Mechanics of Isotropic Half-Space with Shear Modulus Varying Linearly with Depth

Abstract

In this chapter, we study dynamic and static problems stated for a half-space with shear modulus increasing linearly with depth. Further, we consider the case of an isotropic material, when the half-space, having a linearly varying shear modulus, enables a closed solution for the functions $\tilde q(\tilde z,\tilde k),\tilde w(\tilde z,\tilde k),\tilde p(\tilde z,\tilde k)$ to be found in the form of confluent hypergeometric functions. Dynamic problems for a half-space of this kind have been solved, in most cases, by studying free vibrations, i.e. the parameters of Rayleigh and Love waves have been determined [29, 106, 112–114]. Forced vibrations were a subject of concern in [6, 7, 78] for the case of an incompressible medium, and for arbitrary values of Poisson’s ratio in [79].