Showing papers on "Half-space published in 1998"

Elastic solutions for a transversely isotropic half-space subjected to a point load

Abstract: SUMMARY We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the e⁄ect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite di⁄erent for the displacement and stress calculated from isotropic solutions. ( 1998 John Wiley & Sons, Ltd.

Half-space theorems for mean curvature one surfaces in hyperbolic space

Abstract: We give conditions which oblige properly embedded constant mean curvature one surfaces in hyperbolic 3-space to intersect. Our results are inspired by the theorem that two disjoint properly immersed minimal surfaces in R3 must be planes. The half-space theorem says that a properly immersed minimal surface in R that is disjoint from a plane (thus in a half-space) is a plane. The strong halfspace theorem says that two disjoint properly immersed minimal surfaces in R are planes. The latter is deduced from the former by finding a plane between the two surfaces. These theorems are due to D. Hoffman and W. Meeks [H-M]. In this paper we establish results of this nature in hyperbolic 3-space for mean curvature one surfaces and horospheres. Let N denote a horosphere of H, and B the horoball of H bounded by N ; the mean curvature of N is one and the mean curvature vector of N points into B. Let C be the other connected component of H3\N . Theorem 1. Let M be a properly embedded constant mean curvature one surface in H, and assume M ∩N = ∅. If M ⊂ B, then M is a horosphere. If M ⊂ C and the mean curvature vector of M points towards N (i.e. it points into the component of H bounded by N ∪M), then M is a horosphere. In the same spirit we shall establish: Theorem 2. Let N be a catenoid cousin [B] (this is presented in section 1) and let B be the connected component of H to which the mean curvature vector of N points (that is, B is the mean convex domain bounded by N). Let M be a properly embedded constant mean curvature one surface in H, disjoint from N . Then M ∩ B = ∅ and the mean curvature vector of M does not point towards N (notice that such M exist, e.g., any horosphere in H3\B). Theorem 3. Let M1 and M2 be disjoint properly embedded constant mean curvature one surfaces in H. Let W be the connected domain of H bounded by M1 and M2. Then W is not mean convex, i.e. the mean curvature vectors of M1 and M2 do not both point into W . Received by the editors September 10, 1996. 1991 Mathematics Subject Classification. Primary 53A10. c ©1998 American Mathematical Society

High-frequency asymptotic description of head waves and boundary layers surrounding the critical rays in an elastic half-space

Abstract: The high-frequency asymptotic description of head waves as well as the field inside boundary layers surrounding the critical rays are obtained for two cases: (a) a point source, and (b) a circular transducer, both acting normally on an isotropic and homogeneous elastic half-space. The edge head waves underneath a circular transducer are described by the asymptotics of a higher order compared to those of direct compressional, edge compressional, and shear waves, but are still discernible in the radiating near zone and thus might be useful in nondestructive evaluation of industrial materials. The asymptotic formulas produced involve in geometrical zones elementary functions and inside boundary layers well-known special functions. Therefore, they allow us to elucidate the physics of the problem and can be used in writing computer codes which simulate the radiating near field of a circular transducer orders of magnitude faster than full numerical schemes. The formulas have been tested against exact integral solutions evaluated numerically.

L$B!g!>(Bestimate of first-order space derivatives of Stokes flow in a half space

Abstract: We prove the L∞-BMO boundness of first-order space derivatives of Stokes flow in a half space. To show the estimate, we apply the solution formula of Stokes equation in a half space, which is a modified version of Ukai’s formula.

Transient disturbances in a relaxing thermoelastic half space due to moving internal heat source

Abstract: This paper is concerned with the transient waves created by a line heat source that suddenly starts moving with a uniform velocity inside a thermoelastic semi-infinite medium with thermal relaxation ofthe type ofLord and Shulman The source moves parallel to the boundary surface which is traction-free. The problem is reduced to the solution of three differential equations, one involving the elastic vector potential, and the other two coupled, involving the thermoelastic scalar potential and the temperature. Using Fourier and Laplace transforms, the solution for the displacements have been obtained in the transform domain. The displacements have been calculated on the boundary surface for small time

Analytical solution for the electric potential due to a point source in an arbitrarily anisotropic half-space

Abstract: A very large class of important theory and applications in geophysics requires analytical solutions for the determination of the electric potential due to a point source in an arbitrarily anisotropic half-space. In this paper, a very clear and simple solution to the half-space problem has been developed from consideration of an arbitrarily anisotropic whole space. For the first time, the method of images is used to generate the solution for an arbitrarily anisotropic three dimensional half-space. Based on traditional extreme value theory the image source point has been determined and calculated for the half-space case.

