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We perform analytical and numerical studies of aftershock
sequences following abrupt steps of strain in a
rheologically-layered model of the lithosphere. The model consists
of a weak sedimentary layer, over a seismogenic zone governed by a
visco-elastic damage rheology, underlain by a visco-elastic upper
mantle. The damage rheology accounts for fundamental irreversible
aspects of brittle rock deformation and is constrained by
laboratory data of fracture and friction experiments. A 1-D
version of the visco-elastic damage rheology leads to an
exponential analytical solution for aftershock rates. The
corresponding solution for a 3-D volume is expected to be sum of
exponentials. The exponential solution depends primarily on a
material parameter R given by the ratio of timescale for damage
increase to timescale for gradual inelastic deformation, and to a
lesser extent on the initial damage and a threshold strain-state
for material degradation. The parameter R is also inversely
proportional to the degree of seismic coupling across the fault.
Simplifying the governing equations leads to a solution following
the modified Omori power law decay with an analytical exponent p =
1. In addition, the results associated with the general
exponential expression can be fitted for various values of R with
the modified Omori law. The same holds for the decay rates of
aftershocks simulated numerically using the 3-D layered
lithospheric model. The results indicate that low R-values (e.g.,
R 1) corresponding to cold brittle material produce long
Omori-type aftershock sequences with high event productivity,
while high R-values (e.g., R 5) corresponding to hot viscous
material produce short diffuse response with low event
productivity. The frequency-size statistics of aftershocks
simulated in 3-D cases with low R-values follow the
Gutenberg-Richter power law relation, while events simulated for
high R-values are concentrated in a narrow magnitude range.
Increasing thickness of the weak sedimentary cover produces
results that are similar to those associated with higher R-values.
Increasing the assumed geothermal gradient reduces the depth
extent of the simulated earthquakes. The magnitude of the largest
simulated aftershocks is compatible with the Båth law for a range
of dynamic damage-weakening parameter. The results provide a
physical basis for interpreting the main observed features of
aftershock sequences in terms of basic structural and material
properties.
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**Reference:
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- Ben-Zion, Y. and V. Lyakhovsky,
*Analysis of Aftershocks in a Lithospheric Model with Seismogenic Zone Governed by Damage Rheology,*Geophys. J. Int., vol. 164, doi: 10.1111/j.1365-246X.2006.02878.x, 2006.

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