Numerical analysis of isolation of the vibration due to moving loads using pile rows

The isolation of the vibration due to moving loads using pile rows embedded in a poroelastic half-space is investigated in this study. Based on Biot's theory and integral transform method, the free field solution for a moving load applied on the surface of a poroelastic half-space and the fundamental solution for a harmonic circular patch load applied in the poroelastic half-space are derived first. Using Muki and Sternberg's method and the fundamental solution for the circular patch load as well as the obtained free field solution for the moving load, the second kind of Fredholm integral equations in the frequency domain describing the dynamic interaction between pile rows and the poroelastic half-space is developed. Numerical solution of the frequency domain integral equations and numerical inversion of the Fourier transform yield the time domain response of the pile–soil system. Comparison of our results with some known results shows that our results are in a good agreement with existing ones. Numerical results of this study show that velocity of moving loads has an important impact on the vibration isolation effect of pile rows. The same pile row has a better vibration isolation effect for the lower speed moving loads than for the higher speed ones. Also, the optimal length of piles for higher speed moving loads is shorter than that for lower speed moving loads. Moreover, stiff pile rows tend to produce a better vibration isolation effect than flexible pile rows do.     

Dynamic vertical interaction of a foundation–soil system generated by seismic waves

Based on Biot's dynamic poroelastic theory, a foundation–soil interaction model is established to investigate the vertical vibrations of a rigid circular foundation on poroelastic soil excited by incident plane waves, including the fast P waves and SV waves. Scattering waves caused by the foundation and fluid–solid coupling due to the pore water in the soil are also considered in the model. The solution of the vertical vibrations of the foundation subjected to seismic waves are obtained by solving two sets of dual integral equations derived from the mixed boundary-value conditions. The different vertical vibrations of foundation rest on elastic and saturated half-space are compared. The influences of incident angle, permeability of soil and foundation mass on the vertical vibrations of the foundation are then discussed. The results show that resonant phenomenon of the foundation is observed at certain excitation frequencies; the effects of the pore water on the foundation vertical vibrations are significant. In addition, significant differences are found when the foundation is excited by P waves and SV waves, respectively.     

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a fluid-saturated poroelastic medium

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a saturated poroelastic medium is considered. The medium is described by Biot’s constitutive equations, the crack sides are fluid permeable. The problem is reduced to a two-dimensional integral equation for the crack opening vector. Gaussian approximating functions are used for discretization of this equation. For such functions, the elements of the matrix of discretized problem are combinations of four standard one-dimensional integrals that can be tabulated. As a result, numerical integration is not needed. For regular grids of approximating nodes, this matrix has Toeplitz’s structure, and matrix-vector products can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problem. Calculation of crack opening vectors, differential, and total cross-sections of circular and elliptic cracks are performed for longitudinal incident waves orthogonal to the crack surfaces. Dependencies of these characteristics on the medium permeability and wavefrequency are studied. Comparison of a crack in the poroelastic medium and in a dry elastic medium with the same porosity and skeleton elastic properties is presented.     

Prediction of negative dispersion by a nonlocal poroelastic theory

The objective of this work is to show that the negative dispersion of ultrasonic waves propagating in cancellous bone can be explained by a nonlocal version of Biot's theory of poroelasticity. The nonlocal poroelastic formulation is presented in this work and the exact solutions for one- and two-dimensional systems are obtained by the method of Fourier transform. The nonlocal phase speeds for solid- and fluid-borne waves show the desired negative dispersion where the magnitude of dispersion is strongly dependent on the nonlocal parameters and porosity. Dependence of the phase speed and attenuation is studied for both porosity and frequency variation. It is shown that the nonlocal parameter can be easily estimated by comparing the theoretical dispersion rate with experimental observations. It is also shown that the modes of Lamb waves show similar negative dispersion when predicted by the nonlocal poroelastic theory.     

