ESTIMATES OF AVERAGE STRESS FROM LABORATORY MEASUREMENTS OF FRICTION

ROCK FRICTION AND THE STRENGTH OF THE FAULT

We have seen that the average shear stress tau-bar on an earthquake fault can be viewed as the sum of a dynamic part taua and a frictional part taur. The dynamic part is shown to be small from seismic evidence, and so the earthquake stress must be large or small according to the size of taur. We have also seen that taur is small according to geothermal evidence. We now consider a second line of evidence from laboratory measurement of friction which suggests to many that, contrary to the geothermal evidence, taur must be large.

According to these results, rock surfaces will slide when the shear stress on their surface of contact exceeds the static frictional strength tauf, given by

tauf = mu etan', (15a)

where etan' = etan - P (15b)

sigman is the normal pressure pushing the surfaces together, and P is the fluid pressure in the pores and cracks tending to hold the surfaces apart; sigman is called the "effective" normal stress (we generally denote such effective stresses by a prime, " ' "). The proportionality constant mu in equation 15a is the coefficient of static friction; extensive laboratory experiments show that its value is generally in the range 0.6-0.9 for a remarkably large variety of rock types and surface conditions (Byerlee, 1978), although some studies (for example, Wang and others, 1980), reported substantially lower friction coefficients for some geologic materials, including certain types of fault gouge.

We presume that a fault is a fracture with little cohesive strength that remains inactive until the natural shear stress tau resolved along it exceeds its frictional strength tauf given by equation 15. This shear stress, which depends on the magnitudes of the principal stresses and on the angular relation between the fault plane and the principal stress directions (fig. 10.8), is given by (Jaeger, 1956, p. 8)

tau = 1/2, (sigma1 - sigma3) sin 2 theta, (16)

where it is assumed for convenience that the intermediate-principal-stress direction (sigma2) lies in the fault plane (true if mu is independent of the orientation of this plane). In figure 10.8, sigma2 is vertical, and sigma1 and sigma3 are the maximum and minimum horizontal principal stresses. Theta is the angle formed by the fault normal and the direction of least compression (sigma3); it is also the angle between the fault trace and the direction of greatest compression (sigma1). To express the failure criterion (eq. 15) in terms of the stress field and fault orientation, we note that the effective normal stress, sigman', in equation 15 can be written as (Jaeger, 1956, p. 8)

(17)

With equations 15 through 17, the friction stress tauf that must be exceeded on a fault for it to slip can be determined if we know (1) the maximum and minimum principal stresses sigma1 and sigma3, (2) the fluid pressure P, (3) the coefficient of friction mu and (4) the angle theta describing the orientation of the fault relative to the principal-stress axes.

As we increase the stress difference, in what direction (theta) will the Earth ultimately break, and what will be the stress on the failure plane? Clearly, the answer could be influenced by the existence of planes of weakness (McKenzie, 1969); for example, major preexisting faults or foliated country rock might result in directions with anomalously low mu.