/MAT/LAW74
Block Format Keyword This law describes the Thermal Hill orthotropic 3D material and is applicable only to solid elements. The yield stress may depend on strain rate, or on both strain rate and temperature.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW74/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\upsilon $  ${\epsilon}_{p}^{\mathrm{max}}$  ${\epsilon}_{t}$  ${\epsilon}_{m}$  
fct_ID_{E}  E_{inf}  C_{E}  
F_{smooth}  C_{hard}  F_{cut}  
${\sigma}_{11}^{y}$  ${\sigma}_{22}^{y}$  ${\sigma}_{33}^{y}$  
${\sigma}_{12}^{y}$  ${\sigma}_{23}^{y}$  ${\sigma}_{31}^{y}$  
Tab_ID  ${\sigma}_{\mathit{scale}}$  ${\dot{\epsilon}}_{\mathit{scale}}$  
T_{i}  $\rho {\text{\hspace{0.05em}}}_{0}{C}_{p}$ 
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) 
$\left[\text{Pa}\right]$ 
$\upsilon $  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) 

${\epsilon}_{p}^{\mathrm{max}}$  Failure plastic strain. Default = 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) 

F_{smooth}  Smooth strain rate option flag.
(Integer) 

C_{hard}  Hardening coefficient.
(Real) 

F_{cut}  Cutoff frequency for strain rate
filtering. Default = 1.0 × 10^{30} (Real) 
$\text{[Hz]}$ 
${\sigma}_{11}^{y}$  Yield in direction
1. (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{22}^{y}$  Yield in direction
2. (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{33}^{y}$  Yield in direction
3. (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{12}^{y}$  Yield in shear direction
12. (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{23}^{y}$  Yield in shear direction
23. (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{31}^{y}$  Yield in shear direction
31. (Real) 
$\left[\text{Pa}\right]$ 
Tab_ID  Table identifier for yield stress
definition. 6 (Integer) 

