1. Brittle Deformation 2 Lecture 13 – Spring 2016
Structural Geology
Brittle Deformation 2
Lecture 13 – Spring 2016
Prediction of Initiation of Brittle Deformation
An understanding of the conditions necessary for brittle deformation is important to both geologists and engineers, although for somewhat different reasons. Geologists need to know when, where, and why brittle structures, such as joints, faults, veins, and dikes, form. Engineers must estimate the magnitude of stress a building or bridge can withstand before failure. Brittle deformation involves three phenomena:
Tensile Crack Growth
Shear Fracture Development
Frictional Sliding
Conditions for Tensile Crack Growth
W.A. Griffith extended his ideas about mesoscopic cracks developing from preexisting flaws to a mathematical predication of the conditions necessary for tensile crack formation. He modeled a system in which an elastic plate has an elliptical crack.

2. 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

3. 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
This equation assumes that all cracks have high ellipticity
Cracks with lower ellipticity require higher stresses

4. 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)
Since the remote stress depends on fracture toughness, the crack shape, and the length of the crack, if other factors are equal, long cracks propagate before short cracks. Because c increases as the crack starts to grow, crack propagation usually leads to failure of the sample.
Similar equations can be written for Mode II and III cracks. Comparison of the equations shows that if other factors are equal, Mode I cracks, which are perpendicular to σ3, propagate first. In real materials, crack shape and length may allow Mode II or III cracks to propagate first, but they either turn into Mode I cracks or develop as wing cracks.
Criteria for shear fracture
Shear rupture cracking inevitably leads to sample failure, so the criteria for shear rupture are called shear fracture criteria. Charles Coulomb, French naturalist, proposed that if all principal stresses are compressive, such as in the confined compression case, materials fail by formation of a shear fracture, with shear stress parallel to the fracture surface, at the time of failure, related to normal stress

5. 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
The term “coefficient of internal friction” came from studies of friction between grains of sand in unconsolidated samples.
The name has nothing to do with structural geology, and μ is therefore a proportionality constant.
The above equation if called Coulomb’s criterion in his honor.

6. 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
On a Mohr diagram for stress, the Coulomb criterion plots as a straight line. Repeated experiments on cylinders of rock in which we use different confining pressures (σ2 = σ3) and increase the axial load (σ1) until failure occurs can be plotted on a Mohr diagram.

7. 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
Lines drawn from the center of any circle to the Coulomb criterion define the angle 2θ, where θ is the angle between σ1 and the shear fracture plane. This angle is usually about 30̊.

8. 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

9. 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
Coulomb’s criterion is based strictly on experimental evidence, and is thus a type of empirical relation. It does not use knowledge of atomic or crystal scale failure mechanisms. It is not related to physical parameters. It does not define the state at which microcracks begin to propagate. Since the Coulomb criterion does not specify which way the fractures that form dip with respect to the axis of the rock cylinder, it is possible for conjugate shear fractures to develop.

10. Conjugate Shear Fractures
Conjugate fractures always have opposite shear senses, one left-lateral and one right-lateral
Figure 6.17, text

11. 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
Otto Mohr studied the Coulomb criteria and concluded that the straight-line relationship was correct only over a limited range of confining pressures. When he extended studies to lower and higher confining pressures, he found the failure envelope actually represented part of a parabola.

12. Mohr-Coulomb Criterion for Shear Fracturing
Like the Coulomb Failure Criterion, this is also an empirical criterion
Figure 6.18, text

13. Failure Envelope Shaded Area is the failure envelope
Figure 6.19a in text

14. Stable Stress State
Any stress state lying within the envelope is stable
Figure 6.19b in text

15. Defining Stress State
Stress state tangent to the envelope defines the failure state
Figure 6.19c in text

16. 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
Can we extend the concept of the failure envelope to other conditions of confining pressure? If the confining pressure is very high, the sample will very likely fail by ductile flow. Since ductile flow is not brittle failure, the brittle failure envelope clearly does not apply. However, a criterion known as the Von Mises criterion applies to the yield strength for ductile flow.

17. 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

18. 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
The remaining, steeply sloping part of the figure may represent what some geologists call transitional-tensile fractures or hybrid shear fractures. They are supposed to represent fractures somewhere between tensile cracks and shear ruptures. They have not been seen to occur in nature, and laboratory experiments have failed to show these conditions. Many people doubt their existence.
Using the Tensile failure, Mohr-Coulomb, brittle-plastic, and Von Mises criteria, it is possible to construct a composite failure envelope over a wide range of conditions, even when one of the principal stresses is tensile.

