Von Mises yield criterion

Reduced von Mises equation for different stress conditions

Uniaxial (1D) stress

In the case of uniaxial stress or simple tension, ${\displaystyle {\sigma }_1\ne 0,{\sigma }_3={\sigma }_2=0}$, the von Mises criterion simply reduces to

${\displaystyle {\sigma }_1={\sigma }_{\text{y}}}$,

which means the material starts to yield when ${\displaystyle {\sigma }_1}$ reaches the yield strength of the material ${\displaystyle {\sigma }_{\text{y}}}$, in agreement with the definition of tensile (or compressive) yield strength.

Multi-axial (2D or 3D) stress

An equivalent tensile stress or equivalent von-Mises stress, ${\displaystyle {\sigma }_{\text{v}}}$ is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define

${\displaystyle \begin{array}{cc}{\sigma }_{\text{v}}& =\sqrt{3J_2}\\ & =\sqrt{\frac{({\sigma }_{11}-{\sigma }_{22}{)}^2+({\sigma }_{22}-{\sigma }_{33}{)}^2+\left({\sigma }_{33}-{\sigma }_{11}{)}^2+6({\sigma }_{12}^2+{\sigma }_{23}^2+{\sigma }_{31}^2\right)}{2}}\\ & =\sqrt{\frac{({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2}{2}}\\ & =\sqrt{\frac{3}{2}{s}_{ij}{s}_{ij}}\end{array}}$

where ${\displaystyle s_{ij}}$ are components of stress deviator tensor ${\displaystyle {\mathit{\sigma }}^{\text{dev}}}$:

${\displaystyle {\mathit{\sigma }}^{\text{dev}}=\mathit{\sigma }-\frac{\mathrm{tr}\left(\mathit{\sigma }\right)}{3}\mathbf{I}}$.

In this case, yielding occurs when the equivalent stress, ${\displaystyle {\sigma }_{\text{v}}}$, reaches the yield strength of the material in simple tension, ${\displaystyle {\sigma }_{\text{y}}}$. 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, ${\displaystyle {\sigma }_{12}={\sigma }_{21}\ne 0}$, while all other ${\displaystyle {\sigma }_{ij}=0}$, von Mises criterion becomes:

${\displaystyle {\sigma }_{12}=k=\frac{{\sigma }_y}{\sqrt{3}}}$.

This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is ${\displaystyle \sqrt{3}}$ times lower than the yield stress in the case of simple tension. The von Mises yield criterion for pure shear stress, expressed in principal stresses, is

${\displaystyle ({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_1-{\sigma }_3{)}^2=2{\sigma }_y^2}$

In the case of principal plane stress, ${\displaystyle {\sigma }_3=0}$ and ${\displaystyle {\sigma }_{12}={\sigma }_{23}={\sigma }_{31}=0}$, the von Mises criterion becomes:

${\displaystyle {\sigma }_1^2-{\sigma }_1{\sigma }_2+{\sigma }_2^2=3k^2={\sigma }_y^2}$

This equation represents an ellipse in the plane ${\displaystyle {\sigma }_1-{\sigma }_2}$.

Summary

Physical interpretation of the von Mises yield criterion

Hencky (1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value. For this reason, the von Mises criterion is also known as the maximum distortion strain energy criterion. This comes from the relation between ${\displaystyle J_2}$ and the elastic strain energy of distortion ${\displaystyle W_{\text{D}}}$:

${\displaystyle W_{\text{D}}=\frac{J_2}{2G}}$ with the elastic shear modulus ${\displaystyle G=\frac{E}{2(1+\nu )}}$.

In 1937 Arpad L. Nadai suggested that yielding begins when the octahedral shear stress reaches a critical value, i.e. the octahedral shear stress of the material at yield in simple tension. In this case, the von Mises yield criterion is also known as the maximum octahedral shear stress criterion in view of the direct proportionality that exists between ${\displaystyle J_2}$ and the octahedral shear stress, ${\displaystyle {\tau }_{\text{oct}}}$, which by definition is

${\displaystyle {\tau }_{\text{oct}}=\sqrt{\frac{2}{3}{J}_2}}$

thus we have

${\displaystyle {\tau }_{\text{oct}}=\frac{\sqrt{2}}{3}{\sigma }_{\text{y}}}$

Strain energy density consists of two components - volumetric or dialational and distortional. Volumetric component is responsible for change in volume without any change in shape. Distortional component is responsible for shear deformation or change in shape.

Practical engineering usage of the von Mises yield criterion

As shown in the equations above, the use of the von Mises criterion as a yield criterion is only exactly applicable when the following material properties are isotropic, and the ratio of the shear yield strength to the tensile yield strength has the following value:

${\displaystyle \frac{F_{sy}}{F_{ty}}=\frac{1}{\sqrt{3}}\approx 0.577}$

Since no material will have this ratio precisely, in practice it is necessary to use engineering judgement to decide what failure theory is appropriate for a given material. Alternately, for use of the Tresca theory, the same ratio is defined as 1/2.

The yield margin of safety is written as

${\displaystyle M{S}_{\text{yld}}=\frac{F_y}{{\sigma }_{\text{v}}}-1}$