/MAT/LAW80
Block Format Keyword This law allows modeling the ultrahigh strength steel behavior at high temperatures and the phase transformation phenomena from austenite to ferrite, pearlite, bainite and martensite during cooling.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW80/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  fct_ID_{E}  Yscale_{E}  Time_unit  
F_{smooth}  F_{cut}  C_{eps}  P_{eps}  
tab_ID_{Y1}  tab_ID_{Y2}  tab_ID_{Y3}  tab_ID_{Y4}  tab_ID_{Y5}  
Yscale_{1}  Yscale_{2}  Yscale_{3}  Yscale_{4}  Yscale_{5}  
Xscale_{1}  Xscale_{2}  Xscale_{3}  Xscale_{4}  Xscale_{5}  
Θ2  Θ3  Θ4  Θ5  
Alpha1  Alpha2  Iflag_T  fct_ID_T  Iflag_loc  Iflag_tr  Iflag_kin  
QR2  QR3  QR4  Alpha  T_{ref}  
${\tau}_{1}$  ${\tau}_{3}$  Gsize  
KF  KP  Lat1  Lat2  T_{ini}  
B  Mo  Mn  W  Al  
C  Cr  Si  Cu  As  
Co  Ni  V  P  Ti  
Fct_ID_a  Fct_ID_f  Fct_ID_p  Fct_ID_b  Fct_ID_m  
Yscale_{a}  Yscale_{f}  Yscale_{p}  Yscale_{b}  Yscale_{m}  
GFAC_F  PHI_F  PSI_F  CR_F  CF  
GFAC_P  PHI_P  PSI_P  CR_P  CP  
GFAC_B  PHI_B  PSI_B  CR_B  CB  
PHI_M  PSI_M  N_M 
Definitions
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  (Optional) 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 temperature
dependent Young's modulus. (Integer) 

Yscale_{E}  Scale factor for ordinate (Young) for
fct_ID_{E}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Time_unit  Number of time units per hour. Default corresponds to seconds, equals 3600 time units per hour. Defaults = 3600 (Real) 

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

F_{cut}  Cutoff frequency for strain rate
filtering. Default = 10^{30} (Real) 

C_{eps}  Parameter for the effective strain rate
dependency (Cowper Symonds relation). 2 (Real) 

P_{eps}  Parameter for the effective strain rate
dependency (Cowper Symonds relation). 2 (Real) 

tab_ID_{Y1}  Table identifier for yield stress, first
entry effective plastic strain and second temperature, for
austenite. (Integer) 

tab_ID_{Y2}  Table identifier of yield stress for
ferrite. (Integer) 

tab_ID_{Y3}  Table identifier of yield stress for
pearlite. (Integer) 

tab_ID_{Y4}  Table identifier of yield stress for
bainite. (Integer) 

tab_ID_{Y5}  Table identifier of yield stress for
martensite. (Integer) 

Yscale_{1}  Scale factor for ordinate (stress) for
tab_ID_{Y1}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Yscale_{2}  Scale factor for ordinate (stress) for
tab_ID_{Y2}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Yscale_{3}  Scale factor for ordinate (stress) for
tab_ID_{Y3}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Yscale_{4}  Scale factor for ordinate (stress) for
tab_ID_{Y4}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Yscale_{5}  Scale factor for ordinate (stress) for
tab_ID_{Y5}. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Xscale_{1}  Scale factor for third variable strain
rate for tab_ID_{Y1}. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Xscale_{2}  Scale factor for third variable strain
rate for tab_ID_{Y2}. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Xscale_{3}  Scale factor for third variable strain
rate for tab_ID_{Y3}. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Xscale_{4}  Scale factor for third variable strain
rate for tab_ID_{Y4}. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Xscale_{5}  Scale factor for third variable strain
rate for tab_ID_{Y5}. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Θ2  Memory coefficient that determines the
fraction of previous straining in the austenite that will be remembered in the newly
formed ferrite.
(Real) 

Θ3  Memory coefficient that determines the
fraction of previous straining in the austenite that will be remembered in the newly
formed pearlite.
(Real) 

Θ4  Memory coefficient that determines the
fraction of previous straining in the austenite that will be remembered in the newly
formed bainite.
(Real) 

