Attempting to Bridge the Gap Between Laboratory and Seismic Estimates of Fracture Energy

The elastic strain energy released during an earthquake is partitioned into energy radiated by the seismic waves, work expended to overcome frictional resistance, and the fracture energy consumed as the rupture surface expands. Of these three components, the fracture energy G_c is perhaps the least understood and, in particular, attempts to relate laboratory estimates of this component to counterparts measured seismically for earthquakes have been controversial. Because of its important role in the development of dynamic rupture models of earthquakes, it is important to be able to understand the behavior of G_c, especially its dependence on the state of stress and the scale of rupture. Encouraged by success at scaling fault slip and slip rate from biaxial stick-slip friction laboratory experiments to those for earthquakes (McGarr and Fletcher, Bull. Seismol. Soc. Am., in press, 2003), we propose a similar approach for extrapolating G_c measured in the laboratory to investigate its behavior for earthquakes. Fracture energy is commonly represented as the area of a right triangle whose height represents the difference between the yield stress and the dynamic frictional stress (dynamic stress drop) and whose base is the slip-weakening distance D_c. Fracture mechanics analysis suggests that D_c may scale as the dimension of the seismic rupture and the dynamic stress drop is determined by the state of stress in the seismogenic crust. The dimension of rupture is often taken to be the size of the entire earthquake. Slip models developed for major earthquakes, however, indicate that the distribution of slip is quite inhomogeneous, consisting of patches of high slip separated by zones of relatively little offset. This suggests that G_c and D_c scale more appropriately with either the dimension of an asperity or its associated slip; slip turns out to be a more straightforward scaling parameter. Thus, using slip as the parameter to scale from laboratory friction experiments to major well-studied earthquakes typically yields estimates of G_c and D_c that are less than those used in most dynamic rupture models of earthquakes. This new approach to estimating G_c appears to resolve a problem involving calculations of seismic energy radiation from dynamic rupture models of earthquakes; that is, these energies tend to be less than other types of estimates (e.g., teleseismic). We suggest that the high values of D_c used in some recently-published dynamic rupture models resulted in overestimates of the fracture energies at the expense of the seismic energies. We find that distributions of G_c and D_c that are constrained by laboratory results and the distribution of fault slip yield lower fracture energies and higher seismic energies that are in better accord with independent estimates.