Showing papers on "Half-space published in 1982"

Viscoelastic‐gravitational deformation by a rectangular thrust fault in a layered Earth

Abstract: Previous papers in this series have been concerned with developing the numerical techniques required for the evaluation of vertical displacements which are the result of thrust faulting in a layered, elastic-gravitational earth model. This paper extends these methods to the calculation of fully time-dependent vertical surface deformation from a rectangular, dipping thrust fault in an elastic-gravitational layer over a viscoelastic-gravitational half space. The elastic-gravitational solutions are used together with the correspondence principle of linear viscoelasticity to give the solution in the Laplace transform domain. The technique used here to invert the displacements into the time domain is the Prony series technique, wherein the transformed solution is fit to the transformed representation of a truncated series of decaying exponentials. Purely viscoelastic results obtained are checked against results found previously using a different inverse transform method, and agreement is excellent. The major advantage in using the Prony series technique is that deformations can be computed for arbitrary time intervals. A series of results are obtained for a rectangular, 30° dipping thrust fault in an elastic-gravitational layer over viscoelastic-gravitational half space. Time-dependent displacements are calculated out to 50 half space relaxation times τa, or 100 Maxwell times 2τm = τa. Significant effects due to gravity are shown to exist in the solutions as early as several τa. The difference between the purely viscoelastic solution and the viscoelastic-gravitational solutions grows as time progresses. Typically, the solutions with gravity reach an equilibrium value after 10–20 relaxation times, when the purely viscoelastic solutions are still changing significantly. Additionally, the length scaling which was apparent in the purely viscoelastic problem breaks down in the viscoelastic-gravitational problem. Two independent length scales, one of which changes with time, are now seen to characterize the problem.

Diffraction of plane sh waves in a half‐space

Abstract: Scattering of antiplane shear waves (SH) in two dimensions by surface and near-surface defects in a homogeneous, isotropic elastic semi-infinite medium has been studied. Attention has been focused here in the range of medium to long wavelengths. A combined finite element and analytical technique has been used to study the problems of scattering by semi-circular and triangular canyons. The results for the former case are compared with the known exact solution and those for the latter case are compared with some available approximate solutions. Finally a problem of multiple scattering by a triangular canyon and a nearby circular tunnel is studied. Numerical results are presented showing the effects of multiple scattering and different angles of incidence. These results are of interest in earthquake engineering.

Scattering of SH Waves by Subsurface Topography

Abstract: Horizontally polarized shear waves in an elastic alluvial valley of arbitrary shape perfectly bonded to a linearly elastic, homogeneous and isotropic half space are considered. The valley is subjected to a steady state horizontal displacement field. Total displacement field is evaluated along the interface between the half space and alluvial valey. Comparison with known exact solutions for simple geometry provided the following conclusions: 1) the method provides excellent results for wide range of frequencies, 2) for a fixed number of sources, the results are more accurate at lower frequencies, and 3) as the number of sources increases, the accuracy increases.

Scattering of Elastic Waves by an Alluvial Valley

Abstract: Plane strain model for scattering of steady state plane waves by an elastic alluvial valley of arbitrary shape is presented. The valley is presumed to be perfectly bonded to a linearly homogeneous and isotropic half space. Total displacement and stress fields are evaluated along the interface between the alluvial valey and elastic half space using the source method. The results are compared with known exact solutions for simple geometries to determine the accuracy of the proposed method. It was found that the proposed method provides very good results. As an illustrative example diffraction of a plane P-wave by an elastic semi-elliptical alluvial valley is presented in detail.

The image theory electromagnetic fields of a horizontal electric dipole in the presence of a conducting half space

Abstract: The problem considered is a horizontal electric dipole that is located above or below the plane surface of a conducting half space. Expressions for the field components are obtained by employing finitely conducting earth-image theory techniques. The formulas developed are of simple form and are valid from the quasi-static to the far-field ranges as long as the measurement distance is greater than 3 times the burial depth of the source and/or receiver.

