Geoscience Reference
In-Depth Information
respect to the coordinate system
x, y, z
, which will be prin-
cipal planes of stress, and a horizontal surface (
y, x
) of area
A
(Fig. 3.69a). As in the previous case (Fig. 3.68), we will
develop a 2D analysis projecting the prism in the plane
z, x
.
A traction
deviatoric stress, which gives the maximum value for the
shear stress since sin2
cannot be
1 in Equation 29
(Fig. 3.69e). Finally (
1
3
) is the differential stress,
which reflects the eccentricity of the stress ellipse or ellip-
soid. The bigger the differential stress the more eccentric
the ellipse or ellipsoid. This part of the stress tensor will be
responsible for distortions or changes in shape in the
objects.
will be acting over the surface
A
(Fig. 3.69b)
whose normal is inclined at an angle
with respect to the
principal stress
1
. As usual, the traction
can be resolved
into the normal (
) components.
To establish equilibrium conditions the forces have to
be balanced so the sum of all the components acting in
every direction is zero. The force
F
n
) and shear (
3.13.7
The Mohr circle
A
will be the force
pushing over the surface
A
, and can be resolved into the
normal force
F
N
(
F
N
N
A
) and the shear force
F
S
The Mohr circle, named after C. O. Mohr, is a useful tool
to graphically represent the state of stress at a point in 2D
or 3D; all possible values for normal and shear stresses are
confined by the circle. The Mohr circle allows us to visual-
ize and calculate quickly the normal and shear components
of a traction over any surface forming any angle
(
F
S
A
).
F
1
and
F
3
are the forces acting in the directions
of
3
. As before,
F
1
can be resolved into a vertical
F
1
Z
and a horizontal
F
1
X
component (for simplicity in writ-
ing the equations, we will use two subindexes in a force so
the first subindex marks the main force to which the com-
ponent belongs, and the second subindex the direction of
application of the force (for instance
F
1
X
is the force com-
ponent of
F
1
in the
x
, horizontal direction). Similarly,
F
3
can be resolved into the horizontal
F
3
X
and vertical
F
3
Z
components. Starting with the horizontal forces,
F
S
1
and
with
respect to the normal of
3
are known (Fig. 3.70). In other words, having established
the values of
1
, if the magnitudes of
1
and
3
we can choose any surface and read
directly from the diagram the values of the normal and
shear stress components, which is much quicker than cal-
culating the components from the respective equations or
extracting them from the stress tensor by geometric meas-
urements. This kind of diagram has many applications but
is particularly useful in fracture analysis (Section 4.14).
The Mohr diagram consists of a coordinate system, which
represents magnitudes of the shear stress (
1
and
F
3
X
,
both pushing the prism to the left has to be equal to
F
1
X
,
which pushes the prism to the right (Equation 20;
Fig. 3.69d), so that
F
S
F
3
X
. Similarly, the vertical
force component
F
N
, which is trying to move the prism
down, has to be equal to the two forces
F
1
Z
and
F
3
Z
which
try to move the prism up, so
F
N
F
1
X
F
3
Z
(Equation
21; Fig. 3.69c). The final conditions for the derivation
(Fig. 3.69e) has the main stresses and all the traction com-
ponents involved. All surfaces are as a function of
A
, so
that the surface perpendicular to
F
1
Z
) on the verti-
cal axis and normal stress (
n
) on the horizontal axis for
any traction acting over a surface. From the origin, at zero,
the right-hand-side of the
n
axis will represent positive or
compressive stresses, whereas the left side will be the neg-
ative or tensile normal stresses. Considering now the verti-
cal axis,
1
is
A
cos
and the sur-
face perpendicular to
. If the values of the ver-
tical and horizontal force components given in Equations
22-25 as a function of the corresponding tractions are
substituted in Equations 20 and 21 (in the form
F
3
is
A
sin
, the upper part from the origin will be positive,
left-handed, shear and the lower part will be negative,
right-handed, shear. In the general case, the state of stress
is represented by a circle centered at any point along the
horizontal axis, with coordinates (
A
;
for example, for
F
1Z
,
1
cos
is the vertical traction or
stress component and
A
cos
is the surface on which the
traction acts), we obtain Equation 27, for the normal (
0,
so the circle intercepts the axis at two points. The point at
the right end will be the biggest normal stress
1
3
)/2 and
n
)
and Equation 29 for the shear (
) traction components.
Considering the trigonometric identities specified in the
gray boxes in Fig. 3.69e and simplifying, we obtain the
fundamental stress equations for both components. In
the fundamental stress equations the term (
1
and the
point at the left side will be the smaller normal stress
3
.
As you can see, the Mohr circle provides a very intuitive
way to visualize the fundamental stress equations
(Fig. 3.70b). The diameter of the circle is the differential
stress (
1
3
)/2
represents the mean normal stress taking into account all
traction values and may be regarded as a circle in 2D or
a sphere in 3D. The mean normal stress represents a
value for the hydrostatic condition which will be responsi-
ble for changes in area or volume of rock bodies as com-
pression or tension is applied. The term (
1
3
). The larger the circle the bigger will be the
differential stress and the ability to cause distortions. The
center of the circle will have the value of the mean normal
stress
0). Any radius of the
circle will have the value of the deviatoric stress
(
n
(
1
3
)/2 (and
1
3
)/2 is the
1
3
)/2. Note that the segment defined by the
Search WWH ::
Custom Search