垂直地震力作用下弹性半空间上扁壳基础的解析解

用布希涅斯克定义的弹性半空间内的垂直位移包括两项积分,除了积分号前面系数的差别之外,第一项积分是单层位势而第二项积分为双层位势。若扁壳基础是正高斯曲率的几何曲面,则壳底与半空间表面间的挤压强度就是半空间表面作用的分布垂直荷载。当越过边界时,双层势位的函数值和单层势位的法向导数值发生跳跃。利用这些性质,本文得出布希涅斯克积分的反演公式,从而避开要求解偏微分—积分方程组的巨大数学困难而易于得出解析解。以椭园抛物面扁壳为例说明本文方法的应用。     

Two-dimensional scattering of in-plane body waves due to a discontinuity in bedrock

The direct boundary integral equation technique is used to study in-plane surface amplification of in-plane seismic body waves for the case of an inhomogeneity in a bedrock half-space. In the studied soil configuration, a soil layer rests on a rock half-space which includes a rock inclusion. The rock inclusion considered is a semi-infinite horizontal rock layer in which its upper boundary borders the soil layer. Materials in the soil–rock configuration are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to in-plane body waves. The study investigates varying the stiffness and the thickness of the rock inclusion for a range of frequencies and wave incidence angles. Anti-plane waves for this type of soil-rock configuration have been addressed in a previous article by Heymsfield (Earthquake Engng. Struct. Dyn. 28 : 841–855 (1999)). Copyright © 1999 John Wiley & Sons, Ltd.     

Two-dimensional scattering of SH waves in a soil layer underlain with a sloping bedrock

A two-dimensional analysis is applied to examine the effect that a sloping bedrock half-space has on the amplification of an anti-plane shear wave. The direct boundary integral equation method is used for the two-dimensional analysis. The particular soil–rock configuration investigated includes a homogeneous soil layer underlain by a sloping rock half-space. The rock half-space dips for a horizontal distance L and then becomes horizontal so that the overlying soil layer has a thickness H that remains constant from this point to infinity. The materials in the soil–rock configuration are considered viscoelastic except in the rock half-space below soil layer thickness H, which is considered elastic. This limitation in damping is due to the correction used for the truncation of the half-space boundary. Four cases are used to study the relationship between rock slope and surface displacement, vertical, 1:2, 1:4, 1:8. Surface displacements are determined for each of these cases for half-space incidence angles of 90, 75, and 60°. To allow for applicability to a wide range of problems, results are determined as a function of dimensionless parameters. In addition, solutions from a one-dimensional analysis are compared with the results of the two-dimensional analysis to establish limits outside of which a one-dimensional analysis suffices.     

弹性层状半空间中无限长洞室对斜入射 平面SH波的三维散射(Ⅱ) 数值结果与分析

以基岩上单一土层场地为例, 计算分析了在斜入射平面SH波作用下弹性层状半空间中无限长洞室附近的地表位移. 研究表明, 层状半空间中地下洞室对波的散射与均匀半空间情况存在显著差别. 层状场地由于考虑了场地自身的动力特性, 使得洞室附近地表位移幅值的空间变化更为复杂, 基岩与土层刚度比、 土层厚度对散射效应均有着重要影响. 随着基岩与土层刚度比的增大, 地表位移幅值整体上逐渐增大; 随着土层厚度的增大, 土层对地表位移幅值的影响逐渐减小. 在频域解答的基础上, 给出了层状半空间中洞室对斜入射SH波散射的时域解答, 并以Ricker波为例进行了数值计算.      