Boundary element modelling of a coating–substrate composite under an elastic, Hertzian type pressure field: cylinder on flat contact geometry

Abstract: The boundary element model (BEM) approach is established as a surface engineering tool on standard personal computers with reasonable computation times. Modelling of a cylinder–flat contact is presented. The effects of changing material and geometric parameters on stresses is analysed. For homogeneous materials the validity of the useful half space assumption (HSA) is shown to depend on the relative dimensions of the specimen; i.e. the ratio of real sizes over the Hertz contact half width. For coated systems the numerical results reveal the existence of two major local stress maxima whose exact locations depend on the geometry and the materials parameters. In addition, a step discontinuity in sub-surface stress is seen to exist at the coating–substrate interface. The details of the sub-surface stress field depend crucially on the ratio of coating and substrate Young's moduli as well as the relative coating thickness. Another key parameter is the difference between the Poisson ratios of the materials.

The radiating near-field asymptotics of a normal time-harmonic circular ultrasonic transducer in an elastic half-space

Abstract: The modern diffraction theory is applied to analyze the radiating near zone of a normal time-harmonic circular transducer directly coupled to a homogeneous and isotropic solid. The two-tier asymptotic approach is used first to find the far-field asymptotics of a point source and then the radiating near-field asymptotics of the circular transducer. All the known ray-theoretical solutions for body waves, such as the plane P wave and the toroidal edge waves, both P and S, are obtained. The non-ray-theoretical solutions, such as the edge waves present in the axial boundary layer and the total field inside the penumbral boundary layer, are also described. The asymptotic formulas produced all have immediate physical interpretation, give explicit dependence on model parameters and involve in geometrical region elementary functions and inside boundary layers, well-known special functions. It is argued that asymptotic results may be used to write computer codes which simulate the radiating near field of the circular transducer orders of magnitude faster than the exact numerical schemes, but more accurately than other known approximations.

Recognizing 3-D Objects with Linear Support Vector Machines

Abstract: In this paper we propose a method for 3-D object recognition based on linear Support Vector Machines (SVMs). Intuitively, given a set of points which belong to either of two classes, a linear SVM finds the hyperplane leaving the largest possible fraction of points of the same class on the same side, while maximizing the distance of either class from the hyperplane. The hyperplane is determined by a subset of the points of the two classes, named support vectors, and has a number of interesting theoretical properties. The proposed method does not require feature extraction and performs recognition on images regarded as points of a space of high dimension. We illustrate the potential of the recognition system on a database of 7200 images of 100 different objects. The remarkable recognition rates achieved in all the performed experiments indicate that SVMs are well-suited for aspect-based recognition, even in the presence of small amount of occlusions.

Inverse method of identification for three-dimensional subsurface cracks in a half-space

Abstract: A procedure is presented which is well suited for three-dimensional subsurface crack identification in a half-space through the inversion of measured surface displacements. The investigation began with the linear, forward problem of generating contour maps of surface deformation produced by a fracture of known geometry and loading which is embedded in a finite medium. The fundamental solutions for tensile and shear multipoles in a half-space provided an efficient mathematical representation of the three-dimensional fracture. The inverse problem of crack identification centers on the development of a hybrid of the Marquardt–Levenberg algorithm. Initial guesses for the constrained set of search variables were determined heuristically from the correspondences between crack geometry and loading and the resulting uplift at the free surface. Physical measurements of surface deformation were taken for a cube of transparent acrylic polyester in which a fracture was hydraulically pressurized. Displacements induced at the surface of the specimen, which were measured by laser interferometry, had a strong correlation with predictions of the computational model (coupled with a finite element discretization). Numerical tests demonstrate the robustness of the inverse methodology even in the presence of the random and systematic errors corresponding to the experimental interferometric measurements.

