The Coulomb criterion for failure (e.g., Jaeger and Cook, 1979; Scholz, 1990) leads naturally to a quantity or function, called (by various authors) the Coulomb Failure Stress or Coulomb Failure Function (CFF), which provides a measure of the proximity of a fault to failure..
This criterion has been used by a number of authors to study the distribution of aftershocks after an earthquake and to explore earthquake-induced static stress changes on other faults in the vicinity of a mainshock. (For example, Raleigh and others, 1976; Das and Scholz, 1981; Stein and Lisowski, 1983; Mavko, 1982; Mavko and others, 1985; Erickson, 1986; Kato and others, 1987; Li and others, 1987; Oppenheimer and other, 1988; Hudnut and others, 1989; Reasenberg and Simpson, 1992; Harris and Simpson, 1992; Jaume and Sykes, 1992; Stein and others, 1992; King and others, 1994; Simpson and Reasenberg, 1994; Stein and others, 1994.)
There are several ways in which to apply the CFF to earthquake problems, and, to my mind, it is still not a settled question as which of these ways is most appropriate in different cases, or even if the CFF is the appropriate measure of failure to use. Mother nature will have the last word in settling this issue, of course. In the meantime, I think we need to keep an open mind and test the different approaches and different choices of variables against the data from the earth.
In an isotropic medium, planes of all orientations would be equally susceptible to failure, and the optimal failure planes are determined by the orientation of the principal axes of the stress tensor. In the most general case, in order to calculate the change in CFF (Delta_CFF) at a location in the medium caused by an earthquake, it is necessary to know the deviatoric stress tensor before the earthquake and after the earthquake. It is possible to calculate the changes in static stress caused by the earthquake from a model of slip in the earthquake, but this is not enough. One must also know the "regional" state of stress that existed before the earthquake so that one can add the stress changes and determine the final stress field. The pre-existing regional field determines the initial CFF at a location, and the final stress field determines the optimal failure planes, slip direction on them, and the final value of CFF. Note that the CFF values calculated before and after the earthquake may be for differently oriented planes and different directions of slip on those planes. This approach was taken by Oppenheimer and others (1988) to study aftershocks to the 1988 Morgan Hill, California earthquake. It is implemented in program stroop (stress on optimal planes).
A second approach, defines a set of planes with known fixed orientation as the likely planes of failure. If there is a regional fabric (or fabrics) to the geology that introduces a strength anisotropy, this approach would make better sense than the isotropic approach above. If a slip direction (rake) is also specified for the planes, then the regional field can be dispensed with entirely. In the terminology of Armbruster and Seeber (1991), the changes in static stress caused by an earthquake are sufficient to calculate whether failure on such planes has been "encouraged" or "discouraged". If a slip direction (rake) is not specified for the plane (failure is isotropic on the plane), then a regional field (or other information) is still needed to define slip directions. In this case, note that the value of the CFF before an earthquake is calculated using a slip direction defined using just the regional, whereas the final value of CFF is calculated using a slip direction defined by regional plus stress change field. This second direction need not be the same as the first. Both of these approaches for calculating Delta_CFF (rake either specified or not specified) are implemented in program strop (stress on plane).
A hybrid approach has been used by Stein and others (1992, 1994) and King and others (1994) to explore the aftershocks to the 1992 Landers and 1994 Northridge earthquakes in southern California. These authors find Delta_CFF for optimal vertical failure planes (RL or LL slip) and for optimal reverse failure planes in an assumed regional field. They then combined the results (for a specific depth or for combinations of depths) by taking the worst case result from both calculations to get a worst case Delta_CFF.
All of the above calculations assume that a coefficient of friction has been chosen in order to calculate the value of the CFF. A high coefficient of friction implies that changes in normal stress are important in determining proximity to failure, whereas a low coefficient implies that only the shear stresses acting on the failure plane are important. Commonly, the approach taken in many studies has been to use either a high value (0.6 perhaps) consistent with laboratory studies on rock failure, or to try two cases with both low and high values as plausible end members (0.0 and 0.75 perhaps).
It has been suggested that large faults are weaker than laboratory studies would suggest (Zoback and others, 1987), and several studies using static stress calculations seem to find better correlations between calculated stress changes and observed seismicity rate changes if low (0.0-0.2) values of coefficients are assumed (e.g., Reasenberg and Simpson, 1992; Simpson and others, 1994). These results are still weak. However, static stress calculations offer some hope of finding the appropriate values for the coefficient of friction in given tectonic environments. (There may be different values appropriate to different environments.)
Much recent work has been focused on exploring the importance of pore fluids in the earthquake process (e.g., Byerlee, 1990; Rice 1992; Blanpied and others, 1992; Sleep and Blanpied, 1992). Pore-fluid pressures enter into the calculation of the CFF by modulating the normal stress. ("Effective normal stress" is defined as the normal stress minus the pore fluid pressure.) Pore-fluid effects can be incorporated into elastic calculations in an approximate way by estimating the changes in pore pressure that would accompany changes in stress applied to a poro-elastic medium (e.g., Rice and Cleary, 1976; Roeloffs, 1988; Scholz, 1990).
The effectiveness of the coupling is determined by the value of Skemptons coefficient (0 to 1). For programs requiring a coefficient of friction (e.g., stroop, strop, elfpatch), entering a negative value will incorporate the maximum undrained effect for a homogeneous elastic medium (Skemptons coefficient = 1). Rice (1992) presented a model of a compliant fault zone for which the pore pressure effects in the fault zone in the undrained state would be equivalent to lowering the coefficient of friction. One way to incorporate pore effects into the Coulomb law (Byerlee, personal communication) is by introducing an apparent coefficient of friction in place of the usual quantity. This new quantity retains the old form of the Coulomb law, and neatly hides our ignorance of the whole issue.
It also needs to be mentioned that introduce pore effects also introduces time into the picture. The stress changes calculated using the assumption of poro-elastic behavior are instantaneous changes (undrained state). As time passes, if the medium is permeable and the fluid pressures can return to their pre-existing state (drained state) as the fluids migrate in response to the imposed stress changes, then the final state should have the apparent coefficient of friction returning to high (dry) values. Recent attempts to incorporate pore fluids into calculations of earthquake-induced stress changes can be found in Jaume and Sykes (1992) and in Simpson and Reasenberg (1994).
Jaeger, J.C., and Cook, N.G.W., 1979, Fundamentals of rock mechanics (3d ed.): London, Chapman and Hall, 593 p.
Scholz, C.H., 1990, The mechanics of earthquakes and faulting: Cambridge, U.K., Cambridge University Press, 439. p.