Θ5  Memory coefficient that determines the
fraction of previous straining in the austenite that will be remembered in the newly
formed martensite.
(Real) 

Alpha1  Thermal expansion coefficient for
austenite (gamma phase). (Real) 
$\left[\frac{1}{\text{K}}\right]$ 
Alpha2  Thermal expansion coefficient for
products (alpha phase). (Real) 
$\left[\frac{1}{\text{K}}\right]$ 
Iflag_T  Heating process. 5


fct_ID_T  Cooling and heating function identifier.
Only used, if Iflag_T=2. 5
(Integer) 

Iflag_loc  Flag to activate the phase
transformation per element depending on temperature variation. 6
(Integer) 

Iflag_tr  Calculation of transformation strain
flag. 8
(Integer) 

Iflag_kin  Phase transformation kinetics flag.
9
(Integer) 

QR2  Activation energy divided by the
universal gas constant (R=8.314472) for the diffusion reaction of the austenite
ferrite reaction. 1 Default = 11575 (Real) 
$\left[\text{K}\right]$ 
QR3  Activation energy divided by the
universal gas constant (R=8.314472) for the diffusion reaction of the austenite
pearlite reaction. 1 Default = 13840 (Real) 
$\left[\text{K}\right]$ 
QR4  Activation energy divided by the
universal gas constant (R=8.314472) for the diffusion reaction of the austenite
bainite reaction. 1 Default = 13588 (Real) 
$\left[\text{K}\right]$ 
Alpha  Material constant for martensite phase.
3 (Real) 

T_{ref}  Reference temperature for thermal
expansion. (Real) 
$\left[\text{K}\right]$ 
${\tau}_{1}$  Time necessary to start transformation
during heating at temperature T =
$Ae1$
(starting point of austenization). 7 (Real) 
$\left[\text{s}\right]$ 
${\tau}_{3}$  Time necessary to start transformation
during heating at temperature T =
$Ae3$
(final point of austenization). 7 (Real) 
$\left[\text{s}\right]$ 
Gsize  ASTM grain size number for the
austenite. (Real) 

KF  Coefficient of Boron in the composition
of ferrite. 4 (Real) 

KP  Coefficient of Boron in the composition
of pearlite. 4 (Real) 

Lat1  Latent heat for the decomposition of
austenite to ferrite, pearlite, and bainite. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
Lat2  Latent heat for the decomposition of
austenite to martensite. (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
T_{ini}  Initial
temperature. (Real) 
$\left[\text{K}\right]$ 
B  Boron percentage weight in material
(0.0~1.0). (Real) 

Mo  Molybdenum percentage weight in material
(0.0~1.0). (Real) 

Mn  Manganese percentage weight in material
(0.0~1.0). (Real) 

W  Tungsten percentage weight in material
(0.0~1.0). (Real) 

Al  Aluminum percentage weight in material
(0.0~1.0). (Real) 

C  Carbon percentage weight in material
(0.0~1.0). (Real) 

Cr  Chromium percentage weight in material
(0.0~1.0). (Real) 

Si  Silicon percentage weight in material
(0.0~1.0). (Real) 

Cu  Copper percentage weight in material
(0.0~1.0). (Real) 

As  Arsenic percentage weight in material
(0.0~1.0). (Real) 

Co  Cobalt percentage weight in material
(0.0~1.0). (Real) 

Ni  Nickel percentage weight in material
(0.0~1.0). (Real) 

V  Vanadium percentage weight in material
(0.0~1.0). (Real) 

P  Phosphorous percentage weight in
material (0.0~1.0). (Real) 

Ti  Titanium percentage weight in material
(0.0~1.0). (Real) 

Fct_ID_a  Austenite density versus Temperature
function identifier. (Integer) 

Fct_ID_f  Ferrite density versus Temperature
function identifier. (Integer) 

Fct_ID_p  Pearlite density versus Temperature
function identifier. (Integer) 

Fct_ID_b  Bainite density versus Temperature
function identifier. (Integer) 

Fct_ID_m  Martensite density versus Temperature
function identifier. (Integer) 

Yscale_{a}  Scale factor for Austenite
density. Default = 1 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Yscale_{f}  Scale factor for Ferrite
density. Default = 1 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Yscale_{p}  Scale factor for Pearlite
density. Default = 1 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Yscale_{b}  Scale factor for Bainite
density. Default = 1 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
Yscale_{m}  Scale factor for Martensite
density. Default = 1 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
GFAC_F  Ferrite grain size factor
${w}_{f}$
. Default = 0.32 (Real) 

