Tunnel effect of fractal fault and transient S-wave velocity rupture (TSVR) of in-plane shear fault

Transient S-wave velocity rupture (TSVR) means the velocity of fault rupture propagation is between S-wave velocity β and P-wave velocity α. Its existing in the rupture of in-plane (i.e. strike-slip) fault has been proved, but in 2-dimensional classical model, there are two difficulties in transient S-wave velocity rupture, i.e., initialization difficulty and divergence difficulty in interpreting the realization of TSVR. The initialization difficulty means, when v↑v _R (Rayleigh wave velocity), the dynamic stress strength factor K ₂(t)→+0, and changes from positive into negative in the interval (v _R, β). How v transit the forbidden of (v _R, β)? The divergence difficulty means K ₂(t)→+∞ when v↓β. Here we introduce the concept of fractal and tunnel effect that exist everywhere in fault. The structure of all the faults is fractal with multiple cracks. The velocity of fault rupture is differentiate of the length of the fault respect to time, so the rupture velocity is also fractal. The tunnel effect means the dynamic rupture crosses over the interval of the cracks, and the coalescence of the intervals is slower than the propagation of disturbance. Suppose the area of earthquake nucleation is critical or sub-critical propagation everywhere, the arriving of disturbance triggers or accelerates the propagation of cracks tip at once, and the observation system cannot distinguish the front of disturbance and the tip of fracture. Then the speed of disturbance may be identified as fracture velocity, and the phenomenon of TSVR appears, which is an apparent velocity. The real reason of apparent velocity is that the mathematics model of shear rupture is simplified of complex process originally. The dual character of rupture velocity means that the apparent velocity of fault and the real velocity of micro-crack extending, which are different in physics, but are unified in rupture criterion. Introducing the above-mentioned concept to the calculation of K ₂(t), the difficulty of initialization can be overcome, and the integral equation of triggering the initialization of TSVR is given quantitatively. By solving this integral equation, the lower limit of TSVR is 1.105 3β, not β, and the divergence difficulty is overcome. TSVR is unstable solution and may degenerate to sub-Rayleigh wave velocity rupture immediately where the non-critical condition can be measured. The results of this paper show that the initialization and continuum depends on the condition of earthquake nucleation in seismogenic area. Contribution No. 99FC2004, Institute of Geophysics, China Seismological Bureau, China. This study was supported by the State Natural Science Foundation of China (49774217) and the key problem of Chinese Joint Seismological Science Foundation under the grant 9507435.