with such vertical cleavage, and with evidence for
inter-layer slip, we might then conclude that our
model is appropriate. If the cleavage were not vertical – except within the hinge region where symmetry requires it – another kinematic model must
be concocted. We do not claim that geometric,
evolutionary, and kinematic models are the final
goal of our study, but they do provide a means of
organizing observations and sorting out hypotheses as to folding mechanism.
5.4 Velocity fields: the
instantaneous state of motion
We have described the production of the limb of a
chevron fold in terms of the homogeneous deformation, as described by the transformation from
initial to final coordinates of the particles in the
body through relations of the form (5.45). The
quantities F xx , F xy , F yx , and F yy are the components
of a second-order tensor, the deformation gradient tensor. In the models so far developed, the
coefficients have been obtained for two simple
kinds of deformation. Then, we obtained the
coefficients for a sequence of two deformations in
succession, one of each kind. This was consistent
with two observed features of chevron folds: interlayer slip and cleavage. However, this composite
model seems artificial. It suggests a process in
which slip surfaces are initiated between rigid
layers and a fold forms in this manner up to a
certain dip, at which point a completely different
mechanism of folding sets in. The model describes
neither the initiation of slip nor its cessation. The
opposite sequence might also be considered. This
composite process is plausible if the two mechanisms operated during episodes separated in
time, in which the conditions were markedly different. For example, lower temperature and pressure and the presence of fluids might have been
associated with an episode of inter-layer slip and
higher temperature and pressure with an episode
of homogeneous deformation.
Since a parameter such as ␦ changes continuously, the description obtained might be interpreted not only as the description of an initial and
final state, but of the continuous progression of
states between them. The term progressive deformation is often used by structural geologists to denote
such a progression, whether or not it can be
described in detail (Ramsay and Huber, 1983, 1987).
That is, any rock mass may be viewed as having
undergone a progressive deformation, the end
result of which is the suite of structures and
deformed objects that we see in the exposure. On
the other hand, the term progressive deformation
does not appear in the literature or texts of continuum mechanics, which also deal with the deformation of materials, including rocks. This term
seems to be the special invention of structural
geologists. In the context of our discussion, the
description of a progressive deformation would
appear to consist of a specification of the four
coefficients F xx (t), F xy (t), F yx (t), and F yy (t) as functions
of time.
Consider the rate of change in the positions of
particles in the body. This description refers
explicitly to the physical variable time t, and thus
leads toward a consideration of the physical
processes responsible for folding. Taking the
derivatives of the expressions in (5.45) with
respect to time:
(5.48)
Since X and Y are the initial coordinates of the particle, we do not operate on these. But the rates of
change of the current coordinates of the particles
are their velocity components, or:
(5.49)
The velocity field at any time will depend on the
current conditions, properties, and forces
applied to the body. Such a description is generally developed in terms of position and time in
the body, so that in the two-dimensional case
under consideration:
(5.50)
v y ϭ v y (x, y, t, material properties, applied forces)
v x ϭ v x (x, y, t, material properties, applied forces)
dx
dt
ϭ v x ,
dy
dt
ϭ v y
dy
dt
ϭ
dFyx
dt
X ϩ
dFyy
dt
Y
dx
dt
ϭ
dF xx
dt
X ϩ
dFxy
dt
Y
5.4 VELOCITY FIELDS: THE INSTANTANEOUS STATE OF MOTION
177
inter-layer slip, we might then conclude that our
model is appropriate. If the cleavage were not vertical – except within the hinge region where symmetry requires it – another kinematic model must
be concocted. We do not claim that geometric,
evolutionary, and kinematic models are the final
goal of our study, but they do provide a means of
organizing observations and sorting out hypotheses as to folding mechanism.
5.4 Velocity fields: the
instantaneous state of motion
We have described the production of the limb of a
chevron fold in terms of the homogeneous deformation, as described by the transformation from
initial to final coordinates of the particles in the
body through relations of the form (5.45). The
quantities F xx , F xy , F yx , and F yy are the components
of a second-order tensor, the deformation gradient tensor. In the models so far developed, the
coefficients have been obtained for two simple
kinds of deformation. Then, we obtained the
coefficients for a sequence of two deformations in
succession, one of each kind. This was consistent
with two observed features of chevron folds: interlayer slip and cleavage. However, this composite
model seems artificial. It suggests a process in
which slip surfaces are initiated between rigid
layers and a fold forms in this manner up to a
certain dip, at which point a completely different
mechanism of folding sets in. The model describes
neither the initiation of slip nor its cessation. The
opposite sequence might also be considered. This
composite process is plausible if the two mechanisms operated during episodes separated in
time, in which the conditions were markedly different. For example, lower temperature and pressure and the presence of fluids might have been
associated with an episode of inter-layer slip and
higher temperature and pressure with an episode
of homogeneous deformation.
Since a parameter such as ␦ changes continuously, the description obtained might be interpreted not only as the description of an initial and
final state, but of the continuous progression of
states between them. The term progressive deformation is often used by structural geologists to denote
such a progression, whether or not it can be
described in detail (Ramsay and Huber, 1983, 1987).
That is, any rock mass may be viewed as having
undergone a progressive deformation, the end
result of which is the suite of structures and
deformed objects that we see in the exposure. On
the other hand, the term progressive deformation
does not appear in the literature or texts of continuum mechanics, which also deal with the deformation of materials, including rocks. This term
seems to be the special invention of structural
geologists. In the context of our discussion, the
description of a progressive deformation would
appear to consist of a specification of the four
coefficients F xx (t), F xy (t), F yx (t), and F yy (t) as functions
of time.
Consider the rate of change in the positions of
particles in the body. This description refers
explicitly to the physical variable time t, and thus
leads toward a consideration of the physical
processes responsible for folding. Taking the
derivatives of the expressions in (5.45) with
respect to time:
(5.48)
Since X and Y are the initial coordinates of the particle, we do not operate on these. But the rates of
change of the current coordinates of the particles
are their velocity components, or:
(5.49)
The velocity field at any time will depend on the
current conditions, properties, and forces
applied to the body. Such a description is generally developed in terms of position and time in
the body, so that in the two-dimensional case
under consideration:
(5.50)
v y ϭ v y (x, y, t, material properties, applied forces)
v x ϭ v x (x, y, t, material properties, applied forces)
dx
dt
ϭ v x ,
dy
dt
ϭ v y
dy
dt
ϭ
dFyx
dt
X ϩ
dFyy
dt
Y
dx
dt
ϭ
dF xx
dt
X ϩ
dFxy
dt
Y
5.4 VELOCITY FIELDS: THE INSTANTANEOUS STATE OF MOTION
177