/MAT/LAW80

Format

Definitions

Example (Steel)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW80/1/1
steel
#              RHO_I
              7.8E-9
#                  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
             2.51E-5             1.11E-5         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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

1. If Q should be in $\left[\frac{\text{J}}{\text{mol}}\right]$ , then 1 cal =4.1855 J.
2. The strain rate dependency when Cowper Seymonds is used:(1)
   $$\sigma ={\sigma }_y\left(1+{\left(\frac{\dot{\epsilon }}{C_{eps}}\right)}^{\frac{1}{P_{eps}}}\right)$$
3. The martensite volume fraction $x_M$ equation is:(2)
   $$x_M=x_{\gamma }\left(1-\exp \left(-\alpha \left(Ms-T\right)\right)\right)$$
   Where,

   $Ms$

   Temperature of martensite transformation

   $x_{\gamma }$

   Fraction of austenite available when the transformation of martensite starts

4. 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.
5. 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.
6. 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.
7. 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.

   $x_{eq}\left(T\right)=\{\begin{array}{l}0,\text{ if }T\le A{e}_1\\ 1,\text{ if} T\ge A{e}_3\\ \frac{T-A{e}_1}{A{e}_3-A{e}_1}\text{, otherwise}\end{array}$

   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 ${\dot{x}}_{\gamma }$ .

   $\tau \left(T\right)=\{\begin{array}{l}{\tau }_1\text{, if } T\le A{e}_1\\ {\tau }_3\text{, if } T\ge A{e}_3\\ {\tau }_1+ \frac{T-A{e}_1}{A{e}_3-A{e}_1}\left({\tau }_3-{\tau }_1\right)\text{, otherwise}\end{array}$

   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.

8. Two models for transformation strain are available (Iflag_tr):
   • Iflag_tr =1(4)
     $$\text{Δ}{\epsilon }^{tr}=\text{Δ}\left({\displaystyle {\sum }_{i=2}^5{x}_i}\right)\text{Δ}{\epsilon }_{\alpha \gamma }$$
     Where,

     $\text{Δ}{\epsilon }_{\alpha \gamma }$

     Difference in compactness between alpha and gamma phase

     $x_{i=2,5}$

     Product phase fractions

   • Iflag_tr =2(5)
     $$\text{Δ}{\epsilon }^{tr}=\frac{-1}{3\left(\rho +d\rho \right)}{\displaystyle {\sum }_{i=1}^{5}d{x}_i{\rho }_i}$$
     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

9. 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

     Where, n=3 for ferrite and pearlite and 2 for bainite

     $f(x_i)=\frac{{\left(x_i\right)}^{\frac{2\left(1-x_i\right)}{3}}\cdot {\left(1-x_i\right)}^{\frac{2x_i}{3}}}{Y}$

     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

   Where, n=3 for ferrite and pearlite and 2 for bainite

   $f(x_i)=\frac{{\left(x_i\right)}^{{\phi }_i(1-x_i)}\cdot {\left(1-x_i\right)}^{{\phi }_i{x}_i}}{e^{C{r}_i{x}_i^2}}$

   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

10. This law can be used with /HEAT/MAT.
11. This law is compatible with /PROP/TYPE1, /PROP/TYPE9, and /PROP/TYPE10.
12. 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
13. Material phase transformations will occur only during the cooling. There is no material phase transformation due to deformation or heating.