procedure chiefly involves considerations of geometry, although assumptions must be made as to the
initial forms of the objects. However, as the above
quotation suggests, no reference is made to the
underlying physical process that resulted in the
deformation. We begin this chapter by working
through a few examples. Interpretation of the
examples involves many simplifying assumptions
that set aside complicating factors, which we
return to after developing the other parts of a complete mechanics in subsequent chapters.
5.1 Rock deformation: some
observations and a simple
description
5.1.1 Deformed belemnite
The closely lined segments of the stretched belemnite in Fig. 5.1a were initially connected to form an
intact fossil (Badoux, 1963). First quartz and later
calcite was precipitated into the gaps between the
segments as stretching occurred. Measuring the
final span occupied by the segments, l, and initial
length, l 0 , yields a measure of the elongation of a
material line, the belemnite or its centerline. An
assumption is that the segments did not stretch:
the belemnite fractured into pieces that behaved as
rigid objects embedded in a deformable medium,
the rock containing it. The final length, measured
from the figure, is l ϭ 153/60", measured with a
scale marked off in 60ths of an inch, and the sum
of the segment lengths is l 0 ϭ 103/60". The ratio l/l 0
ϭ 1.48 termed the stretch of a material line is a
dimensionless measure of the elongation. Thus,
any units may be used for the measurements. The
use of fossil belemnites to estimate strain goes back
at least to the late-nineteenth century (Cloos, 1946;
Hossain, 1979).
The motivation for study of the belemnite is not
for its sake alone, but for what it might tell us
about the deformation of the rock containing it.
The belemnite lies in the bedding plane and is
inclined at an angle ␣ to cleavage, represented by
the parallel line segments in the figure. Much
research has shown that the plane of cleavage is
normal to the direction in the rock that has undergone the greatest shortening, and contains the
direction in which the rock has undergone the
greatest extension (Wood and Oertel, 1980). A
belemnite oriented parallel to the cleavage would
show the maximum value of the ratio l/l 0 , providing a measure of the bulk rock deformation in that
direction.
Estimation of the maximum stretch from the
information available at the exposure, requires
other assumptions. One is that the stretch of the
belemnite was equal to that of any other linear
material line in the rock with the same orientation, for example, one lying along the line “l” on
the figure, or one much farther from the belemnite. This assumption implies homogeneity of deformation within some volume of rock containing
the belemnite. In detail, the belemnite itself has
not undergone a homogeneous deformation, but
has broken into pieces that have separated by
means of a process involving precipitation of
material into the gaps. The belemnite might have
been more resistant to extension than the surrounding rock, and thus its stretch would underestimate that of the rock.
We have introduced the notion of homogeneous deformation in a simple way. As we show
later in this chapter the technical definition of a
deformation is the transformation that relates
the positions of particles in a body in one state to
that in another. Here we speak of these two states
as the initial state and the final state. A particle is
an element of mass within the rock of small
dimensions relative to the length scale of interest.
Here, the only length scale is the length of the
belemnite, so “small” is taken relative to that. The
position of a particle is given by its position vector
between a coordinate origin and the particle. In
this more formal context homogeneity is defined
as a linear transformation between positions in the
initial and final states.
The intent here is to estimate, from the elongation of the belemnite and the angle between it and
cleavage, the maximum elongation in the plane of
cleavage. We also assume, for this example, that
the plane of cleavage is normal to the bed containing the belemnite. The data given provide no basis
for this assumption, but allow us to consider only
the plane of particles making up the bedding
surface and thereby reduce the problem to two
dimensions.
154
DEFORMATION AND FLOW
initial forms of the objects. However, as the above
quotation suggests, no reference is made to the
underlying physical process that resulted in the
deformation. We begin this chapter by working
through a few examples. Interpretation of the
examples involves many simplifying assumptions
that set aside complicating factors, which we
return to after developing the other parts of a complete mechanics in subsequent chapters.
5.1 Rock deformation: some
observations and a simple
description
5.1.1 Deformed belemnite
The closely lined segments of the stretched belemnite in Fig. 5.1a were initially connected to form an
intact fossil (Badoux, 1963). First quartz and later
calcite was precipitated into the gaps between the
segments as stretching occurred. Measuring the
final span occupied by the segments, l, and initial
length, l 0 , yields a measure of the elongation of a
material line, the belemnite or its centerline. An
assumption is that the segments did not stretch:
the belemnite fractured into pieces that behaved as
rigid objects embedded in a deformable medium,
the rock containing it. The final length, measured
from the figure, is l ϭ 153/60", measured with a
scale marked off in 60ths of an inch, and the sum
of the segment lengths is l 0 ϭ 103/60". The ratio l/l 0
ϭ 1.48 termed the stretch of a material line is a
dimensionless measure of the elongation. Thus,
any units may be used for the measurements. The
use of fossil belemnites to estimate strain goes back
at least to the late-nineteenth century (Cloos, 1946;
Hossain, 1979).
The motivation for study of the belemnite is not
for its sake alone, but for what it might tell us
about the deformation of the rock containing it.
The belemnite lies in the bedding plane and is
inclined at an angle ␣ to cleavage, represented by
the parallel line segments in the figure. Much
research has shown that the plane of cleavage is
normal to the direction in the rock that has undergone the greatest shortening, and contains the
direction in which the rock has undergone the
greatest extension (Wood and Oertel, 1980). A
belemnite oriented parallel to the cleavage would
show the maximum value of the ratio l/l 0 , providing a measure of the bulk rock deformation in that
direction.
Estimation of the maximum stretch from the
information available at the exposure, requires
other assumptions. One is that the stretch of the
belemnite was equal to that of any other linear
material line in the rock with the same orientation, for example, one lying along the line “l” on
the figure, or one much farther from the belemnite. This assumption implies homogeneity of deformation within some volume of rock containing
the belemnite. In detail, the belemnite itself has
not undergone a homogeneous deformation, but
has broken into pieces that have separated by
means of a process involving precipitation of
material into the gaps. The belemnite might have
been more resistant to extension than the surrounding rock, and thus its stretch would underestimate that of the rock.
We have introduced the notion of homogeneous deformation in a simple way. As we show
later in this chapter the technical definition of a
deformation is the transformation that relates
the positions of particles in a body in one state to
that in another. Here we speak of these two states
as the initial state and the final state. A particle is
an element of mass within the rock of small
dimensions relative to the length scale of interest.
Here, the only length scale is the length of the
belemnite, so “small” is taken relative to that. The
position of a particle is given by its position vector
between a coordinate origin and the particle. In
this more formal context homogeneity is defined
as a linear transformation between positions in the
initial and final states.
The intent here is to estimate, from the elongation of the belemnite and the angle between it and
cleavage, the maximum elongation in the plane of
cleavage. We also assume, for this example, that
the plane of cleavage is normal to the bed containing the belemnite. The data given provide no basis
for this assumption, but allow us to consider only
the plane of particles making up the bedding
surface and thereby reduce the problem to two
dimensions.
154
DEFORMATION AND FLOW