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: $\\sigma\_n = \\tfrac{\\sigma\_1+\\sigma\_3}{2} + \\tfrac{\\sigma\_1-\\sigma\_3}{2}\\cos 2\\beta$ and $\\tau = \\tfrac{\\sigma\_1-\\sigma\_3}{2}\\sin 2\\beta$ . Read those two equations as coordinates and the answer is staring at you: as $\\beta$ sweeps, the point $(\\sigma\_n, \\tau)$ traces a circle, centered on the normal-stress axis at the mean stress $\\tfrac{\\sigma\_1+\\sigma\_3}{2}$ with radius $\\tfrac{\\sigma\_1-\\sigma\_3}{2}$ . 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.

Two reading rules make the picture fluent. First, angles double: a plane whose normal sits $\\beta$ from $\\sigma\_1$ appears at arc angle $2\\beta$ 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, $\\tau\_{max} = \\tfrac{\\sigma\_1-\\sigma\_3}{2}$ 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, $S\_v = 67.7$ and $S\_{hmin} = 46$ , it has center $56.9$ and radius $10.85$ 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, and the figure above already carries the two consequences the course will spend its length unpacking. Rock strength is a line in the same axes, and failure is the circle touching it: the solid line is the frictional limit $\\tau = \\mu\\,\\sigma\_n$ with the canon $\\mu = 0.6$ , and the dashed envelope above it is intact rock, which adds a cohesion intercept, $\\tau = S\_0 + \\mu\\,\\sigma\_n$ (Part 3 earns both). And pore pressure, the third slider, slides the circle bodily left without changing its size, which is most of poro-mechanics in one motion: the principal stresses shrink or grow the circle, but only pressure translates it. Try the canon numbers: raise the pore pressure to 35.3 MPa and the effective circle stops about a quarter of an MPa short of the frictional line; at 35.8 it touches, and a well-oriented plane slips. That razor-thin margin is not an artifact of the demo, it is the course thesis, and Section 1.5 makes the sliding rigorous, Part 9 turns the touch into fault reactivation, and Part 10 marches the circle with depletion. Learn to see stress states as circles now; the consequences are already on the canvas.

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.