Coseismic and postseismic subsurface displacements and strains for a vertical strike-slip fault in a three-layer elastic medium

A three-layer elastic-gravitational fault displacement model using dislocation theory has been developed and used to examine the effect of layering of earth elastic moduli on surface and subsurface displacement fields for a vertical strike-slip fault. The model has been used to examine the effect of depth variation of elastic properties at coseismic and postseismic time scales. For pure strike-slip motion the effect of gravity on coseismic and postseismic horizontal deformation is negligible. For coseismic deformation the model predicts that (for constant Poisson's ratio) an increase in elastic moduli with depth attenuates the displacements within the upper layers with respect to displacement distribution for a uniform half-space, while an inclusion of a soft layer between the top layer and lower half-space amplifies upper layer displacements. The effect of variation in Poisson's ratio on surface and subsurface displacements has also been examined.The effect of postseismic stress relaxation on surface and subsurface displacements for a three-layer model has been calculated and compared with that of a uniformly relaxed half-space model. Layer 1 is assumed to correspond to the upper crust, layer 2 the lower crust and layer 3 the upper mantle. The effect of postseismic stress relaxation within a uniform half-space and within just the lower crust and upper mantle has been examined. Stress relaxation within the whole half-space decreases the amplitude and shortens the wavelength of displacements, while stress relaxation within the lower two layers increases the amplitude and broadens the wavelength of displacements. The difference between uniform and layered postseismic relaxation is particularly pronounced at the base of the crust.Coseismic and postseismic normal and volumetric strains for a vertical strike-slip fault have also been examined. For a uniformly relaxed half-space model, an increase in normal strains is shown with respect to the coseismic elastic solution, whereas the postseismic volumetric strain is effectively zero. For a three-layer model with stress relaxation in the lower layers only, the normal and volumetric strains within the top elastic layer resemble coseismic strains, while in the lower layers which suffer a rigidity decrease, the postseismic volumetric strain is effectively zero.     

MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.     

Time-harmonic boussinesq problem for a continuously non-homogeneous soil

An analytical solution is presented for the response of a non-homogeneous, compressible, elastic half-space to a time-harmonic vertical point load on its surface. The shear modulus is assumed to increase continuously with depth. The model is chosen so as to describe uniformly deposited cohesionless soils. Expressions for displacements and stresses in the interior of the half-space medium are derived by means of Hankel transforms and contour integration. Selected numerical results are presented to demonstrate the influence of non-homogeneity. Finally, some effects are pointed out to be used in connection with vibration tests for subsoil investigation.     

Dispersion function of Rayleigh waves in porous layered half-space system

Rayleigh wave exploration is based on an elastic layered half-space model. If practical formations contain porous layers, these layers need to be simplified as an elastic medium. We studied the effects of this simplification on the results of Rayleigh wave exploration. Using a half-space model with coexisting porous and elastic layers, we derived the dispersion functions of Rayleigh waves in a porous layered half-space system with porous layers at different depths, and the problem of transferring variables to matrices of different orders is solved. To solve the significant digit overflow in the multiplication of transfer matrices, we propose a simple, effective method. Results suggest that dispersion curves differ in a lowfrequency region when a porous layer is at the surface; otherwise, the difference is small.     

Rayleigh modes in anisotropic,heterogeneous poroelastic layers

The generalized Rayleigh type surface waves are studied in a multilayered medium consisting of anisotropic poroelastic solid layered stack beneath a fluid layer and overlying a heterogeneous elastic solid half-space. The heterogeneity, considered, is of vertical type. The interface between solid layer and half-space is treated as an imperfect interface and suitable boundary conditions are applied thereat. The technique of transfer matrix is used to obtain the dispersion equation in compact and convenient form. Numerical results are obtained for particular models. The effects of anisotropy and heterogeneity on the surface waves speed are discussed.     

ELASTIC EXTRAPOLATION OF PRIMARY SEISMIC P- AND S-WAVES1

The elastic Kirchhoff-Helmholtz integral expresses the components of the monochromatic displacement vector at any point A in terms of the displacement field and the stress field at any closed surface surrounding A. By introducing Green's functions for P- and S-waves, the elastic Kirchhoff-Helmholtz integral is modified such that it expresses either the P-wave or the S-wave at A in terms of the elastic wavefield at the closed surface. This modified elastic Kirchhoff-Helmholtz integral is transformed into one-way elastic Rayleigh-type integrals for forward extrapolation of downgoing and upgoing P- and S-waves. We also derive one-way elastic Rayleigh-type integrals for inverse extrapolation of downgoing and upgoing P- and S-waves. The one-way elastic extrapolation operators derived in this paper are the basis for a new prestack migration scheme for elastic data.     

