Vertical and torsional vibrations of an embedded rigid circular disc in a transversely isotropic multilayered half-space

Abstract

In this paper, we present a fundamental solution to the mixed boundary-value problem of vertical and torsional vibrations of a rigid circular disc embedded in a transversely isotropic multilayered half-space. We first introduce the cylindrical system of vector functions to express displacements and stresses of the layered half-space induced by the time-harmonic vertical and torsional loads acting within the half-space. We then derive the recursive relation of the expansion coefficients of the displacement and traction vectors from one layer to any other layer via the dual variable and position (DVP) method. Finally, by virtue of the boundary/interface and internal loading conditions, the expansion coefficients of the displacements and tractions are determined, which are integrated back to the physical domains. To satisfy the mixed boundary conditions at the embedded rigid-disc level, we divide the load disc area into many annular load-rings with the corresponding solutions being superposed together via an integral least-square approach. Based on the derived fundamental solutions, various numerical examples on the vertical and torsional dynamic compliances are presented to demonstrate the influence of the embedment depth of the disc, material layering, anisotropy, and input frequency. Furthermore, the derived fundamental solution is applied to a homogeneous half-space with a thin coating to investigate the suitability of two effective boundary conditions on the half-space.