/MAT/LAW73
Block Format Keyword This law describes the Thermal Hill orthotropic material and is applicable only to shell elements.
This law differs from /MAT/LAW43 (HILL_TAB) by the fact that yield stress not only depends on strain rate and plastic strain, but also on temperature (it is defined by a user table).
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW73/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  v  
fct_ID_{E}  E_{inf}  C_{E}  Blank  
r_{00}  r_{45}  r_{90}  C_{hard}  I_{yield0}  
${\epsilon}_{p}^{max}$  ${\epsilon}_{t}$  ${\epsilon}_{m}$  
Tab_ID  ${\sigma}_{scale}$  ${\dot{\epsilon}}_{scale}$  
T_{i}  C_{p} 
Definitions
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
E  Initial Young's
modulus. (Real) 
$\left[\text{Pa}\right]$ 
v  Poisson's
ratio. (Real) 

fct_ID_{E}  Function identifier for
the scale factor of Young's modulus, when Young's modulus is
function of the plastic strain.
(Integer) 

E_{inf}  Saturated Young's modulus
for infinitive plastic strain. (Real) 
$\left[\text{Pa}\right]$ 
C_{E}  Parameter for Young's
modulus evolution. (Real) 

r_{00}  Lankford parameter 0
degree. 3 Default = 1.0 (Real) 

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

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

C_{hard}  Hardening coefficient.
(Read) 

I_{yield0}  Yield stress flag.
(Integer) 

${\epsilon}_{p}^{max}$  Failure plastic
strain. Default = 1.0x10^{30} (Real) 

${\epsilon}_{t}$  Tensile failure strain at
which stress start to reduce. Default = 1.0x10^{30} (Real) 

${\epsilon}_{m}$  Maximum tensile failure
strain at which the stress in element is set to zero. Default = 2.0x10^{30} (Real) 

Tab_ID  Table identifier for yield
stress definition. (Integer) 

