1. A possible mechanism of dynamic earthquake triggering
Yoshihiro Kaneko
(Summary of Johnson and Jia, 2005; Gomberg and Johnson, 2005)

2. Background From Gomberg (2001), we know
Observed finite duration of dynamically triggered earthquakes can not be explained by a population of the “standard” rate-and state-faults, nor faults behaving under subcritical crack growth.
Dieterich rate-state model
Sub-critical crack growth model

3. Background From previous lectures (J.P. Avouac):
the rate-and state-model with an addition of the pore-pressure change may explain the observed duration of dynamically triggered earthquakes.
Rate- and state-friction with variation of pore pressure:
The “shaking” caused by seismic waves results in increase of pore-pressure (the compaction of saturated fault gouge).
The compaction of fault gouge under hydrothermal conditions was observed in laboratory creep experiments (Blanpied et al., 1992).
However, for the range of slip rate (~10-3 – 10-7 m/sec), shear induced dilation has been observed in the lab experiments.
(i.e. Marone et al., 1990; Segall and Rice, 1995)

4. Johnson and Jia (2005) have proposed
dynamic elastic non-linear behavior of fault gouge perturbed by a seismic wave may trigger earthquakes.
They conducted laboratory experiments on granular media (analogous to fault gouge).
3 cm
Diagram: set-up for resonance and pulse-mode experiments in the glass bead pack under applied pressure p.
Excitation of P-wave by transducers.
Measured quantity: displacement on the transducer face (by an optical interferometer)
P-wave excitation at seismic strains (10-6 to 10-4) are used.
1.85 cm

5. Material softening due to non-linear dynamics under resonance conditions (Figure 2)
p fixed at 0.11 MPa
(a) Increase input voltages – decrease in resonance frequency – decrease in P-wave velocity and thus, modulus (r slightly increases during the experiments)
fr = V/(2L), V=(M/r)1/2
These figures show the characteristics’ of non-linear elastic behavior in their experiments.

6. Material softening due to non-linear dynamics under resonance conditions (Figure 2)
p fixed at 0.11 MPa
(a) Increase input voltages – decrease in resonance frequency – decrease in P-wave velocity and thus, modulus (r slightly increases during the experiments)
fr = V/(2L), V=(M/r)1/2
high p
(b) Modulus softening diminishes progressively as the pressure is increased.
(non-linearity decreases as pressure increases).
Below this strain, the granular material behaves as a linear elastic medium
low p

7. Material softening due to non-linear dynamics under resonance conditions (Figure 2)
p fixed at 0.11 MPa
(a) Increase input voltages – decrease in resonance frequency – decrease in P-wave velocity and thus, modulus (r slightly increases during the experiments)
fr = V/(2L), V=(M/r)1/2
high p
(b) Modulus softening diminishes progressively as the pressure is increased.
(non-linearity decreases as pressure increases).
Below this strain, the granular material behaves as a linear elastic medium
low p
(c) Time-dependent behavior (“slow dynamics”):
High amplitude excitation for 7 min following the low amplitude excitation to measure the recovery of the modulus with time

8. Traveling-wave experiments under a pressure of 0.11 MPa (Figure 3)
The source signal is a one-cycle sinusoidal pulse.
Higher strain amplitude is required to produce equivalent modulus softening of the resonance case (source duration matters).

9. Failure model: failure by softening and weakening the fault gouge (Figure 4)
In order for this model to explain observed dynamic triggering,
one needs to assume:
fault is in a critical state to failure.
Fault is weak (or low effective normal stress).
Dynamic strain amplitude is greater than 10-6.

10. Failure model: failure by softening and weakening the fault gouge (Figure 4)
Can explain the observed duration of dynamically triggered earthquakes.

11. Failure model: failure by softening and weakening the fault gouge (Figure 4)
Can explain the observed duration of dynamically triggered earthquakes.

12. Dynamic deformation scaling: in laboratory and seismic observations (Figure from Gomberg, 2005)
Horizontal peak velocity (cm/sec)
Distance normalized by the
square root of the ruptured area
Note: Different symbols - M4.4 to M7.9 earthquakes.
Filled stars - sites of triggered seismicity, open stars (sites of no triggered seismicity)
red bar – Denali triggering PGVs

13. Conclusions
Dynamic elastic nonlinearity of fault gouge may provide an explanation for the occurrence of dynamic triggering in response to seismic waves.
Slow recovery of elastic modulus with time may be responsible for observed duration of dynamically triggered earthquakes.
The seismologically observed dynamic strains are consistent with observed softening strains in the experiments.

14. An alternative explanation?
Quasi-dynamic boundary integral equation (for linear elasticity) combined with rate- and state-friction:
tref - V G/(2Vs)