# von Mises stress

Many laboratory fatigue tests are conducted under uniaxial loading. However, ultrasonic fatigue may involve stresses that act in several directions (multiaxial stress); a simple example is a radially resonant disk which would have both radial stress and hoop stress. The von Mises stress \( \sigma_v \) is an artificial stress (based on theory) that attempts to correlate such a multiaxial fatigue stress to the simpler uniaxial laboratory fatigue stress σ_{u}. Thus, if uniaxial laboratory samples fail (on average) at a fatigue stress \( \sigma_{uf} \), then resonators of the same material with multiaxial stresses should fail (on average) when \( \sigma_v \) equals \( \sigma_{uf} \). The von Mises stress is applicable to isotropic materials that are ductile.

For stresses acting in three perpendicular directions (x, y, and z), the von Mises stress is given by —

\begin{align} \label{eq:13801a} \sigma_v = \left[ \frac{(\sigma_x - \sigma_y)^2 + (\sigma_y - \sigma_z)^2 + (\sigma_z - \sigma_x)^2 + 6 \, (\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)} {2} \right]^{1/2} \end{align}

where —

\( \sigma \) | = normal stress |

\( \tau \) | = shear stress |

If the local stress cube is oriented such that its axes align with the principal axes 1, 2, and 3 (i.e., there are no shear stresses, only tensile or compressive stresses), then the above equation becomes —

\begin{align} \label{eq:13802a} \sigma_v = \left[ \frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2} {2} \right]^{1/2} \end{align}

Notes:

- Also called "von Mises - Hencky stress", "Hencky von Mises stress", "octahedral shear stress", or "distortion energy stress".
- Finite element analysis (FEA) software generally will calculate and display the von Mises stress.
- All stress values on this site are von Mises unless otherwise stated.

See Juvinall, pp. 86 - 89, p. 230, pp. 316 - 317 and Shigley, pp. 152 - 154. Also see Frequently Asked Questions on Von Mises Stress Explained.