19. Composite Failure Envelope
Diagram shows composite failure envelope
Figure 6.21a in text

20. Confining Pressure Effects
The effect of increasing confining pressure
Figure 6.21b in text

21. 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
When shear stress parallel to an inclined surface exceeds the frictional resistance to sliding, frictional sliding results.
Experiments such as those shown in figure 6.22 have led to a series of observations, sometimes known as Amonton’s laws of friction

22. 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
It is also important to distinguish static friction, associated with the initial movement of an object, and dynamic friction, associated with an already moving object. Frictional sliding requires that shear stress parallel to the fault exceed frictional resistance. All surfaces, no matter how polished they may appear, have bumps and depressions.
Amonton reference:
Da Vinci reference:

23. 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
Images:

24. 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)

25. Surface Anchors
In order for movement to occur, asperities must either break off or plow a groove in the opposite surface. The force required to cause either of these to happen is dependent on the real area of contact. As the real area of contact increases, so does the required force.
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)

26. 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
This means the criterion for frictional sliding plots as a straight line on the Mohr diagram.
J. Byerlee proposed two equations in 1978 to fit available experimental data under different conditions of normal stress.

27. Byerlee’s Law For σn < 200 MPa, For 200MPa < σn < 5000MPa
σs = 0.85 σn
For 200MPa < σn < 5000MPa
σs = 50MPa σn
Figure 6.24 in text
Figure 6.24 shows a plot of Byerlee’s law, as it is now known. The coefficient relating normal and shear stress is known as the friction coefficient.
Failure envelopes also allow us to determine whether an existing shear rupture is more likely to slip, or a new shear rupture to form.

28. 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
Failure envelopes allow the determination of whether an existing shear rupture will slip first in a sample, or whether it is more likely that a new shear rupture will form
For B through E, the Mohr circle representing the stress state at failure touches the frictional envelope before it touches the tracture envelope – this means that frictional sliding on a pre-existing fracture will precede the formation the formation of a new fracture. Only for A does the Mohr circle touch the Coulomb envelope, initiating a new fracture.

29. 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
As normal stress increases, the asperities are pushed more deeply into the opposing surface, and increasing resistance to sliding.

30. 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

31. 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
Brittle deformation is most commonly encountered in the upper kilometers of the earth, entirely within the crust.
Within this region, if strain rates are especially low or rocks particularly weak, ductile flow is possible.
Below 15 kilometers, ductile flow is the dominant failure mechanism. Very high strain rates or fluid pressures can cause brittle deformation to occur at greater depths in some cases. Fluid pressures play an especially important role.

32. Fluid Pressure
Fluid pressures are hydrostatic, and are defined by the relation we have previously encountered several times,
Pf = ρgh

33. 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

34. Lithostatic vs. Hydrostatic Gradients
Many rock have densities in the kg/m3 range, so hydrostatic pressure may approach the weight of the overlying rock
Figure compares the different gradients
Figure 6.26 in text
Fluids press outward, opposing confining pressure, and partially supporting the load on the rock.
The ability of water to support a load is amply demonstrated in Florida during prolonged droughts.

35. Collapse Sinkhole Formation 1
No evidence of land subsidence, small- to medium-sized cavities in the rock matrix
Water from surface percolates through to rock, and the erosion process begins
Source:

36. 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
Source:

37. 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
Source:

38. 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
Source:

39. Collapse into Old Mineshaft
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
Pore pressures sometimes exceed the least stress, σ3, and turn it into a tensile stress.
Cracks perpendicular to σ3 will develop tensile cracking, widening the cracks. This leads to hydraulic fracturing of the rock.
Source:
Photo by S. Jerrod Smith, graduate student

40. 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
Water is also capable of interacting with minerals, weakening chemical bonds, and promoting the propagation of cracks.

41. 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)
C = cohesion of the rock, μ is the coefficient of internal friction
The term (σn - Pf) is often labeled σn*, called effective stress

42. 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
Thus, wet rocks may fail when dry rocks do not.
So fluids become very important in determining conditions under which faulting occurs.
The Army demonstrated this some years ago.

43. 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
They attempted to dispose of unwanted chemical warfare agents by deep well injection. The fluids they injected allowed faulting to take place. The resulting earthquakes, although small, attracted attention and quickly put an end to that method of disposal.
More information is available at:
This is a letter to the editor of Seismological research letters by Nano Seeber

44. 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)
The weight f water stored behind a dam can also cause failure. The phenomenon Is known as RIS, for reservoir induced seismicity
Information form:
NORSAR is an independent research foundation specializing in commercial software solutions and research activities within applied geophysics and seismology.
RIS Info:

45. 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
The size of the sample also plays a role. The chance of a larger, correctly oriented Griffith crack being present in the sample somewhere goes up as the sample size increases.
Thus flawless single crystals should be very strong. Indeed turbine blades are often formulated as single crystals for this reason.

46. Other Important Effects
Brittle fracture is the underlying cause of most earthquakes
It also contributes to the formation of regional tectonic features