On the expected relationships among apparent stress, static stress drop, effective shear fracture energy, and efficiency

We consider expected relationships between apparent stress tau(a) and static stress drop Deltatau(s) using a standard energy balance and find tau(a) = Deltatau(s)(0.5 - zeta), where zeta is stress overshoot. A simple implementation of this balance is to assume overshoot is constant; then apparent stress should vary linearly with stress drop, consistent with spectral theories (Brune, 1970) and dynamic crack models (Madariaga, 1976). Normalizing this expression by the static stress drop defines an efficiency eta(sw) = tau(a)/Deltatau(s) as follows from Savage and Wood (1971). We use this measure of efficiency to analyze data from one of a number of observational studies that find apparent stress to increase with seismic moment, namely earthquakes recorded in the Cajon Pass borehole by Abercrombie (1995). Increases in apparent stress with event size could reflect an increase in seismic efficiency; however, eta(sw) for the Cajon earthquakes shows no such increase and is approximately constant over the entire moment range. Thus, apparent stress and stress drop co-vary, as expected from the energy balance at constant overshoot. The median value of eta(sw) for the Cajon earthquakes is four times lower than eta(sw) for laboratory events. Thus, these Cajon-recorded earthquakes have relatively low and approximately constant efficiency. As the energy balance requires eta(sw) = 0.5 - zeta, overshoot can be estimated directly from the Savage-Wood efficiency; overshoot is positive for Cajon Pass earthquakes. Variations in apparent stress with seismic moment for these earthquakes result primarily from systematic variations in static stress drop with seismic moment and do not require a relative decrease in sliding resistance with increasing event size (dynamic weakening). Based on the comparison of field and lab determinations of the Savage-Wood efficiency, we suggest the criterion eta(sw) > 0.3 as a test for dynamic weakening in excess of that seen in the lab.