The Mohr Circle

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

Learning objectives

  • Show that the normal and shear traction over all planes trace a circle: center at the mean stress, radius half the difference
  • Walk a plane around the circle and read the doubled angle between physical planes and circle positions
  • Read the maximum shear at the circle's crest, on the plane at 45 degrees to the principal directions
  • Draw the canon stress circle, center 56.9 and radius 10.85 MPa, that the rest of the course will push around

The Rotation Equations, Drawn

Take the principal frame of the last section and ask, plane by plane, what normal and shear traction each orientation feels: sigman=tfracsigma1+sigma32+tfracsigma1−sigma32cos2beta\sigma_n = \tfrac{\sigma_1+\sigma_3}{2} + \tfrac{\sigma_1-\sigma_3}{2}\cos 2\betan=tfracsigma1+sigma32+tfracsigma1−sigma32cos2beta and tau=tfracsigma1−sigma32sin2beta\tau = \tfrac{\sigma_1-\sigma_3}{2}\sin 2\beta. Read those two equations as coordinates and the answer is staring at you: as beta\beta sweeps, the point (sigman,tau)(\sigma_n, \tau)n,tau) traces a circle, centered on the normal-stress axis at the mean stress tfracsigma1+sigma32\tfrac{\sigma_1+\sigma_3}{2} with radius tfracsigma1−sigma32\tfrac{\sigma_1-\sigma_3}{2}. Every plane through the point is somewhere on that circle; every point on the circle is some plane. One hundred and twenty years after Otto Mohr drew it, it remains the best picture in solid mechanics.

The Mohr CircleInteractive figure, enable JavaScript to interact.

Two reading rules make the picture fluent. First, angles double: a plane whose normal sits beta\beta from sigma1\sigma_1 appears at arc angle 2beta2\beta from the circle's right edge, so perpendicular planes sit diametrically opposite, and the full 180 degrees of physical orientations wraps the circle exactly once. Second, the crest is the maximum shear, taumax=tfracsigma1−sigma32\tau_{max} = \tfrac{\sigma_1-\sigma_3}{2}max=tfracsigma1−sigma32 at 2\beta = 90^\circ: the 45 degree plane of Part 0.2, now visible as the top of an arch. For the teaching state of the last section the circle has center 20 and radius 14.14; for the canon total stresses at our reference depth, Sv=67.7S_v = 67.7v=67.7 and Shmin=46S_{hmin} = 46hmin=46, it has center 56.956.9 and radius 10.8510.85 MPa, and that second circle is the one this course will spend ten parts pushing around.

Why This Picture Runs the Course

The circle earns its keep because everything that matters happens in its plane. Rock strength, in Part 3, is a line drawn in the same axes; failure is the circle touching it. Pore pressure, next section, slides the circle bodily left without changing its size, which is most of poro-mechanics in one motion. Depletion, in Part 10, moves and grows it at once. Fault stability, in Part 9, is a single point on the circle measured against friction. Learn to see stress states as circles now, while the diagram is still empty; the course will spend the rest of its length filling it with consequences.

References

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

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