Structural Geology
Brittle Deformation 2
Lecture 13 – Spring 2016
1
Critical Remote Tensile Stress
• Using thermodynamics and elasticity
theory, W.A. Griffith derived the equation:
 σt = [2Eγ/π(1-ν2)c]½
 where
•
•
•
•
•
σt = critical remote tensile stress
E = Young’s modulus
γ = energy used to create a new crack surface
ν = Poisson’s ratio
c = half-length of the preexisting crack
2
Mode I Crack Criterion
• Engineering studies on linear elastic fracture
mechanics allowed other criteria to be developed
• For Mode I cracks:
• KI = σtY(πc)½
• where
 KI is called the stress intensity factor
 Y is a dimensionless number related to crack
geometry
3
Fracture Toughness
• KI increases as the remote tensile stress increases,
with cracks beginning to grow when KI reaches
the critical stress intensity factor, or Kic
• This is also known as the fracture toughness, and
is a constant for a given material
• The equation can be rewritten in terms of σ:
 σt = Kic/(Y(πc)½)
4
Coulomb’s Criterion
• σs = C + μσn
 where
•
•
•
•
σs = shear stress parallel to the fracture surface
C = cohesion of the rock
μ = coefficient of internal friction and
σn = normal stress across the shear plane
• C is a constant that specifies the shear stress
necessary to cause failure if the normal
stress is zero
5
Mohr Diagram
• Each experiment plots
as a circle
• The further away from
the origin the circle
center is, the larger is
the radius of the circle
• Figure 6.15, text
6
Coulomb Failure Criterion
• Drawing a tangent to
each circle, we find it
intersects the vertical
axis at C, since this is
where σn = 0
• The slope of the line is μ
(= tan φ)
• The straight line thus
represents the Coulomb
failure criterion
7
Shear Stress
Maximum
• Plot of both normal
and shear stress versus
angle α, where α is the
angle between the
shear plane and σ1
• Shear stress reaches a
maximum at α = 45º
• Figure 6.16, text
8
Shear-Normal
Minimum
• However, at α = 45º the
normal stress is still quite
high
• The difference between the
shear and normal stresses
reaches a minimum around α
= 60º, which is equivalent to
a shear plane at a thirty
degree angle to σ1
9
Conjugate Shear Fractures
• Conjugate fractures
always have opposite
shear senses, one leftlateral and one rightlateral
• Figure 6.17, text
10
Dual Tangency Points
• The two fractures occur
at an angle of about 60 º,
and correspond to the
tangency points of the
circle representing the
stress state at failure with
the Coulomb failure
envelope
11
Mohr-Coulomb Criterion for
Shear Fracturing
• Like the Coulomb
Failure Criterion, this
is also an empirical
criterion
• Figure 6.18, text
12
Failure Envelope
• Shaded Area is the
failure envelope
• Figure 6.19a in text
13
Stable Stress State
• Any stress state lying
within the envelope is
stable
• Figure 6.19b in text
14
Defining Stress State
• Stress state tangent to
the envelope defines
the failure state
• Figure 6.19c in text
15
Impossible Stress State
• Any stress state whose circle
lies outside the envelope is
an unstable stress state, and
is not physically possible
• Before stress reaches this
state, the sample would have
failed
• Figure 6.19d in text
16
Yield Strength Independent of
Differential Stress
• Pair of lines parallel to σn,
indicating that the yield
strength is independent of
the differential stress, once
the yield stress is equaled
• Figure 6.20a in text
• If tensile strength is sufficient, the sample fails by cracking
through the entire sample
17
Transitional-Tensile Regime
• Tensile strength is
represented as a range
because it depends on
the size of flaws in the
sample
• Figure 6.20b in text
18
Composite Failure Envelope
• Diagram shows
composite failure
envelope
• Figure 6.21a in text
19
Confining Pressure Effects
• The effect of
increasing confining
pressure
• Figure 6.21b in text
20
Frictional Sliding
• Frictional force does
not depend on the
shape of the object
• Both objects, of the
same mass, have the
same sliding force,
despite having
different areas of
contact
21
Amonton’s Law
• Frictional resistance to sliding  normal stress
component across the surface
• First “published” account this empirical law of
friction was made by the French physicist
Guillaume Amonton in 1699, although Leonardo
da Vinci’s notes indicate he knew of the result
about 200 years earlier
• If normal stress increases, the asperities are
pushed more deeply into the opposing surface, and
increasing resistance to sliding
22
Fracture Surface
• Fracture surface,
showing voids and
asperities (Figure
6.23a, text)
• As another, also
bumpy, surface tries to
slide over the first
surface, their asperities
interact, causing
friction
23
Real Area of
Contact
• The bumps mean that
only a small part of
the surfaces are
actually in contact
• Dark areas are real
area of contact (RAC)
(Figure 6.23c, text)
24
Surface Anchors
• The forces normal to these surfaces will be
concentrated on the small areas in contact
• Asperities cumulatively act as small anchors, retarding
