Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

SECTION V: ANALYTICAL AND NUMERICAL METHODS
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Section V consists of three chapters, of which the first two present the theoretical framework
and equations needed to obtain the practical numerical methods and formulations
included in the third. Thus, Chapters 8 and 9 are for the most part meant as a reference
to Chapter 10, and less so for their own utility. Nonetheless, we decided to include this
material herein in full length because no readily available reference exists containing the
detailed derivation of the powerful stiffness matrix method for all the three major coordinate
systems. Thus, the reader may wish to either browse lightly Chapters 8 and 9 –
especially the example problems in Chapter 9 – or skip these altogether at first and return
later only as may be needed to clarify matters.
Chapter 8 begins with a summary of the solutions to the scalar and vector Helmholtz
equations in three-dimensional space, and then proceeds to give full derivations to these
equations and to the wave equation in all three coordinate systems. Unlike most books on
the theory of elasticity, we use matrix algebra throughout, and manage to express the final
results as products of matrices, each of which depends on one coordinate only (whether
or not the systems are layered). This greatly simplifies applications to layered media.
Chapter 9 makes a compact introduction to the integral transform methods commonly
used to analyze stratified media, such as finite and infinite plates, rods, or spheres. Examples
are also given to illustrate the application of these concepts.
Finally, Chapter 10 contains the detailed equations needed to implement the stiffness
matrix method for layered media in Cartesian, cylindrical, or spherical coordinates (also
called by some the spectral element method). This is a powerful numerical tool that allows
solving dynamically loaded laminated systems consisting of arbitrarily thick layers, or
even infinite media. Examples are included that show what purposes it serves and how the
method is used. A concise description is presented in the introduction and in Section 10.1.
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