9.5 Fracture propagation and fault
growth
Criteria for failure in a homogeneous stress
state, such as the Coulomb criterion (9.50),
inform us about the limiting stress conditions
that are obtained at failure, but do not explicitly
include the structure (e.g. a shear fracture) that
is associated with the process of failure. Criteria
for failure in a heterogeneous stress state, such
as the Griffith criteria, (9.69) and (9.70), include
the stress concentrating structure and describe
the initiation of cracking, but do not address the
propagation of the crack or other possible events
and mechanisms that may be involved in the
evolution of structures such as joints and faults.
By combining solutions to elastic boundary value
problems with principles of fracture mechanics
one can explore the processes of fracturing and
faulting in rock from the initiation stage through
a stage of propagation or development to the
eventual cessation of tectonic activity as the
structure attains the size and configuration we
observe in exposure today.
The evolution of structures in brittle rock is a
large topic with interesting examples that are too
numerous to be described in detail here. Instead
we focus on a few examples that are meant to
provide a summary of the methodology of investigation and some insight concerning the kinds of
results one might expect. The first example is the
propagation of opening fractures such as joints,
veins, and dikes. Earlier in this chapter we
described the loading conditions necessary for
propagation in terms of the stress intensity at the
fracture tip reaching the fracture toughness of
the rock (9.35). Here we show how the loading conditions determine the path that the fracture
follows as it propagates. The second example is the
growth of faults in granite and in sandstone. In
the granite it is the propagation of opening fractures to link adjacent sheared joints that enables
the faults to grow in length. In the sandstone it is
the clustering of deformation bands that enables
the fault to grow in thickness. Through these
examples of fault growth we make the point that
faulting is a process that can involve several different physical mechanisms.
9.5.1 Propagation of joints, veins, and
dikes
Fractures are idealized as two surfaces with
mirror image geometry that are in contact in
the initial unloaded state and are bounded in
extent by a common curve called the tipline.
Sufficiently close to the tipline the shape of this
curve is approximately straight and the surfaces
are approximately planar. We adopt a Cartesian
coordinate system with the y-axis normal to the
plane of the surfaces and the z-axis parallel to the
tipline (Fig. 9.30). Upon loading of the elastic
body the fracture surfaces move relative to one
another and this motion may be classified
according to the coordinate directions we have
chosen (Kanninen and Popelar, 1985). Relative
motion in the y-coordinate direction is referred
to as Mode I or the opening mode; relative motion
in the x-coordinate direction is mode II or the
sliding mode; and relative motion in the z-coordinate direction is mode III or the tearing mode.
Mode I is associated with geologic structures
such as joints, veins, and dikes, whereas modes II
and III are associated with shear fractures and
faults. Modes II and III both involve a shearing
motion of the surfaces with the former being perpendicular to the tipline and the latter being parallel to the tipline. This classification may appear
arbitrary, but because of the symmetry of the
fracture tip the elastic stress fields near the tip
are uniquely distinguished by these modes, and
each mode is associated with a different style of
propagation.
There are many solutions in the literature for
linear elastic problems that involve fractures (Sih,
1973; Tada et al., 1973). A common feature of these
is that the stress components in the vicinity of the
fracture tip are distributed in a way that depends
largely upon the fracture mode (Irwin, 1957;
Williams, 1957). In contrast, the magnitudes of
the stress components depend upon the fracture
geometry away from the tipline and the loading
conditions. This interesting and perhaps nonintuitive fact comes about because the near-tip
stress distributions are dominated by the local
geometry of the fracture tips, which are taken as
identical for all modes, and by the relative motion
of the fracture surfaces, which are uniquely distinguished by the modes (Fig. 9.30).
9.5 FRACTURE PROPAGATION AND FAULT GROWTH
371
growth
Criteria for failure in a homogeneous stress
state, such as the Coulomb criterion (9.50),
inform us about the limiting stress conditions
that are obtained at failure, but do not explicitly
include the structure (e.g. a shear fracture) that
is associated with the process of failure. Criteria
for failure in a heterogeneous stress state, such
as the Griffith criteria, (9.69) and (9.70), include
the stress concentrating structure and describe
the initiation of cracking, but do not address the
propagation of the crack or other possible events
and mechanisms that may be involved in the
evolution of structures such as joints and faults.
By combining solutions to elastic boundary value
problems with principles of fracture mechanics
one can explore the processes of fracturing and
faulting in rock from the initiation stage through
a stage of propagation or development to the
eventual cessation of tectonic activity as the
structure attains the size and configuration we
observe in exposure today.
The evolution of structures in brittle rock is a
large topic with interesting examples that are too
numerous to be described in detail here. Instead
we focus on a few examples that are meant to
provide a summary of the methodology of investigation and some insight concerning the kinds of
results one might expect. The first example is the
propagation of opening fractures such as joints,
veins, and dikes. Earlier in this chapter we
described the loading conditions necessary for
propagation in terms of the stress intensity at the
fracture tip reaching the fracture toughness of
the rock (9.35). Here we show how the loading conditions determine the path that the fracture
follows as it propagates. The second example is the
growth of faults in granite and in sandstone. In
the granite it is the propagation of opening fractures to link adjacent sheared joints that enables
the faults to grow in length. In the sandstone it is
the clustering of deformation bands that enables
the fault to grow in thickness. Through these
examples of fault growth we make the point that
faulting is a process that can involve several different physical mechanisms.
9.5.1 Propagation of joints, veins, and
dikes
Fractures are idealized as two surfaces with
mirror image geometry that are in contact in
the initial unloaded state and are bounded in
extent by a common curve called the tipline.
Sufficiently close to the tipline the shape of this
curve is approximately straight and the surfaces
are approximately planar. We adopt a Cartesian
coordinate system with the y-axis normal to the
plane of the surfaces and the z-axis parallel to the
tipline (Fig. 9.30). Upon loading of the elastic
body the fracture surfaces move relative to one
another and this motion may be classified
according to the coordinate directions we have
chosen (Kanninen and Popelar, 1985). Relative
motion in the y-coordinate direction is referred
to as Mode I or the opening mode; relative motion
in the x-coordinate direction is mode II or the
sliding mode; and relative motion in the z-coordinate direction is mode III or the tearing mode.
Mode I is associated with geologic structures
such as joints, veins, and dikes, whereas modes II
and III are associated with shear fractures and
faults. Modes II and III both involve a shearing
motion of the surfaces with the former being perpendicular to the tipline and the latter being parallel to the tipline. This classification may appear
arbitrary, but because of the symmetry of the
fracture tip the elastic stress fields near the tip
are uniquely distinguished by these modes, and
each mode is associated with a different style of
propagation.
There are many solutions in the literature for
linear elastic problems that involve fractures (Sih,
1973; Tada et al., 1973). A common feature of these
is that the stress components in the vicinity of the
fracture tip are distributed in a way that depends
largely upon the fracture mode (Irwin, 1957;
Williams, 1957). In contrast, the magnitudes of
the stress components depend upon the fracture
geometry away from the tipline and the loading
conditions. This interesting and perhaps nonintuitive fact comes about because the near-tip
stress distributions are dominated by the local
geometry of the fracture tips, which are taken as
identical for all modes, and by the relative motion
of the fracture surfaces, which are uniquely distinguished by the modes (Fig. 9.30).
9.5 FRACTURE PROPAGATION AND FAULT GROWTH
371