Fundamental Solutions in Elastodynamics

Abstract

This 2006 work is a compilation of fundamental solutions (or Green's functions) for classical or canonical problems in elastodynamics presented with a common format and notation. These formulas describe the displacements and stresses elicited by dynamic sources in solid elastic media like full spaces, half-spaces, strata and plates in both two and three dimensions, using the three major coordinate systems (Cartesian, cylindrical and spherical), and also for transient and harmonic motions. Such formulas are useful for numerical methods and practical application to problems of wave propagation in elasticity, soil dynamics, earthquake engineering, mechanical vibration, or geophysics. These formulas were heretofore only found scattered throughout the literature. The solutions are tabulated without proof, but giving reference to appropriate modern papers and books containing full derivations. Most formulas in the book have been programmed and tested within the MATLAB environment. The program listings are available for free download on the book's website.

Garvin’s generalized problem revisited

“…as also given by Kausel (2006). Zipora Alterman (1925-1974) and Dan Loewenthal (1943-2000 -henceforth referred to as A&L for short-were both Professors of Geophysics at Tel Aviv University, and both passed away at the relatively young ages of 49 and 57 years, respectively.…”

“…This powerful strategy has been widely used and is commonly referred to as the Cagniard-de Hoop method, in honor of their inventors (De Hoop, 1960). After this is done, the response functions at the surface are found to be given by the following expressions (see Kausel, 2006, but appropriately modified to account for the different sign convention employed herein for the vertical direction):…”

Complex Frequencies in Elastodynamics, with Application to the Damping-Solvent Extraction Method

“…axis) to a harmonic force in some fixed radial direction, say the horizontal, which causes the cylinder to oscillate in that direction, and the ratio of the applied force to the observed oscillation is the lateral dynamic stiffness or impedance of the cylinder. This is conceptually similar to the problem of a semi-cylindrical foundation at the surface of an elastic half-space undergoing oscillations transverse to the free surface, but the latter is more It can be shown (Kausel, 2006) that the cylindrical displacement components at any arbitrary point and the stresses acting on cylindrical surfaces are of the form…”

“…It can be shown that the torsional response of the sphere is governed by the following equations in spherical coordinates, e.g. see Kausel, 2006:…”

Predictions of the Dynamic Response of Piled Foundations in a Multi-Layered Half-Space Due to Inertial and Railway Induced Loadings

“…From the different available methods based on the wavenumber/frequency domain approach, a hybrid formulation of the Thin Layer Method (TLM) [12,13] and the Direct Stiffness Matrix Method (DSMM) [12,14] is used for the calculation of the response field in the soil. In both these methods, which are similar to the finite element method, element stiffness matrices are used to express the relation between displacements and stresses at the soil element boundaries.…”