Response of an Elastic Half Space to a Decelerating Surface Point Load

1969 ◽  
Vol 36 (4) ◽  
pp. 819-826 ◽  
Author(s):  
K. I. Beitin
Keyword(s):  
Half Space ◽  
Poisson Ratio ◽  
Constant Load ◽  
Elastic Half Space ◽  
Point Load ◽  
Reciprocal Theorem ◽  
Surface Load ◽  
A Value ◽  
Mach Cones ◽  
Rayleigh Wave Speed

The displacement field on the surface of an elastic half space (Poisson ratio = 1/4), caused by the motion of a decelerating point surface load, is investigated by means of the dynamic Betti-Rayleigh reciprocal theorem. The load is applied impulsively and made to move rectilinearly at constant deceleration along the surface. The load speed varies from superseismic to a value less than the Rayleigh wave speed. The results show that significant differences exist between displacements obtained in this problem and those resulting from the usual assumption of constant load speed. The differences are primarily due to the presence of Mach cones which appear at superseismic load speeds and remain ahead of the initial wave fronts even after the speed becomes subseismic.

Nonuniformly Moving Loads on an Anisotropic Elastic Half-Space

2003 ◽  
Vol 19 (1) ◽  
pp. 217-224
Author(s):  
Kuang-Chong Wu
Keyword(s):  
Half Space ◽  
Moving Load ◽  
Constant Load ◽  
Elastic Half Space ◽  
Reciprocal Theorem ◽  
Line Load ◽  
Surface Displacements ◽  
Anisotropic Elastic ◽  
The Reciprocal Theorem ◽  
Rayleigh Wave Speed

ABSTRACTThe transient motion in an anisotropic elastic half-space due to a moving surface line load is considered. The load is applied suddenly on the surface and moves off in a fixed direction with nonuniform speed. Integral expressions for the displacements are derived using the reciprocal theorem. The waves generated by the moving load are discussed. Special attention is paid to the singularities in surface displacements generated as the load moves through the Rayleigh wave speed. Explicit expression is obtained for the particle velocity due to a constant load moving with constant speed.

Impedance of Strip-Traveling Waves on an Elastic Half Space: Asymptotic Solution

1974 ◽  
Vol 41 (2) ◽  
pp. 412-416
Author(s):  
S. H. Crandall ◽  
A. K. Nigam
Keyword(s):  
Asymptotic Solution ◽  
Rayleigh Wave ◽  
Half Space ◽  
Traveling Wave ◽  
Load Distribution ◽  
Wave Speed ◽  
Elastic Half Space ◽  
Surface Load ◽  
Shear Wave Speed ◽  
Rayleigh Wave Speed

The dynamic normal-load distribution across a strip that is required to maintain a plane progressive wave along its length is studied for the case where the strip is of infinite length and lies on the surface of a homogeneous isotropic elastic half space. This configuration is proposed as a preliminary idealized model for analyzing the dynamic interaction between soils and flexible foundations. The surface load distribution across the strip and the motion of the strip are related by a pair of dual integral equations. An asymptotic solution is obtained for the limiting case of small wavelength. The nature of this solution depends importantly on the propagation velocity of the strip-traveling wave in comparison with the Rayleigh wave speed, the shear wave speed and the dilatational wave speed. When the strip-traveling wave propagates faster than the Rayleigh wave speed, a pattern of trailing Rayleigh waves is shed from the strip. The limiting amplitude of the trailing waves is provided by the asymptotic solution.

Response of an Elastic Half Space to Uniformly Moving Circular Surface Load

1970 ◽  
Vol 37 (1) ◽  
pp. 109-115 ◽  
Author(s):  
S. K. Singh ◽  
J. T. Kuo
Keyword(s):  
Half Space ◽  
Hypergeometric Functions ◽  
Integral Form ◽  
Elastic Half Space ◽  
Point Load ◽  
Surface Load ◽  
Static Part ◽  
Circular Load ◽  
Circular Surface ◽  
Velocity Effect

The problem of a uniformly moving circular surface load of a general orientation on an elastic half space for two types of load distribution, viz., “uniform” and “hemispherical,” is considered. The solutions have been obtained in integral form. The displacements on the surface of the half space, in the case in which the load velocity V is smaller than the transverse wave velocity of the medium CT are expressed in a closed form as a sum of two terms by using properties of Gauss’ hypergeometric functions. One of these terms gives the static part of the solution, whereas the other term represents the velocity effect part. At distances greater than about five radii from the center of the moving circular load, a moving point load is found to be a good approximation.

