不同饱和度土层分界面上剪切波的反射与透射

基于多相孔隙介质弹性理论,给出了非饱和土中不同弹性波的传播方程。根据分界面上的边界条件,建立了各势函数波幅值之间的关系式,讨论了入射剪切波在不同饱和度土层分界面上的反射与透射问题。在无限空间非饱和土体中存在3种压缩波和1种剪切波,因此,当剪切波传播到不同饱和度的非饱和土层分界面上将分别在上、下土层激发产生4种反射波和4种透射波。推导出不同反射波和透射波的振幅比例系数和能量比例系数的理论表达式,并且在此基础上进行数值分析。在数值算例中分别研究了各反射波与透射波的能量比例系数（即能量反射率和能量透射率）受入射频率、入射角度以及上、下土层土体饱和度变化的影响情况。计算结果表明：各能量反射率和能量透射率不仅与入射角和入射频率有关,而且其受上、下土层饱和度变化的影响也同样不能忽视。     

REFLECTION AND TRANSMISSION OF SHEAR WAVES IN MONOCLINIC MEDIA

This paper deals with the propagation, reflection and transmission of shear waves in monoclinic media. The dispersion equation for a monoclinic layer overlying a monoclinic half-space has been obtained and curves are plotted. The amplitude ratios for both the reflected and transmitted waves due to reflection of shear waves at the interface of two monoclinic half-spaces have also been computed and the numerical results are presented graphically. The results are compared with the isotropic case. It has been observed that, in monoclinic media, the amplitude ratios for reflected and transmitted wave increases approximately by 25 and 50 per cent respectively, in comparison to the isotropic case. © 1997 John Wiley & Sons, Ltd.     

PROPAGATION OF TORSIONAL SURFACE WAVES IN VISCOELASTIC MEDIUM

It is well known that an elastic homogeneous half-space does not allow torsional surface waves to propagate. The present paper attempts to find out the possibility of propagation of such waves in a viscoelastic half-space. The study reveals that although the homogeneous elastic half-space does not allow torsional surface waves to propagate, a viscoelastic half-space does so. The wave is damped due to the viscoelastic parameter. It has also been found that as the viscoelastic parameter decreases, the medium becomes elastic and the torsional surface waves ceases to propagate.     

饱和孔隙介质中非均匀震电平面波

采用描述地下流体饱和孔隙介质中弹性波与电磁场耦合现象的Pride方程组,利用Helmholtz分解,求解了非均匀震电耦合平面波解,推导了流体饱和孔隙介质中传播的非均匀震电平面波的能流表达式,讨论了非均匀震电平面波的能量特征.结果表明,非均匀震电平面波的平均能流在传播矢量和衰减矢量构成的平面内传播,伴随电磁场对纵波模式的能流没有贡献.当动电耦合系数为0时,震电平面波的能流表达式可退化为Biot弹性波的能流表达式。     

Comparison of transmitting boundaries in dynamic finite element analyses using explicit time integration

Explicit time integration schemes provide an efficient solution to non-linear dynamic finite element analyses of geotechnical problems especially when high frequency response is important. Such explicit time integration schemes require one of two distinct transmitting boundary formulations to overcome the problem of radiation damping. These are the superposition boundary approach, which involves the cancellation of the reflected waves by combining the solutions of two different boundary conditions and the viscous boundary approach, which involves the absorption of incident wave energy by frequency independent viscous dashpots. The theoretical justification of these two approaches and their means of implementation are reviewed. The solutions obtained using the two different boundary approaches to the problem of a rigid massless circular footing vibrating on an elastic half-space are compared with an independent theoretical solution. The performance of the boundaries for problems involving step loading is also examined and the implications for any loading pattern with a non-zero time average are discussed.     

Reflection of<Emphasis Type="Italic">P</Emphasis> and<Emphasis Type="Italic">SV</Emphasis> waves at the free surface of a monoclinic elastic half-space

The propagation of plane waves in an anisotropic elastic medium possessing monoclinic symmetry is discussed. The expressions for the phase velocity ofqP andqSV waves propagating in the plane of elastic symmetry are obtained in terms of the direction cosines of the propagation vector. It is shown that, in general,qP waves are not longitudinal andqSV waves are not transverse. Pure longitudinal and pure transverse waves can propagate only in certain specific directions. Closed-form expressions for the reflection coefficients ofqP andqSV waves incident at the free surface of a homogeneous monoclinic elastic half-space are obtained. These expressions are used for studying numerically the variation of the reflection coefficients with the angle of incidence. The present analysis corrects some fundamental errors appearing in recent papers on the subject.     

Elastic waves along a cylindrical borehole in a poroelastic medium saturated by two immiscible fluids

The propagation of elastic waves along a cylindrical borehole filled with/without liquid and embedded in an infinite porous medium saturated by two immiscible fluids has been studied. The theory of porous media saturated by two immiscible fluids developed by Tuncay and Corapcioglu (1997) is employed. Frequency equations determining the phase velocity of axial symmetric waves are obtained. It is found that the surface waves along cylindrical borehole are dispersive. The dispersion equation of Rayleigh-type surface waves along the boundary of a poroelastic solid half-space saturated by two immiscible fluids is also obtained. Some special cases have been deduced and the dispersion curves are obtained numerically for a peculiar model. It is found that the density of fluids affects the Rayleigh mode.     

