More on Mohr (and other related stuff) Pages , ,

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 σ 1 direction –On a Mohr circle measured COUNTERCLOCKWISE from σ 1 after being DOUBLED (remember 2θ)

Plane σ1σ1 Pole θ σ1σ1 σ3σ3 2θ2θ2θ2θ σNσN σSσS Shear stress on plane Normal stress on plane

σ1σ1 σ3σ3 σNσN σSσS σ1 + σ32σ1 + σ32 σ1 - σ32σ1 - σ32 σ 1 + σ 3 2 = MEAN STRESS or HYDROSTATIC STRESS (pg 120) σ 1 - σ 3 2 = DEVIATORIC STRESS (pg 120)

Hydrostatic (or mean) stress (page 120 Has NO shear stress component All principal stresses are equal (σ 1 = σ 2 = σ 3 ) Changes the volume (or density) of the body under stress As depth increases, the hydrostatic stress on rocks increases

σ1σ1 σ3σ3 σNσN σSσS σ1 + σ32σ1 + σ32 σ1 - σ32σ1 - σ32 Mean stress increases = CENTER of the Mohr Circle shifts towards right σ1/σ1/ σ3/σ3/ σ1/ + σ3/2σ1/ + σ3/2 σ1/- σ3/2σ1/- σ3/2

The size (or the diameter) of the Mohr circle depends on the difference between σ 1 and σ 3 This difference (σ 1 - σ 3 ) 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 σ 1 - σ 3 2

Increased mean stress SHAPE of the body remains the same SIZE changes

Increased DEVIATORIC stress SHAPE of the body changes SIZE remains the same

σ1σ1 σ3σ3 σNσN σSσS σ1 + σ32σ1 + σ32 σ1 - σ32σ1 - σ32 Deviatoric stress increases = RADIUS of the Mohr Circle increases σ1/σ1/ σ3/σ3/ σ1/- σ3/2σ1/- σ3/2

σ 1 ≠ 0 (“nonzero” value) σ 2 = σ 3 = 0 σNσN σSσS UNIAXIAL stress (pages ) = The magnitude of ONE principal stress is not zero (can be either positive or negative). The other two have zero magnitude σ 3 ≠ 0 (“nonzero” value) σ 1 = σ 2 = 0σNσN σSσS - σ N Uniaxial compressive Uniaxial tensile

AXIAL stress (pages ) 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:  σ 1 >σ 2 = σ 3 ≠ 0, or  σ 1 =σ 2 > σ 3 ≠ 0, for both compression and tension

σ1σ1 σ2=σ3σ2=σ3 σNσN σSσS - σ N σ1σ1 σ2=σ3σ2=σ3 Axial compressiveAxial tensile σ3σ3 σ1=σ2σ1=σ2 σNσN σSσS - σ N σ3σ3 σ1=σ2σ1=σ2

The MOST common stress field is TRIAXIAL (page 121) σ 1 >σ 2 > σ 3 ≠ 0 (either compressional or tensile) σ1σ1 σ3σ3 σNσN σSσS σ2σ2

Stress and brittle failure: Why bother?Why bother? The dynamic Coulomb stresses transmitted by seismic wave propagation for the M= earthquake on the North Anatolian fault.

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.

Rock failure: experimental results (pages ) 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

σNσN σSσS 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 σ 3 = the tensile strength of the rock (T 0 ) σ1σ1 σ1σ1 σ3σ3 σ3σ3 Mode I fracture Fracture opens σNσN σSσS - σ N σ 3 =T 0 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 0 Φ m = SLOPE of the line = tan Φ

σNσN σSσS Equation of the Coulomb Failure envelope (pages ) is: σ c = (tan Φ)σ N +σ 0 (equation 5.3, page 234) σ0σ0 Φ σ c = Critical shear stress required for failure (faulting) σcσc σ 0 = Cohesive strength

σNσN σSσS Zooming in the failure envelope… Φ 2θ2θ θ = angle between σ 1 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