Transcript lecture5_frx1

Joints and Shear Fractures
Mohr stress diagram map rock failure.
The diagram illustrates normal and shear stresses, as well as the greatest and
least stresses.
After we test a numerous rocks at different confining pressures, we get a family
of failure values that define a failure envelope.
Creation of Joints & Shear
Fractures in the Lab
There are 2 basic types of rock strength
tests:
1) Tensile strength tests: specimen is
pulled along its axis (s3) with or without
confining pressure is applied to it’s
sides (s1 = s2) until failure, and;
2) Compressive strength tests: specimen
is compressed along its axis (s1) with
or without confining pressure is applied
to it’s sides (s2 = s3) until failure.
At failure, the values of the principal
stresses are noted and so is the
orientation of the plane of failure wrt
either s1 or s3.
These data are plotted in Mohr space.
A single experiment will produce
a single data point that
describes the normal and shear
stress (sn, ss) for the plane of
failure at the instant of failure.
A number of similar experiments
are carried out at different
confining pressures to create a
series of similar data points.
The location of these points
defines a failure envelope.
The envelope separates a
region of Mohr space where
rock is stable - in no danger of
failure - from a region outside
the envelope, where intact rock
cannot exist.
Each bam along the failure
envelope represents rock
failure (e.g., fracture) at
different differential stress.
So each Mohr circle towards
the right of the Mohr plot is a
larger circle.
Rx failure (fracture) at a
specified s3 and s1).
Tensile Strength Tests
 Rocks are typically very weak in tension. Rocks are
typically 2 to 30 times stronger in compression than in
tension.
Remember: in structural geology, tensile stresses are
negative (-) and compressive stresses are positive (+).
 We can visualize tensile failure in Mohr space, and get an
idea of what a tensile failure law might look like.
Tensile Strength Tests
First, compared with compressive tests, rocks are very weak in tension. The
ratios of strength in tension in unconfined compression is about 2:1, by may
exceed 30:1.
Break a Popsicle stick. As we bend it, tension in the outer arc of the bend
and compression in the inner arc. Weaker in tension, the stick snaps (fails)
along the outer arc.
 The state of stress just before the experiment is s1 = s2 = s3 = 0. This is
represented by a single point, where there is not differential stress.
 As tensile stresses build parallel to the length of the sample, differential
stress builds.
Increasing
tensional
stresses, with
increases of
circle
diameter
A beginning of experiment, no differential stress (e.g.,
hydrostatic state of stress).
Tensile failure simply occurs when the tensile strength
of the rock is exceeded, and the plane of failure is
perpendicular to the tensile stress (s3).
Tensile stresses builds up parallel to length of
sample. As differential stress increases, the
diameter of the Mohr circle increases.
Stress perpendicular to the axis of the rock core is
the default direction of s1.
During the test, since tensile stress is negative, it’s
the least principal stress (s3).
When tensile strength of the rock is exceeded, the rock
breaks perpendicular to the direction of tension (e.g., s3).
This is called a mode I fracture.
Tensile Strength Law:
s3 = To
A rock will fail by mode I fracturing if the magnitude of
least principal stress (s3) equals or exceeds the tensile
strength of the rock.
The mode I fracture is parallel to s1 and perpendicular to s3.
In Mohr space, the radius that connects the center of the differential
stress circle with the point of failure lies along the x-axis.
Tensile & Compressive Strength Tests
We can also run triaxial tests (with a small amount of
compressive confining pressure applied to the flanks of the
specimen while at the same time applying a tensile stress
along the axis.
Lets explore the
relations between
differential stress,
confining pressure,
and fracture strength
of a rock in
compression.
10 MPa
If we begin the experiment at confining pressure of 10
MPa. Then we increase the tensile stresses parallel
to the length of the specimen.
When tensile strength of the rock is exceeded, the
rock breaks perpendicular to the direction of tension
(e.g., s3).
A mode I fracture forms.
10 MPa
Here, increasing levels is represented by points (s1) moving to
the left of the origin along the normal stress axis. Ultimately,
the differential stress is sufficient to break the rock.
