Source signature and elastic waves in a half‐space under a momentary shear line impulse

Abstract

SUMMARYThe transient deformation of an elastic half-space under a line-concentrated impulsive vector shear load applied momentarily is disclosed in this paper. While in an earlier work, the author gave an analyticalnumerical method for the solution to this transient boundary-value problem, here, the resultant response of the half-space is presented and interpreted. In particular, a probe is set up for the kinematics of the source signature and wave fronts, both explicitly revealed in the strained half-space by the solution method. The source signature is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Fourteen wave fronts exist behind the precursor shear wave S: four concentric cylindrical, eight plane, and two relativistic cylindrical initiated at propagating centres that are located on the stationary boundaries of the solution domain. A snapshot of the stressed half-space reveals that none of the 14 wave fronts fully extend laterally. Instead, each is enclosed within point bounds. These wave arresting points and the two propagating centres of the relativistic waves constitute the source signature. The obtained 14 wave fronts are further combined into 11 disparate wave fronts that are grouped into four categories: an axis of symmetry wave}so named here by reason of being a wave front that is contiguous to the axis of symmetry, three body waves, five surface waves and two inhibitor waves}so named here by reason that beyond them the material motion dies out. Of the three body waves, the first is an unloading shear wave, the second is a diffracted wave and the third is a reflected longitudinal two-branch wave. Of the two inhibitor waves, the first is a two-joint relativistic wave, while the second is a two-branch wave. The wave system, however, is not the same for all the dependent variables; a wave front that appears in the behaviour of one dependent variable may not exist in the behaviour of another. It is evident from this work that Saint-Venant's principle for wave propagation problems cannot be formulated. Therefore, the above results are valid for the particular proposed model for the momentary line-concentrated shear load. The formulation of the source signature, the wave system, and their role in the half-space transient deformation are presented here.

The response of an elastic half‐space under a momentary shear line impulse

“…History of the shear stress, horizontal particle velocity, and the vertical particle velocity components at ð1; 1; tÞ or ð ffiffi ffi 2 p ; 458; tÞ: The waves that have not been seen in the previous plots are wave 4, which is the reflected longitudinal body wave, and the inhibitor wave 12 (both defined in Reference [24] Figure 16. History of the shear stress and the horizontal normal stress components at ð0:38; 0:94; tÞ or ð1:014; 688; tÞ: The waves that have not been seen in the previous plots are wave 5, which is the head body wave, and the inhibitor wave 14 (both defined in Reference [24] shear wave front S; 14 waves, marked 1-14 in these waveform plots, are revealed by the solution method of the present paper. This wave system, defined and formulated in Reference [24], consists of an axis of symmetry wave, marked 1; four body waves, marked 2-5; five surface waves, marked 6-10; and four inhibitor waves, marked 11-14.…”

“…History of the shear stress and the horizontal normal stress components at ð0:38; 0:94; tÞ or ð1:014; 688; tÞ: The waves that have not been seen in the previous plots are wave 5, which is the head body wave, and the inhibitor wave 14 (both defined in Reference [24] shear wave front S; 14 waves, marked 1-14 in these waveform plots, are revealed by the solution method of the present paper. This wave system, defined and formulated in Reference [24], consists of an axis of symmetry wave, marked 1; four body waves, marked 2-5; five surface waves, marked 6-10; and four inhibitor waves, marked 11-14. Since each critical deformation is explicitly indicated in the plots by a wave front, the interpretation of the response of the halfspace to the applied load is readily available at a glance, as is described below.…”

“…A comprehensive account of the results of this boundaryvalue problem is beyond the scope of the present paper, since the work here was devoted to the establishment of a method of computation. The kinematical formulation of the wave fronts and the kinetic interpretations of their effects on the transient deformation of the half-space are given in Reference [24], where other events that cannot be seen in the waveform plots are also described in detail.…”

“…The small double-circled dots in Figure 4 are the present computational points. The marked numbers are the wave fronts defined and formulated in Reference [24] where waves 6-10 are surface waves and 11 and 13 are inhibitor waves. Table I of Reference [24]).…”

“…Finally, comparable results and a discussion on the solution method are presented. This work is devoted to the development of the solution method, whereas the interpretation of the resultant transient deformation is given in the subsequent paper [24].…”