Download Mechanics - Structural Geology - Lecture Notes and more Study notes Geology in PDF only on Docsity!
Mechanics of Brittle Failure
Ch. 7, p. 126-‐133; 139-‐
1. Mechanics of Brittle Failure: We can think of a material like rock undergoing some sort of structural failure once the stresses reach some critical value for brittle fracturing. Rocks thus have a limit to their strength.
But what is strength and how do we measure it?
This has typically been done in the laboratory using a pressurized vessel containing a rock cylinder placed between hydraulic pistons (called a rock testing machine).
[Figure. Rock testing machine (Pollard & Fletcher, 2005)]
2. Mechanics of Brittle Failure: A range of rock failure styles may occur depending on the loading conditions. For uniaxial loading, opening fractures form parallel to σ 1 (the long axis of the cylinder). This type of failure is called axial splitting or longitudinal splitting.
[Figure. Brittle failure of rock specimens (Twiss & Moores, 2007; Pollard & Fletcher, 2005)]
3. Mechanics of Brittle Failure: For axial compression (cylinder inside a pressurized fluid), shear failure occurs through the development of a diagonal break through the sample. In some cases, conjugate fractures may form. They form at <45° to σ 1 (usually ~30°) so are not planes of maximum shear stress.
These two styles of failure warrant the development of explicit failure criteria to explain brittle fracture of rock.
[Figure. Brittle failure of rock specimens (Twiss & Moores, 2007; Pollard & Fletcher, 2005)]
4. Optimal Fault Orientation for Failure: In nature, we find that failure is always preferred when θ > 45°. In fact, slip is most likely when θ ≈ 60°.
The reason is friction. The likelihood of failure depends on a perfect balance between σs and σn. Some combinations are stable and others are unstable (i.e., sliding). We can think about this in terms of the Mohr circle and what happens on a σs vs σn graph during a rock testing experiment.
[Figure. Cross section through rock cylinder containing a shear fracture] [Figure. Mohr space failure line]
5,6,7. Mohr Circles and Frictional Failure: At the start of the experiment, σ 1 is very small and equal to σ 3. As the pistons push on the sample, σ 1 increases relative to σ 3. This causes the Mohr circle to get bigger through time.
[Figure. Rock testing experiment in Mohr space] [Figure. Rock testing experiment on rock cylinder]
8. Mohr Circles and Frictional Failure: Eventually, the sample will break. But why?
[Figure. Third stage of rock testing experiment in Mohr space] [Figure. Third stage of rock testing experiment on rock cylinder]
9. Mohr Circles and Frictional Failure: The sample broke because the Mohr circle touched the failure line for frictional sliding. This causes the sample to cross into the unstable field and undergo frictional failure. This condition was not true of the other Mohr circles.
This experiment indicates that the strength of rock depends on some critical value of (σ 1 – σ 3 ), or differential stress.
[Figure. Failure stage of rock testing experiment in Mohr space] [Figure. Failure stage of rock testing experiment on rock cylinder]
10,11,12. Mohr Circles and Frictional Failure: With increasing depth and overburden (lithostatic pressure), σ 1 and σ 3 must get bigger and bigger and so must migrate towards the right on the Mohr diagram.
Q: If they increase at the same rate as each other, will frictional failure still be possible? ___________
[Figure. Stable situations in Mohr space]
13. Mohr Circles and Frictional Failure: This tells us that in order for faults to slide frictionally at depth, the Mohr circle must be bigger. In other words, the differential stress (σ 1 – σ 3 ) needs to be greater, implying that the rocks get stronger with depth.
The different combinations of Mohr circles describing frictional failure define the Mohr-‐Coulomb failure line (tangent to the Mohr circles).
[Figure. Definition of a failure line in Mohr space]
14. Mohr-‐Coulomb Frictional Failure: The failure line has the equation: |σs | = C + μσn
This relationship was first determined by Coulomb in 1773, based on earlier work by Amonton in 1699, about frictional sliding. Its relationship to Mohr circles was developed in 1882.
The intercept with the σs axis, C (or S (^) o ), is the inherent shear strength or cohesion of the material. For loose, granular materials, C≈0.
The slope of the failure line is μ, the coefficient of internal friction, which varies between materials but is commonly ~0.6 in rocks.
C and μ are material properties.
