Geology Reference
In-Depth Information
Fig. 10.2
Frictional response (
left
) and state evolution (
right
) for a positive 10 % velocity step, followed by a negative
step.
u
/
L
is the normalized displacement. The plots were built assuming
L
D
10
5
m,
a
D
0.005, and
b
D
2
a
1
C
v
v
0
1
e
v
t=L
state
with constant velocity
v
0
and friction
coefficient
0
, an arbitrary velocity transition
v
0
!
v
will trigger a transient phase during which
the friction coefficient changes continuously
as a consequence of the evolution of a state
variable ™:
L
v
™.t/
D
(10.3)
The constitutive law (
10.1
) shows that the
friction coefficient may change either as a
consequence of velocity variations or as a conse-
quence of state transitions. Figure
10.2
illustrates
the variations of dynamic friction coefficient and
state after positive and negative velocity steps.
After a sudden velocity increase
v
!
'
v
, '>1,
has a positive transition
!
C
a
ln',which
is known as the
direct velocity effect
. Such a dis-
continuous transition is followed by a continuous
decrease in friction, having magnitude
b
ln'.In
fact, by (
10.2
) we have that the state variable ™
decreases exponentially to the asymptotic value
™
1
D
L
/
v
, thereby the third factor at the right-
hand side of (
10.1
) will tend asymptotically to the
value -
b
ln(
v
/
v
0
)(Fig.
10.2
).
As a consequence, the steady state friction
coefficient will be given by:
¢
D
0
C
a ln
v
C
b ln
v
0
™.t/
L
£
.t/
D
v
0
(10.1)
where
a
,and
b
are constants that can be
determined experimentally and ™(0)
™
0
D
L
/
v
0
.
The state variable ™ was interpreted as the age of
a population of contact points supporting the load
¢ across the fault plane. The Dieterich-Ruina for-
mula is based on the assumption that the physical
state of the contact surface can be characterized
at any time by a single variable ™
D
™(
t
), and
that the frictional stress depends only from the
normal stress ¢, the slip rate
v
, and the state
variable ™ (Dieterich
1979
; Ruina
1983
). Several
evolution laws were proposed for the variable
™ (for a review, see Nakatani
2001
). The most
simple of them is (Dieterich and Linker
1992
):
£
¢
D
0
C
.a
b/ln
v
v
0
1
D
(10.4)
This solution implies that for
a
<
b
the steady
state friction decreases with increasing veloc-
ity. This form of friction can be observed for a
wide class of materials, and the reference friction
coefficient
0
results to be nearly independent
from the rock type and from temperature. The
solution (
10.4
) apparently says that the steady
state friction at some velocity
v
depends from
the
previous
steady state pair (
0
,
v
0
). In this
instance, a series of velocity values
v
0
,
v
1
, :::,
v
n
would produce a sequence of friction coefficients
1
L
™.t/
v
P
™.t/
D
1
(10.2)
Experiments showed that the characteristic
distance
L
varies between 2 and 100 mand
increases with the surface roughness and the fault
gouge particle size. Equation
10.2
implies that for
a stationary contact the state variable ™ increases
linearly with time, while for constant
v
> 0we
have:
