Showing papers on "Half-space published in 1971"

Dynamic Shear Cracks with Friction as Models for Shallow Focus Earthquakes

Abstract: Summary In this paper a dynamical model of an earthquake source is investigated. This necessarily idealized model consists of a uniform elastic half space under a shearing pre-stress which tends to produce strike slip on a vertical fault plane. The fault plane is a plane of weakness across which the material is not welded but is initially inhibited from moving by a static frictional resistance which increases with depth. At a certain instant in time and depth in the half space a region of relative slip across the fault plane is initiated which spreads upwards and downwards so as to occupy at all times an infinite strip. Thus we shall be concerned only with two-dimensional SH motion (anti-plane strain). Once slipping occurs only reduced tractions act across the region of slip and it is the resulting stress drop which drives the mechanism. This model is almost the same as that considered by Berg and Weertman but goes further in that the dynamical problem is solved. We here extend previous work by Burridge and Burridge & Willis, in that we now find as part of the solution how the zone of slip spreads as well as the relative displacements, how the increasing friction prevents the crack (zone of slip) from penetrating very deeply, and eventually how it brings the whole mechanism to rest. Finally we calculate the pulse shapes in the far-field radiation and the residual displacements and stresses on the fault plane.

Electromagnetic scattering from cylinders of arbitrary cross-section in a conductive half-space

Abstract: A general numerical technique is presented for solving the problem of electromagnetic scattering by conducting cylinders of arbitrary cross‐section located in a conductive half‐space. Solutions to the electromagnetic wave equation are required for the free space above the half‐space, for the half‐space surrounding the cylinder, and for the cylinder. The problem is formulated by choosing an integral representation for the electromagnetic fields in each of the three homogeneous regions. By enforcing the boundary conditions on tangential E and H, we obtain a set of coupled integral equations which can be solved numerically for the unknown equivalent surface current densities on the interface bounding each homogeneous region. Once these current densities have been estimated, the fields can be calculated at any point from the general integral representations. The following conclusions are among those of importance to AFMAG and VLF surveys: 1) the ratio of Re (H) to Im (H) is a function of traverse position and...

Image Theory for an Arbitrary Quasi‐Static Field in the Presence of a Conducting Half Space

Abstract: The approximate image theory for quasi-static fields in the presence of a conducting half space is treated in full generality by considering an arbitrary periodic source. The general theory is developed in terms of a magnetic Hertz vector aligned perpendicular to the plane surface of the conductor. It is shown that its solution above the conductor can be expressed as the combined Hertz potentials of the source and its image located at a certain complex depth, plus terms that become negligibly small for points somewhat farther than a skin depth from the ordinary mirror image of the source. Image approximations for the individual electric and magnetic field components are derived. The magnetic field is expressed entirely in terms of the magnetic field of the source and its complex image, but the electric field, unless it is everywhere parallel to the surface of the conductor, depends in addition on the mirror image of the source. The general theory is illustrated by its application to the particular examples of magnetic dipole and infinite line current sources.

A rigid disc embedded in an elastic half space

Abstract: A GENERAL METHOD OF SOLUTION IS PRESENTED FOR THE COMPLETE DETERMINATION OF THE STRESS AND DISPLACEMENT FIELDS IN A HOMOGENEOUS, ISOTROPIC ELASTIC HALF-SPACE DUE TO THE VERTICAL DISPLACEMENT OF A BURIED RIGID DISC OF ARBITRARY SHAPE. THE SOLUTION IS DEVELOPED AS AN INTEGRAL EQUATION, BASED ON MINDLIN'S "EMBEDDED POINT LOAD" SOLUTION, WHICH IS THEN SOLVED NUMERICALLY. SPECIFIC RESULTS, FOR A RANGE OF BURIAL DEPTHS, ARE GIVEN FOR CIRCULAR AND RECTANGULAR DISCS. THE RESULTS AGREE CLOSELY WITH ANALYTICAL SOLUTIONS AVAILABLE FOR COMPARABLE SURFACE DISCS. /AUTHOR/

Lateral Structure Interaction With Seismic Waves

Abstract: The interaction of lateral structural inertia forces with horizontal seismic motion is formulated in terms of an integral equation of the Volterra type. By means of normal mode theory the inertia force at the base of the structure is expressed as a function of the foundation motion. After the motion of the two-dimensional elastic half space resulting from a uniform horizontal foundation force varying arbitrarily with time over a specified interval on the boundary of the half space has been determined, the interaction equation is derived. Numerical studies for two free-field acceleration inputs are made for different ground stiffnesses and structural characteristics. The first of these free-field inputs is a ramp sine function and the second is the east-west ground acceleration recorded at Golden Gate Park during the 1957 San Francisco earthquake. The interaction effects for structures similar to nuclear power plants prove to be significant.

Stronger Inequalities for 0, 1 Integer Programming Using Knapsack Functions

Abstract: In deriving the well known cuts for cutting-plane methods in 0, 1 integer programming, the integer points outside the 0,1 space can limit the parallel movement of the hyperplane of the cut toward the solution set. Furthermore it is unnecessarily restrictive to limit the movement of this hyperplane to parallel translations. This paper removes these two limitations in order to derive stronger cuts and reduce the total number of cuts required. Thus, it describes a method based on a special case of the knapsack function that replaces each cut or original constraint by a new inequality whose hyperplane passes through as many integer points in 0, 1 space as possible.

