Plastic criteria

The PlasticCriterion class manages the possibility to replace the default Von Mises plastic criterion by another one, described below.

Isotropic plastic criterion (default in Metafor)

$$ \sqrt{\frac{3}{2}s_{ij}s_{ij}} - (\sigma_{vm} + \sigma_{visq} + \sigma_{grainSize} + ...) = 0 $$

néant

Second order orthotropic plastic criterion

$$ \begin{multline} \sqrt{\frac{1}{2}} \sqrt{F (s_{22}-s_{33})^2 + G (s_{33}-s_{11})^2 + H (s_{11}-s_{22})^2 + 2 (L s_{13}^2 + M s_{23}^2 + N s_{12}^2) } \\- (\sigma_{vm} + \sigma_{visq} + \sigma_{grainSize} + ...) = 0 \end{multline} $$

where stresses are defined in an orthotropic frame.

For sheet metal, the anisotropic parameters can be estimated based on tensile tests (plastic strain of around 10%). Strains are measured along the width ($ \varepsilon_{t} $) and the thickness ($ \varepsilon_{e} $). The plastic anisotropy coefficient is then defined as : $ r = \frac{\varepsilon_{t}}{\varepsilon_{e}} $

This test is done in samples cut along the 0, 45 and 90 degrees axes to define $r_{0}$ , $_{45}$ , $r_{90}$.

A planar average is then defined as : $ r_{moy} = \frac{r_{0} + 2 r_{45} + r_{90}}{4} $

Based on tensile tests, it is not possible to estimate shear through the thickness, so L and M parameters are considered equal to 3.

• $ F = \frac{2}{1+r_0}\frac{r_{0}}{r_{90}} $
• $ G = \frac{2}{1+r_0} $
• $ H = \frac{2r_{0}}{1+r_0} $
• $ L = 3 $
• $ M = 3 $
• $ N = \frac{1+2r_{45}}{1+r_0}\frac{r_{0}+r_{90}}{r_{90}} $