More on Mohr (and other related stuff)

Pages 120-122, 227-245, 304-307

A note on θ

From this point onwards, we will use θ to mean:

– The angle between the POLE of the plane on which the stresses are acting, and the σ1 direction

– On a Mohr circle measured COUNTERCLOCKWISE from σ1 after being DOUBLED (remember 2θ)

Plane

σ1

Pole

θθ

σ1σ3

22θθ

σN

σS

Shear stress

on plane

Normal stress on plane

σ1σ3 σN

σS

σ1 + σ3

2

σ 1 - σ

3

2

σ1 + σ3

2

= MEAN STRESS or HYDROSTATIC STRESS (pg 120)

σ1 - σ3

2

= DEVIATORIC STRESS (pg 120)

Hydrostatic (or mean) stress (page 120

• Has NO shear stress component

• All principal stresses are equal (σ1= σ2= σ3)

• Changes the volume (or density) of the body under stress

• As depth increases, the hydrostatic stress on rocks increases

σ1σ3 σN

σS

σ1 + σ3

2

σ 1 - σ

3

2

Mean stress increases = CENTER of the Mohr Circle shifts towards right

σ1/σ3

/ σ1/ + σ3

/

2

σ 1/ - σ

3/

2

• The size (or the diameter) of the Mohr circle depends on the difference between σ1 and σ3

• This difference (σ1 - σ3) 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

σ1 - σ3

2

Increased mean stress

SHAPE of the body remains the sameSIZE changes

Increased DEVIATORIC stress

SHAPE of the body changesSIZE remains the same

σ1σ3 σN

σS

σ1 + σ3

2σ 1

- σ3

2

Deviatoric stress increases = RADIUS of the Mohr Circle increases

σ1/σ3

/

σ 1/ - σ

3/

2

σ1 ≠ 0 (“nonzero” value)

σ2 = σ3 = 0 σN

σS

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

σ3 ≠ 0 (“nonzero” value)

σ1 = σ2 = 0 σN

σS

- σNUniaxial compressive

Uniaxial tensile

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:σ1 >σ2 = σ3 ≠ 0, or

σ1 =σ2 > σ3 ≠ 0, for both compression and tension

σ1σ2=σ3σN

σS

- σN σ1σ2=σ3

Axial compressiveAxial tensile

σ3 σ1=σ2σN

σS

- σN σ3 σ1=σ2

The MOST common stress field is TRIAXIAL (page 121)

σ1 >σ2 > σ3 ≠ 0 (either compressional or tensile)

σ1σ3 σN

σS

σ2

Stress and brittle failure: Why bother?

The dynamic Coulomb stresses transmitted by seismic wave propagation for the M=7.2 1944 earthquake on the North Anatolian fault.

http://quake.wr.usgs.gov/research/deformation/modeling/animations/

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.html

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

σN

σS

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

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

• Rocks are REALLY weak under tensile stress

• Mode I fractures (i.e. joints) develop when σ3 = the tensile strength of the rock (T0)

σ1

σ1

σ3 σ3

Mode I fracture

Fracture opens

σN

σS

- σN σ3 =T0 σ1 = σ2 = 0

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

x

y

c = intercept on y-axis when x is 0Φ

m = SLOPE of the line = tan Φ

σN

σS

Equation of the Coulomb Failure envelope (pages 233-234) is:

σc = (tan Φ)σN +σ0 (equation 5.3, page 234)

σ0Φ

σc = Critical shear stress required for failure (faulting)

σc

σ0 = Cohesive strength

σN

σS

Zooming in the failure envelope…

Φ

2θ

θ = angle between σ1 and POLE of the fracture plane

90º

Φ = Angle of internal friction = 2θ - 90º (page 235)

180-2θ

180-2θ+Φ+ 90 = 180

tan Φ = coefficient of internal friction