Geoscience Reference
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balanced. The equilibrium equation for the horizontal forces
(Equation 14; Fig. 3.68a) gives the balance between the force
x
A
which is trying to move the prism left and the force
yx
xy
,
yz
zy
. The matrix can be
defined also for a situation in which the axes are coincident
with the directions of the principal normal stresses
zx
xz
, and
1
1
,
2
,
A
cos
which is trying to push the prism right. From this
relation we extract cos
and
3
(Fig. 3.67b). In this particular case the principal
stresses are located in the principal diagonal and all the
shear components are zero, as the tractions are normal to
the corresponding surfaces.
We explained earlier that the stress tensor has the shape
of an ellipse in 2D and an ellipsoid in 3D. To derive the
equation of the stress ellipse we go back to a prism of rock
orientated with respect to a coordinate system
x, y, z
(Fig. 3.68). We will consider the projection of the prism in
the vertical plane
z, x
to analyze the stress conditions in
2D. The two mutually perpendicular faces of the prism
parallel to
z
and
x
will be principal planes of stress, so that
1
is perpendicular to the plane
z, y
(in 2D normal to the
direction
z
) and
1
. Following a similar
approach through Equation 15 (Fig. 3.68a), the vertical
forces are balanced and we extract sin
x
/
z
/
3
. The
trigonometric relations between the directions of
z
and
x
and the angle
are depicted in Fig.
3.68b. Since
sin
2
cos
2
1 we can substitute the values for cos
and
sin
obtained earlier (Equations 16 and 17) which yields the
equation of the stress ellipse (Equation 18; Fig. 3.68b),
which is centered at the origin of the coordinate system. The
major and minor axis of this ellipse are
3
respectively
and are orientated in the
x
- and
z
-directions. Any other radius
of the ellipse will be a traction
1
and
or a stress vector with a mag-
nitude between
1
and
3
. A 3D analysis will result in equa-
3
is perpendicular to the plane
xy
(nor-
mal to the direction of
x
). An inclined surface of area
A
forms an angle
tion 19 (Fig. 3.68b).
with respect to the plane
zy
(normal to
1
). Over the surface
A
the traction
will be resolved into
3.13.6
The fundamental stress equations
two components:
z
parallel to
z
and
x
, parallel to
x
as
depicted in Fig. 3.68.
The fundamental stress equations are derived from the
equations of equilibrium showing the stress balance over a
prism of rock, relating the values of the main stresses
3.13.5
Equations of equilibrium
1
and
3
and the normal (
n
) and shear (
) stress compo-
These have to be defined for the vertical and horizontal
forces acting upon the prism. It is important to remember
that at equilibrium, where no movement of the prism is
allowed, all force components in any given direction sum
to zero. Accordingly, in order to establish these conditions
for the prism in the state of equilibrium, forces acting over
A
in the principal stress directions
nents of a given traction (
) acting over a surface
A
, which
forms any angle
1
. These equations
give the values of the normal and shear components of a
traction for any plane knowing the principal stresses
with the normal to
1
and
3
and the angle
that we choose.
As an example, we consider (Fig. 3.69) a triangular
prism with two surfaces at right angles inclined with
1
and
3
have to be
(a)
(b)
y
s
z
u
z
z
Normal to
A
A
u
since:
x
z
s
1
s
x
sin
2
u
+ cos
2
u
= 1.0
sin
u
u
u
(14)
s
1
A
cos
u
=
s
x
A
;
s
x
= cos
s
x
2
+
s
z
2
=
1
cos
u
u
s
1
(18)
x
s
1
2
s
3
2
(15)
s
3
A
sin
u
=
s
z
A
;
s
z
= sin
cos
u
=
s
x
/
s
1
sin
u
=
s
z
/
s
3
u
s
3
s
3
x
s
x
2
+
s
z
2
+
s
y
2
=
1
(16)
cos
u
(17)
sin
u
(19)
s
1
2
s
3
2
s
3
2
Fig 3.68
(a) Forces at equilibrium for the derivation of the equation of an ellipse with major and minor axis
1
and
3
parallel to the
x
- and
x
and
z
related to the angle
z
-axis; (b) Trigonometric relations for the directions of
and after Pythagoras theorem, the resulting Equation 18
of the stress ellipse (for a 2D analysis). For a 3D analysis a similar approach can be applied resulting in Equation 19 of the stress ellipsoid.
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