Earthquakes as three stage processes

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Relying on basic physics, laboratory and -eld evidence, and accounting for the di-erent time scales involved, we derive an approximate energy function for earthquakes. The fracture term is disregarded since virtually all earthquakes occur on pre-existing faults and since fault gouge has a very low fracture energy. Disregarding also the gravitational term, the importance of which depends on the type of seismic focal mechanism, yields that the energy function has only thermal and radiative terms. The self-similarity ranges in the bulk rocks and in the gouge suggest to take the basic element as two cubes of 10 m side, with in common one face, over which slip occurs. An earthquake is a cascade of such slip processes on a series of neighboring elements, and the volume involved can be self-similarly defined as the two megacubes embedding the set of elements participating in the slip. Dimensional analysis suggests that the slip process is composed of three stages. The first stage consists of a stick slip-episode with an average velocity v ~ 10e-1 m/s over a time t ~ 10e-2 s. In this first stage virtually all energy is converted into heat, with a temperature increase of the order of 10e2 K on the sliding surface. The second stage, in which the occurrence of further slip episodes is hardly relevant, consists of the propagation of the thermal wave generated in stage I to the whole shear zone, with a characteristic time of the order of 10e2 s. In light of the comparatively low permeability of fault gouge with respect to heat di-usivity, this temperature increase induces a pore uid pressure increase. When the transition from hydrostatic to lithostatic pressure is completed on the whole shear zone, a third stage is entered, in which, provided that the pressurization is maintained, virtually frictionless high velocity slip occurs, converting all the available energy into elastic radiation. The duration of this purely radiative stage, the amount of slip, and the size of the earthquake depend on the number of elements cooperatively participating in the cascade slip, which is ruled, just as in the usual single stage cellular automata models, by the correlation length over which the strain level is near the \rupture" threshold. At odds with the classical single stage cellular automata, the model does not require the introduction of strong jumps in stress to be \ignited", and appears thus also capable to explain the Coulomb Failure Stress quandary of very small triggering stresses, with the ignition of large events requiring excess stresses of just 10e2 - 10e3 MPa. The global seismic e-ciency, under the assumption that granular e-ects and viscous resistance are disregarded, is ~ 1. Assuming then statistical self-similarity on the fault plane for the patches that slip cooperatively, and approximating their pattern as a Sierpinski carpet, yields partial and approximately constant stress drop values independent of event size. These emerge from the fractal nature of the slip surface interpreted according to the seismological assumption of constant homogeneous slip on the Euclidean rectangular plane fault which embeds the slipping patches.

Mulargia F., Castellaro S. and Ciccotti M., 2004. Earthquakes as three stage processes. Geophys. J. Int. 158[1], pp. 98-108.


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