The Coulomb Criterion
Learning objectives
- State the Coulomb failure line: shear strength is cohesion plus friction times normal stress
- Find failure as the Mohr circle growing until it just touches the strength line
- Read the failure-plane angle from the tangent point: 45 degrees plus half the friction angle
- Connect cohesion and friction to UCS and the q slope, UCS equal to twice cohesion times root q
A Line and a Circle
Rock fails in shear when the shear stress on some plane overcomes the plane's resistance, and Coulomb's insight was that the resistance has two parts: a constant cohesion that must be broken regardless of load, plus a frictional term proportional to how hard the plane is clamped shut. The failure line is , a straight envelope in the same normal-stress, shear-stress axes as the Mohr circle, and that shared coordinate frame is the whole trick. A stress state is a circle; its strength is a line; and failure is the moment the circle grows to touch the line. Not cross it, touch it: the tangent point names the one plane that fails and the stress at which it does.
Grow the circle in the figure by raising at fixed confinement. It swells until it kisses the envelope, and at that instant the tangent point tells you two things. Its position gives the failure stresses; the angle from the circle's center gives the failure-plane orientation, \beta = 45^\circ + \varphi/2 where . For the crustal , the friction angle is about 31 degrees and the failure plane sits at 60.5 degrees to the least stress, tilted off the pure-shear 45 by friction's preference for a more clamped plane, exactly the oblique break Part 0.2 promised. The tangency also delivers the strength numbers: the unconfined strength is and the confined slope is , so a cohesion of 15 MPa with gives and MPa.
Two Numbers, Every Failure
What makes Coulomb the workhorse is that its two constants, cohesion and friction, are exactly what the triaxial line of the last section measures, and they carry the same that will govern the frictional limit of the crust in Part 5, the breakout width in Part 6, and the fault reactivation of Part 9. The criterion has honest limits the course will name as they arrive: it draws a straight line where real envelopes curve gently (Hoek-Brown, in two sections), it ignores the intermediate stress (the Mohr-3D approximation of Part 1.4), and it says nothing about the tensile side (the Griffith tail, next section). But for the compressive failure of clamped rock, one line against one circle, it is the picture the entire discipline reasons in.
References
- Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.
- Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.
- Fjaer, E., Holt, R. M., Horsrud, P., Raaen, A. M., & Risnes, R. (2008). Petroleum Related Rock Mechanics (2nd ed.). Elsevier.