Effect of initial stress on reflection at the free surface of anisotropic elastic medium

The propagation of plane waves is considered in a general anisotropic elastic medium in the presence of initial stress. The Christoffel equations are solved into a polynomial of degree six. The roots of this polynomial represent the vertical slowness values for the six quasi-waves resulting from the presence of a discontinuity in the medium. Three of these six values are identified with the three quasi-waves traveling in the medium but away from its boundary. Reflection at the free plane surface is studied for partition of energy among the three reflected waves. For post-critical incidence, the reflected waves are inhomogeneous (evanescent) waves. Numerical examples are considered to exhibit the effects of initial stress on the phase direction, attenuation and reflection coefficients of the reflected waves. The phase velocities and energy shares of the reflected waves change significantly with initial stress as well as anisotropic symmetry. The presence of initial stress, however, has a negligible effect on the phase directions of reflected waves.     

Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion

The governing equations for generalized thermodiffusion in an elastic solid are solved. There exists three kinds of dilatational waves and a Shear Vertical (SV) wave in a two-dimensional model of the solid. The reflection phenomena of P and SV waves from free surface of an elastic solid with thermodiffusion is considered. The boundary conditions are solved to obtain a system of four non-homogeneous equations for reflection coefficients. These reflection coefficients are found to depend upon the angle of incidence of P and SV waves, thermodiffusion parameters and other material constants. The numerical values of modulus of the reflection coefficients are presented graphically for different values of thermodiffusion parameters. The dimensional velocities of various plane waves are also computed for different material constants.     

Torsional surface waves in inhomogeneous elastic media

The dispersion of torsional surface waves in two types of inhomogeneous elastic media is discussed (1) in a half-space with shear modulus varying linearly with depth and with constant density (Gibson'S half-space) and (2) in a half-space with shear modulus varying with the square-root of the depth co-ordinate and with constant density. Inhomogeneous media with constant shear wave velocity are also reviewed and the obtained solutions are discussed in relation to appropriate boundary conditions.     

Deformation of a layered half-space due to a very long tensile fault

The problem of the coseismic deformation of an earth model consisting of an elastic layer of uniform thickness overlying an elastic half-space due to a very long tensile fault in the layer is solved analytically. Integral expressions for the surface displacements are obtained for a vertical tensile fault and a horizontal tensile fault. The integrals involved are evaluated approximately by using Sneddon’s method of replacing the integrand by a finite sum of exponential terms. Detailed numerical results showing the variation of the displacements with epicentral distance for various source locations in the layer are presented graphically. The displacement field in the layered half-space is compared with the corresponding field in a uniform half-space to demonstrate the effect of the internal boundary. Relaxed rigidity method is used for computing the postseismic deformation of an earth model consisting of an elastic layer of uniform thickness overlying a viscoelastic half-space.     

Quasi-static deformation of a layered half-space by a long strike-slip fault

Theoretical expressions for the surface displacement and shear stress caused by a long strike-slip dislocation in an elastic layer overlying an elastic half-space are derived and the correspondence principle is used to obtain the quasi-static response when the half-space is Maxwell-viscoelastic. Variation of the surface displacement and shear stress with horizontal distance is studied for various times and vertical extents of the fault. It is seen that the quasi-static response differs significantly from the corresponding elastic response.     

PROPAGATION OF TORSIONAL SURFACE WAVES IN A HOMOGENEOUS SUBSTRATUM OVER A HETEROGENEOUS HALF-SPACE

The paper discusses the propagation of torsional surface wave in a homogeneous substratum over a half-space with linearly varying rigidity and density. The study reveals that under assumed conditions, a torsional surface wave propagates in the medium. The velocities of torsional surface waves have been calculated numerically and are presented in a number of graphs. It is also observed that for a stratum over a homogeneous half-space, the velocity of torsional surface waves coincides with that of Love waves. For a non-homogeneous half-space it is observed that the velocity of torsional surface waves is always higher than that of Love waves propagated in a homogeneous layer over a homogeneous half-space. An attempt is also made to assess the possible propagation of torsional surface waves in a half-space with linearly varying rigidity and density, lacking a superficial layer. It is concluded that such a half-space allows two solutions for torsional waves while a homogeneous half-space has one.     

