Geology Reference
In-Depth Information
the components of
n
are given by:
n
i
D
q
i
/¡.
Substituting into (
7.10
) and taking into account
of (
7.20
)gives:
flow and yielding, is relatively small. The
last important property of the stress tensor is
represented by its
invariants
. The characteristic
Eq. (
7.7
) can be written as follows:
T
N
.n/¡
2
D˙
1
D
¢
i
q
i
(7.21)
det.£
Iœ
i
/
D
œ
3
C
I
1
œ
2
C
I
2
œ
C
I
3
D
0
(7.27)
In this equation, the minus sign is associ-
ated with compressive stresses, thereby, (
7.10
)
reduces to a single triaxial ellipsoidal surface
with semi-axes
j
i
j
1/2
:
where,
I
1
D
11
C
22
C
33
D
Tr
.£/
I
2
D
.
11
22
12
21
/
.
11
33
13
31
/
j
¢
i
j
q
i
D
1
(7.22)
.
22
33
23
32
/
I
3
D
det.£/
The components of the versors normal to this
surface are given by the gradient of (
7.22
)(see
(7.28)
It is possible to prove that the quantities
I
1
,
I
2
,
and
I
3
, which are referred to as the
first
,
second
,
and
third invariant
, are unchanged under coordi-
nate transformations, so that they are invariants of
the stress tensor. While the first invariant simply
states that the hydrostatic pressure is independent
from the selected coordinate system, the second
one plays an important role in the description of
the fluid behaviour of the rocks and non-linear
creep.
@q
i
j
¢
i
j
q
i
D
2›
j
¢
i
j
q
i
m
i
D
›
@
(7.23)
where › is a constant that ensures that
j
m
jD
1.
Now, in the principal axes coordinate system
(
7.3
) reduces to:
1
¡
¢
i
q
i
T
i
.n/
D
¢
i
n
i
D
(7.24)
Therefore,
m
i
D˙
2›
¢
i
q
i
D˙
2›
¡
T
i
.n/
(7.25)
7.2
Displacement Fields
and Strain
This expression proves that
m
and
T
(
n
)are
parallel. As a consequence, Cauchy's stress el-
lipsoid completely determines the state of stress
at a point. Another important general feature of
the stress tensor is that it can be decomposed
into an
isotropic stress
, £
0
, which describes the
hydrostatic component of stress, and a
deviator
,
£
0
, associated essentially but not exclusively with
shear:
Let us face now the problem of describing quan-
titatively the
deformation
of a continuum body
under the action of external forces. When a con-
tinuum body changes its shape, each point in
the undeformed region
R
, with position vector
r
, is subject to a
displacement
u
that depends
from the position. The set of all displacements
for each point
r
2
R
forms a vector field
u
D
u
(
r
)(Fig.
7.6
). Clearly, the existence of a non-
zero displacement field does not imply automat-
ically that a body has been deformed, because
rigid body translations and rotations are also
associated with displacement fields. In principle,
the existence of a deformation can be estab-
lished observing that two neighbor points
r
and
r
C
d
r
have experienced
differential
displacement
£
ij
D
p•
ij
C
£
0
ij
(7.26)
where
p
D
£
kk
/3 is the mean pressure and
£
ij
D
£
ij
£
kk
•
ij
/3. This is often a convenient
separation, because stresses in the deep Earth are
dominated by the large compressive components
associated with the hydrostatic pressure, whereas
the
deviatoric
component,
associated
with
