Week 8 Flashcards
Forming the Mohr Circle
- Equation of a circle with centre (p,0)
(x-p)^2 + y^2 = r^2
x = p + rcosα
y = rsinα
σn = (σ1+σ3)/2 + (σ1-σ3)/2 cos2θ = x
σs = τ = sin2θ(σ1+σ3)/2= y
Using the Mohr Circle diagram
Plane makes an angle at θ between σ1 and σ3
Draw 2θ from σn = work out σn and τ for any given plane (can also use the equations)
Value of σs for planes // to σ1 and σ3
=0
Plot on σn axis
In a triaxial experiment…
σ1 = vertical
σ3 = horizontal
What value of 2θ for τmax?
90
What planes plot onto the Mohr circle?
All other planes containing the σ2 axis
Non zero τ
Diameter of the Mohr circle =
differential stress
= σ1-σ3
Centre of the Mohr circle =
mean stress
= (σ1+σ3)/2
Radius of the Mohr circle =
τmax
Fractures =
planar displacement in displacement and mechanical properties, minerals broken = loss of cohesion
Shear fractures =
faults
Extension/tensile fractures =
joints/fissures/veins/dykes
What happens if pore fluid pressure > σ1 in tensile/mode I fractures?
Liquefaction
Shear fractures, facts
Conjugate pairs Opposed dipping Same displacement type Acute angle ~60' Active same time = mutually cross-cutting
Andersonian Faults
Assume horizontal Earth’s surface
τ=0
= principal stress plane so an axis must be perpendicular
σ1 bisects acute angle
Typical dip of faults (acute angle) in different Andersonian faults
- Reverse/thrust = 30’
- Normal = 60’
- Strike-slip = 90’
Stereonets; Where do the fault planes intersect?
At σ2
- principal stress axis in common
Stereonets; What is the pole to the σ1σ3 plane?
σ2
Stereonets; Which principal stress axis is 90’ from slickenline fibres?
σ2
Stereonets; Which principal stress axis bisects the angle between the fault planes?
σ1
Stereonets; Which principal stress axis do you get to if you count 90’ from σ1?
σ3
Stereonets; Which principal stress axis is ~vertical tells you the type of fault (lithostatic = perpendicular to the Earth’s surface due to g)…
σ1 = normal fault
σ2 = strike-slip fault
σ3 = thrust fault
Stereonets; What does the orientation of the slickenline tell you?
Where the fault planes intersect the σ1σ3 plane
φ =
angle of internal friction
= TOTAL angle between τmax planes and the fault planes (add up)
Controlled by characteristic ratio of σn:τf
How can you find τmax from σ1 or σ3?
Count 45’ from
Triaxial compressional test
σ3 = confining pressure = depth
Initial hydrostatic state (σ1=σ2=σ3)
Compressional stresses loaded = σ1
Triaxial compressional test; ultimate strength =
Differential stress at failure = σ1-σ3
Triaxial compressional test; shearing resistance =
shear stress at failure = radius = τmax
Triaxial compressional test; what happens when you increase confining pressure?
Increase ultimate strength (i.e. rock strength)
I.E STRENGTH OF BRITTLE CRUST INCREASES WITH DEPTH, ITS ABSOLUTE STRENGTH IS DUE TO LITHOLOGY
Triaxial compressional test; envelope of failure
Can draw Mohr circles
θ values ~constant = envelope of failure
- stable
- critical
- unstable
Planes within orientation have τ > that required to fail rock
Coulomb-Naiver failure criterion
τf = f(σn)
τf = S+μσn y = c + mx
τf = shearing resistance S = cohesion of solid/initial shear strength μ = coefficient of internal friction (tanφ) σn = σn on plane at failure
Stress state that leads to brittle failure in compressional conditions
When shear stress = shearing resistance = FRACTURE
Coulomb-Naiver; Typical values of φ
30-40’
Irrespective of rock type
Coulomb-Naiver; Typical values of μ
0.58-0.85
Irrespective of rock type
Coulomb-Naiver; Typical values of θ
50-60’
Irrespective of rock type
Coulomb-Navier; what is S?
Cohesion
= τf on a surface where σn = 0
Coulomb-Navier; where does tensile strength occur?
Where failure envelope intersects normal stress axis (σn = -ve)
Reality check with Mohr failure envelopes
- Envelope generally flattens towards higher differential stresses/confining pressures (approaching ductile regime)
- Assumes materials are isotropic i.e. S and miu don’t vary in different orientations
KEY CONTROL: orientation of anisotropy relative to principal stresses
Von Mises criterion
Failure at higher differential stresses/confining pressures occurs at constant τ and different stress on planes sub// to τmax planes
Plate motion-related stresses
Slab pull
Ridge push
Basal drag
Collisional resistance
Plate motion-related second order stresses
Sediment loading
Glacial loading/unloading
Areas of thin crust + mantle upwelling
Ocean-continent transitions
Orogenic belts
Large weak faults
