Geology Reference
In-Depth Information
are sometimes associated with vector or even
The continuum mechanics representation of
Earth systems also includes
extensive variables
.
These quantities are
global
physical properties,
which depend from the total volume
V
of a sys-
tem through integral expressions involving
den-
sity
functions. A classic example is the total mass
of a rock body. Let
dV
be a volume element cen-
tered at position
r
in the region
R
. The approach
of continuum mechanics is to consider the mass
contained in
dV
as the analog of a point mass, so
that the classic equations of elementary physics
can be easily generalized to the new framework.
To this purpose, we can introduce a new intensive
quantity, the
density of mass
, ¡
D
¡(
r
), such that
the infinitesimal mass contained in the volume
dV
will be given by:
dm
D
¡(
r
)
dV
. In this instance,
the total mass of a body is an extensive property
that can be computed by evaluating the following
integral expression:
the entire system. In elementary mechanics, this
vector is obtained by taking the weighted average
of the individual position vectors, and using the
mass of each particle as a weighting parame-
ter. The continuum mechanics analogue of this
quantity is another extensive variable, which can
be calculated by substituting the sum appear-
ing in the elementary definition by an integral
expression.
Therefore, the center of mass of a continuous
distribution is defined as follows:
Z
1
M
R
D
¡.r/ r
dV
(2.4)
R
where
M
is the total mass. The last extensive vari-
able considered here is the
angular momentum
of
the system, which measures the rotational com-
ponent of motion with respect to an arbitrary ref-
erence point. This quantity is usually calculated
with respect to the origin of the reference frame
or, alternatively, with respect to the center of mass
depending on the problem under consideration. In
the former case, the angular momentum is given
by the following integral expression, which is an
obvious extension of the elementary definition:
Z
M
D
¡.r/
dV
(2.1)
R
Similar expressions can be written for the total
electric charge, magnetization, etc. introducing
appropriate density functions. If a continuous
rock system is subject to an external action-at-a-
distance force field, such as a gravity or magnetic
field, this force operates on each volume element
in
R
. Therefore, we can introduce a
body force
density
(force per unit volume),
f
D
f
(
r
), such
that the infinitesimal force exerted on a volume
element
dV
will be given by:
d
F
D
f
(
r
)
dV
.Using
this definition, the total force,
F
, and the torque,
N
, exerted on the whole body are extensive vari-
ables given respectively by:
Z
L
D
r
¡.r/
v
.r/
dV
(2.5)
R
In this expression, the vector field
v
D
v
(
r
)
represents the velocity of the mass element at
position
r
. In the next section, we shall consider
a special form of expression (
2.5
), which is par-
ticularly useful in plate kinematics, where mass
distributions represent rigid tectonic plates.
Z
2.2
Euler's Theorem and Rigid
Rotations
F
D
f .r/
dV
(2.2)
R
Z
Plate dynamics and kinematics, in short plate
tectonics, cannot be described using a unique
mathematical apparatus and a single physical the-
ory, because the various interacting subsystems of
the solid Earth (plates, slabs, asthenosphere, etc.)
conform to different physical laws, depending
on the time scale of observation (seconds, years,
N
D
r
f .r/
dV
(2.3)
R
An important kinematic parameter of a point
mass distribution is the
center of mass
,which
is a position vector representing the location of
