Crack deflection into an interface

Suppose that a crack extends through the thickness hf of an isotropic elastic film from its free surface to the interface with an isotropic elastic substrate under plane strain conditions. The crack is presumed to have 

In the case of advance of the crack into the substrate, the configuration remains symmetric with respect to the crack plane and the crack advances in mode I. The magnitude of the stress relieved by crack advance is set by ko so, due to requirements of dimensional consistency, the mode I stress intensity factor of the added growth segment must depend on and Aa according to 

For a crack that advances as a kink in the interface, either a single kink 

The quantity is the energy release rate for one of the kinks in the case of double kinking. From (4.28), the stress concentration field phase angle corresponding to (4.86) is 

Note that both the energy release rate Q for crack penetration into the substrate given by (4.85) and the energy release rate k for single or double interface kink formation depend on the increment of crack advance Aa in the same way. Consequently, the ratio of k fo Q, which specifies the relative energy release for kink crack formation of some length Aa to substrate penetration to the same depth, is independent of Aa. This ratio, which is given by 

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