${\sigma}_{scale}$  Yield stress scale
factor. Default set to 1.0 (Real) 
$\left[\text{Pa}\right]$ 
${\dot{\epsilon}}_{scale}$  Strain rate scale
factor. Default set to 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
T_{i}  Initial
temperature. Default set to 293K (Real) 
$\left[\text{K}\right]$ 
${C}_{p}$  Specific heat per mass
unit. (Real) 
$\left[\frac{\text{J}}{\text{kg}\cdot \text{K}}\right]$ 
Example (Steel)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW73/1/1
steel
# RHO_I
0.0078
# E NU
210000.0 0.3
#FUNCT_IDE EINF CE
0 0 0
# R00 R45 R90 C_HARD Iyield0
1.6 1.6 1.6 0.0 0
# EPSP_MAX EPS_T1 EPS_T2
0 0 0
# TABLE SIGMA_SCALE EPSPT_SCALE
10 0 0
# TI CP
273. 4.
#12345678910
/TABLE/1/10
table
3
1011 0.0 273.
1013 0.02 300.
1013 0.04 300.
1012 0.0 300.
1012 0.02 273.
1012 0.04 273.
/FUNCT/1011
1st
0.0 185.0
0.1 339.0
1.0 339.0
/FUNCT/1012
2nd
0.0 190.0
0.1 344.0
1.0 344.0
/FUNCT/1013
3rd
0.0 195.0
0.1 349.0
1.0 349.0
#ENDDATA
/END
#12345678910
Comments
 This material law must be used with property set /PROP/TYPE9 (SH_ORTH) or /PROP/TYPE10 (SH_COMP).
 The yield stress is defined by a
user function and the yield stress is compared to equivalent stress:
(1) $${\sigma}_{eq}=\sqrt{{A}_{1}{{\sigma}_{1}}^{2}+{A}_{2}{{\sigma}_{2}}^{2}{A}_{3}{\sigma}_{1}{\sigma}_{2}+{A}_{\mathrm{12}}{{\sigma}_{\mathrm{12}}}^{2}}$$  Angles for Lankford parameters are defined with
respect to orthotropic direction 1.
$R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}$ $H=\frac{R}{1+R}$ ${A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)$ ${A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)$ ${A}_{3}=2H$ ${A}_{12}=2H({r}_{45}+0.5)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)$ ${r}_{00}=\frac{{A}_{3}}{2{A}_{1}{A}_{3}}$ ${r}_{45}=\frac{1}{2}\left(\frac{{A}_{12}}{{A}_{1}+{A}_{2}{A}_{3}}1\right)$ ${r}_{90}=\frac{{A}_{3}}{2{A}_{2}{A}_{3}}$ The Lankford parameters ${r}_{\alpha}$ is ratio of plastic strain in plane and plastic strain in thickness direction ${\epsilon}_{\mathrm{33}}$ .(2) $${r}_{\alpha}=\frac{d{\epsilon}_{\alpha +\pi /2}}{d{\epsilon}_{33}}$$Where, α is the angle to the orthotropic direction 1.
This Lankford parameters ${r}_{\alpha}$ could be determined from a simple tensile test at an angle α.
A higher value of R means better formability.
 If ${\epsilon}_{p}$ (plastic strain) reaches ${\epsilon}_{p}^{\mathrm{max}}$ , in one integration point, the corresponding shell element is deleted.
 If largest principal strain
${\epsilon}_{1}>{\epsilon}_{t}$
, stress is reduced using the following relation:
(3) $$\sigma =\sigma \left(\frac{{\epsilon}_{m}{\epsilon}_{1}}{{\epsilon}_{m}{\epsilon}_{t}}\right)$$  If ${\epsilon}_{1}>{\epsilon}_{m}$ , the stress is reduced to 0 (but the element is not deleted).
 This law always uses iterative projection for plasticity (I_{plas} from the property set is ignored).
 This law is not available with global formulation for plasticity (N=0 in the property shell is not available).
 The table for yield stress definition must be a 3dimensional table whose parameters respectively represent plastic strain, strain rate, and temperature $\left({\epsilon}^{p},\dot{\epsilon},T\right)$ . Values of the table are yield stress values.
 If ${\epsilon}_{m1}^{p}\le {\epsilon}^{p}\le {\epsilon}_{m}^{p}$ and ${\dot{\epsilon}}_{n1}\le \dot{\epsilon}\le {\dot{\epsilon}}_{n}$ and ${T}_{q1}\le T\le \text{\hspace{0.05em}}\text{\hspace{0.17em}}{T}_{q}$ yield is linearly interpolated between the eight values of the table corresponding to $\left({\epsilon}_{i}^{p},{\dot{\epsilon}}_{j},{T}_{k}\right),i=m1,m;j=n1,n;k=q1,q$ .
 If
$\left({\epsilon}^{p},\dot{\epsilon},T\right)$
falls out of the range of the table, yield
stress is obtained by linear extrapolation. Thus it is necessary to input into
the table the static curves corresponding to zero strain rate (entry
$\dot{\epsilon}=0$
should belong to the table definition). If the /HEAT/MAT option is not associated to the material identifier, adiabatic conditions are assumed and temperature is computed as:
(4) $${\rm T}={T}_{i}+\frac{{E}_{int}}{\rho {C}_{p}\left(Volume\right)}$$Where, ${E}_{\mathrm{int}}$
 Internal energy computed by Radioss,
 $\sigma $
 Volume are the current density, and volume
 ${C}_{p}$
 Heat capacity per mass unit
Otherwise, the finite element formulation for heat transfer must be asked for (I_{form} =1 in option /HEAT/MAT); initial temperature and specific heat input in the option /HEAT/MAT will then be used.
 If the yield stresses have been obtained in the orthotropic direction 1, define I_{yield0} =1; otherwise I_{yield0}=0.