Half-space

About: Half-space is a research topic. Over the lifetime, 2264 publications have been published within this topic receiving 33563 citations.

Continuous and Discrete Inverse‐Scattering Problems in a Stratified Elastic Medium. I. Plane Waves at Normal Incidence

Abstract: This paper investigates the problem of obtaining an analytic solution and practical computational procedures for recovering the properties of an unknown elastic medium from waves that have been reflected by or transmitted through the medium. The medium consists of two homogeneous half‐spaces in contact with a heterogeneous region. The analytic solution is obtained by transforming the equation of motion for the propagation of plane waves at normal incidence in a stratified elastic medium into a one‐dimensional Schrodinger equation for which the inverse‐scattering problem has already been solved. The practical computational procedures are obtained by solving the corresponding discrete inverse‐scattering problem resulting from approximating the heterogeneous region with a sequence of homogeneous layers such that the travel time through each layer is the same. In both the continuous and discrete inverse scattering problems, the impedance of the medium as a function of travel time is recovered from the impulse response of the medium. A discrete analogy of the continuous solution is also developed. Similar results are obtained for a stratified elastic half space bounded by a free surface.

Self Similar Solutions to Adhesive Contact Problems with Incremental Loading

Abstract: This paper is concerned with the calculation of contact stress distributions, surface displacements and contact radii when an elastic half space z > 0 is indented by a rigid body, symmetrical about the z axis and exerting a prescribed force P normal to the interface, with friction sufficient to prevent any slip between the surfaces of the body and the half space during the application of the load. Within the framework of small displacement theory, an exact solution is obtained to the problem of static indentation by a body whose shape is given by a general polynomial z(r), where r is the distance from the z axis. The solution is worked out in detail for indentors of three simple shapes, namely a flat-faced cylinder, a sphere and a cone (the first two of which have previously been considered by rather different methods by Mossakovski (I954, I963). A closed solution is also obtained to another problem closely related to that of the spherical stamp: that of calculating the indentation and resulting velocity defect when a rigid sphere rolls on an elastic half space. The treatment in all cases is elastostatic, i.e. it is assumed for the stamp problems that the load is applied slowly enough for static equilibrium to prevail at all stages, and correspondingly in the case of the rolling sphere, that its velocity is low enough for the same to be true. .The results for rigid indentors are readily extended to cover the more general case of adhesive contact between two elastic bodies of different material properties by means of a straightforward linear transformation noted by Mossakovski (I963) and recapitulated in the present notation as appendix A, which reduces the general elastic-elastic case to that of a rigid-elastic indentation with suitably adjusted [ 55 ]

Green'sFunction for Lamb'sProblem

Abstract: Summary The complete solution to the three-dimensional Lamb'sproblem, the problem of determining the elastic disturbance resulting from a point force in a half space, is derived using the Cagniard-de Hoop method. In addition, spatial derivatives of this solution with respect to both the source co-ordinates and the receiver co-ordinates are derived. The solutions are quite amenable to numerical calculations and a few results of such calculations are given.

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

Abstract: The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh–Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.