Asymmetric interaction of a rigid disc embedded in a transversely isotropic half-space

An analytical treatment has been presented for the lateral response of a rigid circular disc embedded in a transversely isotropic half-space. By means of a complete representation of using displacement potentials, it is shown that the governing equations of equilibrium for this class of problems can be uncoupled into two fourth-order partial differential equations. Using cylindrical co-ordinate system and applying Hankel integral transform in the radial direction and Fourier series, the problem may be changed to a system of four separate integral equations, which, in turn, are reduced to a pair of Fredholm equation of the second kind, whose solutions are then computed. In addition to furnishing a unified view of existing solutions for zero and infinite embedments, the present treatment reveals a severe boundary-layer phenomenon, which is apt to be of interest to this class of problems in general. The present solutions are analytically and numerically in exact agreement with the existing solutions for a half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy. The contact pressure, desired force-displacement relationship, stress and displacement field are explicitly found.