The present paper explores the Airy stress function for a general class of elastic dislocation problems in a doubly connected region. The basic concept of the elastic theory and the dislocation theory, included, are essential to the derivation of the method. Applications deal with two shapes of dislocation cuts; an arbitrary straight cut and a logarithmic spiral cut. For simplicity and illustrative purpose, only an infinite region with a circular hole has been considered. The discussion deals with the singular point and the uniqueness of the general solution.

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