Page 284 - Physical Principles of Sedimentary Basin Analysis

266 Rheology: fracture and flow

x

x 1
t 1 t 2
t
Figure 8.8. Stick–slip fault behavior. The fault moves during a short time interval t 1 and rests during
a long time interval t 2 . The fault accumulates sufficient elastic stress during the time interval t 2 to
overcome the static friction of the fault and to initiate slip.

as shown in Note 8.4. Once the block is at rest it stays at rest until the force of the spring
once more reaches F s = kx s and overcomes static friction. That happens after a time span
t 2 given by

F s = kx s = k x s + v(t 1 + t 2 ) − x 1 (8.16)
or
x 1 − vt 1 2(x s − x d )
t 2 = = (8.17)
v v
where the last equality follows from equation (8.15). The time spans t 1 and t 2 are related by
1 −1 1
λt 1 = π − tan λt 2 . (8.18)
2 2
The movement of the block is periodic – it moves for a time interval t 1 and then rests a time
interval t 2 , before the cycle repeats, as shown in Figure 8.8. The time t 1 is much less than
the time t 2 in the case of faulting. The time of fault movement may be less than a second,
while the fault is at rest for several years. The large difference between the time spans t 1
and t 2 is caused by a very small velocity v. The distance (8.15) the block moves during slip
can therefore be approximated as
mg
x 1 ≈ 2(x s − x d ) = 2(μ s − μ d ) (8.19)
k
which is proportional to the difference between the coefficients of static and dynamic
friction.
Equation (8.17) shows that the time the block is at rest becomes zero if the static and
dynamic friction are equal. Stick–slip behavior of faults therefore requires that the coef-
ficient of static friction is larger than the coefficient of dynamic friction, μ s >μ d . (See
Exercise 8.2.)
Even if this block–spring model of the fault-behavior is simple, it nevertheless captures
the main features of the stick–slip behavior observed in earthquakes. This kind of model
has been the basis for more realistic models, consisting of several blocks connected with
springs, which can reproduce the chaotic and fractal behavior of earthquakes.