Investigation of the phase velocities of guided acoustic waves in soft porous layers

A new experimental method for measuring the phase velocities of guided acoustic waves in soft poroelastic or poroviscoelastic plates is proposed. The method is based on the generation of standing waves in the material and on the spatial Fourier transform of the displacement profile of the upper surface. The plate is glued on a rigid substrate so that it has a free upper surface and a nonmoving lower surface. The displacement is measured with a laser Doppler vibrometer along a line corresponding to the direction of propagation of plane surface waves. A continuous sine with varying frequencies was chosen as excitation signal to maximize the precision of the measurements. The spatial Fourier transform provides the wave numbers, and the phase velocities are obtained from the relationship between wave number and frequency. The phase velocities of several guided modes could be measured in a highly porous foam saturated by air. The modes were also studied theoretically and, from the theoretical results, the experimental results, and a fitting procedure, it was possible to determine the frequency behavior of the complex shear modulus and of the complex Poisson ratio from 200 Hz to 1.4 kHz, in a frequency range higher than the traditional methods.     

Seismic effects of incident P waves on an embedded foundation in poroelastic half-space

Dynamic vibrations of a circular rigid foundation, which is embedded in poroelastic soil and subjected to incident P waves, are studied by semi-analytical methods in this present work. The motion of the soil is governed by Biot's dynamic poroelastic theory. A set of potentials are introduced to represent the incident waves, and the scattering waves caused by the foundation are considered based on the decomposition of the total wave field in soil. The soil along the vertical side of the foundation is assumed to be composed of series of infinitesimally thin poroelastic layers, while the soil under the foundation base is regarded as the poroelastic half-space and to be independent of the overlying soil. The interaction problem is solved by Hankel transforms. Then, combining the boundary conditions along the contact surface between the soil and the foundation and the dynamic equilibrium equation of the foundation, expressions of the vertical and rocking vibration amplitudes of the embedded foundation excited by the incident P waves are acquired. Numerical results are presented to demonstrate the influences of embedded depth, foundation mass, pore water in the soil and incident angle on the vibrations of the foundation.     

Axially symmetric vibrations of fluid-filled poroelastic circular cylindrical shells

Employing Biot's theory of wave propagation in liquid saturated porous media, axially symmetric vibrations of fluid-filled and empty poroelastic circular cylindrical shells of infinite extent are investigated for different wall-thicknesses. Let the poroelastic cylindrical shells are homogeneous and isotropic. The frequency equation of axially symmetric vibrations each for a pervious and an impervious surface is derived. Particular cases when the fluid is absent are considered both for pervious and impervious surfaces. The frequency equation of axially symmetric vibrations propagating in a fluid-filled and an empty poroelastic bore, each for a pervious and an impervious surface is derived as a limiting case when ratio of thickness to inner radius tends to infinity as the outer radius tends to infinity. Cut-off frequencies when the wavenumber is zero are obtained for fluid-filled and empty poroelastic cylindrical shells both for pervious and impervious surfaces. When the wavenumber is zero, the frequency equation of axially symmetric shear vibrations is independent of nature of surface, i.e., pervious or impervious and also it is independent of presence of fluid in the poroelastic cylindrical shell. Non-dimensional phase velocity for propagating modes is computed as a function of ratio of thickness to wavelength in absence of dissipation. These results are presented graphically for two types of poroelastic materials and then discussed. In general, the phase velocity of an empty poroelastic cylindrical shell is higher than that of a fluid-filled poroelastic cylindrical shell.The phase velocity of a fluid-filled bore is higher than that of an empty poroelastic bore. Previous results are shown as a special case of present investigation. Results of purely elastic solid are obtained.     

激光在近表面弹性性质梯度变化的材料中激发超声波的数值分析

激光激发超声波为评价材料近表面弹性性质提供了有效的手段. 考虑到由于冲击硬化、表面热处理、表面氧化等引起的金属材料近表面层弹性性质的变化，建立了一种激光在基底上的梯度材料中激发超声波的理论模型. 用有限元方法模拟了热弹条件下脉冲激光作用于材料上表面激发出的超声波及其传播过程，研究了近表面层离散的层数对超声波的产生和传播的影响，并分别讨论了表面层“变硬”和“变软”两种情况下声表面波的模式变化及用二维傅里叶变换得到各模式的色散曲线. 为进一步研究近表面层的弹性性质建立合理的计算模型及材料性质的反演提供了理论依 关键词： 超声波 有限元法 近表面弹性性质 色散     

Dynamic response of a poroelastic stratum to moving oscillating load

The dynamic response of a poroelastic stratum subjected to moving load is studied. The governing dynamic equations for poroelastic medium are solved by using Fourier transform. The general solutions for the stresses and displacements in the transformed domain are established. Based on the general solutions, with the consideration of boundary conditions, the final expressions of stresses and displacements in physical domain are put forward for the three-dimensional single-layer medium. Some numerical solutions for the stresses, displacements and pore fluid pressure are presented and reveal that the response of a poroelastic stratum varies obviously with the moving velocity.     

