# 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 σu. 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:

1. Also called "von Mises - Hencky stress", "Hencky von Mises stress", "octahedral shear stress", or "distortion energy stress".
2. Finite element analysis (FEA) software generally will calculate and display the von Mises stress.
3. 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.