at the interface. It appears that the distributions of thermoelastic waves in an isotropic slab with one layer are significantly different from those in multilayered slabs. For instance, the negative displacement waves, several stresses with positive or negative signs and temperature distributions produced in the multilayered domains, are quite different from those in a single layer. This method may be generalized to simulate the propagation of thermoelastic waves in various multilayered regions and analyze the behavior of the layered composite structures under the mechanical or thermal impact loads.     

缺陷岩体中弹性波传播的多次散射效应分析

弹性波在岩体中传播时与岩体缺陷相互作用形成复杂的传播图案。为研究缺陷对弹性波多次散射作用的影响,建立了双椭圆缺陷模型,基于Green函数基本解,采用边界积分的计算方法,得到了反映缺陷界面条件的刚度矩阵,分析了弹性波在双椭圆缺陷间的多次散射效应。结果表明：与单椭圆缺陷模型相比,双缺陷的相互作用使得弹性波频散和衰减效应增强,定量给出了缺陷的影响区域,从而明确了多次散射效应的尺度界限。进一步探讨了弹性波传播的多尺度效应,结果表明频散的Rayleigh峰、Mie峰和衰减的峰值频率同椭圆长轴和入射波波长两个尺度密切相关,存在明确的定量关系。相应的数值模拟结果表明,弹性波和缺陷相互作用在缺陷界面上诱发界面波,该界面波也存在频率相关性,影响了弹性波宏观传播的频散和衰减特征。     

Effect of spatially correlated noise on stochastic synchronization in globally coupled FitzHugh-Nagumo neuron systems

The phenomenon of stochastic synchronization in globally coupled FitzHugh-Nagumo (FHN) neuron system subjected to spatially correlated Gaussian noise is investigated based on dynamical mean-field approximation (DMA) and direct simulation (DS). Results from DMA are in good quantitative or qualitative agreement with those from DS for weak noise intensity and larger system size. Whether the consisting single FHN neuron is staying at the resting state, subthreshold oscillatory regime, or the spiking state, our investigation shows that the synchronization ratio of the globally coupled system becomes higher as the noise correlation coefficient increases, and thus we conclude that spatial correlation has an active effect on stochastic synchronization, and the neurons can achieve complete synchronization in the sense of statistics when the noise correlation coefficient tends to one. Our investigation also discloses that the noise spatial correlation plays the same beneficial role as the global coupling strength in enhancing stochastic synchronization in the ensemble. The result might be useful in understanding the information coding mechanism in neural systems.     

The singular function of a surface and physical optics inverse scattering

It is shown how to recover both the location and the reflection coefficient of a scatterer using only high frequency backscattered data. The result is obtained without use of the far field approximation although a separate identity is derived when this approximation is introduced. This latter result improves upon previously derived physical optics far field inverse scattering identities.