/MAT/LAW57 (BARLAT3)
Block Format Keyword This law describes plasticity hardening by a userdefined function and can be used only with shell elements.
This is an elastoplastic orthotropic law for modeling anisotropic materials in forming processes especially aluminum alloys. This material law must be used with property set type /PROP/TYPE9 (SH_ORTH) or /PROP/TYPE10 (SH_COMP).
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW57/mat_ID/unit_ID or /MAT/BARLAT3/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}  m  
${\epsilon}_{p}^{max}$  ${\epsilon}_{t}$  ${\epsilon}_{m}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

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) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
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. 11 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) 

C_{E}  Parameter for Young's modulus
evolution. (Real) 

r_{00}  Lankford parameter 0 degree. 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) 

m  Barlat parameter.
(Real) 

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

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

${\epsilon}_{m}$  Maximum tensile failure damage strain at
which the stress in element is set to zero. Default = 2.0 x 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 (Steel)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW57/1/1
Steel
# RHO_I
.008
# E NU
206000 .300000012
# fct_IDE E_INF CE
0 0 0
# r00 r45 r90 C_hard m
1.79 1.51 2.27 0 0
# EPSP_max EPS_T EPS_M
0 0 0
# fct_ID Fscale_i EPS_i
5 0 0
#12345678910
# 3. FUNCTIONS:
#12345678910
/FUNCT/5
function_5
# X Y
0 157
.1 320
.5 480
1.2 600
#12345678910
#ENDDATA
/END
#12345678910
Comments
 The anisotopic yield criteria
F for plane stress is defined by:
(1) $$F=a{\left{K}_{1}+{K}_{2}\right}^{m}+a{\left{K}_{1}{K}_{2}\right}^{m}+c{\left2{K}_{2}\right}^{m}2{{\sigma}_{y}}^{m}=0$$Where,
${\sigma}_{y}$ is the yield stress
${K}_{1}=\frac{{\sigma}_{xx}+h{\sigma}_{yy}}{2}$ and ${K}_{2}=\sqrt{{\left(\frac{{\sigma}_{xx}h{\sigma}_{yy}}{2}\right)}^{2}+{p}^{2}{\sigma}_{xy}^{2}}$
 Angles for Lankford parameters are defined with respect to orthotropic direction 1.
The material constants a, c, h, and
p are obtained from the three Lankford parameters:
(2) $$\begin{array}{l}a=22\sqrt{\frac{{r}_{00}}{1+{r}_{00}}}\sqrt{\frac{{r}_{90}}{1+{r}_{90}}}\\ c=2a\\ h=\sqrt{\frac{{r}_{00}}{1+{r}_{00}}}\sqrt{\frac{1+{r}_{90}}{{r}_{90}}}\end{array}$$Material constant p is calculated by solving:(3) $$\frac{2m\cdot {\sigma}_{y}{}^{m}}{\left(\frac{\partial F}{\partial {\sigma}_{xx}}+\frac{\partial F}{\partial {\sigma}_{yy}}\right){\sigma}_{45}}1{r}_{45}=0$$  If the last point of the first (static) function equals 0 in stress, the default value of ${\epsilon}_{p}^{max}$ is set to the corresponding value of ${\epsilon}_{p}$ .
 If ${\epsilon}_{p}$ (plastic strain) reaches ${\epsilon}_{p}^{\mathrm{max}}$ , in one integration point, the corresponding shell element is deleted.
 If the largest principal strain
${\epsilon}_{1}>{\epsilon}_{t}$
, the stress is reduced using the following
relation:
(4) $$\sigma =\sigma \left(\frac{{\epsilon}_{m}{\epsilon}_{1}}{{\epsilon}_{m}{\epsilon}_{t}}\right)$$  If ${\epsilon}_{1}>{\epsilon}_{m}$ , the stress is reduced to 0 (but the element is not deleted).
 The maximum number of curves 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.
 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, which means the Young's modulus is scaled by the function
$\mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
:
(5) $$E\left(t\right)=E\cdot \mathrm{f}\left({\overline{\epsilon}}_{p}\right)$$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) $$E\left(t\right)=E\left(E{E}_{inf}\right)\left[1\mathrm{exp}\left({C}_{E}{\overline{\epsilon}}_{p}\right)\right]$$Where, E and E_{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, which means the Young's modulus is scaled by the function
$\mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
: