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Failure and Mohr's CircleWe use a Mohr stress diagram to map the
failure of rocks under stress, by plotting both normal and shear
stresses, as well as the greatest and least stresses on the Mohr
circle. After we test numerous rocks at different confining
pressures, we get a family of failure values that define a failure
envelope.
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Creation of Joints & Shear Fractures in the LabThere are 2
basic types of rock strength tests:Tensile strength tests: specimen
is pulled along its axis (s3). Sometimes confining pressure is
applied to its sides (s1 = s2). The test continues until
failure.Compressive strength tests: specimen is compressed along
its axis (s1) with or without confining pressure applied to its
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.
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A single experiment will produce a circle that describes the
normal and shear stress (sn, ss) for the plane of failure q 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 defines a region of Mohr space where rock is stable -
in no danger of failure. Outside the envelope the rock fails.
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Each red star along the failure envelope represents rock failure
(e.g., fracture) at different differential stress. A larger Mohr
circle represents a greater difference between the largest s1 and
smallest s3 stress. In Geology, tensile stresses are negative.
Rocks are weakest under tension, which plots on the left of zero
for a Geology standard.But it really doesnt matter. For shear t,
aka ss , the plot is symmetrical, and for normal stress sn , both
standards are useful.Rock failure (fracture) at a specified s3 and
s1.
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Tensile Strength Tests Rocks are typically very weak in tension.
Rocks are typically 2 to 30 times stronger in compression than in
tension.In geology (say working for USGS) , tensile stresses are
negative (-) and compressive stresses are positive (+).In
engineering, (say working for a mining or an environmental company)
tensile stresses are positive (+) and compressive stresses are
negative (-). We can visualize tensile failure in Mohr space using
the geology convention, and get an idea of what a tensile failure
law might look like.
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Tensile Strength TestsAgain, 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 pencil. As we bend it, tension occurs in the outer arc
of the bend and compression in the inner arc. Weaker in tension,
the pencil snaps (fails) along the outer arc. DEMO: foam pyroxene
strand, discuss stress concentration
The state of stress before the experiment starts is s1 = s2 = s3
= 0. This is represented by a single point, where there is no
differential stress.
As tensile stresses build parallel to the length of the sample,
differential stress builds.
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At the beginning of the experiment, no differential stress
(e.g., hydrostatic state of stress).Tensile failure simply occurs
when the tensile strength of the rock is exceeded. The plane of
failure is perpendicular to the tensile stress (s3).Increasing
tensional stresses, with increases of circle diameterT0 is the
tensile strength of the rock
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When tensile strength of the rock is exceeded, the rock breaks
perpendicular to the direction of tension (e.g., s3). .Tensile
stresses build 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 for the
geology standard, its the least principal stress (s3).T0 is the
tensile strength of the rock
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Tensile Strength Law:A rock will fail by fracturing if the
magnitude of least principal stress (s3) equals or exceeds the
tensile strength of the rock. s3 = ToThe 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.
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Tensile & Compressive Strength TestsWe can also run triaxial
tests (with 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 and tension, say buried at a divergent margin.
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10 MPaWe begin the experiment at a confining pressure of 10 MPa.
Thats the compressive part. 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). Here, increasing levels of tension
are represented by points (s3) moving further to the left of the
origin along the normal stress axis. In other words, bigger
negative stresses plot further to the left of zero.
Ultimately, the differential stress is sufficient to break the
rock.T0 is the tensile strength of the rock
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As the test goes on, the differential stress (s1 - s3) increases
(the diameter of the Mohr circle) until failure occurs.
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Failure under compressive stress At increasing confining
pressure, we need increased differential stress (s1-s3) for
failure.
The increase of differential stress is shown by an change in the
Mohr circle diameter.
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Coulomb's Law of Failure:Dynamic and mechanical models developed
by Coulomb (1773) and Mohr (1900).
The law describes the height and slope of the linear envelope.
Describes failure of rocks in compression.
Where sc = so + sNtanf
f = angle of internal frictiontanf = coefficient of internal
friction (slope of failure line)sc = critical shear stress required
for faultingso = cohesive strengthsN = normal stressy = b + ax
notice tan f is the slope
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Relationship between stress and fracturing These tests define a
failure envelope for a particular rock.
All of the normal and shearing stresses inside the envelope are
stable no fractures produced.All of the stresses on or outside the
envelope will producing fracturing
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When the Mohr circle becomes tangent to the envelope, then the
sc at that point causes a fracture. 2q there gives the failure q,
and the point gives the sn and t at failure No fractures are
produced by any other combination of sc on the circle.Relationship
between stress and fracturing
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Coulomb's Law of Failure:The slope and straightness of the
envelope reveal that compressive strength of a rock increases
linearly with increasing confining pressure.
The angle of envelope slope is called, the angle of internal
friction (f).
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.
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The point of failure on the Coulomb envelope reveals magnitudes
of sN = 43 MPa and t = ss = 47 MPa.
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.
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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 tanf 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
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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.
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We begin the next experiment at a 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 will flatten slightly as it passes the shear stress axis,
and the failure envelope becomes parabolic (dark line).The two
directions of breaking shown are equally likely. Conjugate
fractures will form under tensionMohr Failure Envelope
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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 straight-line envelope becomes a concave downwards envelope
of lesser slope.Note change in slopeThe 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.Von Mises criterion: brittle to
ductile
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Measured values of tensile strength, cohesive strength, and
internal friction for a few rock types.
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Rock failure envelope for a rock marked by low tensile strength,
low cohesive strength, and low internal angle of friction.
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Rock failure envelope for a rock marked by high tensile
strength, high cohesive strength, and high internal angle of
friction.
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