Geology Reference
In-Depth Information
2
3
2
3
thousands or million years). Even when consid-
ered at a common temporal scale, these subsys-
tems display distinct mechanical behaviors. For
example, the motion of tectonic plates during
the geological time (intervals of Myrs) can be
described in terms of rigid body's kinematics,
whereas the asthenosphere behaves as a fluid at
the same temporal scale. However, both can be
considered as elastic bodies if we are studying
earthquakes and propagation of seismic waves in
the solid Earth. In summary, the sole unifying
framework of plate tectonics is the continuum
mechanics representation illustrated in the previ-
ous section, while both the kinematic description
of the processes and the geodynamic laws that
link forces to kinematics will be different depend-
ing on the subsystem and the temporal scale of
observation.
Observation suggests that tectonic plates can
be considered as
rigid
bodies at first approxi-
mation. Consequently, the volume elements that
fill a region
R
<
x
0
1
x
0
2
x
0
3
y
1
y
2
y
3
z
0
1
z
0
2
z
0
3
x
1
x
2
x
3
y
1
y
2
y
3
z
1
z
2
z
3
4
5
I
T
0
D
4
5
T
D
Now let us define a new 3
3matrix:
A
D
T
0
T
1
(2.6)
The matrix
A
has the property to transform the
original matrix
T
into the new matrix
T
0
:
AT
D
T
0
(2.7)
This equation implies, in turn, that
A
trans-
forms each vector
r
i
into the corresponding ro-
tated vector
r
i
:
Ar
i
D
r
0
i
(2.8)
In general, for any position vector,
r
,thetrans-
formation
A
preserves the distance of the trans-
formed point from the origin, because the sphere
is assumed to be rigid:
3
, representative of a tectonic
plate, are also rigid entities, and the distance
between any pair of volume elements in
R
is an
invariant. This is equivalent to say that the elec-
tromagnetic interaction between adjacent volume
elements is so strong that any external force
is overcome, so that deformation is negligible.
In this instance, an important theorem, due to
Leonhard Euler (1775), can be used as a starting
point for the mathematical description of plate
kinematics. The statement of Euler's theorem is
very simple:
k
Ar
k
D
k
r
k
(2.9)
Squaring this equation gives:
r
T
A
T
Ar
D
r
T
r
(2.10)
thereby
A
T
A
D
I
,where
I
is the 3
3 identity
matrix, and
A
is orthogonal. Now let us take the
determinant of
A
T
A
. It results:
det
A
T
A
D
Œdet.A/
2
D
1
(2.11)
Euler's Theorem
If a sphere S is moved about its center, O, it is
always possible to find a diameter, D, of fixed
points.
Therefore, det(
A
)
D˙
1. If we consider a null
rotation of the sphere from its initial position,
then
A
D
I
and det(
A
)
D
det(
I
)
DC
1. By con-
tinuity, any subsequent infinitesimal rotation or
sequence of rotations must give det(
A
)
DC
1.
Furthermore, by the orthogonality of
A
we have:
A
T
A
A
D
A
T
I
A
D
I
A
Proof
Let
r
1
,
r
2
,
r
3
be three position vectors
pointing to arbitrary points,
P
1
,
P
2
,and
P
3
in
the original sphere. After an arbitrary change of
orientation of the sphere about its center, these
points are moved to new locations, say:
P
0
1
,
P
0
2
,
and
P
0
3
, represented by the position vectors:
r
0
1
,
r
0
2
,
r
0
3
.Let
T
and
T
0
be the 3
3 matrices formed
with the components of these vectors:
(2.12)
det
A
T
I
D
det
.A
I/
T
D
det .A
I/
(2.13)
