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Stress is in general a six-dimensional Tensor quantity (a symmetric 3×3 tensor). Von Mises stress reduces this to a single number (a scalar) for the purposes of calculating yield criteria.

Von Mises stress in three dimensions is

:\sigma_v = \sqrt{ rac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 } {2}}

where

:\sigma_1,\sigma_2,\sigma_3

are the Principal Stresses . In the case of plane stress, \sigma_3 is zero.

Finite Element Analysis results are typically presented as Von Mises stress.


VON MISES CRITERION


Applied mathematician Richard Von Mises came up with the von Mises Criterion in 1913. Also known as the maximum distortion energy criterion, octahedral shear stress theory, or Maxwell - Huber - Hencky -von Mises theory, it is often used to estimate the yield of ductile materials.

The von Mises criterion states that failure occurs when the energy of distortion reaches the same energy for yield/failure in uniaxial tension. Mathematically, this is expressed as,

: rac{1}{2} \Big (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \Big \le \ \sigma_y^2

In the cases of plane stress, \sigma_3 = 0. The von Mises criterion reduces to,

:\sigma_1^2 - \sigma_1 \sigma_2 + \sigma_2^2 \le \ \sigma_y^2

This equation represents a principal stress ellipse as illustrated in the following figure,

Also shown on the figure is the maximum shear stress ( Tresca ) criterion (dashed line). This theory is more conservative than the von Mises criterion since it lies inside the von Mises ellipse.

In addition to bounding the principal stresses to prevent ductile failure, the von Mises criterion also gives a reasonable estimation of fatigue failure, especially in cases of repeated tensile and tensile-shear loading.