As the test goes on, the differential stress (s1 - s3)
increases (the diameter of the Mohr circle) until
failure occurs.
We begin the experiment at confining
pressure of 40 MPa.
If the confining pressures are in the range of
s1 = 3 to 5To (from 3 to 5 times the tensile
strength of the sandstone),
The failure envelope flatten slightly as passes
the shear stress axis and is parabolic (dark
line).
Conguate fractures form
under transitional tensile
behavior.
Failure under compressive stress
 At increasing confining
pressure (s3), we need
increased differential
stress (s1-s3).
 The increase of
differential stress is
shown by an change in
the Mohr circle diameter.
Coulomb's Law of Failure:
s c = s 0  tan  (s N )
Dynamic and mechanical models
developed by Coulomb (1773)
and Mohr (1900).
The law describes the height and
slope of a linear envelope of
failure for rocks in compression.
Where sc = so + sNtan
 = angle of internal friction
tan = coefficient of internal
friction
sc = critical shear stress required
for faulting
so = cohesive strength
sN = normal stress
These tests define a
failure envelope for a
particular rock.
 All of the normal and
shearing stresses
inside the envelop are
stable – no fractures
produced.
 All of the stresses on
or outside the
envelope will
producing fracturing
Relationship between stress
and fracturing
Relationship between stress
and fracturing
 When the Mohr
circle becomes
tangent to the
envelope, then the sc
at that point causes a
fracture.
 No fractures are
produced by any
other combination of
sc on the circle.
Coulomb's Law of Failure:
s c = s 0  tan  (s N )
The slope and straightness of
the envelope reveal that
compressive strength of a
rock increases linearly with
increasing confining
pressure.
The actual angle of slope is
called, the angle of internal
friction ().
The envelope is called the
Coulomb envelope.
A law that describes the
conditions under which a
rock will fail by shear
fracturing under
compressive stress
conditions.
 The point of failure on the
Coulomb envelope reveal
magnitudes of sN = 43 and
ss = 47.
 In terms of Coulomb Law
of failure, the shear stress
value of 47 MPa is the
critical shear stress (sc)
necessary for fracturing to
occur.
 Part of its magnitude is
cohesive strength (s0)
expressing in units of stress,
read directly off of the Mohr
y-intercept of the envelope
of failure.
The rest of critical shear stress
(sc) is the stress required to
overcome internal frictional
resistance to triggering
movement on the fracture.
This component is labeled:
sN tan or the coefficient of
internal friction.
This value is expressed in terms
of the normal stresses acting on
the fault plane and the angle of
internal friction, which is the
slope of the failure envelope
The cohesive strength (s0)
is a small part of critical
shear stress required for
shear fracture.
Most shear fractures form
when shear stresses on a
plane of failure reach a
level slightly over 50% of
the normal shear stresses
acting on the plane.
s c = s 0  tan  (s N )
Note change in slope
What happens with higher
confining pressures
At very high confining pressures,
Coulomb theory is not valid. With
increasing confining pressure, rocks
behave in a less brittle fashion.
This is apparent in our stress/strain
curves, where at higher confining
pressures there is a departure from
the linear relations between stress
and strain.
Analogous to stress/strain, the linear
Coulomb relations between fracture
strength and confining pressure
breaks down at higher confining
pressures – the rock becomes
weaker.
The von Mises criterion describes deformational
behavior above the brittle-ductile transition.
When the critical yield stress is surpassed, the rock will
fail by ductile shear along planes of maximum shear
stress, oriented at 45° to the greatest principal stress.
The straight-line envelope becomes a
concave downwards envelope of
lesser slope.
Measured values of
tensile strength,
cohesive strength,
and internal friction
for a few rock types.
Rock failure envelope
for a rock marked by low
tensile strength, low
cohesive strength, and
low internal angle of
friction.
Rock failure
envelope for a rock
marked by high
tensile strength,
high cohesive
strength, and high
internal angle of
friction.