Analytic resolution of time-domain half-space Green's functions for internal loads by a displacement potential-Laplace-Hankel-Cagniard transform method

Abstract

<jats:p>A refined yet compact analytical formulation is presented for the time-domain elastodynamic response of a three-dimensional half-space subject to an arbitrary internal or surface force distribution. By integrating Laplace and Hankel transforms into a method of displacement potentials and Cagniard's inversion concept, it is shown that the solution can be derived in a straightforward manner for the generalized classical wave propagation problem. For the canonical case of a buried point load with a step time function, the response is proved to be naturally reducible with the aid of a parametrized Bessel function integral representation to six wave-group integrals on finite contours in the complex plane that stay away from all branch points and the Rayleigh pole except possibly at the starting point of the contours. On the latter occasions, the possible singularities of the integrals can be rigorously extracted by an extended method of asymptotic decomposition, rendering the residual numerical computation a simple exercise. With the new solution format, the arrival time of each wave group is derivable by simple criteria on the contour. Typical results for the time-domain response for an internal point force as well as the degenerate case of a surface point source are included for comparison and illustrations.</jats:p>

Unified wavefront singularity characterization of three-dimensional elastodynamic time-domain half-space Green's function under impulsive boundary and internal loads

“…where u is the displacement vector at a material point x with λ, μ and ρ being the two Lamé constants and mass density, respectively, subject to the boundary conditions on the free surface and the radiation condition at infinity. As shown in [27] Case A: a horizontal concentrated point load in e 1 −direction…”

“…with tSP2 being the same as tSS1 in the case of z = 0. In (2.8) to (2.25), tP , tS , tPP , tSS , tPS and tSP are respectively the arrival times of direct P-, direct S-, reflected PP-, SS-, PS-and SP-wave groups and η [27] with…”

“…To explore the possible variety of singular wavefront behaviours in the half-space Green's function, some of the novel features of the adopted integral representation in the preceding section are particularly relevant: with respect to η are all finite in length and stay in the first quadrant of the complex ξ -plane (see illustrations in [27]). They are away from the two branch-points and the Rayleigh pole, except possibly when their starting points ξ ( t, 0), henceforth referred to as ξ0 ( t), are on the upper imaginary axis.…”

“…in terms of only elementary functions. The pair of (3.6) and (3.7) constitutes a dual integral-closed form representation of the singular part of I γ i (r, t), which can, among other usage, serve as the core instrument for an effective application of the method of asymptotic decomposition in dealing with singular integrals as in [16,27]. The specific sets of κ q γ i ( t ), q = 1, 2, 3 in (3.7) for the five fundamental integrals {I γ i } are (1) I h r asym (r, 0; 0, t) radial displacement response due to horizontal load (2) I h θ asymp (r, 0; 0, t) tangential displacement due to horizontal load…”

Research of Dynamic Processes in an Anvil During a Collision with a Sample

“…The problem was taken up again by Sánchez-Sesma et al [31] who provided a full set of formulas with an exact solution for any source and recipient location. Pak and Bai [32] presented an improved but compact analytical formula of the elastodynamic response in the time domain of a threedimensional half-space subjected to an arbitrary distribution of internal or surface forces. For a surface point pulse operating on a 3D medium, Pekeris [33] gave a closed solution for a vertical point source, and Mooney [28] extended the results for vertical loads by any Poisson's ratio, ignoring the radial component.…”