More on Mohr and other related stuff

1
More on Mohr (and other related stuff)

• Pages 120-122, 227-245, 304-307

2
A note on ?

• From this point onwards, we will use ? to mean
• The angle between the POLE of the plane on which
  the stresses are acting, and the s1 direction
• On a Mohr circle measured COUNTERCLOCKWISE from
  s1 after being DOUBLED (remember 2?)

3
s1
?
2?
Pole
4
DEVIATORIC STRESS (pg 120)
sS
s1
s3
sN
MEAN STRESS or HYDROSTATIC STRESS (pg 120)
5
Hydrostatic (or mean) stress (page 120

• Has NO shear stress component
• All principal stresses are equal (s1 s2 s3)
• Changes the volume (or density) of the body under
  stress
• As depth increases, the hydrostatic stress on
  rocks increases

6
sS
s1
s3
sN
Mean stress increases CENTER of the Mohr Circle
shifts towards right
7

• The size (or the diameter) of the Mohr circle
  depends on the difference between s1 and s3
• This difference (s1 - s3) is called DIFFERENTIAL
  stress (page 120)
• This difference controls how much DISTORTION is
  produced on a body under stress
• The radius of the Mohr circle is known as
  DEVIATORIC stress

8
SHAPE of the body remains the same SIZE changes
Increased mean stress
9
SHAPE of the body changes SIZE remains the same
Increased DEVIATORIC stress
10
sS
s1
s3
sN
s1/
s3/
Deviatoric stress increases RADIUS of the Mohr
Circle increases
11
UNIAXIAL stress (pages 120-121) The magnitude
of ONE principal stress is not zero (can be
either positive or negative). The other two have
zero magnitude
Uniaxial tensile
Uniaxial compressive
12
AXIAL stress (pages 120-121)

• NONE of the three principal stresses have a zero
  magnitude (all have a nonzero value)
• Two out of three principal stresses have equal
  magnitude
• So axial stress states can be
• s1 gts2 s3 ? 0, or
• s1 s2 gt s3 ? 0, for both compression and tension

13
Axial compressive
Axial tensile
sS
sN
- sN
14

• The MOST common stress field is TRIAXIAL (page
  121)
• s1 gts2 gt s3 ? 0 (either compressional or tensile)

s1
s3
s2
15
Stress and brittle failure Why bother?

• The dynamic Coulomb stresses transmitted by
  seismic wave propagation for the M7.2 1944
  earthquake on the North Anatolian fault.

http//quake.wr.usgs.gov/research/deformation/mode
ling/animations/
16
Stress and brittle failure Why bother?
This computer simulation depicts the movement of
a deep-seated "slump" type landslide in San Mateo
County. Beginning a few days after the 1997 New
Year's storm, the slump opened a large fissure on
the uphill scarp and created a bulge at the
downhill toe. As movement continued at an average
rate of a few feet per day, the uphill side
dropped further, broke through a retaining wall,
and created a deep depression. At the same time
the toe slipped out across the road. Over 250,000
tons of rock and soil moved in this landslide.
http//elnino.usgs.gov/landslides-sfbay/photos.htm
l
17
Rock failure experimental results (pages 227-238)

• Experiments are conducted under different
  differential stress and mean stress conditions
• Mohr circles are constructed for each stress
  state
• Rocks are stressed until they break (brittle
  failure) under each stress state

18

• The normal and shear stress values of brittle
  failure for the rock is recorded (POINT OF
  FAILURE, page 227)

sS
sN
After a series of tests, the points of failures
are joined together to define a FAILURE ENVELOPE
(fig. 5.34, 5.40)
19

• Rocks are REALLY weak under tensile stress
• Mode I fractures (i.e. joints) develop when s3
  the tensile strength of the rock (T0)

s1
Mode I fracture
s3
s3
Fracture opens
s1
20
Back to the failure envelope

• Under compressive stress, the envelope is LINEAR
• Equation of a line in x y coordinate system can
  be expressed as
• y mx c

y
m SLOPE of the line tan F
c intercept on y-axis when x is 0
F
x
21
Equation of the Coulomb Failure envelope (pages
233-234) is
sc (tan F)sN s0 (equation 5.3, page 234)
s0 Cohesive strength
sc Critical shear stress required for failure
(faulting)
sS
sc
s0
F
sN
22
Zooming in the failure envelope
? angle between s1 and POLE of the fracture
plane
F Angle of internal friction 2? - 90º (page
235)
tan F coefficient of internal friction
sS
90º
2?
180-2?
F
sN
180-2?F 90 180