PHI_F  Ferrite evolution parameter
${\phi}_{f}$
controlling incubation
time. Default = 0.4 (Real) 

PSI_F  Ferrite evolution parameter
${\psi}_{f}$
controlling incubation
time. Default = 0.4 (Real) 

CR_F  Ferrite retardation coefficient
$C{r}_{f}$
. Default = 0.0 (Real) 

CF  Ferrite composition dependent factor
${C}_{f}$
. Default, see Comment 9 (Real) 

GFAC_P  Pearlite grain size factor
${w}_{p}$
. Default = 0.32 (Real) 

PHI_P  Pearlite evolution parameter
${\phi}_{p}$
controlling incubation
time. Default = 0.4 (Real) 

PSI_P  Pearlite evolution parameter
${\psi}_{p}$
controlling incubation time. Default = 0.4 (Real) 

CR_P  Pearlite retardation coefficient
$C{r}_{p}$
. Default = 0.0 (Real) 

CP  Pearlite composition dependent factor
${C}_{p}$
. Default, see Comment 9 (Real) 

GFAC_B  Bainite grain size factor
${w}_{b}$
. Default = 0.32 (Real) 

PHI_B  Bainite evolution parameter
${\phi}_{b}$
controlling incubation
time. Default = 0.4 (Real) 

PSI_B  Bainite evolution parameter
${\psi}_{b}$
. Default = 0.4 (Real) 

CR_B  Bainite retardation coefficient
$C{r}_{b}$
. Default = 0.0 (Real) 

CB  Bainite composition dependent factor
${C}_{b}$
. Default, see Comment 9 (Real) 

PHI_M  Martensite evolution parameter
${\phi}_{m}$
controlling incubation time. Defaut = 0.0428 (Real) 

PSI_M  Martensite evolution parameter
${\psi}_{m}$
. Defaut = 0.382 (Real) 

