Showing papers on "Displacement field published in 1971"

Fracture mechanics of interfacial cracks

Abstract: This paper contains a two-dimensional analysis of the stress field around a crack on the plane interface between two bonded dissimilar anisotropic elastic half-spaces. This analysis is then combined with the usual local form of the Griffith virtual work argument to give an explicit fracture criterion which involves a suitably defined ‘stress concentration vector’ and the specific surface energy of the bonded surfaces. This criterion has a simple structure and reduces to the conventional form of Irwin when the two half-spaces are isotropic and identical. The analysis is then extended to cracks moving uniformly and a local fracture criterion with the same structure as the static criterion is derived by an energy balance argument. The criterion is specialized to isotropic half-spaces for illustration, when it predicts that the speed of a crack on an interface between such media will be limited by a speed Vc which is slightly greater than the smaller of the two Rayleigh wave speeds. A by-product of the analysis is an expression for the displacement field of an arbitrary interfacial dislocation, either stationary or moving uniformly.

The Elasticity Theory of Dislocations in Real Earth Models and Changes in the Rotation of the Earth

Abstract: Summary Steketee's Elasticity Theory of Dislocations is generalized to real Earth models. Taken into account are; (i) self-gravitation, (ii) radial variation of elastic properties, density and gravity, (iii) initial hydrostatic stress, (iv) the presence of the liquid core. Volterra's formula for the displacement field is found to hold in the more general circumstances for slip faults. The dilemma, previously pointed out by Jeffreys and Vicente, which arises when the Adams and Williamson condition is assumed not to hold everywhere perfectly in the core, is resolved. This result also bears on the theory of Earth tides and tidal loading. Changes in the inertia tensor are shown to arise only from spheroidal displacement fields of degree zero and two. These fields have virtually no attenuation with distance from the fault. In the one example in which a direct comparison can be made, the present theory gives a factor of 7.5 increase over a mapped half-space theory and a factor of 2.9 increase over the result for a uniform, spherical Earth, in the contribution to secular polar shift and excitation of Chandler wobble. Calculated and observed levels appear now to be in agreement.

Diffuse Scattering from Defect Clusters near Bragg Reflections

Abstract: The diffuse scattering from crystals containing defect clusters with strong displacement fields has been calculated. The scattering is extremely large and concentrated in small regions near the Bragg reflections. For small concentrations the diffuse intensity is essentially determined by a "cluster form factor" depending explicitly on the displacement field. The scattering is studied in detail for small and large deviations from the Bragg reflection. Exact expressions are given for the moments of the scattering, depending sensitively on the displacement field in the core of the cluster. The diffuse scattering for strong displacement fields shows a pronounced asymmetry; its center does not coincide with the position of the Bragg peak, in agreement with recent experimental observations.

Electron Microscope Image Contrast from Dislocation Loops

Abstract: Experimental electron microscope images of dislocation loops in molybdenum have been compared to theoretical loop images displayed in the form of computer simulated micrographs. The exact displacement field around a circular edge dislocation loop has been calculated using isotropic elasticity theory; this displacement field together with the two-beam dynamical theory of electron diffraction contrast have been used to compute image intensities. Specific results are given for loops of radius ≈ 250 A and ≈ 25 A. Their contrast has been studied over a narrow range of the numerous variables which are known to influence image features. It has been found that the general topographical character of certain prominent experimental images is qualitatively accounted for by the simple “elastic” defect model with the two-beam theory of electron diffraction.

Convergence of lattice sums encountered in computing the scattering of X‐rays by crystal defects

Abstract: The diffuse scattering of X-rays from a crystal containing defects involves a sum over all lattice points. This sum suffers from problems in convergence and truncation. These problems are investigated with a computer using the model of a simple cubic lattice containing a body-centered interstitial with a displacement field μ = Cr/r3. Convergence is discussed in terms of a truncation parameter. Truncating the sum introduces a ripple which is removed when the finite resolution imposed by any measurement is taken into account. Both the truncation parameter and finite resolution determine distances in the lattice. The resolution distance dictates the choice of truncation distance and this is discussed for a Gaussian resolution function. We compare the results of the sum formulation with a commonly used integral approximation. Finally, we find that when the displacements produced by the defect are large the Huang scattering dominates the size-effect scattering and that in this case the diffuse scattering is more intense on the side of a reflection furthest from the origin.

