Most analyses of rock mechanics problems that involve failure or potential failure utilize Mohr's assumption that failure is controlled only by the minimum and maximum principal stresses, σ 1 and σ 3. However, evidence has been accumulating for several decades that the intermediate stress has a "strengthening effect", in the sense that, for a given value of σ 3, the value of σ 1 required to cause failure will be higher if σ 2 3 than it would be if σ 2 =σ 3. Mogi proposed that the failure criterion should be of the form τ oct =f (σ m,2), where τ oct is the octahedral shear stress, and σ m,2 =(σ 1 +σ 3)/2. Analysis of many sets of data from the literature shows that most can be fit reasonably well with a linearized form of Mogi's criterion, τ oct =a +b σ m, 2. This criterion has several advantages: it accounts for the strengthening effect of σ 2, the coefficients a and b can be expressed in terms of the cohesion and coefficient of internal friction that appear in the Mohr-Coulomb criterion, and its linear form allows it to be used in analyzing engineering problems. For the specific problem of the stability of a vertical borehole, this criterion leads to substantially different predictions for the minimum required mud weight, as compared to those found using the Mohr-Coulomb criterion.


Among the large number of shear failure criteria that have been proposed for rocks, the Mohr-Coulomb criterion is the most commonly used in practice. There are two major components of this criterion. The first is the assumption that, at failure, the major principal stress σ 1 is a linearly increasing function of the minor principal stress, σ 3. The second aspect is the assumption that the value of the intermediate principal stress, σ 2, has no influence on the rock strength. Neither of these assumptions is precisely true for most rocks. Failure data for most rocks shows a nonlinear failure envelope with a negative second derivative, although the degree of nonlinearity varies from rock to rock. Much effort has been expended over recent years on deriving nonlinear failure criteria. These are typically fit to traditional failure data in which σ 2 =σ 3, and are then used under the implicit assumption that, if σ 2 3, failure will not depend on the actual numerical of σ 2. However, evidence has been accumulating over the past several decades to suggest that the intermediate principal stress does have an influence on rock strength (Handin 1967, Hoskins 1969, Mogi 1971, Haimson 2002). Many sets of such data have recently been collected by Colmenares & Zoback (2002), and in the monograph by Mogi (2006). A striking example of the extent to which real rock failure data may not comply with the assumption that failure depends only on σ 1 and σ 3 is shown in Figure 1, replotted from Haimson & Song (1995). The data from borehole breakouts, which occur under a true-triaxial stress state, clearly do not fall near the curveobtained from traditional (σ 2 = σ 3) triaxial compression test.

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