Vibration of a rigid disc on a transversely isotropic elastic half space

Abstract: The problem of a massless, rigid disc vibrating harmonically on a ‘constrained’ transversely isotropic elastic half space is considered. The material is called constrained because it is assumed that the elastic constants satisfy a certain equation. The problem for each mode of vibration is reduced to the solution of a Fredholm integral equation of the second kind. Results are presented to show the general effect of the material anisotropy.

Indentation of an elastic half-space by a cooled flat punch

Abstract: SUMMARY A solution is given for the frictionless thermoelastic contact of a cooled flat cylindrical rigid punch and an elastic half-space. If the thermal distortion is sufficiently small, contact is retained over the entire punch face and the solution can be written down from known results. For larger temperature differences the conventional boundary conditions do not permit a solution, but a solution is here developed in which a central circle is in imperfect contact, i.e. a non-zero contact resistance exists between the surfaces. The problem is reduced to a boundary-value problem in potential theory using a solution of the thermoelastic equations in two harmonic functions. These functions are written as the sum of a number of components each of which satisfies two-part boundary conditions. The resulting boundary conditions lead to coupled Abel integral equations which are reduced to a single Fredholm equation by substitution. Results are given for typical contact pressure distributions and for the relationship between load, temperature difference, heat flux and the radius of the imperfect contact region.

B-polarization induction in two generalized thin sheets at the surface of a conducting half-space

Abstract: Summary. A new closed-form solution is obtained analytically for a B-polarization induction problem of geophysical interest, in which a local region of the Earth is represented by a generalized thin sheet at the surface of and in electrical contact with a uniformly conducting half-space. The generalized sheet, first introduced by Ranganayaki & Madden, is a mathematical idealization of a double layer which consists, in this problem, of two adjacent half-planes with distinct conductances representing a surface conductivity discontinuity such as an ocean—coast boundary, underlain by a uniform sheet of finite integrated resistivity representing the lower crust. The resistive sheet exerts a considerable mathematical influence on the solution causing, under certain conditions, an additional pole to appear in one of the forms of contour integral by which the solution can be expressed; it also weakens or eliminates field singularities that would otherwise occur at the conductance discontinuity. A numerical calculation is made for model parameters typifying an ocean—coast boundary underlain by a highly resistive crust. It is found that the residue of the pole associated with the resistive sheet dominates the solution for this example, the main consequence of which is a huge increase in the horizontal range over which the induced currents adjust themselves between the different ‘skin-effect’ distributions at infinity on either side of the model. Moreover the solution shows that this ‘adjustment distance’ has a more complicated dependence on the conductance and integrated resistivity of the sheet than that given simply by the square root of their product which was the length parameter proposed by Ranganayaki & Madden.

Electromagnetic scattering from geophysical targets by means of the T matrix approach: A review of some recent results

Abstract: In the present contribution we review some recent work on scattering of electromagnetic waves from various geophysical targets, by means of P. C. Waterman's approach. As an introduction to the specific applications to be described, a general outline is given of the basic features of this approach, as applied to electromagnetic waves. The algebraic structure of the resulting equations for single and multiple scattering situations is discussed with special emphasis on the case of an infinite interface and an adjacent finite inhomogeneity in one of the half spaces. We review a number of applications relevant for electromagnetic prospecting. The bedrock is modeled as a lossy half space, and various configurations of ore bodies in this half space are considered. The measurements are assumed to be made on the surface or along a borehole into the ground, and the radiation source is again either on the ground or in the borehole. The behavior of the field in the vicinity of the inhomogeneity and interference effects are discussed. Actual ore bodies are often rather thin plates, and the influence of a layered structure of the half space may be of importance. Therefore in other applications the half space is taken to be layered, and the inhomogeneity is taken to be a perfectly conducting spheroid or a perfectly conducting disc.