淤砂层模拟方法对上层理想流体动力反应的影响——平面P波入射

采用解析法,以弹性半空间-淤砂层-理想流体层为对象,分别将淤砂层模拟为固体介质和流体介质,分析了平面P波入射时上层理想流体的动力反应特性。通过算例分析,比较了将淤砂层模拟为弹性固体、黏弹性固体、理想流体和黏性流体介质时与将淤砂层模拟为两相多孔介质时计算结果的差异。研究表明当渗透系数很小时,多孔介质可以近似简化为黏弹性固体,甚至线弹性固体。这一分析研究对于在高坝-库水-淤砂-地基系统地震反应分析的数值模型中采用合理的淤砂层模型,提高计算效率具有重要参考意义,而且还可以用于校核数值模型的合理性。     

The Town Effect: Dynamic Interaction between a Group of Structures and Waves in the Ground

In a conventional approach, the mechanical behaviour of a structure subjected to seismic or blast waves is treated separately from its surroundings, and in many cases, the dynamic coupling effect between multiple structures and the waves propagating in the ground is disregarded. However, if many structures are built densely in a developed urban area, this dynamic interaction may not become negligible. The first purpose of this contribution is to briefly show the effect of multiple interactions between waves and surface buildings in a town. The analysis is based on a recently developed, fully coupled, rigorous mathematical study, and for simplicity, each building in the town is represented by a rigid foundation, a mass at the top and an elastic spring that connects the foundation and mass. The buildings stand at regular spatial intervals on a linear elastic half-space and are subjected to two-dimensional anti-plane vibrations. It is found that the buildings in this model significantly interact with each other through the elastic ground, and the resonant (eigen) frequencies of the collective system (buildings or town) become lower than that of a single building with the same rigid foundation. This phenomenon may be called the “town effect” or “city effect.” Then, second, it is shown that the actual, unique structural damage pattern caused by the 1976 Friuli, Italy, earthquake may better be explained by this “town effect,” rather than by investigating the seismic performance of each damaged building individually. The results suggest that it may also be possible to evaluate the physical characteristics of incident seismic/blast waves “inversely” from the damage patterns induced to structures by the waves.     

Existence of longitudinal waves in pre-stressed anisotropic elastic medium

In a pre-stressed anisotropic elastic medium, three types of quasi-waves propagate along an arbitrary direction. In general, none of the waves is truly longitudinal. The present study finds the specific directions in a pre-stressed anisotropic elastic medium along which longitudinal waves may propagate. This paper demonstrates how the propagation of longitudinal waves is affected by various pre-stresses present in the medium. The study establishes the explicit expressions defining the existence and propagation of longitudinal waves in pre-stressed anisotropic elastic medium. These expressions involve not only the direction and elastic stiffness of the medium, but also the pre-stresses present in the medium. Changes in conditions for the existence of longitudinal waves in orthotropic, monoclinic and triclinic anisotropies are discussed in detail. The most important part of the paper is a practical aspect suggested to calculate the specific directions for the existence of longitudinal waves in pre-stressed anisotropic elastic medium. In this approach, only those parameters are used that can be observed by the receiver in a geophysical experiment of wave propagation. The existence of longitudinal waves has been shown graphically using a numerical example for three types of anisotropic symmetries in elastic medium.     

The interaction between a circular elastic plate and a transversely isotropic elastic half-space

The axisymmetric problem of flexure of a centrally loaded or uniformly loaded thin-elastic plate resting on or buried in a transversely isotropic elastic half-space is solved using a constraint variational scheme. A fundamental solution corresponding to a unit vertical pressure acting over an annular region in the interior of the half-space which is derived through the application of Hankel integral transforms is used to evaluate the strain energy of the elastic half-space. The deflected shape of the plate is represented by a power series of radial coordinate together with a term corresponding to a concentrated load derived from classical plate theory. The coefficients of the power series representation, which could be considered as generalized coordinates corresponding to the interaction problem, are determined through the minimization of a constraint energy functional. The energy functional consists of the strain energies of the plate and half-space, the potential of the applied loads and a term corresponding to plate edge boundary conditions incorporated with a penalty number. The entire formulation is developed as a convenient and flexible matrix solution scheme. The numerical stability and convergence of the solution scheme are established, and the response of the plate is investigated over a range of relative rigidities of practical interest.     

Dispersion law of rayleigh-type waves in a compressible gibson half-space

An analytical-numerical examination has been made of the behavior of harmonic Rayleigh-type waves in a compressible elastic half-space of constant density and whose shear modulus increases linearly with depth. The dispersion law is presented for values of Poisson's ratio of 1/2, 1/3 and 1/4 for wide ranges of degree of inhomogeneity.     

P波在弹性介质与饱和冻土介质分界面上的透反射问题研究

基于弹性波在冻结饱和多孔介质与单相弹性介质中的传播理论,研究了平面P波在饱和冻土介质与单相弹性介质分界面上的透反射问题。利用Helmholtz矢量分解定理,根据分界面上的边界条件,获得了平面P波从单相弹性介质入射到饱和冻土介质分界面上透反射振幅比的理论表达式。通过数值计算,分析了在不同入射频率、胶结参数、孔隙率、饱和度和接触参数下,弹性波的透反射振幅比随入射角变化的关系。研究结果表明：P波从单相弹性介质垂直入射到饱和冻土介质分界面上时只有反射P波和3种透射P波产生,当掠入射时只产生反射而没有透射现象发生。入射频率、胶结参数、孔隙率、饱和度以及接触参数等参数对反射波和透射波的振幅比影响显著。