[Fig. 7.10. Mohr circles for different states of stress at the instant of frictional sliding, given by the Mohr-‐Coulomb failure line]
15. Mohr-‐Coulomb Frictional Failure: The coefficient of internal friction μ = tan φ, where φ is the angle of internal friction. Note, the geometry of the diagram results in every circle being tangent to the failure line at the same angle 2θ.
[Fig. 7.10. Mohr circles for different states of stress at the instant of frictional sliding, given by the Mohr-‐Coulomb failure line]
23. Frictional Failure of Already Fractured Rock: In such cases, there is no cohesion to be overcome to create the fault, so the value of C ≈ 0 and μ is now called the coefficient of sliding friction (or static friction). Its value is typically about the same as the internal friction. The failure criterion in such cases is simply:
|σs | = μ σn (at failure)
This is the same as the original Amonton’s Law (1699), but is typically referred to as the Coulomb criterion.
[Fig. 7.10. Mohr circles for different states of stress at the instant of frictional sliding, given by the Mohr-‐Coulomb failure line]
24. Frictional Failure of Already Fractured Rock: Note that a range of orientations of pre-‐existing faults would be more prone to failure than a new fault forming in the rock.
[Fig. 7.10. Mohr circles for different states of stress at the instant of frictional sliding, given by the Mohr-‐Coulomb failure line]
25. Effect of Fluid Pressure : Fluid pressure p (^) f reduces normal stresses but has no effect on shear stress, so the Mohr circle shifts to the left to reflect the effective stress state. The likelihood of shear failure is thus increased and the failure criterion becomes:
|σs | = C + μσneff^ = C + μ(σn – p (^) f )
[Fig. 7.33. Fluid pressure produces a state of effective stress that moves the Mohr circle to the left]
26. Effect of Fluid Pressure : At low confinement, p (^) f may cause the circle to cross the σs axis (σ 3 eff < 0), resulting in tension cracking if σ 3 eff^ < T. This is called hydraulic fracturing.
Fluid pressure is typically hydrostatic (p (^) f = ρw gh), or about 0.4 of lithostatic (σv = ρr gh). If p (^) f /σv > 0.4 (e.g., confined aquifer), a state of overpressure exists.
[Fig. 7.33. Fluid pressure produces a state of effective stress that moves the Mohr circle to the left]
27. Griffith Criteria for Failure: The Mohr-‐Coulomb criterion provides a quantitative means for establishing when brittle failure should occur, but it provides no insight into the physical mechanisms behind failure. Failure simply “happens.”
Materials are actually a lot weaker than their atomic bond strengths would imply. The reason is the presence of a multitude of tiny cracks or flaws called Griffith flaws (in honor of the engineer who first recognized their significance for weakening materials).
[Fig. 7.18. Weakening of rock in response to the presence of microcracks called Griffith flaws]
28. Griffith Criteria for Failure: Griffith determined that stresses are greatly concentrated at the tips of these microcracks, producing perturbed stresses.
The stress concentration may be many 100s of times greater than the applied stresses (i.e., σ1, σ2, σ3).
An elliptical crack pulled on perpendicular to its long-‐axis length (i.e., σ3 < T < 0) will thus start to lengthen, creating a tension crack.
[Fig. 7.19. Concentrations of stress at the tips of Griffith flaws]
29. Griffith Criteria for Failure: For optimally oriented, obliquely-‐loaded microcracks, shearing creates a stress concentration that results in tiny wing cracks or tailcracks that lengthen parallel to the direction of σ 1.
[Fig. 7.23. Concentrations of stress at the tips of obliquely loaded Griffith flaws]
30. Griffith Criteria for Failure: These wing cracks allow microcracks to link together, ultimately forming a macroscopic through-‐going failure surface (a fault) with an orientation relative to σ 1 and σ 3 as dictated by the Coulomb criterion.
[Fig. 7.23. Concentrations of stress at the tips of obliquely loaded Griffith flaws] [Fig. 7.18. Weakening of rock in response to the presence of microcracks called Griffith flaws]
31. Griffith Criteria for Failure: Griffith thus determined that the production of a shear fracture actually occurs at the microscopic scale by a repeating process of tension fracturing.
The Griffith failure criterion was determined to be:
σs^2 + 4Tσn – 4T 2 = 0
It indicates that C ≈ 2T.
[Fig. 7.17. Merging of Griffith and Coulomb failure criteria]
