It has recently been shown (Burridge, 1973) that in two dimensions plane-strain shear cracks lacking cohesion may run at speeds ranging from the Rayleigh-wave to the S-wave speed or possibly at the P-wave speed. On the other hand, it has long been known that in antiplane strain, cracks lacking cohesion must run at least at the S-wave speed. Since locally at the edge of a three-dimensional crack there is a combination of plane and antiplane strain, we have sought and found solutions for circular shear cracks expanding at the S-wave speed and at the P-wave speed. These have finite shear tractions ahead of the crack and so may correspond to frictional sliding in the absence of cohesion.

The analysis combines the method of Kostrov (1964b) with the results of Burridge (1973).

We carry out a complete evaluation for the displacements and tractions on the fault plane, and far-field radiation for the S-wave-speed crack. The corresponding evaluations for the P-wave speed are not elementary and are not attempted here.

As far as the authors are aware, these are the first analytic solutions of three-dimensional crack problems which satisfy a physically plausible fracture criterion for failure under shear.

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