A finite difference method for a coupled model of wave propagation in poroelastic materials

A computational method for time-domain multi-physics simulation of wave propagation in a poroelastic medium is presented. The medium is composed of an elastic matrix saturated with a Newtonian fluid, and the method operates on a digital representation of the medium where a distinct material phase and properties are specified at each volume cell. The dynamic response to an acoustic excitation is modeled mathematically with a coupled system of equations: elastic wave equation in the solid matrix and linearized Navier-Stokes equation in the fluid. Implementation of the solution is simplified by introducing a common numerical form for both solid and fluid cells and using a rotated-staggered-grid which allows stable solutions without explicitly handling the fluid-solid boundary conditions. A stability analysis is presented which can be used to select gridding and time step size as a function of material properties. The numerical results are shown to agree with the analytical solution for an idealized porous medium of periodically alternating solid and fluid layers.     

傅立叶变换用于电机转速的测量

提出了一种用傅立叶变换进行电机转速测量的新方法,以Labview为平台建立了实验系统,利用采集卡（或计算机声卡）检测被与电机同轴连接的齿盘调制的光电信号,由信号的幅度谱计算电机的转速。使用斩波器对此方法进行了实验验证,当转速范围为100—6000r／m,测量结果和实际值完全吻合。     

Laser-induced ultrasonic waves in steels with gradient changes of elastic property

Taking account of the gradient changes of the near-surface elastic property, a model of multiple gradient layers on a semi-infinite substrate is established. A finite element method (FEM) model is developed to simulate the laser-induced thermo-elastic generation and propagation of acoustic waves in the system. The two-dimensional fast Fourier transform (2-D FFT) method has been used to analyze the ultrasonic signal. The phase velocity dispersion is obtained for a number of waves of different modes obtained and is analyzed in systems with different spatial variations of the material elastic properties and different affected layer depths. The model can be used to extract the elastic parameters of the near-surface regions in an effective way.     

The radiation power of elastic waves excited in a solid half-space by a subsurface time-harmonic source

The method of Fourier transforms is used to solve the problem of excitation of longitudinal, transverse, and Rayleigh surface waves by a time-harmonic point source placed in a homogeneous isotropic, perfectly elastic half-space and acting along the normal surface. Expressions for the time-average radiation powers of the aforementioned waves are obtained by the method of radiation reaction without using any approximations. The distribution of radiation power over different types of waves depending on their velocities and the source’s depth is investigated in detail.     

Verification of surface wave solutions obtained by the reciprocity theorem

Surface wave motions generated by a time-harmonic point load applied at the surface of an isotropic linearly elastic half-space are conventionally solved by the use of integral transform techniques. The inverse transforms, are often complicated and will not always yield closed-form solutions. In this paper expressions for the displacements for surface wave motions radiated from point-load excitation are determined in a simple manner by the use of the elastodynamic reciprocity theorem. It is shown that the radiated amplitudes of the surface displacements obtained by the reciprocity approach are identical to the corresponding results obtained by the use of Hankel transform and by Lamb in his classical paper.     

