Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

The Linear Theory of Elasticity

Steady State Problems in Anisotropic Elasticity

Many of the recently discovered characteristics and applications to electronics of surface elastic waves are discussed. First the propagation of various elastic waves at the surfaces of solids is considered, followed by descriptions of the many ways which have been demonstrated for transduction between surface elastic waves and electromagnetic waves. Surface-wave amplification, primarily in semiconductors, and wave guiding, focusing, and reflection are examined. The properties of these waves suit them for use in a number of applications, which are discussed, ranging from realization of electronic amplifiers, frequency and analog time-domain filters, and coding devices, to the modulation of light beams and the measurement of surface properties of solids. Many references to the recent surface-wave literature are included.

Surface elastic waves

Recent advances in parameter estimation and seismic processing have allowed incorporation of anisotropic models into a wide range of seismic methods. In particular, vertical and tilted transverse isotropy are currently treated as an integral part of velocity fields employed in prestack depth migration algorithms, especially those based on the wave equation. We briefly review the state of the art in modeling, processing, and inversion of seismic data for anisotropic media. Topics include optimal parameterization, body-wave modeling methods, P-wave velocity analysis and imaging, processing in the τ-p domain, anisotropy estimation from vertical-seismic-profiling (VSP) surveys, moveout inversion of wide-azimuth data, amplitude-variation-with-offset (AVO) analysis, processing and applications of shear and mode-converted waves, and fracture characterization. When outlining future trends in anisotropy studies, we emphasize that continued progress in data-acquisition technology is likely to spur transition from t...

Seismic anisotropy in exploration and reservoir characterization: An overview

SUMMARY
3-D and 2-D time-domain elastodynamic fundamental solutions (or Green's functions) for linearly elastic anisotropic materials are obtained by the Radon transform. Fundamental solutions in the frequency domain follow directly by a subsequent evaluation of the Fourier transforms of the time-domain solutions. The solutions are in the form of a surface integral over a unit sphere for 3-D cases and in the form of a contour integral over a unit circle for 2-D cases. The integrals have a simple structure that can be interpreted as a superposition of plane waves. The wavefields can be separated into singular and regular parts. The singular parts correspond to the elastostatic fundamental solutions. The regular parts are bounded continuous functions. The integrals have been evaluated numerically for several examples. The results presented in this paper have direct applications to the formulation of boundary-integral equations for bodies of anisotropic materials and for the subsequent solution of these equations by the boundary-element method.

Elastodynamic fundamental solutions for anisotropic solids

The displacements due to a radiating point source in an infinite anisotropic elastic medium are found in terms of Fourier integrals. The integrals are evaluated asymptotically, yielding explicit expressions for displacements at points far from the source. The relative amplitudes of waves from a point source are thus determined, and it is found that although in general the decay of wave amplitudes is proportional to the distance from the source, it is possible that in certain directions the decay is less than this. The method used in this paper is also shown to be an alternative way of deriving known results concerning the geometry of the propagation of disturbances. As an example, the radiation in a transversely isotropic medium from an isolated force varying harmonically with time is discussed.

V. T. Buchwald

Papers

Elastic waves in anisotropic media