traction vectors and note that stress component
arrows always are drawn in pairs. The normal
stress component yy is defined using a similar
argument.
The combination of the two tangential traction
components, t y (a) and t y (c), in Fig. 6.13a are used
along with the condition (6.30) to define xy as a
shear component of stress. This quantity is represented schematically as two arrows drawn parallel
to sides a and c, with lengths and directions consistent with the respective traction components
(Fig. 6.13c). The two subscripts are chosen because
this stress component is acting on two opposing
planes with normals parallel to the x-axis and it is
acting in an orientation parallel to the y-axis. The
arrows have open heads, but only half of the head
is drawn to distinguish the shear stress from the
normal stress. The shear stress component yx is
defined using a similar argument.
There is an apparent contradiction in algebraic signs between traction components and
stress components. For example, one of the arrows
for xx points in the positive x-direction and the
other arrow points in the negative x-direction (Fig.
6.13b). This stress component is made up of two
traction components that have opposite signs (Fig.
6.13a). On the other hand, we need to give a single
algebraic sign to the stress component. The traditional choice, motivated by considering the sign
conventions for the traction components pointing into the first quadrant on sides a and b (Fig.
6.13a), is to assign positive signs to stress components that correspond to these positive traction
components. This makes a tensile normal stress, represented by outward directed arrows, a positive
quantity, whereas a compressive normal stress, represented by inward directed arrows, is a negative
quantity. Similarly, the two shear stresses are positive as drawn in Fig. 6.13c. This is the convention
used in most of the physics and engineering literature. Because many of the concepts and analysis
methods we use in structural geology are taken
from that literature, this is an attractive choice
and we will use it throughout this book, unless
otherwise stated.
Most practitioners of soil and rock mechanics
(Jaeger and Cook, 1979), as well as many structural
geologists and geophysicists, use the opposite convention for signs of the stress components: compression is positive and tension is negative. Shear
stress components directed as shown in Fig. 6.13c
are given negative signs. This choice is motivated
by the fact that the normal stress typically is compressive at depth in the Earth, because it is related
to the weight of the overlying rock. A disincentive
for choosing this convention is the awkward
result that components of the displacement
vector, when related to the stress components, are
found to be positive in the negative coordinate
directions. This is at variance with standard practice for defining vector components. Students of
structural geology should be able to work with
both sign conventions, because both are used
throughout the relevant literature.
A simple procedure for remembering how to
draw all eight arrows representing the stress components in two dimensions is to consider the
outward normal vector to the plane under consideration. If that normal points in the positive
coordinate direction (as on sides a and b, Fig. 6.13),
a positive stress component arrow should point in
the positive coordinate direction. If that normal
points in the negative coordinate direction (sides
c and d), then a positive arrow should point in the
negative coordinate direction. This produces a set
of arrows consistent with the sign convention for
positive stress components (Means, 1976).
By extension of the reasoning behind the
definition of the Cartesian components of stress
in two dimensions (Fig. 6.13), there are nine
components of stress in three dimensions (Fig.
6.14), one normal component and two shear
components acting on each of the three pairs of
opposing sides of the cubical element. Arrows
representing these nine components are shown
acting on the visible sides of the element. These
components are drawn in their positive orientations and it is understood that arrows drawn in
the opposite directions on the opposing sides of
the element make up the hidden member of
each pair.
The nine components of stress are not independent. This is understood here by treating the
cubical element in Fig. 6.14 as finite in size with
side lengths ␦ x , ␦ y , and ␦ z ; considering the state
of stress to be homogeneous throughout; and
ignoring the effects of body forces. We imagine
cutting the element free of the surroundings and
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FORCE, TRACTION, AND STRESS
arrows always are drawn in pairs. The normal
stress component yy is defined using a similar
argument.
The combination of the two tangential traction
components, t y (a) and t y (c), in Fig. 6.13a are used
along with the condition (6.30) to define xy as a
shear component of stress. This quantity is represented schematically as two arrows drawn parallel
to sides a and c, with lengths and directions consistent with the respective traction components
(Fig. 6.13c). The two subscripts are chosen because
this stress component is acting on two opposing
planes with normals parallel to the x-axis and it is
acting in an orientation parallel to the y-axis. The
arrows have open heads, but only half of the head
is drawn to distinguish the shear stress from the
normal stress. The shear stress component yx is
defined using a similar argument.
There is an apparent contradiction in algebraic signs between traction components and
stress components. For example, one of the arrows
for xx points in the positive x-direction and the
other arrow points in the negative x-direction (Fig.
6.13b). This stress component is made up of two
traction components that have opposite signs (Fig.
6.13a). On the other hand, we need to give a single
algebraic sign to the stress component. The traditional choice, motivated by considering the sign
conventions for the traction components pointing into the first quadrant on sides a and b (Fig.
6.13a), is to assign positive signs to stress components that correspond to these positive traction
components. This makes a tensile normal stress, represented by outward directed arrows, a positive
quantity, whereas a compressive normal stress, represented by inward directed arrows, is a negative
quantity. Similarly, the two shear stresses are positive as drawn in Fig. 6.13c. This is the convention
used in most of the physics and engineering literature. Because many of the concepts and analysis
methods we use in structural geology are taken
from that literature, this is an attractive choice
and we will use it throughout this book, unless
otherwise stated.
Most practitioners of soil and rock mechanics
(Jaeger and Cook, 1979), as well as many structural
geologists and geophysicists, use the opposite convention for signs of the stress components: compression is positive and tension is negative. Shear
stress components directed as shown in Fig. 6.13c
are given negative signs. This choice is motivated
by the fact that the normal stress typically is compressive at depth in the Earth, because it is related
to the weight of the overlying rock. A disincentive
for choosing this convention is the awkward
result that components of the displacement
vector, when related to the stress components, are
found to be positive in the negative coordinate
directions. This is at variance with standard practice for defining vector components. Students of
structural geology should be able to work with
both sign conventions, because both are used
throughout the relevant literature.
A simple procedure for remembering how to
draw all eight arrows representing the stress components in two dimensions is to consider the
outward normal vector to the plane under consideration. If that normal points in the positive
coordinate direction (as on sides a and b, Fig. 6.13),
a positive stress component arrow should point in
the positive coordinate direction. If that normal
points in the negative coordinate direction (sides
c and d), then a positive arrow should point in the
negative coordinate direction. This produces a set
of arrows consistent with the sign convention for
positive stress components (Means, 1976).
By extension of the reasoning behind the
definition of the Cartesian components of stress
in two dimensions (Fig. 6.13), there are nine
components of stress in three dimensions (Fig.
6.14), one normal component and two shear
components acting on each of the three pairs of
opposing sides of the cubical element. Arrows
representing these nine components are shown
acting on the visible sides of the element. These
components are drawn in their positive orientations and it is understood that arrows drawn in
the opposite directions on the opposing sides of
the element make up the hidden member of
each pair.
The nine components of stress are not independent. This is understood here by treating the
cubical element in Fig. 6.14 as finite in size with
side lengths ␦ x , ␦ y , and ␦ z ; considering the state
of stress to be homogeneous throughout; and
ignoring the effects of body forces. We imagine
cutting the element free of the surroundings and
210
FORCE, TRACTION, AND STRESS