Stress Drop Calculator

Stress drop is the reduction in shear stress on a fault produced by the slip of a single earthquake, not the absolute stress the fault was carrying. It follows from elastic crack theory: for a circular rupture in a uniform elastic whole space, average slip is proportional to stress drop times radius divided by rigidity, and combining that with M0 = mu x A x D eliminates the elastic moduli entirely, leaving the drop equal to (7/16) x M0 / r^3. The cubic dependence on radius dominates the calculation. Where the radius is not known directly it is inferred from the corner frequency of the source spectrum, the point at which the flat low-frequency level begins to roll off, using the Brune relation r = 2.34 x Vs / (2 x pi x fc). Observed stress drops for tectonic earthquakes span roughly 0.1 to 100 MPa, with most events between 1 and 10 MPa, and there is no strong trend with magnitude, which is the observational basis for treating rupture as approximately self-similar across many orders of size. Interplate events cluster toward the low end and intraplate events run higher. The result is very sensitive to assumptions: because stress drop varies as the cube of corner frequency, a 20 percent error in picking fc changes the answer by roughly 70 percent, and substituting the Madariaga dynamic-rupture constant for the Brune value shifts computed stress drops by a factor of several. The assumed shear velocity and the attenuation correction applied before fitting the spectrum matter just as much.

The mechanics came from materials science rather than seismology. John Eshelby solved the elastic field of an ellipsoidal inclusion in a 1957 paper in the Proceedings of the Royal Society, a result about inhomogeneities in solids that happened to give exactly the slip distribution needed for a circular shear crack; applied to earthquakes it produces the 7/16 factor. Keiiti Aki supplied the second ingredient in 1967 with his scaling law of the seismic spectrum, showing that source spectra are flat at low frequency at a level proportional to seismic moment and fall off as frequency squared above a corner frequency set by the source dimension. James Brune combined these in a 1970 paper in the Journal of Geophysical Research, deriving the relation between corner frequency and source radius that made stress drop measurable from a routine spectral fit. Raul Madariaga published a dynamic circular fault model in 1976 whose different constant has been a source of systematic disagreement between studies ever since. Hiroo Kanamori and Don Anderson argued in 1975, from empirical scaling relations among magnitude, area and moment, that stress drop is roughly constant across magnitudes. Larger datasets complicated that picture: the global study by Bettina Allmann and Peter Shearer in 2009 found values scattered over about an order of magnitude around a few MPa, with regional patterns correlating with tectonic setting rather than with earthquake size.