====== Plastic criteria ======

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

===== VonMisesPlasticCriterion =====

=== Description ===

Isotropic plastic criterion (default in Metafor)

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

=== Parameters === 
 néant

===== Hill48PlasticCriterion =====

=== Description ===

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.

=== Parameters ===
^   Name                                             ^     Metafor Code     ^  Dependency ^
| $ F $                                       |  ''HILL48_F''  |  néant             |
| $ G $                                       |  ''HILL48_G''  |  néant             |
| $ H $                                       |  ''HILL48_H''  |  néant             |
| $ L $                                       |  ''HILL48_L''  |  néant             |
| $ M $                                       |  ''HILL48_M''  |  néant             |
| $ N $                                       |  ''HILL48_N''  |  néant             |

=== Parameter estimation (for sheet metal) === 

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}} $