Some Problems of a Rigid Elliptical Disk-Inclusion Bonded Inside a Transversely Isotropic Space, Part II: Solutions of the Integral Equations

This is a sequel to the first part of the two-part paper, which addresses the problem of contact of a rigid elliptical disk-inclusion bonded in the interior of a transversely isotropic space under three different types of loading, namely (a) the inclusion is loaded in its plane by a shearing force, whose line of action passes through the center of the inclusion; (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 Part I, the problems corresponding to all three cases of loading have been reduced, in a unified manner, to a system of coupled two-dimensional integral equations. Next, based on Dyson’s theorem and Willis’ generalization of Galin’s theorem, the general structure of solution of the coupled integral equations has been established. In this part, closed-form solutions to these equations are derived by using Dyson’s theorem. Full elastic field in the plane of the inclusion is evaluated and it is shown that the stress field near the edge of the inclusion exhibits the familiar square root singularity in linear fracture mechanics. 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 initial crack propagation for solids containing elliptical inclusions.