Showing papers on "Half-space published in 1985"

The normal modes of a uniform, compressible Maxwell half-space

Abstract: The analytical solution for the load-induced deformation of a uniform, compressible, hydrostatically pre-stressed elastic half-space is derived. The solution is correct to first order in the quantity e, which is inversely proportional to the wave number k of the deformation. Usually e is very small compared with unity for Earth deformations on a scale amenable to the halfspace approximation. Since pre-stress advection is included in the analysis, the correspondence principle allows us to solve the field equations governing the deformation of the associated Maxwell half-space. The viscoelastic solution shows that the relaxation of the Maxwell continuum is characterized by a fundamental mode and a rapidly decaying overtone of much smaller amplitude. In the incompressible limit the overtone is not excited. The significance of the results for the relaxation of the Earth's mantle is briefly discussed.

Mutual effect of a circular surface crack and a free boundary in an axisymmetric problem of the fracture of an incompressible half space in compression along the crack plane

Abstract: In the case where a deformable solid containing a crack which is planar in plan is acted upon only by forces parallel to the crack plane, the stress intensity factors obtained in linear fracture mechanics are equal to zero [6, 8] Fracture criteria of the Griffith-Irwin type cannot be used, and instead use is made of the stability criterion proposed earlier [2] in the framework of the three-dimensional linearized theory of stability [I] A general approach to such problems in the coordinates of the initial strain state was proposed in the monograph [3], where a detailed bibliography was also given The approach is based on the use of general solutions of the equations of the linearized theory of stability with uniform subcritical states

A problem in second order elasticity

Abstract: When an incompressible half space of elastic material is twisted by a rigid circular punch bonded to its surface, to first order the azimuthal displacement is zero. In this paper, second order theory is used to investigate the deviation of the surface of the half space from the undeformed configuration.

Contact Mechanics: Line loading of an elastic half-space

Abstract: The elastic half-space Non-conforming elastic bodies in contact whose deformation is sufficiently small for the linear small strain theory of elasticity to be applicable inevitably make contact over an area whose dimensions are small compared with the radii of curvature of the undeformed surfaces. The contact stresses are highly concentrated close to the contact region and decrease rapidly in intensity with distance from the point of contact, so that the region of practical interest lies close to the contact interface. Thus, provided the dimensions of the bodies themselves are large compared with the dimensions of the contact area, the stresses in this region are not critically dependent upon the shape of the bodies distant from the contact area, nor upon the precise way in which they are supported. The stresses may be calculated to good approximation by considering each body as a semi-infinite elastic solid bounded by a plane surface: i.e. an elastic half-space. This idealisation, in which bodies of arbitrary surface profile are regarded as semi-infinite in extent and having a plane surface, is made almost universally in elastic contact stress theory. It simplifies the boundary conditions and makes available the large body of elasticity theory which has been developed for the elastic half-space. In this chapter, therefore, we shall study the stresses and deformations in an elastic half-space loaded one-dimensionally over a narrow strip (‘line loading’). In our frame of reference the boundary surface is the x–y plane and the z -axis is directed into the solid.

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

Abstract: A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

Propagation of a plane wave in a thermoelastic half-space under smooth heating of its boundary

Abstract: In this paper a closed-form solution to a plane wave propagation problem for a half-space subject to a uniform time-dependent healing of its boundary in the framework of dynamic theory of thermal stresses is presented. The solution is given for two variants of smooth boundary heating. The numerical results are shown in the form of graphs of stress and temperature versus the distance from the loaded boundary.

The Rapid Tearing of a Half Plane

Abstract: The surface of an elastic half space is subjected to sudden antiplane mechanical disturbances. Crack initiation and subsequent crack instability are examined via two idealized problems; the first is concerned with instantaneous crack bifurcation and the second with instantaneous skew crack propagation. In either problem, crack propagation occurs at a constant subsonic velocity under an angle κπ with the normal to the surface. For the externally applied disturbances that are considered here, and for contstant crack-tip velocities, the particle velocity and τθz are functions of r/t and θ only, which allows Chaplygin’s transformation and conformal mapping to be used. The problems can then be solved using analytic function theory. For various values of the angle of crack propagation, the dependence of the elastodynamic stress intensity factors on the crack propagation velocity is investigated. For certain specific geometries, fully analytical solutions are derived to provide check cases.

Formulae of force at a point in the interior of a half space with shear modulus linearly varied with depth

Abstract: According to a lemma and an assumption, this paper presents formulae of force at a point in the interior of a half space with Poisson's ratio ν=constant and shear modulus G linearly varied with depth. These formulae can be used as an approximate basic solution when the integrai equation method is employed for the analysis of piles and other geotechnicai engineering problems.

Electromagnetic Penetration through a Slot into a Rectangular Waveguide.

Abstract: : Electromagnetic penetration into a matched rectangular waveguide through a longitudinal slot in a conducting plane is considered. The problem is divided into two parts, one in half space and another inside the waveguide, by use of the equivalence principle. The resultant equations are the same as those described in a reference. The only difference is that the excitation vector is changed from an incident wave inside the waveguide to an incident plane wave in half space. The magnetic current over the slot is obtained from the solution of the matrix equations. The electromagnetic penetration into a matched rectangular waveguide and the electromagnetic reradiation into half space through a slot are computed from the equivalent magnetic current over the slot. The computed results show that a plane wave at normal incidence and a resonated slot near the side of the waveguide broad wall allow orders of magnitude greater power to penetrate into the waveguide than when the slot is not resonated. Keywords include: Aperture admittance, Electromagnetic penetration, Equivalence principle, Generalized network parameters, Slot coupling, and Waveguide region.

The Acoustic Response of a Sphere in a Two-Layer Half-Space

Abstract: In this paper the acoustic wave equations for a specific cylindrically symmetric model are considered. The model consists of a spherical inhomogeneity in a homogeneous half-space with an overburden layer. A pressure source is located in the top layer, centred on the axis of symmetry. Further details are contained in Section 2. Owing to the geometrical simplicity of the model, the wave equations can be solved analytically to a large extent. Therefore, the results of such a model may be used for the validation of algorithms, solving or dealing with wave equations in more general 3-D models. Moreover, the results for this model give a quantitative impression about the shape and magnitude of the response of a bounded anomaly in comparison with that of a plane layer.

On love wave dispersion in a homogeneous layer lying over a laterally heterogeneous half-space

Abstract: A method is proposed for the determination of the dispersion equation of Love waves propagating in a homogeneous layer lying over a laterally inhomogeneous half-space. The proposed method can be made to work only when the lateral inhomogeneities in the lower half-space are finite in nature so that their Fourier transforms are available. As an illustration the dispersion equation of Love waves is obtained for one of such media in which the shear-wave velocity and the rigidity in the lower half-space either increases or decreases along the direction of propagation of waves according as the parameter of heterogeneity is positive or negative.