Buried moving sources in an elastic halfspace

Summary The surface displacements ofP andS V types of motion produced in an homogeneous isotropic elastic halfspace by line- and couple-sources moving after creation parallel to the free surface with arbitrary speeds are obtained approximately by the method of steepest descent.     

On the buried source in a viscoelastic halfspace

Summary The disturbance due to a buried line source in an homogeneous anelastic halfspace with internal friction is investigated using Fourier transform technique. The displacement of the free surface of the halfspace is calculated by the application of the multiple saddle point approximation.

---

Zusammenfassung Die Störung erzeugt durch eine verborgene Quellenlinie in einem homogenen elastischem isotropen Halbraum mit innerer Reibung wird untersucht mit Hilfe einer Fourier Transformation. Die Verschiebung der freien Oberfläche des Halbraumes ist berechnet unter Verwendung der Sattelpunktsannäherung.

     

Dynamic behaviour of rigid foundations of arbitrary shape on a halfspace

An approximate method for computation of the compliance functions of rigid plates resting on an elastic or visco-elastic halfspace excited by forces and moments in all degrees of freedon is presented. The method is based on a Green's function approach. These functions are given for all degrees of freedom in form of well-behaved integrals. The numerical procedure is described and is used to evaluate the vertical, horizontal, rocking and torsion compliance functions of rectangular plates with side ratios 1 ≤ b/a ≤ 10 and non-dimensional frequency 0≤a₀≤10. It is shown how this method can be extended to problems concerning a linear visco-elastic halfspace and a halfspace with variable stiffness.     

Vibration of viscoelastic foundations

Simple approximate solutions are presented for the steady-state response of a massless, rigid disk, which is supported at the surface of a linear viscoelastic halfspace and is excited by a harmonically varying horizontal or vertical force, or by a harmonically varying moment about an axis in the plane of the disk. The halfspace is idealized either as a standard Voigt solid or as a constant hysteretic solid. The solution for the massless disk is then applied to the analysis of foundations with mass, and for each mode of excitation, comprehensive numerical data are presented to illustrate the effects of viscoelastic action on the response of both the massless disk and of foundations with mass It is shown that the principal effects of material damping are to increase the damping capacity of the foundation and to reduce its ‘natural’ frequency. In the mathematically equivalent representation of the elastically supported massless disk as a spring-dashpot combination without mass, it is shown that the effective stiffness of the spring may become negative over significant ranges of the exciting frequency, and that it is preferable to model the disk-halfspace system as an oscillator with mass, taking its spring constant equal to the static stiffness of the original system.     

Stresses and excess pore pressure induced in saturated poroelastic halfspace by moving line load

This paper examines stresses and excess pore fluid pressure that are induced in a saturated poroelastic soil of halfspace extent by a concentrated line load. The line load is moving at a constant velocity along the surface of the poroelastic halfspace. The governing equations for the proposed analysis are based on the Biot's theory of dynamics in saturated poroelastic soils. The governing partial differential equations are solved using Fourier transforms. The solutions for the stresses and excess pore pressure are expressed in the forms of inverse Fourier transforms. The numerical results are obtained by performing the numerical inversion of the transform integrals. A parametric study is presented to illustrate the influences of the velocity of moving load and the poroelastic material parameters on the stresses and excess pore pressure. At a high velocity, the maximum values of the stresses in a poroelastic halfspace are smaller than those in an elastic solid, whilst at a low velocity the stresses in a poroelastic halfspace are larger than those in an elastic halfspace. The potential of diffusivity has an important influence on the stresses and excess pore pressure.     

Ground subsidence due to uniform fluid extraction over a circular region within an aquifer

This paper examines the ground subsidence caused by the uniform extraction of fluids from a disc-shaped region located within a poroelastic halfspace. Biot’s theory of poroelasticity is used to examine the problem. The fluid extraction over the circular region at the interior of the poroelastic domain induces time-dependent axisymmetric surface settlements. The theoretical results are also used to examine the accuracy of a multi-physics computational scheme that can be used to examine more complex geological settings.     

