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--- doc:user:elements:volumes:plasticity_criterion [2013/07/11 14:48] – created joris
+++ doc:user:elements:volumes:plasticity_criterion [2016/03/30 15:23] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+====== 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}} $