SH Wave Number Green’s Function for a Layered,Elastic Half-Space. Part I: Theory and Dynamic Canyon Response by the Discrete Wave Number Boundary Element Method

We present a closed-form frequency-wave number (ω – k) Green’s function for a layered, elastic half-space under SH wave propagation. It is shown that for every (ω – k) pair, the fundamental solution exhibits two distinctive features: (1) the original layered system can be reduced to a system composed by the uppermost superficial layer over an equivalent half-space; (2) the fundamental solution can be partitioned into three different fundamental solutions, each one carrying out a different physical interpretation, i.e., an equivalent half-space, source image impact, and dispersive wave effect, respectively. Such an interpretation allows the proper use of analytical and numerical integration schemes, and ensures the correct assessment of Cauchy principal value integrals. Our method is based upon a stiffness-matrix scheme, and as a first approach we assume that observation points and the impulsive SH line-source are spatially located within the uppermost superficial layer. We use a discrete wave number boundary element strategy to test the benefits of our fundamental solution. We benchmark our results against reported solutions for an infinitely long circular canyon subjected to oblique incident SH waves within a homogeneous half-space. Our results show an almost exact agreement with previous studies. We further shed light on the impact of horizontal strata by examining the dynamic response of the circular canyon to oblique incident SH waves under different layered half-space configurations and incident angles. Our results show that modifications in the layering structure manifest by larger peak ground responses, and stronger spatial variability due to interactions of the canyon geometry with trapped Love waves in combination with impedance contrast effects.     

多层弹性半空间中的地震波（一）

为了了解地震震源和地球介质的性质,很有必要对地震波的辐射、传播和衰减问题作仔细的分析。作为一种近似,可以暂且忽略地球的曲率,把传播地震波的地球介质视为多层半空间。为简便起见,地震波的衰减问题另作考虑。这样,便需要研究多层、均匀、各向同性和完全弹性半空间中地震震源辐射的地震波传播问题。 用哈斯克尔（Haskell）矩阵法解多层介质中弹性波的传播问题是很方便的。如果     

Seismic waves due to storm over the sea: I

Summary Elastic wave propagation problem has been studied in the composite medium — a gravitating liquid layer overlying an elastic half-space. Assuming moving normal pressure load on the free surface, an attempt has been made to study the order of disturbance on the earth's surface. In the discussion asymptotic values of the displacement, components have been considered for small values of the parameter.     

Ground motion amplification due to elastic inclusions in a half-space

Scattering of plane harmonic SH, P, SV and Rayleigh waves by several inclusions of arbitrary shape, completely embedded into an elastic half-space, is considered. Perfect bonding between the half-space and the inclusions is assumed. The problem is investigated for linear, isotropic and homogeneous elastic materials. The displacement field is evaluated throughout the elastic medium so that the continuity conditions between the half-space and the inclusions are satisfied in mean-square sense. Numerical results of the surface displacement field are evaluated for single and two elliptic inclusions. The results show the following: (a) presence of a subsurface inhomogeneity may lead to large amplifications of the surface ground motion; (2) different surface displacement patterns emerge for different incident waves; (3) the presence of an additional inclusion may change significantly the surface displacement response of a single inclusion; (4) the surface motion extremes strongly depend upon (i) angle of incidence; (ii) frequency of incident field; (iii) embedment depth of the inclusions; (iv) separation distance between the inclusions; (v) material properties of the half-space and the inclusions; and (vi) location of observation point on the surface of the half-space.     

Two-dimensional scattering of SH waves due to a discontinuity in bedrock

The direct boundary integral equation technique is applied to determine the impact on surface amplification caused by an inhomogeneity in a bedrock half-space. The particular soil-rock configuration studied is one in which a soil layer rests on a rock half-space which includes a rock inclusion. The particular rock inclusion considered for this study is a semi-infinite rock layer with its upper boundary bordering the soil layer. Materials are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to a vertically incident shear wave. The study includes varying the stiffness and the thickness of the inclusion for a range of frequencies. Solutions from a one-dimensional analysis are compared with the results of a two-dimensional analysis in order to establish limits inside of which a two-dimensional analysis is required. Copyright © 1999 John Wiley & Sons, Ltd.     

