solid or flowing fluid and this requires an idealization of the rock mass as a material continuum.
For this purpose we consider a sequence of finite
volumes, ␦V i , each with resultant body force, ␦f i ,
and average density, i , and each containing the
point in question (Fig. 6.2b). The finite volumes are
ordered from largest, i ϭ 1, to smallest i ϭ n, such
that the volume approaches zero as n → ϱ. In this
limit the largest dimension of the sample
approaches zero, so the volume converges to the
point P, not to a surface or a curve containing the
point (Malvern, 1969). Furthermore, the resultant
body force varies as one considers successive
volumes in the sequence and approaches zero in
the limit. The formal definition of the body force
per unit mass at the point P in the material continuum is:
(6.3)
Although both the numerator and the denominator in (6.3) approach zero, a fundamental postulate is that the ratio approaches a definite value
at each and every point in the continuum. The
difficulty encountered when rationalizing definitions such as this with our knowledge of the properties of real materials is resolved as follows:
When we pass, however, to the infinitesimal limit dV
(volume) or dS (surface), we are dealing with a hypothetical concept, a continuum or continuous medium,
whose justification depends not on any study of actual
materials in the small, but rather on the efficacy and
utility of the concept in enabling us to describe and
predict the behavior of actual materials in the large,
i.e., the macroscopic behavior (Malvern, 1969).
For rock one must be aware that application of the
material continuum may be problematic at the
grain scale and at the intermolecular scale where
sharp discontinuities break up an otherwise continuous body.
For most problems in structural geology
gravity is the only significant body force, so the
body force per unit mass is the gravitational acceleration, b ϭ g. Furthermore, the magnitude of
this body force, g, usually does not vary significantly over length scales from meters to kilometers, so it may be taken as uniform and equal to
the standard value g* ϭ 9.806 65 m s
Ϫ2 (Mohr and
b ϭ lim
n→ ϱ
␦f n
n ␦V n
Taylor, 2003). For a Cartesian coordinate system
with z-axis vertical and directed upward (Fig. 6.2b)
the components of the body force per unit mass
are:
(6.4)
For the rectangular body of volume ␦V shown in
Fig. 6.2b with side lengths ␦x ϭ x 2 Ϫx 1 , ␦y ϭ y 2 Ϫy 1 ,
and ␦z ϭ z 2 Ϫz 1 the resultant body force magnitude is:
(6.5)
Here density is taken as uniform so the calculation is greatly simplified.
The body force per unit volume is the vector
quantity g. Under conditions where both mass
density and the gravitational acceleration are
taken as known values, constant in time and
uniform in space, the components of the body
force per unit volume with coordinates as in Fig.
6.2 are:
(6.6)
The gravitational body force plays an important
role in many tectonic processes from the buoyant
rise of magma and salt through the crust to the
loading and unloading of buried rock masses
during mountain building and erosion.
