Some Problems of a Rigid Elliptical Disk-Inclusion Bonded Inside a Transversely Isotropic Space: Part I

The paper addresses the problem of contact of an elliptical inclusion in the form of a thin disk, bonded in the interior of a transversely isotropic space. The inclusion is assumed to be absolutely rigid and in perfect contact with the medium. Three different cases of loading are considered, namely, (a) the inclusion is loaded in its plane by a shearing force, whose line of action passes through the center of the disk; (b) the inclusion is rotated by a torque whose axis is perpendicular to the plane of the inclusion; (c) the medium is under uniform stress field at infinity in a plane parallel to the plane of the inclusion. In the first part of the article, the problems corresponding to all three cases are reduced, in a unified manner, to a system of coupled two-dimensional integral equations by using the theory of two-dimensional Fourier transforms. In the second part, closed-form solutions for these equations are obtained by using Dyson’s theorem and Willis’ generalization of Galin’s theorem. Explicit expressions for the stress intensity factors near the edge of the inclusion are extracted from these solutions. Numerical results are plotted illustrating how these coefficients vary with transverse isotropy and the parametric angle of the ellipse. The results can be used to determine the critical failure load and angle of crack initiation for solids containing elliptical inclusions.