Te Scattering By a Cylindrical Dielectric Obstacle Buried in a Half-Space: a H-Field-Based Solution Method

Abstract: Scattering of a transverse electric (TE) wavefield by an inhomogeneous lossy dielectric cylindrical obstacle embedded in a lossy dielectric half-space is investigated. The cylinder is of arbitrary cross-section, and its axis lies parallel to the interface. The formulation involves a domain integral equation of the field obtained by applying the Green's theorem to the appropriate Helmholtz wave equations. The convolution-correlation structure of the formulation is such that the discretized counterpart of the integral equation (provided by the application of a Method of Moments where the obstacle is divided into triangular patches and where the contrast and the H-field are expanded using two different basis functions) can be efficiently handled using a FFT-based conjugate gradient solver. The validity of the approach is illustrated by comparison with results found in the literature.

Elastic Subsurface Stress Analysis for Circular Foundations. I

Abstract: In part I of this analysis an attempt is made to determine a simple estimate of the stresses resulting from a circular foundation subjected to concentric or eccentric loading. It is assumed that the foundation loading can be modeled as combinations of uniform, linear, and quadratic tractions applied over a circular area on the surface of an elastic half space. The present analysis for quadratic and linear loading are combined with a uniform loading solution (normal or shear traction), previously derived by the authors, to provide the requisite loading conditions and resulting internal stress fields. The current analysis consists of using potential functions to derive closed form expressions for the elastic field in the half space. The half space is taken as cross-anisotropic (transversely isotropic), where the planes of isotropy are parallel to the free surface. The x- and y-axes are taken in the plane of the surface with z directed into the half space. Hence the boundary conditions within the circular lo...

Propagation of SH waves in a layered half-space with a frictional contact interface

Abstract: The propagation of SH waves in a layered half-space with a frictional contact interface is considered. The incident wave is assumed to be sufficiently strong so that friction may be broken, and the local slip may take place at the interface. In the stick zones, both the displacements and stresses are continuous, while in the slip zones, the Coulomb friction model is adopted. The mixed boundary conditions lead to recurrence relations for the subcritical angle incidence or singular integral equations for the supercritical angle incidence. The extent and location of slip zones, which are unknown before the solution of the problem, are determined. The local slip velocities and the interface shearing tractions are calculated in detail for the subcritical angle incidence. The results show that the solution of the problem is dependent on the frequency of the incident wave due to the presence of the characteristic length—the thickness of the elastic layer. It is also found that, in some situations, there exist four slip zones instead of two over one representative period. All these features are quite different from those for infinite media.

Inequalities for mixed stationary Poisson hyperplane tessellations

Abstract: Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.

Seismic response to a prescribed seismogram of a body embedded in a multilayered half space

Abstract: While the actual problem is composed of an active fault surface, a soil site and a body embedded at that site; the proposed method provides an alternative smaller linear problem by replacing the propagating rupture on the fault surface by a fictitious focal point and a seismograph station in the vicinity of the given soil site. The Green's function for each of three fundamental problems of isotropic elastic and viscoelastic spaces undergoing harmonic vibration is derived. Infinite elements are adopted in the far field, and finite elements in the near field. The three fundamental problem solutions are used as the shape functions of infinite element nodal lines. The three concentrated orthogonal force components at the focal point are determined in such a way that the Fourier transforms of the three orthogonal acceleration components measured at a seismograph station are checked. For seismic analysis of a finite embedded body, consider the differential between the actual system and the seismic free field, which is the embedding half space without any embedment and being excited by the fictitious focal point forces. All along the analysis has been carried out in the frequency domain. An appropriate inverse Fourier transform algorithm will properly yield all results as time functions.

An exact solution for the first stage of frictional collision of a die and an elastic half-space

Abstract: The first stage of frictional collision of a die and an isotropic linear elastic half-space is studied. The die is a blunt convex rigid body having an arbitrary shape with two orthogonal planes of symmetry, which are both orthogonal to the boundary of the half-space. The problem is investigated using the technique of integral characteristics of solutions to boundary-initial value problems introduced by Borodich. We consider non-frictionless boundary-initial contact problems, for example, adhesive or frictional. Expressions are obtained for the relations between time, depth of indentation, and velocity of the body. In particular cases, when the body is an elliptic paraboloid, a blunt four-sided pyramid or an elliptic cone, some expressions have simple algebraic forms. A proof is given that the expressions are independent of the boundary conditions in the contact region.