${\sigma}_{\mathit{scale}}$  Yield stress scale factor. Default set to 1.0 (Real) 
$\left[\text{Pa}\right]$ 
${\dot{\epsilon}}_{\mathit{scale}}$  Strain rate scale factor. Default set to 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
T_{i}  Initial temperature. Default set to 293 K (Real) 
$\left[\text{K}\right]$ 
$\rho {\text{\hspace{0.05em}}}_{0}{C}_{p}$  Specific heat per volume
unit. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{3}\cdot \text{K}}\right]$ 
Example (Aluminum)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW74/1/1
Aluminum
# RHO_I
.0027
# E NU EPSILON_P_MAX EPSILON_T EPSILON_M
60400 .33 0 0 0
# fct_ID EINF CE
0 0 0
# FSMOOTH C_HARD FCUT
1 0 10
# SIGMA11Y SIGMA22Y SIGMA33Y
1 1 1
# SIGMA12Y SIGMA23Y SIGMA31Y
1 1 1
# TABLE SIGMA_SCALE EPSPT_SCALE
10 0 0
# TI RHO0_CP
0 0
#12345678910
# 3. FUNCTIONS:
#12345678910
/TABLE/1/10
table
#DIMENSION
3
# fct_ID X Z
38 0 293
38 10 293
39 11 293
40 20 293
38 0 400
38 10 400
39 11 400
40 20 400
#12345678910
/FUNCT/38
function_38
# X Y
0 90
2.5E4 100
.001 104.5
.009 121
.01 136
.02 143.5
.04 163
.07 169.5
#12345678910
/FUNCT/39
function_39
# X Y
0 108
2.5E4 120
.001 125.4
.009 145.2
.01 163.2
.02 172.2
.04 195.6
.07 203.4
#12345678910
/FUNCT/40
function_40
# X Y
0 126
2.5E4 140
.001 146.3
.009 169.4
.01 190.4
.02 200.9
.04 228.2
.07 237.3
#12345678910
#ENDDATA
/END
#12345678910
Comments
 This material law must be used with property set /PROP/TYPE6 (SOL_ORTH), /PROP/TYPE14 (SOLID), /PROP/TYPE20 (TSHELL) or /PROP/TYPE21 (TSH_ORTH).
 The yield stress is defined by a user
function and the yield stress is compared to equivalent stress:
(1) $${\sigma}_{\mathit{eq}}=\sqrt{F{\left({\sigma}_{2}{\sigma}_{3}\right)}^{2}+G{\left({\sigma}_{3}{\sigma}_{1}\right)}^{2}+H{\left({\sigma}_{1}{\sigma}_{2}\right)}^{2}+2L{{\sigma}_{23}}^{2}+2M{{\sigma}_{31}}^{2}+2N{{\sigma}_{12}}^{2}}$$Where,(2) $$F=\frac{1}{2}\left(\frac{1}{{\sigma}_{22}^{2}}+\frac{1}{{\sigma}_{33}^{2}}\frac{1}{{\sigma}_{11}^{2}}\right),\text{G}=\frac{1}{2}\left(\frac{1}{{\sigma}_{22}^{2}}+\frac{1}{{\sigma}_{33}^{2}}\frac{1}{{\sigma}_{11}^{2}}\right),\text{H}=\frac{1}{2}\left(\frac{1}{{\sigma}_{22}^{2}}+\frac{1}{{\sigma}_{33}^{2}}\frac{1}{{\sigma}_{11}^{2}}\right)$$(3) $$L=\frac{1}{2{\sigma}_{23}^{2}},\text{M}=\frac{1}{2{\sigma}_{31}^{2}},\text{N}=\frac{1}{2{\sigma}_{12}^{2}}$$Where, ${\sigma}_{11},{\sigma}_{22},{\sigma}_{33},{\sigma}_{12},{\sigma}_{23}$ and ${\sigma}_{31}$ represent the stress components either in the orthotropic frame if an orthotropic property is used, or in the orthogonalized isoparametric frame.
 If ${\epsilon}_{p}$ (plastic strain) reaches ${\epsilon}_{p}^{max}$ , in one integration point, the solid element is deleted.
 If largest principal strain
${\epsilon}_{1}>{\epsilon}_{t}$
, 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 table for yield stress definition can
be 2dimensional or 3dimensional.
 If the table is 2dimensional, its parameters are assumed to represent respectively
plastic strain and strain rate
$\left({\epsilon}^{p},\dot{\epsilon}\right)$
.
Then if ${\epsilon}_{m1}^{p}\le {\epsilon}^{p}\le {\epsilon}_{m}^{p}$ and ${\dot{\epsilon}}_{n1}\le \dot{\epsilon}\le {\dot{\epsilon}}_{n}$ yield is linearly interpolated between the four values of the table corresponding to $\left({\epsilon}_{i}^{p},{\dot{\epsilon}}_{j}\right),i=m1,m;j=n1,n$ .
 If the table is 3dimensional, its parameters are assumed to represent respectively
plastic strain, strain rate, and temperature
$\left({\epsilon}^{p},\dot{\epsilon},T\right)$
.
Then 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 $\left({\epsilon}^{p},\dot{\epsilon}\right)$ or $\left({\epsilon}^{p},\dot{\epsilon},T\right)$ 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).
Values of the table are yield stress values.
 If the table is 2dimensional, its parameters are assumed to represent respectively
plastic strain and strain rate
$\left({\epsilon}^{p},\dot{\epsilon}\right)$
.
 If yield stress also depends on temperature, the
table is 3dimensional:If the /HEAT/MAT option is not associated to the material identifier, adiabatic conditions are assumed and temperature is computed as:
(5) $${\rm T}={T}_{i}+\frac{{E}_{\mathit{int}}}{\rho {C}_{p}\left(\mathit{Volume}\right)}$$Where, E_{int}
 Internal energy computed by $\rho $ ,
 Volume
 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.
 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
$\mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
:
(6) $$E\left(t\right)=E\cdot \text{f}\left({\overline{\epsilon}}_{p}\right)$$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:
(7) $$E\left(t\right)=E\left(E{E}_{\mathit{inf}}\right)\left[1\mathrm{exp}\left({C}_{E}{\overline{\epsilon}}_{p}\right)\right]$$Where, E and E_{inf} are respectively, the initial and asymptotic value of Young's modulus, ${\overline{\epsilon}}_{p}$ 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
$\mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
: