The plane interaction problem for a circular elastic inclusion embedded into an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations  as Green’s functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress-intensity factors representing the strength of the stress singularities are obtained in terms of the asymtotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress-intensity factors are given.
Interaction Between a Circular Inclusion and an Arbitrarily Oriented Crack
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Erdogan, F., Gupta, G. D., and Ratwani, M. (December 1, 1974). "Interaction Between a Circular Inclusion and an Arbitrarily Oriented Crack." ASME. J. Appl. Mech. December 1974; 41(4): 1007–1013. https://doi.org/10.1115/1.3423424
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