Source signature and elastic waves in a half‐space under a sustainable line‐concentrated impulsive normal force

Abstract

SUMMARYFirst, the response of an ideal elastic half-space to a line-concentrated impulsive normal load applied to its surface is obtained by a computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. Then, the geometry is determined of the obtained waves and the source signature}the latter is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Behind the dilatational precursor wave, there exists a pencil of three plane waves extending from the vertex at the impingement point of the precursor wave on the stress-free surface of the half-space to three points located on the other two boundaries of the solution domain. These four wave-arresting points (end points) of the three plane waves constitute the source signature. One wave is an inhibitor front in the behaviour of the normal stress components and the particle velocity, while in the behaviour of the shear stress component, it is a surface-axis wave. The second is a surface wave in the behaviour of the horizontal components of the dependent variables, while the third is an inhibitor wave in the behaviour of the shear stress component. An inhibitor wave is so named, since beyond it, the material motion is dying or becomes uniform. A surface-axis wave is so named, since upon its arrival, like a surface wave, the dependent variable in question features an extreme value, but unlike a surface wave, it exists in the entire depth of the solution domain. It is evident from this work that SaintVenant's principle for wave propagation problems cannot be formulated; therefore, the above results are a consequence of the particular model proposed here for the line-concentrated normal load.

Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism

“…A more definite conclusion regarding the non-validity of SVP to dynamic excitation of a half-space has been put forward by Ziv (2002Ziv ( , 2003: " halfspace response is hypersensitive to the type of loading, to the way it is distributed on the source rim, and to the geometry of the source rim under the load" (Ziv, 2002, p. 402). Therefore (Ziv, 2003, p. 254-255): "Saint-Venant's principle for wave propagation problems cannot be formulated.…”

Elastic half‐space under an oblique line impulse

“…The solution to the present transient boundary-value problem is obtained by superimposing the solution of Reference [1] on that of Reference [2]}or vice versa. As a result, the obtained source signature consists of all the source signature points of the above references plus a new point.…”

“…The objective of this paper is to give the solution to the transient boundary-value problem concerning an elastic half-space under combined line-concentrated impulsive loads. One load is the momentarily applied vector shear force of Reference [1], while the other is the sustainable normal force of Reference [2]. Therefore, the solution for the combined loads is obtained by the superposition principle using the solutions of these references.…”

“…A literature search reveals that the present boundary-value problem where the shear component of the load is applied momentarily was solved in Reference [1]; other works involved sustainable shear loads. Both References [1,2] employed an analytical-numerical method based on the characteristics approach by which the source signature and the wave system emerged explicitly in the solution. The source signature was shown to be the imprint of the spatiotemporal configuration of the excitation source in the resultant response and the resultant wave system was shown to be different from that of the aforementioned traditional one.…”

“…To explain the critical events of the transient deformation of the half-space under the present combined loads, it is necessary to identify the wave fronts traversing the strained half-space. Section 2 is devoted to the source signature [1,2], which is intrinsically affined to the wave system in this boundary-value problem, while the resultant wave system is presented in Section 3. The resultant deformation is presented in Section 4 and the conclusions of the main findings are given in Section 5.…”