Surface wave propagation in a transversely isotropic elastic layer overlying a liquid saturated porous solid half-space and lying under the uniform layer of liquid

Dispersion of Rayleigh-type surface wave is studied in a homogeneous transversely isotropic elastic layer overlying a nondissipative liquid-saturated porous solid half-space and lying under a uniform layer of homogeneous liquid. The frequency equation in the form of ninth-order determinant is obtained.Special cases have been deduced by reducing the depth of the layers to zero and by changing the transverse isotropic layer to an isotropic layer. Dispersion curves for the phase velocity have been plotted for a particular model.     

Axisymmetric transient waves in transversely isotropic half-space

A transversely isotropic material in the sense of Green is considered. Using a series of potential functions proposed in [Eskandari-Ghadi M. A complete solution of the wave equations for transversely isotropic media. J Elasticity 2005; 81:1–19], the solutions of the transient wave equations within a half-space under surface load are obtained in the Laplace–Hankel domain for axisymmetric problems. The solutions are investigated in detail in the special case of a surface point force pulse varying with time as Heaviside function. Using Cagniard–De Hoop method, the inverse Laplace transform and inverse Hankel transform of the solutions are then obtained in the form of integrals with finite limits. For validity of the analytical results, the final formulations for surface waves are degenerated for an isotropic material and compared with the existing formulation obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80], to show that they are exactly the same. The numerical evaluations of the integrals for some transversely isotropic materials as well as an isotropic one are obtained. The present approach is then numerically verified by comparing a particular case of displacements for the surface of an isotropic half-space subjected to a point load of Heaviside function with the solutions obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80]. In addition, the wave equations for the mentioned medium are obtained on the vertical line directly under the applied surface load. The final formulations are degenerated for an isotropic material and compared with the existing formulation given in Graff [Wave motion in elastic solids. New York: Dover Publications Inc; 1975 [New Ed edition, November 1991]], to show that they are also exactly the same. Then equations are presented in graphical forms using an appropriate numerical evaluation.     

Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space

Based on the analytical layer-element method, an analytical solution is proposed to determine the dynamic interaction between the elastic circular plate and transversely isotropic multilayered half-space. The dynamic response of the elastic circular plate is governed by the classical thin-plate theory with the assumption that the contact surface between the plate and soil is frictionless. The total stiffness matrix of the transversely isotropic multilayered half-space is acquired by assembling the analytical layer-element of each soil layer with the aid of the continuity conditions between adjacent layers. According to the displacement condition of coordination between the plate and soil, the dynamic interaction problem is reduced to that of multilayered transversely isotropic half-space subjected to axisymmetric harmonic vertical loading. Some numerical examples are given to study the vertical vibration of the plate, and the results indicate that the dynamic response of elastic circular plate depends strongly on the material properties of the soils, the rigidity of the plate, the frequency of excitation and the external load form.     

有薄疏松复盖层的弹性半空间中地震波的传播问题

本文将地表疏松风化层看作是附在弹性介貭上的一种有慣性而无弹性的薄层,当扰动在弹性介貭中传播吋,风化薄层跟随振动;討論了弹性半空間內点震源产生的地震波和疏松薄层对地震波反射的影响。     

Static deformation of a multilayered half-space by a very long strike-slip fault

Summary The formulas for the elastic residual field of a very long strike-slip fault in a multilayered elastic half-space are given. It is assumed that the medium is made up ofn parallel, isotropic and homogeneous layers lying over an isotropic, homogeneous half-space and being in welded contact. The formulas which allow the calculation of the displacement and stress fields due to an arbitrary very long strike-slip fault are derived. The explicit expressions of coefficients which appear in these formulas are listed for the casesn=1 andn=2.