Static and Dynamic Problems for Homogeneous Transversely Isotropic Half-Space

Abstract

Forced vibrations occurring in an anisotropic half-space have been the object of study in a small number of reports [61, 102, 124]. In most cases, free vibrations or solutions in the wave number space have been considered [11–13, 29, 59, 60]. A possibility of employing potentials to separate the equations of motion for a medium has been studied in [19]. In this report, the author presents an additional condition for the elastic constants, which enables separated wave equations for potentials to be found. Kirkner [61] considered vibrations of a circular stiff disk on a transversely isotropic half-space that satisfies the condition stated in [19]. However, construction of a solution for problems dealing with forced vibrations in a homogeneous transversely isotropic half-space is relatively simple (no additional constraints on elastic coefficients are needed) when using the equations presented in Chap. 1, or via formulations that employ stiffness matrices for a half-space, or for layers which form a layered half-space [11–13, 59, 60, 102, 124]. Solutions of static problems for the homogeneous transversely isotropic half-space are well represented in the literature, starting with the work of Michell [74]. Numerous solutions related to a half-space subjected to loads applied to its surface are presented in [32, 36, 65, 130]. Static Green’s functions for the case of forces applied within the half-space have been constructed in [86, 87, 107]. In the present chapter, we consider some problems of the dynamics and statics for a homogeneous transversely isotropic half-space: namely, the action of vertical and horizontal forces applied to the surface of a half-space and in its depth; and vibrations of a stiff disk on a half-space.