A RECTANGULAR LOOP SOURCE OF CURRENT ON A TWO-LAYERED EARTH*

Using the concept of reciprocity and the known solution for the electric field of a vertical oscillating magnetic dipole source placed over a two-layered halfspace, an integral expression for the vertical magnetic field produced by a horizontal rectangular loop, carrying an oscillating current and placed on the surface of the same halfspace, is deduced. This expression is such that it can be evaluated by a combination of straightforward numerical integration and digital linear filter techniques. Displacement currents everywhere in space are neglected. Fields both inside and outside the loop are presented as sounding curves. A little over a decade of frequencies (100–3000 Hz) appears to be the minimum requirement and two decades (100–10000 Hz) are sufficient for most situations. Parametric and geometric sounding curves both show good resolution of subsurface layering. Phase shows better resolution than amplitude. Similarly, resolution is higher when conductivity increases with depth than when it decreases. Besides being useful for the electromagnetic depth sounding for layered earth structures, such computations can be applied to determine normal corrections to Turam observations, whenever a rectangular loop of finite size is used as a transmitter.     

The response of a horizontal conducting cylinder embedded in a uniform earth for uniform and line current sources

The response of a horizontal conducting cylinder embedded in a uniform conducting earth is studied using mathematical models of uniform and line current source excitation for the period range 10 to 10⁴ s. The line current source is located at heights ranging from 100–750 km above the surface of the earth. From the calculated results, it is shown that for periods greater than 10³ s the ratioE _x /H _y at the surface of the earth for localized fields, such as the auroral and equatorial electrojet normally situated at heights of about 100 km, is considerably different from that for a uniform source. The results presented also show that the magneto-telluric method of geophysical prospecting for ore bodies in regions of the electrojet may not be very practicable for periods exceeding 10³ s.     

Temporal variations in the tidal response of the medium in the vicinities of the sources of catastrophic earthquakes

The idea of predicting earthquakes by continuously monitoring temporal variations in the tidal response of the medium was suggested by Beaumont and Berger in 1974. However, it became possible to implement the idea only recently. This possibility has arisen due to the deployment of Global Seismic Network (GSN), which collects the data on tidal tilts and gravity in the epicenters of strong earthquakes before and after the strongest events. In this paper, we present the results of model analytical and numerical calculations of the elastic displacements of the Earth’s surface caused by the earthquakes and their preparatory processes. The analytical calculations are limited to the model of a uniform elastic halfspace; the numerical calculations, in addition to this model, also cover the models with radially heterogeneous distributions of elastic moduli in the crust and in the upper mantle, which are determined by the PREM model. We describe the results of modeling temporal variations in the tidal response of the medium in the vicinity of the source of a catastrophic earthquake. The model of seismic source is specified by the length and the orientations of the fault plane and by the value of the discontinuity in the tangential component of the displacement vector on the opposite sides of the fault. The model is based on the GPS data on the horizontal and vertical displacements of the Earth’s surface. We suggest the method for determining temporal changes in the tidal response of the medium in the seismically active regions. This method improves the sensitivity and time resolution of the standard techniques of sliding-window analysis by more than an order of magnitude. The comparative analysis of temporal variations in the tidal response of the medium in the zones of the magnitude 9 earthquake in Japan (March 9, 2011) illustrates the described approach.

     

Approximate dynamic model of embedded foundation in time domain

A discrete model to represent the unbounded soil (halfspace) in a soil–structure interaction analysis in the time domain is developed. For each dynamic degree of freedom of the foundation node, the discrete model consists of a mass M₀ which is attached to a rigid support with a spring K and with a damper C₀. In addition, a free node with the mass M₁ is introduced, which is connected to the foundation node with a damper C₁. All coefficients are frequency-independent. The discrete model is semi-empirical. It is based on a semi-infinite truncated cone, whereby, after enforcing the static stiffness, the remaining parameters are modified to achieve an optimal fit of the dynamic-stiffness coefficient in the frequency domain. The spring K is equal to the static stiffness. The coefficients appearing in the equations for the dampers C₀, C₁ and the masses M₀, M₁ are specified (assuming a homogeneous halfspace) for the disc, the embedded cylinder, the rectangle (also embedded) and the strip. A square on a layer whose stiffness increases with depth resting on a homogeneous halfspace is also treated. For an embedded foundation, eccentricities arise. Material damping increases the damper C₀ and the mass M₀.     

