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.
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). Sometimes
confining pressure is applied to it’s
sides (s1 = s2). The test continues until
failure.
2) Compressive strength tests: specimen
is compressed along its axis (s1) with
or without confining pressure 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 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.
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 doesn’t matter. For
shear t, aka ss , the plot is
symmetrical, and for normal stress
sn , both standards are useful.
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.
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 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.
T0 is the tensile
strength of the rock
Increasing
tensional
stresses, with
increases of
circle
diameter
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).
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, 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).
.
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.
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.
T0 is the tensile
strength of the rock
We begin the experiment at a confining pressure of 10 MPa.
That’s 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.
As the test goes on, the differential stress (s1 - s3)
increases (the diameter of the Mohr circle) until
failure occurs.
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.
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 the linear envelope.
Describes failure of rocks in
compression.
Where sc = so + sNtan
 = angle of internal friction
tan = 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  is the slope
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
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. 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.
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 angle of envelope 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 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 yintercept 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 )
We begin the next experiment at a confining pressure
of 40 MPa.
Mohr Failure Envelope
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
tension
Von Mises criterion: brittle to ductile
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.