Failure and Mohr's Circle

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Failure and Mohr's Circle
We 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 Lab

• There 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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Rock failure (fracture) at a specified s3 and s1.
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.
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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 Tests

• Again, compared with compressive tests, rocks are
  very weak in tension. The ratios of strength in
  tension in unconfined compression is about 21,
  by may exceed 301.
• 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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T0 is the tensile strength of the rock
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 diameter
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T0 is the tensile strength of the rock
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).
When tensile strength of the rock is exceeded,
the rock breaks perpendicular to the direction of
tension (e.g., s3). .
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Tensile Strength Law
s3 To
A rock will fail by fracturing if the magnitude
of least principal stress (s3) equals or exceeds
the tensile strength of the rock.
The 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 Tests We 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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T0 is the tensile strength of the rock
We 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).
10 MPa
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.
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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 friction tanf coefficient of
internal friction (slope of failure line) sc
critical shear stress required for faulting so
cohesive strength sN normal stress
y 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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Relationship between stress and fracturing

• 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.

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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).
Mohr Failure Envelope
The two directions of breaking shown are equally
likely. Conjugate fractures will form under
tension
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Von Mises criterion brittle to ductile
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 slope
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.
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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.