# Response of an elastic half-space to normal pressure over an elliptic area

TL;DR: In this paper, a displacement field is obtained by the use of Cagniard-De-Hoop technique and different wave fronts expected are identified and nature of approximate form of displacement near wave fronts are discussed.

Abstract: Transient response of an elastic half-space subjected to a uniform normal pressure acting over an elliptic area on the boundary is obtained. This displacement field is obtained by the use of Cagniard-De-Hoop technique. Different wave fronts expected are identified and nature of approximate form of displacement near wave fronts are discussed.

##### Citations

More filters

••

TL;DR: In this paper, the authors considered a buried elliptic source distributed over an inclined plane and presented some graphical plots of the surface displacement for various inclination of the fault plane, where the Cagniard method has been used.

Abstract: Usually an exact solution to the surface displacement in an elastic half space is available for sources parallel to the surface.
Here we consider a buried elliptic source distributed over an inclined plane. Circular and Point sources have been considered as
particular cases of an elliptic source. Cagniard method has been used. We present some graphical plots of the surface displacement for various inclination of the fault plane.

2 citations

••

TL;DR: In this paper, the Cagniard De-Hoop technique has been used to obtain the two dimensional exact transient response due to the slip in the vertical mode via body force equivalent.

Abstract: The application of the property of dynamic similarity is useful to the solution which admits a self-similarity or homogeneous form. One independent variable has been dropped in the present equivalent set of the governing equations. The displacement discontinuity on the crack face and also the displacement field on the surface due to an in-plane shear model over an expanding zone of slippage of arbitrary dip have been obtained. The moving slip edge extends towards the surface with a constant velocity. Cagniard De-Hoop technique has been used here to obtain the two dimensional exact transient response due to the slip in the vertical mode via body force equivalent. The results of the present paper are valid at least up to the time when the diffracted waves from the crack edge have not reached the receiving station. The spectral behavior of the source time function has also been discussed.

##### References

More filters

••

TL;DR: The Knopoff-deHoop representation theorem has been used to calculate the form of the body waves radiated from an elliptical fault as discussed by the authors, where Rupture is assumed to initiate at one focus of the ellipse and then spread out radially on the fault plane.

Abstract: The Knopoff-deHoop representation theorem has been used to calculate the form of the body waves radiated from an elliptical fault. Rupture is assumed to initiate at one focus of the ellipse and then spread out radially on the fault plane. Two cases are considered: 1) constant slip everywhere on the fault surface and 2) a variable slip which approaches zero at the fault edge. The radiation is calculated for distances from the fault which are large compared to the fault dimensions. The body waves are described by the product of two factors, one of which is the familiar equivalent-force system radiation pattern. The other factor includes the time dependence of the signal; it does not depend upon the direction of slip. The body waves exhibit two stopping phases. The theory is used to estimate the fault dimensions associated with six deep-focus earthquakes studied by Kasahara. The estimated fault dimensions are about twice the dimensions of the focal sphere as found by Kasahara. Finally, the difference between the phase spectrums of shallow and deep-focus earthquake radiation observed by Kishimoto is shown to be related to a difference in shape of the two fault surfaces; shallow-focus earthquakes appear to be associated with elongated fault surfaces, whereas deep-focus earthquakes are associated with more circular fault surfaces.

196 citations

••

TL;DR: In this article, the displacement field is analyzed using multi-integral transforms and an inversion scheme based on the well-known Cagniard technique, which reduces the displacements to single integral and algebraic contributions, each of which is identified as the disturbance behind a specific wave front.

Abstract: The propagation of transient waves in an elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are computed for all points of the half-space as well as for all load speeds.
The disturbance is analyzed by using multi-integral transforms and an inversion scheme based on the well-known Cagniard technique. This reduces the displacements to single integral and algebraic contributions, each of which is identified as the disturbance behind a specific wave front. The same solution is valid for all load speeds, even though the wave front geometry varies greatly, depending on the speed of the load relative to the body wave speeds. Moreover, the surface displacements are obtained from the interior ones, but only after the Rayleigh waves are computed by a separate calculation. Then, by taking advantage of the form of the exact solution, wave front expansions and Rayleigh wave approximations are computed for all load speeds.
Several other analytical results are obtained for restricted values of the load speed. In particular, when it exceeds both of the body wave speeds the steady-state displacement field is separated from the transient one and reduced to algebraic form. Also, for the limit case of zero load speed a new representation of the interior displacements for Lamb's point load problem is displayed in terms of single integrals.

147 citations

••

TL;DR: In this article, the Cagniard De-Hoop technique was used to generate a pulse in an elastic half space by impulsive normal pressure over a circular area on the surface.

Abstract: The generation of pulse in an elastic half space by impulsive normal pressure over a circular area on the surface has been investigated by Cagniard De-Hoop technique. The approximate representations of the displacement field near the times of arrival of various wave fronts have been derived by a limiting process.

9 citations

••

TL;DR: In this article, the Laplace transforms of the displacement components of an axisymmetrical poloidal pulse were derived for a semi-infinite, homogeneous, isotropic elastic solid by applying a uniform time-dependent normal pressure over a circular portion of the surface of the half-space.

Abstract: In this paper we consider the generation of an elastic pulse in a semi-infinite, homogeneous, isotropic elastic solid by the application of a uniform time-dependent normal pressure over a circular portion of the surface of the half-space. Formal expressions are presented for the Laplace transforms of the displacement components which describe the axisymmetrical poloidal pulse. The pattern of wavefronts which develops consists of a direct wavefront, edge dilatational and rotational wavefronts, and head wavefronts. The discontinuous behaviour associated with the arrival of the various wavefronts is examined with the aid of approximate representations of the displacement field.

7 citations