ESTIMATES OF AVERAGE STRESS FROM FAULT ENERGETICS

ENERGY BALANCE

In figure 10.2, an earthquake is viewed, according to Reid's (1910) rebound theory, as a strained patch of fault surface of area A that suddenly breaks, permitting points initially in contact to be displaced from one another by an average amount u. The breakage is like the sudden failure of an overloaded leaf spring. We are interested in the average shear stress acting parallel to the wall in the failed section of the fault surface. We denote its initial value by tau1 and its final value by tau2. The inclined line in figure 10.3 represents the elastic unloading of the medium as the earthquake displacement increases to its final value is. The area under this line, which represents the total elastic energy released by the earthquake per unit area of faulted surface, can be written as

E/A = tau-bar mu , (1a)

where tau-bar = 1/2 (tau1 + tau2. (1b)

The energy E must supply the work Ea of generating seismic waves and the work Er, converted to heat in overcoming frictional resisting forces. Thus,

E = Ea + Er + ? , (2)

where the question mark is a reminder (which we shall forget for the moment) that there may be other significant sinks of earthquake energy, such as the surface energy consumed in creating new fractures. We can now write

Ea/A = tauau (3)

Ea/A = tauau (4)

where taua, the "apparent stress" of seismology, is the portion of the earthquake stress tau-bar allocated to the production of seismic waves, and taur is the average frictional resisting stress allocated to the production of heat. The individual areas represented by equations 3 and 4 are shown in figure 10.3 by contrasting patterns.

This interpretation of the areas in figure 10.3 is fairly general, as long as we define tau1, tau2, and taur, respectively, as the weighted averages of initial stress, final stress, and friction over the faulted surface, the weighting function being the local fault slip (see Savage and Wood, 1971; Lachenbruch and Sass, 1980). Combining equations 1 through 4 yields

tau-bar = taua + tuar + ? , (5)

which states that unless the question mark represents something important that we've neglected, the average earthquake stress tau-bar is the sum of the apparent stress tuaa, to be estimated from seismic measurements of Ea (eq. 3), and the resisting stress taua, to be estimated from thermal measurements of frictional heat Er (eq. 4).