any slippage along the surface (Figure 6.23b, text) 25
Criteria for Frictional Sliding
• Before the initiation of frictional sliding, enough
shear must be present to overcome friction
• We can define a criterion for frictional sliding to
represent the necessary shear
• Experimental work has shown that, independent of
rock type, the following criterion holds
 σs/σn = constant
26
Byerlee’s Law
• For σn < 200 MPa,
 σs = 0.85 σn
• For 200MPa < σn <
5000MPa
 σs = 50MPa + 0.6 σn
• Figure 6.24 in text
27
Blair Dolomite
• Data for the Blair dolomite, showing both a frictional sliding
line with φ = 40º and σs = 45 + σntan 45º
• For values of θ between 15º < θ < 75º, the preexisting
fracture would slide first
• Figure 6.25a in text
28
Low θ Values
• At very low values of θ, the shear stress is very low,
and sliding is less likely so initiation of a new fracture
is more likely
29
High θ
Values
• If θ is very high, the normal stress component across
the existing fracture pins the fracture, preventing
movement, so initiation of a new fracture is again
more likely
30
Relationship Between New and
Existing Fracture Planes
• Line B represents line for Coulomb
shear fracture initiation in an intact rock
• Surface A is a Coulomb shear fracture
that would form in an intact rock, prior
to slippage along B
• B through E are surfaces that would slip
with decreasing friction coefficients
• Figure 6.25b in text
31
Fluid Pressure
• Fluid pressures are hydrostatic, and are
defined by the relation we have previously
encountered several times,
 Pf = ρgh
32
Pore Pressure
• For water, ρ = 1000 kg/m3, g = 9.8 msec-2,
and h is the depth
• When permeability is restricted, pore
pressure may exceed the hydrostatic
pressure, known as overpressurization
• This happens when the pore spaces are not
interconnected, so that hydrostatic pressure
beings to approach lithostatic pressure
33
Lithostatic vs. Hydrostatic
Gradients
• Many rock have densities
in the 2500-3000 kg/m3
range, so hydrostatic
pressure may approach
the weight of the
overlying rock
• Figure compares the
different gradients
• Figure 6.26 in text
34
Collapse Sinkhole Formation 1
• No evidence of land subsidence, small- to mediumsized cavities in the rock matrix
• Water from surface percolates through to rock, and
the erosion process begins
35
Collapse Sinkhole Formation 2
• Cavities in limestone continue to grow larger
• Missing confining layer allows more water to flow
through to the rock matrix
• Roof of the cavern is thinner, weaker
36
Collapse Sinkhole Formation 3
• Groundwater levels drop during the dry season
• Weight of the overburden exceeds the strength of the
cavern roof
• Overburden collapses into the cavern, forming a
sinkhole
37
Winter Park Sinkhole
• Collapse sinkholes,
such as this one in
Winter Park, Florida
(1981), may develop
abruptly (over a period
of hours) and cause
catastrophic damage
• Photo: USGS
38
Collapse into Old Mineshaft
Photo by S. Jerrod
Smith, graduate student
• Old mineshafts can also
cause collapse
• Photo shows collapse
into an abandoned mine
along Tar Creek in
Oklahoma
• This is a Superfund
cleanup site in
Oklahoma
39
Effect of Hydration
• A) Si-O bonds in dry
sample are strong
• B) Si-OH bonds in wet
sample are weaker,
and easier to break
• Figure 6.28 in text
40
Pore Pressure and Shear
Fracturing
• Pore pressure also plays a role on shear
fracturing
• Since pore pressure counteracts the
confining pressure, we can rewrite the
Coulomb Failure Criterion equation for
shear stress to take pore pressure into
account:
 σs = C + μ(σn - Pf)
41
Movement of Stress Along σ3 Axis
• When represented on a
Mohr diagram, the
Mohr circle moves to
the left along the
normal stress axis as
Pf increases, where is
may encounter the
failure envelope
• Figure 6.27 in text
42
Rocky Mountain Arsenal
• In 1967, an earthquake of magnitude 5.5
followed a series of smaller earthquakes
• Injection had been discontinued at the site
in the previous year once the link between
the fluid injection and the earlier series of
earthquakes was established
43
Reservoir Induced Seismicity (RIS)
• Reservoirs can also trigger seismicity
• It is repeatedly observed how seismic activity in
both space and time closely follows the changes in
the reservoir level
• The reservoir increases water pressure by the
height of the reservoir water column
• RIS occurrence was associated with about 120
reservoirs at the International Symposium on RIS,
held in Beijing China (November 1995)
44
Importance of Fracturing in Geology
• Fracturing affects many processes in geology - among
these are:
 Permeability of rocks
 Strength of rocks
 Resistance of a rock to erosion
 Slope stability
 Suitability of an area to serve as a reservoir
 Safety of mine shafts
 The location of dams
 Velocity and direction of subsurface fluid
flow, including toxic waste
45
Other Important Effects
• Brittle fracture is the underlying cause of
most earthquakes
• It also contributes to the formation of
regional tectonic features
46