Octahedral Shear Stress⚠ unverified

Understanding the Parameters

• Von Mises stress $\sigma_{vm}$ — the distortion-energy equivalent stress (Von Mises Stress). $\tau_{oct}$ is just a rescaling of it, so the two carry the same information and predict the same yield.

• Octahedral planes — imagine the plane whose normal makes equal angles with all three principal axes ($\cos^{-1}(1/\sqrt3) \approx 54.7^\circ$); there are eight such planes forming an octahedron. The shear on them is the same by symmetry and represents the "average" shearing that distorts (not just compresses) the material.

• Why it equals distortion energy — the stress on an octahedral plane splits into a normal part (the hydrostatic mean, which only changes volume) and the shear part $\tau_{oct}$ (which only changes shape). Yielding is a shape-change phenomenon, so $\tau_{oct}$ is the natural yield measure.

• The correct factor — $\sqrt2/3 \approx 0.471$, not $1/\sqrt2 \approx 0.707$. The distinction matters whenever the octahedral shear is compared against a physical limit.

Derivation (Approaching a Proof)

Resolve the traction on a plane whose unit normal has equal direction cosines $(1/\sqrt3, 1/\sqrt3, 1/\sqrt3)$ with the principal axes. The normal stress on it is the hydrostatic mean $\sigma_{oct} = (\sigma_1+\sigma_2+\sigma_3)/3$. The shear component is obtained from $\tau_{oct}^2 = |\mathbf{t}|^2 - \sigma_{oct}^2$, where $\mathbf{t}$ is the traction vector; carrying out the algebra gives

$$\tau_{oct} = \frac{1}{3}\sqrt{(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2}.$$

The von Mises stress is the same radical scaled differently:

$$\sigma_{vm} = \frac{1}{\sqrt2}\sqrt{(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2}.$$

Dividing, the common radical cancels:

$$\frac{\tau_{oct}}{\sigma_{vm}} = \frac{1/3}{1/\sqrt2} = \frac{\sqrt2}{3} \approx 0.471 \;\Longrightarrow\; \tau_{oct} = \frac{\sqrt2}{3}\,\sigma_{vm}.$$

This is the exact factor — the registry's $1/\sqrt2$ omits the $1/3$-vs-$1/\sqrt2$ bookkeeping and lands $\tfrac32\times$ too high.

Dimensional check. $\tau_{oct} = \tfrac{\sqrt2}{3}\,\sigma_{vm} = (\text{–})\cdot\text{Pa} = \text{Pa}$ — a stress, as required ($\sqrt2/3$ is a pure number).

History and Development

The octahedral-shear-stress theory, developed by Nádai (1930s), gives the von Mises / distortion-energy criterion (von Mises 1913; Huber 1904) its physical interpretation: yielding is governed by the shear on the octahedral planes, i.e. by the energy of distortion rather than of volume change. The two formulations are mathematically identical, which is why the octahedral, distortion-energy, and von Mises criteria are three names for one theory.

Related Concepts: Von Mises Stress, Strain Energy Density, Principal Stresses, Tresca Stress, Static Failure Theories, Factor of Safety yield

Notes: Correct $\tau_{oct} = \tfrac{\sqrt2}{3}\sigma_{vm} \approx 0.471\,\sigma_{vm}$; registry's $\sigma_{vm}/\sqrt2 \approx 0.707\,\sigma_{vm}$ is $\tfrac32\times$ too high (bug). Octahedral shear ≡ distortion-energy measure ≡ von Mises, rescaled.