ABSTRACT
The propagation of mechanical disturbances in solids is of profound importance in many disciplines including physical sciences and engineering. In these kinds of problems, the loading or disturbance is applied at such a “fast” rate that the effect of inertia cannot be ignored (as we have been doing in this book so far). These loadings are described as suddenly applied like a dropping mass sticking to the ground (modeled as Heaviside step function in time) or applied impulsively like in a dynamic impact (modeled by Dirac delta function in time). These loadings are referred to as dynamic loadings instead of quasi-static loadings. Energy of disturbance or waves may propagate in the solids at different wave speeds, depending on the nature of the disturbances and on whether they are dilatational or shear in nature. Both displacement and stress responses of the solids are functions of time (rather than in the sense of the viscoelastic type of creeping or relaxation discussed in Chapter 7). For isotropic solids of infinite extent, these waves are either dilatational or compressional waves (also called P-waves) or shear waves (also called S-waves). The phase velocities of the particle movements are parallel and perpendicular to the direction of wave propagation for dilatational and shear waves, respectively. Referring to a fixed coordinate system, we can further decompose shear waves as SH-waves or SV-waves, corresponding to the polarized components along the “horizontal” and “vertical” directions, respectively. For anisotropic solids, these waves would no longer be purely dilatational or purely transverse. The mathematical techniques used in solving dynamic problems are more tedious and lengthy; only some simple situations have been solved analytically. In geomechanics applications, seismic wave propagation induced by earthquakes is of major concerns. Manmade structures are vulnerable to ground shakings. Both responses and failure mechanisms in soils or rocks can be highly sensitive to dynamic loadings. Many important topics need to be included in a chapter on dynamics and wave propagations. In view of the size limitations of this book, only some selected topics in dynamics and wave propagations in geomaterials are included in this chapter. Although one-dimensional wave propagations and dynamic problems provide very useful insight into the problems of dynamics, in view of its limited usefulness in geomechanics we will start with 3-D formulations of wave equations in solids. Surface and interfacial waves, including Rayleigh waves, Love waves, and Stoneley waves, will be discussed because of their relevance to seismic wave propagations. The nature of elastic-plastic waves in geomaterials will be discussed.