N_M  Martensite exponent
${n}_{m}$
. Defaut = 0.191 (Real) 
Example (Steel)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW80/1/1
steel
# RHO_I
7.8E9
# E Nu Fct_IDE YscaleE Time_unit
210000 .3 0 0 3600
# Fsmooth Fcut Ceps Peps
0 0 0 0
# TAB_IDY1 TAB_IDY2 TAB_IDY3 TAB_IDY4 TAB_IDY5
10 10 10 10 10
# Yscale1 Yscale2 Yscale3 Yscale4 Yscale5
0 0 0 0 0
# Xscale1 Xscale2 Xscale3 Xscale4 Xscale5
0 0 0 0 0
# Theta2 Theta3 Theta4 Theta5
0 0 0 0
# Alpha1 Alpha2 Iflag_T fct_ID_T Iflag_loc Iflag_tr Iflag_kin
2.51E5 1.11E5 0 0 0 0 0
# QR2 QR3 QR4 Alpha Tref
13022 15569 15287 .011 298.14999
# tau1 tau3 Gsize
0 0 8
# KF KP Lat1 Lat2 Tini
190000 31000 590 640 1083
# B Mo Mn W Al
.0025 0 1.23 0 0
# C Cr Si Cu As
.248 .24 .29 0 0
# Co Ni V P Ti
0 0 0 .015 0
# Fct_ID_a Fct_ID_f Fct_ID_p Fct_ID_b Fct_ID_m
0 0 0 0 0
# YScaleA YScaleF YScaleP YScaleB YScaleM
0 0 0 0 0
# GFAC_F PHI_F PSI_F CR_F CF
0 0 0 0 0
# GFAC_P PHI_P PSI_P CR_P CP
0 0 0 0 0
# GFAC_B PHI_B PSI_B CR_B CB
0 0 0 0 0
# PHI_M PSI_M N_M
0 0 0
#12345678910
/TABLE/1/10
table
3
2011 0.0 273.
2013 0.02 300.
2013 0.04 300.
2012 0.0 300.
2012 0.02 273.
2012 0.04 273.
/FUNCT/2011
1st
0.0 185.0
0.1 339.0
1.0 339.0
/FUNCT/2012
2nd
0.0 190.0
0.1 344.0
1.0 344.0
/FUNCT/2013
3rd
0.0 195.0
0.1 349.0
1.0 349.0
#12345678910
#ENDDATA
/END
#12345678910
Comments
 If Q should be in $\left[\frac{\text{J}}{\text{mol}}\right]$ , then 1 cal =4.1855 J.
 The strain rate dependency when Cowper
Seymonds is used:
(1) $$\sigma ={\sigma}_{y}\left(1+{\left(\frac{\dot{\epsilon}}{{C}_{eps}}\right)}^{\frac{1}{{P}_{eps}}}\right)$$  The martensite volume fraction
${x}_{M}$
equation is:
(2) $${x}_{M}={x}_{\gamma}\left(1\mathrm{exp}\left(\alpha \left(MsT\right)\right)\right)$$Where, $Ms$
 Temperature of martensite transformation
 ${x}_{\gamma}$
 Fraction of austenite available when the transformation of martensite starts
 In order to take into account the Boron added in the composition of the material, the functions of ferrite and pearlite are modified: the coefficients KF and KP, multiplies the weight percentage of Boron (B), respectively in ferrite and pearlite composition functions.
 By default, this law considers dealing
with a cooling process. Iflag_T can be used to define if heating or
cooling is simulated as:
 Iflag_T = 0: Cooling  Austenite transforms to product phase (martensite)
 Iflag_T = 1: Heating  Austenite is formed from ferrite
 Iflag_T = 2: Cooling and heating is flag is defined as a function of time with using fct_ID_T. Cooling occurs when the function is 0 and heating occurs when the function is 1.
 Flag for global or local phase
transformation:
 Iflag_loc = 2 (default) phase change is global per part depending on the Iflag_T.
 Iflag_loc = 1 phase change is treated automatically per element respecting its temperature variation in time. In this case, Iflag_T is used only for initialization of phase fractions values and only at time=0.
 The Austenization model is based on a
modified Leblond model. ^{1}
(3) $${\dot{x}}_{\gamma}=\frac{{x}_{eq}\left(T\right){x}_{\gamma}}{\tau \left(T\right)}$$Where, ${x}_{\gamma}$ is the fraction of austenite.
${x}_{eq}\left(T\right)=\{\begin{array}{l}0,\text{if}T\le A{e}_{1}\\ 1,\text{if}T\ge A{e}_{3}\\ \frac{TA{e}_{1}}{A{e}_{3}A{e}_{1}}\text{,otherwise}\end{array}$
Where, ${x}_{eq}$ is the evolution of the austenite fraction for very law heating rates (quasi isothermal). For a given temperature, ${x}_{eq}$ is the asymptotic value which tends to the solution of the equation ${\dot{x}}_{\gamma}$ .
$\tau \left(T\right)=\{\begin{array}{l}{\tau}_{1}\text{,if}T\le A{e}_{1}\\ {\tau}_{3}\text{,if}T\ge A{e}_{3}\\ {\tau}_{1}+\frac{TA{e}_{1}}{A{e}_{3}A{e}_{1}}\left({\tau}_{3}{\tau}_{1}\right)\text{,otherwise}\end{array}$
Where, $T$
 Temperature.
 $A{e}_{1}$
 Starting temperature of austenization.
 $A{e}_{3}$
