Subsonic semi-infinite crack with a finite friction zone in a bimaterial

Propagation of a semi-infinite crack along the interface between an elastic half-plane and a rigid half-plane is analyzed. The crack advances at constant subsonic speed. It is assumed that, ahead of the crack, there is a finite segment where the conditions of Coulomb friction law are satisfied. The contact zone of unknown a priori length propagates with the same speed as the crack. The problem reduces to a vector Riemann–Hilbert problem with a piece-wise constant matrix coefficient discontinuous at three points, 0, 1, and ∞ . The problem is solved exactly in terms of Kummerʹs solutions of the associated hypergeometric differential equation. Numerical results are reported for the length of the contact friction zone, the stress singularity factor, the normal displacement u 2 , and the dynamic energy release rate G. It is found that in the case of frictionless contact for both the sub-Rayleigh and super-Rayleigh regimes, G is positive and the stress intensity factor K II does not vanish. In the sub-Rayleigh case, the normal displacement is positive everywhere in the opening zone. In the super-Rayleigh regime, there is a small neighborhood of the ending point of the open zone where the normal displacement is negative.