Response of an Elastic Half Space to Expanding Surface Loads

Abstract

A class of elastic half-space problems involving axisymmetric, normally applied, surface loads is investigated. Each load is assumed to suddenly emanate from a point on the surface and expand radially at a constant rate. The cases of loads shaped like a ring and a disk are considered in detail. Exact solutions are derived for the displacements at every point in the half space in terms of single integrals. Each integral is identified as a specific wave. The integrals are evaluated analytically and numerically for different depths in the half space, for loads expanding at superseismic, transeismic, and subseismic rates, and for different values of Poisson’s ratio. Moreover, the interaction of the loads and the Rayleigh wave is described. Then solutions are obtained for loads of other shapes by convoluting the ring and disk load solutions.

Discussion: “Response of an Elastic Half Space to a Decelerating Surface Point Load” (Beitin, K. I., 1969, ASME J. Appl. Mech., 36, pp. 819–826)

“…Similar couplings of impulsive surface loads traveling at constant velocities with Rayleigh waves have been observed by Ang [6] and Payton [7] for moving line loads, and by the discusser [8] for an expanding ring load.…”

“…In conclusion, a load traveling steadily at the Rayleigh wave speed does not cause a singular response of the entire half space, if a transient problem is considered as in [1,[5][6][7][8]. Rather, such a singular response may only occur for steady-state problems, as in [9,10].…”

Theoretical Effect of Yield and Burst Height of Atmospheric Explosions on Rayleigh Wave Amplitudes

“…The solutions to an expanding ring and an expanding disk surface load on an isotropic half space have been obtained by Gakenheimer & Miklowitz (1969), and Gakenheimer (1969Gakenheimer ( , 1971), among others. Rayleigh and body waves due to an atmospheric explosion modelled by an equivalent moving normal force applied to a multilayered half space, have been formulated by Hudson (1969) and computed for an equivalent stationary point source at the free surface by Douglas,Hudson & Blamey (1 972).…”

Theoretical Effect of Yield and Burst Height of Atmospheric Explosions on Rayleigh Wave Amplitudes

“…Toksoz and Ben-Menahem (1965) assumed a point source in a homogeneous liquid halfspace overlying a homogeneous solid hilfspace; they made use of an equation of Cagniard (1962) The solutions to an expanding ring and an expanding disk surface load on an Isotropie solid half space have been obtained by Gakenheimer and Mlklowitz (1969), i-jikenheimer (1969), and Gakenheimer (1971 among others. Rayleigh waves and body waves due to an atmosphere explosion modeled by an equivalent moving normal force applied to a multilayered half space have been formulated by Hudson (1969) and computed for an equivalent stationary point source at the free surface by Douglas et al (1972).…”