Three Circles: Mohr in 3D

Part 1, Part 1: Stress, the Tensor and the Circle

Learning objectives

  • Add the intermediate principal stress and draw the three Mohr circles it creates
  • Locate any plane, described by its direction cosines, inside the shaded region between the circles
  • Show that the largest shear anywhere still lives on the outer circle, indifferent to the intermediate stress
  • Draw the canon triple, 67.7 over 62 over 46 MPa, as its three circles for the first time

The Third Player

Real stress states have three principal values, sigma1gesigma2gesigma_3\sigma_1 \ge \sigma_2 \ge \sigma_3, and the plane picture generalizes with more grace than it has any right to. Each principal pair draws its own circle, so a 3D state appears as three nested circles: the big one spanning sigma1\sigma_1 to sigma3\sigma_3, and two smaller ones sharing its ends, sigma1\sigma_1 to sigma2\sigma_2 and sigma2\sigma_2 to sigma3\sigma_3. A plane whose normal lies along a principal axis plots on the rim of the corresponding circle, exactly as in 2D. And a general plane, tilted toward all three axes and described by its direction cosines (l,m,n)(l, m, n), lands somewhere in the crescent-shaped region between the small circles and the big one. No stress state can put a plane outside the big circle or inside the small ones; the shaded lune is the entire admissible world.

Mohr In 3dInteractive figure, enable JavaScript to interact.

Steer the plane in the figure. Tilt it in the sigma1\sigma_1-sigma3\sigma_3 plane and its point rides the outer rim; swing it toward sigma2\sigma_2 and the point dives into the lune's interior. Then drag sigma2\sigma_2 itself: as it approaches sigma3\sigma_3 the lower small circle shrinks to a point, the state becomes axially symmetric, and the picture collapses toward the single circle of the last section. The defaults draw the canon triple for the first time: Sv=67.7S_v = 67.7v=67.7 over SHmax=62S_{Hmax} = 62Hmax=62 over Shmin=46S_{hmin} = 46hmin=46 MPa, the vertical stress on top, which Part 5 will name a normal-faulting state.

What the Middle Stress Does, and Does Not

Look where the maximum shear lives: at the crest of the outer circle, taumax=tfracsigma1−sigma32=10.85\tau_{max} = \tfrac{\sigma_1 - \sigma_3}{2} = 10.85max=tfracsigma_1−sigma32=10.85 MPa for the canon triple, on the plane at 45 degrees between the largest and smallest stresses, containing the sigma2\sigma_2 axis. Slide sigma2\sigma_2 anywhere between its neighbors and that number does not move. This is the geometric seed of a modeling choice the whole course leans on: the Coulomb failure criterion of Part 3 judges rocks by sigma1\sigma_1 and sigma3\sigma_3 alone, treating the intermediate stress as a spectator. It is not exactly true, laboratory rocks show a modest sigma2\sigma_2 effect, but it is close, honest about being an approximation, and it buys the two-dimensional clarity every diagram from here to Part 10 will exploit.

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.
  • Mohr, O. (1900). Welche Umstaende bedingen die Elastizitaetsgrenze und den Bruch eines Materials? Zeitschrift des Vereines deutscher Ingenieure, 44, 1524-1530.

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