/MAT/LAW32 (HILL)
Block Format Keyword This law describes the Hill orthotropic plastic material. It is applicable only to shell elements. This law differs from LAW43 (HILL_TAB) only in the input of yield stress.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW32/mat_ID/unit_ID or /MAT/HILL/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  
a  ${\epsilon}_{0}$  n  ${\epsilon}_{p}^{max}$  ${\sigma}_{\mathrm{max}\text{}0}$  
${\dot{\epsilon}}_{0}$  m  
r_{00}  r_{45}  r_{90}  I_{yield0} 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  Unit identifier. (Integer, maximum 10 digits) 

mat_title  Material title. (Character, maximum 100 characters) 

${\rho}_{i}$  Initial density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
E  Young's modulus. (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  Poisson's ratio. (Real) 

a  Yield parameter. (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{0}$  Hardening parameter. (Real) 

n  Hardening exponent. (Real) 

${\epsilon}_{p}^{max}$  Failure plastic strain. Default = 10^{30} (Real) 

${\sigma}_{\mathrm{max}\text{}0}$  Maximum stress. Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
${\dot{\epsilon}}_{0}$  Minimum strain rate. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
m  Strain rate exponent. Default = 0.0 (Real) 

r_{00}  Lankford parameter 0 degree. 5
Default = 1.0 (Real) 

r_{45}  Lankford parameter 45 degrees. Default = 1.0 (Real) 

r_{90}  Lankford parameter 90 degrees. Default = 1.0 (Real) 

I_{yield0}  Yield stress flag.
(Integer) 
Example (Steel)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
kg mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/HILL/1/1
void_steel
# RHO_I
7.8E6
# E NU
210 .3
# A EPSILON_0 n EPS_max SIGMA_max0
.17 .2 .45 0 0
# EPS_DOT_0 m
0 0
# r00 r45 r90 Iyield0
.75 1 1.25 0
#12345678910
#ENDDATA
/END
#12345678910
Comments
 The yield stress is defined
as:$${\sigma}_{y}=a{\left({\epsilon}_{0}+{\epsilon}_{p}\right)}^{n}\mathrm{max}{\left(\dot{\epsilon},{\dot{\epsilon}}_{0}\right)}^{m}$$
The elastic limit is given by:
$${\sigma}_{0}=a{\left({\epsilon}_{0}\right)}^{n}{\left({\dot{\epsilon}}_{0}\right)}^{m}$$Where, ${\epsilon}_{p}$
 Plastic strain
 $\dot{\epsilon}$
 Strain rate
 The yield stress is compared to the
equivalent stress:$${\sigma}_{eq}=\sqrt{{A}_{1}{\sigma}_{1}^{2}+{A}_{2}{\sigma}_{2}^{2}{A}_{3}{\sigma}_{1}{\sigma}_{2}+{A}_{12}{\sigma}_{12}^{2}}$$
 This material law must be used with property set type /PROP/TYPE10 (SH_COMP) or /PROP/TYPE9 (SH_ORTH).
 Iterative projection (I_{plas} =1) and radial return (I_{plas} =2) for shell plane stress plasticity are available.
 Angles for Lankford parameters are
defined with respect to orthotropic direction 1.$$\begin{array}{ll}R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}& H=\frac{R}{1+R}\\ {A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)& {A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)\\ {A}_{3}=2H& {A}_{12}=2H({r}_{45}+0.5)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)\\ {r}_{00}=\frac{{A}_{3}}{2{A}_{1}{A}_{3}}& {r}_{45}=\frac{1}{2}\left(\frac{{A}_{12}}{{A}_{1}+{A}_{2}{A}_{3}}1\right)\\ {r}_{90}=\frac{{A}_{3}}{2{A}_{2}{A}_{3}}& \end{array}$$
The Lankford parameters ${r}_{\alpha}$ is the ratio of plastic strain in plane and plastic strain in thickness direction ${\epsilon}_{33}$ .
$${r}_{\alpha}=\frac{d{\epsilon}_{\alpha +\pi /2}}{d{\epsilon}_{33}}$$Where, $\alpha $ is the angle to the orthotropic direction 1.
This Lankford parameters ${r}_{\alpha}$ could be determined from a simple tensile test at an angle $\alpha $ .
A higher value of R means better formability.
 If the yield stresses have been obtained in the orthotropic direction 1, define I_{yield0} =1; otherwise I_{yield0} =0.
 When ${\epsilon}_{p}$ reaches the value of ${\epsilon}_{p}^{max}$ , in one integration point, then the corresponding shell element is deleted.