Body Wave Excitations of Embedded Hemisphere

Abstract: Solutions of three-dimensional vibrations of an embedded hemispherical foundation are presented for excitation by incident plane P, SV, and SH waves with arbitrary angles of incidence The effects of embedment are illustrated by comparing the motion of the embedded hemispherical foundation with those of a rigid disc bonded to a half space of equal radius and excited by the same waves The impedence matrix and the foundation motion are analyzed and compared with similar and reLated approximate solutions In the context of Fourier synthesis, the results can be viewed as representing the transfer functions for the forces and motions of the foundation For arbitrary transient excitation which can be represented by plane body waves, the results can be used to calculate the transient motion of the foundation

Thermal shock in a viscoelastic half-space

Abstract: Thermal shock due to sudden heating of the bounding surface of a viscoelastic half-space is discussed. Coupling of the temperature and strain fields has been taken into account, and the effects of thermal relaxation on the shock propagation have also been considered. The linear standard model has been used to represent the viscoelastic behavior of the solid in shear which is elastic in hulk deformation. The analysis, although valid for small times, shows that the stress and temperature fields have discontinuities at two points at a given instant of lime. It is further noted that attenuation takes place more rapidly than in the corresponding case of an elastic continuum. The stress and temperature fields are the result of superposition of two shock waves: one is mechanical in origin, the other is due to thermal relaxation. An estimate of the finite speed of the heat signal is also obtained for a specific case.

Torsional Dynamic Response of Solid Systems

Abstract: A practical numerical method which solves the multidimensional axisymmetric torsional wave equation for linear and nonlinear inelastic solids was developed by Henke. The application of this procedure is the prediction of the dynamic response of soil-foundation systems subjected to torsional loads having a broad range of intensities. Here the method is further validated. Two solutions giving the steady state response of a linear half space to different loadings are found to agree with analytical solutions. A third study considers the transient response of a rigid disk on a nonlinear as well as linear half space. Slip is permitted at the interface between the disk and half space. Since analytical solutions were not found, examples are validated using an energy balance. In all studies physical and mathematical aspects of the solutions are discussed. It is concluded that to predict accurately the dynamic response of a soil-foundation system, nonlinear inelasticity must be taken into account if large shearing strains are developed and that the developed numerical method is practical and accurate.

Transient coupling between finite circuits on an anisotropic conducting half-space

Abstract: The frequency domain response of a grounded linear circuit is deduced. The model of the ground is a homogeneous conducting half-space whose horizontal and vertical conductivities are specified. The source itself is formulated as an insulated current-carrying wire of finite length lying on the surface that is grounded at its end points. The tangential electric field expressions are used to calculate the induced voltage between two additional electrodes that are connected by an insulated wire. Particular attention is paid to the transient coupling response in the potential circuit when the source current is a step-function of time. Among other things, it is shown that the voltage response is a step-function only if the circuits are at right-angles and if the ground is effectively isotropic and nonpolarizable.

Analysis of a moving heated punch over the surface of a thermoelastic half-space

Abstract: Analysis of a healed punch moving over the surface of a thermoelastic half-space is carried out for temperature and stress distribution in the medium. The formulation results in a mixed boundary-value problem independent of time in a moving nondimensional coordinate system. Exponential Fourier transforms are applied and the resulting system solved to obtain a general solution in terms of the functions expressed in the form of inverse Fourier transforms. Specifically, closed-form solutions are obtained for flat, cylindrical, and wedge punches. For a uniform temperature distribution the results derived agree with those available in the literature. Also, in the case of a concentrated force moving on the surface of an elastic half-space the solution obtained agrees with that in the literature.

Laterally Loaded Cylinders in Half Space

Abstract: The extended Neuber solution which has the curl of a harmonic vector added to Papkovich-Neuber solution is proposed as a useful solution to asymmetric problems referring to cylindrical coordinates. A three-dimensional analysis for a laterally loaded semi-infinite cylinder embedded in an elastic half-space is presented on the basis of that solution. The potentials of displacement for a semi-infinite cylinder and an elastic half-space are given. The components of displacement and stress are expressed in those potentials of displacement. The Fourier, Bessel, and Dini expansion are used in order to satisfy the boundary and continuity conditions. Numerical calculations are made for various values of Poisson’s ratios and the shear moduli of a semi-infinite cylinder and an elastic half space. Some available data for the behavior of such a structure as single piles in a foundation, anchor bolts, and reinforcements subjected to a shearing force in a concrete with cracks are presented.