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

/MAT/LAW43/mat_ID/unit_ID or /MAT/HILL_TAB/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  
fct_ID_{E}  E_{inf}  C_{E}  
r_{00}  r_{45}  r_{90}  C_{hard}_{}  I_{yield0}  
${\epsilon}_{p}^{\mathrm{max}}$  ${\epsilon}_{t}$  ${\epsilon}_{m}$  
fct_ID_{i}  Fscale_{i}  ${\dot{\epsilon}}_{i}$ 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
E  Young's modulus (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  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. 12 Default = 0: in this case the evolution of Young's modulus depends on E_{inf} and C_{E}. (Integer) 

E_{inf}  Saturated Young's modulus for infinitive
plastic strain. (Real) 
$\left[\text{Pa}\right]$ 
C_{E}  Parameter for Young's modulus evolution.
12 (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.
(Real) 

I_{yield0}  Yield stress flag.
(Integer) 

${\epsilon}_{p}^{max}$  Failure plastic strain. Default = 1.0 × 10^{30} (Real) 

${\epsilon}_{t}$  Tensile failure strain at which stress
starts to reduce. Default = 1.0 × 10^{30} (Real) 

${\epsilon}_{m}$  Maximum tensile failure strain at which
the stress in element is set to zero. Default = 2.0 × 10^{30} (Real) 

fct_ID_{i}  Plasticity curves
.i^{th} function identifier. (Integer) 

Fscale_{i}  Scale factor for
i^{th} function. Default set to 1.0 (Real) 

${\dot{\epsilon}}_{i}$  Strain rate for
i^{th} function. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Example (Metal)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/HILL_TAB/1/1
metal
# RHO_I
80
# E NU
206000 .3
#FUNCT_IDE EINF CE
0 0 0
# r00 r45 r90 C_hard Iyield0
1.73 1.34 2.24 0 0
# EPSP_max EPS_t1 EPS_m
0 0 0
# func_IDi Fscale_i EPS_dot_i
5 0 0
#12345678910
# 3. FUNCTIONS:
#12345678910
/FUNCT/5
metal
# X Y
0 260
.002 270
.005 280
.01 297
.02 322
.05 370
.1 422
.15 457
.2 485
.3 528
#12345678910
#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) $${\sigma}_{eq}=\sqrt{{A}_{1}{\sigma}_{1}^{2}+{A}_{2}{\sigma}_{2}^{2}{A}_{3}{\sigma}_{1}{\sigma}_{2}+{A}_{12}{\sigma}_{12}^{2}}$$  Angles for Lankford parameters are
defined with respect to orthotropic direction 1.
(2) $$\begin{array}{ll}R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}& H=\frac{R}{1+R}\\ {A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)& {A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)\\ {A}_{3}=2H& {A}_{12}=2H({r}_{45}+0.5)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)\\ {r}_{00}=\frac{{A}_{3}}{2{A}_{1}{A}_{3}}& {r}_{45}=\frac{1}{2}\left(\frac{{A}_{12}}{{A}_{1}+{A}_{2}{A}_{3}}1\right)\\ {r}_{90}=\frac{{A}_{3}}{2{A}_{2}{A}_{3}}& \end{array}$$The Lankford parameters r_{a} is the ratio of plastic strain in plane and plastic strain in thickness direction ${\epsilon}_{33}$ .
(3) $${r}_{\alpha}=\frac{d{\epsilon}_{\alpha +\pi /2}}{d{\epsilon}_{33}}$$Where, α is the angle to the orthotropic direction 1.
This Lankford parameters r_{a} could be determined from a simple tensile test at an angle α.
A higher value of R means better formability.
 If the last point of the first (static) function equals 0 in stress, default value of ${\epsilon}_{p}^{max}$ is set to the corresponding value of ${\epsilon}_{p}$ .
 Element deletion:
 Once ${\epsilon}_{p}$ (plastic strain) reaches ${\epsilon}_{p}^{max}$ , in one integration point, the element is deleted.
 If
${\epsilon}_{1}$
reaches
${\epsilon}_{t}$
, the stress is reduced using the following relation:
(4) $$\sigma =\sigma \left(\frac{{\epsilon}_{m}{\epsilon}_{1}}{{\epsilon}_{m}{\epsilon}_{t}}\right)$$  If ${\epsilon}_{1}$ (largest principal strain) reaches ${\epsilon}_{m}$ ( ${\epsilon}_{1}>{\epsilon}_{m}$ ), the stress in element is reduced to 0 (but the element is not deleted).
 Once ${\epsilon}_{1}$ (largest principal strain) reaches ${\epsilon}_{f}$ (maximum tensile failure strain), the element is deleted.
 The maximum number of curves that can be input is 10.
 If $\dot{\epsilon}\le {\dot{\epsilon}}_{n}$ , the yield is interpolated between ${f}_{n}$ and ${f}_{n1}$ .
 If $\dot{\epsilon}\le {\dot{\epsilon}}_{1}$ , function ${f}_{1}$ is used.
 Above ${\dot{\epsilon}}_{\mathrm{max}}$ , yield is extrapolated.
 Radial return is not available (only iterative plasticity).
 If the yield stresses have been obtained in the orthotropic direction 1, define I_{yield0} =1; otherwise I_{yield0} =0.
 The evolution of Young's
modulus:
 If fct_ID_{E} > 0, the curve defines a scale factor for Young's modulus
evolution with equivalent plastic strain
${\overline{\epsilon}}_{p}$
, which means the Young's modulus is scaled by the
function
$\text{f}\left({\overline{\epsilon}}_{p}\right)$
:
(5) $$\text{E}\left(t\right)=\text{f}\left({\overline{\epsilon}}_{p}\right)E$$The initial value of the scale factor should be equal to 1 and it decreases.
 If fct_ID_{E} = 0, the Young's modulus is calculated as:
(6) $$\mathrm{E}\left(t\right)=E(E{E}_{\mathrm{inf}})\left(1\mathrm{exp}({C}_{E}{\overline{\epsilon}}_{p})\right)$$Where, $E$ and ${E}_{\mathrm{inf}}$ are respectively the initial and asymptotic value of Young's modulus, and ${\overline{\epsilon}}_{p}$ is the accumulated equivalent plastic strain.
Note:If fct_ID_{E} = 0 and C_{E} = 0, Young's modulus E is kept constant.
 If fct_ID_{E} > 0, the curve defines a scale factor for Young's modulus
evolution with equivalent plastic strain
${\overline{\epsilon}}_{p}$
, which means the Young's modulus is scaled by the
function
$\text{f}\left({\overline{\epsilon}}_{p}\right)$
: