The
Frictional Regime
Processes in Structural Geology & Tectonics
Ben van der Pluijm
© WW Norton+Authors, unless noted otherwise
1/25/2016 10:08 AM
We Discuss …
The Frictional Regime
• Processes of Brittle
Deformation
• Tensile Cracking
• Axial Experiment
• Formation of Shear
Fractures
• Failure Criteria
• Faults and Stress
• Andersonian Theory
• Frictional Sliding
• Byerlee’s Law
• Stress and Sliding
• Role of Fluids
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Generalized Fault Model and Deformation Regimes
Pl/sd
T(oC)
Rock
Regime
San Andreas Fault, Carrizo
Plain
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Extra: Frictional and Plastic Deformation Regimes
Frictional regime where deformation is
localized and involves fractures.
•
•
•
normal stress and pressure dependent
temperature and strain insensitive
shear stress is function of normal stress
Plastic regime where deformation is
distributed and involves crystal flow.
•
•
•
normal stress and pressure insensitive
temperature and strain dependent
shear stress is function of temperature and
strain rate
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Brittle Deformation and Structures
Brittle deformation is permanent change in a solid material due to formation
of cracks and fractures and/or due to sliding on fractures once they formed.
Categories of brittle
deformation:
• Tensile cracking
• Shear rupture
• Frictional sliding
• Cataclastic flow
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Atomic View: Elasticity and Failure
Atomic
bonds are
elastic, like
springs
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Rock Strength Paradox
Strength Paradox:
Rocks allow only few % elastic strain
(e) before permanent deformation
(failure), which occurs at low
stresses.
s=E.e
(next)
= 5x1010 . 0.1 = 5x109 Pa
= 5x103 MPa
Thus, theoretical strength of rock is
1000’s MPa, but the observed tensile
strength is only 10’s MPa.
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Elasticity
Elastic Behavior:
s=E.e
E = Young’s Modulus
e = strain = (l-lo)/lo
(rubber band)
ss = G. g
G=shear modulus
g = shear strain
Other elastic parameters:
Bulk Modulus (K): K = s/DV;
Shear Modulus (Rigidity, G): G = ss/g
Poisson’s ratio: n = eperpendicular to s/eparallel
to s
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Tensile Cracking and Stress Concentration
Remote v. local stress: stress concentration, C, is: (2b/a) +1
Crack 1 x .02 mm: C = 100 ! Greater as cracks grow, so larger is weaker.
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Axial Experiments: Griffith Cracks
Extension
Compression
Effect of preexisting (or Griffith) cracks: preferred activation,
then longest cracks grow because of stress concentration.
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Crack Modes
Tensile cracks (Mode I)
Shear cracks (Mode II and III)
Shear cracks are not faults. When
they propagate, they rotate into
Mode I orientation (“wing cracks”)
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Formation of Shear Fractures
Shear fracture is
surface across which
rock loses continuity
when shear stress
parallel to surface is
sufficiently large.
Shear fractures are not
same as large tensile
cracks, as their
eventual orientation
shows.
(I, II)
Initial state
I, II
III
(III)
dilatancy
IV
Differential stress at
failure is called failure
stress (σf).
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Failure Criteria in Mohr Space
•
•
•
•
Tensile crack: Griffith criterion (A)
Shear fracture: Mohr-Coulomb criterion
(B-C)
Frictional-Plastic transition (D)
Plastic shear zone (E)
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Shear Failure Criteria 1 – Coulomb Criterion
Coulomb failure criterion:
ss = C + msn
σs is shear stress parallel to fracture
surface at failure
C is cohesion, a constant that specifies
shear stress necessary to cause
failure if normal stress across
potential fracture plane equals zero
σn is normal stress across shear
fracture at instant of failure
μ is a constant: coefficient of internal
friction
Fracture surfaces (two !) at ~30o to
s1 and ~60o to s3. Yet, ss is
maximum at 45o. Why?
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Why 30o instead of 45o Fracture Angle with s1?
Fracture surfaces at ~30o to s1 and ~60o
to s3. Yet, ss is maximum at 45o.
s1
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Shear Failure Criteria 2 – Tensile Stress
Parabolic failure envelope: steeper
near tensile field and shallower at
high sn (Mohr criterion)
Fractures toward 90o in tension,
fractures around 30o in
compression, relative to σ1
Combined:
Mohr-Coulomb failure criterion
failure envelope
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Shear Failure Criteria 3 – High Stress
At high confining stresses:
Von Mises criterion
Plastic deformation occurs at high
(normal) stress, which is shear
stress independent
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Composite failure envelope
Note the changing
angle of rupture with
respect to σ1
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Fault Types: Normal, Reverse, Lateral-slip Faults
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Predictions from Anderson’s Theory of Faulting
Earth solid-air contact is free surface, meaning no shear
stresses, so orthogonal principal stresses.
