An earthquake model is derived by considering the effective stress available to accelerate the sides of the fault. The model describes near- and far-field displacement-time functions and spectra and includes the effect of fractional stress drop. It successfully explains the near- and far-field spectra observed for earthquakes and indicates that effective stresses are of the order of 100 bars. For this stress, the estimated upper limit of near-fault particle velocity is 100 cm/sec, and the estimated upper limit for accelerations is approximately 2g at 10 Hz and proportionally lower for lower frequencies. The near field displacement u is approximately given by u(t) = (σ/μ) βr(1 - e−t/r) where. σ is the effective stress, μ is the rigidity, β is the shear wave velocity, and τ is of the order of the dimension of the fault divided by the shear-wave velocity. The corresponding spectrum is

Ω(ω)=σβμ1ω(ω2+τ−2)1/2(1)

The rms average far-field spectrum is given by

〈 Ω(ω) 〉=〈 Rθϕ 〉σβμrRF(e)1ω2+α2(2)

where 〈Rθϕ〉 is the rms average of the radiation pattern; r is the radius of an equivalent circular dislocation surface; R is the distance; F(e) = {[2 – 2e][1 – cos (1.21 eω/α)] +e2}1/2; e is the fraction of stress drop; and α = 2.21 β/r. The rms spectrum falls off as (ω/α)−2 at very high frequencies. For values of ω/α between 1 and 10 the rms spectrum falls off as (ω/α)−1 for e < ∼0.1. At low frequencies the spectrum reduces to the spectrum for a double-couple point source of appropriate moment. Effective stress, stress drop and source dimensions may be estimated by comparing observed seismic spectra with the theoretical spectra.

http://www.openseismo.org/contributors/Lee/MoWorking_Backups/Mo2012_0424backup/MoWorking/Papers_NotUsed/0_Theory/928_Bru70_Brune_JGR1970_p4997.pdf

Tectonic stress and the spectra of seismic shear waves from earthquakes

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

The conventional magnitude scale M suffers saturation when the rupture dimension of the earthquake exceeds the wavelength of the seismic waves used for the magnitude determination (usually 5–50 km). This saturation leads to an inaccurate estimate of energy released in great earthquakes. To circumvent this problem the strain energy drop W (difference in strain energy before and after an earthquake) in great earthquakes is estimated from the seismic moment M_0. If the stress drop Δσ is complete, W = W_0 = (Δσ/2μ)M_0 ∼ M_0/(2×10^4), where μ is the rigidity; if it is partial, W_0 gives the minimum estimate of the strain energy drop. Furthermore, if Orowan's condition, i.e., that frictional stress equal final stress, is met, W_0 represents the seismic wave energy. A new magnitude scale M_w is defined in terms of W_0 through the standard energy-magnitude relation log W_0 = 1.5M_w + 11.8. M_w is as large as 9.5 for the 1960 Chilean earthquake and connects smoothly to M_s (surface wave magnitude) for earthquakes with a rupture dimension of about 100 km or less. The M_w scale does not suffer saturation and is a more adequate magnitude scale for great earthquakes. The seismic energy release curve defined by W_0 is entirely different from that previously estimated from Ms. During the 15-year period from 1950 to 1965 the annual average of W_0 is more than 1 order of magnitude larger than that during the periods from 1920 to 1950 and from 1965 to 1976. The temporal variation of the amplitude of the Chandler wobble correlates very well with the variation of W_0, with a slight indication of the former preceding the latter. In contrast, the number N of moderate to large earthquakes increased very sharply as the Chandler wobble amplitude increased but decreased very sharply during the period from 1945 to 1965, when W_0 was largest. One possible explanation for these correlations is that the increase in the wobble amplitude triggers worldwide seismic activity and accelerates plate motion which eventually leads to great decoupling earthquakes. This decoupling causes the decline of moderate to large earthquake activity. Changes in the rotation rate of the earth may be an important element in this mechanism.

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The energy release in great earthquakes

Physical factors likely to affect the genesis of the various fault rocks—frictional properties, temperature, effective stress normal to the fault and differential stress—are examined in relation to the energy budget of fault zones, the main velocity modes of faulting and the type of faulting, whether thrust, wrench, or normal. In a conceptual model of a major fault zone cutting crystalline quartzo-feldspathic crust, a zone of elastico-frictional (EF) behaviour generating random-fabric fault rocks (gouge—breccia—cataclasite series—pseudotachylyte) overlies a region where quasi-plastic (QP) processes of rock deformation operate in ductile shear zones with the production of mylonite series rocks possessing strong tectonite fabrics. In some cases, fault rocks developed by transient seismic faulting can be distinguished from those generated by slow aseismic shear. Random-fabric fault rocks may form as a result of seismic faulting within the ductile shear zones from time to time, but tend to be obliterated by continued shearing. Resistance to shear within the fault zone reaches a peak value (greatest for thrusts and least for normal faults) around the EF/QP transition level, which for normal geothermal gradients and an adequate supply of water, occurs at depths of 10–15 km.

Fault rocks and fault mechanisms

A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones

The absence of a heat flow anomaly greater than ∼0.3 µcal/cm2/sec associated with the San Andreas fault is used to estimate the upper limit for the steady state or initial shear stress. Under the assumption that the long-term rate of motion along the fault is 5 cm/yr and occurs primarily in the form of creep, this upper limit is about 100 bars. If the motion is primarily accomplished by faulting during large earthquakes and if the frictional stress is equal to the final stress as suggested by E. Orowan (1960), the upper limit is estimated to be about 200 bars. Without Orowan's assumption, the estimation of the upper limit is about 250 bars, based on earthquake energy-magnitude-moment relations. If the long-term rate of motion along the San Andreas fault is only ∼2 cm/yr, these results are increased to 250, 350, and 400 bars, respectively.

Thomas L. Henyey

Papers

Heat flow, stress, and rate of slip along the San Andreas Fault, California