For a center frictional interfacial crack in a bimaterial medium under remote shear loading, one crack tip is `open' and the other closed. Near the closed crack tip, the crack surfaces are in contact over a sizeable region, and the stress singularity is weaker than r −1/2. At the `open' tip, the crack-tip stress field is well approximated by the classical oscillatory solution obtained by assuming traction-free crack surfaces although theoretically there is still an extremely small contact zone near the tip. Under a combined shear and compression loading, the open crack zone may become small, and the near-tip stress field approaches the characteristic of a field with a singularity stronger than r −1/2. The strong singularity is obtained from a closed-form solution for a bimaterial medium with a frictional interfacial crack under combined shear and compression loading based on the assumption of a completely closed crack. It is found that the classical definition of strain-energy release rate becomes zero (or unbounded) if the singularity is weaker (or stronger) than r −1/2 and, thus, cannot be used as a fracture criterion. The finite crack-extension strain-energy release rate, which is uniquely related to the generalized mode-II stress-intensity factor, is proposed to establish a fracture criterion for frictional interfacial cracks with weak as well as strong singularities. Finally, a numerical example simulating a fiber pull-out problem is presented to demonstrate the fiber/matrix debonding crack tip behavior during pull-out.