MohrCoulomb failure

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Mohr-Coulomb failure
Goal To understand relationship between stress,
brittle failure, and frictional faulting and to
use this relationship to predict rock behavior
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Stress review

• Stress Force/Area
• 3 principal vectors s1, s2, and s3 at right
  angles to each other
• s1 s2 s3
• s1 is the maximum principal stress direction, s2
  is the intermediate principal stress direction,
  and s3 is the minimum principal stress direction

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We also define

• Static stress as s1 s2 s3
• Lithostatic stress as static stress generated by
  mass of overlying rocks
• Differential stress (sd) as (s1 - s3)
• Confining pressure as s2 s3 for the conditions
  s1 gt s2 s3

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Shear stress and normal stress

• For any plane in a stress field defined by s1,
  s2, and s3 with strike parallel with s2

s1
?
s3
s3
s1
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• The stress is resolved into 2 components
• Shear stress (ss), acting parallel with the plane
• Normal stress (sn), acting perpendicular to the
  plane

s1
sn
?
ss
ss
s3
s3
sn
s1
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• Stress components are related by
• ss ½(s1 - s3)sin(2?)
• sn ½(s1 s3) - ½(s1 - s3)cos(2?)
• where ? angle between plane and s1

s1
sn
?
ss
ss
s3
s3
sn
s1
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Mohr diagram for stress

• Relationship between s1, s3, ss, and sn is
  plotted graphically in Cartesian coordinates

ss
sn
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• Mohr circle for stress circle with diameter sd
  plotted on mohr diagram
• Center on the sn-axis at point ½(s1 s3)

ss
sn
s1
s3
½(s1 s3)
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Finding ss, and sn

• Can use a Mohr circle to find ss, and sn for any
  plane

ss
sn
s1
s3
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• Plot a line from center to edge of circle at
  angle 2?-clockwise from sn-axis

ss
2?
sn
s1
s3
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• X- and y-coordinates of intersection of line and
  circle define ss and sn for the plane

(ss, sn) of plane
ss
sn
s1
s3
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Coulombs failure criterion

• Every homogeneous material has a characteristic
  failure envelope for brittle shear fracturing
• Combinations of ss and sn outside of the envelope
  result in fracture

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Determining failure envelope

• Experimental rock deformation

Holger Stunitz in the lab at Basel University
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The Coulomb envelope
ss
Shear Fracture
Stable
2?
Tensile Fracture
sn
s1
s3
Stable
Shear Fracture
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Coulomb law of failure
sc s0 tan(f)sn
ss
f
s0
sn
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sc s0 tan(f)sn

• Formula defines shear stress under which rocks
  will fracture
• sc critical shear stress ss at failure
• s0 cohesive strength ss when sn 0
• f angle of internal friction f 90 - 2?

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• For most rocks, angle of internal friction 30
• Therefore, ? at failure is also 30
• ss is greatest when ? 45

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Failure envelopes for different rocks
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Slip on pre-existing fractures

• Pre-existing fractures have no cohesive strength,
  s0 0
• Failure envelopes for pre-existing fractures
  derived experimentally

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Envelope of sliding friction
ss
ff angle of sliding friction
sn
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Byerlees law

• Describes frictional sliding envelope
• sc tan(ff)sn
• ff 40 for low confining pressures and 35
  for high confining pressures

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Byerlees law for different rock types
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Effect of pore-fluid pressure

• Pore fluid pressure (Pf) effectively lowers the
  stress in all directions
• The effective stresses (s1eff, s2eff, and s3eff)
  principal stresses - Pf
• s1eff s1 - Pf s2eff s2 - Pf s3eff
  s3 - Pf

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Stable stress conditions
ss
s1
sn
s3
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Increase in pore fluid pressure can drive
faulting!!
ss
s1eff
s1
sn
s3
s3eff