Abstract: The branch cracks problems of circular region in elastic longitudinal shear are investigated by the dislocations analysis method. Based on the complex potential of a point dislocation in an infinite region, a complementary term was introduced to satisfy the traction-free condition along the circular boundary, and then the elementary solution for branch crack problems in the circular region was obtained. By matching the traction along the cracks, Cauchy singular integral equations for the branch cracks in the circular region were derived. A constraint equation was formulated for the displacement single value condition. By using a semi-open quadrature rule, the singular integral equations were transformed to algebraic equations, and finally the stress intensity factors at the crack tips were obtained. This semi-analytical method is accurate and widely used. It overcomes the limitation of the analytical solution such as conformal mapping. The position of cracks can be arbitrary. The results can be applied to actual projects.

Key words: stress intensity factor , singular integral equation, circular region, longitudinal shear, branch crack