Fig. 5 Schematic illustration of slip-weakening friction law. Cy, co, and Cf indicate yielding stress, initial stress and frictional stress, respectively. Dc and Dmax correspond to the slip-weakening distance and final amount of slip, respectively critical strength drop (Atb) is defined as Cy — Cf.

c max are shear modules and shear wave velocity, respectively. K,ljjl is the kernel that represents ij-component of stress at (x, t) when ^-component of unit slip occurs at (£, t) on the fault. It should be noted that although Eq. (1) is derived by the boundary integral equation method, this relation is not limited to a particular numerical method but is a universal feature, because this relation is obtained based on the continuities and symmetries of stress and dislocation on the fault. Therefore, Eq. (1) explicitly shows that the spatio-temporal distribution of slip is uniquely related to the spatio-temporal distribution of stress change.

Using this idea, we are theoretically able to estimate the shape of slip weakening curves from the spatio-temporal evolution of slip velocity on the fault [Ide and Takeo (1997)] although we still cannot estimate the absolute stress by this method. But it should be noted that there is a resolution problem in this technique [Spudich and Guatteri (2004)]. Since the temporal resolution of slip velocity is generally not sufficient due to the band-limited nature in the waveforms which was caused by attenuation and scattering of the high frequency waves during the propagation. This limitation makes the estimation of small Dc values difficult [Guatteri and Spudich (2000)].

To overcome this situation, a new technique has been proposed [Fukuyama et al. (2003b); Mikumo et al. (2003)] in which temporal variation of stress is not required. In this method, stress is assumed to drop at frictional stress level (of) at peak slip velocity time [Mikumo et al., 2003] as shown in Fig. 6. This assumption is correct as long as the rupture propagates smoothly without strong barriers and asperities [Fukuyama et al. (2003b)].

Fig. 6 Schematic illustration of a conventional method to estimate slip-weakening distance (Dc). Dc is defined as the amount of slip when the stress drops to the frictional level (of). D'c is an approximation of Dc which is defined as the amount of slip where slip velocity becomes maximum. This approximation is valid if the peak slip velocity time is close to the break down time [Modified from Mikumo et al. (2003)].

Time (sec)

Fig. 6 Schematic illustration of a conventional method to estimate slip-weakening distance (Dc). Dc is defined as the amount of slip when the stress drops to the frictional level (of). D'c is an approximation of Dc which is defined as the amount of slip where slip velocity becomes maximum. This approximation is valid if the peak slip velocity time is close to the break down time [Modified from Mikumo et al. (2003)].

As long as the slip velocity function is obtained accurately enough we will be able to determine the range of the slip-weakening parameters. Even using the above technique, we cannot completely avoid the band-limited effect of the observation [Spudich and Guatteri (2004)]. Thus our discussion is still restricted to the order of these parameters and they are considered to be an upper bound of the real values.

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