Von Mises Yield Criterion - Reduced Von Mises Equation For Different Stress Conditions

Reduced Von Mises Equation For Different Stress Conditions

The above equation can be reduced and reorganized for practical use in different loading scenarios.

In the case of uniaxial stress or simple tension, the von Mises criterion simply reduces to

,

which means the material starts to yield when reaches the yield strength of the material, and is agreement with the definition of tensile (or compressive) yield strength.

It is also convenient to define an Equivalent tensile stress or von Mises stress, which is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define

\begin{align} \sigma_v &= \sqrt{3J_2} \\ &= \sqrt{\frac{(\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{11} - \sigma_{33})^2 + 6(\sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2)}{2}} \\ &= \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2 } {2}} \\ &= \sqrt{\textstyle{\frac{3}{2}}\;s_{ij}s_{ij}} \end{align} \,\!

where are the components of the stress deviator tensor :

\boldsymbol{\sigma}^{dev} = \boldsymbol{\sigma} - \frac{1}{3} \left(\mbox{tr} \ \boldsymbol{\sigma} \right) \mathbf{I} \,\!.

In this case, yielding occurs when the equivalent stress, reaches the yield strength of the material in simple tension, . As an example, the stress state of a steel beam in compression differs from the stress state of a steel axle under torsion, even if both specimens are of the same material. In view of the stress tensor, which fully describes the stress state, this difference manifests in six degrees of freedom, because the stress tensor has six independent components. Therefore, it is difficult to tell which of the two specimens is closer to the yield point or has even reached it. However, by means of the von Mises yield criterion, which depends solely on the value of the scalar von Mises stress, i.e., one degree of freedom, this comparison is straightforward: A larger von Mises value implies that the material is closer to the yield point.

In the case of pure shear stress, while all other, von Mises criterion becomes:

.

This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is times lower than the tensile stress in the case of simple tension. The von Mises yield criterion for pure shear stress, expressed in principal stresses, is

In the case of plane stress, the von Mises criterion becomes:

This equation represents an ellipse in the plane, as shown in the Figure above.

The following table summarizes von Mises yield criterion for the different stress conditions.

Load scenario Restrictions Simplified von Mises equation
General No restrictions
Principal stresses
Plane stress

Pure shear

Uniaxial

Notes:

  • subscripts 1,2,3 can be replaced with x,y,z, or other orthogonal coordinate system

Read more about this topic:  Von Mises Yield Criterion

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