scholarly journals The near-resonant regimes of a moving load in a three-dimensional problem for a coated elastic half-space

2016 ◽  
Vol 22 (1) ◽  
pp. 89-100 ◽  
Author(s):  
Bariş Erbaş ◽  
Julius Kaplunov ◽  
Danila A Prikazchikov ◽  
Onur Şahin
Keyword(s):  
Half Space ◽  
Moving Load ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Point Load ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Dispersive Effect ◽  
Mach Cones ◽  
Integral Solutions

This paper deals with the three-dimensional analysis of the near-resonant regimes of a point load, moving steadily along the surface of a coated elastic half-space. The approach developed relies on a specialized hyperbolic–elliptic formulation for the wave field, established earlier by the authors. Straightforward integral solutions of the two-dimensional perturbed wave equation describing wave propagation along the surface are derived along with their far-field asymptotic expansions obtained using the uniform stationary phase method. Both sub-Rayleigh and super-Rayleigh cases are studied. It is shown that the singularities arising at the contour of the Mach cones typical of the super-Rayleigh case, are smoothed due to the dispersive effect of the coating.

scholarly journals An analytical solution to investigate the dynamic impact of a moving surface load on a shallowly-buried tunnel

2021 ◽  
Author(s):  
Xueyu Geng
Keyword(s):  
Half Space ◽  
Ground Surface ◽  
Elastic Half Space ◽  
Cylindrical Wave ◽  
Point Load ◽  
Surface Load ◽  
Critical Depth ◽  
Dynamic Impact ◽  
Moving Surface ◽  
Moving Speed

In order to investigate the dynamic impact of a moving surface load on a shallow-buried tunnel, an analytical model of a tunnel embedded in an elastic half-space was proposed. The half-space and the tunnel structure were modeled as visco-elastic media and the moving surface load was simplified as a moving point load on the ground surface. Based on the fundamental solution for the isotropic elastic half-space system in Cartesian and cylindrical coordinates, the dynamic response of a shallowly-buried tunnel in a half-space generated by a moving surface load was obtained. The transformations between plane wave and cylindrical wave functions were used to facilitate the application of boundary conditions at the ground surface and the tunnel interface. It was found that the vibration of the shallowly-buried tunnel increases significantly as the load moving speed increases, and reaches a maximum value at a critical load velocity. The tunnel vibration can be greatly reduced as the buried depth increases, and can satisfy the requirement of vibration specification (ISO 04866-2010) after it exceeds the critical depth. The critical depth increases exponentially with the increase of the moving speed of the surface load.

scholarly journals Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 505-515 ◽  
Author(s):  
D. C. Gakenheimer ◽  
J. Miklowitz
Keyword(s):  
Exact Solution ◽  
Free Surface ◽  
Steady State ◽  
Wave Front ◽  
Half Space ◽  
Displacement Field ◽  
Elastic Half Space ◽  
Point Load ◽  
Limit Case ◽  
Transient Waves

The propagation of transient waves in a homogeneous, isotropic, linearly 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 derived for the interior of the half space and for all load speeds. Wave-front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb’s point load problem.

The azimuthal and polar radiation patterns obtained from a horizontal stress applied at the surface of an elastic half space

1962 ◽  
Vol 52 (1) ◽  
pp. 27-36
Author(s):  
J. T. Cherry
Keyword(s):  
Surface Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Poisson’S Ratio ◽  
Horizontal Stress ◽  
Elastic Half Space ◽  
The Body ◽  
Body Waves ◽  
Poisson's Ratio ◽  
A Value

Abstract The body waves and surface waves radiating from a horizontal stress applied at the free surface of an elastic half space are obtained. The SV wave suffers a phase shift of π at 45 degrees from the vertical. Also, a surface wave that is SH in character but travels with the Rayleigh velocity is shown to exist. This surface wave attenuates as r−3/2. For a value of Poisson's ratio of 0.25 or 0.33, the amplitude of the Rayleigh waves from a horizontal source should be smaller than the amplitude of the Rayleigh waves from a vertical source. The ratio of vertical to horizontal amplitude for the Rayleigh waves from the horizontal source is the same as the corresponding ratio for the vertical source for all values of Poisson's ratio.

On an Incompressible Nonlinearly Elastic Half-Space under a Compressive Point Load

2004 ◽  
Vol 9 (1) ◽  
pp. 97-117 ◽  
Author(s):  
Mary R Lee ◽  
Debra A Polignone Warne ◽  
Paul G Warne
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Point Load ◽  
Nonlinearly Elastic
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