Rayleigh wave and detection of low-velocity layers in a stratified half-space

The excitation and propagation of the guided waves in a stratified half-space and a Rayleigh wave exploration method in shallow engineering seismic exploration are studied in this paper. All the modes of the guided waves are calculated by the bisection method in the case where the low velocity layers are contained in a stratified half-space. Cases when the formation shear wave velocity gradually decreases from the top to the bottom layers are also studied. The dispersion curves obtained in actual Rayleigh wave exploration are usually noncontinual zigzag curves, but the dispersion curves given by the elastic theory for given modes of the guided waves are smooth and continual curves. In this paper, the mechanism of zigzag dispersion curves in Rayleigh wave exploration is investigated and analyzed thoroughly. The zigzag dispersion curves can give not only the possible positions of the low-velocity layers but also the other information on the formation structure (fractures, oil, gas, etc.). It is found that the zigzag dispersion curves of the Rayleigh wave are the result of the leap of the modes and the existence of low velocity layers in a stratified half-space. The effects of the compressional wave velocity, shear wave velocity, and density of each layer on zigzag dispersion curves and the relationship of the low velocity layers to zigzag dispersion curves are also investigated in detail. Finally, the exploration depth of the Rayleigh wave is discussed. The exploration depth of the Rayleigh wave is equal to the wavelength multiplied by a coefficient that is variable and usually given by the work experience and the formation properties of the local work area.     

Seismic wave propagation in laterally inhomogeneous geological region via a new hybrid approach

2D seismic wave propagation in a local multilayered geological region rested in an inhomogeneous half-space with a seismic source is studied. Plane strain state is suggested. The vertical variation of the soil properties in the half-space is modelled by a set of horizontal flat isotropic, elastic and homogeneous layers. The finite local region is with non-parallel layers and free surface relief. Efficient hybrid wavenumber integration-boundary integral equation method (WNI-BIEM) is proposed, validated and applied for synthesis of seismic signals in the finite soil stratum. The numerical simulation reveals that the developed hybrid method is able to demonstrate the sensitivity of the obtained synthetic signals to the seismic source properties, to the heterogeneous character of the wave path and to the relief peculiarities of the local stratified geological deposit. The advantages and disadvantages of the proposed method are discussed.     

3D analysis of in-filled trench as passive barriers for ground vibration isolation

The three-dimensional (3D) problem of the ground vibration isolation by an in-filled trench as a passive barrier is studied theoretically. Integral equations governing Rayleigh wave scattering are derived based on the Green’s solution of Lamb problem. The integral equations are solved accurately and efficiently with an iteration technique. They are used to evaluate the complicated Rayleigh wave field generated by irregular scatterers embedded in an elastic half-space solid. The passive isolation effectiveness of ground vibration by the in-filled trench for screening Rayleigh wave is further studied in detail. Effects of relevant parameters on the effectiveness of vibration isolation are investigated and presented. The results show that a trench filled with stiff backfill material gets a better isolation effect than a soft one, and increasing the depth or width of the in-filled trench also improves its screening effectiveness. The effectiveness and the area of the screened zone are surging with the increase in the length of the in-filled trench. Supported by the National Natural Science Foundation of China (Grant Nos. 50678128 and 50538010) and the Research Fund for PhD Student of Chinese College (Grant No. 20050247030)     

Analysis of multiscale scattering and poroelastic attenuation in a real sedimentary rock sequence

Compressional waves in heterogeneous permeable media experience attenuation from both scattering and induced pore scale flow of the viscous saturating fluid. For a real, finely sampled sedimentary sequence consisting of 255 layers and covering 30 meters of depth, elastic and poroelastic computer models are applied to investigate the relative importance of scattering and fluid-flow attenuation. The computer models incorporate the known porosity, permeability, and elastic properties of the sand/shale sequence in a binary medium, plane layered structure. The modeled elastic scattering attenuation is well described by stochastic medium theory if two-length scale statistics are applied to reflect the relative thickness of the shale layers when compared to the sand layers. Under the poroelastic Biot/squirt flow model, fluid-flow attenuation from the moderate permeability (10(-14) m2) sands may be separated in the frequency domain from the attenuation due to the low permeability (5 x 10(-17) m2) shale layers. Based on these models, the overall attenuation is well approximated by the sum of the scattering attenuation from stochastic medium theory and the volume weighted average of the attenuations of the sequence member rocks. These results suggest that a high permeability network of sediments or fractures in a lower permeability host rock may have a distinct separable attenuation signature, even if the overall volume of high permeability material is low. Depending on the viscosity of the saturating fluid, the magnitude of the flow-based attenuation can dominate or be dominated by the scattering attenuation at typical sonic logging frequencies (approximately 10 kHz).     

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.     

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.