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Kinking of arc cracks

Hi, I wanted to ask a question regarding the crack path of an arc crack at the interface. While Toya's solution provides the condition of crack propagation along the interface, I am interested to find what happens when the arc crack kinks out of the interface. If we think of an arc crack which has kinked out of the interface (between a circular inclusion and the matrix), the geometry resembles to a wing crack. The wing crack model has been extensively used for compressive loading and has been developed for the same. Can someone please help me with literature which provides some analytical basis of treating propagation of a kinked interfacial arc crack? Thanks you.

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