Temporal variations in the tidal response of the medium in the vicinities of the sources of catastrophic earthquakes

The idea of predicting earthquakes by continuously monitoring temporal variations in the tidal response of the medium was suggested by Beaumont and Berger in 1974. However, it became possible to implement the idea only recently. This possibility has arisen due to the deployment of Global Seismic Network (GSN), which collects the data on tidal tilts and gravity in the epicenters of strong earthquakes before and after the strongest events. In this paper, we present the results of model analytical and numerical calculations of the elastic displacements of the Earth??s surface caused by the earthquakes and their preparatory processes. The analytical calculations are limited to the model of a uniform elastic halfspace; the numerical calculations, in addition to this model, also cover the models with radially heterogeneous distributions of elastic moduli in the crust and in the upper mantle, which are determined by the PREM model. We describe the results of modeling temporal variations in the tidal response of the medium in the vicinity of the source of a catastrophic earthquake. The model of seismic source is specified by the length and the orientations of the fault plane and by the value of the discontinuity in the tangential component of the displacement vector on the opposite sides of the fault. The model is based on the GPS data on the horizontal and vertical displacements of the Earth??s surface. We suggest the method for determining temporal changes in the tidal response of the medium in the seismically active regions. This method improves the sensitivity and time resolution of the standard techniques of sliding-window analysis by more than an order of magnitude. The comparative analysis of temporal variations in the tidal response of the medium in the zones of the magnitude 9 earthquake in Japan (March 9, 2011) illustrates the described approach.     

Seismic behaviour of girder bridges for horizontally propagating waves

The results obtained from a parametric study on the influence of horizontally propagating waves for the earthquake behaviour of continuous girder bridges are presented. The investigation is performed for floating-supported bridges excited by an earthquake acting in their vertical planes. The shallow foundations are assumed to be supported on the surface of either a visco-elastic halfspace or a visco-elastic layer resting on an elastic halfspace. Steady-state response for harmonic excitation and transient response for an artificial time history are investigated. Approximate equations describing the influence of horizontally propagating waves are presented and criteria are derived indicating when an increase of the internal forces and of the relative motion between the girder and the abutments can be expected.     

Electromagnetic response of a buried cylindrical conductor for sheet and line current sources

A major problem in electromagnetic induction studies in regions of localized source fields, such as the auroral and equatorial electrojet regions, is the source effect. Using an analytical model, the electromagnetic response of a buried conducting cylinder to sheet current and line current excitations has been studied for the period rangeT=5 s to 24 h. The validity of the numerical results obtained from the analytical model are compared with the numerical results obtained from a finite difference model. The results show that for periods less than 30 min, there is no significant difference in the response of the cylinder to both source fields. However, significant differences are observed at longer periods. It was also observed that the equivalent height at which a uniform sheet current at 100 km above the earth's surface can be approximated by a line current varies as a function of the source period.     

Vertical vibration of ring foundations

A comprehensive analysis is made of the harmonic response of vertically excited, massless, rigid ring foundations supported at the surface of an homogeneous elastic halfspace. The parameters considered include the thickness to radius ratio for the ring, the frequency of the exciting force and of the resulting steady-state response, and Poisson's ratio for the supporting medium. The response quantities examined include the stiffness and damping coefficients of the system in an equivalent spring–dashpot representation of the supporting medium, the displacements of the foundation and of points on the ground surface away from the foundation, and the normal pressure at the foundation–medium interface. The results in each case are compared with those obtained for a solid disk having the same radius as the outer radius of the ring, and a simple approximate model is used to interpret the results and to provide insight into the action of the system. The reported data are computed by a method of analysis that takes due account of the mixed boundary conditions at the surface of the halfspace, and are believed to be of high accuracy.     