Andersonian predictions:
• High-angle normal faults
• Low-angle reverse faults
• Lateral-slip faults
Unexplained:
• Low-angle normal faults
• Non-conjugate fault systems
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Sliding on a Surface (Friction Criterion)
Frictional sliding refers to movement on a pre-existing surface when
shear parallel to surface exceeds sliding resistance.
Amonton’s Laws of Friction:
• Frictional force is function of
normal force.
• Frictional force is
independent of
(apparent) area of contact.
• Frictional force is (mostly)
independent of material used.
(17th C)
(15th C da Vinci experiments)
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Surface Roughness (Asperities)
The bumps and irregularities (roughness) that protrude on natural
surfaces are called asperities, which “anchor” surfaces.
Real v. Apparent
Area of Contact
Larger mass (larger F),
deeper penetration
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Surface Roughness
Sliding has to exceed
resistance (strength) of
asperity (+dilation).
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Frictional Sliding Criteria (Byerlee’s Law)
ss / sn = constant = m
= coefficient of friction
Byerlee’s Law depends on σn.
For σn < 200 MPa, the
best-fitting criterion is
σs = 0.85.σn.
For 200 MPa <σn< 2000 MPa,
the best-fitting criterion is
σs = 50 MPa + 0.6.σn.
m = 0.6-0.85, or ~0.75 (static
friction)
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Sliding on Existing Fractures or Create New Fractures ?
s3
s1
Instead of new failure surface (A), sliding on existing surfaces (gray
area). Preexisting surfaces from B to E will slide with decreasing
friction coefficients.
(Blair Dolomite experiment)
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Principal Stresses and Fault Types
= r.g.h
Normal Fault
= r.g.h
= r.g.h
Reverse Fault
Lateral-Slip Fault
ss
Differential stress for
faulting is increasingly
greater from normal, to
lateral slip to reverse
faulting.
Normal
Reverse
sn
ρ⋅g⋅h
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Stress Conditions for Sliding
Consider m = 0.75 (Friction Law)
s1 = s3 . 4
ss = m . sn
Normal fault:
Substitute for principal
stresses, fracture angle
and rearrange gives:
s1 = s3 .
[m2+1)1/2
+
m]2
s1 = (ρ⋅g⋅h), so
(ρ⋅g⋅h) = s3 x 4
(s1–s3) = (ρ⋅g⋅h) – 0.25(ρ⋅g⋅h) = 0.75(ρ⋅g⋅h)
Reverse fault:
s3 = (ρ⋅g⋅h)
s1 = (ρ⋅g⋅h) x 4
(s1–s3) = 4(ρ⋅g⋅h) - (ρ⋅g⋅h) = 3(ρ⋅g⋅h)
σd ≥ β (ρ⋅g⋅h)
β is 3, 1.2, and 0.75 for reverse, strike-slip,
and normal faulting
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Stress Conditions for Sliding
(s1–s3)
m = 0.75
σd ≥ β (ρ⋅g⋅h)
β is 3, 1.2, and 0.75 for
reverse, strike-slip, and
normal faulting
borehole measurement,
sliding on strike-slip fault,
but stable on reverse fault
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Fluids and Fracturing
Hydrostatic (fluid) pressure
Pf = ρ ⋅ g ⋅ h, where ρ is density
of water (1000 kg/m3), g is gravitational
constant (9.8 m/s2), and h is depth.
Lithostatic pressure
Pl = ρ ⋅ g ⋅ h, weight of overlying column of
rock (ρ = 2500–3000 kg/m3).
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Fluid Pressure and Effective Stress
“outward push”
σs = C + μ.(σn – Pf) [fracturing]
σs = μ.(σn – Pf) [sliding]
(σn – Pf) is commonly labeled
σn*, the effective stress.
(hydraulic fracturing or “fracking”)
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So, meffective = m (1 – Pf/sn)
meffective ≤ m
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Limiting Stress Conditions for Sliding, with Pore-fluid
(s1–s3)
σd ≥ β (ρ⋅g⋅h) . (1 – λ)
l = 0.9 (90% of lithostatic pressure)
β is 3, 1.2, and 0.75 for reverse, strike-slip,
and normal faulting (with m = 0.75)
λ = Pf/Pl , ratio of pore-fluid pressure and
lithostatic pressure
(λ ranges from ∼0.35 (1000/2750) for
hydrostatic fluid pressure to 1 for lithostatic
fluid pressure)
Increasing λ increases fracturing potential:
“fracking”, injection well EQs
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Extra: Friction Coefficient from Stress Measurement
KTB, Germany (9100m)
Calculate μeff from KTB deep borehole
measurements at strike-slip
conditions:
s1 = s3 . [m2+1)1/2 + m]2
meff = 0.6
s1 = s3 . 3.15
σd ≥ β (ρ⋅g⋅h)
So, crust critically stressed for strikeslip faulting
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