Interaction of rocking vibration of a buried rigid circular disc and a two-layer transversely isotropic half-space

A linear elastic layer of finite thickness bonded on a half-space each containing a transversely isotropic material is considered. A rigid circular disc attached to the interface of these two domains is considered to be affected by a rocking vibration of constant amplitude. With the aid of a scalar potential function, the equations of motion in each domain are solved using Fourier series and Hankel integral transforms. Because of the involved integral transforms, the mixed boundary value problem is changed to dual integral equations; which are reduced to Fredholm integral equations. Because of the complex integrand function existing in the dynamic case, analytical solution cannot be given in general. However, a closed-form solution is introduced for the static case, which itself degenerates to the solution for an isotropic case existing in the literature. The contact stress in between the rigid disc and the surrounding media, and the related impedance function are analytically determined in the static case. With the help of contour integration, the governing Fredholm integral equations are numerically evaluated in the dynamic case. The dynamic contact pressure and the impedance function are numerically evaluated in a general dynamic case. The shape induced singularity in the contact pressure is investigated in detail. Some numerical evaluations are given for different transversely isotropic materials to show the effect of anisotropy.