Earthquakes - Quantitative Plate Tectonics - page 276

0 , 1 , :::, n , such that the steady state coeffi-
cient for the stage k C 1 would be given by:

The resulting plots of frictional stress, velocity,
and state are illustrated in Fig. 10.3 . We note a
prominent peak of the state variable ™ for v D
0, and the frictional stress peak at t D t 1 ,which
represents the static friction stress ¢ s that must
be applied to resume sliding. In this numerical
experiment, the duration of the stick interval is
initially zero, so that the initial value of static
friction can be defined as the dynamic friction
coefficient at the onset of slipping:
s .0/ D 0 C b ln v 0 ™.t 1 /
L

kC1 D k C .a b/ln v k C 1

v k I k

D 0;1;:::;n 1 (10.5)

However, this recurrence formula can be eas-
ily solved in terms of ( 0 , v 0 ), so that the steady
state coefficient for the k -th stage can be calcu-
lated easily from ( 0 , v 0 ). It results:

(10.8)

k D 0 C .a b/ln v k

v 0 I k D 0;1;:::;n
(10.6) Alternatively, if we keep the slider at rest for

some finite time interval t stick D t - t 1 , we will ob-
serve a logarithmic increase of the static friction
s ( t stick ) as a consequence of the linear growth of
the state variable ™ for v D 0(Eq. 10.2 ). Such an
increase of the friction coefficient during station-
ary contact is known as healing (e.g., Berthoud
et al. 1999 ). The curves in Fig. 10.3 show that
static friction is not a separate phenomenon but
can be explained in terms of rate-and-state evolu-
tion. Furthermore, they show that the initial value
of s does not represent an absolute threshold
determined exclusively by the physical properties
of the materials. In fact, by ( 10.8 ) we see that
s (0) also depends from the current value of the
state variable ™ at the stop time and, apparently,
from the velocity at which we resume sliding af-
ter the stick interval ( v 0 in this example). Clearly,
this dependence from the restarting velocity is not
actual, because the system does not know a priori
at which velocity the slider will be moved. In
reality, the appearance of a velocity dependence
arises from the fact that the order of magnitude of
the dynamic friction coefficient does not change
for a wide range of velocities. For example,
using the parameters of Fig. 10.3 and Eq. ( 10.4 ),
we see that for a transition v 0 ! 10 3 v 0 ,the
friction coefficient will change from D 0.60 to
Š 0.63. Figure 10.4 shows the predicted steady
state friction coefficient for a wide interval of
relative velocities v / v 0 .
Therefore, if we apply a small push to the
slider when it is at rest, we will effectively ac-
celerate the mass to a very small velocity v ,

Therefore, in the calculation of the dynamic
friction coefficient through the rate-and-state law
( 10.1 ), 0 and v 0 can be considered as reference
values, so that the resulting steady state value of
will be independent from the previous state of
the system. Now we may wonder if the existence
of a static friction coefficient s can be ex-
plained through the rate-and-state law ( 10.1 ). An
interesting numerical experiment, which can also
be performed as real laboratory experiment (see
Nakatani 2001 ), will help clarifying the concept
of static friction.
In a slide - hold - slide experiment, a sliding
mass M is initially moved applying a constant
shear stress £ 0 . In steady state conditions, this
stress is balanced by a frictional stress 0 ¢,so
that £ 0 D 0 ¢ and the mass moves at constant
velocity v ( t ) D v 0 . Now let us assume that at
some time t D t 0 the applied stress is reduced to a
lower value £<£ 0 (possibly zero). At this point,
the velocity will start decreasing according to the
following equation of motion:

1
M Œ£ .t/¢

P v .t/ D (10.7)

where M is mass per unit surface and ( t )is
the dynamic friction coefficient at time t ,which
evolves according to the rate-and-state law ( 10.1 )
and the state evolution law ( 10.2 ). We also as-
sume that at some later time t 1 D t 0 , as soon as
the velocity has decreased to zero, the slider is
restarted and moved again at constant velocity v 0 .