Von Mises Yield Criterion

The von Mises yield criterion, also known as the maximum distortion energy criterion, is a widely used model for predicting the onset of plastic deformation in ductile materials. It postulates that yielding begins when the elastic strain energy of distortion per unit volume in a material reaches a critical value. This is distinct from the hydrostatic energy (associated with volume change), which is assumed not to contribute to yielding in ductile metals.

Mathematically, this is equivalent to stating that the second invariant of the deviatoric stress tensor, \(J_2\), reaches a constant value. The deviatoric stress tensor is the total stress tensor minus its hydrostatic component. The criterion is often expressed through the von Mises equivalent stress, \(\sigma_v\), which is a scalar combination of the six components of the stress tensor. For a general 3D state of stress, it is calculated as: \(\sigma_v = \sqrt{\frac{1}{2}[(\sigma_{11}-\sigma_{22})^2 + (\sigma_{22}-\sigma_{33})^2 + (\sigma_{33}-\sigma_{11})^2 + 6(\sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2)]}\).

Yielding is predicted to occur when \(\sigma_v\) equals the material’s yield strength, \(\sigma_y\), which is typically determined from a simple uniaxial tensile test. In principal stress space, the von Mises criterion defines a smooth, circular cylinder whose axis is the hydrostatic line (\(\sigma_1 = \sigma_2 = \sigma_3\)). This contrasts with the Tresca criterion, which defines a hexagonal prism. The von Mises criterion generally provides a better fit with experimental data for most ductile metals and is continuously differentiable, which is advantageous in numerical computations.