Nearly Circular Connections of Elastic Half Spaces

In this paper we solve the elasticity problem of two elastic half spaces that are joined together over a region that does not differ much from a circle, i.e., the problem of an external planar crack leaving a nearly circular uncracked connection. The method we use is based on the perturbation technique developed by Rice (1985) for solving the elastic field of a crack whose front deviates slightly from some reference geometry. Quantities such as crack opening displacement and stress intensity factor are derived in detail to the first order of accuracy in the deviation of the shape of the connection from a circle. In addition, some results such as the crack face weight functions and Green's functions for a perfectly circular connection are also discussed under various boundary conditions at infinity. The formulae derived are used to study the configurational stability problem for quasistatic growth of an external circular crack. The results, derived when the crack front is perturbed from circular in a harmonic waveform and is subjected to axisymmetric loading, suggest that a perturbation of wavenumber higher than one is configurationally stable under all boundary conditions at infinity. The perturbation with wavenumber equal to one, which corresponds to a translational shift of the geometric center of the circular connection, turns out to be configurationally stable if any rotation in the remote field is suppressed and configurationally unstable if there is no such restraint.