Geology Reference
In-Depth Information
elasto-plastic rheology
. When the stress is below
the yield strength, the system stretches the spring
and there is no slip along the contact surface. In
these conditions, the material deforms elastically.
As soon as the stress attains the critical value, the
mass is pulled at constant stress and deformation
proceeds plastically until the load falls below the
yield stress £
c
. Apparently, this behaviour is quite
similar to the viscous flow. However, differently
from visco-elastic deformation, plastic flow oc-
curs only when the yield stress is attained. It is
termed
ductile deformation
when the continuity
within the material persists despite strain local-
ization. Conversely, when the material loses its
cohesion through the development of fractures or
faults we will use the term
brittle deformation
.
The mode of failure of Earth's rocks mainly
depends from the confining hydrostatic pressure
and from temperature, while the yield point is a
decreasing function of the strain rate. The effect
of increased hydrostatic pressure is to inhibit
fracturing and cracking. Therefore, in so far as
the hydrostatic pressure is increased, the me-
chanical behaviour changes suddenly from brittle
to ductile. The range of depths at which this
transition occurs is called the
brittle-ductile tran-
sition zone
and depends not only from (
P
,
T
)
conditions but also from the strain rate and the
presence of water. In general, the rheology of
Earth's lithosphere rocks changes with increas-
ing depth from brittle to ductile to visco-elastic.
The mechanical behaviour is usually represented
graphically through strength (yield stress) versus
depth profiles that are called
rheological profiles
or
yield
-
strength envelopes
. Now we are going
to describe the general methods adopted to build
these profiles.
The static frictional resistance along fault
planes is generally constant, so that by
Amonton's
law
required for seismic slip is independent from
the contact area and increases linearly with the
confining pressure. Byerlee (
1968
) determined
experimentally the following linear relations
over a range of normal stresses from 3 MPa
to 1.7 GPa:
0:85¢
N
I
3<¢
N
< 200 MPa
60
˙
10
C
0:6
¢
N
I
¢
N
> 200 MPa
(7.77)
£
S
D
These simple relations hold for all geologic
materials except certain clay minerals (e.g., Brace
and Kohlstedt
1980
). They can be used to build
the upper part of a rheological profile, assuming
that ¢
N
corresponds to the hydrostatic pressure.
In this instance, the strength increases linearly
with depth and is independent from the specific
kind of rocks. The lower part of a rheological pro-
file for the lithosphere is built taking into account
that Earth's rocks exhibit non-linear viscous be-
haviour at relatively low temperatures. In fact,
the linear visco-elastic rheological models dis-
cussed above provide good approximations of the
real mechanical behaviour of Earth's rocks only
at lower lithosphere and asthenosphere condi-
tions, that is, temperatures between 1,000
ı
Cand
1,500
ı
C and slow strain rates (10
12
-10
14
s
1
).
Conversely, in the case of upper lithospheric man-
tle and crustal rocks, we observe steady-state flow
even at small stresses, but the relation between
stress and strain rate is nonlinear. In this instance,
the viscosity is a decreasing exponential function
of temperature through an Arrhenius relationship
and the stress raised to some power (usually
between 3 and 5) is proportional to the strain rate.
The most important empirical non-linear relation
between stress and strain rate is known as the
power
-
law creep
or
Dorn equation
:
the
coefficient
of
static
friction,
s
,
P
©
D
A£
S
e
E=RT
(7.78)
determines the
shear
stress
required to
have
seismic slip:
where
n
is determined experimentally and de-
pends from the material,
A
is a constant depend-
ing from the material and the (
P
,
T
) conditions,
T
is the temperature,
R
is the gas constant, and
the
thermal activation energy E
is of the order of
100-600 kJ mol
1
£
S
D
s
¢
N
(7.76)
where £
S
and ¢
N
are respectively the shear stress
and the normal stress
along the fault plane.
depending from the material.
Amonton's law
implies
that the
shear
stress
