/MAT/LAW93 (ORTH_HILL) or (CONVERSE)

It could be used with property set /PROP/TYPE11, /PROP/TYPE17, /PROP/TYPE51, /PROP/PCOMPP for shell and /PROP/TYPE6 for solid.

Format

Definition

Comments

1. The yield stress is compared to an equivalent stress in the orthotropic frame. For solid elements, this equivalent stress is defined as:(1)
   $${\sigma }_{eq}=\sqrt{F\left({\sigma }_{22}^2-{\sigma }_{33}^2\right)+G\left({\sigma }_{33}^2-{\sigma }_{11}^2\right)+H\left({\sigma }_{11}^2-{\sigma }_{22}^2\right)+2L{\sigma }_{23}^2+2M{\sigma }_{31}^2+2N{\sigma }_{12}^2}$$
   Where,

   $F=\frac{1}{2}(\frac{1}{R_{22}^2}+\frac{1}{R_{33}^2}-\frac{1}{R_{11}^2})$

   $G=\frac{1}{2}(\frac{1}{R_{33}^2}+\frac{1}{R_{11}^2}-\frac{1}{R_{22}^2})$

   $H=\frac{1}{2}(\frac{1}{R_{22}^2}+\frac{1}{R_{11}^2}-\frac{1}{R_{33}^2})$

   $L=\frac{3}{2R_{23}^2}$

   $M=\frac{3}{2R_{31}^2}$

   $N=\frac{3}{2R_{12}^2}$

   $R_{ii}^{}=\frac{{\sigma }_F^{ii}}{{\sigma }_F}$

   Normal directions

   $R_{ij}^{}=\frac{\sqrt{3}{\sigma }_F^{ij}}{{\sigma }_F}$

   Shear directions

   ${\sigma }_F^{ij}$

   Yield stress in direction $ij$

   ${\sigma }_F^{}$

   Global flow stress that can either be defined with a sum of Voce hardening, or can be tabulated (see below).

   Under plane-stress conditions, for shell elements, the equivalent yield stress becomes:(2)

   $${\sigma }_{eq}=\sqrt{\left(G+H\right){\sigma }_{11}^2+\left(F+H\right){\sigma }_{22}^2-2H{\sigma }_{11}{\sigma }_{22}+2N{\sigma }_{12}^2}$$

2. The yield function $\Phi$ will compare the Hill’s equivalent stress ${\sigma }_{eq}$ to the flow stress ${\sigma }_F^{}$ as:(3)
   $$\Phi ={\sigma }_{eq}-{\sigma }_F$$
   The two different ways to define the flow stress ${\sigma }_F^{}$ are: parameter input or curve input

   • For parameter input, the flow stress is defined with an initial yield stress and a double Voce hardening as:(4)
     $${\sigma }_F={\sigma }_Y^0+R({\epsilon }_p)$$

     With $R({\epsilon }_p)={\displaystyle \sum _i^{2}Q{R}_i\cdot \left(1-e^{-C{R}_i\cdot {\epsilon }_p}\right)}$ .

   • For curve input, the parameters input values will be ignored.
     The yield can be defined with using stress versus plastic strain curve taking in account the strain rate effect. When the stress versus strain curves are defined, this is the default method for defining the hardening.

     1. If $\dot{\epsilon }\le {\dot{\epsilon }}_n$ , the yield is interpolated between ${\mathrm{f}}_n$ and ${\mathrm{f}}_{n-1}$ .
     2. If $\dot{\epsilon }\le {\dot{\epsilon }}_1$ function, ${\mathrm{f}}_1$ is used.
     3. Above ${\dot{\epsilon }}_{\max }$ , yield is extrapolated.

        
        
        Figure 1.

3. For tabulated flow stress, the strain rate $\dot{\epsilon }$ computation depends on the value of the flag VP.
   • If VP= 1, the plastic strain rate is used
   • If VP= 2, the total strain rate is used
   • If VP= 3, the total strain rate is used

   In all cases the strain-rate computation includes a filtering. The cutoff frequency is automatically set for VP = 1. However, for VP = 1 or 3, you can input a cutoff frequency F_cut; otherwise, a default value will be set.