Transformation of earthquake displacement field for spherical earth

Abstract: An addition theorem for spherical waves is used to transform the displacement field due to an arbitrary shear dislocation in an unbounded homogeneous medium with origin at the focus to that in a coordinate system with origin at the center of the Earth. The final results are expressed in terms of the eigenvector solutions of the vector Navier equation of dynamical elasticity. Some serious mistakes in recent papers on the subject have been pointed out.

Transformation of the earthquake displacement field for spherical earth—Static case

Abstract: Explicit expressions for the static displacement field for a Volterra dislocation and a center of compression in an infinite homogeneous medium are obtained. Using an addition theorem, the field is transformed to a polar coordinate system with origin at the center of the Earth. Expressions for the discontinuity in the motion stress vector across the concentric spherical surface through the source are then obtained. These results can be used in studying the deformation of a multilayered spherical earth model induced by internal sources by the Thomson-Haskell matrix method which has so far been mostly applied to dynamic problems.

Response of a semi-infinite elastic solid to an arbitrary torque along the axis

Abstract: The response of an otherwise stress-free elastic half space to an arbitrarily distributed static torque along the axis of symmetry is obtained. As an example, numerical results are found for the stress and displacement fields due to the special case of a concentrated torque acting at a finite distance from the free surface.

SH waves from torsional sources in semi-infinite heterogeneous media

Abstract: The theory of Hankel transforms is used to obtain the displacement field forSH waves generated by time harmonic, buried, torsional sources in semi-infinite heterogenous media in which material properties are functions of the depth coordinate. An application to a heterogeneous medium with exponentially varying properties has been discussed. Surface displacements from a surface source have been evaluated at large distances from the source. The results have been compared with those for the homogeneous medium.

Displacement at the Centre of the Earth Induced by an Earthquake or an Underground Explosion

Abstract: Ben-Menahem & Singh (1968) gave the basic theory for the deformation of a homogeneous, isotropic, non-gravitating, elastic sphere due to a buried tangential dislocation, tensile dislocation or centre of explosion. In general, the displacement field can be evaluated by applying the condition that the stresses vanish at the surface of the sphere. Ben-Menahem and Singh found that the evaluation of the static field for a sphere associated with the Legendre polynomial of the first degree (1 = 1) poses some problems. In this case one must incorporate additional conditions, namely, that the angular momentum of the sphere about its centre is zero and that the centre of mass of the sphere is not displaced. The aim of the present note is to determine the displacement at the centre of a sphere due to buried sources. It is interesting to note that out of all the values of 1 (the degree of the Legendre polynomial) only one value, namely 1 = 1, gives non-zero contribution to the angular momentum of the sphere about its centre, the displacement of its centre of mass or the displacement of its centre itself. We consider a homogeneous elastic sphere of radius a and Poisson ratio CT. Let there be a centre of explosion or a localized tangential dislocation of fault area dS, dip angle 6, slip angle A and displacement jump U at the point (r = ro, 8 = 0) inside the sphere. The origin of a spherical co-ordinate system (r,8,4) is chosen at the centre of the sphere. Recently Ben-Menahem & Israel (1970) calculated the displacement field at an arbitrary point within the sphere induced by buried sources. To get the residual displacement at the centre of the sphere, we put r = 0 in the expressions of Ben-Menahem and Israel and obtain the following results for various ,ources.

Plane near-field response of an elastic plate

Abstract: The plane near-field response of an elastic plate is investigated both experimentally and theoretically. Experimental observations of the displacement field within the plate are obtained by means of high-speed photography of grid lines imbedded in a clear specimen molded from a low-modulus material. The analytical solution is obtained by using a recently proposed numerical procedure developed for plane problems in dynamic elasticity.