 Final temperature of austenization.
Leblond defines this time variable as follow: "t constant temperature $T$ , ${x}_{\gamma}$ tends exponentially towards ${x}_{eq}$ with a time constant equal to $\tau $ ".
In fact, ${\tau}_{1}$ and ${\tau}_{3}$ should be identified in a way to correctly describe the beginning and ending of transformation, respectively.
The starting and final temperatures of austenization are calculated automatically based on the composition of the steel and written to the Starter output file.
 Two models for transformation strain are
available (Iflag_tr):
 Iflag_tr =1
(4) $$\text{\Delta}{\epsilon}^{tr}=\text{\Delta}\left({\displaystyle {\sum}_{i=2}^{5}{x}_{i}}\right)\text{\Delta}{\epsilon}_{\alpha \gamma}$$Where, $\text{\Delta}{\epsilon}_{\alpha \gamma}$
 Difference in compactness between alpha and gamma phase
 ${x}_{i=2,5}$
 Product phase fractions
 Iflag_tr =2
(5) $$\text{\Delta}{\epsilon}^{tr}=\frac{1}{3\left(\rho +d\rho \right)}{\displaystyle {\sum}_{i=1}^{5}d{x}_{i}{\rho}_{i}}$$Where, $d\rho $
 Change of density from fcc to bcc
 ${\rho}_{i}$
 Density of the phases given in the functions Fct_ID_a, Fct_ID_f, Fct_ID_p, Fct_ID_b, Fct_ID_m
 Iflag_tr =1
 Two transformation kinetics models are
available (Iflag_kin):Iflag_kin = 1: the transformation kinetics are based on the model of Kirkaldy ^{2} for ferrite, pearlite and bainite and on Koistinen and Marburger ^{3} model for martensite.
 Kirkaldy:
(6) $$\frac{d{x}_{i}}{dt}=f(G)\cdot f(C)\cdot f(T)\cdot f({x}_{i})$$Where, $f(G)={2}^{G\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.}$
 Effect of grain size
 $f(T)={\left({T}_{cr}T\right)}^{n}\cdot {e}^{\frac{{Q}_{i}}{RT}}$
 Effect of temperature
 $f({x}_{i})=\frac{{\left({x}_{i}\right)}^{\frac{2\left(1{x}_{i}\right)}{3}}\cdot {\left(1{x}_{i}\right)}^{\frac{2{x}_{i}}{3}}}{Y}$
 Effect of current fraction formed
 $f(C)$
 Alloy composition dependent factor computed internally
 $i=f$
 For ferrite
 $i=p$
 For pearlite
 $i=b$
 For bainite
 Martensite:
(7) $${x}_{m}={x}_{\gamma}\left(1{e}^{\alpha ({M}_{s}T)}\right)$$Where, ${M}_{s}$
 Temperature of martensite transformation
 ${x}_{\gamma}$
 Fraction of austenite available when the transformation of martensite starts
Iflag_kin = 2: the transformation is modified according to Hippchen ^{4} as:(8) $$\frac{d{x}_{i}}{dt}=f(G)\cdot f(C)\cdot f(T)\cdot f({x}_{i})$$Where, $f(G)={2}^{{w}_{i}G}$
 Effect of grain size adding parameter ${w}_{i}$
 $f(T)={\left({T}_{cr}T\right)}^{n}\cdot {e}^{\frac{{Q}_{i}}{RT}}$
 Effect of temperature
 $f({x}_{i})=\frac{{\left({x}_{i}\right)}^{{\phi}_{i}(1{x}_{i})}\cdot {\left(1{x}_{i}\right)}^{{\phi}_{i}{x}_{i}}}{{e}^{C{r}_{i}{x}_{i}^{2}}}$
 Effect of current fraction formed
 $f(C)={C}_{i}$
 $i=f$
 For ferrite
 $i=p$
 For pearlite
 $i=b$
 For bainite
If $f(C)=0$ , then, by default, use the function $f(C)$ computed internally as for Iflag_kin =1.
Depending on temperature rate, martensite fraction is calculated as:(9) $$\frac{d{x}_{m}}{dT}=\alpha {\left({M}_{s}T\right)}^{{n}_{m}}\cdot {x}_{m}^{{\phi}_{m}}{\left(1{x}_{m}\right)}^{{\psi}_{m}\left(2{x}_{\gamma}\right)}$$Where, ${M}_{s}$
 Temperature of martensite transformation
 ${x}_{\gamma}$
 Fraction of austenite available when the transformation of martensite starts
 Kirkaldy:
 This law can be used with /HEAT/MAT.
 This law is compatible with /PROP/TYPE1, /PROP/TYPE9, and /PROP/TYPE10.
 List of Animation output (/ANIM/SHELL/USRII/JJ):
 USR 2= Austenite Phase Fraction
 USR 3= Ferrite Phase Fraction
 USR 4= Pearlite Phase Fraction
 USR 5= Bainite Phase Fraction
 USR 6= Martensite Phase Fraction
 USR 7= Hardness
 USR 8= Temperature
 USR 9= Yield
 USR 10= XGAMA in martensite equation
 Material phase transformations will occur only during the cooling. There is no material phase transformation due to deformation or heating.