The Stress Tensor

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

Learning objectives

Write Cauchy's relation: the traction on any plane is the stress tensor acting on the plane's normal
Read the nine components as three tractions on three reference faces, reduced to six by moment balance
Drive a two-dimensional stress element and its probe plane, and watch the traction respond to every component
Verify on the element that the sum of the normal stresses does not care how you orient the axes

Three Faces, Nine Numbers

Part 0 ended with a promise: one object that knows the traction on every plane through a point. Here is how it is built. Cut three small reference faces through the point, one perpendicular to each coordinate axis, and record the traction vector on each. Three faces, three components each: nine numbers, written $\sigma_{ij}$ ij, the stress on the $i$ face in the $j$ direction. The diagonal entries $\sigma_{xx}, \sigma_{yy}, \sigma_{zz}$ xx,sigmayy,sigmazz are normal stresses, pushing straight on their faces; the off-diagonal entries are shear stresses, dragging along them. Balance of moments on the vanishing element forces $\sigma_{ij} = \sigma_{ji}$ , a fact you can feel: if the shears on adjacent faces did not match, the element would spin ever faster as it shrank. Nine numbers become six independent ones.

Cauchy's Relation

The payoff for recording those particular nine numbers is Cauchy's relation: the traction on any plane with unit normal $\mathbf{n}$ is simply $\mathbf{t} = \boldsymbol{\sigma}\,\mathbf{n}$ , the tensor applied to the normal. Nothing else about the plane matters, and no new measurements are needed: the three reference faces already contain every plane's answer. This is the machine of Part 0.2 made explicit, and it is worth pausing on how strong a statement it is. An infinity of planes, each with its own traction vector, all generated by six numbers.

Drive the element above. The face arrows draw the components you have dialed in; the probe plane sweeps through the point and reports its traction, split into normal and shear parts, by Cauchy's relation. Two things reward attention. First, with any shear present, the probe finds angles where the shear traction vanishes entirely; those special directions are the next section's subject. Second, watch the readout's last line: however you rotate, $\sigma_{xx} + \sigma_{yy}$ holds constant. Quantities like that, indifferent to your choice of axes, are called invariants, and they are the tensor's way of telling you what is physically real versus what is bookkeeping.

The Convention, Once More

Everything in this course is written with compression positive, and stresses at depth are overwhelmingly compressive: at our canon depth the smallest stress in the ground, 46 MPa, still presses harder than four hundred atmospheres. Tensile states will appear exactly twice, at the wall of an over-pressured borehole in Part 6 and inside a propagating hydraulic fracture in Part 7, and both times the sign will carry the entire story. Next, we hunt the planes on which the shear vanishes: the principal directions, where the tensor shows its simplest face.

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.