6.1.2 Surface force: the traction vector
In order to discuss deformation in the Earth we
need a way of talking about and quantifying the
distribution of forces acting on an arbitrary
surface within a rock mass. For example, we
might ask what were the forces distributed on the
surfaces of a fault that would cause it to slip? Or,
what are the forces distributed on the surfaces of
a dike that would cause it to open? Questions like
these require us to understand and use the traction vector. The traction vector also is used to
define the distribution of forces (or lack thereof)
acting on any external surfaces of a rock mass. For
example, the Earth’s surface is characterized as
being a traction-free surface. Of course large buildings, dams, and other engineering structures
impose non-zero tractions and the filling of large
b x ϭ 0, b y ϭ 0, b z ϭ Ϫg*
b z Ύ
z 2
z 1
Ύ
y 2
y 1
Ύ
x 2
x 1
dx dy dz ϭ Ϫg*␦x␦y␦z ϭ Ϫg*␦V
b x ϭ 0, b y ϭ 0, b z ϭ Ϫg*
6.1 CONCEPTS OF FORCE AND TRACTION
197
For this purpose we consider a sequence of finite
volumes, ␦V i , each with resultant body force, ␦f i ,
and average density, i , and each containing the
point in question (Fig. 6.2b). The finite volumes are
ordered from largest, i ϭ 1, to smallest i ϭ n, such
that the volume approaches zero as n → ϱ. In this
limit the largest dimension of the sample
approaches zero, so the volume converges to the
point P, not to a surface or a curve containing the
point (Malvern, 1969). Furthermore, the resultant
body force varies as one considers successive
volumes in the sequence and approaches zero in
the limit. The formal definition of the body force
per unit mass at the point P in the material continuum is:
(6.3)
Although both the numerator and the denominator in (6.3) approach zero, a fundamental postulate is that the ratio approaches a definite value
at each and every point in the continuum. The
difficulty encountered when rationalizing definitions such as this with our knowledge of the properties of real materials is resolved as follows:
When we pass, however, to the infinitesimal limit dV
(volume) or dS (surface), we are dealing with a hypothetical concept, a continuum or continuous medium,
whose justification depends not on any study of actual
materials in the small, but rather on the efficacy and
utility of the concept in enabling us to describe and
predict the behavior of actual materials in the large,
i.e., the macroscopic behavior (Malvern, 1969).
For rock one must be aware that application of the
material continuum may be problematic at the
grain scale and at the intermolecular scale where
sharp discontinuities break up an otherwise continuous body.
For most problems in structural geology
gravity is the only significant body force, so the
body force per unit mass is the gravitational acceleration, b ϭ g. Furthermore, the magnitude of
this body force, g, usually does not vary significantly over length scales from meters to kilometers, so it may be taken as uniform and equal to
the standard value g* ϭ 9.806 65 m s
Ϫ2 (Mohr and
b ϭ lim
n→ ϱ
␦f n
n ␦V n
Taylor, 2003). For a Cartesian coordinate system
with z-axis vertical and directed upward (Fig. 6.2b)
the components of the body force per unit mass
are:
(6.4)
For the rectangular body of volume ␦V shown in
Fig. 6.2b with side lengths ␦x ϭ x 2 Ϫx 1 , ␦y ϭ y 2 Ϫy 1 ,
and ␦z ϭ z 2 Ϫz 1 the resultant body force magnitude is:
(6.5)
Here density is taken as uniform so the calculation is greatly simplified.
The body force per unit volume is the vector
quantity g. Under conditions where both mass
density and the gravitational acceleration are
taken as known values, constant in time and
uniform in space, the components of the body
force per unit volume with coordinates as in Fig.
6.2 are:
(6.6)
The gravitational body force plays an important
role in many tectonic processes from the buoyant
rise of magma and salt through the crust to the
loading and unloading of buried rock masses
during mountain building and erosion.
6.1.2 Surface force: the traction vector
In order to discuss deformation in the Earth we
need a way of talking about and quantifying the
distribution of forces acting on an arbitrary
surface within a rock mass. For example, we
might ask what were the forces distributed on the
surfaces of a fault that would cause it to slip? Or,
what are the forces distributed on the surfaces of
a dike that would cause it to open? Questions like
these require us to understand and use the traction vector. The traction vector also is used to
define the distribution of forces (or lack thereof)
acting on any external surfaces of a rock mass. For
example, the Earth’s surface is characterized as
being a traction-free surface. Of course large buildings, dams, and other engineering structures
impose non-zero tractions and the filling of large
b x ϭ 0, b y ϭ 0, b z ϭ Ϫg*
b z Ύ
z 2
z 1
Ύ
y 2
y 1
Ύ
x 2
x 1
dx dy dz ϭ Ϫg*␦x␦y␦z ϭ Ϫg*␦V
b x ϭ 0, b y ϭ 0, b z ϭ Ϫg*
6.1 CONCEPTS OF FORCE AND TRACTION
197