Von Mises Failure Criteria

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Von Mises stress, $\boldsymbol{\sigma_v}$ , or simply Mises stress, is a scalar function of the components of the stress tensor that gives an appreciation of the overall 'magnitude' of the tensor. This allows the onset and amount of plastic deformation under triaxial loading to be predicted from the results of a simple uniaxial tensile test. It is most applicable to ductile materials.

Plastic deformation, or yielding, initiates when the Mises stress reaches the initial yield stress in uniaxial tension and, for hardening materials, will continue provided the Mises stress is equal to the current yield stress and tending to increase. Mises stress can then be used to predict failure by ductile tearing. It is not appropriate for failure by crack propagation or fatigue, which depend on the maximum principal stress.

In 3-D, the Mises stress can be expressed as:

$\sigma_v = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 } {2}}$

where $σ 1,σ 2,σ 3$ are the principal stresses. In 1-D, this reduces to the uniaxial stress.

or, in terms of a local coordinate system:

$\sigma_v = {\frac 1 \sqrt{2}} \sqrt{(\sigma_x - \sigma_y)^2 + (\sigma_y - \sigma_z)^2 + (\sigma_z - \sigma_x)^2 + 6(\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)}$

[edit] Von Mises yield criterion

Von Mises stress in two dimensions.

This criterion for the onset of yield in ductile materials was first formulated by Maxwell in 1865 but is generally attributed to von Mises in 1913.^[1] Originally suggested by Maxwell purely on the grounds of mathematical simplicity, the corresponding yield function $Φ$ is the simplest function that meets certain physical requirements for yielding, taking the form:

Φ = J 2 - k 2

where $J 2$ is the second deviatoric invariant of the stress tensor and k is the yield stress in shear. The use of von Mises yield criterion is therefore sometimes called $J 2$ flow theory.

Von Mises yield criterion can be interpreted physically in terms of the maximum distortion strain energy, octahedral shear stress theory, or Maxwell-Huber-Hencky-von Mises theory. This states that yielding in 3-D occurs when the distortion strain energy reaches that required for yielding in uniaxial loading. Mathematically, this is expressed as:

$\frac{1}{2} \Big[ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \Big] \le \ \sigma_y^2$

In the case of plane stress, $σ 3 = 0$ , von Mises criterion reduces to:

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

In the 2-D stress space shown in the figure above, this equation represents the interior of an ellipse. Stress states $σ 1,σ 2$ not touching the boundary of the ellipse produce only elastic deformation. Yielding initiates when the stress state pushes against the boundary. This ellipse is the projection, onto a plane, of the yield surface in 3-D stress space, which takes the form of a cylinder equiaxial to the three stress axes.

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

In addition to bounding the principal stresses to prevent ductile failure, von Mises' criterion sometimes gives a reasonable estimation of fatigue life, especially with complex loading (mode II & III loading).