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Brittle Failure Criteria
Ch. 7, p. 129-Āā133; 139-Āā
1. Frictional Failure of Already Fractured Rock: The Mohr-ĀāCoulomb criterion predicts the frictional failure of pristine, intact rock. Where rocks are already faulted (typically the case in nature), sliding occurs when the frictional strength of the existing fault surface is overcome.
[Fig. 7.10. Mohr circles for different states of stress at the instant of frictional sliding, given by the Mohr-ĀāCoulomb failure line]
2. 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 ā 0 and Ī¼ 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:
|Ļs | = Ī¼ Ļn (at failure)
This is the same as the original Amontonās 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-ĀāCoulomb failure line]
3. Frictional Failure of Already Fractured Rock: Note that a range of orientations of pre-Āāexisting 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-ĀāCoulomb failure line]
4. 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:
|Ļs | = C + Ī¼Ļn eff = C + Ī¼(Ļn ā p (^) f )
[Fig. 7.33. Fluid pressure produces a state of effective stress that moves the Mohr circle to the left]
5. Effect of Fluid Pressure : At low confinement, p (^) f may cause the circle to cross the Ļs axis (Ļ 3 eff^ < 0), resulting in tension cracking if Ļ 3 eff^ < T. This is called hydraulic fracturing.
Fluid pressure is typically hydrostatic (p (^) f = Ļw gh), or about 0.4 of lithostatic (Ļv = Ļr gh). If p (^) f /Ļv > 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]
6. Griffith Criteria for Failure: The Mohr-ĀāCoulomb 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 āhappens.ā
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]
7. 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., Ļ1, Ļ2, Ļ3).
An elliptical crack pulled on perpendicular to its long-Āāaxis length (i.e., Ļ3 < T < 0) will thus start to lengthen, creating a tension crack.
[Fig. 7.19. Concentrations of stress at the tips of Griffith flaws]
8. Griffith Criteria for Failure: For optimally oriented, obliquely-Āāloaded microcracks, shearing creates a stress concentration that results in tiny wing cracks or tailcracks that lengthen parallel to the direction of Ļ 1.
[Fig. 7.23. Concentrations of stress at the tips of obliquely loaded Griffith flaws]
9. Griffith Criteria for Failure: These wing cracks allow microcracks to link together, ultimately forming a macroscopic through-Āāgoing failure surface (a fault) with an orientation relative to Ļ 1 and Ļ 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]
10. 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:
Ļs^2 + 4TĻn ā 4T 2 = 0
It indicates that C ā 2T.
[Fig. 7.17. Merging of Griffith and Coulomb failure criteria]
7. Source of Stress: Strain is the end-Āāresult of the cumulative stress state; however, many factors contribute to creating stress. These include: (1) overburden (lithostatic stress, rgh); (2) tectonic stress (slab pull; ridge push; plate collision; lateral sliding; basal drag); (3) plate bending (vertical loads; subduction; curvature effects on a non-Āā spherical planet); (4) thermal effects; (5) fluid pressure (water; oil; gas).
[Fig. 5.14. Tectonic contributions to the overall stress state]
8. Limits to Stress Magnitudes: The Mohr-ĀāCoulomb criterion also indicates that there is a limit to how big the stress magnitudes can be, particularly the differential stress at any particular depth.
[Fig. 7.10. Mohr circles for different states of stress at the instant of frictional sliding, given by the Mohr-ĀāCoulomb failure line]
9. Limits to Stress Magnitudes: We generally assume that the vertical stress (Ļv ) is always a principal stress and is equal to the overburden or lithostatic stress, Ļr gh (dry rocks) or (Ļr ā Ļw)gh (wet rocks). In the absence of other stress components, the lithostatic reference state is isotropic: Ļ 1 = Ļ 2 = Ļ 3 = Ļv. A typical gradient is 25-Āā30 MPa/km.
[Fig. 5.6. Measurements of vertical stress and fluid pressures as a function of depth in (a) Norway and (b) North Sea and other locations]
10. Limits to Stress Magnitudes: Fluid pressure in rocks will be hydrostatic from the water table down if the pores and fractures are fully connected up to the surface. Confined units may experience overpressure (p (^) f > hydrostatic), which is more prone to fracturing rock.
[Fig. 5.6. Measurements of vertical stress and fluid pressures as a function of depth in (a) Norway and (b) North Sea and other locations]
11. Limits to Stress Magnitudes: In general, horizontal stresses differ from Ļv , indicating other contributors to the stress state. We differentiate the maximum horizontal stress (ĻH or S (^) H ) from the minimum horizontal stress (Ļh or S (^) h ). Which principal stress these represent depends on the tectonic setting.
[Figure. Horizontal stresses S (^) h in a sedimentary basin in the U.S. (Twiss & Moores, 2007)]
12. Limits to Stress Magnitudes: If Ļv = Ļ 3 and S (^) H = Ļ 1 (contraction), stresses fall to the right of the lithostat (limited by green line).
If Ļv = Ļ 1 and S (^) h = Ļ 3 (extension), stresses fall to the left of the lithostat (limited by the purple line).
Rocks are stronger in compression than tension.
[Figure. limits to the value of horizontal stresses in extensional and contractional tectonic settings. Assumes C= MPa and Ī¼ = 0.6 (Twiss & Moores, 2007)] [Fig. 5.16. Rock strength versus depth for different lithologies]
