The von Mises criterion suggests that yielding begins when the second deviatoric stress invariant <math>\ J_2</math> reaches a critical value <math>\ k</math>. For this reason, it is sometimes called the <math>\ J_2</math>-plasticity or <math>\ J_2</math> flow theory. Mathematically the yield function for the von Mises condition is expressed as:
- <math>\ f(J_2) = \sqrt{J_2} - k = 0</math>
An alternative form is:
- <math>\ f(J_2) = J_2 - k^2 = 0</math>
Although formulated by Maxwell in 1865, it is generally attributed to von Mises (1913).[1] Huber (1904), in a paper in polish, anticipated to some extent this criterion.[2] This criterion is referred also as the Maxwell-Huber-Hencky-von Mises theory. Because the von Mises yield criterion is independent of the first stress invariant, <math>\ I_1</math>, it is applicable for the analysis of plastic deformation for ductile materials such as metals, as the onset of yield for these materials do not depend on the volumetric component of the stress tensor (mean stress or hydrostatic stress). The Von-Mises yield criterion, as a function of the principal stresses, is defined as:
- <math>\ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2 = 6k^2</math>
or
- <math>\ (\sigma_1^2 + \sigma_2^2 + \sigma_3^2) - \sigma_1\sigma_2 - \sigma_2\sigma_3 - \sigma_1\sigma_3 = 3k^2</math>
or
- <math>\ (\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{11} - \sigma_{33})^2 + 6(\sigma_{23}^2 + \sigma_{31}^2 + \sigma_{12}^2) = 6k^2</math>
This equation defines the yield surface as a circular cylinder (See Figure) whose yield curve, or intersection with the deviatoric plane, is a circle with radius <math>\ \sqrt{2}k</math>. This implies that the yield condition is independent of hydrostatic stresses.
Hencky (1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value. [3] For this, the von Mises criterion is also known as the maximum distortion strain energy criterion. This comes from the relation between <math>\ J_2</math> and the elastic strain energy of distortion <math>\ W_D</math>:
- <math>\ W_D = \frac{J_2}{2G}</math>, where <math>\ G</math> is the elastic shear modulus defined as:
- <math>\ G = \frac{E}{2(1+\nu)}</math>
Von Mises criterion is also known as the maximum octahedral shear stress in view of the relation between <math>\ J_2</math> and the octahedral shear stress, <math>\ \tau_{oct}</math>. In this sense, the criterion suggests that yielding begins when the octahedral shear stress reaches a critical value <math>\ k</math>:
- <math>\tau_{oct} = \sqrt{\frac{2}{3}J_2} = \sqrt{\frac{2}{3}}k</math>
which reduces to
- <math>\ J_2 - k^2=0</math>
In the case of pure shear stress, <math>\ \sigma_{12} = \sigma_{21}\neq0</math>, while all others <math>\ \sigma_{ij} = 0</math>, von Mises criterion becomes:
- <math>\ J_2=\sigma_{12}^2</math>, therefore <math>\ k = \sigma_{12}</math>. This means <math>\ k</math> is the magnitude of the shear stress at yielding in pure shear.
In the case of uniaxial stress or simple tension, <math>\ \sigma_3 = \sigma_2=0</math>, the von Mises criterion becomes:
- <math>\ J_2=\frac{\sigma_1^2}{3}</math>, where <math>\ \sigma_1</math> is the yield stress in simple tension <math>\ \sigma_y</math>.
Replacing into the yield function equation gives:
- <math>\ k=\frac{\sigma_y}{\sqrt{3}}</math>
Using this equation, which gives the relation between the uniaxial yield stress, <math>\ \sigma_y</math>, and the pure shear yield stress, <math>\ k</math>, the von Mises criterion can also be expressed as:
- <math>\ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2 = 2\sigma_y^2</math>
For this equation the radius of the yield curve is <math>\ \sqrt{\frac{2}{3}}\sigma_y</math>.
In the case of plane stress, <math>\ \sigma_3 = 0</math>, the von Mises criterion becomes:
- <math>\ \sigma_1^2- \sigma_1\sigma_2+ \sigma_2^2 = 3k^2</math>
or
- <math>\ \sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2 = \sigma_y^2</math>
This equation represents an ellipse in the plane <math>\ \sigma_1-\sigma_2</math>, as shown in the Figure above.
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).