Double wave of Stoneley type on the interface of a stratified fluid layer and an elastic solid half-space

Abstract: Sound propagation from a point time-harmonic source in a stratified water layer lying over an elastic solid half-space is investigated. It is assumed that the sound speed in the layer is less than the shear speed in the solid bottom, and that it increases with the depth. Numerical examples are given which show that the dependence of the wave field on the range between the source and the receiver can sharply change the character under rather small variations of the frequency. Namely, for some particular frequencies, the sound amplitude shows a periodical dependence on the range, while for other frequencies there is no periodicity. A theoretical explanation of this phenomenon is given in a mathematical development using the normal modes theory and high-frequency asymptotic approximations. The dispersion phase curves are found to have “quasi-intersections,” i.e., small domains where two adjacent curves almost intersect. The corresponding frequencies are called the “specific” frequencies. For any nonspecific ...

Elastodynamic stress intensity factors for a semi-infinite crack due to three-dimensional concentrated shear loadings on the crack faces

Abstract: The dynamic stress intensity factors for a semi-infinite crack in an otherwise unbounded elastic body is investigated. The crack is subjected to a pair of suddenly-applied shear point loads on its faces at a distance l away from the crack tip. This problem is treated as the superposition of two problems. The first problem considers the disturbance by a concentrated shear force acting on the surface of an elastic half space, while the second problem discusses a half space with its surface subjected to the negative of the tangential surface displacements induced by the first problem in the front of the crack edge. A fundamental problem is proposed and solved by means of integral transforms together with the application of the Wiener-Hopf technique and Cagniard-de Hoop method. Exact expressions are then derived for the mode II and III dynamic stress intensity factors by taking integration over the fundamental solution. Some features of the solutions are discussed.

Pulsed Reflection and Transmission for a Dispersive Half Space

Abstract: : One-dimensional propagation of a normally incident, pulsed (finite-cycle sine), electromagnetic plane wave on an isotropic, spatially homogeneous, Lorentz half space is investigated analytically. Detailed examinations of the reflected and transmitted fields are made. The inversion integral for the time-domain reflected field is expressed in terms of pole contributions and branch-cut integrals, which are computed numerically; whereas the uniform asymptotic methodology of Oughstun and Sherman is applied to the transmitted field. Only the contributions from the distant saddle points to the transmitted field are studied thoroughly. An example is provided that shows that the reflection and transmission coefficients may not be ignored. Specifically, for Brillouin's choice of the medium's physical parameters, the reflected field has a peak value that is 21% of the incident field's amplitude and that corresponds to a 21% decrease in the main signal (pole contributions) of the transmitted field when the transmission coefficient is unity. This work generalizes past formulations by accounting for reflection from the medium and by addressing how inclusion of frequency-dependent transmission and reflection coefficients affects the fields.

Dynamic Response of a Hemispherical Foundation Embedded in an Elastic Half-Space

Abstract: The dynamic response of a massless rigid hemispherical foundation embedded in a uniform homogeneous elastic half-space is considered in this study The foundation is subjected to external forces, moments, plane harmonic P and SH waves, respectively The series solutions are constructed by three sequences of Lamb’s singular solutions which satisfy the traction-free conditions on ground surface and radiation conditions at infinity, automatically, and their coefficients are determined by the boundary conditions along the soil-foundation interface in the least square sense The fictitious eigen-frequencies, which arise in integral equation method, will not appear in the numerical calculation by the proposed method The impedance functions which characterize the response of the foundation to external harmonic forces and moments at low and intermediate frequencies are calculated and the translational and rocking responses of the foundation when subjected to plane P and SH waves are also presented and discussed in detail

Synthetic Seismograms in Transversely Isotropic Plane Layered Media

Abstract: The purpose of the work presented in the paper, is to compute the response of transversely isotropic plane layered media excited by a buried or surface source, in order to obtain a method which can help interpreting real seismograms. The computation in frequency f and wavenumber k domain based on the Kennett's reflectivity method is exposed. Displacements in space and time are calculated by numerical integration (Fourier-Hankel transform) of the response in frequency and wavenumber domain. The numerical surface response of a transverse isotropic two-layered half space excited by a surface source is presented. The mechanical properties of the proposed two-layered halfspace falls within the range of marine sediments. The frequency and offset proposed domain correspond with geotechnical surveys. Effects of anisotropy are put forward by comparing the responses in anisotropic case to the responses in isotropic case.