Mathematical analysis for an axissymmetric disc-shaped crack in transversely isotropic half-space

A half-space of linear elastic transversely isotropic material containing a disc-shaped crack with a small initial crack opening between the crack faces at an arbitrary depth from the surface of the half-space is considered such that the crack surfaces are parallel to the free surface of the half-space and perpendicular to the axis of material symmetry. The crack surfaces are affected by time-harmonic axissymmetric tractions parallel to the axis of material symmetry. The equations of motion are solved with the use of a simple potential function and applying Hankel integral transforms. Then, the stresses and displacements are determined with the aid of the relations with the potential functions and the theorem of inverse of Hankel integral transforms. The displacements and stresses are analytically determined for the static case of transversely isotropic full-space containing a crack in it as a degeneration of the main goal of the paper. The numerical results are, in general case, evaluated by utilizing the contour integration, where very accurate results are developed. The analysis given here is used for deep understanding of fracture mechanics of anisotropic material, which is now a day known as a main engineering material.