Vertical S waves in soil stratum over halfspace

An analytical model of a soil layer resting on a halfspace with vertically propagating shear waves is studied under a harmonic excitation on the surface. The soil is treated as a Kelvin-Voigt continuum, using the one-dimensional wave equation. The material parameters can be strain dependent. A hyperbolic model proposed by Hardin & Drnevich is used for describing the nonlinear stress-strain relationship. Based on this, a solution to a problem where thirty thousand people were jumping to the rhythm of rock music on the field in a stadium is given. The soil stratum is a 5–60m thick uniform soft high-plastic clay. The particle velocity was estimated to range from 22·5 mm/s to 45 mm/s in the first overtone. These velocity values exceeded the critical value, of 20 mm/s, of the German building code DIN 4150 which may explain the occurence of damages.     

Stationary approximation of the magnetotelluric field in a halfspace with a vertical boundary and three-dimensional perturbing body

Summary Based on the generalized potential of a double layer, integral formulae have been derived for calculating the stationary approximation of the magnetotelluric field in a halfspace divided by a vertical boundary into two quarterspaces, one of which contains a three-dimensional perturbing body. The appropriate boundary integral equation and other surface integrals have been computed for a perturbing body in the shape of the three-dimensional prism located at the vertical boundary, or in contact with the said boundary. The exciting electrical field is assumed to be homogeneous and perpendicular to the vertical boundary. Isoline graphs of the electrical and magnetic fields on the surface of the halfspace have been plotted and their anomalies are discussed.     

Diffraction of a compressional wave by a rigid barrier in a liquid layer — Part II

Summary The problem of diffraction of compressional waves striking obliquely at a rigid barrier fixed in the surface of a liquid layer lying over a solid halfspace has been studied. Exact values for the amplitudes of the waves reflected in the solid halfspace have been obtained. The method of solution is based on the Wiener-Hopf technique.     

A semi-analytical model in time domain for moving loads

A model for investigations of ground motions due to continuously moving loads with constant and time-varying amplitudes is presented. The vertical displacements excited by moving load areas are obtained at a fixed observation point at the surface of the three-dimensional halfspace in time domain. The load is moving along a straight line with constant speed. To solve this nonaxisymmetric, initial boundary value problem a semi-analytical, discretized model is developed. It is based on Green's functions for a suddenly applied, stationary surface point load with Heaviside time dependency. These functions, also called influence functions of the halfspace, are valid for any homogeneous, isotropic and linear-elastic medium. The principle of superposition is used. Results are shown for the transient and the steady-state ground motions, and they are compared with analytical solutions. The load speed is varied in the subcritical range up to the propagation velocity of Rayleigh-waves.     

Diffraction of seismic waves in an elastic, cracked halfplane using a boundary integral formulation

The presence of subsurface cracks in a halfspace excited by elastic waves may give rise to scattered body and surface waves. For many engineering applications, such as non-destructive testing or oil exploration, the scattered field may yield valuable information to detect cracks and other scatterers. We use the Indirect Boundary Element Method (IBEM) to study the diffraction of P, SV waves with various incidence angles and Rayleigh surface waves. This approximate boundary integral technique is based upon the integral representation for scattered elastic waves using single-layer boundary sources. Our approach is usually called indirect BEM as the sources' strengths should be obtained as an intermediate step. This indirect formulation can give to the analyst a deep physical insight on the generated diffracted waves because it is closer to the physical reality and can be regarded as a realization of Huygens' Principle. In any event, mathematically it is fully equivalent to the classical Somigliana's representation theorem. In order to gauge accuracy we test our method by comparing with previous results in the literature. Various crack configurations, including multiple cracks, are investigated. Results in frequency and time domains are displayed. Under certain conditions the amplitude spectra of those waves clearly show conspicuous resonance peaks.     

柱坐标系下任意埋藏源层状弹性半空间问题的奇异解

通过方位角的Fourier变换及关于径向的Hankel变换，将柱坐标系下一般形式的弹性力学基本方程和本构方程转化为波数域内非齐次的二阶常微分方程组，利用初参数法，建立了介质层传递矩阵．利用层间完全接触条件，给出了任意埋藏源层状弹性半空间频域奇异解，时域奇异解可通过关于频率的Fourier积分得到，该方法可应用到固体、流体层的情况.