Torsional vibration of a finite cylindrical cavity in a two-layer transversely isotropic half-space

• A two-layered transversely isotropic half-space with a cylindrical cavity of finite length is considered.
• The wall and the base of cavity are affected by time harmonic shear stresses making torsion.
• A combination of Fourier cosine and Hankel integral transforms are used to solve the problem.
• A singular integral equation (SIE) for the unknown shear stress is obtained.
• The SIE is solved for some collocation points with distances, which are gradually decreased.

The aim of this paper is to present a rigorous investigation for a two-layered transversely isotropic linear elastic half-space containing a circular cylindrical cavity of length equal to the top layer undergoing mono-harmonic ring shape shear stress applied either on the vertical cylindrical surface or on the base of the cavity. To this end, a combination of Fourier cosine integral transform for depth and Hankel integral transform for radial distance are used, which translate the boundary value problem to a singular integral equation for the shear stress comes out from the continuity of two layers. The integral equation is solved for some collocation points with a smoothed variable of distance, which is adapted with the use of a free parameter. It is shown that, although the shear stress is highly singular, it does not highly depend on this free parameter. Both the analytical and numerical results are verified with both the static isotropic and dynamic transversely isotropic homogeneous cases. In addition, some new graphical results are presented for more understanding in engineering point of view.