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An interaction between an anti-plane crack with a three-phase circular composite by using complex potential methods is considered in this paper. The solution procedures for solving this problem consist of two parts. In the first part, based on complex potential methods in conjunction with analytical continuation theorem and alternating technique, the complex potential functions of a screw dislocation interacting with three-phase circular composites are obtained. The second part consists of the derivation of logarithmic singular integral equations by introducing the complex potential functions of screw dislocation along the crack border together with superposition technique. The logarithmic singular integral equations are then solved numerically by modeling a crack in place of several segments. Linear interpolation formulae with undetermined coefficients are applied to approximate the dislocation distribution along the elements, except at vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. The mode-III stress intensity factors are then obtained numerically in terms of the values of the dislocation density functions of the logarithmic singular integral equations.

Complex Potential Methods for a Crack and Three-phase Circular Composite in Anti-plane Elasticity