Complex frequencies in elastodynamics, with application to the Damping-Solvent Extraction method

This paper addresses the use of complex frequencies in problems of wave propagation and structural vibrations. The most common form of application is as artificial damping that is extracted after the response in the time domain has been obtained. Then again a rather unorthodox application is in the simulation of systems of infinite spatial extent by means of finite systems modeled with discrete methods such as finite elements, a task that can be accomplished even when no transmitting or absorbing boundaries are used. This latter application of complex frequencies, which goes by the name damping-solvent extraction method or its acronym DSE, is assessed herein by means of exact solutions to canonical problems that are used to establish the conditions that must be met by the finite models to work as intended, especially the size of the models, the magnitude of the imaginary component of frequency, and the limitations of the method

Comments on ‘The Reissner-Sagoci problem for a transversely isotropic half-space’ by A. K. Ghorbani-Tanha and M. Eskandari-Ghadi, International Journal for Numerical and Analytical Methods in Geomechanics 2006; 30 (11):1063-1074

See erratum for this manuscript at: http://hdl.handle.net/1721.1/92752The writers of this paper chose to revisit the classical problem of torsional oscillations of a rigid, circular footing welded to an elastic half-space, and provided an extension to a medium with transverse isotropy, but then again restricted it to purely static loads. As it turns out, not only is this limitation unnecessary, but the complete derivation can be fully reduced into the same form of the classical problem. Thus, a dynamic solution to the transverse isotropy problem follows directly from that of the isotropic problem. It suffices for this purpose to scale appropriately the spatial coordinates

On the frequencies of inhomogeneous soil strata: Dobry's paradox

This brief article elaborates on some clearly unlikely predictions made some four decades ago by R. Dobry within the context of his doctoral dissertation, which concerns the resonant frequencies of soil strata whose stiffness starts at zero at ground level and then increases continuously as some power of the depth. Although the problem was eventually traced to subtle changes in the boundary conditions which take place when the soil parameter exceeds a threshold value, and the discrepancy was ultimately fully resolved, Dobry initially countered to the technical objections of the writer by presenting an extremely elegant counter-proof based on dimensional analysis whose strength emanated from the fact that it was free from the issue of the boundary conditions, and thus forced the writer to seek alternative explanations for the contradictory results. It is this fruitful exchange of ideas which provides the motivation for this brief technical note expounding on an apparent paradox, which may prove useful to the soil dynamics community for its potential didactical value, not to mention as an outright tool

Early history of soil–structure interaction

Soil–structure interaction is an interdisciplinary field of endeavor which lies at the intersection of soil and structural mechanics, soil and structural dynamics, earthquake engineering, geophysics and geomechanics, material science, computational and numerical methods, and diverse other technical disciplines. Its origins trace back to the late 19th century, evolved and matured gradually in the ensuing decades and during the first half of the 20th century, and progressed rapidly in the second half stimulated mainly by the needs of the nuclear power and offshore industries, by the debut of powerful computers and simulation tools such as finite elements, and by the needs for improvements in seismic safety. The pages that follow provide a concise review of some of the leading developments that paved the way for the state of the art as it is known today. Inasmuch as static foundation stiffnesses are also widely used in engineering analyses and code formulas for SSI effects, this work includes a brief survey of such static solutions

Number and location of zero-group-velocity modes

The frequency-wavenumber spectra of laminated media often exhibit anomalous modes with descending branches whose group velocity is negative, and these terminate at a minimum point at which the group velocity transitions from negative to positive. These minima are associated with resonant conditions in the medium, which have been clearly observed in experiments and in the nondestructive testing of laminated plates. Starting from first principles, this paper provides a theoretical analysis on the number and location of such zero-group-velocity (ZGV) modes for horizontally layered media bounded by any arbitrary combination of external boundaries. It is found that these ZGV points are few in number and show up mostly in low-order modes which are characterized by a negative second derivative at the cutoff frequencies, a condition that can readily be tested. It is also shown that the effective number of ZGVs is small even when the ratio of the dilatational and shear wave velocity is a rational number and there exist coincidences in cutoff frequencies, a condition that at first would suggest that the number of ZGVs is infinite. Finally, it is shown that the number of ZGVs decreases with the Poisson’s ratio

Effects of Local Soil Conditions on the Topographic Aggravation of Seismic Motion: Parametric Investigation and Recorded Field Evidence from the 1999 Athens Earthquake

During the 1999 Athens earthquake, the town of Adàmes, located on the eastern side of the Kifissos river canyon, experienced unexpectedly heavy damage. Despite the particular geometry of the slope that caused significant motion amplification, topography effects alone cannot explain the uneven damage distribution within a 300-m zone parallel to the canyon’s crest, which is characterized by a rather uniform structural quality. In this article, we illustrate the important role of soil stratigraphy and material heterogeneity on the topographic aggravation of surface ground motion. For this purpose, we first conduct an extensive time-domain parametric study using idealized stratified profiles and Gaussian stochastic fields to characterize the spatial distribution of soil properties, and using Ricker wavelets to describe the seismic input motion; the results show that both topography and local soil conditions significantly affect the spatial variability of seismic motion. We next perform elastic two-dimensional wave propagation analyses based on available local geotechnical and seismological data and validate our results by comparison with aftershock recordings

On the long-time limit of the Garvin–Alterman–Loewenthal solution for a buried blast load

The Garvin–Alterman–Loewenthal solution refers to the problem of a line blast load suddenly applied in the interior of an elastic half-space. It is expected that the long-time asymptotic limit of this solution should be equal to the solution of a related static problem. This expectation is justified here. First, the solution of the static problem is constructed. Then, the asymptotic limit of the transient problem is found, correcting previously published results

Laplace Transform of Products of Bessel Functions: A Visitation of Earlier Formulas

This note deals with the Laplace transforms of integrands of the form ${x^\lambda }{J_\alpha }\left( {ax} \right){J_\beta }\left( {bx} \right)$, which are found in numerous fields of application. Specifically, we provide herein both a correction and a supplement to the list of integrals given in 1997 by Hanson and Puja, who in turn extended the formulas of Eason, Noble and Sneddon of 1956. The paper concludes with an extensive tabulation for particular cases and range of parameters

Dynamic stresses in an elastic half-space

This paper deals with the problem of time-varying point loads applied onto the surface of an elastic half-space and the stresses that such loads elicit within that medium. The emphasis is on the evaluation of the isobaric contours for all six of the stress components at various frequencies of engineering interest and for a full range of Poisson’s ratios. The extensive set of pressure bulbs presented herein may be of help in predicting the severity of dynamic effects in common practical situations in engineering—or even the lack thereof

Off-Center Monopole And Dipole Sources In Fluid-Filled Boreholes

A variety of seismic testing techniques rely on the use of seismic sources, detectors, or both, placed at some depth below the ground surface; these are often installed within fluid-filled boreholes. The interpretation of the records obtained in the course of such explorations requires a thorough understanding of how waves propagate in the borehole and its immediate vicinity. Depending on the distance between the source and the receiver as well as their placement and orientation relative to the axis of the borehole, it is known that very complex wave patterns may arise. In this paper, analytical-numerical solutions are used to study the wave-field elicited by monopole or dipole sources within a fluid-filled cylindrical cavity drilled through an unbounded homogeneous elastic medium. This model is used to assess the effects of a source-receiver tool, placed in an off-centered and/or tilted position inside the fluid-filled borehole, on the propagation of both axisymmetric and non-axisymmetric wave modes