Deformation of a multilayered half-space by stress dislocations and concentrated forces

Abstract: The problem of the static and dynamic response of a nonhomogeneous, isotropic, elastic half-space to stress dislocations and concentrated forces is discussed. Integral expressions for the displacements are obtained in the case of a two-layered half-space. Results of Singh (1970) are used to study the static deformation of a multilayered half-space caused by a point source placed at an arbitrary depth below the free surface. The source is represented as a discontinuity in the z-dependent coefficients of the displacement and stress integrands at the source level.

Normal Mode Expansion in Neutron Thermalization Theory with a Simple Scattering Kernel

Abstract: Elementary solutions to the energy-dependent Boltzmann equation with a one-term degenerate scattering kernel are derived in plane geometry, and the weight function W (z) is obtained which makes these solutions mutually orthogonal over the half-range interval of the continuum. The weight function greatly facilitates determination of the expansion coefficients in general solutions and is applied to the problems in infinite half space. The diffusion length (discrete space eigenvalue) υ0 is exactly expressed by using the halfrange characteristic function X(z). In a 1/υ-absorbing medium, as the absorption concentration q increases from zero to a critical value g*, the diffusion length decreases from infinity to the end of the continuum 1/σmin. For q≥q*, v0 vanishes and the neutron density can be represented by the transient term alone, whose exact expression is obtained.

Formulation of Two-Dimensional and Axisymmetric Problems for an Elastic Half-Space.

Abstract: : Interrelationships are developed between various formulations of the equations relating stresses and displacements for an elastic half-space. There is a parallelism between the axisymmetric and two-dimensional cases. (Author)

On stresses and displacements due to a moving line load over the plane boundary of a heterogeneous elastic half-space

Abstract: The problem of concentrated line load moving with supersonic speed along the boundary of an isotropic heterogeneous medium has been solved as a plane strain problem. The stresses and displacements in the heterogeneous case considered are found to decay exponentially with distance.

Interaction of an Obliquely Incident p‐Polarized Plane Electromagnetic Wave with a Hot Plasma Half‐Space

Abstract: Obliquely incident p-polarized plane electromagnetic wave interaction with hot plasma half space, using linearized relativistic Vlasov equation and Laplace transform technique

Surface‐Wave Propagation over an Elastic Cosserat Half‐Space

Abstract: The equations describing an elastic isotropic Cosserat continuum are presented. Solutions of these equations for the various modes of plane harmonic waves in an infinite medium are briefly discussed. The analysis then concentrates on the development and interpretation of the surface‐wave solution for straight crested waves on a Cosserat half‐space. It is found that a wave analogous to the Rayleigh wave of the classical elasticity theory exists, except that it is a dispersive wave in this theory. The phase velocity of the surface wave may either increase or decrease with frequency, depending on the relative magnitude of the micromaterial moduli.

Propagation of Elastic Waves Caused by an Impulsive Source in a Half Space with a Corrugated Surface

Abstract: Summary The motion caused by a point source and a source of finite extent in an elastic half space with corrugated boundary is obtained and compared with the motion in a flat half space. The method of solution is based upon a combination of a perturbation theory and a finite difference method. The effect of corrugation on body and surface waves is investigated.

Propagation of Rayleigh waves in a layer lying over a heterogeneous half space

Abstract: Propagation of Rayleigh waves in a layer lying over a heterogeneous incompressible half space has been studied. The dispersion equations have been derived and displacements due to an infinite line source have also been investigated.

Surface Wave Dispersion in a Mass‐Loaded Half‐Space

Abstract: The effect of a layer on surface wave propagation on a half‐space can be separated into terms arising from elastic and from inertial restraining forces. In a number of situations of practical importance, it turns out that the inertial terms are dominant. The period equations governing the dispersion for this pure mass loading are obtained in suitable form and numerical solutions for Rayleigh waves for both solid and fluid loading are obtained. It is shown that by presenting the results in terms of appropriate variables, extremely simple, approximately linear, dispersion characteristics are found. A single computation suffices for substrate materials having the same Poisson ratio. The validity of this approximation is explored for modified Rayleigh waves for both solid and liquid layers, and for Love waves.

Response of a Semi-Infinite Elastic Solid to an Arbitrary Line Load Along the Axis

Abstract: The axisymmetric problem of a line load acting along the axis of a semiinfinite elastic solid is solved using Hankel transforms. In this solution the line load is interpreted as a body force loading and by assuming the line load to be of the form of a Dirac delta function the solution of Mindlin's problem of a point load within the interior of the half space is obtained. Solutions of this problem presented in the literature have been obtained using semiinverse techniques whereas the solution given here is obtained in a systematic step-by-step manner.

Integer points on the Gomory fractional cut (hyperplane)

Abstract: In this paper we show that the Gomory fractional cut (hyperplane) for the integer program is either void of integer points or contains an infinite number of them. The conditions for each case are presented. Also, we derive a stronger cut from the hyperplane which does not intersect integer points.